Advances in Boundary Element Techniques XI

Advances in Boundary Element Techniques XI

ISBN 978-0-9547783-7-8 Publish by EC Ltd, United Kingdom

ECltd

Advances in Boundary Element Techniques XI Proceedings of the 11th International Conference Berlin, Germany 12-14 July 2010

Edited by Ch Zhang MH Aliabadi M Schanz

Advances In Boundary Element Techniques XI

Advances In Boundary Element Techniques XI Edited by Ch Zhang M H Aliabadi M Schanz

EC

ltd

Published by EC, Ltd, UK Copyright © 2010, Published by EC Ltd, P.O.Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

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ISBN: 978-0-9547783-7-8

The text of the papers in this book was set individually by the authors or under their supervision. No responsibility is assumed by the editors or the publishers for any injury and/or damage to person or property as a matter of product liability, negligence or other wise, or from any used or operation of any method, instructions or ideas contained in the material herein.

International Conference on Boundary Element Techniques XI 12-14 July 2010, Berlin, Germany

Organising Committee: Prof. Dr.-Ing. Chuanzeng Zhang, University of Siegen, Germany [email protected] Prof. Ferri M.H. Aliabadi Department of Aeronautics Imperial College London E-mail: [email protected] Prof. Martin Schanz Graz University of Technology Graz, Austria [email protected] International Scientific Advisory Committee Abascal R (Spain) Abe K (Japan) Albuquerque EL (Brazil) Baiz P (UK) Baker G (USA) Beskos D (Greece) Blasquez A (Spain) Bonnet M (France) Chen JT (Taiwan) Chen Weiqiu (China) Chen Wen (China) Cheng A (USA) Cisilino A (Argentina) Davies A (UK) Denda M (USA) Dong C (China) Dumont N (Brazil) Estorff Ov (Germany) Gao XW (China) Garcia-Sanchez F (Spain) Gaul L (Germany) Gatmiri B (France) Gray L (USA) Gospodinov G (Bulgaria) Gumerov N (USA) Han X (China) Harbrecht H (Germany) Hartmann F (Germany) Hematiyan MR (Iran) Hirose S (Japan) Kinnas S (USA) Kuna M (Germany)

Langer S (Germany) Liu,G-R (Singapore) Mallardo V (Italy) Mansur WJ (Brazil) Mantic V (Spain) Marburg S (Germany) Marin L (Romania)) Matsumoto T (Japan) Mattheij RMM (The Netherlands) Mesquita E (Brazil) Millazo A (Italy) Minutolo V (Italy) Mohamad Ibrahim MN (Malaysia) Nishimura N (Japan) Niu Z (China) Ochiai Y (Japan) Pan E (USA) Panzeca T (Italy) Phan AV (USA) Partridge P (Brazil) Perez Gavilan JJ (Mexico) Pineda E (Mexico) Prochazka P (Czech Republic) Qin T (China) Qin Q (Australia) Rjasanow S (Germany) Saez A (Spain) Salvadori A (Italy) Sändig,A-M (Germany) Sapountzakis EJ (Greece) Sarler B (Slovenia) Schneider R (Germany) Sellier A (France) Seok Soon Lee (Korea) Shiah Y (Taiwan) Sladek J (Slovakia) Sollero P (Brazil) Stephan EP (Germany) Taigbenu A (South Africa) Tan CL (Canada) Tao W (China) Telles JCF (Brazil) Venturini WS (Brazil) Wang Y (China) Wen PH (UK) Wendland W (Germany) Wrobel LC (UK) Yao Z (China) Ye W (Hong Kong) Zhao MH (China)

PREFACE The Conferences in Boundary Element Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques in both boundary element methods (BEM) and boundary integral equation methods (BIEM). Previous successful conferences devoted to Boundary Element Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008) and Athens, Greece (2009). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the Maritim Hotel Berlin, Germany, during 12th-14th July 2010. Research papers received from 18 counties formed the basis for the Technical Program. The themes considered for the technical program included solid mechanics, fluid mechanics, potential theory, composite materials, fracture mechanics, damage mechanics, contact and wear, optimization, heat transfer, dynamics and vibrations, acoustics and geomechanics. A symposium “Recent Advances in Theory and Application of BEM” was organized at the conference in honor of Professor Zhenhan Yao (Tsinghua University, Beijing. PR China), who is working on BEM for many years and has made many significant contributions to the Computational Mechanics especially to BEM. We would like thanks the organizers of the symposium (Prof. Ch. Zhang, Prof. C.Y.Dong and Prof. Y.H.Liu) for their effort. The conference organizers would also like to express their appreciation to the International Scientific Advisory Board for their assistance in supporting and promoting the objectives of the meeting and for their assistance in the form of reviews of the submitted papers. Editors July 2010

Contents Study of contact stress evolution on fretting problems using a 3D boundary elements formulation L Rodriguez-Tembleque, R Abascal Shape optimization with topological derivative and its application to noise barrier for railway viaducts K Abe, T Fujiu and K Koro On the transient response of actively repaired damaged structures by the boundary element method A Alaimo, G Davì, A Milazzo Computation of moments in thin plates of composite materials under dynamic load using the boundary element method K R Sousa, A P Santana, E L Albuquerque, and P Sollero Drilling rotations in BEM P Baiz Blob regularization of boundary integrals G Baker, H Zhang On the accuracy of the fast hierarchical DBEM for the analysis of static and dynamic elastic crack problems I Benedetti, A Alaimo, M H Aliabadi A boundary knot method for three-dimensional harmonic viscoelastic problems B Sensale, A Canelas Non-Incremental boundary element discretization of non-linear heat equation based on the use of the proper generalized decompositions G Bonithon, P Joyot, F Chinesta and P. Villon Three-dimensional boundary elements for the analysis of anisotropic solids F C Buroni, J E Ortiz, A Sáez Sensitivity analysis of cracked structures with static and dynamic Green’s functions O Carl, Ch Zhang A D-BEM approach with constant time weighting function applied to the solution of the scalar wave equation J A M Carrer and W J Mansur A novel boundary meshless method for radiation and scattering problems Z Fu, W Chen Anti-plane shear Green’s function for an isotropic elastic layer on a substrate with a material surface W. Q. Chen and Ch Zhang Stress intensity factor formulas for a rectangular interfacial crack in threedimensional bimaterials C-H Xu, T-Y Qin, Ch Zhang, N-A Noda Iterative optimization methodology for sound scattering using the topological derivative approach and the boundary element method, A Sisamon, S C Beck, A P Cisilino, S Langer A Laplace transform boundary element solution for the Cahn-Hilliard equation A J Davies and D Crann

1

7

13

20

26 32 38

46

54

62 69

77

83 91

97

104

110

Strategy for writing general scalable parallel boundary-element codes F C de Araújo, E F d'Azevedo, and L J Gray Incomplete LU preconditioning of BEM systems of equations based upon the generic substructuring algorithm F C de Araújo, E F d'Azevedo, and L J Gray Hypersingular BEM analysis of semipermeable cracks in magnetoelectroelastic solids R Rojas-Dıaz, M Denda, F Garcıa-Sanchez, A Saez Boundary element analysis of cracked transversely isotropic and inhomogeneous materials C Y Dong, X Yang and E Pan A family of 2D and 3D hybrid finite elements for strain gradient elasticity N A Dumont, D H Mosqueira Transient thermoelastic crack analysis in functionally graded materials by a BDEM A Ekhlakov, O Khay, Ch Zhang Time-Domain boundary element analysis of semicircular hill on viscoelastic media under vertically incident SV wave A Eslami Haghighat, S A Anvar, M Jahanandish, A Ghahramani HEDD-FS method for numerical analysis of cracks in 2D finite smart materials C-Y Fan, G-Tao Xu and M-Hao Zhao Recent developments of radial integration boundary element method in solving nonlinear and nonhomogeneous multi-size problems X W Gao, M Cui and Ch Zhang A meshless boundary interpolation technique for solving the Stokes equations C Gáspá A boundary element formulation based on the convolution quadrature method for the quasi-static behaviour analysis of the unsaturated soils P Maghoul, B Gatmiri, D Duhamel

118 124

130

136

144 154

162

168 174

184 190

Elastodynamic laminate element method for lengthy structures E V Glushkov, N V Glushkova and A A Eremin

196

Three-dimensional eigenstrain formulation of boundary integral equation method for spheroidal particle-reinforced materials H Ma, Q-H Qin Green’s functions, boundary elements and finite elements F Hartmann

202

Crack identification in magneto-electro-elastic materials using neural networks and boundary element method G Hattori and A Saez The singular nodal integration method for evaluation of domain integrals in the BEM M R Hematiyan, A Khosravifard, M Mohammadi Application of convolution quadrature method to electromagnetic acoustic wave analysis S Hirose, Y Temma and T Saitoh Boundary integral equations for unsymmetric laminated Composites C Hwu

215

208

221

227

231

BEM analysis of dynamic effects of microcracks and inclusions on a main crack J Lei, Ch Zhang, Q Yang, Y-S Wang Nonlinear transient thermo-mechanical analysis of functionally graded materials by an improved meshless radial point interpolation method A Khosravifard, M R Hematiyan Adaptive-hybrid meshfree method Leevan Ling Analysis of acoustic wave propagation in a two-dimensional sonic crystal based on the boundary element method F Li, Y-S Wang, Ch Zhang Analysis of two intersecting three-dimensional cracks L N Zhang, T Qin, Ch Zhang Reconstruction of elasticity fields in isotropic materials via a relaxation of the alternating procedure L Marin and B T Johansson Dual reciprocity boundary element formulation applied to the non-linear Darcian diffusive-advective problems C F Loeffler, F P Neves, P C Oliveira Analysis of the dynamic response of deep foundations with inclined piles by a BEM-FEM model L A Padrón, J J Aznárez, O Maeso, A Santana

237

Fast Multipole Boundary Element Method (FMBEM) for acoustic scattering in coupled fluid-fluidlike problems V Mallardo, C Alessandri, M H Aliabadi Galerkin projection for the potential gradient recovery on the boundary in 2D BEM V Mantic-Lugo, L J Gray, V Mantic, E Graciani, F Parıs Shape sensitivity analysis of 3-D acoustic problems based on direct differentiation of hypersingular boundary integral formulation C J Zheng, T Matsumoto, T Takahashi and H B Chen BEM and the Stoke system with a slip boundary condition D Medkova The BEM on general purpose graphics processing units (GPGPU): a study on three distinct implementations J Labaki, E Mesquita, L O Saraiva Ferreira Dynamic analysis of damaged magnetoelectroelastic laminated structures A Alaimo, A Milazzo, C Orlando Seismic behaviour of structures on elastic footing, BEM-FEM analysis. S Ciaramella, V Minutolo, E Ruocco Boundary element analysis of uncoupled transient thermo-elasticity involving non-uniform heat sources M Mohammadi, M R Hematiyan, L Marin Three-dimensional thermo-elastoplastic analysis by triple-reciprocity boundary element method, Y Ochiai Elastoplastic analysis for active macro-zones via multidomain symmetric BEM T Panzeca, E Parlavecchio, S Terravecchia, L Zito

292

245

252 258

266 272

280

286

298

306

312 316

324 330 334

340

346

Interaction problems between in-plane and out-plane loaded plates by SBEM. T Panzeca, F Cucco, A La Mantia, M Salerno Genetic algorithm with boundary elements for simultaneous solution of minimum solution of minimum weight and shape optimization problems Li Chong Lee, Bacelar de Castro, P W Partridge

353

New boundary integral equations for evaluating the static and dynamic Tstresses, A.-V. Phan The boundary element method applied to visco-plastic analysis E Pineda, M H Aliabadi, J Zapata Optimal shape of fibers in composites with various ratios of phase stiffnesses P P Prochazka Extended stress intensity factors of a three-dimensional crack in electromagnetothermoelastic solid T Y Qin, X J Li, L N Zhang Adaptive cross approximation and its applications R Grzhibovskis and S Rjasanow Nonlinear analysis of shear deformable beam-columns partially supported on tensionless Winkler foundation E J Sapountzakis and A E Kampitsis Solution of hot shape rolling by the local radial basis function collocation method B Šarler, Siraj-ul-Islam, U Hanoglu Regularization for a poroelastodynamic collocation BEM M Messner, M Schanz A Fast BEM for the dynamic analysis of plates with bonded piezoelectric patches I Benedetti, Z S Khodaei, M H Aliabadi On the displacement derivatives of the three-dimensional Green’s function for generally anisotropic bodies Y C Shiah, C L Tan, W X Sun, Y H Chen Coupled thermoelastic analysis for interface crack problems J Sladek, V Sladek, P Stanak Local integral equations combined with mesh free implementations and time stepping techniques for diffusion problems V Sladek, J Sladek, Ch Zhang Computation of moments in thin plates of composite materials under dynamic load using the boundary element method K R Sousaa, A P Santanaa, E L Albuquerqueb, P Sollero Meshless boundary element methods for exterior problems on spheroids E P Stephan, A Costea ,Q T Le Gia, T Tran 3-D Green element method for potential flows E Nyirenda, A Taigbenu BEM Fracture Mechanics Analysis of 3D Generally Anisotropic Solids C L Tan, Y C Shiah, J R Armitage, W C Hsia

365

A BEM analysis of the fibre size effect on the debond growth along the fibrematrix interface

359

373 381 387

392 398

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412 418

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433 441

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455 462 468 474

L Tavara, V Mantic, E Graciani, F Parıs Nonlinear nonuniform torsional vibrations of shear deformable bars application to torsional postbuckling configurations E J Sapountzakis and V.J Tsipiras Harmonic analysis of spatial assembled plate structures coupled with acoustic fluids using the boundary element method J Useche, E L Albuquerque, S Shoefel On the numerical analysis of damage phenomena in saturated porous media E T Lima Junior, W S Venturini, A Benallal Efficient solution of acoustic radiation problems by boundary elements and interpolated transfer functions O von Estorff, O Zalesk A fast solver for boundary element elastostatic analysis J O Watson Stress analysis of cracked structures considering crack surface contact by the boundary element method W Weber, K Willner, P Steinmann, G Kuhn Fatigue crack growth in functional graded materials by meshless Method P H Wen, M H Aliabadi An analysis of elastic plates under concentrated loads by non-singular boundary integral equations K-C Wu, Z-M Chang A time-domain BEM for dynamic crack analysis in piezoelectric solids using non-linear crack-face boundary conditions M Wünsche, Ch Zhang, F García-Sánchez, A Sáez Fast bEM for 3-D elastodynamics based on pFFT acceleration Technique Z Yan, J Zhang, W Ye A new time domain boundary integral equation of elastodynamics Z H Yao Regularization of the divergent integrals in boundary integral equations V.V. Zozulya On Levi Functions W L Wendland Domain integrals in a boundary element algorithm S Nintcheu Fata, L J Gray Meshfree micro-scale modelling and stress analysis of 3D orthogonal woven composites L Li, P H Wen, M H Aliabadi

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526 534

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549 555 561

569 570 571

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz

1

Study of contact stress evolution on fretting problems using a 3D Boundary Elements formulation L. Rodr´ıguez-Tembleque1 , R. Abascal2∗ Departamento de Mecánica de los Medios Continuos, Escuela Técnica Superior de Ingenieros, Camino de los descubrimientos s/n, E41092 Sevilla, SPAIN 1 [email protected], 2 [email protected]

Keywords: Fretting Wear, Contact Mechanics, Boundary Elements Method.

Abstract. A Boundary Elements Method wear formulation is applied to simulate fretting wear on a cylinder-on-flat configuration for gross sliding and partial slip conditions. The present formulation applies the Boundary Elements Method to approximate the solids elastic response, and an Augmented Lagrangian formulation to solve the contact problem. Wear on contact surfaces is computed using the Archard wear law. The numerical methodology is based on previous works [1, 2], and the works of Str¨ omberg [3], and Sfantos and Aliabadi [4, 5]. The evolution of the solids contact geometries, the contact pressure and the solids stresses can be predicted using this numerical tool. Also this formulation allows to investigate the evolution of subsurface stress fields due to material removal by fretting wear. Introduction A Boundary Elements Method fretting wear formulation is applied to simulate contact tractions evolution on a cylinder-on-flat configuration for gross sliding and partial slip conditions. The Boundary Elements Method is applied to approximate the solids elastic response, and an Augmented Lagrangian formulation to solve the contact problem. Wear on contact surfaces is computed using the Archard wear law. The numerical methodology is based on previous works [1, 2], and the works of Str¨ omberg [3], and Sfantos and Aliabadi [4, 5]. The evolution of the solids contact geometries, the contact pressure and the solids stresses can be predicted using this numerical tool. Also this formulation allows to investigate the evolution of subsurface stress fields due to material removal by fretting wear. In the literature, some recent works have studied this kind of problem using a finite elements 2D model: McColl et al.[6], Ding et al. [7] and Madge et al. [8]. This work uses the boundary elements method, which allow to have analogous results a a very good accuracy using a very low number of elements, compared with the finite elements 2D models, even using a general 3D fretting wear model. Contact equations The gap variable is defined for the pair I ≡ {P 1 , P 2 } of points (P α ∈ Ωα , α = 1, 2) at all times (τ ), as [9]: g = BT (X2 − X1 ) + BT (u2o − u1o ) + BT (u2 − u1 ) (1) being BT (X2 − X1 ) the geometric gap between two solids in the reference configuration (g g ), and BT (u2o − u1o ) the gap originated due to the rigid body movements (go ). Therefore, the gap of the I pair remains as follows: (2) g = ggo + BT (u2 − u1 ) where ggo = gg + go . In this work, the reference configuration for each solid (Xα ) that will be considered is the initial configuration (before applying load). Consequently, gg may also be termed initial geometric gap. In the expression (2) two components can be identified: the normal gap, gn = ggo,n + u2n − u1n , and the tangential gap or slip, gt = ggo,t + u2t − u1t , being uαn and uαt the normal and tangential components of the displacements

2


The unilateral contact condition and the law of friction defined for any pair I ≡ {P 1 , P 2 } ∈ Γc (Γc : Contact Zone) of points in contact can be compiled as follows, according to their contact status: tn ≤ 0 ; gn = 0 ; g˙ t = 0

Contact-Adhesion: ⎧ ⎨ Contact-Slip:

No contact :

⎩

tn ≤ 0 ; g n = 0 (3) tt = µ |tn | ; g˙ t · tt = −g˙ t tt t n = 0 ; gn ≥ 0 ; tt = 0

In the expression above gn is the pair I normal gap, and tn is the normal contact traction defined as: tn = BTn t1 = −BTn t2 , where tα is the traction of point P α ∈ Γαc expressed in the global system of reference, and Bn = [n] is the first column in the change of base matrix: B = [Bt |Bn ] = [t1 |t2 |n]. The normal tractions acting upon the pair I points are of the same value and opposite signs, in accordance with Newton’s third law. Wear equations The Holm-Archard’s wear law allow to compute the total volume of solid particles worn (W ) by adhesion wear, as Fn W = kad Ds (4) H where Fn is the contact normal load, H is the surface hardness, Ds is the sliding distance, and kad is the nondimensional wear coefficient, which represents the probability of forming a substantial wear particle. Expression (4) can be written locally as gw = kw tn Ds

(5)

being gw the wear depth, tn the normal contact pressure, and kw = kad /H the dimensional wear coefficient. The total volume worn (W ) can be computed integrating the state variable, g w , on the contact zone: W = gw dΓ (6) Γc

Wear process evolves over time, so equation (5) can be expressed in a differential for g˙ w = kw tn D˙ s

(7)

where D˙ s is the tangential slip velocity module: D˙ s = g˙ t . Considering wear on the contact surfaces, governed by the Holm-Archard’s law, the normal contact gap (gn ) is rewritten as gn = ggo,n + (u2n − u1n ) + gw (8) for an instant τi . For quasi-static contact problems, wear depth defined on instant τi , is computed as gw = gw (τk−1 ) + kw tn ∆gt

(9)

being tn and ∆gt the normal contact pressure and the sliding distance (∆gt = gt (τk ) − gt (τk−1 )), respectively, calculated on the same instant, and gw (τk−1 ) the internal variable value on instant τk−1 . To obtain wear on each solid surface from the wear depth computed g w , we can apply two criteria: α (α = 1, 2), is known, wear depth on each If the wear dimensional coefficient of each surface, kw surface is: gw gw 1 2 gw ; gw (10) = = 2 /k 1 ) 1 /k 2 ) 1 + (kw 1 + (kw w w


3

Boundary integral discrete equations The boundary integral equations for a body Ω, can be written in a matrix form as: Hu − Gp = F

(11)

where the vector u represents the nodal displacements, and F contains the applied boundary conditions. These equations are well known and can be found in many books like [10] or [11]. For the two bodies Ωα (α = 1, 2) in contact, the Equations (11) have to be regrouped as: Aαx xα + Aαp pαc = Fα

(12)

xα

where is the nodal unknowns vector which contains the unknowns outside the potential contact zone (xαp ), and the nodal displacements vector on the potential contact zone (uαc ). pαc is the nodal contact tractions vector, Aαx are the columns of Hα and Gα matrices, and Aαp are the columns of Gα matrix corresponding to the contact nodes. The equations (12), according to [9], can be rewritten for two solids contact, in the following way: R1 x1 + R2 x2 + Rλ Λ + Rg k − F = 0, being,

⎡ R1

=⎣

⎤ A1 0 ⎦ (C1 )T

⎤ 0 ⎣ 0 ⎦ Rg = Cg ⎡

⎡ R2

=⎣

⎤ 0 2 ⎦ A −(C2 )T

(13) ⎡

⎤ ˜1 A1p C 2 ⎣ ˜ Rλ = −Ap C2 ⎦ 0 (14)

⎤ ⎡ ⎤ 0 b1 2 ⎦ ⎦ ⎣ ⎣ 0 b F= + Cg kgo Cg n w(k) ⎡

Cα (α = 1, 2) is a boolean matrix which allows to extract the contact nodes displacements from x α , and Cg is the identity matrix (Cg = I). Wear discrete equations The discrete form of kinematic equation (8) for I pair, on instant k, is: 2 (k(k) )I = (k(k) go )I + (d

(k)

)I − (d1

(k)

)I + (Cg n w(k) )I

(15)

where matrix Cg n is constituted of the Cg columns which affect the normal gap of contact pairs, and w(k) is a vector which contains the contact pairs wear depth. According to the Holm-Archard’s law (7), wear is caused by the tangential slip ratio or the tangential slip velocity. In case of a contact problem, the discrete for of Expression (9) can be expressed for I pair as (w(k) )I = (w(k−1) )I + (∆w(k) )I (16) (k) (k−1) (∆w(k) )I = kw (Λn )I (kkt )I − (kt )I (k)

where Λn is a vector which contains the normal traction components of contact pairs at instant k. Contact discrete equations The contact restrictions for every I pair, at instant k can be expressed as: (Λ∗n (k) )I − PR− ( (Λ∗n (k) )I ) = 0 ; (Λ∗t (k) )I − PCg ( (Λ∗t (k) )I ) = 0 (k)

(k)

(17) (k)

(k)

The augmented contact variables are defines as: Λ∗n (k) = Λn + rn kn and Λ∗t (k) = Λt − rt (kt − (k−1) ). The value of g for the tangential projection region, on I pair, is: g = µ|P R− ( (Λ∗n(k) )I )| or kt (k) g = µ|PR− ( (Λn )I )|.

4


(a)

(b)

Figure 1: (a) Short cylinder over a flat. (b) Normal an tangential.

Results The short cylinder-against-flat 3D fretting wear problem, see Fig.1(a), is considered in this work. The cylinder of radii R = 6 mm and thickness e = 1 mm is subjected to fixed normal force per unit length, F , with superimposed cyclic tangential displacement, δ, as it is presented schematically in Fig.1(b), and collected on Table 1. The flat specimen geometrical dimensions are: e = 1 mm and L = 6 mm. Both solids have the same material properties, presented on Table 2. Case 1: Gross slip fretting problem Fig.2(a) shows the contact surface profiles versus number of fretting wear cycles for gross slip (Case 1). Note that horizontal and vertical positions refer, respectively, to the y and z coordinates. As frettingwear proceeds, the contact surface profiles are modified. The contact tends towards conforming. Fig.2(b) shows the BE predicted contact pressure distribution versus the number of fretting cycles. Case 2: Partial slip fretting problem Using the same wear coefficient as in Case 1, the predicted development of the contact profiles for the partial slip conditions of Case 2 are illustrated in Fig.3. There is no wear predicted in the stick region due to the absence of slip. Slight wear occurs in the slip regions, resulting in an increased gap for the initial configuration on these regions (Fig.3(a)). Fig.3(b) shows the BE predicted contact pressure distribution versus the number of fretting cycles. It can be observed that the initial pressure distribution is consistent with the Hertz solution. However, after 10000 wear cycles, the normal pressure in the stick zone increases significantly, particulary at the stick-slip boundaries where sharp peaks develop. In contrast, in the slip zones is reduced to negligible values, due to the increased gap caused by wear. Case

Normal load (F [N/mm])

1 2

120 120

Peak-to-Peak stroke amplitude (δ) ±10 ±2.5

Table 1: Fretting cases parameters.


Material Young’s nodulus (Ec , Ef [N/mm2 ]) Poisson’s ratio (ν) Friction coefficient (µ) c = k f [mm2 /N ]) Wear coefficient (kw w

5

Nitrided CrMoV high strength steel 200 · 103 0.3 0.6 1.0 · 10−7

Table 2: Material properties.

(a)

(b)

Figure 2: (a) Surface profiles versus the number of fretting severe wear cycles. (b) Contact pressure.

(a)

(b)

Figure 3: (a) Surface profiles versus the number of fretting slight wear cycles. (b) Contact pressure.

6


Summary and conclusions This work applies the BEM wear methodology developed by authors [1, 2], and based on Str¨ omberg [3], and Sfantos and Aliabadi [4, 5], for wear simulation wear on 3D contact problems to simulate fretting wear on a cylinder-on-flat configuration for gross sliding and partial slip conditions. The comparison with the previous models such as [6] and [7] has a very good agreement. The evolution of the solids contact geometries and the contact pressures are predicted using this numerical tool. For the gross slip regime, the high wear leads to the contact edges moving rapidly outwards, leaving the material in a permanently compressive state, which prohibits fretting-wear initiation. For the partial slip regime, wear increases the maximum contact pressure and shift its location to the boundaries between the stick-slip zones. Acknowledgments This work was co-funded by the DGICYT of Ministerio de Ciencia y Tecnolog´ıa, Spain, research project DPI2006-04598, and by the Conseger´ıa de Innovaci´ on Ciencia y Empresa de la Junta de Andaluc´ıa, Spain, research projects P05-TEP-00882 and P08-TEP-03804. References [1] L. Rodr´ıguez-Tembleque, R. Abascal, and Aliabadi M.H. A boundary element formulation for 3d wear simulation in rolling-contact problems. In Abascal R. and Aliabadi M.H., editors, Advances in Boundary Elements Techniques IX. EC, Ltd., UK, 2008. [2] L. Rodr´ıguez-Tembleque, R. Abascal, and Aliabadi M.H. Wear prediction in tribometers using a 3d boundary elements formulation. In Sapountzakis E.J. and Aliabadi M.H., editors, Advances in Boundary Elements Techniques IX. EC, Ltd., UK, 2009. [3] N. Str¨ omberg. An augmented lagragian method for fretting problems. Eur. J. Mech. A/Solids, 16(4):573–593, 1997. [4] G.K. Sfantos and M.H. Aliabadi. Wear simulation using an incremental sliding boundary element method. Wear, 260(9-10):1119–1128, 2006. [5] G.K. Sfantos and M.H. Aliabadi. A boundary element formulation for three-dimensional sliding wear simulation. Wear, 5-6(262):672–683, 2007. [6] I.R. McColl, J. Ding, and S.B. Leen. Finite element simulation and experimental validation of fretting wear. Wear, 256:1114–1127, 2004. [7] J. Ding, S.B. Leen, and I.R. McColl. The effect of slip regime on fretting wear-induced stress evolution. International Journal of Fatigue, 26:521–531, 2004. [8] J.J. Madge, S.B. Leen, I.R. McColl, and P.H. Shipway. Contact-evolution based prediction of fretting fatigue life: Effect of clip amplitude. Wear, 262:1159–1170, 2007. [9] L. Rodr´ıguez-Tembleque and R. Abascal. A 3d fem-bem rolling contact formulation for unstructured meshes. Int. J. Solids Struct., 47:330–353, 2010. [10] C.A. Brebbia and J. Dom´ınguez. Boundary Elements: An Introductory Course (second edition). Computational Mechanics Publications, 1992. [11] M.H. Aliabadi. The Boundary Element Method Vol 2: Application in Solids and Structures. Wiley, London, 2002.


Shape Optimization with Topological Derivative and Its Application to Noise Barrier for Railway Viaducts Kazuhisa Abe1,a , Takasuke Fujiu2 and Kazuhiro Koro3,b 1

3

Department of Civil Engineering and Architecture, Niigata University 8050 Igarashi 2-Nocho, Nishi-ku, Niigata 950-2181, JAPAN, 2 Graduate School of Science and Technology, Niigata University 8050 Igarashi 2-Nocho, Nishi-ku, Niigata 950-2181, JAPAN Department of Civil Engineering and Architecture, Niigata University 8050 Igarashi 2-Nocho, Nishi-ku, Niigata 950-2181, JAPAN a [email protected], b [email protected]

Keywords: BEM. Shape optimization. Topological derivative. Noise barrier.

Abstract. A shape optimization method is developed for sound insulating walls and applied to design of noise barrier for railway viaducts. The optimization is achieved by the BE-based topology optimization method which has been proposed by the authors. To cope with local minima, the topological change which will improve the performance is realized by the nucleation of small scatterers located around the main wall. The location and the number of scatterers are determined based on the topological derivative. This value is formulated within the framework of the boundary element analysis. After the nucleation the shape is tuned by the shape optimization process. The developed method is applied to the design of noise barrier installed in a railway viaduct. Through numerical results, capabilities of the method are demonstrated.

Introduction Noise in urban area originating from vehicles is of serious issues which should be coped with. In general, to insulate the traffic noise, highways and railways are equipped with walls. Since the capability of such a structure strongly depends on its profile, it is worth to explore effective shapes. Therefore, the shape optimization method can be a practical tool for this task. Abe et al. [1] have proposed the BE-based shape optimization method for sound scattering problems in which the coordinates of all boundary element junctions are used as design variables. This method enables the shape to be released from any restriction. While the complexity can thus be attained in the boundary shape, there is a possibility of branch breaking during the optimization process. To cope with this anomaly, the topological change was allowed in the context of shape optimization analysis. This was realized by the aid of the level set method [2]. Within the framework of the gradient-based shape optimization methods, the improvement of shape is to be achieved only in the neighborhood of the current form. Therefore, the shape may converge to a local optimal. Even if the above optimization method allows any topological change, still the obtained optimal shapes depend on the initial shapes. In this paper, in order to cope with this problem, an optimization method is developed with the aid of the topological derivative. Based on its value, small scattering bodies are allocated in the design domain so that the objective function will be decreased. The topological derivatives are given by the sensitivity to the growth of an infinitesimal void in accordance with the definition by Novotny et al. [3]. The BE-based formulation given in Ref.[4] for elastostatic analysis is applied to sound scattering problems. Distribution of the topological derivative enables us to determine the required number and allocation of small obstacles introduced in the current shape. Therefore, it will be possible to relieve the optimal shape from local minima. The developed method is applied to the design of noise barrier installed in a railway viaduct. Through numerical results, capabilities of the method are demonstrated.

Design Sensitivity Analysis with Boundary Element Method Boundary Element Equation. Let us consider a section of a railway viaduct placed in a half-plane sound scattering field as illustrated in Fig.1. The noises are originating from the wheels and the pantograph. The

7

8


y(m) source points observation points

design domain 1.5

R.L. 0.5 0.5 3.5 7.0

2.0 40.0 20.0

5.0 0.5

... (0,0)

x(m)

Fig.1 Outline of the problem

noise barrier is to be arranged in the design domain which is set above the wall. Several observation points are allocated at a height of 0.5m from the ground. The boundary element equation for this problem is given by [H]{P} = {P∗ },

(1)

where [H] is the coefficient matrix calculated with the fundamental solution for a half-plane, {P} is a vector of nodal sound pressure and {P∗ } is a vector given by the point noises. Notice that the Neumann condition is assumed on the boundary. Since in this study the Green’s function of Helmholtz equation in a half-plane is used for the boundary integral equation, the element discretization is needed only on the viaduct, the car body and the noise barriers. Moreover, in order to avoid the fictitious eigenfrequencies, the Burton-Miller formulation [5] is employed. Design Sensitivity Analysis. The shape optimization problem for noise barrier is defined by min J(P; Xb ) := F(P; Xb ) + [λ]T {HP − P∗ } + λ+ (V − Vmax ), Xb

subject to

[λ]T {HP − P∗ } = 0

for ∀{λ},

λ+ (V − Vmax ) = 0,

(2)

λ+ ≥ 0,

where J is the objective function, F is a cost function estimating the sound pressure. The second term of the right-hand side stands for the condition associated with the boundary element equation, and {λ} is a vector of Lagrange multiplier. The third term is the volume restriction. V is the volume (area) of the barrier, Vmax is an allowable limit and λ+ is a Lagrange multiplier. In eq(2) the all co-ordinates of element junctions {Xb } on the barrier are used as the design variables. The variation of J due to the geometrical change {∆Xb } is given by ∂F T ∂F T ] {∆P} + [ ] {∆Xb } ∂P ∂Xb ∂P∗ ∂V T ∂H + [λ]T [ · ∆Xb ]{P} + [H]{∆P} − { · ∆Xb } + λ+ [ ] {∆Xb }, ∂Xb ∂Xb ∂Xb

∆J =[

(3)

where {∆P} is the change of pressure resulting from the shape change {∆Xb }. As mentioned above, in this study every element junction on the barrier is used to represent the boundary shape. In order to save the computational cost, the following adjoint equation is introduced, [H]{λ} = −{

∂F }. ∂P

(4)


grid point

9

contour−line segment

element junction

boundary element

Fig.2 Boundary element discretization with level set function

Substituting the solution {λ} of eq(4) into eq(3), we can obtain the following expression ∆J = [Re(β) + λ+

∂V T ] {∆Xb }, ∂Xb

(5)

where {β} is a vector given by {β} = {

∂H ∂P∗ ∂F } + [λ]T { P− }. ∂Xb ∂Xb ∂Xb

(6)

The velocity which leads to an optimal shape is given by {V} = −{Re(β) + λ+

∂V }. ∂Xb

(7)

In this paper the nodal velocity {V} is directed to the outward normal at each boundary element junction locating on the barrier.

Shape Updating Process with Level Set Method In this paper, to cope with the topological change which may happen during the shape optimization process, the shape is updated based on the Eulerian frame. The topological change is captured with the aid of the level set method [2]. The level set function ψ is assigned at each fixed grid point (Fig.2). The boundary is implicitly defined by the zero contour of the level set function. Once the contour line is drawn on the background grid, the element junctions are equidistantly distributed along the contour. The boundary element discretization is then accomplished by connecting the element junctions with each other. Shape change results from the advection of the level set function governed by the Hamilton-Jacobi equation, ∂ψ = −v · ∇ψ ∂t

(8)

where v is the convective vector defined on the grid points. It is evaluated from the nodal velocity {V} defined on the boundary element junctions[6].

Topological Derivative for Sound Scattering Problems Formulation of Topological Derivative. The topological derivative is defined by the rate of cost function resulting from the growth of an infinitesimal hole [3]. To derive this within the framework of the BE analysis, let us consider a sound scattering field with a small obstacle of radius a centered at xˆ . In this case the objective function is modified as ˆ ˆ a) := F + [λ]T {HP − P∗ + P}, J(P, P;

(9)

10


ˆ is a vector given by the integration on the boundary of scattering body Γa , i.e., where {P} ∂G(xi , y) pˆ i = p(y) dΓy , ∂ny Γa

(10)

here p is the sound pressure, G is the fundamental solution and ny stands for the outward normal. pˆ i is evaluated at a collocation point xi allocated on the boundary Γ except on Γa . We assume that the topological derivative DT can be regularized for the objective function by DT (ˆx) = lim

a→0 ∆a → 0

∆J , ∆Va

(11)

where Va is the volume of the circular obstacle, therefore, ∆Va = 2πa∆a. In this paper we consider a cost function given by F=

N

|pzi |,

(12)

i

where pzi is sound pressure at the ith observation point zi and N is the number of observation points. pzi is evaluated by the integral representation, ∂G∗ (zi , y) ∂G∗ (zi , y) pzi = − p(y) dΓy − p(y) dΓy + p∗zi , (13) ∂n ∂ny y Γ Γa where G∗ is the half-plane fundamental solution with weak singularity. Variation ∆J due to the increment of radius ∆a is given by ⎛ N ⎞ N ⎜⎜⎜ p¯zi ∂pzi ⎟⎟⎟ p¯zi ∂pzi T T ⎜ ˆ ∆J = Re ⎜⎜⎝[ · ] {∆P} + ( · )∆a + [λ] {H∆P + ∆P}⎟⎟⎟⎠ , |pzi | ∂P |pzi | ∂a i

(14)

i

ˆ are variations resulting from the increment ∆a. In the following, where (¯) denotes conjugate, and {∆P}, {∆P} for the sake of brevity, Re( ) is omitted. In order to eliminate the terms concerning ∆P in eq(14), the following adjoint equation is introduced, [H]T {λ} = −{

N p¯zi ∂pzi · }. |p ∂P zi | i

(15)

Substituting the solution of eq(15) into eq(14), from eq(11) the topological derivative can be expressed by DT (ˆx) =

N p¯zi ˙ˆ p˙ zi + [λ]T {P}, |p zi | i

1 ∂pzi p˙ zi := lim , a→0 2πa ∂a

˙ˆ := lim 1 ∂Pˆ . {P} a→0 2πa ∂a

(16)

˙ˆ From eq(16), p˙ can be obtained by calculating ∂p /∂a. Since p is given by eq(13), Derivation of p˙ zi , {P}. zi zi zi we can evaluate ∂pzi /∂a from ∂G∗ ∂ ∂pzi p dΓy . =− (17) ∂a ∂a Γa ∂ny Let us consider a small scattering body of radius a embedded in waves propagating with wavenumber k in an infinite field. By letting a → 0, the sound pressure on Γa can be given as, p(y) = p s (ˆx) + 2(y − xˆ ) · ∇p s

y on Γa

(18)


11

[mm]

[dB] 600

design domain 3400

2000

1000

150

1000

100 500 300

718

500

1500

3700 2890

sound pressure level

1300

600

60

50

R.L. 200

3500

Fig.3 Car body, viaduct and initial shape of noise barrier

0

200

400

optimization step

Fig.4 Time history of sound pressure level

where p s is the sound pressure observed before the nucleation. Substituting eq(18) into eq(17), we can evaluate explicitly ∂pzi /∂a with the aid of far field approximation, and then obtain p˙ zi from eq(16) as p˙ zi = −

∂p ik2 ik ∂p s p s (ˆx){H0(1) (kr s ) + H0(1) (krs )} − {H1(1) (kr s ) + H1(1) (krs ) s }, 4 2 ∂s ∂s

(19)

where Hn(m) is mth Hankel function of order n, ∂p s /∂s is the directional derivative of p s in the direction of r s = xˆ − zi , r s = |r s |, and ( ) stands for a quantity concerning the mirror image point with respect to the ground surface. ˙ˆ in the similar manner. We can derive {P}

Numerical Example Analytical Conditions. As an example, a viaduct for the Sinkansen railway is considered. Outline of the analysis region is illustrated in Fig.1. Detail around the design domain of the noise barrier is shown in Fig.3. The sources of noise originating from rolling wheels and pantograph of a train running at a speed of 200km/h are set to 9.5Pa and 2.0Pa [7], respectively, with a frequency at 500Hz. Five observation points are placed at distances of 20, 25, 30, 35 and 40 (m) from the center of the track. A rectangular design domain of 2×1.5 (m) is set at the top of wall. Notice that only the upper part of the wall is optimized. The fixed grid of 0.02m size is embedded in the design domain for the level set analysis. The boundary of noise barrier is discretized with constant elements of about 0.02m length. The volume of the barrier is allowed up to twice of the initial volume V0 , i.e., Vmax = 2V0 . Optimal Size of Scattering Body. A new scatterer will be nucleated at a portion having a negative DT . Therefore, the cost function will decrease with growing the hole to a certain extent. However excessive size may produce reverse effect. It implies the existence of optimal size for the nucleated hole. Based on numerical experiments, we approximate the cost function F in terms of the volume of hole Va as F(Va ) ≈

N i

1 |Fi0 + F˙ i0 Va + F¨ i0 Va2 | 2

(20)

where Fi0 is the sound pressure at zi , F˙ i0 and F¨ i0 are topological derivatives of first and second orders. These values are evaluated under the limit a → 0. Based on eq(20), the optimal size which will minimize the cost function F is determined.

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9.50

9.50

9.25

9.25

9.00

9.00

8.75

8.75

8.50

8.50

8.25

8.25

8.00

8.00 2.40

2.65

2.90

3.15

3.40

3.65

3.90

4.15

4.40

2.40

2.65

2.90

0th step

3.15

3.40

3.65

3.90

4.15

4.40

85th step

9.50

9.50

9.25

9.25

9.00

9.00

8.75

8.75

8.50

8.50

8.25

8.25

(m)

8.00

8.00 2.40

2.65

2.90

3.15

3.40

3.65

225th step

3.90

4.15

4.40

2.40

2.65

2.90

3.15

3.40

3.65

3.90

4.15

4.40

500th step

Fig.5 Shape optimization of the noise barrier

Numerical Results. A small scattering body is located at a position where the minimum topological derivative takes place. Time history of the sound pressure is shown in Fig.4. The nucleation is performed when the reduction of noise resulting from the shape optimization is relaxed. In this example new holes were introduced at the 85th and 225th steps. It can be found that the nucleation activates the optimization again, and noise reduction of about 15dB was achieved. After the nucleation the shape optimization with the level set method is resumed. The lower limit of the radius is determined by the resolution of the background grid. The nucleation is stopped when the optimal radius estimated from eq(20) is smaller than the grid size. Under this criterion, in this case two holes were created. Fig.5 shows the shapes at the 0th, 85th, 225th and 500th steps. From the figure we can see that the second scatterer is smaller than the first one.

Conclusion A shape optimization method has been developed for sound insulating wall installed in railway viaducts. In order to cope with the local minima, the nucleation of scattering bodies was attempted by the aid of the topological derivative. This strategy enables us to discover an optimum topology. After the nucleation the profile is updated within the framework of the shape optimization. New holes are located based on the topological derivative. The optimal size of the nucleated obstacle is determined by eq(20). Through numerical example, the capabilities of the proposed method were proved. References [1] K.Abe, S.Kazama and K.Koro Advances in Bound Elem Tech 2007, 8, 379-384 (2007). [2] G.Allaire, F.Jouve and A-M.Toader J Comput Phys, 194, 363-393 (2004). [3] A.A.Novotny, R.A.Feijóo, E.Taroco and C.Padra Comput Methods Appl Mech Engrg, 192, 803-829 (2003). [4] K.Abe, T.Fujiu and K.Koro Advances in Bound Elem Tech 2008, 9, 235-240 (2008). [5] AJ.Burton and GF.Miller Proc Roy Soc Lond A, 323, 201-210 (1971). [6] K.Abe, S.Kazama and K.Koro Commun Numer Meth Engng, 23, 405-416 (2007). [7] K.Nagakura QR of RTRI, 37, No.4, 210-215 (1996).


On the Transient Response of Actively Repaired Damaged Structures by the Boundary Element Method A. Alaimo1, G. Davì2, A. Milazzo3 1

University of Palermo, Dipartimento di Ingegneria Strutturale Aerospaziale e Geotecnica, Viale delle Scienze Edificio 8, 90128 Palermo Italy, [email protected]

2


3


Keywords: Active repair, Piezoelectric patch, Fracture mechanics, Boundary Element Method, Transient analysis.

Abstract. The transient fracture mechanics behavior of damaged structures repaired through active piezoelectric patches is presented in this paper. The analyses have been performed through a boundary element code implemented in the framework of piezoelectricity to take account of the coupling between the elastic and the electric fields, which represents the peculiar feature of piezoelectric media. The multidomain technique has been also involved to assemble the host structures and the active patches and to model the cracks. Moreover, the patches have been considered elastically bonded to the damaged structure by means of a zero thickness adhesive layer. This has been achieved through the implementation of an interface spring model which has allowed, coupled with an iterative procedure, to prevent overlapping at the interface between the host structure and the active patch as well as between the crack surfaces. The Dual Reciprocity Method (DRM) has been used in the present time dependent application for the approximation of the domain inertia terms. Numerical analyses have been carried out in order to characterize the dynamic repairing mechanism of the assembled structure by means of the computation of the dynamic Stress Intensity Factors.

Introduction. The unique feature of piezoelectric media to couple the elastic and the electric fields gives rise to the opportunity of employing this kind of material in the field of “Active Repair technology”. In fact, the actuating capability of piezoelectric materials, achieved through the converse piezoelectric effect [1], can be used to avoid the failure of a damaged structure by reducing the crack opening displacements [2, 3]. It follows that piezoelectric patches can effectively be used to repair flawed structures by replacing the most commonly used repair methods involving bonded or riveted metallic or composite patches, which act in a passive manner [4, 5]. The design of active patches, that can be arranged by bonding or embedding piezoelectric layers into the host damaged structures, needs the development of analysis tools in order to understand their repairing mechanisms and the overall fracture mechanics behavior of the repaired structures. Several analytical and numerical strategies have been developed to study the static electromechanical response of active repairs. Analytical models for cracked beam actively repaired through piezoelectric actuators have been developed by Wang et al. [6, 7]. The main idea proposed in the aforementioned works is to reduce the singularity at the crack tip by inducing, via the active repairs, a local moment. Finite element procedures have been developed by Duan et al. [8] and Liu et al. [3, 9] to analyze the active patch behavior applied on isotropic host damaged structures, while the boundary element method has been proposed by Alaimo et al. for the analysis of the piezoelectric patch activity in the repair of both isotropic and composite damaged structures [2, 10]. Alaimo et al. [11] have also analyzed the effect of the

13

14


Coulomb’s frictional contact on the fracture mechanics behavior of actively repaired delaminated composite structures. To the authors’ knowledge, although the static behavior of the active piezoelectric patch has been widely characterized by both numerical and analytical formulations, few works on the transient response of piezoelectric active patches have been presented yet. Among this Wang et al. [12] have proposed an analytical model to study the repair of cracked beam with piezoelectric patches under the effect of dynamical loadings. A repair criterion, based on the restoring of the natural frequency of the healthy beam through a suitable external voltage applied on the piezoelectric actuator, has been adopted in the aforementioned paper. A methodology for the optimal design of the voltage to be applied on the piezoelectric patch for the repair of vibrating delaminated beam has been also proposed by Wu et al. [13]. In the present work the boundary element method is used to model the dynamic fracture mechanics behavior of isotropic damaged structure actively repaired through piezoelectric patches. The BI formulation is developed by using the elastostatics fundamental solutions for two-dimensional anisotropic media, properly reformulated for the piezoelectric problem in terms of generalized variables. The inertia terms are considered as body forces and the Dual Reciprocity BEM is used to compute the mass matrix. The multidomain technique [14], provided with an interface spring model, allows the assembling between the host structures and the patches as well as the modeling of the adhesive layer at the interface between contiguous domains. The transient behavior of the repaired structures is characterized in terms of the dynamic Stress Intensity Factors (dSIFs) directly computed from the crack tip opening displacements [14]. Numerical analyses are performed on a cracked isotropic beam under two different dynamic repairing voltage applied across the single layered piezoelectric repair.

Basic equations. The boundary integral procedure is formulated for the piezoelectricity problem under the assumption of plane strain conditions. According to Barnett and Lothe's generalized formalism for piezoelectricity [15], it is possible to write the governing equations of the piezoelectric problem as generalized Navier-like equation and, by applying the Betti's reciprocity theorem with the static electroelastic fundamental solutions, the Somigliana identity for the electromechanical problem is obtained in terms of generalized variables [16],

c* U P0

³ T U U T d w: ³ *

*

w:

:

U* Fd :

(1)

In Eq. (1) the generalized body force vector F is given by the inertial force components only, since the electric field is considered to be as quasi static. Then the generalized body force vector writes

F = UU

(2)

where Udenotes the 4 by 4 inertia matrix obtained from the scalar matrix U,by replacing the last diagonal term with zero.The piezoelectric dynamic problem is then solved numerically by means of the Boundary Element Method [17], in such a way the following equations of motion are obtained

H = GP M

(3)

In Eq. (3), ' and P are the vectors of the generalized displacements and boundary tractions nodal values, respectively while H and G are influence matrices computed by integrating the kernel fundamental solutions weighted by linear shape functions, employed to express the generalized displacements and tractions on the boundary. The mass matrix M is instead approximated through the Dual Reciprocity Method [18,19], by transforming the domain inertia integral, represented by the right hand side of Eq. (1), into boundary integral.


15

The Houbolt method [20] is used to proceed the time integration of the equation of motion, Eq. (3). It at the instant t ' t as allows to approximate the acceleration

= 1 2 5 4 t 't t 't t t 't t 2 't 't 2

(4)

By substituting Eq. (4) into the equations of motion, Eq.(3), the following system of algebraic equations is obtained H = GP h

(5)

where ' and P are representative of the displacements and tractions at the instant t ' t , the term h take into account inertial effects related to the displacement history and is defined as h=

M 5t 4t 't t 2 't 't 2

(6)

while the influence matrix H depends on the integration time step 't as H=H

2 M 't 2

(7)

Host Structure/Active Patch assembling and crack modeling strategy. The assembling between the host structure and the piezoelectric patch as well as the modeling of the crack is achieved through the multidomain technique [16]. The equations of motion for each of the N homogeneous sub-region are then written as k k k k k k M H = G P

k 1, 2..., N

(8)

and the global system of equation pertaining the overall assembled structure is then obtained by applying the compatibility and equilibrium conditions along all the sub-region interfaces w: ij i

w: ij ; j

In Eq. (9), the subscript w:ij

Pw: ij i

Pw: ij j

i 1,..., N 1;

j

i 1,..., N

(9)

indicates quantities associated with the nodes belonging to the interface

between the i-th and j-th sub-regions. The crack is modeled by providing the multi-domain technique with an interface spring model. By so doing, a zero thickness elastic layer, having vanishing stiffness, is considered between the crack surfaces and by means of an iterative procedure, deeply discussed in Alaimo et al. [2], the inconsistence of the overlap is also avoided. The spring model also allows the modeling of the adhesive layer among the host structure and the active repair. Since the elastic interface conditions [2] represent an uncoupled behavior between interlayer tractions and displacements jumps components, characterized by the compliance constants kN and kT, the modeling of the bonding layer through an equivalent zero-thickness elastic interface may be achieved by linking the mechanical properties of the adhesive to the compliance interface constants. For the interested reader the aforementioned procedure is fully described in [2]. Finally, the dynamic stress intensity factors, allowing the fracture mechanics characterization of the repaired structure, are directly computed from the crack opening displacements [14].

16


Numerical Results. The results obtained for a cantilever isotropic beam with a vertical crack actively repaired through a piezoelectric patch, shown in Fig. 1, are discussed in the following. The cracked beam is characterized by having length L = 0.08 m, height H = 0.02 m and a crack length a = 0.01 m. The beam is clamped on the left side while the crack is centered with respect to the span of the host structure. The material of the host damaged beam is PMMA plastics, having Young modulus E = 3.3 GPa, Poisson ratio = 0.35 and mass density = 1200 kg/m3. The active patch, made up by a single layer of PZT-4 piezoelectric ceramic whose material properties are those used by Liu [3], is bonded on the bottom side of the beam and across the crack, as depicted in Fig.1. The geometry of the patch is characterized by the following dimensions, LP = H / 0.75 and h = 0.25 H. Firstly, the natural frequencies of both the un-cracked and cracked beam are computed and the results compared with those obtained by the finite element computations performed through Comsol Multiphysics®.

L F crack H

x2

h

P

V

x1 Figure 1: Active repair configuration obtained with a single layered patch.

As shown in Table 1, the results obtained from the two different numerical analyses are in good agreement with the only exception for the fourth natural frequency which evidences a maximum percentage discrepancy of -12.9 % and -12.6 % for the healthy and cracked beam respectively. It is worth nothing that the presence of the crack leads to a reduction of the natural frequencies of the patched beam due to the less stiffness introduced by the damage. Mode

Healthy beam

fBEM 1 2 3 4

[Hz]

fFEM

[Hz]

Cracked beam

% difference

fBEM [Hz]

fFEM [Hz]

% difference

749

760

-1.47 %

727

735

3953

3922

0.78 %

3212

3067

-1.1 % 4.5 %

4631

4552

1.7 %

4400

4287

2.6 %

7503

8476

-12.9 %

7363

8295

-12.6 %

Table 1: Natural frequencies for both the healthy and the cracked beam.

The characterization of the free vibration behavior of the repaired beam has been followed by the analysis of its transient response for two different electric loads, applied on the piezoelectric patch in order to reduce the crack opening displacements corresponding to the static deformed configuration under the static transverse load F, see Fig. 1. Both the dynamical analyses are then performed by considering as initial


17

conditions the static solution of the cracked repaired beam loaded by a transverse concentrated load F = 100 N/m, as shown in Fig. 1. 0.4

1

KI(t)/KIU Applied Voltage 0.8

0.3

0.6

0.25

0.4

0.2

0.2

KI/KIU

V(t)/V0

0.35

0.15

0 0

0.005

0.01

0.015

0.02

0.025

Time [s]

Figure 2: Dynamic Mode I SIF corresponding to quasi-static repairing voltage.

As previously highlighted by Liu [3] and by Alaimo et al. [2], for this particular mechanical load case, mode I is predominant. For this reason, the dynamic fracture mechanics behavior and consequently the repairing mechanism induced by the actuated patch is characterized in terms of KI only. The first electric load considered refers to a quasi-static voltage, see Fig. 2, whose time dependence is expressed as t § V (t ) V0 ¨1 e W ¨ ©

· ¸ ¸ ¹

(10)

where V0 is set to 2000 V while the time constant is 8 ms. In Fig. 2 the dynamic KI/KIU behavior, being KIU the stress intensity factor characterizing the cracked beam without repair, is shown. It can be observed that, by increasing the voltage, the KI/KIU approaches quasi-statically its minimum value 0.173, which is very close to the KI/KIU value obtained through the static analysis corresponding to the repairing voltage VR = 1290 V, see Alaimo et al. [2]. Once reached its minimum value, the SIF turns to increase tending to KI/KIU = 0.2, corresponding to its static value at V=V0. The second analysis deals with the same repaired configuration under an Heaviside electrical loads having amplitude V = 1290 V, corresponding to the static repairing voltage VR [2]. Fig. 3 shows the time history of the dimensionless mode I stress intensity factor for both perfect

and adhesive interface conditions between the host structure and the piezoelectric repair. The last condition is obtained by setting kN = 1.56*105 m/GPa and kT = 9.33*105 m/GPa in order to model an adhesive layer having Young modulus E = 3 GPa, Poisson ratio = 0.4 and thickness t = 0.1 mm. The results obtained point out that, after the transient behavior, the SIFs, characterizing both the perfect and imperfect bonding conditions, oscillates at the first natural frequency of the cracked structure. Moreover, the effect of the adhesive is to increase the amplitude of the dynamic SIF, while the effect of the inertia force is to drive the SIFs behind the limiting value, KI/KIU = 0.178 [2], characterizing the static repairing behavior of the piezoelectric patch.

18


Perfect Adhesive Static SIF at V=VR

0.4

KI/KIU

0.3

0.2

0.1

0

0.002

0.004

0.006

0.008

Time [s]

Figure 3: Dynamic Mode I SIF corresponding to the Heaviside electric loads. Conclusion. The transient dynamic response of an actively repaired isotropic cracked beam has been analyzed in this paper. A boundary element code, implemented in the framework of piezoelectricity, has been used for the analyses and the multi-domain technique has been involved to interface the host damaged structure with the piezoelectric repair. The bonding layer has been modeled through the implementation of an interface spring model while the mass matrix has been approximated by means of the Dual Reciprocity Method. Numerical analyses have been performed on a cantilever cracked beam repaired with a single layered PZT-4 patch. The repairing mechanisms of the assembled structure under two different repairing voltage have been described in terms of the mode I dynamic stress intensity factor. References

[1] I. Chopra AIAA Journal, 40 (10), 2145-87 (2002). [2] A. Alaimo, A. Milazzo, C. Orlando Engineering Fracture Mechanics, 76, 500-511 (2009). [3] T.J.C. Liu Theoretcial and Applied Fracture Mechanics, 47, 120-132 (2007). [4] A.A. Baker, R. Jones Bonded Repair of Aircraft Structure, Dordrecht: Martinus Nijhoff (1988). [5] C.H. Chue, L.A. Lin, S.C. Wang Engineering Fracture Mechanics, 48, 91-101 (1994). [6] Q. Wang, S.T. Quek Smart Materials and Structures, 13, 1222-1229 (2004). [7] Q. Wang, S.T. Quek, K.M. Liew Smart Materials and Structures, 11, 404-410 (2002). [8] W.H. Duan, S.T. Quek, Q. Wang Smart Materials and Structures, 17, 0150017 (2008). [9] T.J.C. Liu Engineering Fracture Mechanics, 75, 2566-2574 (2008). [10] A. Alaimo, A. Milazzo, C. Orlando ICCES, 11 (1), 9-16 (2009). [11] A. Alaimo, G. Davì, C. Orlando, Advances in Boundary Element Technique IX, 489-495, (2009).


[12] Q. Wang, W.H.Duan, S.T. Quek, International Journal of Mechanical Sciences, 46, 1517-1533 (2004). [13] N. Wu, Q. Wang, Smart Materials and Structures, 19, 8pp. (2010). [14] M.H. Aliabadi The boundary element method. Application in solids and structures. Vol.2, Chirchester: Wiley (2002). [15] D.M. Barnett, J. Lothe, Physics State Solid (b), 67, 105-111 (1975). [16] G. Davì, A. Milazzo, International Journal of solids and Structures, 38, 7065-7078 (2001). [17] E.L. Albuquerque, P. Sollero, M.H. Aliabadi, International Journal of solids and Structures, 39, 14051422 (2002). [18] P.W. Partridge, C.A. Brebbia, L.C. Wrobel, The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, and Elsevier, London, 1992. [19] G. Dziatkiewich, P. Fedelinski, CMES, 17(1), 35-46 (2007). [20] J.C. Houbolt, J. Aeronaut. Sci., 17, 540-550 (1950).

19

20


Computation of Moments in thin Plates of Composite Materials under Dynamic Load using the Boundary Element Method K. R. Sousaa , A. P. Santanaa , E. L. Albuquerqueb , and P. Solleroc a Federal Institute of Maranhao ˜ Department of Mechanical and Materials ˜ Luis, MA, Brazil 65025-000, Sao

{kerlles,andre}@ifma.edu.br b University

of Brazilia - UNB Faculty of Technology 70910-900, Brasilia, Bsb, Brazil [email protected] c University

of Campinas - UNICAMP Faculty of Mechanical Engineering 13083-970, Campinas, SP, Brazil [email protected]

Keywords: Boundary element method, radial integration method, dual reciprocity boundary element method, plates, composite materials, dynamic of plate, and stress analyses.

Abstract. This work presents a dynamic formulation of the boundary element method for moments of anisotropic thin plates. The elastostatic fundamental solution for anisotropic thin plates is used and inertia terms are treated as body forces. Domain integrals that come from body forces are transformed into boundary integrals using the radial integration method (RIM). In this method, the inertia term is approximated as a sum of approximation functions times coefficients to be determined. In this work, the augmented thin plate spline is used as the approximation function. The time integration is carried out using the Houbolt method. Only the boundary is discretized. Numerical results show good agreement with results available in literature. Introduction. Nowadays, BEM is a well-established numerical technique to deal with an enormous number of engineering complex problems. Analysis of plate bending problems using the BEM has attracted the attention of many researchers during the past years, proving to be a particularly adequate field of applications for that technique. In recent years, the boundary element formulation for plate bending has included the analysis of anisotropic problems. Shi and Bezine [10] presented a boundary element analysis of plate bending problems using fundamental solutions proposed by [16] based on Kirchhoff plate bending assumptions. Rajamohan and Raamachandran [6] proposed a formulation where singularities were avoided by placing source points outside the domain. Albuquerque et al [2] presented a method to transform domain integrals into boundary integrals in the classical plate theory for composite laminate materials. The transformation follows the radial integration method, as proposed by Gao [3]. In [1], this formulation was extended for dynamic problems. Shear deformable shells have been analyzed using the boundary element method by [13] with the analytical fundamental solution proposed by [14]. Wang and Huang [12] presented a boundary element formulation


21

for orthotropic shear deformable plates. Later, in Wang and Schweizerhof [15], the previous formulation was extended to laminate composite plates. Stress and moment computation by the BEM has been addressed by some works in literature. For example, Zao [18] and Zao and Lan [17] have discussed the computation of stresses in plane elastic problems, Knopke [4] presented and discussed the integral formulation for computation of stresses in isotropic thin plate, Rashed et al [7] presented an stress integral formulation in the BEM fo Reissner plate bending problems. To the best of authors knowledge, the computation of moments by the BEM in anisotropic plates have still not been addressed in literature. This paper proposes a numerical procedure to compute moments at internal points and at the boundary of composite laminated plates using a dynamic boundary element plate formulation that follows the Kirchhoff hypotheses. Boundary integral equations. The boundary integral formulation for anisotropic thin plate problems uses two integral equations, for displacement and rotation (see [2]). The transversal displacement equation is given by:

Kw(Q) +

Γ

Vn∗ (Q, P)w(P) − m∗n (Q, P)

Nc

∂ w(P) dΓ(P) + ∂n

Nc

∑ R∗c (Q, P)wc (P) = ∑ Rc (P)w∗c (Q, P) + i

i

i

i=1

i=1

Γ

Vn (P)w∗ (Q, P) − mn (P)

i

Ωg

b(P)w∗ (Q, P)dΩ +

∂ w∗ (Q, P) dΓ(P), ∂n

(1)

where P is the field point; Q is the source point; Γ is the boundary of the domain Ω of the plate; Ωg is the part of the domain Ω where the body force b is applied; the constant K is introduced in order to consider that the source point Q can be placed in the domain, on the boundary, or outside the domain (if the point Q is on a smooth boundary, then K = 1/2); ∂∂()n is the derivative to the outward unity vector n that is normal to the boundary Γ at the field point P; mn and Vn are, respectively, the normal bending moment and the Kirchhoff’s equivalent shear force on the boundary Γ; Rc is the thin plate reaction of corners; wc is the transversal displacement of corners; Nc is the number of corners; and the symbol * stands for fundamental solutions. The rotation equation is given by: 1 ∂ w(Q) + 2 ∂ n1 Nc

∂ R∗ci

Γ

∂V ∗ ∂ m∗n ∂w (P) dΓ(P) + (Q, P)w(P) − (Q, P) ∂ n1 ∂ n1 ∂n

∂ w∗i

i

i=1

∂ w∗ (Q, P)dΩ + ∂ n1 Ωg i=1 ∂ w∗ ∂ ∂ w∗ Vn (P) (Q, P) dΓ(P), (Q, P) − mn (P) ∂ n1 ∂ n1 ∂ n Γ Nc

∑ ∂ n1 (Q, P)wc (P) = ∑ Rc (P) ∂ n1c (Q, P) + i

b(P)

(2)

where ∂∂ n()1 is the derivative to the outward unity vector n1 that is normal to the boundary Γ at the source point Q.

22


As can be seen, domain integrals arise in the formulation owing to the presence of the body force b. In order to transform these integrals into boundary integrals, consider, as in the DRM, that the body force b is approximated over the domain Ωg as a sum of M products between approximation functions fm and unknown coefficients γm , that is: M

b(P) =

∑ γm fm + ax + by + c

(3)

m=1

with M

M

M

m=1

m=1

m=1

∑ γm xm = ∑ γm ym = ∑ γm = 0

(4)

The approximation functions used in this work is the well known thin plate spline given by: fm3 = R2 log(R),

(5)

used with the augmentation function given by equations (3) and (4) . It has been shown in some works from literature that this approximation function can give excellent results for many different formulations (see Partridge [5]). Equations (3) and (4) can be written in a matrix form, considering all source points, as: b = Fγ

(6)

γ = F−1 b

(7)

Thus, γ can be evaluated as:

For transient analysis, the body force vector is given by: ¨ b = ρ hw.

(8)

¨ is the acceleration (double dots stands for second where ρ is the material density, h is the plate thickness, w time derivative). To carry out the time integration during a interval T , this interval is divided into N equal intervals (time steps) of size ∆τ (T = N∆τ ). The acceleration for the time step τ + ∆τ is given by: ¨ τ +∆τ = w

1 (2wτ +∆τ − 5wτ + 4wτ −∆τ − wτ −2∆τ ) . ∆τ 2

(9)

Provided that wτ , wτ −∆τ , and wτ −2∆τ are known, we can compute wτ +∆τ by doing: Axτ +∆τ = yτ +∆τ

(10)

where xτ +∆τ is the vector of unknown variables and yτ +∆τ is the vector of known variables in which the elements are computed taking into account boundary conditions and computed values for prior time steps. Moments are written in terms of transversal displacement as:


∂ 2w ∂ 2w ∂ 2w mx = − D11 2 + D12 2 + 2D16 , ∂x ∂y ∂ x∂ y ∂ 2w ∂ 2w ∂ 2w , my = − D12 2 + D22 2 + 2D26 ∂x ∂y ∂ x∂ y ∂ 2w ∂ 2w ∂ 2w mxy = − D16 2 + D26 2 + 2D66 , ∂x ∂y ∂ x∂ y

23

(11)

So, in order to compute moments, it is necessary to calculate second derivatives of transverse displacement w. These derivatives are given by (see [9]):

∂ 2 w(Q) ∂ x2

=

2 ∗ ∂ V n ∂ x2

Γ

Vn (P)

Γ

Ω

∂ 2 w(Q) ∂ y2

=

b(P)

n ∂ y2

Vn (P)

Γ

Ω

∂ 2 w(Q) ∂ x∂ y

=

b(P)

n

∂ x∂ y

Γ

Vn (P)

Ω

b(P)

Nc ∂ 2 w∗ci ∂ 2 w∗ ∂ 3 w∗ (Q, P) − mn (P) (Q, P) dΓ(P) + ∑ Rci (P) (Q, P) + 2 2 ∂x ∂ n∂ x ∂ x2 i=1 (12)

(Q, P)w(P) −

Nc ∂ 2 R∗ ∂ 2 m∗n ∂ w(P) ci dΓ(P) + ∑ (Q, P) (Q, P)wci (P) − 2 2 ∂y ∂n i=1 ∂ y

Nc ∂ 2 w∗ci ∂ 2 w∗ ∂ 3 w∗ (Q, P) − m (P) (Q, P) dΓ(P) + ∑ Rci (P) (Q, P) + n ∂ y2 ∂ n∂ y2 ∂ y2 i=1

∂ 2 w∗ (Q, P)dΩ ∂ y2

2 ∗ ∂ V Γ

Nc ∂ 2 R∗ ∂ 2 m∗n ∂ w(P) ci (Q, P) (Q, P)wci (P) − dΓ(P) + ∑ 2 2 ∂x ∂n i=1 ∂ x

∂ 2 w∗ (Q, P)dΩ ∂ x2

2 ∗ ∂ V Γ

(Q, P)w(P) −

(Q, P)w(P) −

(13) Nc ∂ 2 R∗ ∂ 2 m∗n ∂ w(P) ci (Q, P) dΓ(P) + ∑ (Q, P)wci (P) − ∂ x∂ y ∂n i=1 ∂ x∂ y

Nc ∂ 2 w∗ci ∂ 2 w∗ ∂ 3 w∗ (Q, P) − mn (P) (Q, P) dΓ(P) + ∑ Rci (P) (Q, P) + ∂ x∂ y ∂ n∂ x∂ y ∂ x∂ y i=1

∂ 2 w∗ (Q, P)dΩ ∂ x∂ y

(14)

Numerical results. Consider a square clamped-plate under a uniformely distributed step load applied at time τ0 = 0 with amplitude q = 2, 07.106 N/m2 .The plate is orthotropic with the following material properties: E2 = 6895 MPa, E1 = 2E2 , G12 = 2651.9 MPa, ν12 = 0.3, ρ = 7166 kg/m3 . The edges of the plate is a = 254 mm and thickness h = 12.7 mm. This problem is equivalent to problem proposed by Sladek et al. (2007) which was analyzed using the MPLG. The static moment of the central node of the plate is given 3 2 by mstat x = 9, 54 × 10 N.m and the normalization factor of time by to = a /(4 ρ h/D). Twelve quadratic

24


discontinuous boundary elements (three per edge) with equal length and time steps ∆τ = 3.9447.10−5 s are used in the discretization of space and time, respectively. Results are obtained using 1, 9, and 25 internal points. They are shown in figurue 1. These moments are compared with a meshless Petrov Galerkin formulation ([8]) and the finite element method ([11]). Figure 1 shows moments mx at the central node of the plate as a function of time. 2.5

2

m x /m est. x

1.5

1 1 internal point 9 internal points 25 internal points FEM Sladek et al. (2006)

0.5

0

−0.5 0

0.1

0.2

t/to

0.3

0.4

0.5

Figure 1: Moment at the centre of the plate using different internal points. Results obtained with 25 internal points were closer to the solution of the finite element method and meshless than results with 1 and 9 internal points. However, results with 9 and 25 points are very close, indicating convergency. Results with one internal point are very smooth. So, the use of internal points are needed to obtain better precision. Conclusions. This paper analysed the use the radial integration method applied to transient analysis of anisotropic plates. From results, we can conclude that Acknowledgment. The authors would like to thank the State of Maranhão Research Foundation (FAPEMA) and the National Council for Scientific and Technological Development (CNPq) for the financial support of this work.

References [1] E. L. Albuquerque, P. Sollero, and W. P. Paiva. The radial integration method applied to dynamic problems of anisotropic plates. Communications in Numerical Methods in Engineering, 23:805–818, 2007. [2] E. L. Albuquerque, P. Sollero, W. Venturini and M. H. Aliabadi. Boundary element analysis of anisotropic Kirchhoff plates. International Journal of Solids and Structures, 43:4029–4046, 2006.


[3] X.W.Gao. The radial integration method for evaluation of domain integrals with boundary only discretization, Engineering Analysis with Boundary Elements, Vol. 26, pp. 905–916, (2002). [4] B. Knopke. The hypersingular integral equation for the bending moments mxx, mxy, and myy of Kirchhoff plates, Computational Mechanics, Vol. 15, pp. 19-30, (1994). [5] P. W. Partridge. Towards criteria for selection approximation functions in the dual reciprocity method. Engineering Analysis with Boundary Elements, 24:519–529, 2000. [6] C. Rajamohan and J. Raamachandran. Bending of anisotropic plates charge simulation method, Advances in Engineering Software, Vol. 30, pp. 369–373, (1999). [7] Y. F. Rashed, M. H, Aliabadi, C. A. Brebbia. On the evaluation of stress in the BEM for Reissner plate bending problems, Applied Mathematica Modeling, Vol. 21, pp. 155-163, (1997). [8] J. Sladek, V. Sladek, Ch. Zhang, J. Krivacek and P.H. Wen. Analysis of orthotropic thick plates by meshless local PetrovGalerkin (MLPG) method. International Journal for Numerical Methods in Engineering, v. 67, p. 1830–1850, 2006. [9] K. R. P. Sousa. Analysis of Stress in thin Plates of Composite Materials under Dynamic Load using the Boundary Element Method. Master Thesis, Faculty of Mechanical Engineering, University of Campinas., 2009. [10] G. Shi and G. Bezine. A general boundary integral formulation for the anisotropic plate bending problems, Journal of Composite Material, Vol. 22, pp. 694–716, (1988). [11] J. Useche. Shellcomp v3.4: Finite Element Analysis Program for Linear Static and Dynamic Analysis of Composite Shell Structures. Universidade Tecnolgica de Bolivar, Cartagena, Colmbia, 2008. [12] J. Wang and M. Huang. Boundary element method for ortotropic thick plates, Acta Mechanica Sinica, Vol. 7 (3), pp. 258–266,(1991). [13] J. Wang and K. Schweizerhof. Free vibration of laminated anisotropic shallow shells including transverse shear deformation by the boundary-domain element method, Computers and Structures, Vol. 62, pp. 151–156, (1997). [14] J. Wang and K. Schweizerhof. The fundamental solution of moderately thick laminated anisotropic shallow shells, International Journal of Engineering Science, Vol. 33, pp. 995–1004, (1995). [15] J. Wang and K. Schweizerhof. Fundamental solutions and boundary integral equations of moderately thick symmetrically laminated anisotropic plates, Communications in Numerical Methods in Engineering, Vol. 12, pp. 383–394, (1996). [16] B.C. Wu and N.J. Altiero. A new numerical method for the analysis of anisotropic thin plate bending problems,Computer Methods in Applied Mechanics and Engineering, Vol. 25, pp. 343–353, (1981). [17] Z. Zhao and S. Lan. Boundary stress calculation - a comparison study, Computers & Structures, Vol. 71, pp. 77-85, (1999). [18] Z. Zhao. On the calculation of boundary stress in boundary elements, Engineering Analysis with Boundary Elements, Vol. 16, pp. 317-322, (1995).

25

26


Drilling Rotations in BEM P. M. Baiz Department of Aeronautics, Imperial College London, London SW7 2AZ, UK [email protected]

Keywords: Drilling Rotation, Partition of Unity, 2D Elasticity. Abstract. This paper presents an approach to include drilling rotations in the two-dimensional boundary integral equation. The approach is based on a simple partition of unity strategy that gives rise to a fictitious rotational degree of freedom. A functional based on the rotational residual ties the average fictitious drilling rotation field to the true rotation field induced from the twodimensional elasticity problem. The approach maintains the boundary only character of BEM and the partition of unity enrichment makes it more general and efficient, as just certain areas of the boundary could be enriched. The accuracy of the proposed method is assessed with a well known benchmark problem (Cook’s membrane). Introduction When modelling thin walled structures (plate and shell assemblies) by BEM, similar to 2D and 3D problems with domains of different material properties, the multi-domain BEM technique is required (see Figure 1 (a)). Now, contrary to 2D and 3D problems, when assembling plate segments as the ones shown in Figure 1 (a), a difficulty will arise on the assignment of the stiffness in the out-of-plane rotation of each plate. This is because classical plate bending formulations (see Figure 1 (c)) do not produce equations associated with this rotational degree of freedom (d.o.f.), also known as the ’drilling rotation’. Figure 1 (b), shows the classical d.o.f. in 2D elasticity (u1 and u2 ) and the out of plane rotational d.o.f. (drilling rotation). Within the FEM community this has been a topic of intensive study [2]. The most common approaches employ various special devices to develop successful elements. The first of such techniques was proposed by Allman [3], who introduced a quadratic displacement approximation to supplement drilling d.o.f. to nodes. More rigorous mathematical developments [4] have also been proposed (based on variational principles employing independent rotation fields). Unfortunately, this has been an area totally neglected in the BEM community. The current approach [5] relies on the assumption that the plate flexural rigidity in its own plane is so large that it is possible to ignore its associated deformation, in other words, there is no drilling rotation. This is a good approximation for wide plates, but not for narrow ones that behave like beams. This paper aims to propose an approach to include drilling rotations in the classical BEM 2D elasticity formulation. The approach is based on a simple partition of unity strategy that gives rise to a fictitious rotational d.o.f. (quadratic displacement approximation to supplement drilling as proposed by [3]). Because of the local character of the approach (approximation on linear boundary elements), the main characteristic of BEM will be maintained. Classical BEM for 2D Elasticity Lets consider flat isotropic sheets of thickness h, Young’s modulus E, Poisson’s ratio ν with a boundary Γ. The two-dimensional boundary integral equation for displacements at the boundary point x ∈ Γ in the absence of body forces can be written as [1],


27

Figure 1: Typical Thin Walled Assembly (a) and Degrees of Freedom in 2D (b) and Plate Bending (c).

cαβ (x )uβ (x ) =

Γ

∗ ∗ Uαβ (x , x)tβ (x)dΓ − − Tαβ (x , x)uβ (x)dΓ Γ

(1)

where − denotes a Cauchy principal-value integral and cαβ (x ) is a function of the geometry at the collocation points equal to 1/2δαβ for a smooth boundary. The boundary displacements and ∗ (x , x) and T ∗ (x , x) are displacement and tractions are denoted by uα and tα , respectively. Uαβ αβ traction fundamental solutions for 2D elasticity and can be found in [1]. Compatible quadratic displacements with vertex connectors As shown by Allman [3], it is possible to write the normal and tangential components of displacement (un and ut ) along a typical linear boundary element as, un = a1 + a2 s + a3 s2 ,

ut = a4 + a5 s

(2)

where the coordinate s is measured from one end of the side and where the coefficients a1 ,..,a5 are to be evaluated in terms of the connectors at both ends of the side. Since there are five coefficients and six d.o.f., this problem is mathematically overdetermined and the following strategy was adopted in [3]: four boundary conditions are used to define the end-displacements and the fifth boundary condition is chosen to define the difference of the derivatives of the end displacements: un |s=0 = un1 ,

un |s=len = un2 ,

ut |s=0 = ut1 ,

ut |s=len = ut2 ,

∂un ∂un (3) |s=0 − |s=len = −w2 + w1 ∂s ∂s Clearly, w1 and w2 are not true rotations in the context of plane elasticity analysis, but they are closely related to the true rotations at the ends of the element. Following this approximation leads to the well known ’Allman triangle’ which rotational d.o.f.. have been employed by many researches to design quadrilateral and advanced solid elements. This type of approximation was recently re-examined and reformulated in an extremely simple manner. Tian and Yagawa [7] found that, although developed two decades ago, this type of elements (Allman) takes a typical form of the nowadays partition of unity (PU) based approximation, as follows:

28


ue1 ue2

=

n

Ni

i=1

1 0 −λyî 0 1 λxî

⎛

⎞

ui1 ⎜ i ⎟ ⎝ u2 ⎠ , wi

λ = 0

(4)

PU [6] is a set of functions that sum to unit at an arbitrary point of the domain, such as usual FE and many meshfree shape functions. According to the property of a PU method, any PU can be used as the functions Ni . As pointed in [7], this theoretically explains why these rotation formulae can be extended to different finite elements with translations only. This approximation has also been extended to meshfree methods and, as shown in this work, to boundary element methods. In equation (4), for the present work Ni represents the classical linear shape functions for boundary elements N1 = 0.5(1 − ξ) and N1 = 0.5(1 + ξ), ueα represents the discretized in plane displacements and uiα , wi the nodal d.o.f.. Following exactly the same implementation in [3] λ is taken as 1/2 and the functions yî and xî are given as, yî =

2

Nj (yj − yi )

xî =

j=1

2

Nj (xj − xi )

(5)

j=1

Enriched BEM formulation To apply enrichment to BEM, equation (4) must be introduced in equation (1) as follows: cαβ (x )

2 i=1

Ni (ξp )unβ¯ i +

−

Ne 2 n=1 i=1

2

Ni (ξp ) (−λˆ yi (ξp )δβ1 + λˆ xi (ξp )δβ2 ) wn¯ i

i=1

ni ni Pαβ uβ −

Ne 2 n=1 i=1

=

2 Ne n=1 i=1

ni Pαβ (−λˆ yi (ξ)δβ1 + λˆ xi (ξ)δβ2 ) wni

ni Qni αβ tβ

(6)

where n ¯ is the number of the element containing x and ξp refers to the local coordinate of the source point. The first and second term in the right hand side of equation (6) remain the same as in standard BEM while the third term contains the enriched term (drilling rotation). A similar equation to (6) was recently presented by Simpson and Trevelyan [10] for mode I and II fracture analysis. Relation between true and assumed rotation fields As mentioned in [10], implementing the above enrichment introduce an additional unknown to the classical BEM. In their work, a technique based on additional collocations points was implemented. In the present work a more physically based approach will be used to obtain the additional equations necessary to have a well posed problem. Following a similar approach continuously used in FEM, the following functional could be considered: Π(u, w) ¯ =

γ 2

Γ

(Ψ − w) ¯ 2 dΓ

(7)

The expression in (7) was recently used within the functional of the total potential energy by Choi et al. [9] for the hybrid Trefftz method (a type of boundary element method). This penalty term ties the average drilling rotation field w ¯ to the true rotation Ψ induced from uα , and is given as Ψ = 1/2(u2,1 − u1,2 ) [2, 3, 9]. The penalty parameter γ in equation (7) was determined through some numerical tests in [9]. In the present case such parameter (γ) can take any value without affecting the solution. This


29

parameter becomes only important when (7) is used within the functional of the total potential energy in order to balance its contribution to the classical two-dimensional problem. Taking the first variation of equation (7) with respect to the drilling rotation field w ¯ and minimizing gives: δΠ(u, w) ¯ ¯ T (Ψ − w)dΓ ¯ =0 (8) = δw δw ¯ Γ Introducing equation (4) in (8) gives the following expression,

δwn1 δwn2

ξ=+1 N1 (ξ) 1×2 ξ=−1

N2 (ξ)

2×1

1 1 1 − N1,ξ (ξ)un1 N1,ξ (ξ)un1 1 + 2 2 Jy (ξ) Jx (ξ)

+ (lenλ (N1,ξ (ξ)N2 (ξ) + N2,ξ (ξ)N1 (ξ)) − N1 (ξ)) wn1 −

1 N2,ξ (ξ)un2 1 Jy (ξ)

+

1 n2 N2,ξ (ξ)un2 J(ξ)dξ = 0 2 − (lenλ (N1,ξ (ξ)N2 (ξ) + N2,ξ (ξ)N1 (ξ)) + N2 (ξ)) w Jx (ξ)

(9)

Equation 9 can be seen also as the final matrix representation of a weak form of the rotation residual given in equation 7. In other words, equation 9 permits the element to satisfy local equilibrium of out of plane rotations in a weak sense. Numerical Results (Cook’s Membrane) The enriched BEM formulation proposed above was verified using the most common benchmark example in FEM for assessment of new elements (particularly those with drilling rotation capabilities). The reason for its popularity among FEM is because it shows the element’s ability to model membrane situations with distorted meshes. This problem was first proposed by Cook [11] as a test case for non-rectangular quadrilateral elements. There is no known analytical solution but the most refined case of several references will be used for comparison. The material properties, geometry and boundary conditions are shown in Figure 2. The tip displacements and rotations at point A (see Figure 2) for different meshes are shown in Table 1. Table 1: Tip Displacements for Cook’s membrane problem.

16x16Q (AL) [9] 32x32Q (Green strain-Allman) [8] 32x32Q (Right stretch strain-Allman) [8] 32x32Q (FEAP shell 6 dof) in [8] based on [12] 64x64x2T (OPT) [13] Present 16 Linear (4 per side) Present 64 Linear (16 per side) Present 128 Linear (32 per side) Present Uniform 172 Linear

u1 — -10.675 -10.663 -10.678 — -8.05093 -10.85637 -10.75679 -10.74402

u2 23.86350 23.916 23.876 23.922 23.95 23.20510 23.99848 24.14329 24.07600

w — 0.92553 0.89046 0.85404 — 1.61773 0.77387 0.81364 0.81050

Figures 3 presents the deformed shapes for the 16 Linear Boundary Elements (4 per side) and for the more refined uniform 172 Linear Boundary Elements. As shown in Table 1, excellent agreement is obtained with the reported literature with less than 1% difference for deflection and 5% for rotation.

30


Figure 2: Cook’s membrane problem.

Figure 3: Deformed Shape for more refined and less refined meshes.


Conclusions This paper presented the first attempt to develop a BEM formulation that includes drilling rotations based on Allman’s triangular finite element. This rotations are necessary for accurate simulations of complex multi-domain plate assemblies (formulation that include 6 degrees of freedom per node: 3 displacements and 3 rotations). Some of the most important advantages of the approach are: its boundary only character and the PUM approximation that could lead to great efficiency by just adding drilling rotations along the boundary that needs them (junction boundary).

References [1] M.H. Aliabadi,The Boundary Element Method, vol II: Application to Solids and Structures, Chichester, Wiley (2002). [2] 0.C. Zienkiewicz, L.R. Taylor, The Finite Element Method (vol. 2, Solid Mechanics), Butterworth-Heinemann, 5 edition (2000). [3] D.J. Allman, A compatible triangular element including vertex rotations for plane elasticity analysis, Computers and Structures, v57, pp. 1-8 (1984). [4] T.R.J. Hughes, F.Brezzi, On drilling degrees of freedom, Comput. Methods Appl. Mech. Engrg., v72, pp. 105-121 (1989). [5] P.M. Baiz, M.H. Aliabadi, Local buckling of thin-walled structures by the boundary element method, Eng. Anal. Bound. Elem., v33, pp. 302-313 (2009). [6] I. Babuska, G. Caloz, J. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM Journal of Numerical Analysis, v31, pp. 945-98 (1994). [7] R. Tian, G. Yagawa, Allmans triangle, rotational DOF and partition of unity, Int. J. Numer. Meth. Engng, v69, pp. 837 - 858 (2007). [8] K. Wisniewski, E. Turska, Enhanced Allman quadrilateral for finite drilling rotations, Comput. Methods Appl. Mech. Engrg., v195, pp. 6086 - 6109 (2006). [9] N. Choi, Y. S. Choo, B. C. Lee, A hybrid Trefftz plane elasticity element with drilling degrees of freedom, Comput. Methods Appl. Mech. Engrg., v195, pp. 4095 - 4105 (2006). [10] R. Simpson, J. Trevelyan, Enrichment of the Boundary Element Method through the partition of unity method for mode I and II fracture analysis, Advances in Boundary Element Techniques X, Athens, Greece (2009). [11] R.D. Cook, Improved two-dimensional finite element, J. Struct. Div., ASCE 100 (ST6), pp. 1851 - 1863 (1974). [12] O.C. Zienkiewicz, R.L. Taylor, The finite element method, 4th ed. Basic Formulation and Linear Problems, vol. 1, McGraw-Hill, 1989. [13] C.A. Felippa, A study of optimal membrane triangles with drilling freedoms, Comput. Methods Appl. Mech. Engrg., v192, pp. 21252168 (2003).

31

32


Blob regularization of boundary integrals Gregory Baker, Huaijian Zhang Department of Mathematics, The Ohio State University 231 W. 18th Ave, Columbus, OH43210, [email protected] Abstract: Boundary integral methods have proved very useful in the simulation of free surface motion, in part, because only information at the surface is necessary to track its motion. However, the velocity of the surface must be calculated quite accurately, and the error must be reasonably smooth, otherwise the surface buckles as numerical inaccuracies grow, leading to a failure in the simulation. For two-dimensional motion, the surface is just a curve and the boundary integrals are simple poles that may be removed, allowing spectrally accurate numerical integration. For three-dimensional motion, the singularity in the integrand, although weak, prevents highly accurate numerical methods, and the intuitive requirement that the errors be smooth can be difficult to achieve. By smoothing the integrand, or equivalently the Green’s function, by multiplication with an appropriate function, the singularity may be removed at a cost in accuracy. This loss of accuracy can be balanced with the accuracy of the numerical integration to produce an overall third-order accuracy. We demonstrate the results with some test cases. Key–Words: Free Surface Flows, Singular Integrals, Vortex Methods, Blob Regulariztion.

1

Introduction

Blob methods have been developed primarily for tracking vorticity in incompressible, inviscid flows numerically [1]. The Euler equations of motion in vorticity form are: ∂ω + u · ∇ω = ω · ∇u , ∂t ∇ · u = 0,

(1) (2)

where the velocity is u and the vorticity ω = ∇ × u. The velocity may be determined from the vorticity by the Biot-Savart integral [2], (3) u(x, t) = − ω(x , t) × ∇G(x − x ) dx , where G is the free-surface Green’s function for Laplace’s equation. The result is valid in the absence of solid boundaries, but additional contribution to the velocity can be added to account for them. If ω is known at some time t, then (3) can be integrated to determine u and then (1) can be advanced in time to update ω. Clearly, the method is well-suited for representing the vorticity at a collection of Lagrangian markers that then track with the fluid velocity. Only points where the vorticity is non-zero need be tracked. An important special case is when the vorticity is distributed as a delta function along a surface ω = γ δ(n) ,

subject to n · γ = 0 ,

(4)

where n is the distance along the normal n to the surface and γ is called the vortex sheet strength. Since the vorticity must point along a tangent to the surface, we may write γ = γ t, where t is a unit tangent vector. As a consequence, the fluid velocity can be expressed as u(x, t) = − γ(p, q) t(p, q) × ∇G x − x(p, q) dS(p, q) ,

(5)

(6)


33

where the surface location is written in parametric form x(p, q). There are several important properties of vortex sheets that are obtained by taking the limit as x → x(η, ζ), a point on the surface, along the normal there. The result is u± (η, ζ) = uP (η, ζ) ±

γ(η, ζ) n(η, ζ) × t(η, ζ) , 2

where the negative subscript is from the side into which the normal points and x(η, ζ) − x(p, q) 1 uP (η, ζ) = γ(p, q) t(p, q) × 3 dS(p, q) x(η, ζ) − x(p, q) 4π

(7)

(8)

is a principal-valued integral. Thus the vortex sheet strength measures the jump in tangential components of the velocity across the surface, (9) γ(p, q) n(p, q) × t(p, q) = u+ (p, q) − u− (p, q) , while the normal components are continuous, n(p, q) · u+ = n(p, q) · u− = n(p, q) · uP .

(10)

The nature of vortex sheets lends itself to a representation for free surfaces between immiscible fluids because the normal component of the fluid velocity is automatically continuous and the jump in tangential component captures the generation of vorticity in the presence of a jump in densities. Indeed, a vortex sheet can be viewed as a free surface between two immiscible fluids of equal densities where there is no generation of vorticity and the vortex sheet merely advects with the average fluid velocity at the surface. The extension to fluids with different densities on either side of the surface has been derived in two-dimensional motion [3] and in three-dimensional motion [4]; a comprehensive treatment is also available [5]. What these studies also show is that spectrally accurate methods are easily obtainable in two-dimensional motion because the pole singularity in boundary integrals can be removed leaving the integrands analytic. The 3/2-power singularity in (8) cannot be removed completely by analytic techniques, although it can be weaken further [4]. The challenge is to find accurate numerical methods for the integration (8) that give errors that are smooth, avoiding artificial buckling of the surface as it moves with the calculated velocity. One way forward has been suggested by an appropriate regularization of the Green’s function and tested thoroughly for two-dimensional motion with good success [6]. A version has been proposed for three-dimensional motion [7] with a specific application to deep water waves. Here, we are interested only in testing the success of regularization of the Green’s function in three-dimensional boundary integral methods.

2

Blob Regularization

Blob regularization is based on modifying the Green’s with a smoothing function. The approach adopted here is given in [7]. A smoothed Green’s function takes the form r 1 Gδ (x) = − S , with r = |x| . (11) 4πr δ The smoothing function S(r) is chosen so that S(r) → 1 rapidly as r → ∞. Since 1 1 d2 S r r ≡ , ψ ∇2 Gδ (r) = − 4πrδ 2 dr2 δ δ3 δ the result may be interpreted as an approximation to the delta function as δ → 0. The smoothing function S(r) must satisfy additional conditions to ensure accuracy in using Gδ in place of G. The error is dominated by the integral, (12) Gδ (x) − G(x) pn (x) dS(x) ,

34


where pn is a homogeneous polynomial of degree n. Expressed in polar coordinates, (12) takes the form, Cn δ n+1 S(r) − 1 rn dr .

(13)

The accuracy is determined by how many moment conditions (13) are satisfied. The choice, 2r 2 S(r) = erf(r) + √ e−r , π

(14)

ensures (13) is satisfied for both n = 0, 1. The error in using the blob regularization is then O δ 3 . Since the integration is now smooth with the regularized Green’s function, a standard integration such as the trapezoidal rule can by applied. The error can be shown to be O h3 provided δ/h is kept fixed [6, 7]. Here h is a measure of the spacing between the Lagrangian points. Thus altogether, the error is O h3 with δ/h kept fixed. A simple case will be used to confirm these estimates.

3

Test Case: A Cylindrical Vortex Sheet

The surface is given by x(p, q) = R cos(p) ,

y(p, q) = R sin(p) ,

z(p, q) = q ,

(15)

n = (− cos(p), − sin(p), 0) .

(16)

with corresponding surface vectors, t1 = (0, 0, 1) ,

t2 = (− sin(p), cos(p), 0) ,

The easiest way to construct velocity components (uP , vP , wP ) that correspond to a vortex sheet, γ = γ1 (p, q) t1 + γ2 (p, q) t2 ,

(17)

u = ∇φ ,

(18)

is to introduce the velocity potential,

since ∇ × u = 0 away from the surface. By invoking (2), φ must satisfy Laplace’s equation inside and outside the vortex sheet. Now simply make a choice, for example,

where f (r) must satisfy r

φ = f (r) cos(nθ) cos(αz) ,

(19)

d df r − (n2 + α2 r2 ) f = 0 , dr dr

(20)

the modified Bessel equation of integer order n. The appropriate choice of solutions, suitably normalized, are In (αr) , αIn (αR) Kn (αr) f (r) = B , αKn (αR)

f (r) = A

r < R,

(21)

r > R.

(22)

The requirement that the normal component of the velocity is continuous on the surface (10) makes A = B. The jump in velocity across the surface is u+ − u− =

AΓn −n sin(p) sin(np) cos(αq), n cos(p) sin(np) cos(αq), αR cos(np) sin(αq) , αR

(23)

which must match vortex sheet strength through (9). Here, Γn =

In (αR) Kn (αR) − . In (αR) Kn (αR)

(24)


35

Finally, nAΓn sin(np) cos(αq) , αR γ2 = −AΓn cos(np) sin(αq) . γ1 =

(25) (26)

The vortex sheet distribution generates the surface velocity up by the surface integral (8). In this specific example, sin(p) − sin(η), cos(η) − cos(p), 0 R2 ∞ 2π γ1 (p, q) up (η, ζ) = 3/2 dp dq 4π −∞ 0 R2 (cos(η) − cos(p))2 + R2 (sin(η) − sin(p))2 + (ζ − q)2 ∞ 2π cos(p) (ζ − q), sin(p) (ζ − q), R (1 − cos(η − p) R + γ2 (p, q) 3/2 dp dq . (27) 4π −∞ 0 R2 (cos(η) − cos(p))2 + R2 (sin(η) − sin(p))2 + (ζ − q)2 On the other hand, (7) means that 2up = u+ + u− , or up = (uP , vP , wP ) where nAVn sin(η) sin(nη) cos(αζ) , αR nAVn cos(η) sin(nη) cos(αζ) , 2vP (η, ζ) = 2A sin(η) cos(nη) cos(αζ) − αR 2wp (η, ζ) = −AVn cos(nη) sin(αζ) ,

2uP (η, ζ) = 2A cos(η) cos(nη) cos(αζ) +

where Vn =

In (αR) Kn (αR) + In (αR) Kn (αR)

(28) (29) (30)

(31)

The choice for γ1 (25) and γ2 (26) will produce the velocity components given above. This, then, provides a test case for the numerical integration of the boundary integrals (27) using the regularized Greens function (11).

4

Numerical Results

A specific test case is chosen with A = R = n = 1. The surface variables p and q are divided into N and M evenly spaced intervals respectively. This means the spacing can have different values, h1 = 2π/N and h2 = 2π/(αM ) depending the integration variable. By making different choices of N, M , we can assess the behavior of the error on either δ/h1 or δ/h2 . The numerical integration is performed by applying the standard trapezoidal rule in both integrations. The infinite range of integration in q may be replaced by a finite range of integration [0, 2π/α] since the vortex sheet strength is periodic and the method of images may be applied. Specifically, 2π/α ∞ ∞ F (q) dq = F (q + 2kπ/α) dq . (32) −∞

0

k=−∞

Bearing in mind that the integrands in (27) must be multiplied by H(r/δ) = S(r/δ) − (r/δ) S (r/δ) to account for the influence of the smoothing function, the parts of the integrands that must be summed are ∞ H(rk /δ) rk3

k=−∞

and

∞ −∞

(ζ − q − 2kπ/α)

H(rk /δ) , rk3

(33)

where rk2 = 2R2 (1 − cos(η − p)) + (ζ − q − 2kπ/α)2 is the denominator. The convergence of these sums can be improved by using the symmetries in the sums for large |k| and subtracting a common part whose sum is known. For example, the first sum can be written as (without the contribution from k = 0) ∞

∞ α3 1 α3 1 H(rk /δ) H(r−k /δ) + 3 + − . (34) 3 2 4π 3 |k|3 4π k3 rk3 r−k k=−∞ k=0

k=1


6

6

5

5

−log10(error)

−log10(error)

36

4

3

2

1

0

4

3

2

1

0

0.5

1

δ/h1

1.5

0

2

4.5

0

0.5

1

1.5

δ/h1

2

6

4 5

−log10(error)

−log10(error)

3.5 3 2.5 2 1.5

4

3

2

1 1 0.5 0

0

0.5

1

δ/h

1

1.5

2

0

0

0.5

1

1.5

2

δ/h

2.5

3

3.5

4

1

Figure 1: The absolute error in the velocity for different ratios h1 /h2 : top left figure h1 /h2 = 1; top right figure h1 /h2 = 2; bottom left figure h1 /h2 = 4; bottom right corner h1 /h2 = 0.5. Each curve corresponds to a different resolution, starting at N = 16 and increasing up to N = 512 in powers of 2. The first sum in (34) converges as 1/K 4 if the sum is truncated (−K, K) and the second sum has a known analytic result. The second sum in (33) can be treated similarly. First, let’s consider the relative role of h1 and h2 on the absolute error in the velocity at the point θ = 0 and z = 0 for the case α = 1. The ratio h1 /h2 is kept fixed while δ/h1 is varied. In this way we can assess the impact of the smoothed Green’s function for various choices of resolution. The results are shown in Fig. 1 for the ratios h1 /h2 = 1, 2, 4, 0.5. Each curve corresponds to a different resolution, starting at N = 16 and increasing in powers of 2 up to N = 512. All the results show a peak in accuracy although this peak depends on the resolution. For h1 = h2 , this peak is around δ = h1 . To the left of the peak, the accuracy drops to first-order as the effective range of the smoothing on the Green’s function δ falls below the grid spacing. To the right of the peak, the accuracy downgrades slowly but retains third-order convergence. A curious feature appears in the figure for h1 /h2 = 4 where an improvement in accuracy spikes around the value δ/h1 = 0.25. At this value, δ/h2 = 1. The interpretation is clear; there is improvement when δ exceeds the spacing h2 , but full accuracy is only reached when δ also exceeds h1 . When h1 and h2 are close, as in the results for h1 = 2h2 , the different in scales is less noticeable. The results for h1 = 0.5h2 confirm the interpretation; here δ/h1 = 2 or δ/h2 = 1 must be exceeded for full accuracy. The conclusion is that min(δ/h1 , δ/h2 ) should exceed 1 to obtain the best accuracy. A good choice seems to be h1 = h2 with δ/h1 between 1.0 and 1.5. Besides creating accurate results for boundary integrals associated with free surface motion in incompressible, inviscid fluids, the expectation is that the errors will prove to be smooth functions of the surface variables. This expectation is confirmed in Fig. 2, which shows the absolute error in the velocity as a function of p = θ and q = z for the case h1 = h2 , δ = 1.5h1 and N=256. Results have been presented for the choice α = 1. We have confirmed that the results are similar for other choices of α, specifically α = 2, 0.5.

5

Conclusion

Blob regularization of boundary integrals can produce very accurate results provided the blob size δ exceeds the largest grid spacing. The numerical evidence suggests that the errors are smooth functions of the surface


37

−5

x 10 1.5

error

1

0.5

0 6 6

4 4 2 z

2 0

0

θ

Figure 2: The absolute error in the velocity as a function of the surface coordinatesθ and z. parameters. Acknowledgements: The research was supported by NSF (grant OCE-0620885). References: [1] J.T. Beale and A.J. Majda, High order accurate vortex methods with explicit velocity kernels, J. Comput. Phys., 58, 1985, p. 188. [2] P.G. Saffman, Vortex Dynamics, Cambridge University Press, 1992. [3] G.R. Baker, D.I. Meiron and S.A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech., 123, 1982, pp. 477–501. [4] G. R. Baker, D. I. Meiron and S. A. Orszag, Boundary integral methods for axi-symmetric and threedimensional Rayleigh-Taylor instability problems, Physica 12D, 1984. [5] G.R. Baker, Boundary Element Methods in Engineering and Sciences, Chapter 8, Imperial College Press, 2010. [6] G.R. Baker and J.T. Beale, Vortex blob methods applied to interfacial motion, J. Comput. Phys., 196, 2004, pp. 233–258. [7] J.T. Beale, A convergent boundary integral method for three-dimensional water waves, Math. Comput., 70, 2001, p. 977.

38


On the accuracy of the fast hierarchical DBEM for the analysis of static and dynamic elastic crack problems I. Benedetti1, A. Alaimo1, M.H. Aliabadi2 1 Dipartimento di Ingegneria Strutturale Aerospaziale e Geotecnica, Università di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy, [email protected], [email protected] 2

Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW72AZ, London, UK, [email protected]

Keywords: DBEM, Adaptive Cross Approximation, Hierarchical Matrices, Fast BEM solvers, Elastodynamics, Laplace Transform Method, Stress Intensity Factors. Abstract. In this paper the main features of a fast dual boundary element method based on the use of hierarchical matrices and iterative solvers are described and its effectiveness for fracture mechanics problems, both in the static and dynamic case, is demonstrated. The fast solver is built by representing the collocation matrix in hierarchical format and by using a preconditioned GMRES for the solution of the algebraic system. The preconditioner is computed in hierarchical format by LU decomposition of a coarse hierarchical representation of the collocation matrix. The method is applied to elastostatic problems and to elastodynamic cases represented in the Laplace transform domain. The application of the hierarchical format in the Laplace domain is straightforward and offers some interesting advantages related to the use of some local preconditioners. The accuracy in the determination of both static and dynamic stress intensity factors is assessed and the effectiveness of the technique is successfully demonstrated. Introduction The Boundary Element Method (BEM) is nowadays a powerful numerical tool for the analysis and solution of many physical and engineering problems and represents a sensible alternative to other numerical approaches, such as the Finite Element Method (FEM), especially in some fields such as Fracture Mechanics [1, 2]. In general the main advantage offered by the BEM is related to the reduction in the degrees of freedom needed to model a given physical system. Such reduction relies on the underlying boundary integral formulation which requires, for its numerical solution, only the discretization of the boundary of the analyzed domain. This results not only in the reduced size of the solving systems, but also in faster data preparation. Although these appealing advantages, as the size of the analyzed problems increases BEM techniques lose part of their attractiveness, mainly due to the time required to solve the final system of equations. The solution matrix produced by BEM is in fact generally fully populated and neither symmetric nor definite. This circumstance results in the main drawbacks of the method, that is increased memory requirements as well as increased solution time with respect to other numerical approaches for problems of comparable numerical size. Such considerations have limited the size of the problems that could be effectively tackled on common computers using the standard BEM and have hindered for many years the industrial development of the method and confining its use to the analysis of small or medium size problems. However, in the recent years, considerable efforts have been devoted to the development of strategies aimed at reducing the computational complexities of the BEM, reducing both memory requirements and time consumption. Many investigations have been carried out and different techniques have been developed such as the fast multipole method (FMM) [3, 4], the panel clustering method [5], the mosaic-skeleton approximation [6] and the methods based on the use of hierarchical matrices [7]. The general aim of such techniques is to reduce the computational complexity of the matrix-vector multiplication which is the core operation in iterative solvers for linear systems. However, while FMMs and panel clustering tackle the problem from an analytical point of view and require the knowledge of some kernel expansion in advance to carry out the integration, mosaic-skeleton approximations and hierarchical matrices provide purely


39

algebraic tools for the approximation of the boundary element matrices, thus proving particularly suitable for problems where analytic closed form expressions of the kernels are not available or difficult to expand. In the present work the use of hierarchical matrices and iterative solvers for the rapid solution of threedimensional elastostatic and elastodynamic dual boundary element crack problems is described. The use of hierarchical matrices allows a noticeable reduction of the storage memory requirements for a given problem as well as a solution time reduction. In the following sections the basic equations of the dual boundary element method (DBEM) are briefly reviewed for both the static case and the dynamic formulation in the Laplace transform domain. The main features of the fast hierarchical solution strategy for both cases are then described. Finally, some problems are analyzed with a focus on the accuracy of the hierarchical solution in the determination of the crack parameters. The DBEM for the static and dynamic analysis of elastic crack problems The boundary integral equations governing the static behavior of an elastic body can be written as

ci j x0 u j x0 ³ Ti j x0 , x u j x d *

³U x , x t x d * ij

*

(1)

j

0

*

where U i j and Ti j are the fundamental solution of elastostatic problems. If an elastic dynamic problem is tackled by using the BEM in the Laplace transform domain, the following boundary integral equations can be written ci j x0 u j x0 ³ Ti j x0 , x, s u j x d *

³ U x , x, s t x d * ij

*

0

(2)

j

*

where the tilde indicates transformed quantities and s is the Laplace parameter. The boundary integral representation of the elastodynamic problem in the Laplace domain has the same form as that of the elastostatic problem. Eq.(2) is to be used in conjunction with the transformed boundary conditions to solve any specific problem. If cracks are present in the analyzed domain, also the following equations have to be used to close the problem cij x0 u j x0 cij x0 u j x0 ³ Tij x0 , x u j x d * *

³U x ij

*

cij x0 t j x0 cij x0 t j x0 n j x0 =³ Tijk x0 , x uk x d * *

0

, x t j x d*

n j x0 ³ U ijk x0 , x t j x d *

(3)

*

The previous equations are the displacement and traction boundary integral equations collocated at opposite crack surfaces [8]. Their form is the same for both static and dynamic problems in the Laplace transform domain, with the obvious difference that, in the last case, they involve the transformed kernels and transformed quantities depending on the Laplace parameter. After discretization and application of suitable boundary conditions, Eqs.(1-3) lead to a linear system of equations of the form A x

y

(4)

in the static case and of the form A ( s ) x

y

(5)

in the dynamic case in the Laplace transform domain, where the dependence on the complex parameter has been highlighted. To analyze a general elastodynamic problem by using the Laplace transform technique, one has generally to compute the solution of the system (5) for a set of Laplace parameters sk , with k 1,..., L , in order to calculate the time-dependent values of any relevant variable by means of some Laplace inverse transformation technique. Wen et al. [9] obtained for example accurate results for long durations in the time domain by using

40

sk


V

2S ki T

k

0,..., 25

(6)

with V T 5 and T t0 20 , where t0 is the unit time and by using the Durbin’s inversion technique to get time domain quantities. The static and dynamic stress intensity factors are computed by means of crack-front interpolation functions, which are used on the crack surface elements adjacent to the crack front in order to catch the r behavior of the displacements field. Discontinuous eight node quadrilateral elements are then used to represent the crack surface and the corresponding special shape functions, derived for eight-noded and ninenoded discontinuous Lagrangian elements by Mi et al. [19] and Cisilino et al.[18], are considered to compute the crack tip opening displacements. Once the crack displacement field is computed, the SIFs at a point P of the crack front can be directly obtained from the crack tip opening displacements

K IP

E S Pu ub ubPl 4(1 Q 2 ) 2r

K IIP

S Pu E un unPl 4(1 Q 2 ) 2r

K IIIP

E S Pu ut utPl 4(1 Q 2 ) 2r

(7)

where Pu and Pl are the nodes of the upper and lower crack surfaces behind the crack front while ub , un , and ut are the displacement components with respect to a local crack coordinate system [1]. The dynamic SIFs are computed directly from the transformed displacements of the crack surfaces and the corresponding time dependent values are obtained by the Durbin's Laplace transform inversion [9]. Hierarchical matrices for static and dynamic DBEM crack problems

To reduce both storage memory and solution time required by the elastostatic and elastodynamic BEM analysis in the Laplace domain, systems (4) and (5) are represented in hierarchical format. It is worth noting that, in the Laplace domain, system (5) has to be set for each value of the Laplace parameter. The hierarchical or low rank representation of a BEM matrix is built by generating the matrix itself as a collection of sub blocks, some of which admit a special approximated and compressed format. Such blocks, referred to as low rank blocks, can be stored in the form

B # Bk

k

¦u

i

vTi

U VT

(8)

i 1

The block B of order m u n is approximately generated through the product of U, of order m u k , and V T , of order k u n . If k, i.e. the rank of the block, is low, then the representation (8) allows to reduce both memory storage and the computational cost of the matrix-vector multiplication, which is the bottleneck of any iterative solver. The approximation of the low rank blocks (8) is built by computing only some of the entries of the original blocks through adaptive algorithms known as Adaptive Cross Approximation (ACA) [10, 11], that allow to reach an initially selected accuracy H. Low rank blocks represent the numerical interaction, through asymptotic smooth kernels, between sets of collocation points and clusters of integration elements which are sufficiently far apart from each other. The distance between clusters of elements enters a certain admissibility condition of the form

min diam : coll , diam : int d K dist (: coll , : int )

(9)

where : coll and : int are clusters of elements and K ! 0 is a parameter influencing the number of admissible blocks on one hand and the convergence speed of the adaptive approximation of the low rank blocks on the other hand [12]. The blocks that do not satisfy such condition are called full rank blocks and they need to be computed and stored entirely, without approximation. Once low and full rank blocks have been generated, some recompression techniques can be used to further reduce the storage memory and


41

computational complexity of the single blocks and of the overall hierarchical matrix (reduced SVD [13] and coarsening [14]). As an almost optimal representation is obtained, the solution of the system can be tackled either directly, through hierarchical matrix inversion [15], or indirectly, through iterative methods [16]. In both cases, the efficiency of the solution relies on the use of a special arithmetic, i.e. a set of algorithms that implement the operations on matrices represented in hierarchical format, such as addition, matrix-vector multiplication, matrix-matrix multiplication, inversion and hierarchical LU decomposition. A collection of algorithms that implement many of such operations is given in [12] while the hierarchical LU decomposition is discussed in [16]. The use of iterative methods takes full advantages of the hierarchical representation, exploiting the efficiency of the low-rank matrix-vector multiplication. The convergence of iterative solvers can be improved by using suitable preconditioners. In this work a hierarchical LU preconditioner is built starting from a coarse approximation of accuracy H p of the collocation matrix. In this work, an iterative GMRES algorithm is used in conjunction with the hierarchical preconditioner for solving the static system and the system set up for each value of the Laplace parameter of interest. Even using the hierarchical format, the setup of a preconditioner is an expensive procedure. In the dynamic analysis a new preconditioner should be built for each value sk of the Laplace parameter. In order to further speed up the overall dynamic analysis, local preconditioners are then used. For the interested reader, the full description of the use of local preconditioners can be found in the work of Benedetti et al.[17]. Here is only mentioned that a preconditioner computed for a certain Laplace parameter sk can be successfully used to precondition the system set up for other close Laplace parameters sk j , thus eliminating the need to compute a new preconditioner for each new Laplace parameter P sk , H p A sk j , H c x sk j P sk , H p y sk j

(10)

Numerical experiments

To assess the accuracy of the hierarchical DBEM in the determination of the relevant crack parameters, the stress intensity factors for a static problem and two dynamic cases are computed. Static analysis. An inclined embedded penny crack in a cylindrical bar subjected to static load acting over the bases is first considered, Fig.1. The crack is inclined of an angle D 45D with respect to the bases of the bar and has radius a. The bar has radius R 5a and height H 6 R . The analyzed mesh is comprised of 800 elements and 3652 nodes. The analytical solution of this problem is known in the case of infinite domain, that can be simulated with the abovementioned dimensions, and the expression for the stress intensity factors K I , K II and K III is reported in the literature [1].

Figure 1 Penny crack in a cylindrical bar and non-dimensional stress intensity factor for mode I.

42


Figure 2 Non-dimensional stress intensity factors for mode II and mode III.

The hierarchical solution is computed by setting: nmin

36 , K

3, Hp 4

102 and H GMRES

108 . Three

3

different accuracies are evaluated for the collocation matrix, H c 10 , H c 10 and H c 102 , and the effect on the accuracy of the SIFs is checked. In Fig.1 and Fig.2 the non-dimensional stress intensity factors for mode I, II and III, as computed by the hierarchical scheme for the different accuracies, are compared to the exact values and to the values obtained by the standard DBEM. All the SIFs are made dimensionless by dividing by Ki 0 i I , II , III , Ki 0 being the maximum value of the analytical SIFs. As it can be noted, the hierarchical analysis gives very accurate results in all the considered cases, and the error of the hierarchical solution with respect to the analytical values is in the range of 2-4%. The achieved speed-up ratios (hierarchical over standard time) are 0.34, 0.18 and 0.16 going from the highest to the lowest accuracy, while the required storage memory for the collocation matrix is 18.09%, 13.20% and 8.84% of the original allocation.

Dynamic analysis. The dynamic stress intensity factor (DSIF) for two crack problems has also been computed. The transformed DSIFs are computed directly from the transformed displacements on the crack surface and the time-dependent values are then obtained by the Durbin's Laplace transform inversion, following the procedure used in [9], to which the reader is referred for further details. An embedded crack case is considered first. A bar of size 2w1 u 2 w2 u 2h , containing a central pennyshaped crack, is subjected to the Heaviside traction load V (t ) V 0 H (t ) acting on the ends, as shown in Fig.3. The radius of the crack is a and w1 w2 , a 0.5w , h 2w . The Poisson's ratio is Q 0.2 . This problem has been considered by Wen et al. [9] and it has been dealt with by using the standard dynamic DBEM on which the present work is based. The standard DBEM has been validated in [9] by comparison with the displacement discontinuity method (DDM) [20] and it has been shown to produce highly accurate results. Here the results obtained by using the original code developed by Wen et al. are compared with the results given by the hierarchical solver. Fig.3 shows the normalized DSIF for mode I, i.e. K I t K 0 with K 0 2V 0 a / S , at the crack front point shown in Fig.3 with the thick marker. The DSIF given by the standard DBEM is compared with that computed by prescribing for the collocation matrix, in the hierarchical scheme, the accuracies H c 103 and H c 102 and setting the other hierarchical parameters to: nmin

36 , K

3, Hp

101 , N max

200 . In both cases the approximated solution, computed by using the

fast solver, is not distinguishable from that computed by using the standard direct solver. The present results have been obtained from the mesh shown in Fig.3, which is comprised of 530 elements and 1792 nodes. Each crack surface is modeled with 20 eight-node discontinuous elements. It is worth noting that, on the external boundary, this mesh is finer than that used by Wen et al. to capture the DSIF. However, it has been used here to force the computation of large matrix blocks through ACA and to investigate the effects of the hierarchical approximation on the quality of the computed crack parameters. It was observed that also the DSIF satisfies, in terms of L2 norm, the accuracy required to the hierarchical scheme.


Figure 3 Embedded crack in a prismatic bar and dynamic SIF for mode I.

It may be interesting to know that, for H c

103 , the overall assembly, solution and total speed up ratios

were 0.81, 0.16 and 0.26 respectively. For H c 102 , the same ratios were 0.73, 0.15 and 0.24. Moreover, in both cases, only around 75% of the original memory allocation was used for storing both the collocation matrix and the preconditioner. As second case, a surface breaking semi-circular crack of radius a is contained in a plate of size 2w u 2h u b subjected to the Heaviside load V (t ) V 0 H (t ) acting as shown in Fig.4. The dimensions of the plate are: a 1 cm , w 3 cm , h 3 cm , b 2.5 cm . The Young’s modulus is E 105 N / m 2 , the Poisson’s ratio Q 0.3 and the mass density U 103 kg / m 2 . The load amplitude is V 0 100 N / m 2 . A nonuniform mesh with 462 elements and 1533 nodes is used for the computation. The normalized DSIF for mode I at the thick point shown in Fig.4 is computed. This case has already been considered by Zhang and Shi [21] and Zhong and Zhang [22] and the results obtained in the present work by using the standard DBEM are in good agreement with their results, although some differences in detail may exist. Fig.4 shows the comparison of the DSIFs computed by the standard DBEM, using the code developed by Wen et al. [9], and those obtained by using the hierarchical solver. Two hierarchical solutions have been considered, setting the prescribed accuracy to H c 103 and H c 102 and the other hierarchical parameters to: nmin

36 , K

3, Hp

101 , N max

200 . In both cases the hierarchical solution is not graphically

distinguishable from the direct solution. The existing differences can be seen in terms of L2 norm for the DSIF values, and they are within the accuracy set for the hierarchical scheme. Overall assembly, solution and total speed up ratios were: 1.01, 0.20 and 0.39 for H c 103 ; 0.92, 0.18 and 0.36 for H c 102 . Between 75% and 85% of the original memory allocation was used for storing the collocation matrix. The hierarchical technique proves then to be a powerful, accurate and reliable tool for the dynamic analysis of crack problems, either in the case of embedded or surface breaking cracks, with uniform and non-uniform meshes. For further details about the structure of the hierarchical matrix with respect to the distance of the crack from the external surface the reader is referred to the previous works of the authors [23, 24].

43

44


Figure 4 Surface breaking crack in a prismatic bar and dynamic SIF for mode I.

Summary

In this work the use of a fast DBEM solver for the analysis of static and dynamic elastic crack problems has been described and its effectiveness in the determination of the static and dynamic stress intensity factors has been demonstrated. In particular, it has been shown that accurate SIFs can be obtained also setting a moderate accuracy for the hierarchical representation of the collocation matrix, obtaining then relevant savings in terms of storage memory and solution time. References

[1] M.H. Aliabadi, The Boundary Element Method: Applications in Solids and Structures, vol. 2. John Wiley & Sons Ltd, 2002. [2] M.H. Aliabadi, Applied Mechanics Reviews, 50, 83–96, 1997. [3] H. Rokhlin, J Comp Phys, 60, 187-207, 1985. [4] V. Popov, H. Power, Eng. An. Bound. Elem., 25, 7–18, 2001. [5] W. Hackbusch and Z.P. Nowak, Numerische Mathematik, 73, 207-243, 1989. [6] E. E. Tyrtyshnikov, Calcolo, 33, 47-57, 1996. [7] W. Hackbusch, Computing, 62, 89-108, 1999. [8] M.H. Aliabadi, International Journal of Fracture, 86, 91-125, 1997. [9] P.H. Wen, M.H. Aliabadi and D.P. Rooke, Comp Meth App Mech Eng, 167, 139-151, 1998. [10] M. Bebendorf, Numerische Mathematik, 86, 565-589, 2000. [11] M. Bebendorf, S. Rjasanow, Computing, 70, 1-24, 2003. [12] S. Börm, L. Grasedyck and W. Hackbusch, Eng. An. Bound. Elem., 27, 405–422, 2003. [13] M. Bebendorf, Effiziente numerische Lösung von Randintegralgleichungen unter Verwendung von Niedrigrang-Matrizen, Ph.D. Thesis, Universität Saarbrücken, 2000. dissertation.de, Verlag im Internet, ISBN 3-89825-183-7, 2001.


[14] L. Grasedyck, Computing, 74, 205-223, 2005. [15] L. Grasedyck, W. Hackbush, Computing, 70, 295-334, 2003. [16] M. Bebendorf, Computing, 74, 225-247, 2005. [17] I. Benedetti, M.H. Aliabadi, Int J Num Meth Eng, In Press (ref. NME-Aug-09-0529.R1), 2010. [18] A.P. Cisilino, M.H. Alaibadi, International Journal of Pressure Vessel and Piping, 70, 135-144, 1997. [19] Y. Mi, M.H. Alaibadi, International Journal of Fracture, 67, R67-R71, 1994. [20] P.H. Wen, Dynamic Fracture Mechanics: Displacement Discontinuity Method, Computational Mechanics Publications, Southampton and Boston, 1996. [21] Y.Y. Zhang, W. Shi, Engineering Fracture Mechanics, 47(5), 715-722, 1994. [22] M. Zhong, Y.Y. Zhang, Applied Mathematics and Mechanics, 22(11), 1344-1351, 2001. [23] I. Benedetti, M.H. Aliabadi, G.Davì, Int. J. Solids Structures, 45, 2355-2376, 2008. [24] I. Benedetti, A. Milazzo, M.H. Aliabadi, Int J Num Meth Eng, 80(10), 1356-1378, 2009.

45

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54


Non-Incremental Boundary Element Discretization of non-linear heat equation based on the use of the Proper Generalized Decompositions G. Bonithon1,4 , P. Joyot1 , F. Chinesta2 and P. Villon3 1 2

ESTIA-Recherche, technopole izarbel, 64210 Bidart, France, [email protected]

EADS Corporate Foundation International Chair, GEM CNRS-ECN, 1 rue de la Noë BP 92101, 44321 Nantes cedex 3, France, [email protected] 3 4

UTC-Roberval UMR 6253, 60200 Compiègne, France, [email protected]

EPSILON Ingénierie, 10 rue Jean Bart, BP 97431, France, 31674 Labège Cedex

Keywords: Boundary element method, Separated representations, Proper Generalized Decomposition

Abstract. In this work, we propose a new approach for solving the heat equation within the Boundary Elements method framework. This technique lies in the use of a separated representation of the unknown field that allows decoupling the space problem (that results steady state) from the temporal one (one dimensional that only involves the time coordinate). Introduction The Boundary Elements Method (BEM) allows efficient solution of partial differential equations whose kernel functions are known. The heat equation is one of these candidates when the thermal parameters are assumed constant (linear model). When the model involves large physical domains and time simulation intervals the amount of information that must be stored increases significantly. We propose here an alternative strategy able to change the nature of the problem. Thus, the temperature field involved by the so called heat equation is approximated using a separated representation involving products of space and time functions. This kind of approximation is not new, in fact, proper orthogonal decomposition [1] allows such one decomposition, but in this case this decomposition must be applied a posteriori, i.e. on the transient solution of the considered model. The technique that we propose in this paper allows to transform the transient model in a sequence of space problems (all of them steady state) and time problems (that only involve the time coordinate). This iteration procedure leads to a proper space-time generalized decomposition of the model solution. The efficiency of such one approach was proven in [2, 3, 4]. In our knowledge, this technique has never been coupled with a BEM for solving the resulting steady problem defined in the physical domain. We start summarizing the main ideas of the Proper Generalized Decomposition and we will focus on the application of such technique in the context of the BEM. Finally, numerical example, with a non linear source term, will be presented and discussed. A Proper Generalized Decomposition Boundary Element Method Let us consider the heat equation ∂u − a∆u = f (u) ∂t

in Ω × (0, Tmax ]

(1)

with homogeneous initial and boundary conditions, where a is the diffusion coefficient, Ω ⊂ Rd , d ≥ 1, Tmax > 0. The aim of the separated representation method is to compute N couples of functions


55

{(Xi , Ti )}i=1,...,N such that {Xi }i=1,...,N and {Ti }i=1,...,N are defined respectively in Ω and (0, Tmax ] and the solution u of this problem can be written in the separate form N

u(x,t) ≈ ∑ Ti (t) · Xi (x)

(2)

i=1

The weak formulation yields: Find u(x,t) such that Tmax Ω

0

u

∂u − a∆u − f (x,t) dx dt = 0 ∂t

(3)

for all the functions u (x,t) in an appropriate functional space. We compute now the functions involved in the sum (eq (2)). We suppose that the set of functional couples {(Xi , Ti )}i=1,...,n with 0 ≤ n < N are already known (they have been previously computed) and that at the present iteration we search the enrichment couple (R(t), S(x)) by applying an alternating directions fixed point algorithm that after convergence will constitute the next functional couple (Xn+1 , Tn+1 ). Hence, at the present iteration, n, we assume the separated representation n

u(x,t) ≈ ∑ Ti (t) · Xi (x) + R(t) · S(x)

(4)

i=1

The weighting function u is then assumed as u = S · R + R · S

(5)

Introducing (eq (4)) and (eq (5)) into (eq (3)) it results ∂R − a∆S · R dx dt = (S · R + R · S ) · S · ∂t Ω

Tmax 0

=

Tmax 0

Ω

(S · R + R · S ) ·

n ∂ Ti + a ∑ ∆Xi · Ti f (x,t) − ∑ Xi · ∂t i=1 i=1 n

dx dt

(6)

We apply an alternating directions fixed point algorithm to compute the couple of functions (R, S): • Computing the function S(x). First, we suppose that R is known, implying that R vanishes in (eq (5)). Thus, eq (6) writes Ω

S · (αt S − aβt ∆S) dx =

Ω

n

n

i=1

i=1

S · γt (x) − ∑ αti Xi + a ∑ βti ∆Xi

dx

(7)

56


where ⎧ Tmax ∂R ⎪ ⎪ ⎪ α = R(t) · (t) dt t ⎪ ⎪ ∂t 0 ⎪ Tmax ⎪ ⎪ ∂ Ti ⎪ i= ⎪ α (t) dt R(t) · ⎪ t ⎪ ⎪ ∂t ⎨ 0Tmax R2 (t) dt βt = ⎪ ⎪ 0 ⎪ Tmax ⎪ ⎪ ⎪ R(t) · Ti (t) dt βti = ⎪ ⎪ ⎪ 0 ⎪ ⎪ T max ⎪ ⎪ ⎩ γt (x) = R(t) · f (x,t) dt; ∀x ∈ Ω

(8)

0

The weak formulation (eq (7)) is satisfied for all S , therefore we could come back to the associated strong formulation n

n

i=1

i=1

αt S − aβt ∆S = γt − ∑ αti Xi + a ∑ βti ∆Xi

(9)

that one could solve by using any appropriate discretization technique for computing the space function S(x). • Computing the function R(t). From the function S(x) just computed, we search R(t). In this case S vanishes in (eq (5)) and eq (6) reduces to Tmax ∂R − a∆S · R dx dt = (S · R ) · S · ∂t 0 Ω Tmax n n ∂ Ti + a ∑ ∆Xi · Ti dx dt (S · R ) · f (x,t) − ∑ Xi · (10) = ∂t 0 Ω i=1 i=1 where all the spatial functions can be integrated in Ω. Thus, by using the following notations ⎧ ⎪ ⎪ = S(x) · ∆S(x) dx α x ⎪ ⎪ Ω ⎪ ⎪ ⎪ ⎪ αi = S(x) · ∆Xi (x) dx ⎪ x ⎪ ⎪ Ω ⎨ βx = S2 (x) dx (11) ⎪ Ω ⎪ ⎪ i ⎪ ⎪ βx = S(x) · Xi (x) dx ⎪ ⎪ ⎪ Ω ⎪ ⎪ ⎪ ⎩ γx (t) = S(x) · f (x,t) dx; ∀t Ω

equation (eq (10)) reads Tmax 0

n n ∂R i ∂ Ti i − aαx R − γx (t) + ∑ βx − ∑ aαx · Ti dt = 0 R · βx ∂t ∂t i=1 i=1

As eq (12) holds for all S , we could come back to the strong formulation

(12)


βx

n ∂ Ti n ∂R = a · αx · R + γx (t) − ∑ βxi · + ∑ a · αxi · Ti ∂t ∂t i=1 i=1

57

(13)

which is a first order ordinary differential equation that can be solved easily (even for extremely small times steps) from its initial condition. These two steps must be repeated until convergence, that is, until verifying that both functions reach a fixed point. The BEM is used to solve eq (9). We can notice that this equation defines a steady state elliptic equation with constant coefficients. Numerical example We considered a simple rectangular domain Ω = (0, 1) × (0, 1) and a time interval I = (0, 1]. The source term is set to f (u) = u2 (1 − u), the boundary conditions and the initial condition is set to an exact solution of this problem given by : ure f (x,t) = with η (x,t) =

√1 2

x + √t 2

eη(x,t) 2 + eη(x,t)

The domain boundary Γ consists of nΓ × nΓ segments Γi . The time interval I is discretized by using nτ nodes, uniformly distributed. First we are analyzing the convergence rates as a function of the space discretization (i.e. nΓ ). For all the space meshes the time discretization (i.e. nτ ) is adapted in order to reach the maximum precision. Figure 1 show the evolution of the L2 error in time and space as a function of the level of approximation, that is, as a function of the number of functional couples Xi (x) · Ti (t) involved in the approximation of u (x,t) for different meshes. This error is defined by: n re f ∑ Xi (x) · Ti (t) − u (x,t) i=1 en =

ure f (x,t)

2 LΩ×I

2 LΩ×I

We can notice that for a given number of functional couples the error en decreases when nΓ increases, reaching an asymptotic value. For reducing the value of the error we must increase nΓ as well as the number of functional couples Xi (x) · Ti (t) involved in the functional approximation. In the case of the example here addressed we must consider 4 functional couples for reaching a quatratic convergence rate for 4 nΓ 16. Figure 3 depicts functions {X1 (x) , T1 (t)}, {X2 (x) , T2 (t)}, {X3 (x) , T3 (t)} for nΓ = 16 and nτ = 256. Finally, figure 2 depicts the unknown field u (x,t). Conclusion The proposed approach transforms an incremental BEM procedure into a decoupled one that needs the solution of some steady problems defined in space (Poisson equation in the case here addressed),

58


3.0

3.5

log10(en )

4.0

4.5

2

5.0

nRS =1 nRS =2 nRS =3 nRS =4 5.5 1.3

1.2

1.1

1.0

0.9

log10(1/nΓ )

0.8

0.7

0.6

Fig. 1: Evolution of the error en versus the space discretization for different levels of approximation n. and some ordinary differential equations that only involve the time coordinate. Significant reduction of CPU time is expected due to the non-incremental nature of the proposed technique, as well as a significant reduction of the amount of information to be stored. As shown by the numerical exemple, this technique seems specially well adapted for the treatment of non-linear transient BEM models.


t =2.4e−01

59

t =5.0e−01 1.000 0.900

0.800

1 000

1 000

0.750

0.700

0.950

0.900

0.800

0.750

0.700

1.000

0.950

0.850

0.850

[min =0.68, max =1.00] × 5.3e−01 t =7.6e−01

[min =0.69, max =1.00] × 5.7e−01 t =1.0e +00

1 000

0.960

0.920

0.880

0.840

0.800

0.760 1 000

0.950

0.900

0.800

0.750

1.000

1.000

0.850

[min =0.71, max =1.00] × 6.0e−01

[min =0.72, max =1.00] × 6.3e−01

Fig. 2: u (x,t) and ure f (x,t) (dashed line) for t = {0.24s, 0.5s, 0.76s, 1s}, nΓ = 4 and nτ = 256

60


X1(

x) 1.2

×1 7 −03 . e

T1(t)

1.0

0.8 -0

-0.150 -0.300 -0.450

.9

00

0.6

0.4

0

5 .7

-0

0.2

0.0

00

.6

-0

0.2

0.4 0

X2(

x) 1.2

1.0

×6 7 −04 . e

T2(t)

-0.200

1.0

0.8 00

0.8

0

0.200

0.60

0.400

0.000

0.6

0.4

0.2

0.0

0.2

0.4 0

X3(

x)

0.6

1.0

×5 5 −05 . e

T3(t)

0.

0.2

50

-0

-0.1

-0.450

0

00

0.4

.9

0.0

-0.300

00

0.2

0 75

. -0

0.4

0.6

00

0

.0

0

0

.6

-0

0.8

1.0 0

Fig. 3: Functional couples {Xi (x) , Ti (t)} for nΓ = 16 and nτ = 256

1.0

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz References [1] F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchene, and F. Boust. Alleviating mesh constraints: Model reduction, parallel time integration and high resolution homogenization. Computer Methods in Applied Mechanics and Engineering, 197(5):400 – 413, 2008. [2] A. Ammar, F. Chinesta, and P. Joyot. The nanometric and micrometric scales of the structure and mechanics of materials revisited: An introduction to the challenges of fully deterministic numerical descriptions. International Journal for Multiscale Computational Engineering, 6(3):191–213, 2008. [3] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Journal of Non-Newtonian Fluid Mechanics, 139(3):153 – 176, 2006. [4] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids: Part ii: Transient simulation using space-time separated representations. Journal of Non-Newtonian Fluid Mechanics, 144(2-3):98 – 121, 2007.

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62


Three-dimensional Boundary Elements for the Analysis of Anisotropic Solids Federico C. Buroni†,1, Jhonny E. Ortiz2, Andrés Sáez3 School of Engineering, University of Seville, Camino de los descubrimientos s/n, E41092 Sevilla, SPAIN 1 [email protected], 2 [email protected], 3 [email protected] Keywords:Green’s function; Fundamental solutions; Cauchy’s residue theory; Degenerate materials; Anisotropy; Boundary element method (BEM)

Abstract. An alternative boundary element formulation for the analysis of anisotropic three-dimensional (3D) elastic solids is presented. This numerical procedure uses explicit expressions for the fundamental solution displacements and tractions developed by others authors by means of Stroh formalism and Cauchy’s residue theory, plus it implements a multiple pole residue approach to additionally account for the case of mathematically degenerate materials when Stroh’s eigenvalues are coincident. Meanwhile, the numerical instabilities that may be observed in quasi-degenerate cases when Stroh’s eigenvalues are nearly equal are addresed as well. Thus, an explicit Boundary Element Method (BEM) approach for the numerical solution of 3D linear elastic problems for solids with general anisotropic behavior is developed and validated with some numerical examples. Introduction Meanwhile for the two-dimensional anisotropic case many works have been reported in the literature, the three-dimensional (3D) modeling of general anisotropic elastic solids with BEM has not been studied so profoundly. The main difficulties deal with the numerical treatment of rather complex fundamental solutions. Ting & Lee [1] deduced a unique generic explicit expression in terms of the Stroh’s eigenvalues which is valid for mathematical degenerate cases, but the computation of the Green’s functions derivatives was not addressed in their work. Sales & Gray [2] and Phan et al. [3] later presented a multiple pole residue approach for the numerical evaluation of the Green’s function and its derivative which covers all the mathematical degenerate cases. However, this set of solutions has never been implemented into a BEM code to the best of the authors’ knowledge. By considering the integral expressions developed by Barnett [4] and applying the Cauchy’s residue theory, Lee [5] deduced explicit expressions for the first- and second-order derivatives of the Green’s function in terms of the Stroh’s eigenvalues for non-degenerate materials. The resulting expressions for the first-order derivative have been recently implemented and validated in the work of Shiah et al. [6] and even more recently implemented in a BEM code by Tan et al. [7]. In these works the degenerate problem is overcome by introducing a small perturbation to the repeated Stroh’s eigenvalues. The aim of the present work is to develop an alternative approach to numerically evaluate explicit 3D anisotropic displacement and traction fundamental solutions, both for mathematical degenerate and non-degenerate materials, and validate their implementation into a general BEM code. In this work, a multiple pole residue approach is proposed to treat mathematical degenerate cases based in the Ting & Lee’s [1] and Lee’s solution [5]. First the integral-form of the fundamental solutions are presented. Then, and starting from these integral-form solutions, expressions for the explicit evaluation of the displacement and traction fundamental solutions in mathematical degenerate and non-degenerate materials are developed. Validation of the proposed fundamental solutions in degenerate cases are presented. The boundary element formulation is illustrated by solving a benchmark example and finally the conclusions of the work are presented. Integral-form of displacement fundamental solution ∗ (x, x ) the displacement fundamental solution (free-space Green’s function) at a point x due to Let Ujk ∗ depends on the relative vector x − x so, henceforth a unit force applied at point x . The tensor Ujk


63

it is considered that the Cartesian coordinate system {xi } has the origin at the point x for simplicity. Following Ting & Lee’s approach [1], the Green’s function can be expressed as a singular term by a modulation function H as 1 ∗ Ujk (x) = (1) Hjk (x) 4πr where x = rˆ e with r = |x|. The modulation function Hjk (x) depends on the direction of x but not on its e) and that is one of the three Barnett-Lothe tensors which is symmetric, modulus, so Hjk (x) = Hjk (ˆ positive-definite and H(ˆ e) = H(−ˆ e). Hence, U∗ (x) is also symmetric and U∗ (x) = U∗ (−x). The tensor Hjk can be evaluated as [1] Hjk (ˆ e) = with

1 π

+∞ −∞

Γ−1 jk (p)dp

(2)

Γjk (p) = Qjk + (Rjk + Rkj )p + Tjk p2

expressed in terms of the parameter p, and being Qjk = Cijkl ni nl ;

Rjk = Cijkl ni ml ;

Tjk = Cijkl mi ml

(3)

e} where ni and mi are the components of any two mutually orthogonal unit vectors such that {n, m, ˆ is a right-handed triad. Note that Qjk and Tjk are symmetric and positive-definite. It is well-known that the kernel in (2) is a single-valued analytic function except for three complex poles with positive imaginary part and their conjugates. These poles correspond to the roots of the sixth-order polynomial equation |Γ(p)| = 0 (4) and in the Stroh’s formalism context are the Stroh’s eigenvalues pα [8]. The determinant in (4) can be factorized as [1] |Γ(p)| = |T| 3ξ=1 (p − pξ )(p − p¯ξ ) with T as defined in (3), the overbar ¯· denoting √ complex conjugate and pξ = αξ + iβξ , βξ > 0 ∧ αξ , βξ ∈ R with i = −1. ˆ jk = |Γ(p)|δpk where δpk is the Kronecker delta, ˆ jk be the adjoint of Γjk , defined as Γpj (p)Γ Let Γ then Hjk can be expressed as [1] Hjk (ˆ e) =

1 π|T|

ˆ jk (p) Γ

+∞

−∞

3

dp

(5)

(p − pξ )(p − p¯ξ )

ξ=1

In Ting & Lee’s work [1] the above integral for the Green’s function is further reduced to a general explicit expression in terms of the Stroh’s eigenvalues. However, direct derivation of the resulting expression is somehow cumbersome, so that an alternative approach is needed for the numerical implementation of the derivatives, as next shown [5]. Integral-form of traction fundamental solution The traction fundamental solution follows easily from the derivative of the free-space Green’s function as ∗ ∗ Tik = Cijlm Ulk,m ηj (6) where ηj are the components of the external unit normal vector to the boundary ∂Ω at point x and Cijkm denote the components of the fourth-rank elasticity tensor. The first-order derivative of the Green’s function may be expressed in a similar way to equation (1), as a singular term by a modulation ˆ as function which only depends on e ∗ Upj,q (x) =

1 ˜∗ e) U (ˆ 4πr 2 pj,q

(7)

64


where, according to Lee’s approach [5], the modulation function is given by Cpqrs ˜ ∗ (ˆ el Hij + (Mlqiprj eˆs + Msliprj eˆq ) U ij,l e) = −ˆ π

(8)

where the Msliprj integrals have the following representation in terms of the parameter p [5] Mijklmn =

1 |T|2

+∞

−∞

Φijklmn (p) dp (p − p1 )2 (p − p2 )2 (p − p3 )2

(9)

where T has been previously defined in (3), pα are the Stroh’s eigenvalues and the function Φijklmn (p) :=

ˆ kl (p)Γ ˆ mn (p) Bij (p)Γ (p − p¯1 )2 (p − p¯2 )2 (p − p¯3 )2

(10)

has been introduced together with the definition of Bij := ni nj + (ni mj + mi nj )p + mi mj p2

(11)

The Φijklmn (p) function is analytic everywhere in the upper half-plane ((p) > 0) and the kernel in the integral (9) has three complex-double poles with positive imaginary part corresponding to the roots of |Γ(p)|2 = 0. ˆ the components Mijklmn satisfy the following Because of the symmetry of the adjoint matrix Γ, symmetry conditions Mijklmn = Mijmnkl = Mijlkmn = Mijklnm. The matrix Bij is also symmetric, resulting in an additional symmetry Mijklmn = Mjiklmn . This leads to a considerable reduction in the number of components Mijklmn to be calculated, and it must be considered in the numerical implementation. Mijklmn represents 729 components, but only 126 of them are different. It is necessary to remark that in the Lee’s work [5] explicit expressions in terms of Stroh’s eigenvalues are further obtained, but only for mathematical non-degenerate cases (p1 = p2 = p3 ). Multiple pole residue approach Both the integrals in the Barnett-Lothe tensor Hij (ˆ e), Equation (5), and the integrals Mijklmn (ˆ e) in the modulation function of the derivative of the Green’s function, Equation (9), can be evaluated by Cauchy’s residue theory. This approach leads to explicit expressions in terms of the Stroh’s eigenvalues. From a numerical point of view, it is actually necessary to provide a scheme for Green’s function evaluation of general validity. Such scheme should be able to deal with both mathematical degenerate and non-degenerate materials. The degenerate case occurs when repeated Stroh’s eigenvalues are obtained, and this may happen due both to the material properties Cijkm and the direction of the vector x. Furthermore, numerical instabilities should be avoided in quasi-degenerate cases as well, when Stroh’s eigenvalues are sufficiently close. In this work, a multiple pole residue approach is proposed in order to overcome such degeneracies and obtain accurate results, leading to new expressions for the explicit evaluation of the derivative of the Green’s function. +∞ In order to compute an integral of a rational function of complex variable of the form −∞ f (p)dp, calculus of residues yields +∞

−∞

f (p)dp = 2πiσ

(12)

where σ is the sum of the residues Res(po ) of f (p) at the poles po which lie in the upper half plane. Let the point po be a pole of m-order of f (p). A formula for evaluating the residue at this pole is given as (see, e.g., [3, 10]) 1 dm−1 [(p − po )m f (p)] (13) lim Res(po ) = (m − 1)! p→po dpm−1 Equation (13) allows to write explicit solutions in terms of the Stroh’s eigenvalues for degenerate and non-degenerate cases.


65

Table 1: Material A. Transversely isotropic symmetry Elastic constants 109 mN2 C1111 C1122 C3333 C2323 C1133 √ C1111 C3333 −C1133 1133 2 49.4 34.6 30( C 9.7 C1122 ) 2 Although in Ting & Lee’s work [1] a solution valid for degenerate cases has been derived for the displacement fundamental solution, in the present work a multiple pole residue approach is proposed also for the displacement solution for the sake of completeness, in a similar fashion to the works by Wang [9] and Phan et al. [3]. At most, there are N (1 N 3) distinct Stroh’s eigenvalues pα of mα -multiplicity. Hence, a general expression, valid for degenerate and non-degenerate materials, of the Barnett-Lothe tensor Hjk may be obtained as ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ mα −1 N ˆ jk (p) ⎥ ⎢d Γ 2i 1 ⎥ ⎢ Hjk (ˆ e) = { } ⎥ N |T| (mα − 1)! ⎢ dpmα −1 ⎢ m α=1 (p − p¯α )mα [(p − pξ )(p − p¯ξ )] ξ ⎥ ⎦ ⎣ ξ=1 ξ=α

(14) at p=pα

Similarly, general expressions both for degenerate and non-degenerate materials may be derived for the Mijpkmn components to yield ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ 2mα −1 N ⎥ ⎢d (p) Φ 2πi 1 ijpkmn ⎢ Mijpkmn (ˆ e) = { N }⎥ ⎥ ⎢ 2 2m −1 α |T| (2mα − 1)! ⎢ dp α=1 [(p − pξ )]2mξ ⎥ ⎦ ⎣ ξ=1 ξ=α

(15) at p=pα

Validation and a BEM example In order to assess and validate the proposed scheme, a material with transverserly isotropic symmetry is next considered, since it exhibit degeneracy [11] and analytic closed-form solutions are available [12]. An artificial Material A is considered which has two repeated Stroh’s eigenvalues for any direction of the evaluation vector x, except for evaluation points x along the isotropy axis (x3 -axis) where the degeneracy (p1 = p2 = p3 ) is observed. The elastic constants for Material A are listed in Table 1122 1. Moreover, for transversely isotropic materials it is satisfied C1212 = C1111 −C , C1313 = C2323 , 2 C2222 = C1111 , C2233 = C1133 . Observation points x = {0, sin φ, cos φ} are considered. In Figure 1(a) the modulus of the difference between the Stroh’s eigenvalues |pα −pβ | (α = β), in function of the φ-angle with respect to the x3 -axis, are shown for the Material A. The degeneracy behavior previously mentioned is clearly observed. Hence, ∗ this material is degenerate anywhere and non-degenerate solutions are not defined. Arbitrarily, U22 ∗ components are considered. Figures 1(b) and 1(c) show these components of the fundamental and T22 solutions, respectively, for r = 1 and an outward unit normal defined by η = {0, sin φ, cos φ}. The displacement fundamental solution component U22 is evaluated by using Equations (14) (assuming that p1 = p2 = p3 ) and (assuming that p1 = p2 = p3 ). In the same way, the traction ∗ is evaluated assuming the p = p = p degeneracy and assuming fundamental solution component T22 1 2 3 that p1 = p2 = p3 by using Equations (14) and (15). The analytic solution by Pan & Chou [12] is considered for comparison and plotted together with the present solution. The proposed degenerate solutions provide a stable and accurate solution in function of the kind of the mathematical degeneracy. The proposed multiple pole residue approach is implemented in a three-dimensional collocationBEM code. Nine-node quadratic boundary elements are used. A simple example is presented in order

66


*$$ *

*$ $ * *$$ *

%&

"#(&#!('#"

%&"')#% $ $ $

%&"')#% $ $ $

%&

"#(&#!('#" %&"')#% $ $ $

%&"')#% $ $ $

%&

∗ and (c) T ∗ components vs. Figure 1: (a) Absolute values of difference between eigenvalues, (b) U22 22 φ-angle for Material A.

to futher validate the correctness of the expressions for the displacement fundamental solutions Uij∗ and the traction fundamental solutions Tij∗ developed in the present work. A cube of length a subjected to uniaxial tension σ0 is considered, as sketched in Figure 2. Symmetry boundary conditions are imposed on the coordinate planes, i.e., u1|x1 =0 = 0, u2|x2 =0 = 0 and u3|x3 =0 = 0. The cube is discretized using one element per face, totalizing 6 elements and 26 nodes. Two set of materials are considered. First, an isotropic material (Material B) with the elastic constants listed in Table 2. In such a case, a triple eigenvalue p0 = i, ∀x exists. The previously defined Material A is considered as well. The present formulation is robust enough to model both degeneracies. The displacements values obtained at points A, B, C, D and E shown in Figure 2 are listed in Tables 3 and 4 for the two materials B and A, respectively. The values are normalized and compared with the corresponding analytic solutions in both Tables. Excellent accuracy is observed for all the cases.

Figure 2: Cube under tension stress.


67

Table 2: Material B. Isotropic material Elastic constants 109 mN2 C1111 C1122 15.7264 6.4957 Table 3: Results of BEM example. Cube under simple tension with isotropic material properties (Material B). Point A

Coordinates (a, 0.5a, 0.5a)

B

(a, a, 0.5a)

C

(0.5a, a, a)

D

(a, 0.5a, a)

E

(a, a, a)

Result Present work Analytic solution Present work Analytic solution Present work Analytic solution Present work Analytic solution Present work Analytic solution

u1 /ua3 -0.14615 -0.14615 -0.29231 -0.29231 -0.58463 -0.58463 -0.14615 -0.14615 -0.29231 -0.29231

u2 /ua3 -0.14615 -0.14615 -0.14615 -0.14616 -0.58463 -0.58463 -0.29231 -0.29231 -0.29231 -0.29231

u3 /ua3 1.00001 1.00000 1.00001 1.00000 1.00002 1.00000 1.00001 1.00000 1.00001 1.00000

Summary and conclusions A multiple pole residue approach has been proposed to derive explicit expressions for the numerical evaluation of the 3D anisotropic fundamental solutions. These expressions are based on Ting & Lee’s [1] and Lee’s [5] solutions for the Green’s function and its derivative, and result in terms of the Stroh’s eigenvalues, so there is no integration needed in the proposed approach. The solution covers all the possible mathematically degenerate and non-degenerate materials, while being numerically stable for quasi-degenerate cases when Stroh’s eigenvalues are nearly equal. Subsequently, an explicit BEM implementation for modeling three-dimensional materials with general anisotropic behavior has been presented. In contrast with previous works, the media is allowed to be both mathematically degenerate or non-degenerate, and the formulation is independent of the approximation order of the physical variables and geometry.

Table 4: Results of BEM example. Cube under simple tension with degenerate transversely isotropic material properties (Material A). Point A

Coordinates (a, 0.5a, 0.5a)

B

(a, a, 0.5a)

C

(0.5a, a, a)

D

(a, 0.5a, a)

E

(a, a, a)

Result Present work Analytic solution Present work Analytic solution Present work Analytic solution Present work Analytic solution Present work Analytic solution

u1 /ua3 -0.23061 -0.23095 -0.23074 -0.23095 -0.05772 -0.05774 -0.11539 -0.11547 -0.11553 -0.11547

u2 /ua3 -0.11535 -0.11547 -0.23079 -0.23095 -0.11546 -0.11547 -0.05774 -0.05774 -0.11551 -0.11548

u3 /ua3 0.99929 1.00000 0.99929 1.00000 0.99936 1.00000 0.99924 1.00000 0.99995 1.00000

68


Acknowledgments This work was supported by the Ministerio de Ciencia e Innovación of Spain and the Consejería de Innovación, Ciencia y Empresa of Andaluca (Spain) under projects DPI2007- 66792-C02-02 and P06TEP-02355. J.E. Ortiz was supported by the Ramón y Cajal Program of the Ministerio de Ciencia e Innovación of Spain. F.C. Buroni is gratefully acknowledged to the Junta de Andalucía of Spain for financial support throughout the Excellence Scholarship Program. References [1] Ting TCT, Lee VG. The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids Q. J. Mech. Appl. Math.1997; 50:407-426. [2] Sales MA, Gray LJ. Evaluation of the anisotropic Green’s function and its derivatives Computers and Structures 1998; 69:247-254. [3] Phan AV, Gray LJ, Kaplan T. Residue approach for evaluating the 3D anisotropic elastic Green’s function: multiple roots. Engineering Analysis with Boundary Elements 2005; 29:570-576. [4] Barnett DM. The precise evaluation of derivatives of the anisotropic elastic Green’s functions. Phys. Stat. Sol. (b) 1972; 49, 741-748. [5] Lee VG. Explicit expressions of derivatives of elastic Greens functions for general anisotropic materials. Mech. Res. Comm. 2003; 30, 241-249. [6] Shiah YC, Tan, CL, Lee VG. Evaluation of Explicit-form Fundamental Solutions for Displacements and Stresses in 3D Anisotropic Elastic Solids. Computer Modelling in Engineering and Sciences 2008; 34:205-226. [7] Tan CL, Shiah YC, Lin CW. Stress analysis of 3D generally anisotropic elastic solids using the boundary element method Computer Modelling in Engineering and Sciences 2009; 41:195-214. [8] Ting TCT. 1996 Anisotropic Elasticity, Oxford University Press, Oxford. [9] Wang CY. Elastic fields produced by a point source in solids of general anisotropy. Journal of Engineering Mathematical 1997; 32:41-52. [10] Sveshnikov A, Tikhonov A. 1978 The theory of functions of a complex variable, 2nd edn. Moscow: Mir Publishers [11] Tanuma K. Surface-impedance tensors of transversely isotropic elastic materials.Quarterly Journal of Mechanics and Applied Mathematics1996;49:29-48. [12] Pan Y, Chou T. Point force solution for an infinite transversely isotropic solid. Journal of Applied Mechanics and Transactions ASME 1976; 98:608-612.


Sensitivity analysis of cracked structures with static and dynamic Green’s functions Oliver Carl, Chuanzeng Zhang Department of Civil Engineering, University of Siegen, Paul-Bonatz-Str. 9-11, 57076 Siegen, Germany, E-mails: [email protected], [email protected] Keywords:

Green’s functions, sensitivity analysis, linear elastic fracture mechanics, spring model

Abstract. In this paper, a simplified analytical method for sensitivity analysis of cracked or damaged structures is presented. The method enables the prediction of the difference between the solutions for an uncracked and a cracked structure by considering only the cracked or damaged region of a structure, which results in a local analysis and consequently lower computational effort. The method is based on the comparison between the strain energies of an uncracked and a cracked structure as well as the exact or approximate Green's functions of the problem.

Introduction From the point of view of structural mechanics, a static or dynamic analysis of a structure is nothing else than the establishment of a correlation between the external loading (action or input) and the structural response to the loading (reaction or output). For linear problems, the correlation can be described by the socalled Green's functions. If a structural or material change appears in a part of a structure, such as stiffness changes and cracking of the structural components during the manufacturing process or under in-service loading conditions, the static and dynamic responses or the problem solutions of the structures will be also altered. In the linear structural analysis, the corresponding new solutions can be found by using modified Green's functions. However, such Green’s functions have to be formulated and derived for the cracked or modified structures, which requires a new global analysis of the structures and will be hence computationally expensive. In this paper, sensitivity analysis by the Green’s function (SAGF) approach [4, 12], derived for the local analysis of stiffness modifications and loss of supports, is applied to static and dynamic problems. Furthermore stiffness modifications in structures due to damage and cracking are modeled by the concept of linear elastic fracture mechanics and applied to beam-like structures. For analytical solutions, cracked regions of beam or truss structures are approximated by spring models based on linear elastic fracture mechanics. The analytical solutions are suitable for sensitivity analysis of weakened or cracked beam and truss structures consisting of homogeneous materials or fiber-reinforced composite like reinforced concrete beams and bars. For more complicated problems, sensitivity analysis is applied to derive cracked finite elements based on cracked or weakened Green's functions. Then, the cracked finite elements can be utilized for the numerical solutions of the more complicated problems in engineering applications.

Global analysis by Betti’s theorem and Green’s functions A two-dimensional linear elastic problem namely the plane stress state in a bounded body with a continuous boundary * * D * N is considered. The displacement vector u >ui @ satisfies the following partial differential equations Lij u j U ui P ui , jj O P u j , ji U ui pi , i, j 1, 2 , (1) with Lamé operator L ª¬ Lij º¼ , inertia force U ui according to d’Alembert and body force vector pi . Additionally we consider the following boundary conditions ui (x, t ) ui* on * D , V ij n j ti* on * N , (2)

69

70


and to complete the problem formulation the following initial conditions at time t ui (x,0) ui0 and ui (x,0) vi0 . The stress tensor

0 (3)

ª¬V ij º¼ is defined by Hooke’s law V ij O H kk G ij 2P H ij ,

and the linearized strain tensor

(4)

ª¬H ij º¼ by

1 (5) ui, j u j ,i 2 with the isotropic elastic constants O and P . An equivalent formulation of the initial boundary value

H ij

problem is the variational equation of motion [2] with the test function uˆ u V for all t such that * ³ V ij Hîj d : ³ U ui uî d : ³ pi uî d : ³ ti uî d * N . :

:

:

ûî `

for finding a function

(6)

*N

Considering a structure under time-harmonic loading with an excitation frequency Z q x, t q x eiZt ,

(7) the displacements are also time-harmonic or in a steady state u x, t u x eiZt . (8) From eq (6) we obtain Green’s first identity or the principle of virtual displacements allowing for homogeneous initial conditions 2 * ³ V ij Hîj d : U Z ³ ui uî d : ³ pi uî d : ³ ti uî d * N . (9) :

:

:

*N

ª¬ Lij º¼ leads to a global analysis, in which we have to consider the complete structure regardless whether the domain is uncracked or cracked B u, uˆ ³ ( Lu U Z 2 u) uˆ d : ³ (t uˆ u tˆ) d * ³ u (Luˆ U Z 2 uˆ ) d : 0 . (10) In this case, Betti’s reciprocal theorem for the Lamé operator L

:

*

:

To obtain an integral representation for the displacements u x at a point x

x1 , x2 , we substitute for

uˆ

the associated Green’s functions G y , x into the reciprocal theorem. The Green’s functions are the solution of the dual problem L y G ij y, x U Z 2 G ij y , x G i j y x I on : (11) j i

and satisfy homogeneous boundary conditions on * D and * N . To represent a point load ( i 0 ) in

x j direction acting at point x , we introduce the Dirac delta function G i j y x and the identity matrix I .

Substituting for u the solution of eq (1) with u 0 on * D , we find from Betti’s reciprocal theorem

u j x, t

Z ³ G y, x q y e d : j 0

i t

:

y

Z ³ G y, x t y e d * j 0

*

i t

y

*N

To obtain an integral expression for the stresses we find analogously V jk x, t ³ G1jk y , x q y eiZt d : y ³ G1jk y , x t* y eiZt d * y :

*N

(12)

(13)

where the lower subscript i 1 denotes that G y , x are the Green’s functions based on first-order derivatives of the Dirac delta function [12]. Exact closed-form Green’s functions from eq (11) can be obtained only for some special problems using for example Laplace- or Fourier transform. For more complicated problems we have to use approximate Green’s functions Gi ,jh y, x . A tool for this purpose is the boundary element method [11] or the finite element method. Working with the finite element method the displacement fields representing the Green’s functions are computed by nodal loads based on Dirac delta function or their derivatives [12]. To improve the approximated Green’s functions from FE analysis the Green’s function decomposition approach is suggested [9]. The method is originally developed for point-wise error estimation and adaptivity jk 1


71

in finite element method. The Green’s functions are decomposed into the known fundamental solutions and the unknown regular solutions, which will be found by a FE calculation G ij,h y , x g ij y , x u Rj , h y , x . (14)

Sensitivity analysis by Green’s functions for the local analysis of damaged regions In this section, we derive an analytical method for sensitivity analysis of cracked or weakened structures, in which we only need to consider the damaged regions of a structure to predict changes in the deformations, stresses, eigenfrequencies or mode shapes. This method leads to a lower computational effort in contrast to the classical procedure in structural mechanics. The basic idea of sensitivity analysis by the Green’s functions (SAGF) approach is based on the weak-form eq (9) for an uncracked system 2 * ³ V ij Hîj d : U Z ³ ui uî d : ³ qi uî d : ³ ti uî d * N . (15) :

:

:

*N

Taking the stiffness modifications due to damage into account, the weak-form for a cracked system with additional symmetric terms of the damaged region :c can be written as

³ O H

:

c kk

G ij 2P H ijc Hîj d :

³ 'O H

c kk

:c

G ij 2'P H ijc Hîj d :c U Z 2 ³ uic uî d : Z 2 ³ 'U uic uî d :c :

:c

³ q uˆ d : ³ t i

i

:

* i

uî d * N .

(16)

*N

The displacement vector of the cracked system is denoted by u c u 'u , and the mass and stiffness modifications are represented by 'U and the modified elastic constants 'O and 'P . After some mathematical manipulations we obtain the difference of the strain energies and inertia terms of the original (uncracked) system and the modified (cracked) system 2 c 2 c ³ 'V ij Hîj d : U Z ³ 'ui uî d : ³ V ij Hîj d :c Z ³ 'U ui uî d :c , (17) :

with 'V ij

:

:c

O 'H kk G ij 2P 'H ij and V ijc

:c

'O H kkc G ij 2'P H ijc . Substituting the virtual displacements uˆ with

the associated Green’s functions G ij we obtain the central equation of the approach

J 'u 'Es 'Ek

³ ' <

Gi j

:

d : U Z 2 ³ 'u < G ij d : :

³ c < Gi d : c Z 2 ³ 'U u c < G ij d : c , j

:c

:c

(18)

where 'Es and 'Ek represent the changes in strain and kinetic energy. Moreover we know from [12], that the changes in deformations or internal forces at a position x can be expressed by a linear functional J 'u J uc J u . Equation (18) is superior to Betti’s reciprocal theorems eq (12) and (13). It should be noted that eq (18) can also be rewritten as ³ c < Gi d : c Z 2 ³ 'U u c < G ij d : c j

:c

:c

³ < Gc i d : c Z 2 ³ 'U u < G ij,c d : c , j

:c

:c

(19)

ª¬V ij º¼ 'O H kk G ij 2'P H ij . To apply the derived method we must know the solution u c of a cracked problem and the Green’s functions for the uncracked system or vice versa. But this is not very practical, because if we solve the boundary value problems for both systems, we are also able to compare the two solutions u and u c directly. Using the following approximation, we only consider terms of the original (uncracked) system to predict response changes due cracks or damages

in which a dot denotes the scalar product of two vectors or matrices and

J 'u | ³ < Gi d :c Z 2 ³ 'U u < G ij d :c . j

:c

:c

(20)

If we apply the SAGF approach to crack problems, then the mass density will not change in the damaged region and with 'U | 0 we have

J 'u | ³ < Gi d :c . j

:c

(21)

72


Beam problems and approximation of cracks with spring models In beam problems the cracks are approximated by spring models. For mode-I cracks we use rotational springs and for mode-II cracks translational springs as depicted in Fig. 1. The spring stiffnesses cM and cw are obtained by using the concept of linear elastic fracture mechanics and derived from the energy release rate [10]. The compliances or inverse spring stiffnesses are 1 2b 1 2 cw cM E 1 2Q b2 M 2 ³ K I2 da and ³ K II da , E 1 Q 2 Q 2 a a

(22)

with elastic modulus E , Poisson’s ratio Q , beam width b, bending moment M , transverse shear force Q , and the stress intensity factors depending on the crack configuration [14]. M

M

Q

h

h

a

a

(mode I) M

M

Q

Q

(mode II)

cw

cM

Q

Fig. 1 Approximation of cracks for different kinds of loading and crack configurations by spring models. Cracking in fiber-reinforced composites like reinforced concrete beams can also be approximated by spring models, Fig. 2. The spring stiffness of a rotational spring used for bending cracks is split into a concrete and a reinforcement term cM , j c j ,concrete c j ,reinforcement , (23) in which the stiffness of the cracked concrete is modeled by eq (22) also taking the interaction of multiple cracks into account [6, 15]. The spring stiffness of the reinforcement depends on some geometrical considerations in combination with the steel’s strain, the cross-sectional area, the details of the reinforcement, and the crack width [5]. M

M

M

M cM ,1

cM ,2

cM ,3

Fig. 2 Reinforced concrete beam with bending cracks approximated by rotational springs. Applying eq (18) to cracked beam-like structures with spring models, we obtain the change in deflection, slope, bending moment or transverse shear force ( i 0, 1, 2, 3 ) due to cracking in an initially uncracked system as w i 'w w i wc w i w 1 1 i ¦ M xc , j M Gi ,c xc , j Q xc , j QGi ,c xc , j (24) wxi wxi wx cw, j j cM , j

with the bending moment and the shear force resulting from the load case of the uncracked system, and the bending moment and the shear force from the associated Green’s functions Gi ,c for the cracked system, which are, similar to eq (21), approximated by the Green’s functions Gi of the uncracked system at the position xc of the crack number j . Closed-form solutions. For Euler-Bernoulli beams, closed-form analytical solutions are obtained by using static or dynamic Green's functions for uncracked beams, which are modified and include terms taking


73

cracks into account. Applying the SAGF approach in combination with the superposition principle it is also possible to take stiffness jumps into account and add new supports if they are elastic and modeled by springs. Furthermore for computations with the theory of plastic limit analysis of continuous beams in which plastic hinges are modeled by rotational springs with spring stiffness cM o 0. This means an extension of the closed-form solutions for continuous beams under static loadings from [8] and for dynamic problems from [1]. Numerical solutions. For numerical solutions of more complicated beam-like structures, modified beam elements based on Green’s functions of a cracked clamped-clamped beam as shown in Fig. 3 are derived and can be easily integrated into standard FE codes. Considering a mode-I crack, e.g. the entry k33c of the modified element stiffness matrix is given by 3EI xc2 cM l 3 4 EI . k33c D 33 k33 (25) 2 4 EI l 3 xc2 3 l xc cM l 3 l Comparing the modified element stiffness matrix with that from [3], in which cracks are also approximated by spring models and their degrees of freedom are eliminated by Guyan’s reduction, we find that the two solutions are identical. M2 M1

k23c

Q1

w1 1

Q2

cM

c k26

modified stiffness matrix

c k25

c k22

cracked Green's functions M1

k33c

Q1

M2

k32c

M1 1

cM

Q2

xc

k35c

k36c

Kc

ª k11c « «0 « « « « « ¬«

0 c k22

sym.

0 k23c k33c

k14c 0 0 k44c

0 k25c k35c 0 k55c

0º » k26c » k36c » » 0» c » k56 » k66c ¼»

l xc

Fig. 3 Cracked Green’s functions to obtain the modified stiffness matrix of a cracked beam element.

Numerical Examples Example 1: Cracked dynamic Green’s functions of a two-span beam for closed-form analytical solutions The cracked static or dynamic Green’s functions of a two-span beam can be derived from the Green’s functions Gi y , x of an uncracked single-span beam extended by terms taking cracks into account as shown in eq (24) 1 1 Gi ,c ( y, x) Gi ( y , x) ¦ M xc , j M Gi xc , j Q xc , j QGi xc , j , (26) c c j M, j w, j where y denotes the field point and x indicates the source point of the load. By applying the geometrical compatibility condition w xB 0 G0,c ( xB , x) G0,c ( xB , xB ) B at the position

xB of the support B, we obtain the support reaction force as G ( x , x) B 0,c B . (27) G0,c ( xB , xB ) Now the cracked Green’s functions of a two-span beam (TSB) using the superposition principle are obtained as GiTSB Gi ,c ( y , x) B Gi ,c ( y, xB ) . (28) ,c ( y, x)

74


To verify the presented method we consider a two-span beam with three equal parallel edge cracks under a time-harmonic loading with the frequency : 2S f 2S 3 Hz, F0 50 kN, rectangular cross-section and span length L 10 m. F (t )

F (t ) = F0 ei t

crack distance s = crack depth a

3 equal parallel edge cracks

cK1 cK 2 cK 3 L 2

L 2

EI = const. L

Fig. 4 A two-span beam with three equal parallel edge cracks under time-harmonic loading. Fig. 4 shows the first and second eigenfrequency for a ratio of the crack-depth to the beam size a / h 0, 2 from closed-form analytical solutions in comparison to a two-dimensional FE calculation with ANSYS using crack-tip elements. Furthermore the results of deflection, bending moment and transverse shear force over the beam length for a / h 0,5 obtained by using closed-form cracked Green’s functions in eq (28) (exact) and approximate solutions in eq (21) (approximate) are presented in Fig. 4. In this example the translational springs are neglected, because their influences on deformations and internal forces are very small (< 1%).


The improvement of the approximate solutions in Fig. 4 (dashed lines) for more complicated problems is in progress. To achieve a higher accuracy of the approximate solutions a recurrence relation based on the SAGF approach could be applied, which is similar to applications for linear discrete systems, where the inverse of a modified stiffness matrix is approximated by a Taylor expansion [7].

Example 2: Two-dimensional cracked cantilever plate under a self-weight load crack

a

h = 5,0 m

1

1

y x

l = 10, 0 m

Fig. 5 Cracked cantilever plate under a self-weight load and stress changes 'V xx and 'V yy in section 1-1. To show the sensitivity of the stresses V xx and V yy in section 1-1 to a crack of length a as shown in Fig. 5, we model the cracked region as a loss of support stiffness. From the SAGF approach eq (18) we obtain

75

76


'V xx x

³ u y A y d* c

G1

a

,

(29)

a

with the support reactions A G1 from the uncracked approximate Green’s functions calculated by a twodimensional FE analysis, and the displacement vector u c of the cracked system approximated by beam models with springs ( cM – only mode-I, [ cM cw ]– mode-I and mode-II) to approximate the vertical and the horizontal nodal displacements.

Summary In this paper, a simplified analytical method for sensitivity analysis of cracked or weakened structures subjected to static or dynamic loading is presented. The basic idea of the method is to predict changes in deformations, stresses, eigenfrequencies and mode shapes from stiffness modifications by using the differences of strain energies of an uncracked and a cracked structure in combination with static or dynamic Green’s functions. For a given structure, the intensity of theses changes in deformations or internal forces depends on the number, size and location of cracks, and the external loads. If the Green’s functions are obtained exactly or approximately, then they can be used to obtain solutions for different kinds of load cases. The present approach has the advantage that we only need to consider the cracked regions of a structure to calculate its response changes, which leads to a lower computational effort. The extension of the approach to more complicated problems is in progress. In principle, the method can also be extended to nonlinear problems as presented in references [12, 13]. Potential applications of structural sensitivity analysis by the Green’s functions approach could be found for example in damage monitoring, crack identification and predictive maintenance of structures or structural elements.

References [1] M. Abu-Hilal Journal of Sound and Vibration, 267, 191-207 (2003). [2] J.D. Achenbach Wave Propagation in Elastic Solids, North-Holland (1973). [3] A.S. Bouboulas and N.K. Anifantis Engineering Structures, 30, 894-901 (2008). [4] O. Carl and F. Hartmann 3rd MIT Conference on Computational Fluid and Solid Mechanics, Boston, USA (2005). [5] O. Carl and Ch. Zhang 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2010, Karlsruhe, Germany (2010). [6] M.B. Civelek and F. Erdogan International Journal of Fracture, 19, 139-159 (1982). [7] A. Deif Sensitivity Analysis in Linear Systems, Springer (1986). [8] G. Failla and A. Santini International Journal of Solids and Structures, 44, 7666-7687 (2007). [9] T. Grätsch and F. Hartmann Computational Mechanics, 37, 394-407 (2006). [10] D. Gross and T. Seelig Fracture Mechanics, Springer (2006). [11] F. Hartmann Introduction to Boundary Elements, Springer (1989). [12] F. Hartmann and C. Katz Structural Analysis with Finite Elements, Springer (2007). [13] F. Hartmann and T. Kunow Proceedings of the Ninth International Conference on Computational Structures Technology, Civil-Comp Press (2008). [14] Y. Murakami Stress Intensity Factors Handbook, Volume 1 and 2, Pergamon (1987). [15] L. Rohde, R. Kienzler and G. Herrmann Philosophical Magazine, 85, 4231-4244 (2005).


A D-BEM Approach with Constant Time Weighting Function Applied to the Solution of the Scalar Wave Equation J. A. M. Carrer1 and W. J. Mansur2 1

PPGMNE: Programa de Pós-Graduação em Métodos Numéricos em Engenharia, Universidade Federal do Paraná, Caixa Postal 19011, CEP 81531-990, Curitiba, PR, Brasil email: [email protected] 2

Programa de Engenharia Civil, COPPE/UFRJ, Universidade Federal do Rio de Janeiro, Caixa Postal 68506, CEP 21945-970, Rio de Janeiro, Brasil email: [email protected] Keywords: scalar wave equation, DS-BEM, time weighting D-BEM

Abstract. A D-BEM approach, based on time weighting residuals, is developed for the solution of 2D scalar wave propagation problems. The basic equation of the proposed formulation is generated by weighting, with respect to time, the basic D-BEM equation, under the assumption of linear and cubic time variation for the potential and for the flux. A constant time weighting function is adopted. As the time integration reduces the order of the time derivative that appears in the domain integral, the initial conditions are directly taken into account. The potentialities of the proposed formulation are verified by the examples included at the end of the work. Introduction The solution of time dependent problems by the Boundary Element Method (BEM) offers to the researcher a vast range of possibilities, according to the way the problem will be solved. In this work the scalar wave equation in 2D is solved by employing the so-called D-BEM formulation, D meaning domain. This formulation employs the static fundamental solution, instead of the time dependent fundamental solution. As a consequence, it is characterized by the presence of a domain integral whose kernel is constituted by the static fundamental solution multiplied by the second order time derivative of the potential (acceleration), e.g. Carrer et al. [1], Hatzigeorgiou and Beskos [2]. The selection of an adequate approximation to the acceleration, in order to perform the march in time, is a task that deserves attention. Although the Houbolt method [3] has been widely used and seems to be a good choice, alternative time marching schemes can be found in Carrer and Mansur [4], Souza et al. [5], Chien et al. [6]. Here, another approach is developed, in which the basic D-BEM equation is integrated from the initial time, say t0, to a specified final time, say tF. Over each time interval tn t tn+1 a constant time weighting function is adopted in a feature that can be identified as the subdomain collocation or the first approximation of the method of moments, e. g. Zienkiewicz and Morgan [7], Finlayson [8]. Initially, linear time approximation was assumed to both the potential and to the flux in the interval tn t tn+1. This approach, called DSL-BEM (D for the D-BEM formulation, S for the subdomain method and L for the linear time variation), did not produce reliable results, see Carrer and Mansur [9]. For this reason, another attempt to improve the results was carried out, in which a cubic time variation was assumed, in the interval tn2 t tn+1, for the potential and for the flux, thus generating the DSC-BEM approach (C meaning cubic). In order to avoid handling with values related to times previous to t0, this approach is not employed at the beginning of the analysis; in this way, the time marching process begins with the DSL-BEM approach and, after some time has elapsed, employs the DSCBEM approach. The use of the DSL-BEM and DSC-BEM approaches constitutes the DS-BEM formulation, which produced the expected reliable results. It is important to mention that the contribution of the nonhomogeneous initial conditions is directly taken into account, as the domain integral now contains only the first order time derivative of the potential. At the end of the article two examples are presented and discussed. In the examples, the DS-BEM results are compared with the corresponding analytical solutions, with the aim of verifying the potentialities of the proposed formulation.

77

78


Constant Time Weighting of the D-BEM Equation The basic integral equation of the D-BEM formulation is written as follows:

´ µ ¶*

´ µ ¶*

c([) u([,t) = µ u*([,X)p(X,t) d*(X) µ p*([,X)u(X,t) d*(X)

1 ´ µ u*([,X)u..(X,t) d:(X) c2 µ ¶:

(1)

where u*([,X) is the fundamental solution, p*([,X) is the normal derivative of the fundamental solution and the coefficient c([) is the same of the static problem, see Carrer et al. [1]. In order to solve eq (1), initially it is assumed a linear time variation for both the potential and the flux at each time interval tn t tn+1. Subsequently, integration on time, interpreted as a time weighting statement, is carried out from the initial time t0 = 0 to the final time of analysis, say tF. With these comments in mind, noting that W(t) is the time weighting function, eq (1) is rewritten as:

´ ´tF ´tF c([)µ W(t)u([,t)dt = µ u*([,X)µ W(t)p(X,t)dt d*(X) µ ¶t0

µ ¶*

µ ¶t0

´ ´ ´tF ´tF µ p*([,X)µ W(t)u(X,t)dt d*(X) 12 µ u*([,X) µ W(t)u..(X,t)dt d:(X) c µ µ µ µ ¶t0 ¶t0 ¶* ¶:

(2)

In the subdomain collocation or the first approximation of the method of moments, e. g. Zienkiewicz and Morgan [7], Finlayson [8], the time weighting function W(t) is chosen to satisfy: °

1

if

tn t tn+1

¯°

0

if

t tn or t ! tn+1

W(t) = ®

(3)

With the use of the time weighting function defined by eq (3), the time integration is restricted to the interval tn t tn+1. In this interval one has: u(X,t) = )n(t)un(X) + )n+1(t)un+1(X)

and

p(X,t) = )n(t)pn(X) + )n+1(t)pn+1(X)

(4)

where )n(t) and )n+1(t) are the linear interpolation functions: )n(t) =

tn+1 t 't

and

)n+1(t) =

t tn 't

(5)

In this work, only time intervals with constant length, 't = tn+1 tn, were used. The substitution of eqs (3, 4) in eq (2), followed by the time integration, gives:

´ 't 't c([) 2 ª¬un+1([) + un([)º¼ = µ u*([,X) 2 ª¬pn+1(X) + pn(X)º¼ d*(X) µ ¶* ´ ´ µ p*([,X) 't ªun+1(X) + un(X)º d*(X) 12 µ u*([,X) ª u. n+1(X) u. n(X)º d:(X) ¬ ¼ ¼ 2 ¬ c µ µ ¶* ¶:

(6)

In eq (6), the subscripts (n + 1) and n represent the time tn+1 = (n + 1)'t and tn = n't. Note that the initial conditions can be directly applied in eq (6) at the beginning of the analysis, i.e. when the subscript n = 0. In this work, the boundary is approximated by linear elements and the domain, by triangular linear cells. After performing the boundary and the domain integrations, the matrix form of eq (6) is: b b bb ª Hbb 0 º ° un+1 + un ½° ª G º b b ¾ = « db » ® pn+1 + pn « db » ® d d ¬ H I ¼ °¯ un+1 + un ¿° ¬ G ¼ ¯

½ ¾ ¿

.b .b bb Mbd º °un+1 un 2 ªM 2 « db »® c 't ¬ M .d Mdd ¼ ° . d ¯ un+1 un

½° ¾ °¿

(7)


79

in which the superscripts b and d correspond to the boundary and to the domain, respectively. In the submatrices, the first superscript corresponds to the position of the source point and the second superscript, to the position of the field point. The identity matrix is related to the coefficients c([) = 1 of the internal points. .b .d In eq (7), the derivatives un+1 and un+1 are approximated by employing the backward finite difference formula below, see Zienkiewicz and Morgan [7]: un+1 un . (8) un+1 = 't After substituting eq (8) in eq (7) and rearranging, yields: 2Mbd ª §©(c't)2Hbb + 2Mbb·¹ º ° ubn+1 °½ ª (c't)2Gbb º b ½ « § 2 db » ® d ¾ = « 2 db » ®¯ pn+1 ¾¿ ¬ ©(c't) H + 2Mdb·¹ §©(c't)2I + 2Mdd·¹ ¼ ¯° un+1 ¿° ¬ (c't) G ¼ 0 ª (c't)2Hbb º ° ubn °½ ª (c't)2Gbb º b ½ ª 2Mbb 2Mbd º ° unb + 't u.bn « 2 db » ®° ud ¾° + «¬ (c't)2Gdb »¼ ®¯ pn ¾¿ + «¬ 2Mdb 2Mdd »¼ ® d . d °¯ un + 't un n ¿ (c't)2I ¼ ¯ ¬ (c't) H

½° ¾ °¿

(9)

As usual in D-BEM formulations, the unknowns in eq (9) are the potential and the flux at the boundary * and the potential at the domain :. All the development described so far can be identified as the DSL-BEM formulation. The results provided by this formulation are not reliable, due to a significant presence of numerical damping, see Carrer and Mansur [9]. In this way, the search for another formulation became imperative. With the subdomain collocation, a natural choice was to assume a quadratic behaviour for u(X,t) and p(X,t) in an interval tn1 t tn+1; this assumption, however, was not useful, being characterized by instability in the results (not presented here). The adoption of the cubic Lagrange interpolation for u(X,t) and p(X,t), in an interval tn2 t tn+1, produced accurate results. This formulation, named DSC-BEM, is described in what follows. The time weighting function W(t), now, is chosen to satisfy: °

1

if

tn2 t tn+1

¯°

0

if

t tn2 or t ! tn+1

W(t) = ®

(10)

and the time integration is restricted to the interval tn2 t tn+1. In this interval one has: u(X,t) = )n2(t)un2(X) + )n1(t)un1(X) + )n(t)un(X) + )n+1(t)un+1(X) and p(X,t) = )n2(t)pn2(X) + )n1(t)pn1(X) + )n(t)pn(X) + )n+1(t)pn+1(X)

(11) (12)

where the )i(t) functions, i = (n 2), (n 1), n, (n + 1), are the Lagrange interpolation functions, see [9]. After substitution of eqs (10, 11, 12) in eq (2), and bearing in mind that the time integration is carried out from time tn2 to time tn+1, one has: 9 9 3 3 c([) 't ª8 un2([) + 8 un1([) + 8 un([)+ 8 un+1([)º =

¬ ¼ ´ 9 9 3 µ u*([,X) 't ª pn2(X) + pn1(X) + pn(X)+ 3 pn+1(X)º d*(X) 8 8 8 ¬8 ¼ µ ¶* ´ µ p*([,X) 't ª3 un2(X) + 9 un1(X) + 9 un(X)+ 3 un+1(X)º d*(X) 8 8 8 ¬8 ¼ µ ¶* 1 ´ µ u*([,X) ª u. n+1(X) u. n2(X)º d:(X) ¬ ¼ c2 µ ¶:

(13)

80


. . The first order time derivatives un+1 and un2 that appear in the domain integral in eq (13) are given by: . 1 ª un+1 = 11un+1 18un + 9un1 2un2º¼ 6't ¬

and

. 1 ª un2 = 2u 9un + 18un1 11un2º¼ 6't ¬ n+1

(14)

One must note that the first expression in eq (14) is the well-known Houbolt approximation for the first order time derivative, see Houbolt [3]. This result was already expected, as the Houbolt method is obtained by cubic Lagrange interpolation of u = u(t) in the interval tn2 d t d tn+1. After substituting eq (14) in eq (13) and integrating over the boundary and the domain, the matrix form for the DSC-BEM formulation is written as: 4Mbd ª §©(c't)2Hbb + 4Mbb·¹ º ° ubn+1 « § 2 db »® d ¬ ©(c't) H + 4Mdb·¹ §©(c't)2I + 4Mdd·¹ ¼ ¯° un+1 0 ª (c't)2Hbb º ° 3ubn + 3ubn1 + ubn2 ½° « 2 db » ®° 3ud + 3ud + ud ¾° + 2 H I (c't) (c't) ¬ ¼ ¯ n n1 n2 ¿

ª (c't)2Gbb º b b b « » ®¯ 3pn + 3pn1 + pn2 ¬ (c't)2Gdb ¼

½ ¾+ ¿

ª 4Mbb « ¬ 4Mdb

½° ª (c't)2Gbb º b ¾=« » ® pn+1 2 db ¿° ¬ (c't) G ¼ ¯

½ ¾ ¿

b b b ° un + un1 un2 ½° 4Mbd º

»®

¾

d d d 4Mdd ¼ ° ¯ un + un1 un2 °¿

(15)

The DSL-BEM, eq (9), is employed initially, as it takes into account the initial conditions directly and avoids handling with values related to times previous to t0; then, after a few time steps, the DSC-BEM, eq (15), is used. As mentioned at the Introduction to this work, the use of both formulations is referred to as DS-BEM approach. A measure of the time step is provided by the dimensionless variable E't defined as follows: c't (16) E't = " in which c is the wave propagation velocity, 't is the time interval and " is the length of the smallest element used in the boundary discretization. It is important to mention that the best values for E't are determined empirically and differs according to the formulation employed. Examples In what follows, the DS-BEM results are compared with the corresponding analytical solutions, see Stephenson [10], Kreyszig [11]. One-dimensional bar. This example consists of a one-dimensional bar defined in the domain 0 d x d a, 0 d y d a/2, fixed at one side (x = 0) and free at the other (x = a), subjected to the initial conditions given by: . u0(x) = Ux and u0(x) = 0 (17) The mesh, constituted of 48 boundary elements and 256 cells, is depicted in Fig. 1.

Figure 1. One-dimensional bar: boundary and domain discretization.


81

Results corresponding to the potential at boundary node A(a,a/4) and to the flux at boundary node B(0,a/4) are depicted, respectively, in Fig. 2 and Fig. 3. The time interval was selected by taking E't = 2/3. The analytical solution is: f ª(1)n+1 cos §(2n 1)Sct·º sin §(2n 1)Sx· 2a 2a ¬ © ¹¼ © ¹ 8Ua (18) u(x,t) = 2 (2n 1)2 S

¦

n=1 u/Ua

p/U

analytical DS-BEM

1.2

analytical DS-BEM

1.2

0.8

0.8

0.4

0.4

0.0

0.0

-0.4

-0.4

-0.8

-0.8

-1.2

-1.2 0.0

4.0

8.0

12.0

16.0

20.0 ct/a

Figure 2. One-dimensional bar: potential at boundary node A(a,a/4).

0.0

4.0

8.0

12.0

16.0

20.0 ct/a

Figure 3. One-dimensional bar: flux at boundary node B(0,a/4).

Square membrane. In this example a square membrane, defined in the domain 0 d x d a, 0 d y d a, is subjected to the initial conditions given by: . (19) u0(x,y) = U x(x a)y(y a) and u0(x,y) = 0 The boundary discretization employed 80 elements and the square domain, 800 cells, see Fig. 4.

Figure 4. Square membrane: boundary and domain discretization. The general analytical solution to this problem, for a rectangular membrane with dimensions a and b, is: f f 64Ua2b2 1 m2 n2 §mSx· §nSy· § · u(x,y,t) = + (20) 6 3 3 sin © a ¹ sin © b ¹ cos © OmnSct¹ with Omn = S a2 b2 mn

¦¦

m=1 n=1

82


Results corresponding to the displacement at point A(a/2,a/2) and to the support reaction at boundary node B(a,a/2) are depicted, respectively, in Fig. 5 and Fig. 6. For this example, E't = 3/10. u/U

p/U

analytical DS-BEM

1.5

analytical DS-BEM

0.4

1.0

0.2 0.5

0.0

0.0

-0.5

-0.2 -1.0

-1.5 0.0

4.0

8.0

12.0

16.0

20.0 ct/a

Figure 5. Square membrane: displacement at point A(a/2,a/2).

-0.4 0.0

4.0

8.0

12.0

16.0

20.0 ct/a

Figure 6. Square membrane: support reaction at boundary node B(a,a/2).

Conclusions The D-BEM formulation presented here, called DS-BEM, confirms the assertion done at the beginning of the work, i.e. the solution of time dependent problems by the BEM offers to researchers a vast range of possibilities. In the proposed approach, the drawback represented by the domain integration is compensated by the easy computational implementation and, mainly, by the accurate results obtained even for large time values, see Figs. 2,3,5,6. Besides, due to the time integration, the initial conditions are imposed directly, with no need of further developments. For this reason, the authors’ conclusion is that the proposed approach looks very promising and, consequently, some research work concerning its development can be done in the near future, which includes a possible extension to elastodynamics. References [1] J.A.M.Carrer, W.J.Mansur, R.J.Vanzuit. Scalar Wave Equation by the Boundary Element Method: a DBEM Approach with Non-homogeneous Initial Conditions. Computational Mechanics, 44, 31-44 (2009). [2] G.D.Hatzigeorgiou, D.E.Beskos. Dynamic Elastoplastic Analysis of 3-D Structures by the Domain/Boundary Element Method. Computers & Structures, 80, 339-347 (2002). [3] J.C.Houbolt. A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft. Journal of the Aeronautical Sciences, 17, 540-550 (1950). [4] J.A.M.Carrer and W.J.Mansur. Alternative Time-marching Schemes for Elastodynamic Analysis with the Domain Boundary Element Method Formulation. Computational Mechanics, 34, 387-399 (2004). [5] L.A.Souza, J.A.M.Carrer, C.J.Martins. A Fourth Order Finite Difference Method Applied to Elastodynamics: Finite Element and Boundary Element Formulations. Structural Engineering and Mechanics, 17, 735-749 (2004). [6] C.C.Chien, Y.H.Chen, C.C.Chuang. Dual Reciprocity BEM Analysis of 2D Transient Elastodynamic Problems by Time-Discontinuous Galerkin FEM. Engineering Analysis with Boundary Elements, 27, 611624 (2003). [7] O.C.Zienkiewicz and K.Morgan. Finite Elements & Approximation. John Wiley & Sons, Inc. (1983). [8] B.A.Finlayson. The Method of Weighted Residuals and Variational Principles. Academic Press (1972). [9] J.A.M.Carrer and W.J.Mansur. Scalar Wave Equation by the Boundary Element Method: a D-BEM Approach with Constant Time-Weighting Functions. International Journal for Numerical Methods in Engineering, 81, 1281-1297 (2010). [10] G.Stephenson. An Introduction to Partial Differential Equations for Science Students. Longman (1970). [11] E.Kreyszig. Advanced Engineering Mathematics. John Wiley & Sons, Inc., 8th edition (1999).


83

A novel boundary meshless method for radiation and scattering problems Zhuojia Fu1, Wen Chen2 1

Department of Engineering Mechanics, Hohai University, Nanjing 210098, P.R. China, [email protected]

2

Department of Engineering Mechanics, Hohai University, Nanjing 210098, P.R. China, [email protected]

Keywords: Singular boundary method, meshless, singular fundamental solution, unbounded domain, radiation, scattering

Abstract. This paper proposes a novel meshless singular boundary method (SBM) to solve time-harmonic exterior acoustic problems. Compared with the other boundary-type meshless methods, the innovative point of the SBM is to employ a novel inverse interpolation technique to circumvent the singularity of the fundamental solution at origin. The method is mathematically simple, easy-to-program, meshless and integration-free. This study tests the method to three benchmark radiation and scattering problems under unbounded domains. Our numerical experiments reveal that the SBM is a competitive numerical technique to the exterior acoustic problems. 1. Introduction The finite element method (FEM) [3-5] is one of the most popular methods in numerical acoustics but requires the effective treatment of unbounded domains, among which are the local and nonlocal absorbing boundary conditions [6-8], infinite elements [9], and absorbing layers [10,11]. These boundary treatments could be very tricky and arbitrary and are largely based on trial-error experiences. On the other hand, the boundary element method (BEM) [12-17] appears very attractive to handle the unbounded domain problems because its basis function is the fundamental solution which satisfies the governing equation and the Sommerfeld radiation condition at infinity [9]. And no special treatment for unbounded domains is required. However, the treatment of singularity and hyper-singularity [17] is mathematically complex and computationally very expensive. To avoid the singularities of fundamental solutions, the method of fundamental solutions (MFS) [18-20] distributes the boundary knots on a fictitious boundary outside the physical domain, and the location of fictitious boundary is vital for the accuracy and reliability. However, despite great effort of decades, the optimal placement of fictitious boundary is still arbitrary and tricky and is largely based on experiences. Recently, Young et al. [21] proposed an alternative meshless method, called regularized meshless method (RMM) [22], to remedy this drawback. By employing the desingularization of subtracting and adding-back technique, the RMM places the source points on the real physical boundary. In addition, the ill-conditioned interpolation matrix of BEM and MFS is also remedied. However, the original RMM requires the uniform distribution of nodes and severely reduces its applicability to complex-shaped boundary problems. Similar to the RMM, Sarler [23] proposes the modified method of fundamental solution (MMFS) to solve potential flow problems. However, the MMFS demands a complex calculation of the diagonal elements of interpolation matrix. It is worthy of noting that the MFS, RMM and MMFS do not require any mesh and are all truly meshless. This paper proposes a novel numerical method, called singular boundary method (SBM), to calculate the exterior acoustic problems. The SBM is developed to overcome the above-mentioned major shortcomings in the MFS, RMM, and MMFS while retaining their merits. The key point of the SBM is to use a simple numerical approach to calculate diagonal elements when the collocation and source nodes are coincident and are all placed on the physical boundary. This study also examines the efficiency, stability, and accuracy of the proposed technique in testing three benchmark exterior radiation and scattering problems. Based on the results reported here, some remarks will be concluded in section 4.

2. Singular boundary method for exterior Helmholtz problems The problem under consideration is the Helmholtz equation in the domain D exterior to a closed bounded curve

84


S . To be precise, we consider propagation of time-harmonic acoustic waves in a homogeneous isotropic acoustic medium which is described by the Helmholtz equation 2u ( x) k 2u ( x) 0, xD, (1) subjected to the boundary conditions: u x u x *D (2a)

wu x wn

t ( x)

x *N

t

(2b)

where u is the total acoustic wave ( velocity potential or acoustic pressure), k Z / c the wave number, Z the angular frequency, c the wave speed in the exterior acoustic medium D, and n denotes the unit inward normal on physical boundary. * D , * N denote the essential boundary (Dirichlet) and the natural boundary (Neumann) conditions, respectively, which construct the whole closed bounded curve S . For the exterior acoustic problems, it requires ensuring the physical requirement that all scattered and radiated waves are outgoing. This is accomplished by imposing an appropriate radiation condition at infinity, which is termed as the Sommerfeld radiation condition [9]: 1

lim r 2

(dim 1)

r of

§ wu · ¨ iku ¸ 0 , w r © ¹

(2c)

1 . where dim is the dimension of the acoustic problems (dim=2 in this study), and i The solution u(x) of the acoustics problem (Eqs. (1) and (2)) can be approximated by a linear combination of the two-dimensional fundamental solution G N

¦D G x

u xm

j

m

j 1

, s j ,

xD

(3)

where N denotes the number of source points, D j is the jth unknown coefficient, and the fundamental solution

G x, s j

iS (1) H0 k x s j 2

2

, H

(1) n

is the nth order Hankel function of the first kind. We can find that the

fundamental solution G satisfies both the governing equation (1) and the Sommerfeld radiation condition (2c). Thus, the formulation (3) only requires satisfying the boundary conditions (2a) and (2b). If the collocation points xm and source points sj coincide, i.e., xm=sj, we will encounter well-known singularity at

origin, i.e., G xm , s j

iS (1) H 0 0 . In order to remedy this troublesome problem, the MFS places the source 2

nodes on an artificial boundary outside the physical domain. However, despite of great effort, the placement of this artificial boundary remains a perplexing issue when dealing with complex-shaped boundary or multiply-connected domain problems. The SBM places all source and boundary collocation nodes on the same physical boundary. Moreover, the source points and the boundary collocation points are the same set of boundary nodes. The SBM formulation is given by N

¦D G x

u xm

j

m

j 1

u xm

t xm

N

¦

j 1, j z m

j

j 1

N

t ( xm )

¦

j 1, j z m

xm : e , xm * D

D j G xm , s j D m G S (m),

wG xm , s j

N

¦D

, s j ,

Dj

wn

,

wG xm , s j wn

xm * D

xm :e , xm * N D m G S (m),

xm * N

(4a) (4b)

(4c)

(4d)

where GS and GS are defined as the source intensity factors, namely, the diagonal elements of the SBM interpolation matrix. This study employs a simple numerical technique, called the inverse interpolation technique (IIT), to determine the source intensity factors. In the first step, the IIT requires choosing a known sample solution


85

uS of the Helmholtz acoustic problem and locating some sample points yk inside the physical domain. It is noted that the sample points yk do not coincide with the source points sj, and the sample points number NK should not be fewer than the source node number N on physical boundary. By using the interpolation formula (3), we can then determine the influence coefficients E j and E j by the following linear equations

^G y , s `Ê ` û y ` k

j

j

S

(5a)

k

°wG yk , s j ½° wuS yk ½ ® ¾Ê j ` ® ¾ n w ¯ wn ¿ ¯° ¿°

(5b)

Replacing the sample points yk with the boundary collocation points xm, the SBM interpolation matrix of the Helmholtz problem (Eqs. (1) and (2)) can be written as G x1, s2 G x1, sN º ª GS (1) « » G x , s GS (2) G x2 , sN » 2 1 « (6a) E ûS xm ` « » j « » ¬«G xN , s1 G xN , s2 GS ( N ) ¼»

^ `

wG x1, s2 wG x1, sN º ª « GS (1) » w wn n « » wG x2 , sN » « wG x2 , s1 wuS xm ½ GS (2) « » E wn wn « » ^ j ` ®¯ wn ¿¾ « » « wG x , s » x s w G , N N 1 2 « GS ( N ) » wn wn ¬« ¼» The source intensity factors can be calculated by the following formulations:

uS xm GS (m)

GS (m)

N

¦

j 1,s j zxm

E jG xm , s j

xm s j , xm *D

Ej N wG xm , s j wuS xm ¦ Ej wn wn j 1,s j zxm

Ej

xm s j , xm *N

(6b)

(7a)

(7b)

It is stressed that the source intensity factors only depends on the distribution of the source points, the fundamental solution of the governing equation and the boundary conditions. Theoretically speaking, the source intensity factors remain unchanged with different sample solutions in the IIT. Therefore, by employing this novel inverse interpolation technique, we circumvent the singularity of the fundamental solution upon the coincidence of the source and collocation points. It is noted that like the MFS, the SBM does not require considering the Sommerfeld radiation condition (2c) and is a truly meshless numerical technique; unlike the MFS, the SBM avoids the perplexing issue of the fictitious boundary.

3. Numerical results and discussions In this section, the efficiency, accuracy and convergence of the present SBM are tested to the exterior acoustics problems. It is stressed that the boundary conditions are discontinuous in Cases 1 and 2. The present SBM is compared with the exact solution, the RMM and the MFS. Lerr(u) represents the L2 norm error, which are defined as below

Lerr (u )

1 NT

NT

¦ u k u k

2

,

(8)

k 1

where u k and u k are the analytical and numerical solutions at xi, respectively, and NT is the total number

86


of points in the interest domain which are used to test the solution accuracy, the sample solution

us ( r , T )

H 2(1) (kr ) cos 2T for Dirichlet boundary problem, us (r ,T ) H 2(1) (ka )

H1(1) (kr ) iT e for Neumann boundary H1(1)c (ka)

problem. The number of inner sample points is equal to the boundary knots, and the distribution of sample points depends on the shape of the physical domain. In the MFS, according to the boundary shape of the physical domain, we typically place the source points outside physical domain with a parameter d defined as

d

xi si xi op

(9)

in which op is the geometric center, namely the origin point in this paper.

3.1 Radiation problems Case 1: Nonuniform radiation problem (Dirichlet boundary condition) for a circular cylinder. We first consider a nonuniform radiation problem (Dirichlet) from a sector of a cylinder as shown in Fig. 1(a). The boundary condition has a constant inhomogeneous value on the arc D 2 d T d D 2 and vanishing elsewhere. Two discontinuous boundary points can be found on the physical boundary. The analytical solution [8] is

u (r ,T )

D H 0(1) (kr ) 1 f sin nD H n(1) (kr ) cos nT ¦ 2S H 0(1) (ka ) S n 1 n H n(1) (ka )

(10)

where H n(1) (kr ) is the first kind Hankel function of the n order. We choose the parameters D

5S , ka=1. The 32

analytical solution is obtained by using 20 terms in the series representations. Fig. 2(a) shows the comparison of the L2 norm errors between the MFS with different fictitious boundary parameters (d=0.01,0.2,0.5) and the present SBM. It can be observed that the arbitrary placing of the off-set boundary points may cause numerical stability. The present SBM avoids such trial-error placement of the fictitious boundary and is more efficient than the MFS with the boundary nodes of the same number.

( 2 k 2 )u (r , T)

D u( a,T )

T

0, (r , T) D 5S 32

( 2 k 2 )u (r , T)

*f

D u(a,T ) 1, D 2 dT d D 2

0

t ( a,T )

0, (r , T) D

S

*f

9

T

t (a ,T ) 1,

D 2 dT d D 2

0

(a) Dirichlet (Case 1) (b) Neumann (Case 2) Fig. 1 Nonuniform radiation (a) Dirichlet (Case 1) and (b) Neumann (Case 2) problem of a circular cylinder Case 2: Nonuniform radiation problem (Neumann boundary condition) of a circular cylinder. A nonuniform radiation problem (Neumann) from a sector of a cylinder is considered as shown in Fig. 1(b) [24]. The discontinuous boundary condition is

1, D 2 d T d D 2 t a, T ® otherwise ¯0,

(11)

The analytical solution [24] is

u (r ,T )

D H 0(1) (kr ) 1 f sin nD H n(1) (kr ) ¦ cos nT 2S k H 0(1)c (ka) S k n 1 n H n(1)c (ka )

(12)


Here we choose the parameters D

S 9

87

, ka=1. The analytical solution is obtained by using 20 terms in the series

representations. Fig. 2(a) shows the convergence curves of the MFS with different fictitious boundary parameters (d=0.01,0.3,0.5) and the present SBM. It can be observed that the fictitious boundary has a big influence on the MFS solution and its optimal placement is problem-dependent. The present SBM can obtain the acceptable results by using only 40 boundary nodes and outperforms the MFS in computational accuracy.

(a) Dirichlet (Case 1) (b) Neumann (Case 2) Fig.2 The accuracy variation of Case 1 and 2 against the number of interpolation knots by the MFS with d=0.01,0.2,0.5 for Case 1 and d=0.01,0.3,0.5 for Case 2 and the present SBM.

3.2 Scattering problems The scattering problem with the incident wave can be divided into two parts, (a) incident wave field and (b) radiation field. And the radiation boundary condition in part (b) can be obtained as the minus value of the incident wave function, i.e. tR= -tI for hard scatter or uR=-uI for soft scatter, where the superscripts R and I denote radiation and incidence, respectively.

Fig. 3 The problem of a plane wave scattered by a rigid infinite circular cylinder (Neumann) in Case 3 Case 3: Scattering problem (Neumann boundary condition) of a rigid infinite circular cylinder We consider a plane wave scattered by a rigid infinite circular cylinder as shown in Fig. 3 [20]. The analytical solution of this scattering field [20] is

u (r ,T )

f J 0c (ka) (1) J c (ka ) (1) H 0 (kr ) 2 ¦ i n n H n (kr ) cos nT n 1 H 0(1)c (ka) H n(1)c (ka)

(13)

The analytical solution in the following figures is calculated by using the first 20 terms in the above series representation (13). Figs. 4(a) and 4(b) plot both the real and imaginary parts of u on r=2a for ka 4S by using the SBM and the MFS (d=0.01,0.2) with 100 boundary nodes. It can be found that both the SBM and the MFS

88


with fictitious boundary parameter d=0.2 agree the analytical solution very well. However, the MFS with d=0.01 can not obtain the right result. Thus, the determination of such a parameter d is very tricky and delicate in applications. It is noted that the present SBM avoids the headachy choice of the optimal fictitious boundary and is superior to the MFS. Figs. 5, 6(a) and 6(b) display the contour plot of the real-part potential by using the analytical solution, the present SBM and the MFS with 100 boundary nodes. It can be seen from Fig. 10 that the SBM solution matches the analytical solution very well.

(a) Real part (b) Imaginary part Fig. 4 Plane wave scattered by a rigid infinite circular cylinder (Neumann) in case 3 for ka part, (b) Imaginary part

4S ,r=2a: (a) Real

4. Conclusions

This study proposes a novel singular boundary method formulation to calculate the exterior radiation and scattering problems. Numerical results demonstrate that the SBM performs more stably than the MFS and more accurate than the RMM, while retaining their merits. The present SBM appears a promising numerical technique to the exterior acoustic problems. In addition, the present SBM is mathematically simple, easy-to-program, accurate, meshless and integration-free and avoids the controversy of the fictitious boundary in the MFS, the uniform boundary node requirement of the RMM, and the expensive calculation of diagonal elements in the MMFS.

Fig. 5 The contour plot of the real-part analytical solution of a plane wave scattered by a rigid infinite circular cylinder in Case 3


(a) SBM solution (b) MFS (d=0.2) solution Fig. 6 The contour plot of the real-part (a) SBM and (b) MFS (d=0.2) solution of a plane wave scattered by a rigid infinite circular cylinder in Case 3

References [1] P. Morse, K. Ingard, Theoretical acoustics, McGraw-Hill, New York, 1968. [2] A. Pierce, Acoustics: an introduction to its physical principles and applications. McGraw-Hill series in mechanical engineering, McGraw-Hill, New York, 1981. [3] F. Ihlenburg, Finite element analysis of acoustic scattering, Applied Mathematical Sciences(132). Springer, New York, 1998. [4] I. Harari, A survey of finite element methods for time-harmonic acoustics, Comput Methods Appl Mech Eng. 195 (2006) 1594-1607. [5] L.L. Thompson, A review of finite element methods for time-harmonic acoustics, J Acoust Soc Am. 119 (2006) 1315-1330. [6] D. Givoli, Recent advances in the DtN finite element method for unbounded domains, Archives of Computational Methods in Engineering 6 (1999) 71-116. [7] M.J. Grote, C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys. 201 (2004) 630-650. [8] J.R. Stewart, T.J.R. Hughes, h-adaptive finite element computation of time-harmonic exterior acoustics problems in two dimensions, Comput. Methods Appl. Mech. Eng. 146 (1997) 65-89. [9] I. Harari, P.E. Barbone, M. Slavutin, R. Shalom, Boundary infinite elements for the Helmholtz equation in exterior domains, Int. J. Numer. Methods Eng. 41 (1998) 1105-1131. [10] Q. Qi, T.L. Geers, Evaluation of the perfectly matched layer for computational acoustics, J. Comput. Phys. 139 (1998) 166-183. [11] A. Bermúdez, L. Hervella-Nieto, A. Prieto, R. Rodríguez, An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems, J. Comput. Phys. 223 (2007) 469-488. [12] C.A. Brebbia, J.C.F. Telles, L.C.L. Wrobel, Boundary element techniques: theory and applications in engineering, Springer, New York, 1984.

89

90


[13] J.T. Chen, K.H. Chen, I.L. Chen, L.W. Liu, A new concept of modal participation factor for numerical instability in the dual BEM for exterior acoustics, Mech. Res. Comm. 26 (2003) 161-174. [14] R.D. Ciskowski, C.A. Brebbia, Boundary element methods in acoustics, Computational mechanics publications, Elsevier Applied Science, 1991. [15] S. Kirkup, The boundary element method in acoustics, Integrated Sound Software, 1998. [16] Von Estorff, Boundary elements in acoustics: advances and applications, WIT Press, 2000. [17] V. Sladek, J. Sladek, M. Tanaka, Optimal transformations of the integration variables in computation of singular integrals in BEM, International Journal for Numerical Methods in Engineering, 47 (2000) 1263-1283. [18] G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1998) 69-95. [19] G. Fairweather, A. Karageorghis, P.A. Martin, The method of fundamental solutions for scattering and radiation problems, Engineering Analysis with Boundary Elements 27 (2003) 759-769 [20] I.L. Chen, Using the method of fundamental solutions in conjunction with the degenerate kernel in cylindrical acoustic problems, J. Chinese Inst. Engrg. 29 (2006) 445-457. [21] D.L. Young, K.H. Chen, C.W. Lee, Novel meshless method for solving the potential problems with arbitrary domain, J. Comput. Phys. 209 (2005) 290-321. [22] D.L. Young, K.H. Chen and C.W. Lee, Singular meshless method using double layer potentials for exterior acoustics, J Acoust Soc Am 119 (2006) 96-107 [23] B. Šarler, Chapter 15: Modified method of fundamental solutions for potential flow problems, in: C.S. Chen, A. Karageorghis, Y.S. Smyrlis (Eds.), The Method of Fundamental Solutions- A Meshless Method, Dynamic Publisher, 2008. [24] J.T. Chen, C.T. Chen, P.Y. Chen, I.L. Chen, A semi-analytical approach for radiation and scattering problems with circular boundaries, Comput. Meth. Appl. Mech. Engng, 196 (2008) 2751-2764.


Anti-plane shear Green’s function for an isotropic elastic layer on a substrate with a material surface W. Q. Chen1 and Ch. Zhang2 1

Department of Engineering Mechanics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China, [email protected] 2

Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany, [email protected]

Keywords: Green’s functions, elastic layer, material surface, Gurtin-Murdoch theory.

Abstract. The anti-plane shear Green’s function solution for an isotropic elastic layer on a rigid substrate with a material surface subject to a point force is derived using Fourier cosine transform. The elasticity and residual stress of the upper surface of the layer are taken into consideration by adopting the Gurtin-Murdoch theory, while the lower surface of the layer is assumed to be fixed on the rigid substrate. For two limiting cases of a half-plane, analytical expressions are obtained, which coincide with those in literature. Introduction With the miniaturization of electromechanical devices and systems, the size of structural components becomes smaller and smaller, while the surface-to-volume ratio increases significantly and hence the surface/interface effect becomes more important [1]. Recently, there appear some interesting works in which the significant effect of surface elasticity and residual stress on various mechanical responses of materials and structures has been clarified [2-10]. These works are all based on the continuum theory of a deformable material surface suggested by Gurtin and Murdoch [11] (hereafter, referred as the GM theory) as early in 1975. Gurtin and Murdoch [12] themselves have already shown that the surface stress has an important influence on both the static and dynamical problems of elastic bodies with appropriate surfaces. Green’s functions play an important role in solving boundary-value problems in elasticity. They are especially very useful in the boundary element method. Based on the GM theory, but with the simplifications that the surface material has the same elastic property as the bulk material and that the bulk material is incompressible, He and Lim [13] obtained the surface Green’s function of a half-space using the double Fourier transforms. Wang and Feng [14] presented analytical solutions of a half-plane with a material boundary subject to uniform as well as concentrated loads acting on the boundary; they used the Fourier transform technique and considered only the effect of residual surface tension. Koguchi [15] derived the surface Green’s functions in complex integral forms for an anisotropic half-space with material surface using Stroh’s formulism. Zhao and Rajapakse [16] investigated the plane problem of a surface-loaded isotropic elastic layer with surface effects by the Fourier transform. Recently, Chen and Zhang [17] derived analytical expressions for the point force solution of an isotropic elastic half-plane with a material surface subject to anti-plane shear deformation. They also showed that the obtained Green’s functions can be utilized to construct appropriate boundary integral equation so that the boundary element method could be used to deal with more complicated problems involving surface effects. In this paper, we reconsider the layer model in Zhao and Rajapakse [16], but confine ourselves to the case of anti-plane deformation induced by an interior point shear force. The upper boundary of the layer is modeled as a material surface which possesses both residual surface tension and elasticity, for which the GM theory is employed. The lower boundary is fixed on the rigid substrate. Because of the symmetry of the problem, Fourier cosine transform is employed. The Green’s function is obtained and expressed in an integral form. It is found that for two limiting cases, both corresponding to a half-plane model, either classical results for the anti-plane elasticity or that obtained in Ref. [17] are recovered.

91

92


Basic formulations Consider the anti-plane shear deformation of an isotropic elastic layer on a rigid substrate as shown in Fig. 1. The layer is subject to a point shear force of magnitude p0 at an interior point, which coincides with the origin of the coordinate system (x, y). The distances between the x-axis and the upper surface and the lower interface are denoted as h1 and h2, respectively, and hence the total thickness of the layer is H h1 h2 . The bulk material is isotropic, obeying the same laws as a conventional elastic material. For the anti-plane deformation, we therefore have the following nonzero strain components 1 ww 1 ww , H yz (1) H xz 2 wx 2 wy where w is the out-of-plane displacement component which depends on both x and y. The nonzero stress components are calculated from the Hooke’s law as ww ww V xz P , V yz P (2) wx wy where P is the shear modulus. The equilibrium equation is wV xz wV yz f 0 (3) wx wy where f is the z-component of the body force vector. Substituting Eqs. (1) and (2) into Eq. (3) gives f 2 w 0 (4)

P

where 2

w 2 / wx 2 w 2 / wy 2 is the two-dimensional Laplacian.

surface: Ws,Os, Ps

bulk: O, P

1

p0

2

h1

x

h2 y

Fig. 1. An elastic layer on a rigid substrate with a material boundary at y

h1 .

The upper surface of the layer is assumed to have different material properties from the bulk, and the GM theory [11] is employed. Hence, we have wV zxs V yz 0 (5) wx ww ww , V xzs ( P s W s ) (6) V zxs P s wx wx


at y

93

h1 , where the superscript s designates the quantity associated with the surface, W s is the residual

surface tension, and P s is the surface shear modulus. Note that the displacement compatibility between the surface and bulk at y h1 has been implied. The lower interface is assumed to be fixed, i.e. we have w 0 (7) at y h2 . The configuration in Fig. 1 has been considered by Zhao and Rajapakse [16], who however paid their attention to the in-plane deformation due to a concentrated force applied on the upper surface only. Green’s function solution

First, we divide the layer into two regions, region 1 ( h1 d y d 0 ) and region 2 ( 0 d y d h2 ), as shown in Fig. 1. The governing equation in each region becomes homogeneous as w 2 w( N ) w 2 w( N ) 0 (N 1, 2) (8) wx 2 wy 2 where the superscript N denotes the respective region. At y 0 , we have the following continuity/equilibrium conditions: ww(2) ww(1) p0G ( x) P (9) w(1) w(2) , P wy wy The following condition at the material surface ( y h1 ) can be derived from Eqs. (5) and (6): w 2 w(1) ww(1) P (10) 0 2 wx wy On the other hand, the condition at the interface between the layer and substrate is still given by Eq. (7), but with w replaced by w(2) . We use the following Fourier cosine transform: 2 f (11a) W (k , y ) ³ w( x, y) cos(kx) d x

Ps

S

w( x, y )

2

S

0

³

f

0

(11b)

W (k , y ) cos( kx) d k

Now, by applying the Fourier cosine transform defined by Eq. (11a) to Eq. (8) and the conditions (9), (10) and (7), we get d 2 W (N ) k 2W (N ) 0 (N 1, 2) (12) d y2 d W (1) rP k 2W (1) dy W (1)

W (2) ,

d W (2) p0 dy P

W (2)

0 1 2S at y

at y

h1

d W (1) dy

(13) at y

0

(14)

h2 (15) 0 P / P , the shear modulus ratio between the surface and bulk, has the dimension of length, and can be regarded as an intrinsic length parameter of the problem. The solutions to Eq. (12) in the two regions are (16) W (1) A eky B e ky , W (2) C eky D e ky where rP

s

94


Substituting into Eqs. (13) through (15) yields Ak e kh1 Bk e kh1 rP k 2 ( A e kh1 B e kh1 ) Ak Bk Ck Bk

p0

1

P

2S

0, A B C D

, C e kh2 D e kh2

0

0

(17)

which in turn gives A B C

D

(1 rP k )(e 2 kH e 2 kh1 )

1 p0 k[(1 rP k ) (1 rP k ) e 2 kH ] 2 2S P (1 rP k )(e 2 kh2 1) k[(1 rP k ) (1 rP k ) e

1 2 kH

p0

] 2 2S P

(1 rP k ) (1 rP k ) e 2 kh1

1 p0 k[(1 rP k ) (1 rP k ) e 2 kH ] 2 2S P

(1 rP k ) e 2 kh2 (1 rP k ) e 2 kH

1

p0

k[(1 rP k ) (1 rP k ) e ] 2 2S P Substituting into Eq. (16) and in view of Eq. (11b), we get 2 kh ky ky 1 p0 f [(1 rP k ) e 1 e (1 rP k ) e ] 2 kh2 w(1) (e 1) cos(kx) d k kH 2 ³ k[(1 rP k ) (1 rP k ) e ] 2S P 0 w(2)

1 p0 2S P

³

f

0

2 kH

e ky e 2 kh2 e ky [(1 rP k ) (1 rP k ) e 2 kh1 ]cos(kx) d k k[(1 rP k ) (1 rP k ) e 2 kH ]

(18)

(19a) (19b)

It seems difficult to obtain the closed-form solution due to the complicated integrands involved in Eqs. (19a) and (19b). However, one may verify that 1 p0 f 1 ky ky [e e ]cos(kx) d k w(2) w(2) (20) w(1) 2S P ³0 k Thus, the anti-plane shear displacement in the elastic layer can be written in a unified way as 2 kh ky ky 1 p0 f [(1 rP k ) e 1 e (1 rP k ) e ] 2 kh2 (21) w (e 1) cos(kx) d k 2 kH ³ 0 k[(1 rP k ) (1 rP k ) e ] 2S P The integration generally can be performed by an appropriate numerical scheme. Two limiting cases

We now consider two special cases. The first is for h1 o f , which corresponds to an infinite half-plane with a fixed surface at y h2 subject to a point shear force at y 0 . Taking the limit h1 o f , we obtain from Eq. (21) 1 p0 f e ky (1 e 2 kh2 ) cos(kx) d k w 2S P ³0 k (22) 1 p0 ª x 2 ( y 2h2 ) 2 º ln « » 4S P ¬ x2 y 2 ¼ It is readily seen that when y h2 , w 0 , i.e. the boundary condition at the fixed surface is satisfied. As expected, no effect of the material surface is involved in the displacement field given by Eq. (22), from which, the corresponding shear stress components in the half-plane can be calculated according to Eq. (2). These results should be the same as the classical ones in elasticity for anti-plane deformation of an isotropic half-plane with a fixed surface, and hence are not presented in the following.


y

95

The second is to let h2 o f , which corresponds to an infinite half-plane with a material surface at h1 [17]. In this case, we obtain from Eq. (21) that w

1 p0 2S P

³

f

[(1 rP k ) e ky (1 rP k ) e2 kh1 e ky ] k (1 rP k )

0

cos(kx) d k

(23) 1 p0 1 p0 ln ^( x 2 y 2 )[ x 2 ( y 2h1 ) 2 ]` J c ( x, y 2h1 ; rP ) 4S P S P where a constant, which contributes nothing to the stress/strain field, has been neglected in the above expression, and ky f ae (24) J c ( x , y; a ) ³ cos(kx) d k Re ª¬e z E1 ( z ) º¼ 0 1 ak

where z

³

( y i x) / a , and E1 ( z )

f

z

(e t / t ) d t ( Arg( z ) S ) is the exponential integral [18]. The

expression in Eq. (23), which is obtained through the limiting procedure, is identical to that obtained in Chen and Zhang [17]. The first term in Eq. (23) is identical to the classical result of a half-plane with a free surface, while the second is induced by the surface elasticity, which vanishes identically as rP 0 . In fact, it is the

right difference between the classical solution and the one including the surface effect. Thus, the Jcintegral defined in Eq. (24) represents the influence of the surface elasticity. Its characteristic is shown in Fig. 2 for several values of y. As can be seen, the Jc-integral decreases monotonously with x; that is to say, when the field point is away from the load point, the surface effect term in Eq. (23) also diminishes. The reader is referred to Chen and Zhang [17] for a more detailed discussion on the stress field in the half-plane as well as other aspects of the analytical Green’s functions.

0.6 y=1

0.5

y=2 y=4

0.4

Jc 0.3 0.2 0.1 0 0

Fig. 2.

1

2

x

3

4

5

Integral J c ( x, y; a) for a 1 . Since J c ( x, y; a) J c ( x / a, y / a;1) , the drawings for a z 1 can be readily imagined.

96


Summary

We present in this paper an analysis of an isotropic elastic layer subject to anti-plane shear deformation under the action of a point shear force. One surface of the layer is assumed to have different material properties from the bulk and the Gurtin-Murdoch theory is therefore employed. The other surface is assumed to be fixed on a rigid substrate. Fourier cosine transform is employed to derive the Green’s function in an integral form. By a limiting analysis, two special cases for a half-plane are discussed, with analytical Green’s functions presented. In particular, in the case of a half-plane with a material surface, the results are found identical to those derived by Chen and Zhang [17]. Acknowledgements

The work was sponsored by the National Natural Science Foundation of China (Nos. 10725210 and 10832009). Financial support from the German Research Foundation (DFG, Project-No.: ZH 15/15-1) is also gratefully acknowledged. References

[1]

R.C. Cammarata Progress in Surface Science 46, 1-38 (1994).

[2]

R.E. Miller and V.B. Shenoy Nanotechnology 11, 139-147 (2000).

[3]

L.H. He, C.W. Lim and B.S. Wu International Journal of Solids and Structures 41, 847-857 (2004).

[4]

C.W. Lim and L.H. He International Journal of Mechanical Science 46, 1715-1726 (2004).

[5]

C.W. Lim, Z.R. Li and L.H. He International Journal of Solids and Structures 43, 5055-5065 (2006).

[6]

L.H. He and Z.R. Li International Journal of Solids and Structures 43, 6208-6219 (2006).

[7]

Z.Y. Ou, G.F. Wang and T.J. Wang European Journal of Mechanics, A/Solids 28, 110-120 (2009).

[8]

C.F. Lü, C.W. Lim and W.Q. Chen International Journal of Solids and Structures 46, 1176-1185 (2009).

[9]

C.F. Lü, W.Q. Chen and C.W. Lim Composites Science and Technology 69, 1124-1130 (2009).

[10] C.I. Kim, P. Schiavone and C.Q. Ru Journal of Applied Mechanics 77, 021011 (2010). [11] M.E. Gurtin and A.I. Murdoch Archive for Rational Mechanics and Analysis 57, 291-323 (1975). [12] M.E. Gurtin and A.I. Murdoch International Journal of Solids and Structures 14, 431-440 (1978). [13] L.H. He and C.W. Lim International Journal of Solids and Structures 43, 132-143 (2006). [14] G.F. Wang and X.Q. Feng Journal of Applied Physics 101, 013510 (2007). [15] H. Koguchi Journal of Applied Mechanics 75, 061014 (2008). [16] X.J. Zhao and R.K.N.D. Rajapakse International Journal of Engineering Science 47, 1433-1444 (2009). [17] W.Q. Chen and Ch. Zhang International Journal of Solids and Structures 47, DOI: 10.1016/j.ijsolstr.2010.03.007 (2010). [18] M. Abramovitz and I. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964).


97

Stress intensity factor formulas for a rectangular interfacial crack in three-dimensional bimaterials 1

ChunHui Xu , TaiYan Qin1, Chuanzeng Zhang2, Nao-Aki Noda3 1

College of Science, China Agricultural University, Beijing 100083, PR China, E-mails: [email protected]; [email protected]

2

Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany, E-mail: [email protected]

3

Department of Mechanical Engineering, Kyushu Institute of Technology, Kitakyushu 8048550, Japan, E-mail: [email protected]

Keywords: Boundary integral equations; Fracture mechanics; Stress intensity factors; Interface cracks; Composite materials

Abstract Numerical solutions of hypersingular boundary integral equations (BIEs) are presented for the analysis of a planar rectangular interfacial crack in three-dimensional bimaterials. The problem is formulated as a system of hypersingular BIEs on the basis of the body force method. Based on the numerical results of the BIEs, the stress intensity factor formulas in terms of an area parameter to evaluate rectangular interfacial cracks in three-dimensional bimaterials are considered. Here “area” denotes the projected area of the defects or cracks. In the cases of mode-I and mode-II cracks, the formulas for homogeneous materials are available for interfacial cracks. For the interfacial cracks subjected to tension at the infinity, the stress intensity factors are expressed as a function of the bimaterial constant. For the cracks subjected to shear at the infinity, the stress intensity factors are investigated for varying Poisson's ratio and the aspect ratio of the crack. It is found that the maximum stress intensity factors normalized by the area are always insensitive to the crack aspect ratio. The fitting formulas presented in this paper are useful for engineering applications.

1. Introduction In recent years, composite materials and adhesives or bonded joints are being used in a wide range of engineering sciences. With the rapidly increasing use of composite materials and adhesives, much attention has been paid to the interface because the fracture is usually originated from the interfacial region. Since almost all structural materials contain some types of defects in the form of cracks, cavities, and inclusions, three-dimensional crack solutions may be useful for evaluating the strength of the structures. In the previous studies, stress intensity factor formulas were proposed for evaluating the maximum stress intensity factors for arbitrarily shaped internal cracks subjected to tension V zf at infinity for the coordinate system as shown in Fig. 1 [1-6]. For a crack subjected to tension V zf we have [6, 7]

K I m ax

0.50V zf S S .

(1)

For a crack subjected to shear one has [8-11] K II m ax

f 0.55W yz S S,

f K III m ax 0.45W yz S S,

(2)

(3)

where “ S ” is the projected area of the crack or defect. For example, in Fig. 1(a) area S ab , and in Fig. 1(b) area 4ab . However, it should be noted that area 20b 2 when a / b t 5 , and area 20b 2 when a / b d 0.2 . To confirm the accuracy of the formulas (1)-(3), the exact maximum stress intensity factors of elliptical [3, 4] and rectangular cracks [5, 6] at A and B subjected to V zf and W yzf at infinity are shown in Table 1 [5-8].

98


Evaluation formula

Elliptical crack

Rectangular crack

0.5

0.47-0.52 [6]

0.47-0.52 [8]

0.55, a/b1

0.46-0.64 [7]

0.47-0.64 [9]

0.45, a/b1

0.32-0.52 [7]

0.39-0.54 [9]

FI* FII* FIII*

Tab. 1 Maximum stress intensity factors In Table 1, normalized stress intensity factors are introduced as

FI*

K Imax

V zf S S

K II max

, FII*

W

f yz

S S

, FIII*

K III m ax

W

f yz

S S

.

y

y B

B

A

2b

x

z

2b

A

z

2a

x

2a

Fig. 1(a) An elliptical crack

Fig. 1(b) A rectangular crack

It should be noted that F is independent of Poisson’s ratio Q , but FII and FIII are depending on Q . Therefore Table 1 shows the range of the maximum stress intensity factors for Q 0 0.5 . In previous papers [11-13], the authors studied the problem of an interfacial crack under tension or shear loads at the infinity by solving the corresponding hypersigular boundary integral equations for computing the stress intensity factors, obtained fast convergence and high precision of the numerical results. In this paper, we will discuss the applicability of the formulas (1)-(3) for a rectangular interface crack. For this purpose, a boundary element method (BEM) based on hypersingluar BIEs is developed, which are briefly described in the next section.

I

2. Hypersingular boundary integral equations for a planar interfacial crack Consider two dissimilar elastic half-spaces bonded together along the x–y plane (see Fig. 2) with a fixed rectangular Cartesian coordinate system xi ( i x, y, z ). Suppose that the upper half-space is occupied by an elastic medium with constants ( P1 ,Q 1 ) , while the lower half-space by an elastic medium with constants ( P2 ,Q 2 ) . Here, P1 and P2 are shear moduli for space I and space II, and Q 1 and Q 2 are Poisson’s ratios for space I and space II. The crack is assumed to be located on the bimaterial interface. Hypersingular intergro-differential equations for three dimensional cracks on a bimateral interface as shown in Fig. 2 were derived by Chen–Noda-Tang [4] and can be expressed as follows P1 / 2 /1 3P1

w'u z x, y wx

P1

2/ /1 / 2

/1 / 2 / ° x [ 2S

®³ °¯ S

r5

2S

1

³r

3

'u x ([ , K )dS ([ , K )

S

2

'u x ([ , K )dS ([ , K ) ³ S

x [ y K r5

½° 'u y ([ , K )dS ([ , K ) ¾ °¿

(4a) px ( x, y ),


P1 / 2 /1 3P1

w'u z x, y wy

P1

2/ /1 / 2 2S

/1 / 2 / ° x [ y K ®³ ¯° S

2S

r5

1

³r

3

'u y ([ ,K )dS ([ , K )

S

'u x ([ , K )dS ([ , K ) ³

y K

S

r5

2

½° 'u y ([ , K )dS ([ , K ) ¾ ¿°

§ w'u x x, y w'u y x, y · /1 / 2 1 ¸ P1 ³S r 3 'uz ([ ,K )dS ([ ,K ) ¨ ¸ w w x y 2S © ¹

P1 /1 / 2 ¨ /

P2 , /1 P1 P 2

N1

3 4Q 1 , N 2

P2 , /2 P1 N1 P 2 3 4Q 2 , r 2

^

'u x x, y

u x ( x, y , 0 ) u x ( x, y , 0 )

'u y x, y 'u z x, y

2

`

S

(4b) p y ( x, y ),

p z x, y ,

(4c)

P2 , P 2 N 2 P1

x [

x, y S ,

99

y K , 2

(4d)

( x, y ) x d a, y d b ,

u y ( x, y , 0 ) u y ( x , y , 0 )

2

1

¦P

l

2

1

¦P l 1

u z ( x, y , 0 ) u z ( x, y , 0 )

f zx x, y ,

l 1

2

f yz x, y ,

Nl 1

¦ P (N l 1

(4e)

l

l

l

1)

f zz x, y .

In Eq. (4), the unknown functions are the crack-opening-displacements, in other words, displacement discontinuities 'u x , 'u y , 'u z defined in Eq. (4e), which are equivalent to the body force densities f zx x, y , f yz x, y , f zz x, y as given in Eq. (4e). Here, ([ ,K , ] ) is a rectangular coordinate system where the f , V zf at infinity. displacement discontinuities are distributed, and px , p y , pz are the stresses W zxf , W yz

Since the integrals have a hypersingularity of the form r 3 when x [ and y K , the integrals should be interpreted in a sense of the Hadamard finite-part integrals in the region S. Outside the region of S ,

'u x

0, 'u y

0, 'u z

0 , which means that the displacement field is continuous.

Space I

P1 , Q 1

W zyf

z, 9

V zf W zxf y, K

x, [

O Space II

P2 , Q 2

W zyf

W zxf V zf (a)

Fig. 2 Problem configuration The hypersingular BIEs (4a)-(4c) have been solved numerically by using the procedure of the BEM. In the numerical solution procedure, it is necessary to express the oscillatory singular stresses, which are present at the crack-tip or on the crack-front of interface cracks [12-14]. Once the crack-opening-displacements have been obtained numerically, the stress intensity factors at a point Q on the crack-front can be computed by using the following relations

100


K (Q)

K I (Q) iK II (Q) lim 2r1 / 2 iH V zI ( r , T ) iV yzI (r , T ) r o0

K III (Q) lim 2r V (r , T ) 1/ 2

I zx

r o0

T 0

T 0

, (5)

.

3. Maximum stress intensity factors of an interface crack under tension

Let us consider a rectangular interface crack under tension at infinity. If a t b , the maximum stress intensity factors K I and K II would appear at points x, y

0, rb . The

K III values are smaller than the values of

K I and K II , and the maximum value of K III appears at a point which is very close to the corner of the

rectangle. The dimensionless stress intensity factors can be expressed as K I max K II max FI max , FII max , FIII max V zf S b V zf S b FI max

K I max

V zf S S

, FII max

K II max

V zf S S

, FIII max

K III max

V zf S b

,

K III max

V zf S S

(6)

.

Suppose that a t b , the variations of the stress intensity factors FI , FII , FIII and FI , FII , FIII are given in Tables 2-4. It is noted that in Tables 2 and 3, the dimensionless stress intensity factors FI and FII are dependent on the bi-material parameter H only, but the values of FIII are determined by Poisson’s ratios also. The numerical errors are given in Table 2; it shows that the maximum error is less than 8%. The numerical error is defined by FI m ax 0.5 G u 100%. FI m ax The definition of the bi-material parameter H is 1 § P 2N1 P1 · (7) ln ¨ H ¸ , N l 3 4vl , l 1, 2 . 2S © P1N 2 P2 ¹ By applying the least-square method to the numerical results obtained by the BEM, the following approximate or fitting formula can be obtained FI m ax (H ) =0.507+0.030 H -2.294 H 2 +12.67 H 3 , 0 d H d 0.1 . (8) The comparing curves of the numerical results and the fitting formula are given in Fig. 3. In particular, when H 0 , FI m ax 0.507 , the value is almost the same as the approximate formula (1), and the following two approximate formulas can be applied to calculate the values of FII m ax and FIII m ax FII max | H ,

FIII max d 0.5 FII max

0.5H .

(9)

FI *

FI a/b= 1

a/b = 2

a/b = 4

a/b = 8

a/b = 1

G (%)

a/b = 2

G (%)

a/b = 4

G (%)

a/b = 8

H =0[10]

0.753

0.906

0.977

0.995

0.532 5

6.1

0.538 6

7.2

0.488 5

-2.4

0.470 4

-6.3

H =0.02

0.752 8

0.905 2

0.976 0

0.994 7

0.532 3

6.1

0.538 1

7.1

0.488 0

-2.5

0.470 2

-6.3

H =0.04

0.750 9

0.903 8

0.975 0

0.993 8

0.531 0

5.8

0.537 3

6.9

0.487 5

-2.6

0.469 8

-6.4

H =0.06

0.747 8

0.901 3

0.973 0

0.992 0

0.528 8

5.4

0.535 8

6.7

0.486 5

-2.8

0.468 9

-6.6

H =0.08

0.743 3

0.897 5

0.969 9

0.989 1

0.525 7

4.9

0.533 5

6.3

0.484 9

-3.1

0.467 6

-6.9

H =0.10

0.737 3

0.892 1

0.965 4

0.984 8

0.521 4

4.1

0.530 3

5.7

0.482 7

-3.6

0.465 6

-7.4

Tab. 2 Stress intensity factor FI and FI at (0,b)

G (%)


101

FII *

FII a/b = 1

a/b = 2

a/b = 4

a/b = 8

a/b = 1

a/b = 2

a/b = 4

H =0

0

0

0

0

0

0

0

a/b = 8 0

H =0.02

0.027 4

0.035 2

0.038 8

0.039 7

0.019

0.021

0.019

0.019

H =0.04

0.054 2

0.069 6

0.076 8

0.078 6

0.038

0.041

0.038

0.037

H =0.06

0.079 8

0.102 7

0.113 4

0.116 0

0.056

0.061

0.056

0.055

H =0.08

0.104 0

0.133 8

0.147 9

0.151 5

0.073

0.079

0.074

0.072

H =0.10

0.126 3

0.162 7

0.180 1

0.184 5

0.089

0.096

0.090

0.087

Tab. 3 Stress intensity factor FII and FII at (0,b)

FIII

FIII *

a/b = 1

a/b = 2

a/b = 4

a/b = 8

a/b = 1

a/b = 2

a/b = 4

H =0

0

0

0

0

0

0

0

0

H =0.02

0.012 0

0.011 9

0.010 1

6.83×10-3

0.008

7.07×10-3

5.05×10-3

3.23×10-3

H =0.04

0.023 8

0.023 5

0.020 0

1.36×10-2

0.017

0.013 9

0.010

6.43×10-3

0.029 5

2.01×10

-2

0.025

0.020 5

0.014 8

9.50×10-3

2.63×10

-2

0.032

0.026 6

0.019 1

1.24×10-2

3.21×10

-2

0.039

0.032 2

0.023 2

1.52×10-2

H =0.06

0.035 1

H =0.08

0.045 6

H =0.10

0.055 3

0.034 5 0.044 8

0.038 3

0.054 2

0.046 4

a/b = 8

Tab. 4 Stress intensity factor FIII and FIII at ( a / b,0.91* b) for Q 1 Q 2 0.3

0.508

0.8

0.6

0.507 0.506 0.505

(5){(1)+(2)+(3)+(4)}/4

(6) Eq.(9)

FI

0.7

(1)a/b=1 (2)a/b=2

0.504

FI

0.503 0.5

0.502 0.4

0.3 0.00

(3)a/b=4

0.501

(4)a/b=8

0.500

0.02

0.04

0.06

0.08

0.00

0.10

{(1)+(2)+(3)+(4)}/4

FI HH H 0.02

0.04 H

H

Fig. 3(a) Stress intensity factors at ( 0, b) for different aspect ratios of the crack subjected to a load V zf

0.06

0.08

0.10

Fig. 3(b) Comparison of the average values of the stress intensity factors with approximate formula

4. Maximum stress intensity factors of an interface crack under shear

The dimensionless stress intensity factors under shear are defined as K I max K II max FI max , FII max , FIII max W yzf S b W yzf S b FI max

K I max

W yzf S S

,

FII max

K II max

W yzf S S

,

FIII max

K III max

W yzf S b K III max

W yzf S S

, .

(10)

102


The stress intensity factors K II and K III are always insensitive to the varying ratio of the shear modulus, and are mainly determined by Poisson's ratio [15]. Table 5 shows the range of the maximum stress intensity factors when Q 0 0.5 . We found that F,,* varies in the range of 0.47 0.64 # 0.55 even when the rectangular shape ratios are changed extremely from a / b 1 to a / b o f , and the value is almost the same as predicted by the formula (2). However, for FIII , the range 0.39 0.52 # 0.45 is applicable only for a / b 1 , so we cannot use formula (3) except for a / b 1 . When a / b ! 1 , the derived values of FIII are less

than the case for a / b 1 , where formula (3) is not applicable. For the stress intensity factor FI , its values can be approximated by FI # H . Besides, with the presence of the shear stress, the stress intensity factors are dependent on Poisson’s ratios. When Poisson’s ratios v1 and v1 vary from 0 to 0.5, the errors of F,,* and FIII are 9.7% and 18% respectively in the case of v1 v2 0.3 . Therefore, we usually assume v1 v2 0.3 in our approximate calculations. Q1

Q2

H

FII

FIII

FI

0

0

0.053 6

0.760 3

0.742 1

0.074 0

0

0.5

0.134 9

0.827 6

0.632 5

0.188 7

0.585 3

0.447 3

0.133 5

0.1

0.1

0.047 5

0.785 8

0.716 0

0.068 2

0.555 7

0.506 4

0.048 2

0.1

0.5

0.115 5

0.840 1

0.626 8

0.167 9

0.594 1

0.443 3

0.118 7

0.2

0.2

0.040 0

0.813 2

0.685 8

0.059 8

0.575 1

0.485 0

0.042 3

0.2

0.5

0.093 5

0.854 0

0.618 5

0.141 6

0.604 0

0.437 4

0.100 1

0.3

0.3

0.030 4

0.842 8

0.650 7

0.047 5

0.596 0

0.460 2

0.033 6

0.3

0.5

0.068 0

0.869 6

0.606 2

0.107 5

0.615 0

0.428 7

0.076 0

0.499 9

0

0.557 0

-6

0.643 4

0.393 9

5h10-6

0.499 9

0.909 8

7h10

FII * 0.537 7

FIII * 0.524 8

FI * 0.052 3

Tab. 5 Stress intensity factors at ( 0, b) for a / b 1

5. Conclusions By using a BEM based on hypersingular boundary integrals equations, the stress intensity factor formulas in terms of an area parameter to evaluate rectangular interfacial cracks in three-dimensional bimaterials are considered in this paper, and two conclusions can be made as follows: 1) When the infinity is subjected to a tensile stress V zf , the maximum error between the numerical results and formula (1) is less than 8%. Fitting formulas for stress intensity factors with the bimaterial parameter H as a variable are derived, and they are given by: FI m ax (H ) =0.507+0.030 H -2.294 H 2 +12.67 H 3 , FII max | H and FIII max d 0.5 FII max

0.5H .

2) When the infinity is subjected to a shear stress W yzf and the rectangular shape ratio varies from a / b 1 to a / b o f , the range of FII is 0.47 0.64 # 0.55 , which is close to formula (2). However, for FIII , the applicable range is only for a / b 1 , so we cannot use formula (3) except when a / b 1 . When a / b ! 1 , the derived values of FIII are less than the case a / b 1 , where formula (3) is not applicable. For the

stress intensity factor FI , its values can be approximated by FI # H .

Acknowledgments The project is supported by the National Natural Science Foundation of China (No. 10872213) and the personnel exchange program of China Scholarship Council (CSC) and German Academic Exchange Service (DAAD).

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz References [1 ] England F.J., Appl. Mech., 1965, 32(3): 829-836. [2 ] Shibuya T., Koizumi T., Iwamoto T., JSME International Journal: Series A, 1989, 32(4): 485-491. [3 ] Noda N.A., Kagita M., Chen M.C., Int. Solid.& Struct., 2003, 40(24): 6577-6592. [4 ] Chen M.C., Noda N.A., Tang R.J., J. Appl. Mech., 1999, 66(6): 885-890. [5 ] Zhao M.H., Guo C. J., Fang Z.P., J. Mech. Strength, 2002, 24(4): 535-538 (In Chinese). [6 ] Murakami Y.S., Nemat-Nasser S., Engng. Fract. Mech., 1983, 17(3): 193-210. [7 ] Murakami Y., Engng. Frac. Mech., 1985, 22(1): 101-114. [8 ] Irwin G.R., J. Appl. Mech., 1962, 29: 651-654. [9 ] Kassir M.K., Sih G.C., J. Appl. Mech., 1966, 33: 601-611. [10 ] Wang Q., Noda N.A., Honda M.A., Chen M.C., Int. J. Fract., 2001, 108(2): 119-131. [11 ] Noda N.A., Kihara T.A., Archive of Applied Mechanics, 2002, 72(8): 599-614. [12 ] Noda N.A., Xu C.H., Takase Y., JSME,Series A, 2007, 73(4): 379-386 (In Japanese). [13 ] Noda N.A., Xu C,H., Takase Y., JSME,Series A, 2007, 73(4): 468-474 (In Japanese). [14 ] Xu C.H., Noda N.A., Takase Y., JSME,Series A, 2007, 73(7): 768-774 (In Japanese).

103

104


Iterative Optimization Methodology for Sound Scattering using the Topological Derivative Approach and the Boundary Element Method Agustín Sisamon1, Silja C. Beck2, Adrián P. Cisilino3, Sabine Langer4 1

2

Institut für Angewandte Mechanik, Technische Universität Braunschweig, Spielmannstr. 11, 38106 Braunschweig, Germany, [email protected], http://www.infam.tu-bs.de 3

4

INTEMA, Universidad Nacional de Mar del Plata ± CONICET. Av. Juan B. Justo 4302, 7600 Mar del Plata, Argentina, asisamon@ fi.mdp.edu.ar, http://www.intema.gov.ar

INTEMA, Universidad Nacional de Mar del Plata ± CONICET. Av. Juan B. Justo 4302, 7600 Mar del Plata, Argentina, [email protected], http://www.intema.gov.ar

Institut für Angewandte Mechanik, Technische Universität Braunschweig, Spielmannstr. 11, 38106 Braunschweig, Germany, [email protected], http://www.infam.tu-bs.de

Keywords: 2D acoustics, topological derivative, iterative optimization process Abstract. Today, reduction of sound emission plays a vital role while designing objects of any kind. Desirable aspects might include decreased radiation in certain directions of such an object. This work shows an approach to iteratively compute the shape of an obstacle which fulfils best to prescribed design variables using the framework provided by the Topological Derivative and the Boundary Element Method. At the beginning of the process an empty design space is defined in which in iterative steps the shape will be developed. A regular array of points is set over the entire design space. The objective function is given by a set of prescribed pressure values for the scatter pattern on a circle around this design space. The object, which acts as a scatterer, is considered acoustically rigid. The shape of the object builds up cumulatively, adding in each iterative step a rigid inclusion at the position that the Topological Derivative identifies as the most effective to achieve the prescribed design values. The procedure is repeated until a given stopping criteria is satisfied. The proposed method requires the computation of a forward problem and an adjoint problem for each step. The first is solved using a standard BEM for 2D acoustics, while the latter is solved backwards using the prescribed pressure values. The insertion of the rigid inclusions in each step is done by removing points from the design space. The BEM model geometry is updated automatically using a weighted Delaunay triangularization DOJRULWKP FDSDEOH RI GHWHFWLQJ µKROHV¶ at those positions where the points have been eliminated. The capabilities of the proposed strategy are demonstrated by solving an example. Introduction The desire for a quiet environment has lead to including acoustic optimization into the design process of objects of any kind whether they emit acoustic radiation, transport or receive it. Especially when structures are to act as noise barriers, a lower sound level in certain areas around these objects is the target. To determine sound propagation around and along bounded entities the Boundary Element Method (BEM) is used. It has proven to be effective and has become a well-established technique. Employing the BEM directly means that the boundary of an object is known in both terms of geometry and boundary conditions. For the case of the geometry of an object not being known but at the same time having information about the field surrounding the object, Feijoó [1] has proposed a method to inversely reconstruct this boundary. The solution of this inverse scattering problem is based on the topological derivative. An optimization problem is posed which aims to minimize the difference between the scattering pattern acquired when placing small scattering objects in the region of interest and the known scattering pattern. The rate of change of this difference is the topological derivative field. High values in the topological derivate field


105

indicate positions for placing inclusions to converge to the objective and thus define the sought-after boundary of the object. In this work it is intended to combine the BEM and the topological derivative approach to iteratively identify the optimized shape of an object for a given objective. The framework here is strictly 2D. In the following sections the topological derivative approach of Feijoó will be outlined, and the iterative process will be sketched in detail. The example is followed by concluding remarks. The Topological Derivate Approach The functional value. The problem outlined in [1] consists of a domain : which encloses a scatterer of unknown shape :o (see Fig. 1). On an oblique virtual surface *s in : a set of values of the wavefield ‫݉ݑ‬ are known (e. g. from measurements). This set of values corresponds to illuminating the object by an incident wave from a given angle. The question now is where to put the boundary *o of the scattering object :o, so that the results of the wavefield u in the chosen configuration of :o fit best to the known wavefield values um.

um

*0 :0 n

*s d

Figure 1: The inverse scattering problem. ෡ to this optimization problem is of a least square type: The solution ȍ ෡ = arg min ݆ሺȍሻ ȍ (1) with 1 ݆ሺȳሻ = ‫ܬ‬൫‫ݑ‬ሺȳሻ൯ = ‫*׬‬s ȁ‫ ݑ‬െ ‫ ݉ݑ‬ȁ2 ݀Ȟ (2) 2 The values of u correspond to the solution of e. g. a plane wave uinc (x) exp(i kx d) (k being the wavenumber and d the direction of propagation) interacting with the medium and the scattering object. The solution u uinc u s needs to fulfill ‫׏‬2 ‫ ݑ‬+ ݇ 2 ‫ = ݑ‬0 in Թ2 ‫ ך‬:o (3) ‫ ݑ׏‬ή ‫ = ܖ‬0 on Ȟo (4) ߲‫ݑ‬ lim‫ݎ‬՜λ ‫ ݎ‬1Τ2 ቀ ߲‫ ݏݎ‬െ ݅݇‫ ݏݑ‬ቁ = 0. (5) Eq. (3) is the Helmholtz equation for a homogeneous medium (without any attenuation: Im(k) = 0), Eq. (4) is the boundary condition defining the scattering object as being acoustically rigid and Eq. (5) states the Sommerfeld condition, allowing scattered waves only to travel into infinity. The topological derivative. The idea is now to place a small circular scattering object of radius H into the domain at point x WKXV FUHDWLQJ D µKROH¶ BH(x) . This leads to a new domain :ɂ = :\BH(x) and a new functional value ݆ሺ:ɂ ሻ. If ݂ሺɂሻ FRUUHVSRQGVWRWKHQHJDWLYHYDOXHRIWKHFLUFOH¶Vextent the new functional value takes a form of ݆ሺ:ɂ ሻ = ݆ሺȳሻ + ݂ሺɂሻDT ሺ‫ܠ‬ሻ + o൫݂ሺɂሻ൯ (6) with DT ሺ‫ܠ‬ሻ = limߝ՜0

݆ ሺ:ɂ ሻെ݆ ሺȳሻ ݂ሺɂሻ

as

limߝ՜0

o൫݂ሺɂሻ൯ ݂ሺɂሻ

=0

(7)

106


The topological derivative DT ሺ‫ܠ‬ሻ is an indication for the rate of change in the functional value with UHVSHFWWRWKHµVL]H¶RIWKHscatterer. When evaluating DT ሺ‫ܠ‬ሻ at all points of the domain :, a scalar field is obtained, called the topological derivative. Reconstruction now can be done by placing small scatterers at points with high values of DT . The final expression for the topological derivative is of the form (see [1]) DT ሺ‫ܠ‬ሻ = Reൣ2‫׏‬ɉതሺ‫ܠ‬ሻ ή ‫ݑ׏‬ሺ‫ܠ‬ሻ െ ݇ 2 ɉതሺ‫ܠ‬ሻ‫ݑ‬ሺ‫ܠ‬ሻ൧ (8) where u is the solution of the so-called forward problem (see Eq. (3)-(5)), and ɉത is the conjugate complex of the solution of the adjoint problem given by (9) ‫׏‬2 ɉ + ݇ 2 ɉ = 0 in Թ2 ‫ ך‬:o ‫׏‬ɉ ή ‫ = ܖ‬െሺ‫ ݑ‬െ ‫ ݉ݑ‬ሻ on *s (10) ߲ɉ lim‫ݎ‬՜λ ‫ ݎ‬1Τ2 ቀ ߲‫ ݏݎ‬െ ݅݇ɉ‫ ݏ‬ቁ = 0. (11) The adjoint problem solves field ɉ0 to be specified on the boundary of the scatterer, *o, which results in the mismatch between the forward problem solution u and the known values um on the virtual surface *s. The Iterative Process In [1] the topological derivative is computed once to find the most probable shape of the unknown scatterer. The algorithm proposed in this work allows starting the optimization processes using an initial geometry for the scatterer. The objective of the optimization is given by prescribed pressure values um on a virtual surface *s that surrounds the design domain (see Fig. 2a). The iterative algorithm can be summarized as follows: 1) Solve the direct problem (see Eqs. (3) to (5)) for the actual geometry of the scatter using BEM for the incident planar wave (Fig. 2a) 2) Compute the sound pressure field ‫ݑ‬ሺ‫ܠ‬ሻ and its gradient ‫ݑ׏‬ሺ‫ܠ‬ሻ at the internal points and at the points along the virtual surface *s. 3) Compute the mismatch between the forward problem solution u and the known values ‫ ݉ݑ‬on the virtual surface *s, ሺ‫ ݑ‬െ ‫ ݉ݑ‬ሻ. 4) Solve the adjoint problem (see Eqs. (9) to (11)) to find the pressure field ɉ on the boundary of the scatterer. 5) Compute the sound pressure field ɉሺ‫ܠ‬ሻ and its gradient ‫׏‬ɉሺ‫ܠ‬ሻ at the internal points. 6) Compute the DT at the internal points using expression (8). 7) Remove the internal points with the maximum values of DT (a few percent of the total number of points). See Fig. 2b. 8) Remesh the model. 9) Check stopping criterion (typically a limit value for the difference ሺ‫ ݑ‬െ ‫ ݉ݑ‬ሻ). If necessary repeat

from step 1) 10) At this stage the desired final geometry is obtained. um

um

um

*s

*s

*s

d

Figure 2: BEM implementation: (a) Initial BEM model, (b) Elimination of internal points, (c) BEM model remeshing.


107

Solution of the direct problem (step 1). Since the DT needs the solution of the field ‫ݑ‬ሺ‫ܠ‬ሻ and its gradient ‫ݑ׏‬ሺ‫ܠ‬ሻ, the solution of the direct problem is done using a standard BEM formulation. The geometry of the scatterer is discretized using two-node linear continuous elements. Sound-hard boundary condition, ߲‫ݑ‬/ ߲݊=0, is specified along the complete model boundary. The design space is filled with a regular array of internal points following the pattern depicted in Fig. 2a. Solution of the adjoint problem (step 4). The adjoint problem is solved by setting a system of equations with the nodal values of the pressure field ɉ on the boundary of the scatterer as unknowns. To this end, the points on the virtual surface *s are assimilated as internal points, and their values of the mismatch ሺ‫ ݑ‬െ ‫ ݉ݑ‬ሻ are expressed in terms of the boundary values of ɉ. This results in a system of equations of the form

൦

݄11 ݄21

݄12 ݄22

‫ڭ‬

‫ڭ‬

݄ܰ1

݄ܰ2

ሺ‫ ݑ‬െ ‫ ݉ݑ‬ሻ1 ǥ ݄1݊ ߣ1 ሺ‫ ݑ‬െ ‫ ݉ݑ‬ሻ2 ǥ ݄2݊ ߣ2 ൪൦ ൪ = ൦ ൪, ‫ڭ‬ ‫ڰ‬ ‫ڭ‬ ‫ڭ‬ ǥ ݄ܰ݊ ߣ݊ ሺ‫ ݑ‬െ ‫ ݉ݑ‬ሻܰ

(12)

where the coefficients ݄݆݅ contain the integrals of the fundamental solution ߲‫ כݑ‬/߲݊ on the boundary elements, ߣ݆ are the values of the pressure at the ݊ nodes on the boundary and ሺ‫ ݑ‬െ ‫ ݉ݑ‬ሻ݅ are the values of the pressure mismatch at the ܰ points along the virtual surface *s. Following standard procedures (see reference [2]), the system of equations is set using ܰ = 3݊, this is, the number of points along the virtual surface is chosen three times the number of nodes used for the model discretization. The system of Eq. (12) is solved using a single value decomposition (SVD) algorithm. Model remeshing (step 8). The removal of internal points is followed by a model remeshing. For this purpose, the program MeshSuite based on an D-shapes algorithm is employed [3]. Upon the input of the coordinates of the boundary nodes and internal points after each optimization step (see Fig. 2b), MeshSuite outputs the connectivity of the new model boundary (see Fig. 2c). Thus, those points not used as boundary nodes are assimilated to internal points in the new discretization for the next iteration. The new boundary element mesh is checked for multi-connected boundary points and smoothed using a simple relaxation DOJRULWKP 7KH VPRRWKLQJ SURFHVV LV LQWHQGHG WR DYRLG ³QRLV\´ VROXWLRQV GXH WR WKH SUHVHQFH RI WKH ORFDO irregularities on the model boundary as a consequence of the spatial disposition of the internal points. Further details about the remeshing procedure can be found in reference [4]. Example The proposed optimization strategy is illustrated by means of an example. We attempt to reconstruct the shape of a circular scatterer of radius ܴ = 2 ݉ starting from a square scatterer of side ‫ = ܮ‬2 ݉ (see Figure 3a). The radius of the virtual surface *s is ‫ = ݎ‬3.5 ݉. The object is illuminated in the direction ‫ ݔ‬by a plane wave with wavenumber ݇ = 32 and an amplitude of 1 Pa. The objective values of the wavefield ‫ ݉ݑ‬are specified at ܰ = 800 points evenly distributed along *s. This large number of points guarantees the fulfillment of the condition ܰ ൒ 3݊ when solving the adjoint problem (see Eq (12)). The distance between the internal points (which it is also element length of the BEM discretization) is ݈ = 0.05 ݉, being approximately four times smaller than the wave length. Figures 4a and 4b illustrate contour plots for the pressure solutions ‫ ݑ‬and ߣ for the direct (Eqs. (3) to (5)) and adjoint problems (Eqs. (9) to (11)) for the initial geometry. The topological derivative result after computing Eq. (8) is plotted in Figure 4c. It can be seen that maximum values for the DT are at the top and bottom sides of the object. It is from those zones that the internal points are removed to update the geometry of the scatterer for the second step. The resulting geometries for the subsequent steps are plotted in Figure 3. Figure 5 depicts the evolution of the pressure results along the *s with the optimization process. In every case the resulting pressure values are plotted together with the objective values, ‫ ݉ݑ‬. It can be seen from Figs. 3 and 4 that the pressure results converge towards their objective values as the shape of the scatterer approaches that of a circle.

108


4

4

Objetive

*s

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

(a) Initial geometry -4

-3

-2

-1

0

-4

1

2

3

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

(c) Step #6 -4

-3

-2

-4

-1

0

1

2

3

4

(b) Step #1 -4

4

-3

-2

-1

0

1

2

3

4

-1

0

1

2

3

4

(d) Step #9 -4

-3

-2

Figure 3: Evolution of the shape of the scatterer during the optimization process.

(a)

(b)

(c)

Figure 4: Contour plots for the (a) direct and (b) adjoint pressure fields and (c) the associated topological derivative result.

Pressure (Module)


1.4

1.4

1.4

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1.2

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200

400

600

800

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109

400

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Node

Node

(a)

(b)

(c)

600

800

Figure 5: Evolution of the pressure results: (a) initial geometry, (b) first step (c) ninth step (Objective function is plotted in black).

Conclusions In this work an iterative optimization strategy based on the topological derivative and BEM has been presented. The proposed strategy has the ability to compute the shape of an obstacle which fulfils best to a prescribed sound pressure field specified along a virtual surface which encloses the design space. The given example shows strategy provides is a promising approach to design the shape of scattering objects. Its implementation can be easily adapted to optimize the shape of existent objects in order to decrease its radiation in certain directions. The discretization and remeshing schemes showed to be flexible and reliable, allowing dealing with problems of arbitrary shape. However, the effect of the geometric irregularities induced on the model boundary due to the spatial disposition of the internal points can seriously affect the performance of the optimization process. Thus, it is important to perform a smoothing of the model boundary in order to guarantee the performance of the algorithm. Acknowledgements This work has been supported by the Project DA0806 sponsored by the MINCYT (Argentina) and the DAAD (Germany). References [1] G. R. Feijoo ³$ QHZ PHWKRG LQ LQYHUVH VFDWWHULQJ EDVHG RQ WKH WRSRORJLFDO GHULYDWLYH´ Inverse Problems, 20, 1819-1840 (2004). [2] S. Hampel, S. Langer and A.P. Cisilino³Coupling Boundary Elements to a Ray Tracing Procedure´ International Journal for Numerical Methods in Engineering. Vol. 73/3, pp. 427-445 (2008). [3] N. Calvo, S.R. Idelsohn and E. Oñate. ³The extended Delaunay tessellation´ Engineering Computations, 20/5-6 (2003). [4] L. Carretero Neches and A.P. Cisilino. ³Topology Optimization of 2D Elastic Structures Using Boundary Elements.´ Engineering Analysis with Boundary Elements, Vol. 32, pp. 533-544 (2008).

110


A Laplace transform boundary element solution for the Cahn-Hilliard equation A. J. Davies and D. Crann School of Physics, Astronomy and Mathematics University of Hertfordshire, Hatfield, Herts. AL10 9AB, U.K.

Keywords: Laplace transform, boundary elements, Cahn-Hilliard equation, biharmonic diffusion. Abstract The Cahn-Hilliard (C-H) equation describes the time development of the concentration of the components of a fluid during phase separation. The process is described by a non-linear biharmonic diffusion equation. Numerical solutions of the C-H equation have used a finite difference approach for the time variable with finite elements for the space variation. In previous papers the current authors have shown that the Laplace transform provides a suitable alternative approach to deal with the time variation. In particular they have shown that it works very well with the boundary element method for a variety of diffusion problems including nonlinear equations and those of biharmonic form. In this paper we shall show how the introduction of a chemical potential reduces the fourth order partial differential equation into a pair of coupled second order equations, one a parabolic equation and the other an elliptic equation. The Laplace transform boundary element method can then be used to solve the coupled system. Introduction In a previous paper [1] the authors have considered a Laplace transform boundary element approach to the solution of the biharmonic diffusion equation, ∇4 u = α1 ∂u ∂t . The approach involves the introduction of a second dependent variable, v, such that ∇2 u = v leading to a pair of coupled second order partial differential equations. The C-H equation takes the form ∇4 u = −

1 ∂u + β∇2 φ α ∂t

(1)

whereAdv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz φ(u) = au + bu2 + cu3

111

(2)

and a, b and c are constants. Clearly, there are two differences introduced in the C-H equation: (i) the negative coefficient of ∂u ∂t gives a backward form of the diffusion equation (ii) the functional form of φ leads to a non-linear equation. In this paper we consider a linearised form in which b = c = 0 to investigate the suitability of the Laplace transform approach. In the modelling of thin oil films [2] the backward biharmonic diffusion equation ∇4 u = − α1 ∂u ∂t is developed. In a preliminary study, the current authors [3] have shown that, even though the backward Laplacian diffusion problem ∇2 u = − α1 ∂u ∂t is ill-posed [4] e.g. the solution does not exist for most initial data and even if a solution exists it is very likely to blow up. However the equivalent biharmonic problem does not appear to suffer from these difficulties. This leads us to consider a more serious study of the C-H equation. The Cahn-Hilliard equation The C-H equation is used to model two-phase fluid flow and was first described by Cahn and Hilliard [5, 6, 7]. Some mathematical features of the equation have been discussed by, among others, Elliott et al. [8, 9, 10]. We shall consider the linear form of equations (1) and (2) with b = c = 0 so we can write equation (1) in the form 1 ∂u + k∇2 u (3) α ∂t to be solved in some region Ω subject to the boundary conditions on Γ ∇4 u = −

u = u(s, t) and q ≡

∂u = q(s, t) ∂n

(4)

and the initial condition u(x, y, 0) = u0 (x, y)

(5)

We follow the approach of Toutip et al. [11] and write v = ∇2 u so that equation (3) becomes the pair of coupled equations ∇2 u = v

(6)

1 ∂u + kv (7) α ∂t When applying the boundary conditions we note that we have a pair of Laplacian operators in equations (6) and (7). To ensure the problem is properly-posed we must apply either u or q, but not both, at each point of Γ. We consider Γ comprised of two sections, Γ1 and Γ2 and write u1 (s, t) s ǫ Γ1 u(s, t) = u2 (s, t) s ǫ Γ2 ∇2 u = −

112

q1 (s, sǫΓ Adv. Bound. Elem. Tech. Cht)Zhang, M1H Aliabadi, M Schanz q(s, t) = Eds: q2 (s, t) s ǫ Γ2

then we choose u = u1 on Γ1 and q = q2 on Γ2 p≡

∂v = p1 ≡ ∇2 q1 on Γ1 , ∂n

v = v2 = ∇2 u2 on Γ2

(8) (9)

The Laplace transform We denote by u ¯(x, y; λ) and v¯(x, y; λ) the Laplace transforms of u(x, y, t) and v(x, y, t) respectively. Then equations (6) and (7) become, in transform space, ∇2 u ¯ = v¯ (10) 1 ∇2 v¯ = − (λ¯ u − u0 ) + k¯ v (11) α The solutions of equations (10) and (11) are obtained using the following iterative scheme:

with

1 u(n) − u0 ) + k¯ v (n) ∇2 v¯(n+1) = − (λ¯ α

(12)

∇2 u ¯(n+1) = v¯(n+1)

(13)

u ¯(0) = u ¯0 and v¯(0) = v¯0 = ∇2 u ¯0

We use the dual reciprocity boundary element method to solve the coupled elliptic equations (12) and (13). Equation (12) is solved subject to the boundary condition v¯ = v¯2 on Γ2 and p¯ = p¯1 on Γ1

(14)

and equation (13) is solved subject to the boundary condition u ¯=u ¯1 on Γ1 and q¯ = q¯2 on Γ2

(15)

The dual reciprocity boundary method We divide Γ into N elements, Γk , and choose L points inside Ω in the usual boundary element manner. Now, we can write each of the equations (12) and (13) in the form ∇2 u ¯=b (16) The integral equation equivalent to equation (16) is Z I I ∗ ∗ bu∗ dΩ ¯ dΓ − u q¯ dΓ = cΓ u ¯Γ + q u Γ

Γ

Ω

(17)

1 whereAdv. u∗ Bound. = − 2π ln R Tech. is theEds: usual solution for the Laplacian113 Elem. Ch fundamental Zhang, M H Aliabadi, M Schanz ∂u∗ ∗ operator and q = ∂n . We expand the domain function b as a linear combination of radial basis functions N +L X αj fj (R) (18) b≈ j=1

where the fj (R) are chosen such that, for some u ˆj , we have [12] ∇2 u ˆj = fj (R)

(19)

Hence, using equations (18) and (19) together with Green’s theorem, equation (17) becomes ci u ¯i +

N Z X k=1

N +L X

αj

j=1

cj u îj +

Γk

q u ¯ dΓ − ∗

N Z X k=1

Γk

N Z X k=1 Γk

q u ˆj dΓ − ∗

u∗ q¯ dΓ = !

N Z X k=1

u qˆj dΓ ∗

Γk

for i = 1, . . . , N . We now write this system of equations in the usual matrix form where we use the subscript B to denote that the corresponding quantity is associated with a boundary node: h i ¯ B − GB Q ¯ B = HB U ˆ − GB Q ˆ α HB U (20) where the values of the coefficients, αi , are obtained by collocating at the N + L points giving the usual system of equations b = Fα

(21)

α = F−1 b

(22)

and α is given by Internal values may be obtained in a similar manner [1] and written in the usual matrix form h i ¯ I = GI Q ¯ B − HI U ¯ B + HI U ˆ − GI Q ˆ α+IUα ˆ IU (23) Finally, then equations (20) and (23) can be combined in the form h i ¯ − GQ ¯ = HU ˆ − GQ ˆ F−1 b HU Now we define

h i ˆ − GQ ˆ F−1 S = HU

(24)

(25)

¯BSchanz ¯ B and to 114 obtain theAdv. system ofElem. equations the and Q Bound. Tech. for Eds: Chboundary Zhang, M Hsolution Aliabadi, U M ¯I the internal solution U ¯ − GQ ¯ = Sb1 HU (26) In a similar manner we obtain for v ¯ − GP ¯ = Sb2 HV

(27)

Hence the discrete systems of equations associated with the partial differential equations (10) and (11) are ¯ (n+1) − GP ¯ (n+1) = b1 HV ¯ (n+1) − GQ ¯ (n+1) = b2 HU

¯ (n) , V(n) U ¯ (n+1) V

(28)

¯ (0) = U ¯ 0 and V ¯ (0) = V ¯0 with U On application of the boundary conditions equation (28) may be written in the form ¯ (n+1) ¯ (n) , V ¯ (n) and A2 y(n+1) = F2 V A1 x(n+1) = F1 U h iT h iT ¯ (n+1) P ¯ (n+1) ¯ (n+1) Q ¯ (n+1) and the where x(n+1) = V and y(n+1) = U iteration is terminated using a suitable stopping criterion. Finally then, the approximate transform is inverted to obtain the approximate solution U. Numerical inversion of the Laplace transform We use the Stehfest numerical procedure [13] which has been shown to be well-suited to the solution of diffusion-type problems [14]. Choose a specific time value, τ , at which we seek a solution and define a set of transform parameters ln 2 λj = j : j = 1, 2, . . . , m; m even τ The dual reciprocity boundary element method is used for each λj to obtain sets of approximate boundary and internal values given respectively by ¯B, ij ; i = 1, . . . , N ; j = 1, . . . , m and U ¯I, kj ; k = 1, . . . , L; j = 1, . . . , m U The inverse transforms are then given by ln 2 X ¯ ≈ wj UB,rj τ m

UB, r

j=1

ln 2 X ¯ wj UI,rj ≈ τ m

and

UI, r

j=1

whereAdv. r = Bound. 1, . . . , Elem. N forTech. boundary points and r Aliabadi, = 1, . . . ,MLSchanz for internal points.115 Eds: Ch Zhang, MH The weights, wj , are given by Stehfest [13] and tabulated in [15]. Results We illustrate the process with the following example 1 ∂u + k∇2 u + h(x, y, t) α ∂t in the unit square {(x, y); 0 < x < 1, 0 < y < 1} subject to Dirichlet and Neumann boundary conditions appropriate to the exact solution in the case α=1 u(x, y, t) = (1 + x4 + y 4 )e−t ∇4 u = −

We use 36 linear boundary elements and 9 internal nodes with f (R) = 1+R. In Figure 1 we show the time development at three points and in Figure 2 we show the space variation for three times, in both cases the points are along the line of symmetry. In both cases we see that the approximate solutions compare very well with the analytic solution. Finally we note here that, as reported in [1], the choice of Γ1 and Γ2 is somewhat arbitrary and a change in this choice does not affect the accuracy of the solutions. u(x, y, t) 2 LT approx. analytic

1.8 1.6 1.4 1.2

x=y =0.2 x=y =0.5

1

x=y =0.8

0.8 0.6 0.4 0.2 0

0

1

2

3

4

5

t

Figure 1: Time development at three points Conclusions The Laplace transform dual reciprocity method has been shown to provide a suitable approach to the solution of a backward biharmonic diffusion

116

t) Adv.u(l, Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz 2.5

LT approx. analytic

2

t =0.2 t =0.5 t =1.0

1.5

1

0.5

t =5.0 0

0

0.2

0.4

0.6

0.8

1

√ l/ 2

Figure 2: Space variation along the line of symmetry, l is the distance from the origin problem, the linearised C-H equation. It is very pleasing to report that even though it is posed as a backward problem we find no evidence of illconditioning. This particular problem is rather stylised since the C-H equation is non-linear. Nevertheless it gives us confidence to move on and tackle the non-linearity.

References [1] Davies AJ and Crann D. A Laplace transform solution of the biharmonic diffusion equation. Boundary Elements XXVIII, 243–252 (2006). [2] Tanner LH and Berry MV. Dynamics and optics of oil hills and oilscapes. J. Phys. D; Appl. Phys., 18: 1037–1061 (1985). [3] Crann D and Davies AJ. A Laplace transform boundary element solution for the biharmonic diffusion equation. University of Hertfordshire Department of Physics, Astronomy and Mathematics Technical Report 97 (2006). [4] Wilmott P, Howison S and Dewynne J. The mathematics of financial derivatives. Cambridge University Press (1995). [5] Cahn JW and Hilliard JE. Free energy of a non-uniform system-I: Interfacial free energy. J. Chem. Phys. 28: 258–267 (1958). [6] Cahn JW. Free energy of a non-uniform system-II: thermodynamic basis. J. Chem. Phys. 30: 1121–1124 (1959).

[7] Cahn JW and Hilliard JE. Ch Free energy of a non-uniform Adv. Bound. Elem. Tech. Eds: Zhang, M H Aliabadi, M Schanz system-III: 117 Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31: 688–699 (1959). [8] Elliott CM and French DA. Numerical studies of the Cahn-Hilliard equation for phase separation. IMA J. App. Math. 38: 97–128 (1987). [9] Blowey JF and Elliott CM. The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part 1: Mathematical Analysis. Euro. J. App. Meth. 2: 233–280 (1991). [10] Blowey JF and Elliott CM. The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part 2: Numerical Analysis. Euro. J. App. Meth. 3: 147–179 (1992). [11] Toutip W, Davies AJ and Kane SJ. The dual reciprocity method for solving biharmonic problems. Boundary Elements XXIV, 373-380 (2004). [12] Partridge PW, Brebbia CA and Wrobel LC. The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications (1992). [13] Stehfest H. Numerical inversion of Laplace transforms. Comm. ACM., 13: 47–49 and 624 (1970). [14] Crann D. The Laplace transform boundary element method for diffusion-type problems. PhD Thesis, University of Hertfordshire (2005). [15] Davies AJ and Crann D. A handbook of essential mathematical formulae. University of Hertfordshire Press (2008).

118


Strategy for writing general scalable parallel boundary-element codes F.C. de Araújo¹, E. F. d'Azevedo²,†, and L.J. Gray²,‡ ¹Dept Civil Eng, UFOP, Ouro Preto, MG, Brazil; [email protected] ‡ ²CSMD, ORNL, Oak Ridge, P.O. Box 2008, USA; †[email protected]; [email protected] Keywords: 3D standard BE formulations, parallel processing, CNT-based composites, thin-walled elements, subregion-by-subregion technique.

Abstract. In this work, a strategy based on a generic subregion-by-subregion (SBS) algorithm is employed to develop general scalable BE parallel codes. In this algorithm, the interactions between the subdomains are taken into account only during the solution of the system by a Krylov iterative method. Thereby, significant reduction of memory and CPU-time consumption is achieved as the global system matrix is not explicitly treated. On the other hand, special integration procedures for calculating nearly-stronglysingular integrals make the use of discontinuous boundary elements (in which quasi-singular integrals occur) possible. In fact, discontinuous elements are very useful for establishing BE models for complex heterogeneous domains. A matrix-copy option, useful for modeling systems with repeated parts, as identical fiber reinforcements, is also available. To verify the performance of the code, the 3D microstructural analysis of carbon-nanotube-reinforced composites (CNT composites) is considered. Particularly, mechanical properties of composites are measured. The representative volume elements (RVEs) adopted consist of carbon-nanotubes (shell-like elements) coupled with a polymeric material matrix. Introduction The Finite-Element Method (FEM) is still the tool of choice in engineering analysis of structures and solids, however its application to thin-walled solids and composites has been accompanied with a series of issues as element-distortion sensitivity, locking phenomena, and the non-fulfillment of stress continuity between layers or at matrix-fiber interfaces. Especially in the case of composites, the mesh generation itself is a bottleneck in finite-element (FE) analysis. To escape these difficulties, in recent works, the direct application of 3D standard boundary-element formulations has then been considered as an alternative to solve general composites and thin-domain problems [1, 2, 3]. Besides advantages as high accuracy, fulfillment of radiation conditions, and easier mesh generation, the Boundary Element Method (BEM) also presents the following interesting characteristic: it is derived from the exact integral representation of the problem response and does not require any interelement compatibilty (in the FE sense) for assuring solution convergence. This actually allows more flexibility for generating boundary-element models as long as the integrals involved are accurately evaluated. Indeed, this is the basis of discontinuous boundary elements, very useful for the BE subregion-by-subregion (BE-SBS) algorithm [4], considered in this work for the development of the parallel code. In many of the works concerning the development of BE parallel codes [5, 6, 7], either for symmetric multiprocessor (SMP) or massively parallel processor (MPP) architectures, the parallelism has been based on different ways to generate and to scatter the global boundary-element (BE) system of equations onto the available processors, its solution, in most cases, being carried out by applying available high-performance packages as LAPACK or ScaLAPACK. Unlike these works, [8] used a domain decomposition method (DDM) for solving 2D potential problems. They directly scattered the subdomain systems onto the processors and got the solution for the whole problem from the independent solution of each subdomain, wherein iterative schemes were used to introduce the coupling conditions. In [9], a DDM-based strategy is also considered to solve 2D elasticity problems with cracks. This procedure allows the independent assembling of the subdomain matrices, and is based on the condensation of the problem response to the interface tractions. In fact, this strategy may be cumbersome and time-consuming for 3D problems with complex composite morphology. In the present paper, the parallel version of the BE subregion-by-subregion (BE-SBS) algorithm [3, 10], a generic non-overlapping domain decomposition method, is presented. Besides being a fundamental technique in BE formulations, substructuring techniques are a spontaneous way to develop parallel codes, irrespective of the computer architecture. However, differently from displacement-based FE formulations, wherein one should never worry about traction discontinuity at the element corners or edges, BE


119

formulations are mixed formulations and require simulating traction discontinuity at some corner/edge nodes of the subdomains. Certainly, this is a fact that makes the modeling with subregions, in case of solids with complex internal geometries, a somehow tedious task. This was verified in [11], where continuous boundary elements are used to model 3D frequency-dependent elastodynamic problems. To get rid of this issue, discontinuous boundary elements have been adopted [2, 4, 10]. Doing so, coupling conditions can be directly imposed, and BE-subdomain models are then a lot easier generated. The other issue related to BE-subdomain algorithms (in parallel or serial versions) is how to optimally deal with the highly sparse resulting matrices. Kamiya et al. [8] proposed iterative coupling procedures that perfectly treat the matrix sparsity but are not reliable concerning convergence. In [9], the coupling conditions are directly introduced, but condensing the system unknowns to the interface tractions is awkward, requires additional memory beyond that necessary for allocating the isolated subsystems, and may be time-consuming for complex models. In this work, the BE-SBS algorithm [3, 10] is adopted. Employing some iterative solver, a solution strategy for general coupled problems is derived wherein no explicit global matrix have to be assembled, and only memory space for strictly allocating the subregion subsystems is needed. In this work, a simple diagonal-preconditioned Bi-CG solver is applied. However it is emphasized that the coupling conditions are directly enforced. Structured matrix-vector product (SMVP) and matrix-copy options are also implemented to increase the efficiency the code [4]. 3D simulations of CNT-based composites are carried out to show its performance. The BE-SBS-based parallel algorithm The BE parallel code is based on the BE-SBS algorithm detailed in previous papers [3, 10]. This algorithm considers a substructuring technique (non-overlapping domain decomposition method, DDM), and makes use of iterative solvers, similarly as done in element-by-element-based (EBE-based) finite-element formulations, to solve the global BE system of equations without explicitly assembling it. In general, after the boundary conditions have been introduced at each BE subregion separately, a set of n s algebraic systems of equations given by i 1

¦ H

im u mi

G im p im A ii x i

n

¦ H

m i 1

m 1

im u im

G im p mi B ii y i

, i 1, n s ,

(1)

where n s is the number of subregions, has to be solved by enforcing continuity and equilibrium conditions at the interfaces: °u ij ® °¯p ij

u ji

p ji

at *ij .

(2)

In Eq. (1), H ij and G ij denote the usual BE matrices obtained for source points pertaining to subregion : i and associated respectively with the boundary vectors u ij and p ij at *ij . Note that if i z j , *ij corresponds to the interface between : i and : j , which denote the i-th and j-th subregion respectively; *ii is the outer boundary of : i . The global system in Eq. (1) is then conveniently solved by applying an iterative solver. Here particularly, the diagonal-preconditioned biconjugate gradient (J-BiCG) solver. As in the BE SBS algorithm there is no overlapping of coefficients belonging to edges or corners shared by different subregions, as it happens in finite-element models, the data structure in Eq. (1) does not need any further optimization. All zero blocks present in the highly-sparse global system matrix are perfectly excluded. Besides, the following techniques/strategies are especially important for increasing the efficiency of the BE-SBS-based code: discontinuous boundary elements, structured matrix-vector products (SMVP), special integration quadratures, and the matrix-copy option. In the references [2, 3, 4, 10], the BE SBS algorithm is thoroughly described. In Fig. 1, the flowchart of the BE-SBS-based parallel code is presented, wherein it is assumed that k processes is considered. In fact, as in the BE SBS algorithm the subdomains are independently treated during the entire analysis, its implementation for running in a parallel-processing platform is immediate. Assembling the algebraic systems needs no information from other processes. Only during its solution, communication between the processes is needed for updating the boundary values in all subregions (Fig. 2).

120


parallel processes

ip=0

ip=1

ip=k

data input (read)

data input (broadcast from ip=0)

data input (broadcast from ip=0)

matrix assembly

matrix assembly

matrix assembly

boundary conditions

boundary conditions

boundary conditions

coupling search

coupling search

most-CPU-timeconsuming parts of the code

coupling search

Krylov solver

Figure 1. Flowchart of the BE-SBS-based parallel code. ip=0

ip=1

ip=k

iter = 1,2,...,until convergence

boundary data

boundary data

boundary data

SMVP

SMVP

SMVP

solution

solution

solution

boundary-data transfer between substructures in different processes structured matrix-vector products (SMVP)

Figure 2. Solution phase (Krylov solver). Applications and discussions The performance of the BE SBS-based parallel code detailed above is observed by determining engineering constants for the CNT-based composites shown in Fig. 3, which consider hexagonal fiber-packing patterns for arranging the CNTs inside the matrix material. In all representative volume elements (RVEs), constructed coupling together single unit cells with dimensions l1 10 nm , and l 2 l 3 20 nm (see Fig. 3a), the following phase constants, adopted in [12], are considered:


CNT:

E CNT

Matrix:

Em

1,000 nN 100 nN

nm 2

nm 2

(GPa); Q CNT

(GPa); Q CNT

121

0.30 ,

0.30 .

3

3 3

2 1

2

(a) 1

(d)

(b)

1

2 (c)

(e)

Figure 3. RVEs based on hexagonal-packed CNTs The long CNT fibers are geometrically defined by cylindrical tubes having outer radius r0 radius ri

4.6 nm , and length l f

5.0 nm , inner

10 nm (equal to the RVE thickness). In Table 1, additional data for the

models shown in Fig.1 are provided. When needed, discontinuous boundary elements are automatically generated by shifting the nodes interior to the elements a distance of d 0.10 (measured in the natural coordinate system). The matrix-copy option is also conveniently considered to replicate physically and geometrically identical subdomains. The boundary element adopted is an 8-node quadrilateral one, and the tolerance for the iterative solver (J-BiCG) is taken as ] 10 5 . The analyses were carried out at ORNL Institutional Cluster, consisting of 80 usable nodes, each one having Dual Intel 3.4GHz Xeon EM64T processors, 4GB of memory, and dual Gigabit Ethernet Interconnects. In Table 2, the engineering parameters obtained by employing the present code are confronted with results given in [12], and estimated by the rule of mixture [4, 12, 13]. As seen, values estimated by the rule of mixture and by refined 3D FE models [12] are in very good agreement with the ones calculated with the present method. No significant change in the values is also observed as a function of the number of unit cells per RVE. As sample results for checking the parallel-processing performance, CPU-time and memory-use scalability curves for the 10 u 10 -unit-cell RVE (largest model) under strain state 1 are shown in Fig. 4.

122


Table 1. Model data for the RVEs model

nsub 6 17 34 86 321

1u 1 2u 2 3u 3 5u5 10 u 10 *

*

nel** 138 656 1,464 4,040 16,080

nnodes† 856 3,456 7,800 21,720 87,040

ndof‡ 2,568 10,368 23,400 65,160 261,120

sparsity (%) 72 86 93 97 99

n. of subregions; **n. of elements; †n. of functional nodes; ‡n. of degrees of freedom

Table 2. Engineering constants for the hexagonal-packed long-CNT RVEs

†

model

E1 /E m

1u 1 2u 2 3u 3 5u5 10 u 10

1.8081 1.8074 1.8074 1.8126 1.8014 1.8131

rule of mixture† RVE volume fraction is V f

E2 /Em , E3 /Em

1.0889 1.0839 1.0916 1.0813 1.0805 -

Q 12 ,Q 13

Q 23

0.2943 0.2936 0.2931 0.2927 0.2926 -

0.5107 0.5107 0.5185 0.4997 0.5103 -

9.035% 240

0.6

Memory scaling (strain state 1)

CPU time/niter (sec./iter.)

measured values logarithmic fit 0.5

0.45

used memory (real-valued array, Mbytes)

CPU-time scaling (strain state 1) 0.55

220

200

measured values logarithmic fit

180

160

140

120

100

0.4 20

30

40

number of processors

(a)

50

20

30

40

50

number of processors

(b)

Figure 4. Scalability curves for CPU time and storage memory Concerning memory use, the scalabilty is very good, practically following the logarithmic fit (see Fig. 4b). Concerning the CPU-time, it is observed that the speedup tends to decrease when the number of processors is incremented (e.g. from 30 to 50 processors; see Fig. 4a). An explanation for that is a relative increase on the interprocessor communication compared to the load per processor. Conclusions A robust BE-SBS technique is used to derive a general 3D BE parallel code. Particular applications of the code concern the evaluation of effective engineering constants for 3D CNT-reinforced composites. First, it is observed that as a consequence of the special quadratures available in the code, discontinuous and disproportionate boundary elements can be employed. In this way, the modeling of complex coupled solids, as composites, is greatly simplified. In addition, the matrix-copy option, which avoids the repeated mesh generation and calculation of coefficient matrices for identical substructures, considerably facilitates the modeling of very complex periodic composites. In the particular applications shown above, no efficiency


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gain has been actually observed during the assembly phase as the corresponding CPU-time measurements were insignificant compared to the solver CPU time (dominant). However, for large identical subregions, this option might increase the computational efficiency. Moreover, for complex composites, boundary-integralbased models are simpler to generate than volume-based ones. Thus, the strategy proposed is believed to be very convenient for analyzing general composites. Among others, a contribution of this study is certainly the proposal of a general strategy for developing parallel-processing BE codes, readily applicable to any BIE-based methods. We notice that the algorithm proposed presents the following interesting general characteristics: (1) the BE models are independently generated, stored, and manipulated (no explicit global matrix assembly takes place), (2) no variable condensing is carried out, avoiding then the calculation of Schur complements, (3) the interface conditions are directly imposed, avoiding then the use of some iterative strategy, (4) discontinuous boundary elements are used to make the generation of coupled models easier, (5) an iterative (Krylov) solver is employed, (6) the high sparsity of the system is perfectly exploited. Obviously, as the models are independently stored, the memory-use scalability of the code is excellent, as we see from the results in the previous section. On the other hand, if the number of processors is incremented, the interprocessor communication will be more intense, decreasing then the processing speedup after a certain critical number of processors. In this work, indeed focused on the BE-SBS-based parallel code, no special attention has been properly paid to the Krylov solver itself. As noted, just a plain diagonal-preconditioned BiCG solver has been employed. In fact, it is known that this particular solver presents irregular convergence behavior, sometimes even not converging, depending on the system-matrix spectrum. Anyway, considering all the development brought about on iterative solvers and preconditioning techniques in the last two decades [14], we do believe that the BE-SBS algorithm is the optimal way to solve complex coupled BE models, and a promising alternative to develop general BE parallel codes, accounting for scalability of memory requirements and processing time. Acknowledgement This research was sponsored by the Brazilian Research Council (CNPq), and by the Research Foundation

for the State of Minas Gerais (FAPEMIG). References [1] X.L.Chen and Y.J.Liu Eng. Anal. Boundary Elements, 29, 513-523 (2005). [2] F.C.Araújo, L.J.Gray Comput. Mechanics, 41, 633-645 (2008). [3] F.C.Araújo, K.I.Silva, J.C.F.Telles Comm. Num. Methods Eng., 23, 771-785 (2007). [4] F.C.Araújo, L.J.Gray Comp. Mod. Eng. Sci., 24(2), 103-121 (2008). [5] R.Natarajan, D.Krishnaswamy Eng. Anal. Boundary Elements, 18, 183-193 (1996). [6] S.W.Song and R.E.Baddour Eng. Anal. Boundary Elements, 19, 73-84 (1997). [7] M.T.F.Cunha, J.C.F.Telles, A.L.G.A.Coutinho Adv. Eng. Software, 35, 453–460 (2004). [8] N.Kamiya, H.lwase, E.Kita, Eng. Anal. Boundary Elements, 18, 209-216 (1996). [9] X.Lu and W.-L.Wu Eng. Anal. Boundary Elements, 29, 944–952 (2005). [10] F.C.Araújo, K.I.Silva, J.C.F.Telles Int. J. Numer. Methods Engrg. 68, 448-472 (2006). [11] F.C.Araújo, C.Dors, C.J. Martins, W.J.Mansur J. Braz. Soc. Mech. Sci. Eng., 26(2), 231-248 (2004). [12] X.L.Chen and Y.J.Liu Comput. Mat. Sci., 29, 1–11 ( 2004). [13] M.W.Hyer, Stress Analysis of Fiber-Reinforced Composite Materials, 1st ed., McGraw-Hill (1998). [14] R.Barett, M.Berry, T. F.Chan, J.Demmel, J.Donato, J.Dongarra, V.Eijkhout, R.Pozo, C.Romine, H.van der Vorst Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Ed., SIAM (1994).

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Incomplete LU preconditioning of BEM systems of equations based upon the generic substructuring algorithm F.C. de Araújo¹, E. F. d'Azevedo²,†, and L.J. Gray²,‡ ¹Dept Civil Eng, UFOP, Ouro Preto, MG, Brazil; [email protected] ‡ ²CSMD, ORNL, Oak Ridge, P.O. Box 2008, USA; †[email protected]; [email protected]

keywords: 3D boundary-element models, Krylov solvers, subregion-by-subregion algorithm, incomplete LU preconditioners.

Abstract. The generic substructuring algorithm, developed in previous works, is directly employed to

construct global incomplete LU preconditioners for BEM systems of equations. As the BE matrices for each BE subregion are independently assembled, the corresponding L and U factors are easily calculated, and incomplete-LU-based preconditioners for particular BE models are immediately formed. So as to highlight the efficiency of the preconditioning proposed, the Bi-CG solver, known to present a quite erratic convergence behavior, is considered. Complex 3D representative volume elements (RVEs) of carbonnanotube (CNT) composites are analyzed to show the performance of the preconditioned iterative solver. The models contain up to several tens of thousands of degrees of freedom. The relevance of the preconditioning technique is also discussed in the context of developing general (parallel) BE codes. Introduction

The parallelism embedded in fast reliable Krylov solvers combined with the today's parallel computer architectures have definitely contributed for the devising of efficient scalable parallel codes, becoming then in the last decades an appealing alternative for solving large-order engineering problems [1, 2]. In these cases, in general, direct solvers present the following disadvantages: they may be exceedingly CPU time-consuming and memory-consuming, and their parallel implementation is awkward. For general non-symmetric matrices, like BE matrices, the very difficult problem is how to devise reliable iterative solvers. Indeed, reliability regarding convergence in this case (when the matrices are non-symmetric or indefinite), despite the number of outstanding scientific contributions across the last six decades, is a still open question [3]. Knowing that basic iterative solvers, for instance, the Jacobi or Gauss-Seidel methods, are convergent only in very special cases, e.g. when the spectral radius of the corresponding iteration matrix is less than 1, so the possible alternatives for dealing with non-symmetric are then Krylov solvers, which in this case can be subdivided in two broad classes of algorithms: long-recurrence algorithms (GMRES and variants), and short-recurrence ones (Bi-CG and variants). Because of the memory requirements for large problems, and non-rare convergence stagnation in practice, long-term recurrence methods should indeed be avoided, and so our universe of possible efficient iterative solvers for large BE models, at last, reduces to short-recurrence ones such as the Bi-CG. However, the Bi-CG method presents an erratic convergence behavior, and typically fails to find the solution for non-Hermitian systems. Thus, to smooth out possible convergence irregularities, modified hybrid solvers has been to generate by combining the Bi-CG solver with residualminimization methods, as the GMRES. Following this idea, culminated then in developing solvers such as the transpose-free Bi-CGSTAB(l) [4] and the GPBi-CG (generalized product Bi-CG) [5]. Additionally, preconditioners may be employed to accelerate the iterative process [1]. For BEM solvers, a series of preconditioners have been reported in the technical literature [6-8]. In general, the splitting matrix of basic iterative methods as the Jacobi, Gauss-Seidel or incomplete LU decomposition methods can be used to construct preconditioners. Roughly speaking,


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preconditioners are a way to state a relationship between direct and iterative solvers, in the sense that if the preconditioning matrix becomes the system matrix, so the iterative method at hand becomes a direct solver (converging then to the system solution at one single iteration step.) Furthermore, domain decomposition methods (DDM) allied with direct methods may also be employed to construct global preconditioners. This will be very important e.g. to parallelize incomplete LU-based preconditioner, which is among the most efficient ones, but not easily parallelizable. In this paper, the BE substructuring algorithm [9-10] is employed to construct a global incomplete LU preconditioner for the BE model at hand. In other words, a DDM-based technique, applied to decompose a certain problem domain into a generic number of coupled BE models, is considered to define the allowable fill-in positions for the incomplete LU decomposition preconditioner. In the algorithm, the coupling conditions between the subdomains are imposed in a direct (non-iterative) way, and the subsystems are independently assembled, so that the blockdiagonal matrices corresponding to each subregion can be easily decomposed in their L and U factors. For the applications here, the preconditioning proposed is incorporated into the Bi-CG solver. An important point in this respect is that, as the Bi-CG solver is expected to fail in practice, as commented above, then in the numerical experiments the efficiency of the preconditioner itself will be highlighted. Several models, with thousands of degrees of freedom, employed to simulate complex carbon-nanotube (CNT) composites, are considered to show the performance of the preconditioning. The efficiency and relevance of the preconditioning proposed is also discussed in the context of ideas for developing general scalable BE parallel codes. 2. The BE-SBS algorithm and the associated preconditioner As well reported in previous works [9, 10], the boundary-element substructuring-by-substructuring (BE-SBS) algorithm is comparable to the element-by-element (EBE) technique, developed to finiteelement analysis (FEA) [27] while a subregion or substructure corresponds to a finite element. Thus, if needed, we can have a subregion mesh as fine as a finite-element mesh, and if the BE global system matrix were explicitly assembled, it would be highly sparse as well. It is also noted that the BE-SBS algorithm can be compared to Finite Element Tearing and Interconnecting (FETI) methods [11] as well, where a given problem domain is decomposed (torn) into non-overlapping subdomains, and posteriorly interconnected by imposing the corresponding continuity conditions at the interfaces. The BE-SBS algorithm embeds Krylov iterative solvers, and the global response for a problem is obtained by working exclusively with its local full-populated subsystems of equations. No global explicit system matrix is assembled; no zero blocks are stored or handled. The boundary conditions are introduced during the matrix assembly for each subsystem, and the interface conditions (between the subdomains), given by

°u ij ® °¯p ij

u ji p ji

(1)

at *ij

are directly (not iteratively) imposed in the matrix-vector products during the iterative solution process. For n s subregions, after introducing the boundary conditions, the BE global system of equations is then given by i 1

¦ H m 1

im

u mi G im p im A ii x i

ns

¦ H

m i 1

im

u im G im p mi B ii y i , i 1, n s ,

(2)

126


where H ij and G ij denote the regular BE matrices obtained for source points pertaining to subregion : i and associated respectively with the boundary vectors u ij and p ij at *ij . Note that if i z j , *ij denotes the interface between : i and : j ; *ii is the outer boundary of : i . If the system of equations in (2) were, say for n s would have the following general aspect:

:

: : :

:

A 11 H12 H13 H14 G12 H21

:

:

G13

G21 A 22

H23 H24 H32

H31

:

4 (four subregions), explicitly assembled, it

H41

G14 G23

G31 G32 A 33 H42

G24 H34 H43

G34 G41 G42 G43 A 44

:

:

:

x 11 u 12 u 13 u 14 p 21 x 22 u 23 u 24 p 31 p 32 x 33 u 34 p 41 p 42 p 43 x 44

:

B 11 B 22 B 33 B 44

y1

:

y2

:

y3

:

y4

:

.

(3)

H ji G ij G ji 0 if the there is no coupling between i and j subdomains. However, as commented previously, we do not have any explicit system of equations. Instead, the working subsystems are those ones shown in expression (2). The matrix-vector and transpose-matrix-vector products are then calculated from the separate contributions from each subsystem, while as already commented above, during the solver iterations, the interface conditions are imposed in a direct way. In this study, the allowable fill-in positions for the incomplete LU decomposition are taken as those of the diagonal blocks of the coupled system, i.e., for the particular (explicit) system of equations shown in (3), the subsets of positions highlighted in gray. Inferring from Eq. (3) that, for a generic number of subregions, the diagonal blocks of the coupled system are given by In this system, note that H ij

Qi

> G

i1

G i ,i 1

A ii

H i ,i 1 H in @

, i 1, n s

(4)

where the Q i matrices are straightforwardly formed having the subregion matrices of the model at hand, the construction of the global SBS-based ILU preconditioner for the coupled system of equations (3) is then immediate. However, as the subdomain submatrices, this global preconditioner is not explicitly assembled either; it is separately stored per subregion at an additional memory space of the size (nno u ndofn) u (nno u ndofn) , where nno is the number of nodes of the model, and ndofn is the number of degrees of freedom per node. In the code, the BE-SBS-based preconditioner is employed to accelerate the Bi-CG iterations.


127

4. Results and discussions The performance of the ILU SBS-based preconditioner detailed above is measured by analyzing the complex CNT-based composites shown in Fig. 1, wherein representative volume elements (RVEs) based on 1u 1 , 2 u 2 , and 5u 5 unit cells are employed. The long CNT fibers are geometrically defined by cylindrical tubes having outer radius r0 5.0 nm and inner radius ri 4.6 nm , and length l f 10 nm . In general, when needed, discontinuous boundary elements are automatically

generated by shifting the nodes interior to the elements a distance of d 0.10 (measured in the natural coordinate system). The matrix-copy option is also conveniently considered to replicate physically and geometrically identical subdomains, avoiding then assembling repeatedly their corresponding matrices. The 8-node quadrilateral boundary element is employed, and in all analyses, 8 u 8 and 6 integration points are used for evaluating all surface and line integrals involved, respectively, in the special integration quadratures embedded in the code. In all (RVEs), the following pure phase constants are adopted [12]: CNT:

E CNT

Matrix:

Em

1,000 nN 100 nN

nm

(GPa); Q CNT 0.30 , nm 2 0.30 . 2 (GPa); Q CNT

The tolerance for the iterative solver (Bi-CG) is taken as ] 10 8 . The diagonal preconditioning (Jacobi) and the preconditioning proposed in this paper (BE-SBS-based ILU decomposition) are then contrasted to show the efficiency brought about by the latter preconditioner. The analyses were carried out at a notebook with dual intel 2.26GHz processor, and 3GB of random access memory. Important model data are provided in Table 1. In Table 2, the engineering parameters extracted from the analysis of all the RVEs shown in Fig. 2 are confronted with results calculated by Liu and Chen [12] via finite-element analysis, and estimated (when possible) by the rules of mixture [12]. As seen, very good agreement between the results is obtained. Furthermore, no significant change in the constant values is also observed as the number of unit cells per RVE increases. Table 1. Model data for the square-packed long-CNT RVEs nel** nnodes† ndof‡ sparsity (%) nsub* 2 128 608 1,824 29 1u1 8 512 2,660 7,980 81 2u 2 50 1,344 17,456 52,368 97 5u5 * n. of subregions; **n. of elements; †n. of nodes; ‡n. of degrees of freedom model

3

3

3

1

2

2

(a)

2

1

(b)

1

Figure 1. Square-packed long-CNT-based RVEs.

(c)

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Table 2. Engineering constants for the square-packed long-CNT RVEs model

E1 /E m

1u1 2u 2 5u5

1.3227 1.3228 1.3228 1.3255 1.3255

Chen & Liu (3D FE) rule of mixture† † RVE volume fraction is V f

E2 /Em , E3 /Em

Q 12 ,Q 13

Q 23

0.2974 0.2973 0.2972 0.3000 -

0.3595 0.3600 0.3580 0.3799 -

0.8302 0.8319 0.8319 0.8492 -

3.617%

Table 3. Performance data for the square-packed long-CNT RVEs; tol 1.0 u 10 8 model system order n. of n. of CPU time (s) CPU time (s) iterations iterations (BE SBS(Jacobi) based ILU)† (BE SBS(Jacobi) based ILU) 1x1 unit cell, 1,824 57 561 2 5 strain state 1 1x1 unit cell, 1,824 73 621 2 6 strain state 2 2x2 unit cells, 7,980 81 2241 11 104 strain state 1 2x2 unit cells, 7,980 104 1805 12 84 strain state 2 5x5 unit cells, 52,368 116 8920 119 2917 strain state 1 5x5 unit cells, 52,368 157 5983 142 2,084 strain state 2 † Including the LU decomposition CPU time

In Table 3, results showing the performance of the preconditioners are presented. Compared to the Jacobi preconditioner, a considerable acceleration of Bi-CG solver is observed when the BE SBSbased ILU one is applied (e.g. the Bi-CG solver becomes about 24 times faster for the 5u 5 -unitcell RVE under strain state 1). The decaying of the Euclidean residual norm, 2 , as a function of the iteration order for both preconditioners is also shown in Fig. 2. This graph clearly shows the superiority of the BE SBS-based ILU preconditioning. 1000 100

1000

10

Preconditioner Jacobi BE SBS-based ILU

0.1 0.01 0.001 0.0001 1E-005 1E-006

Preconditioner Jacobi BE SBS-based ILU

100 10

residual Euclidean norm

residual Euclidean norm

1

1 0.1 0.01 0.001 0.0001 1E-005 1E-006

1E-007

1E-007

1E-008

1E-008

1E-009

1E-009

0

1000

2000

3000

4000

5000

iteration

6000

7000

8000

0

1000

2000

3000

4000

iteration

(b) Strain state 2 (a) Strain state 1 Figure 2. Residual norm vs. iteration: 5 u 5 -unit-cell, square-packed long CNT

5000


129

5. Conclusions and prospects The BE SBS technique proposed in previous papers ([9], [10]) is straightforwardly used to construct

incomplete-LU-based preconditioners for BE systems of equations. The performance of this preconditioning was verified by analyzing complex composite RVEs. Observing the Table 3, and graphs in Figure 2, we see that the BE-SBS-based ILU preconditioning, compared to the Jacobi (diagonal) one, is considerably more efficient. In fact, the BE-SBS-based ILU preconditioning states a transition (or connection) between direct and iterative solvers, in the sense that the less the number of interfaces, the closer to the global system matrix the preconditioning matrix, Q , is. In addition, knowing that the global coupled system is highly sparse, we can well conclude that the preconditioner proposed will be certainly a good approximation of the global system matrix, which is one of the requirements for finding good preconditioners. Generally speaking, the larger the size of the subsystems, the higher the cost for constructing the preconditioner, however, on the other hand, a better approximation for the global system is achieved, reducing then the number of iterations. Furthermore, being this preconditioner based on the BE-SBS algorithm, its parallelization is immediate. In general, solver-convergence reliability and parallel-processing suitability are attained. Acknowledgements This research was sponsored by the Brazilian Research Council (CNPq), and by the Research Foundation for the State of Minas Gerais (FAPEMIG). References [1] H. A. van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge University Press (2003). [2] Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2003). [3] R. Barrett, M. Berry, J. Dongarra, V. Eijkhout, C. Romine, J. Comp. Appl. Mathematics 74, 91-109 (1996) . [4] G.L.G. Sleijpen, D.R. Fokkema Electronic Trans. Num. Methods Anal., 1, 11-32 (1993). [5] S.-L. Zhang Comp. and Appl. Math. 149, 297–305 (2002). [6] S.A. Vavasis SIAM Journal on Matrix Analysis and Applications 13, 905-925 (1992). [7] K. Davey, S. Bounds Applied Numerical Mathematics 23, 443-456 (1997). [8] M. Merkel, V. Bulgakov, R. Bialecki, G. Kuhn Eng. Anal. Boundary Elements 22, 183-197 (1998). [9] F.C. Araújo, K.I. Silva, J.C.F. Telles Int. J. Numer. Methods Engrg. 68, 448-472 (2006) . [10] F.C. Araújo, L.J. Gray Comp. Mod. Eng. Sci. 24(2), 103-121 (2008). [11] C. Farhat, F.-X. Roux SIAM J. Sci. Statist. Comput. 13, 379–396 (1992). [12] X.L. Chen, Y.J. Liu Comput. Mat. Sci. 29, 1–11 (2004).

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Hypersingular BEM analysis of semipermeable cracks in magnetoelectroelastic solids R. Rojas-D´ıaz1∗ , M. Denda2 , F. Garc´ıa-Sánchez3 , A. Sáez1∗ 1

2

Departamento de Mecánica de los Medios Continuos, Escuela Técnica Superior de Ingenieros, Camino de los descubrimientos s/n, E41092 Sevilla, SPAIN [email protected],[email protected] Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Rd., Piscataway, NJ 08854-8058, U.S.A. [email protected] 3 Departamento de Ingenier´ ıa Civil, de Materiales y Fabricaci´ on, Escuela de Ingenier´ıas (Ampliaci´ on Campus Teatinos), Universidad de Málaga, C/ Dr. Ortiz Ramos, 29071-Málaga, Spain [email protected]

Keywords: BEM, magnetoelectroelastic solids, semipermeable cracks, fracture mechanics.

Abstract. In this work, an efficient numerical tool based on the dual formulation of the BEM presented in [1] is developed for the analysis of different electric/magnetic crack faces boundary conditions in magnetoelectroelastic solids. A new algorithm for the resolution of multiple semipermeable cracks in magnetoelectroelastic media, based on the one developed in [2] for piezoelectric solids, is designed and implemented. Results for a Griffith crack in a magnetoelectroelastic media will be presented and compared with the analytical solution, for different mechanical, electric and magnetic loadings. Introduction Piezoelectric/piezomagnetic multiphase composites represent a new class of smart materials which present a fully coupling between the mechanical, electric and magnetic fields. In them, a new electromagnetic coupling which did not appear in each phase is present [3]. Their ability to convert energy between mechanical, electric and magnetic fields makes them very interesting for their use in smart structures applications. The analysis of crack face boundary conditions in magnetoelectroelastic fracture is not a closed topic. Two ideal conditions (impermeable and permeable) and a more realistic one (semipermeable) are usually considered. Impermeable conditions establishes that the crack is isolated of the electromagnetic fields, while permeable condition implies that cracks conduct electric and magnetic fields. However, both assumptions are not completely realistic, and it is possible to say that a consistent crack face boundary condition will be between them. In this paper, a new algorithm to solve semipermeable crack problems using the hypersingular BEM formulation and based in the one developed by Denda [2] for piezoelectric materials is presented. This formulation will be validated by the comparison with analytical solutions available in literature. Governing equations for linearl magnetoelectroelasticity The behavior of a homogeneous and linear magnetoelectroelastic solid under in-plane mechanical, magnetic and electric loading can be described in an elastic-like way by introducing some extended variables. In particular, the displacement vector is extended with the electric and magnetic potentials, while the stress tensor is extended with the electric displacements and the magnetic inductions ⎧ ⎧ ⎨ ui , I=1,2 ⎨ σij , J=1,2 φ, I=4 Di , J=4 ; σiJ = (1) uI = ⎩ ⎩ ϕ, I=5, Bi , J=5,


131

where the lowercase subscripts (elastic) vary from 1 to 2, whereas the uppercase ones (extended) take the values 1,2,4,5. An associated generalized traction vector corresponding to a unit normal n = (n1 , n2 ) can also be defined as ⎧ ⎨ pj = σij ni , J=1,2 Dn = Di ni , J=4 pJ = σiJ ni = (2) ⎩ Bn = Bi ni , J=5, And now, the constitutive equations can be expressed as σiJ = CiJKl uK,l where the material properties have been grouped as ⎧ elij ⎨ cijkl J, K = 1, 2 eikl J = 4; K = 1, 2 −il CiJKl = ⎩ hikl J = 5; K = 1, 2 −βil

(3)

together into a generalized elasticity tensor defined J = 1, 2; K = 4 hlij J = 1, 2; K = 5 J, K = 4 −βil J = 4; K = 5 J = 5; K = 4 −γil J, K = 5

(4)

being cijkl , il and γil the elastic stiffness, the dielectric permittivities and the magnetic permeabilities tensors, respectively, whereas elij , hlij and βil are the piezoelectric, piezomagnetic and electromagnetic coupling coefficients, respectively. Dual BEM formulation and implementation The dual BEM formulation for bidimensional problems in MEE solids was presented in [1]. If cracks are self-equilibrated and the extended displacements and tractions in the crack are expressed as − + − ∆uJ = u+ J − uJ ; ∆pJ = pJ + pJ

(5)

then, the hypersingular formulation of the BEM consists in the application of the extended displacement boundary equation (EDBIE), equation (6), for collocation points which does not belong to the crack, and the application of the extended traction boundary equation (ETBIE), equation (7), which can be obtained by derivation of the EDBIE, when the collocation point belongs to the crack. Both boundary integral equations can be then expressed as

p∗IJ ∆uJ dΓ = u∗IJ pJ dΓ Γ Γ+ ΓB B pJ + Nr s∗rIJ uJ dΓ + Nr s∗rIJ ∆uJ dΓ = Nr d∗rIJ pJ dΓ p∗IJ uJ dΓ +

cIJ uJ +

ΓB

Γ+

(6) (7)

ΓB

where ΓB is any external boundary and Γ+ is one of the crack surfaces. cIJ are the so-called free terms and u∗IJ and p∗IJ are the Green’s functions corresponding to the response of an infinite homogeneous 2-D linear magnetoelectroelastic solid due to the application of a static generalized point force, which is available in the literature [4] and is expressed in terms of the extended Stroh’s formalism. In ETBIE, N denotes the outward unit normal to the boundary at the collocation point and the kernels s∗rIJ and d∗rIJ are obtained by differentiation of p∗IJ and u∗IJ , respectively, with the following expressions d∗rIJ = CrIM l u∗M J,l ; s∗rIJ = CrIM l p∗M J,l

(8)

Meshing strategy and integration of the hypesingular kernels in equations (6-7) follow the approach developed by [1]. In particular, discontinuous quadratic elements are used to mesh the cracks in order to fulfill the C 1 continuity of the displacements required to compute the ETBIE, and quarter point discontinuous elements are used to ensure a proper representation of the square-root behavior of the displacements around the crack tip. And, as done in [1], extended field intensity factors (stress intensity factors -SIF-, electric displacement intensity factor -EDIF- and magnetic induction intensity

132


factor -MIIF) can be directly obtained from the nodal values of the crack opening displacements and the electric and magnetic potential jumps across the crack from ⎞ ⎛ ⎛ ⎞ ∆u1 KII ⎜ ⎜ KI ⎟ ⎟ π −1 ⎜ ∆u2 ⎟ ⎟ ⎜ (9) ⎝ KIV ⎠ = 8r Y ⎝ ∆φ ⎠ , ∆ϕ KV where r is the distance between the crack tip and the point where extended crack opening displacements are evaluated and Y is the Irwin Matrix, which depends on the material constants [1]. Crack faces boundary conditions In fracture mechanics analysis of magnetoelectroelastic solids, three different boundary conditions on open crack surfaces can be considered. While the mechanical boundary conditions on the crack surface is always traction free, the electric and magnetic ones comes in different degrees of shielding the electric displacement and magnetic induction defined, respectively, by the electric permittivity and by the magnetic permeability. Thus, a crack along the x1 −axis can be considered as (i) Fully impermeable crack, if the normal electric displacement and magnetic induction on the crack surfaces are zero, so D2+ = D2− = 0 (10) B2+ = B2− = 0 which means that the crack is extended tractions free on its surface. (ii) Fully permeable crack if the crack does not obstruct any electric or magnetic field. This condition can be expressed as φ+ − φ− = 0 D2+ = D2− ; (11) B2+ = B2− ; ϕ+ − ϕ− = 0 (iii) Semipermeable crack. This condition, which gives a more realistic boundary condition for opened cracks, was proposed in [5] as an extension of the one previously proposed [6] for piezoelectric solids. − + − D2+ (u+ D2+ = D2− ; 2 − u2 ) = −c (φ − φ ) (12) − + − ϕ− ) B2+ = B2− ; B2+ (u+ − u ) = −γ (ϕ c 2 2 where c is the permittivity of the medium between the crack faces and γc , the permeability. The solution of the problem in which either impermeable or permeable (ideal) crack faces boundary conditions are considered, is carried out by a direct evaluation of both boundary integral equations (67), but imposing different boundary conditions in the crack. Thus, if a permeable crack is assumed, ∆u4 = 0 and ∆u5 = 0 must be imposed and, in the other hand, if impermeable crack faces boundary condition is considered, the conditions p4 = 0 and p5 = 0 must be applied to both boundary integral equations. In the next section, it will be introduced a procedure to analyze cracks under the semipermeable crack face boundary condition assumption. Numerical solution algorithm for semipermeable cracks The more realistic semipermeable condition is given by a non-linear equation so, for solving that problem, an iterative algorithm will be proposed and implemented. This algorithm is a generalization of the one proposed by [2] for piezoelectric cracked solids. Let us call the jumps of the electric and magnetic potentials in the crack as ∆u4 = (φ+ − φ− ) and ∆u5 = (ϕ+ − ϕ− ). The semipermeable solution implies that the electromagnetic potentials, the electric displacement and the magnetic induction on the crack faces are generally different to zero. Since the semipermeable crack solution is somewhere in between the two ideal crack surface boundary condition, then the following iteration procedure for multiple cracks problem is proposed.


[0]

133

[0]

1. Get the impermeable solution ∆u4 , ∆u5 , which will be used as the starting point of the iteration procedure. The number between brackets denotes number of iteration step. ki 2. Define, for each crack k, two pairs of proportionality parameters hki e and hm (i=1,2), which vary in the interval (0,1) k1 3. (a) Take, for each crack, hk1 e and hm slightly bigger than zero (what would correspond to the k2 quasi-permeable solution and, thus, values for c and γc tend to infinity) and hk2 e and hm slightly lower than one (what would correspond to the quasi-impermeable solution). Then, k[1] [0] k[2] [0] k[1] [0] k[2] [0] set ∆u4 = hk1 = hk2 = hk1 = hk2 e ∆u4 , ∆u4 e ∆u4 , ∆u5 m ∆u5 and ∆u5 m ∆u5 .

(b) Calculate the mechanical crack opening displacement, the electric displacement and the magnetic induction based on the set values of the previous item. (c) Calculate for each crack at M sample collocation points ξj ki ki j = Dn (ξj )

∆uki ∆uki 2 (ξj ) 2 (ξj ) ; γjki = Bnki (ξj ) ki (ξ ) ∆uki (ξ ) ∆u j j 4 5

(13)

which are obtained by the substitution in equation (12) of the corresponding ECOD and the electric (Dn ) and magnetic (Bn ) tractions previously obtained in step (3b). ki (d) Calculate the averages for each crack and each pair of parameters hki e and hm of the parameteres defined in section (3c). M ki M ki j=1 j j=1 γj ki = ; γ ki = (14) M M

This parameter are the so-called electric permittivity in the crack and magnetic permeability in the crack, respectively. (e) While the electric permittivity and magnetic permeability of any crack is not equal to the values for the medium between the crack surfaces, 0 and γ0 iterate using a procedure to k[n] k[n] solve non-linear equations, until a pair of values he and hm for each crack is obtained. Let us remark that all those values may be different. k[n]

ki[n]

[0]

k[n]

4. After setting ∆u4 = he ∆u4 and ∆u5 the semipermeable solution searched.

ki[n]

[0]

= hm ∆u5 , solve the problem required to get

Validation of the algorithm In this section, a single horizontal crack of length 2a in an infinite magnetoelectroelastic domain will be analyzed for different crack faces boundary conditions and loading combinations. A BaT iO3 − CoF e2 O4 magnetoelectroelastic solid with a Vf = 0.5, which properties can be found, e.g., in [1], will be considered. In all cases, the medium between both crack faces is air, what implies that the electric permittivity and magnetic permeability are, respectively, 0 = 8.8542 · 10−12 N/V 2 and γ0 = 4π · 10−7 N/A2 . The analytical solution for this problem, was first obtained by Wang and Mai [5], and will be now briefly presented. The extended crack opening displacements ∆uI , I = 1, ..., 5; are given by

− ∞ c 2 2 ∆uI = u+ (15) I − uI = 2YIJ (σJ2 − σJ2 ) a − x1 ∞ are the components of the extended stress tensor where Y is the compliance (Irwin) matrix, σJ2 c applied at infinity, σJ2 are the components of the extended stress tensor on the crack surfaces, and the summation rule over repeated indices is applied. The different crack face boundary conditions that may be considered for a crack along the x1 -axis.

134


(i) Fully impermeable crack. In this case, the crack is extended traction free, what implies that D2c = 0 ; B2c = 0 D2+

where, since crack surfaces.

=

D2−

and

B2+

=

B2− ,

(16)

the upperindex c has been used to denote either of the

(ii) Fully permeable crack. For fully permeable cracks no jump in the electromagnetic potential appear. This condition can be expressed as ∆u4 = 0 ; ∆u5 = 0

(17)

The substitution of that condition in (15) will lead to a system of equation whose solution provides the analytical expressions of the extended tractions on the crack faces D2c =

Y4J Y55 − Y5J Y45 ∞ Y5J Y44 − Y4J Y54 ∞ σ ; B2c = σ Y44 Y55 − Y54 Y45 J2 Y44 Y55 − Y54 Y45 J2

(18)

(iii) Semipermeable crack. The semipermeable crack conditions are D2c ∆u2 = −c ∆u4 ; B2c ∆u2 = −γc ∆u5

(19)

Substituting now (19) in (15) and operating a non-linear system of equations which defines the extended tractions in a semipermeable crack, it will be obtained. D2c = −c

∞ − Y Dc − Y B c ∞ − Y Dc − Y B c Y4J σJ2 Y5J σJ2 44 2 45 2 54 2 55 2 ; B2c = −γc ∞ c c ∞ − Y Dc − Y B c Y2J σJ2 − Y24 D2 − Y25 B2 Y2J σJ2 24 2 25 2

(20)

where the summation rule over repeated is applied.

1.8 1.6

0.8

1.4

perm | 2 x =0

1 0.9

1

0.7 0.6

∆ u /∆ u

0.5 0.4 0.3

1 0.8

0.1 −0.5

0

x /a 1

0.5

0.6 0.4

Impermeable Semipermeable Permeable

0.2

0 −1

1.2

2

∆ u2/∆ uperm |x 2

1

=0

The analytical solution previously deduced will be compared with the results obtained with the proposed formulation. In figure 1 the mechanical opening displacement are shown for the case in which only a mechanical loading is applied and in the case in which a combination of loads defined by ∞ = 1N/m2 , D ∞ = 10−9 C/N and B ∞ = 10−8 A−1 · m is applied. The analytical solution is plotted σ22 2 2 in lines, comparing them with the results obtained numerically (points), and those magnitudes are normalized with their respective value under permeable conditions in x1 = 0.


0.2

1

0 −1

−0.5

0

0.5

1

x /a 1

Figure 1: Crack opening displacement when only a mechanical loading is applied (left) and a full combination of electromagnetomechanic loading is applied (right).


135

In figure 2 the analytically obtained jumps in the electric and magnetic potentials are compared with the results obtained numerically (points). Those magnitudes are normalized with their respective values under impermeable conditions in the center of the crack (x1 = 0). In all cases, an excellent agreement between analytical and numerical results is observed 0

0

Impermeable Semipermeable Permeable 1

=0

1

−0.4

−0.4

5

∆ u /|∆ uimp|

−0.6

−0.6

5

∆ u4/|∆ uimp | 4 x


−0.2

x =0

−0.2

−0.8

−1

−1

−1

−0.8

−0.5

0

x /a 1

0.5

1

−1

−0.5

0

0.5

1

x1/a

Figure 2: Jumps in the electric (left) and magnetic (right) potentials on the crack.

Summary and conclusions A new algorithm for the study of the different crack faces boundary conditions in magnetoelectroelastic solids has been designed and implemented in a hypersingular boundary elements code. The accuracy of this numerical tool has been proved by comparing with analytical results available in literature. References [1] F. Garc´ıa-S´ anchez, R. Rojas-D´ıaz, A. Sáez, and Ch. Zhang. Fracture of magnetoelectroelastic composite materials using boundary element method (BEM). Theoretical and Applied Fracture Mechanics, 47(3):192–204, 2007. [2] M. Denda. BEM analysis of semipermeable piezoelectric cracks. Key Engineering Materials, 383:67–84, 2008. [3] Y. Beneviste. Magnetoelectric effect in fibrous composites with piezoelectric and piezomagnetic phases. Physical Review, B 51:16424–16427, 1995. [4] J.X. Liu, X. Liu, and Y. Zhao. Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack. International Journal of Engineering Science, 39:1405–1418, 2001. [5] B. Wang and Y.W. Mai. Applicability of the crack-face boundary conditions for fracture of magnetoelectroelastic materials. International Journal of Solids and Structures, 44:387–398, 2006. [6] V.Z. Parton and B.A Kudryatsev. Electromagnetoelasticity. Gordon and Breach science publisher, New York, 1988.

136


Boundary Element Analysis of Cracked Transversely Isotropic and Inhomogeneous Materials C.Y. Dong1, X Yang1 and E Pan2 1

Department of Mechanics, School of Aerospace Engineering, Beijing Institute of Technology, China 2 Department of Civil Engineering, University of Akron, Akron, Ohio, USA

Keywords boundary element method, transversely isotropic materials, inclusion, crack, stress intensity factors

Abstract. In this paper, the boundary element method (BEM) is utilized to study the effect of the transversely isotropic inclusion on the mixed-mode stress intensity factors (SIFs) of a rectangular crack. Both conventional and three special nine-node quadrilateral elements are used to discretize the inclusion-matrix interface and the square crack surface. Displacement and traction integral equations are applied to the inclusion-matrix interface and the square crack surface. Once the displacements and tractions over the interface and the crack opening displacements over the crack are obtained, the mixed-mode SIFs are calculated using a well-known formulation. In the numerical calculation, the distance between the inclusion and the crack as well as the orientation of the isotropic plane of transversely isotropic medium are varied to show their influences on the mixed-mode SIFs along the crack fronts. Introduction The mechanical behavior of heterogeneous materials of isotropy such as composites, rock structures, porous and cracked medium has been widely investigated using various methods, e.g. finite element method [1] and integral equation method [2]. However, only a few studies exist when the inhomogeneous material is of anisotropy, e.g., transverse isotropy. Huang and Liu [3] used the eigenstrain method to obtain the elastic fields around the inclusion and further studied the interactive energy between the inclusion and the applied loads. Pan and Young [4] investigated the fracture mechanics problems in three-dimensional (3D) anisotropic solids using the combined displacement and traction integral representations. Ariza and Dominguez [5] obtained the boundary traction integral equation for cracked 3D transversely isotropic bodies in which explicit expressions for the fundamental solution traction derivatives are presented. Yue et al. [6] calculated the 3D stress intensity factors (SIFs) of an inclined square crack within a bi-material cuboid using the dual BEM. Chen et al. [7] studied the fracture behavior of a cracked transversely isotropic cuboid using 3D BEM. Benedetti et al. [8] presented a fast dual BEM for cracked 3D problems. In existing literature, the interaction between the inclusions and cracks embedded in a transversely isotropic medium has not been researched yet. Therefore, in this paper, the effect of a spherical inclusion on the SIFs of a square-shaped crack, both being embedded in a transversely isotropic matrix, is studied using the single-domain BEM. The influence of the distance between the inclusion and the square-shaped crack and the material orientation on the SIFs of the crack fronts


137

is discussed. Boundary integral equations We consider a transversely isotropic inclusion embedded in a cracked infinite matrix of transverse isotropy. In order to study the effect of the inclusion on the SIFs of the rectangular crack, the single-domain dual BEM is used. In other words, the displacement and traction boundary integral equations for a cracked medium of transverse isotropy are as follows [4]

bij u j ( y S )

³U

ij

S

³T

ij

*

( y S , xS ) t j ( xS ) dS ( xS ) ³ Tij ( y S , xS ) u j ( xS ) dS ( xS ) S

( y S , x* )[u j ( x* ) u j ( x* )]d * ( x* ) ui0 ( y S )

(1)

[tl ( y * ) tl ( y * )] / 2 n m ( y * ) ³ clmik Tij , k ( y * , x S )u j ( x S ) dS ( x S ) S

n m ( y * ) ³ clmik Tij , k ( y * , x* )[u j ( x* ) u j ( x* )]d * ( x* ) *

n m ( y * ) ³ clmik U

* ij , k

(2)

( y * , x S ) t j ( x S ) dS ( x S ) [ t ( y * ) t ( y * )] / 2 0 l

0 l

S

where bij are coefficients that depend only on the local geometry of the inclusion–matrix interface S at yS . A point on the positive (or negative) side of the crack is denoted by x* (or x* ), and on the inclusion–matrix interface S by both xS and yS ; nm is the unit outward normal of the positive side of the crack surface at y* ; clmik is the fourth-order stiffness tensor of the material transverse isotropy; ui0 ys is the displacement component along the i-direction at point yS caused by a

given remote uniform loading, and tl0 y*

and tl0 y*

are the corresponding traction

components along l-direction at the points y* and y* ; ui and ti are the displacements and tractions on the inclusion–matrix interface S (or the crack surface * ); U ij and Tij are the Green’s functions of displacements and tractions; U ij , k and Tij , k are, respectively, the derivatives of the Green’s displacements and tractions respect to the source point. The displacement integral equation for a transversely isotropic inclusion is as follows

138


bij u j ( y S )

³U S

ij

( y S , xS ) t j ( xS ) dS ( xS ) ³ Tij ( y S , xS ) u j ( xS ) dS ( xS )

(3)

S

Equations (1), (2) and (3) can be used to investigate the effect of the inclusion on the SIFs of the crack embedded in a transversely isotropic medium. In discretization of these equations, we apply nine-node quadrilateral curved elements as shown in Fig.1 to the inclusion-matrix interface and the crack surface in which the crack front surface is meshed into special elements. Taking each node in turn as the collocation point, and performing various kinds of integrals, we finally obtain the compact forms of the discretized equations from Eqs. (1), (2) and (3) as

K

K 7

8

7

9

Type I 4

5

6

4

[

9

2

5

6

2

3

2/3 1

1

8 Type II

[

3 crack front K

7

K

8

9

7

8

9

Type IV

Type III 5 4

6

1

3

[

2/3 2

4

5

6

1

2

3

2/3 [

2/3 crack front

2/3 crack front

Fig. 1: Four element types for the crack surface.

ª H11 H12 º ª U m º ª B1 º »« » «H »« ¬ 21 H 22 ¼ ¬ 'U c ¼ ¬B 2 ¼

ª G11 G12 º ªTm º «G »« » ¬ 21 G 22 ¼ ¬ Tc ¼

(4)

and Hi Ui

G i Ti

(5)

where the subscripts i and m represent the inclusion and the matrix, respectively; H and G are


139

respectively the influence coefficient matrices containing integrals of the fundamental Green’s solutions; B1 and B2 are the displacement and traction vectors, respectively, induced by the remote loading; U m ( U i ) and Tm ( Ti ) are the node displacement and traction vectors, respectively, over the matrix side (inclusion side) of the inclusion-matrix interface; 'U c and Tc are, respectively, the discontinuous displacement and traction vectors over the crack surface. In this paper, we assume that the tractions on both sides of the crack are equal and opposite. Therefore Tc is equal to zero. Using the continuity condition of the displacement and traction vectors along the interface between the inclusion and matrix, we can combine Eqs. (4) and (5) into ª H11 G11G i1H i « 1 ¬ H 21 G 21G i H i

H12 º U m ½ B1 ½ ¾ ® ¾ »® H 22 ¼ ¯'U c ¿ ¯B 2 ¿

(6)

Therefore, once the unknowns U m and 'U c are solved, the SIFs along the crack front can be evaluated using the following equation [4, 9] 'u1 ½ K II ½ 2r 1 ° ° ° ° ' L 2 u ® 2¾ ® KI ¾ S ° 'u ° °K ° ¯ 3¿ ¯ III ¿

(7)

where r is the distance behind the crack front; L is the Barnett-Lothe tensor [9] which depends only on the anisotropic properties of the solid in the local crack-front coordinates; and 'u1 , 'u2 and 'u3 are the relative crack opening displacements in the local crack-front coordinates. Numerical examples An embedded spherical inclusion and a square-shaped crack in an infinite matrix are shown in Fig. 2. The radius of the sphere is R=1.0m and it is made of transversely isotropic marble with the following elastic properties: E X length are E X

of

the

12GPa , EZ

square 4GPa ,Q XY

90GPa , EZ is

2a

Q YZ

55GPa ,Q XY

(=2.0m).

0.3 , GYZ

The

Q YZ

0.3 , GYZ matrix

21GPa . The side

material

properties

1.6GPa . Here, we should note that all these

material properties are with respect to the local coordinates and that the local X, Y and Z represent the longitudinal, transverse and normal directions, respectively. The plane X-Y is the plane of isotropy. On the other hand, x, y and z refer to the space-fixed global coordinates, and the relation between these two coordinates (X,Y,Z) and (x,y,z) is shown in Fig. 3 with E and < being the

140


orientation and inclined angles. The crack center and the inclusion center are located on the x-axis, separated by a distance d. A far-field stress V f

1.0GPa is applied in the z-direction.

Vzz=1GPa

D

C

L=2m R=1m z

y

d

d A

x

B

x

Fig.2 (b): Planeform in x-y plane.

Fig.2 (a): A spherical inclusion and a square shaped crack in an infinite domain

z Z

X

\

Y y

E

x Fig. 3: The relationship between the principal material coordinates (X,Y,Z) and the space-fixed global coordinates (x,y,z) [4] Twenty-four nine-node quadrilateral elements with 98 nodes (Fig. 4(a)) and one hundred nine-node quadrilateral elements with 441 nodes (Fig. 4(b)) are used to discretize the inclusion-matrix interface and the crack surface, respectively. For various values of d and fixed E and
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168


HEDD-FS Method for Numerical Analysis of Cracks in 2D Finite Smart Materials 1

CuiYing Fan , GuangTao Xu2 and MingHao Zhao3 1,2,3

The School of Mechanical Engineering, Zhengzhou University, No. 100 Science Road, Zhengzhou, 450001, China E-mail: [email protected], [email protected], [email protected]

Abstract In this paper, the Hybrid Extended Displacement Discontinuity-Fundamental Solution Method (HEDD-FSM) is presented for numerical analysis of cracks in two-dimensional (2D) finite smart media by combing the extended displacement discontinuity method (EDDM) and the fundamental solution method (FSM). In the HEDD-FSM, the solution is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with the extended displacement discontinuities placed on the crack. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy the prescribed conditions on the boundary of the domain and on the crack face. The HEDD-FSM and the calculation of the extended intensity factors at crack tips are revisited for elastic, piezoelectric and magnetoelectroelastic problems. Keywords: HEDD-FSM, smart material, crack, extended displacement discontinuity method, fundamental solution method, stress intensity factor 1. Introduction Piezoelectric and magnetoelectroelastic materials, two kinds of smart materials, are finding more and more applications in smart structures and systems due to the mechanical-electrical-magnetic coupling effect. Defects, such as cracks, inclusions and voids in the materials greatly influence the performance of the structures and systems. Many efforts have been made to study the fracture problems of smart material [1-8]. Analytical solutions are usually difficult to obtain especially for cracks in finite domains. It is necessary to resort to numerical methods. As is well known, the Extended Displacement Discontinuity Method (EDDM) [9] is the preferable technique in dealing with problems involving stress singularities and has proved to be one of the most powerful methods in fracture mechanics of purely elastic media, piezoelectric media, as well as magnetoelectroelastic media[10-15]. On the other hand, the Fundamental Solution Method (FSM) is similar to the Charge Simulation Method (CSM). They share all the advantages of the BEM over domain discretization methods and have been used to solve various problems [16-21]. Combing the EDDM and the FSM, the Hybrid Extended Displacement Discontinuity-Fundamental Solution Method (HEDD-FSM) is proposed to solve the fracture problems of finite piezoelectric media [19,20] and magnetoelectroelastic media [21]. In this paper, we summarize briefly the HEDD-FSM for fracture analysis of two-dimensional (2D) problems of finite elastic, piezoelectric and magnetoelectroelastic media. 2. Basic Equations In the absence of body force, electric charge and electric current, the governing equations for a threedimensional magnetoelectroelastic medium are given by (1b) i, j 1, 2, 3 ( x, y , z ) , ij , j 0, Di ,i 0, Bi ,i 0, where V ij , Di and Bi denote the stress, electric displacement and magnetic induction components, respectively, and a subscript comma denotes the partial differentiation with respect to the coordinate. The constitutive equations can be expressed in terms of the displacement components ui ( u1 u x , u2 and u3

uy

u z ), the electric potential M and the magnetic potential \

V ij

cijkl (u k ,l u l ,k ) / 2 ekij M ,k f kij\ ,k ,

(2a)


169

Di

eikl (u k ,l u l ,k ) / 2 H ik M ,k g ik\ ,k ,

(2b)

Bi

f ikl (u k ,l u l ,k ) / 2 g ik M ,k P ik\ ,k ,

(2c)

where cij , eij , f ij , H ij , g ij and P ij are the elastic constants, piezoelectric constants, piezomagnetic constants, dielectric permittivity, electromagnetic constants and magnetic permeability, respectively. On letting f ij 0, g ij 0 and P ij 0 , the first two equations in eq (1) and eq (2) are reduced to the governing equations for a piezoelectric medium. 3. Boundary Conditions

z

There is an arbitrarily shaped plate occupying finite planar 1 2 domain V bounded by * , as show in Fig. 1. The Cartesian coordinate system oxz is set up, such that the polarization N1 3 N2 direction of the smart material is along the z-axis. There is a line crack S on the x-axis. Because electric field and magnetic field x 8 can exist everywhere, the electric and magnetic boundary conditions on crack faces are more complex compared with -a o a 4 those in purely elastic problems. S * 7 3.1 Boundary conditions for piezoelectric material 5 For piezoelectric material, there are two kinds of boundary 6 conditions on boundary and on crack faces S. One is the Fig.1 Source and collocation points for a mechanical condition and the other the electric condition. The finite domain of smart plate mechanical boundary conditions on boundary and on crack face S are given by (3) u x u x , u z u z , or t x { V xx nx V xz nz t x , t z { V xz nx V zz nz t z , where the over bar “-” denotes the prescribed values on the boundary, ti is the traction on the boundary, and ni is the unit outward normal vector on boundary and on crack face S. The electric boundary conditions on boundary is given by (4) M M , or Z { D x n x D z n z Z , where Z is the electric displacement boundary value. And the electric boundary condition on crack faces S takes one of the following electric boundary conditions [3], i.e., (5a) Dz Dz Z , for electrically impermeable condition, where superscripts “+” and “-” denote the quantities on the upper and lower crack faces, respectively, or (5b) Dz Dzc Dz Dzc Z , M M , for electrically permeable condition, where Dzc denotes the electric displacement in the crack cavity in the zaxis direction, or (5c) Dz Dzc Dz Dzc Z , Dzc H c [M M ] /[uz uz ], for crack opening model of piezoelectric media, where H c is the dielectric constant of the material in the crack cavity. 3.2 Boundary conditions for magnetoelectroelastic material There are three categories of boundary condition on boundary and on crack faces S, namely the mechanical condition, the electric condition and the magnetic condition. The mechanical boundary conditions have the same form as that for piezoelectric material given in eq (3). The electric and magnetic boundary conditions on boundary are given by (6) M M , or Z { Dx nx Dz nz Z and \ \ , or J { Bx nx Bz nz J , where is the magnetic induction boundary value. However, there are five kinds of electric and magnetic boundary condition on the crack faces S as described below [14], (7a) Dz Dz Z , and Bz Bz J , for electrically and magnetically impermeable condition, or (7b) Dz Dzc Dz Dzc Z , M M , and Bz Bzc B z B zc J , \ \ ,

170


for electrically and magnetically permeable condition, or Dz Dz Z , and B z Bzc B z Bzc J , \ \ , for electrically impermeable and magnetically permeable condition, or Dz Dzc Dz Dzc Z , M M , and Bz Bz J , for electrically permeable and magnetically impermeable condition, or D z D zc D z D zc Z , D zc H c [M M ] /[u z u z ],

(7c) (7d)

(7e) B z B zc B z B zc J , B zc P c [\ \ ] /[u z u z ], for crack opening model of magnetoelectroelastic media, where Bzc is the magnetic induction in the zdirection in the crack cavity and P c is the magnetic permittivity of the material in the crack cavity.

4. HEDD-FSM for Analysis of Cracks in Finite Magnetoelectroelastic Media The HEDD-FSM for smart media [19-21] is used to analyze the crack in a finite magnetoelectroelstic medium, as shown in Fig 1. First, N 1 collocation points are selected on the boundary, and an equal number of source points N 1 are taken accordingly outside the domain V. An unknown extended concentrated load Pkj (k 1, 2 N 1 ; j 1 4) is applied at source point k , where Pk1 and Pk 2 are the mechanical loads along the x- and z-axis, respectively, and Pk 3 and Pk 4 denote the point electric charge and electric current, respectively. Secondly, the crack S is divided into N 2 elements represented by the middle point. Constant elements are used and unknown extended displacement discontinuities u kj { u kj u kj (k 1, 2, ..., N 2 ) are uniformly distributed on each element. Using the extended point force fundamental solutions given in [8], the extended Crouch fundamental solutions given in [15], and the superposition principle, the extended displacement and the extended stress at any field point X can be expressed by [19-21] N1

ui ( X )

4

¦¦ u

* ij

k 1 j 1

V i (X )

N1

k 1 j 1

4

(8a)

k 1 j 1

4

¦¦ V

N2

( X , X P ) Pkj ¦¦ u kjc ( X , X S ) u kj , (i 1, 2, 3,4),

* ij

N2

4

( X , X P ) Pkj ¦¦ V ijc ( X , X S ) u kj , (i

1, 2, 7),

(8b)

k 1 j 1

where u i denote the extended displacements u x , u z , M and \ , respectively; V i denote the extended stresses V xx , V zz , V xz , Dx , Dz , Bx and Bz , respectively; u ij* and V * ij are the fundamental solutions corresponding to extended point forces given in [8], and u ijc and V ijc are the extended Crouch fundamental solutions given in [15]. X P and X S respectively denote the source point outside the domain and the source point on crack. On letting eq (8) satisfy the given boundary conditions at the collocation points on boundary * and every element on the crack, one obtains 2(N 1 +N 2 ) linear algebraic equations for purely elastic media, 3(N 1 +N 2 ) linear algebraic equations for piezoelectric media and 4(N 1 +N 2 ) linear algebraic equations for magnetoelectroelastic media to determine the unknown extended loads Pki and the unknown extended discontinuity displacements u kj . Solving the equations gives the unknown quantities. Fitting the calculated extended displacement discontinuity using the corresponding values of the N c - and (N c +1)-th elements from the crack tip, the asymptotic behavior of the extended displacement discontinuity near the crack tip can be expressed as (9) ui ki1r 1/ 2 ki 2 r 3 / 2 , where r is the distance of the point on crack faces from the crack tip, k i1 and k i 2 are the fitting coefficients. Furthermore, one has (10) lim u i / r k i1 . r o0

Finally, the extended intensity factor can be calculated by using the extended displacement discontinuity KI 2S S lim [ L31 u z L32 M ] r , K II 2S S lim L11 u x r , KD 2S S lim [ L41 u z L42 M ] r , (11a) r o0

for cracks in piezoelectric media [3], and

r o0

r o0


KI

2S S lim [ L31 u z L32 M L33 \ ]

r,

K II

2S S lim L11 u x

KD

2S S lim [ L41 u z L42 M L43 \ ]

r,

KB

2S S lim [ L51 u z L52 M L53 \ ]

r o0

r o0

171

r,

r o0

(11b) r,

r o0

for cracks in magnetoelectroelastic media [14], where Lij are material constants given in [3,14]. 5. HEDD-FSM for the Crack Opening Model in Finite Magnetoelectroelastic Media Based on eq (8) and the boundary condition in eq (7e), 4(N 1 +N 2 ) algebraic equations can be obtained to determine the unknown extended loads acting at the source points and the unknown extended discontinuity displacements on a crack face in a magnetoelectroelastic media, while the electric and magnetic boundary conditions on crack faces are expressed as N1

4

¦¦ V k 1 j 1 N1

4

* 5j

N2

4

( X , X P ) Pkj ¦¦ V 5c j ( X , X S ) u kj Dzc N2

4

¦¦ V 7* j ( X , X P ) Pkj ¦¦ V 7c j ( X , X S ) ukj Bzc k 1 j 1

Z ,

(12)

k 1 j 1

J .

k 1 j 1

Eq (5c) and eq (7e) are nonlinear algebraic equations, and the solution is difficult to obtain. The iterative method [14] is adopted here to solve the nonlinear problem. At first, the crack is treated as an impermeable one with the boundary condition given in eq (5a) and eq (7a): (13) Dzc(0) ( x, z) 0, Bzc(0) ( x, z) 0, where the number “l” in the parenthesis in superscript “(l)” denotes the l-th iteration as described below. Solving the 4(N 1 +N 2 ) nonlinear algebraic equations, we obtain the solution of the extended loads acting at (1) source points and the extended discontinuity displacement on crack face denoted by u i . Then substituting the value of the obtained extended discontinuity displacement into eq (7e), new values of electric displacement and magnetic induction in the crack cavity can be calculated: (1) (1) (1) (1) (14) Dzc (1) H c M / u z , Bzc (1) P c \ / u z . The first round iteration has been completed. The iteration continues until the final solution is obtained when the preset accuracy G i (i 1,2) is satisfied Dzc ( l ) Dzc ( l 1) / Dzc ( l ) G1 ,

Bzc ( l ) Bzc ( l 1) / Bzc ( l ) G 2 .

(15)

Substituting the extended discontinuity displacements of the final iteration results into eq (11), one can obtain the extended intensity factors, and the electric displacement and the magnetic induction fields in the crack cavity in the crack opening model. 6. Numerical Examples Consider a center crack of length 2a in a smart plate of 2h length and 2b width, as shown in Fig.2. The poling direction is along the z-axis. Based on the HEDD-FSM [19-21], N 1 =200 collocation points are on the boundary of domain V, and an equal number of N 1 =200 source points are taken outside the domain V. And N 2 =100 elements are on the line crack. The ratio of AA 1 /AB is 2.3, where AA 1 is the distance between collocation point A on boundary and the corresponding source point A 1 outside the domain, while AB is the distance between collocation point A and its adjacent collocation point B. By using the HEDD-FS method, the intensity factor of a center crack in an elastic plate is calculated under p z 100MPa and plotted versus crack length a/b in Fig. 3, where the normalized stress intensity factors F e and F I respectively based on the analytical solution [22] and the HEDD-FSM are given by (16) Fe sec Sa /(2b) , FI K , /( p z Sa ).

pz , Z,J

z

b

b

h S

O a

a

A

A1

B

B1

x

h

Fig.2 A center crack in a smart plate

172


Normalized intensity factors FI, Fe

It can be seen that the HEDD-FSM is effective to study the fracture problems with finite domain. Fig.4a and Fig.4b show the dimensionless intensity factors by the impermeable crack model and crack opening model for center cracks in piezoelectric medium PZT-6B under the applied load p z 100MPa and

1.8 1.6

FI impermeable crack model FD impermeable crack model

1.4

pz

1.2

100MPa, Z

0.1C/m 2

1.0 0.8 0.0

0.2

0.4

0.6

1.8

HEDD-FSM FI Analytical solution Fe

1.6 1.4 1.2 1.0 0.8 0.0

0.2

0.4

0.6

0.8

1.0

a/b

Fig. 3 Intensity factor comparison in elastic plate between HEDD-FSM and analytical solution

Normalized intensity factor FI,FD

Normalized intensity factor FI,FD

Z 10C/m 2 , where the normalized electric displacement intensity factor F D is given by (17) FD K D /(Z Sa ). The results demonstrate F I and F D increase with increasing a/b, but the effect of the geometric size on F I and F D is different under different electric boundary condition.

2.0

1.8 1.6

FI crack opening model FD crack opening model

1.4 1.2 1.0

pz

0.8

100MPa, Z

0.1C/m 2

0.6 0.4 0.0

0.8

0.2

0.4

0.6

0.8

a/b

a/b

Fig.4a Intensity factors versus crack length a / b for impermeable crack in a piezoelectric plate

Fig.4b Intensity factors versus crack length a / b for crack opening model in a piezoelectric plate

1.8 FI impermeable crack model

1.6

FD impermeable crack model FB impermeable crack model

1.4 1.2 1.0 0.8

pz

100 MPa , Z 1.0C/m, J

10N/Am

0.6 0.0

0.2

0.4

0.6

0.8

a/b

Fig.5a Intensity factors versus crack length a / b for impermeable crack in a magnetoelectroelastic plate

Normalized intensity factors FI,FD,FB

Normalized intensity factors FI,FD,FB

Under the combing mechanical-electric-magnetic loading p z 100MPa , Z 1.0C/m and J 10N/Am , the dimensionless intensity factors of a center crack in the magnetoelectroelastic medium are displayed in Fig.5a and Fig.5b by using respectively the impermeable crack model and the crack opening model, where the normalized magnetic induction intensity factor F B is given by (18) FB K B /(J Sa ). The magnetoelectroelastic medium is a composite material made of BaTiO 3 and CoFe 2 O 3 with the volume fraction of the piezoelectric inclusion Vi 0.5 . It can be seen that the three normalized intensity factors increase with crack length. For impermeable boundary condition, the three normalized intensity factors are equal for a small crack. However, the three normalized intensity factors are unequal considering the crack opening and the electric and magnetic fields in the crack cavity. 1.8

pz

100 MPa , Z 1.0C/m, J

10N/Am

1.6 FI crack opening model FD crack opening model

1.4

FB crack opening model

1.2 1.0 0.8 0.6 0.0

0.2

0.4

0.6

0.8

a/b

Fig.5b Intensity factors versus crack length a / b for crack opening model in a magnetoelectroelastic plate


173

Similarly, the intensity factors for other electric and magnetic boundary conditions can be obtained by using the HEDD-FSM. The results demonstrate that the electric and magnetic boundary conditions, as well as the geometry of the medium, greatly influence the solutions. 7. Conclusion The HEDD-FSM composes the merits of both the extended displacement discontinuity method and the charge simulation method in analyzing cracks in 2D finite piezoelectric and magnetoelectroelastic media. Different electric and magnetic boundary conditions on crack face, such as the impermeable, permeable and semi-permeable conditions, can be easily incorporated in this method. This method is of higher computing speed and more efficiency for analysis of cracks in twodimensional smart media. Numerical examples show that the freedom used in this method is two orders less than that used in the finite element method for analyzing cracks in smart plates. This method can be used to solve problems including multi-cracks, mixed-mode cracks, and can be further extended to solve crack problems of 3D finite smart media. Acknowledgement The work was supported by the National Natural Science Foundation of China (10872184), the Innovation Scientists Technicians Troop Construction Projects of Henan Province (084200510004) and the Program for Innovative Research Team (in Science and Technology) in University of Henan Province (2010IRTSTHN013). References [1] Z. Suo, C.M. Kuo, D.M. Barnett and J.R.Willis Journal of the Mechanics and Physics of Solids, 40,739-765 (1992). [2] Q.H. Qin Southampyon: WIT Press (2001). [3] T.Y. Zhang, M.H. Zhao and P.Tong Advances in Applied Mechanics, 38, 147-289 (2002). [4] K. Meinhard Engineering Fracture Mechanics, 77, 309-326 (2010). [5] G.C. Sih, R. Jones and Z.F. Song Theoretical and Applied Fracture Mechanics, 40, 161-186 (2003). [6] B.L. Wang and Y.W. Mai European Journal of Mechanics, A/Solids, 22, 591-602 (2003). [7] X. Wang and Y.P. Shen International Journal of Engineering Science, 40, 1069-1080 (2002). [8] H.J. Ding, A.M. Jiang, P.F. Hou and W.Q. Chen Engineering Analysis with Boundary Elements, 29, 551-561 (2005). [9] S.L. Crouch International Journal for Numerical Methods in Engineering, 10, 301-343 (1976). [10] E. Pan Engineering Analysis with Boundary Elements, 23, 67-76 (1999). [11] C.Y. Dong, S.H. Lo and H. Antes Computational Mechanics, 41, 207-217 (2008). [12] B.J. Zhu and T.Y. Qin Theoretical and Applied Fracture Mechanics, 47, 219-232 (2007). [13] J.A. Sanz, M.P. Ariza and J. Dominguez Engineering Analysis with Boundary Elements, 2, 586-596 (2005). [14] M.H. Zhao, C.Y. Fan, F. Yang and T. Liu International Journal of Solids and Structures, 44, 45054523 (2007). [15] M.H. Zhao, C.Y. Fan, T. Liu and F. Yang Engineering Analysis with Boundary Elements, 31, 547-558 (2007). [16] J.R. Berger and A. Karageorghis Engineering Analysis with Boundary Elements, 25, 877-886 (2001). [17] S. Chantasiriwan, B.T. Johansson and D. Lesnic Engineering Analysis with Boundary Elements, 33, 529-538 (2009). [18] X.C. Li and W.A. Yao Engineering Analysis with Boundary Elements, 30, 709-717 (2006). [19] C.Y. Fan, M.H. Zhao and Y.H. Zhou Journal of the Mechanics and Physics of Solids, 57, 1527-1544 (2009). [20] M.H. Zhao, G.T. Xu and C.Y. Fan Engineering Analysis with Boundary Elements, 33, 592–600 (2009). [21] M.H. Zhao, G.T. Xu and C.Y. Fan Computational Mechanics, 45, 401-413 (2010).

[22] H. Tada, P.C. Paris and G.R. Irwin Del Research Corp, Hellertown, PA, 1973.

174


Recent developments of radial integration boundary element method in solving nonlinear and nonhomogeneous multi-size problems X. W. Gao 1, M. Cui 1 and Ch. Zhang 2 1

School of Aeronautics and Astronautics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, PR China, E-mails: [email protected]; [email protected] 2

Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany, E-mail: [email protected]

Keywords: Boundary element method; Radial integration method; Nearly singular integrals, Thin structures

Abstract This paper presents some recent developments of the meshless boundary element method based on radial integration method (RIM) for solving 2D and 3D heat transfer and thermoelasticity problems. Special attention is paid to the consideration of the effects such as structural multi-sizes, nonlinear, nonhomogeneous, and anisotropic problems. Firstly, general boundary-domain integral equations for heat conduction and stress analysis are derived using the weighted residual method based on the source point isolation technique, in which fundamental solutions for the corresponding linear homogeneous problems are adopted. The use of linear isotropic fundamental solutions for anisotropic, nonlinear and nonhomogeneous problems results in domain integrals appearing in the basic integral equations. The domain integrals appearing in the integral equations, then, are transformed into equivalent boundary integrals using RIM, resulting in a pure boundary element analysis algorithm without the need of internal cells. The thermal and mechanical material properties can be the functions of both temperature (resulting in nonlinear heat transfer) and spatial coordinates (for non-homogeneous materials). The Newton-Raphson iteration scheme is applied to solve the resulting nonlinear equation set. The nearly singular boundary integrals stemming from treating thin-structures using BEM are evaluated using the non-equally spaced element sub-division technique. The three-step solver of multi-domain BEM is employed to solve composite structural problems consisting of different materials. Finally, numerical examples are given to demonstrate the accuracy and efficiency of the presented method.

1. Introduction Thin structures are frequently used in aerospace engineering [1], such as multi-layered coatings, laminated structures, honeycomb structures etc. The investigation shows that the thermal stresses induced in laminated structures are the main cause of structural failure [2]. Therefore, the thermal stress analysis of composite structures is significantly important in aerospace engineering. The boundary element method (BEM) has distinctive advantages in solving problems of fracture mechanics [3] and thin structural problems [4], since it only needs to discretize the boundary of the problem into elements. However, the conventional BEM is not so attractive in solving nonhomogeneous, nonlinear and thermoelasticity problems, since domain integrals are inevitably introduced in the resulting integral equations [5]. A direct evaluation of domain integrals requires the discretization of the domain into internal cells. This severely eliminates the advantage of BEM. To overcome this disadvantage, Nardini and Brebbia [6] developed the dual reciprocity method (DRM) to transform the domain integrals into equivalent boundary integrals. To avoid using particular solutions required in the DRM, Gao proposed the radial integration method (RIM) [7] which can transform any domain integrals to the boundary based on a pure mathematical manipulation. RIM has been successfully applied to solve thermoelasticity [8], elastic inclusion [9], and creep damage mechanics problems [10]. In view of the robustness and simplicity of RIM in evaluating domain integrals without using internal cells, Hematiyan [11] gave a very good assessment to RIM, and Albuquerque et al. [12] compared RIM to DRM numerically through applications to dynamic problems with a more positive conclusion. Although thermoelasticity problems with constant material properties have been solved using the boundaryonly element method based on RIM [8], this methodology has yet not been applied to solve heat conduction


175

and thermoelastic thin structure problems with varying material properties. This paper is an attempt for this purpose. First, boundary-domain integral equations for temperature and displacements are derived from the weighted residual forms of governing equations. Then, the domain integrals arising in the integral equations are transformed into equivalent boundary integrals using RIM, resulting in a pure boundary element solution algorithm. Material properties are allowed to be any type of functions of spatial coordinates. The treatment of nearly singular integrals is a challenge issue in solving thin structural problems using BEM [13]. A nonequally spaced element sub-division technique is presented for evaluating the nearly singular integrals. Numerical examples are given to demonstrate the correctness and efficiency of the presented method.

2. Boundary-domain integral equations for general nonlinear and nonhomogenous heat conduction problems 2.1. Formulations for general heat conduction problems The governing equation for general heat conduction problems can be expressed as

w wxi

§ wT · ¨ k ij ¸Q ¨ wx ¸ j ¹ ©

0,

(1)

where k ij and Q are the thermal property tensor and the source term, respectively, and T denotes the temperature. kij and T both may be functions of spatial coordinates for non-homogeneous problems or functions of the temperature for non-linear problems. It is noted that the thermal property tensor is symmetric, i.e., k ij k ji . The repeated subscripts in eq (1) represent summation over the ranges of their values. Using a weight function G to multiply both sides of eq (1) and integrating over the entire domain :, the following weak-form can be written. w § wT · (2) ³: G wxi ¨¨ k ij wx j ¸¸d: :³ GQd: 0 . ¹ © The first integral can be manipulated as follows

w § wT · ¨¨ kij ¸ d: wx j ¸¹ i ©

³ G wx

:

³ Gk

ij

*

wT wG wT ni d * ³ kij d: wx j wxi wx j :

w xj w :

³ Gqd * ³ q Td * ³ *

*

§ wG · ¨ kij ¸ Td : © wxi ¹

(3)

³ Gqd * ³ q Td * I : , *

*

where

q

k ij

I:

wT ni , wx j

wG k ijn j , wxi

w § wG · ¸Td: , ¨¨ k ij wxi ¸¹ j ©

³ wx

:

q

(4)

(5)

in which * denotes the boundary of the domain :, ni is the i-th component of the outward normal vector to *, and q is the heat flux. It is noted that the domain integral in eq (5) may be strongly singular (depending on the choice of G) and, therefore, a different integral symbol is used to denote this. We assume that the weight function G is a fundamental solution of either isotropic or anisotropic problems. Usually, it is a function of the distance r between the source point p and the field point q [14]. When r o 0 , G may be singular and, therefore, an infinitesimal circular domain : H centered at the source point p with radius H can be isolated from : (Fig. 1).

176


*

: H :H

*H

p

Fig. 1 An infinitesimal domain : H isolated from : The last term in eq (3) now can be written as

w § wG · ¨ kij ¸ Td : wxi ¹ j ©

³ wx

I:

:

T ( p ) lim H o0

³

:H

w wx j

lim H o0

w § wG · w § wG · ¨ kij ¸ Td : lim ¨ kij ¸ Td : ³ o H 0 wxi ¹ wx j © wxi ¹ j © ::H

³ wx

:H

§ wG · w ¨ kij ¸d : ³ w w x xj i ¹ © :

§ wG · ¨ kij ¸ Td : © wxi ¹

(6)

k ( p)T ( p) ³ VTd :, :

where

k ( p)

lim H o0

w § wG · ¨¨ kij ¸d : wx j ¸¹ i ©

³ wx

:H

V

w wxi

lim ³ kij H o0

*H

§ wG · ¨ k ij ¸ ¨ wx ¸ j ¹ ©

wG ni d * wx j

kij ( p) lim ³ H o0

*H

wG n j d *, , wxi

(7)

wk ij wG w 2G k ij . wxi wx j wxi wx j

(8)

It is noted that the last domain integral in eq (6) is interpreted in the Cauchy principal value sense. Substituting eq (6) into eq (3) and the result into eq (2) yields

kT ³ q Td* *

³ G qd* ³ V Td: ³ GQd: . *

:

(9)

:

Equation (9) is a boundary-domain integral equation valid for isotropic, anisotropic, linear and nonlinear heat conduction problems. The weight function G can be any type of functions. If G is a regular function, the coefficient k as determined by eq (7) will be zero since the radius H o 0 ; if G is chosen as the Green’s function [14] which is weakly singular when the source point p approaches the field point q under integration, k has a finite value; and if G is chosen as a higher singular function than the Green’s function, k is infinite and, therefore, this type of G doesn’t make sense. Once the weight function G and thermal property tensor k ij are given, all coefficients and kernel functions in eq (9) are known, and the unknown quantities be computed using the standard BEM discretization procedure [5]. It is noted that although eq (9) is derived for an internal source point p, it can also be used for boundary nodes since it is actually not necessary to compute the coefficient k directly using eq (7). This is based on the fact that the contribution of k to the final system of equations is in the diagonal term, which can be determined using a more efficient way, i.e., the “rigid body motion condition” [5].

2.2 Using isotropic fundamental solutions for general anisotropic heat conduction problems In principle, the weight function G can be any function. However, the simplest way is to choose G as the Green’s function for isotropic heat conduction problems, i.e.,

G

° ° ® ° °¯

1 1 ln( ), 2 D, 2S r 1 , 3 D, 4S r

(10)


177

where r is the distance between the source point p and field point q. The derivatives of G can be expressed as

1 r,i , 2SDr D

wG wxi

1 (G ij Er,i r, j ) , 2SDr E

w 2G wxi wx j where r,i

(11) (12)

wr / wxi , E=2 for 2D and E=3 for 3D problems, and D=E-1. Since *H is a circle (2D) or a sphere

(3D), we have n, j

r, j . Thus, from eq (7) it follows that k

k ij

2SD ³ *H

r,i n j r

D

kij

2SD ³

d*

r,i r, j

*H

rD

d* .

(13)

For an internal point, using the following relationship [5]

³

*H

SG ij , 2 D, ° ® 4S ° 3 G ij , 3D, ¯

r,i r, j r

d* D

(14)

equation (13) can be integrated as

k

k ii / E .

(15)

It can be seen that k is the average value of the diagonal term of k ij . This is helpful for understanding the coefficient k in eq (9). It is also pointed out that if the problem is isotropic, the last term in eq (8) is zero and eq (9) is reduced to the result in [14].

3. Boundary-domain integral equations for thermoelasticity with variable coefficients The governing equations of the thermoelasticity problems can be expressed as [8]

V ij

0 P Cijkl uk ,l G ijD~T ,

where

D~

2(1 Q ) PD , 1 2Q

(16)

(17)

2Q (18) G ijG kl G ik G jl G il G jk , 1 2Q in which P represents the shear modulus, Q the Poisson’s ratio, D the thermal expansion coefficient, and uk the displacement components. Both P and D are functions of temperature and spatial coordinates. 0 Cijkl 䋽

Through applying the weighted residual method [8,15] to eq (16), the following boundary-domain integral equations for the displacements and the stresses can be obtained

³U

ui ( y )

*

ij

( x , y )t j ( x )d *( x ) ³ Tij ( x, y )u j ( x )d *( x ) *

³ Vij ( x , y )u j ( x )d :( x ) ³ U ij , j ( x, y )Tˆ ( x )d :( x ), :

V ij ( y )

³U *

ijk

:

( x , y )tk ( x )d *( x ) ³ Tijk ( x, y )uk ( x )d *( x ) *

³ Vijk ( x, y )uk ( x )d :( x ) ³ < ij ( x , y )[Tˆ ( x ) Tˆ ( y )] d :( x ) :

(19)

:

wr Tˆ ( y ) ³ r ln r < ij ( x , y )d *( x ) G ij hTˆ ( y ) Fijk ( y )uk ( y ), wn *

(20)

178


where U ij , Tij , Vij , U ijk , Tijk , Vijk and Fijk can be found in [15], and other quantities are given by

u~ j

Pu j ,

(21)

2(1 Q ) PD T, 1 2Q (1 2Q ) (G ij E r,i r, j ) , 2DS (1 Q ) r E Tˆ

@

(26)

2m

in which f i denotes the 2mi -th derivative of the function f, Li is the length of the element in the i-th direction. From eq (25) it can be seen that the integration error relies on the number of Gaussian points and the element length. Therefore, to ensure a desired accuracy, a big element must be divided into small subelements. Based on the analysis of an upper bound of the relative error, Mustoe [17] presented the following approximate formula for the specified accuracy tolerance e

§L · 2¨ i ¸ © 4R ¹

2 mi

( 2mi p 1)! de, ( 2mi )!( p 1)!

(27)

where p is the singularity order of the integral kernel characterized by r p , R is the minimum distance from the source point to the element. Equation (27) shows that, to retain the specified accuracy e and the number of Gaussian points cannot exceed a given number, the value of Li / R needs to be reduced by dividing the big element into a number of sub-elements. Based on the numerical investigation, Gao and Davies [16, 5] proposed the following formula for determining the number of Gaussian points.

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz p' log e ( e / 2) , 2 log e >Li /( 4 R )@

mi

179 (28)

where

p'

2 2 p . 3 5

(29)

After a rearrangement eq (28) yields p'

§ e · 2 mi L i 4 R¨ ¸ . ©2¹

(30)

Equation (28) can be used to determine the minimum number of Gaussian points for a given accuracy tolerance e, while eq (30) can be used to determine the length of each sub-element for the specified values of allowed maximum number of Gaussian points, singularity order and the minimum distance. The non-equally spaced element sub-division technique can be summarized as follows: 1) Compute length Li of the boundary element under integration and the minimum distance R from the source point y to the element. Detailed Fortran subroutines for determining Li and R can be found in [5]. 2) Calculate the required number of Gaussian points mi using eq (28) in terms of the values of Li and R. 3) If mi d m max ( mmax being the specified maximum number of Gaussian points), evaluate integrals using Gaussian quadrature formulas. 4) If mi ! m max , let mi m max and compute the length Lni of each sub-element along the integration direction i using the value of R and eq (30). 5) As shown in Fig. 2, mark the graduations in two integration directions in terms of L1n and Ln2 and form all sub-elements (enclosed by dashed lines) from these graduations. 6) Evaluate integrals over each sub-element using Gaussian quadrature formulas. For understanding this process easily, Fig. 2 gives the schematic show of a big boundary element divided into 12 sub-elements by partitioning the line along [1 direction into 4 segments and the line along [ 2 direction into 3 segments. 3

6XEHOHPHQWV

L2

2

L2 1

L2

R y

1

L1

2

L1 3

L1 4

L1

Fig. 2 Schematic show of the element sub-division

5. Numerical examples Based on the method described in the paper, a computer code named BERIM (Boundary Element analysis based on Radial Integration Method) has been developed. In the code, all domain integrals appearing in eqs (9), (19) and (20) have been transformed into boundary integrals using RIM [7], resulting in a pure boundary element analysis algorithm without need of internal cells. When the material properties are functions of the temperature, the Newton-Raphson iteration scheme is applied to solve the nonlinear heat conduction equations. To solve composite structure problems, the three-step multi-domain BEM (MDBEM) solver proposed in [18] is adopted for both heat conduction and thermoelasticity problems. Corner and edge points are treated using the discontinuous elements to model the discontinuity of the heat flux and the traction. Two

180


numerical examples are presented in the following. 5.1 Thermal stress analysis over a honeycomb structure The first example is a honeycomb structure which is commonly used in thermal protection system (TPS) [1]. The structure consists of upper and lower cover plates with a thickness of 0.125mm. The honeycomb core has a wall thickness of 0.035mm, a wall height of 7.11mm, and a width of 4.76mm. The structure has a total of 100 honeycombs with the global dimension of 49.98mm u 42.56mm u 7.36mm. Figure 3 shows the BEM model consisting of 8946 four-noded boundary and interface elements with 8484 nodes. In the BEM model, the upper and lower plates, the honeycomb wall, and the hollow volume filling with air are treated as different sub-domains. The heat conductivities are 7.8u 10 5 W/(mmK) for the upper and lower plates, 1.7u 10 4 W/(mmK) for the honeycomb wall, and 2.3u 10 5 W/(mmK) for the filled air. The thermal boundary conditions are given as follows: the top and bottom surfaces are specified with the temperature distribution of

T ( x, y, z )

225 xz 75 x 600 z 200 (K) and the side surfaces are adiabatic. 49.98 u 7.36 49.98 7.36

Firstly, the heat conduction computation is performed to obtain the temperature distribution in the structure, and then the thermoelasticity computation is carried out using the obtained temperature. In the thermoelasticity computation, the material parameters are taken as P=280GPa, Q=0.25, and the thermal expansion coefficient is D 2.47 u 10 -6 mm/K . The top surface is uniformly imposed by traction conditions of W x =0.05MPa and W z =-0.5MPa, the bottom surface is fixed, and the side surface is tractionfree. Figure 5 shows the computed temperatures at points shown in Fig. 4, which are located below the inner surface of the upper plate with a distance of 0.01mm to the inner surface. The computed heat flux qz , displacement u x and stress V xx at these selected points are presented in Figs. 6, 7 and 8, respectively. Wz

0.5Mpa

Wx

z

0.05Mpa

T=500K

T=800K

T=200K

q=0

x

T=125K

Fig. 4 Points for results plotting

2

Heat flux q z (W/mm )

Temperature (K)

Fig. 3 BEM model and boundary conditions

x-coordinate (mm)

Fig. 5 Distribution of temperature

x-coordinate (mm)

Fig. 6 Distribution of heat flux qz

181

xx (0Sa)

Displacement ux (mm)


x-coordinate (mm)

x-coordinate (mm)

Fig. 7 Displacement u x

Fig. 8 Stress V xx

From Fig. 6, we can see that the heat flux q z has a much larger value on the honeycomb side wall than in the hollow air. This important phenomenon captured in the example is attributed to the discretization of the two surfaces of the honeycomb wall into boundary elements. Figure 8 shows that the computed stress V xx is much larger than the imposed tractions. This indicates that the thermal stress is an important factor in TPS failure analysis. 5.2 Rectangular plate with a crack under tensile loading The second example analyzed is a rectangular plate with an edge crack, which is subjected to a uniform tensile loading as depicted in Fig. 9. The geometry of the cracked plate is described by: plate width b=10, plate length 2h=30 and crack-length a=0.4b. To demonstrate the capability of the presented method to treat the crack problem, a single computational domain is used in our computation. The upper and lower surfaces of the crack is very close, but not completely in contact, measured with the width ratio of the opening distance to the crack-length a. The boundary of the plate including the crack surfaces is discretized into 115 quadratic boundary elements with 254 boundary nodes. Two nodes are defined at the crack-tip for utilizing the discontinuous element [18] to model the discontinuity of the traction across the tip. Plain strain condition is assumed in our computation.

7RS

8SSHU

Q 0.25 P 4000

Fig. 9 A plate with an edge crack

/RZHU

Fig. 10 Deformed plate

Figure 10 shows the deformed plate plotted using the computed displacements multiplied by a factor of 200 for the case of the width ratio being 0.1%. To verify the correctness, this problem is also computed using the multi-domain BEM code [18], in which lower and upper parts of the plate are treated as two sub-domains along the crack. The displacements in y-direction computed using the present single domain method denoted by “1-Domain” and using the multi-domain BEM denoted by “2-Domain” are listed in Table 1 for three corner nodes as shown in Fig. 10. Comparison of the results from the two methods shows that the relative errors are rather small. To investigate the convergence of the computational results to the width of the crack, computations are carried out using different ratios of the crack-width to the crack-length. The computed results for the three selected corner nodes (see Fig. 10) are shown in Fig. 11. From Fig. 11, it can be seen

182


that the convergence is achieved when the width ratio is larger than 0.02% which is small enough to model a real crack-width. Figure 11 also shows that the usual single domain BEM combined with the non-equally spaced element sub-division technique described in this paper can solve the crack problems efficiently without the use of other complicated methods [3, 18]. Table 1 Computed displacement u y at three selected nodes 1-Domain 2-Domain Error (%)

Lower 8.66874E-4 8.75949E-4 -1.04

Upper 5.7889E-3 5.7325E-3 0.984

Top 6.6526E-3 6.6045E-3 0.728

-2

Displacement u y (X10 )

Lower Upper Top

Width ratio of crack (%)

Fig. 11 Displacement computed using different values of the crack-width

6. Conclusions A boundary element technique is presented for solving 2D and 3D nonlinear and non-homogeneous heat transfer and thermoelasticity problems. A non-equally spaced element sub-division technique is proposed for evaluating nearly singular boundary integrals in the analysis of thin structures using BEM. Numerical results show that the presented method can not only effectively solve the thin-wall structure problems, but can also solve crack problems in the usual way. This is very convenient to solve complicated engineering problems.

References [1] Myers D.E., Martin C,J., Blosser M.L., Parametric weight comparison of advanced metallic, ceramic tile, and ceramic blanket thermal protection systems. NASA/TM-2000-210289, 2000. [2] Hohe J., Goswami S., Becker W., Assessment of interface stress concentrations in layered composites with application to sandwich panels. Computational Materials Science, 2002, 26: 71-79. [3] Chen J.T., Hong H.K., Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Applied Mechanics Reviews, ASME, 1999, 52(1): 17-33. [4] Luo J.F., Liu Y.J., Berger E.J., Analysis of two-dimensional thin structures (from micro- to nano-scales) using the boundary element method. Computational Mechanics, 1998, 22: 404-412. [5] Gao X.W., Davies T.G., Boundary Element Programming in Mechanics. Cambridge, London, New York: Cambridge University Press, 2002. [6] Nardini D., Brebbia C.A., A new approach for free vibration analysis using boundary elements. In: Brebbia C.A., editor. Boundary Element Methods in Engineering. Berlin: Springer, 1982, 312-326. [7] Gao X.W., The radial integration method for evaluation of domain integrals with boundary-only discretization. Engineering Analysis with Boundary Elements, 2002, 26: 905-916. [8] Gao X.W., Boundary element analysis in thermoelasticity with and without internal cells. International Journal for Numerical Methods in Engineering, 2003, 57: 975-990. [9] Dong C.Y., Lo S.H., Cheung Y.K., Numerical solution for elastic inclusion problems by domain integral equation with integration by means of radial basis functions. Engineering Analysis with Boundary Elements, 2004, 28: 623-632.


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[10] Gun H., 3D boundary element analysis of creep continuum damage mechanics problems. Engineering Analysis with Boundary Elements, 2005, 29: 749-755. [11] Hematiyan M.R., A general method for evaluation of 2D and 3D domain integrals without domain discretization and its application in BEM. Computational Mechanics, 2007, 39: 509-520. [12] Albuquerque E.L., Sollero P., Paiva W.P., The radial integration method applied to dynamic problems of anisotropic plates. Commun. Numer. Meth. Eng., 2007, 23: 805-818. [13] Niu Z., Cheng C., Zhou H., Hu Z., Analytic formulations for calculating nearly singular integrals in two-dimensional BEM. Engineering Analysis with Boundary Elements, 2007, 31: 949-964. [14] Gao X.W., A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity. Int. J. Numer. Meth. Engng., 2008, 66: 1411-1431. [15] Gao X.W., Zhang Ch., Guo L., Boundary-only element solutions of 2D and 3D nonlinear and nonhomogeneous elastic Problems. Engineering Analysis with Boundary Elements, 2007, 31: 974-982. [16] Gao X.W., Davies T.G., Adaptive algorithm in elasto-plastic boundary element analysis. Journal of the Chinese Institute of Engineers (English edition), 2000, 23(3): 349-356. [17] Mustoe G.G.W., Advanced integration schemes over boundary elements and volume cells for two- and three-dimensional non-linear analysis. In: Development in Boundary Element Methods, Elsevier, London, 1984. [18] Gao X.W., Guo L., Zhang Ch., Three-step multi-domain BEM solver for nonhomogeneous material problems. Engineering Analysis with Boundary Elements, 2007, 31: 965-973.

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A Meshless Boundary Interpolation Technique for Solving the Stokes Equations Csaba Gáspár1 1

Széchenyi István University, Department of Mathematics, P.O.Box 701, H-9007 GyĘr, Hungary, email: [email protected]

Keywords: Stokes equations, regularized method of fundamental solutions

Abstract. A boundary-only meshless method for the Stokes equations based on the regularized method of fundamental solutions is presented. The second-order Stokes operator is approximated by a fourth-order operator. The fundamental solution of this new operator is continuous at the origin, thus, the Method of Fundamental Solutions can be applied without difficulty; moreover, the source and collocation points are allowed to coincide and can be located on the boundary. The method converts the original problem to a far less ill-conditioned system of equations than the original method of fundamental solutions. In fact, the method is a boundary multi-elliptic interpolation technique, which produces a divergence-free vector field. An error estimate is also derived which shows that a careful choice of the scaling factor appearing in the definition of the method results in quite good approximate solution. At the same time, however, the use of severely ill-conditioned matrices is avoided. The Stokes equations and the Method of Fundamental Solutions Consider the permanent 2D Stokes equations written in dimensionless form: − ∆u +

∂p = 0, ∂x

− ∆v +

∂p = 0, ∂y

div (u , v) = 0 ,

(1)

where u, v are the dimensionless velocity components, p is the dimensionless pressure. Eq. (1) is required in a bounded domain Ω . In this paper, Equation (1) is supplied with Dirichlet boundary condition: u |Γ = u 0 , v |Γ = v0

(2)

where Γ denotes the boundary of the domain Ω . It is well known that Eqs. (1)-(2) have a unique solution e.g. in the function space H 1 (Ω) × H 1 (Ω) × L2 (Ω) provided that the boundary conditions (u 0 , v 0 ) satisfy the compatibility condition ³ (u 0 , v 0 ) ⋅ n dΓ = 0 and p is assumed to satisfy ³ p dΩ = 0 . Γ

Ω

A popular and efficient method for solving Problem (1)-(2) is the Method of Fundamental Solutions (MFS, [1,2]). Here the boundary is discretized by the collocation points ( x k , y k ) ∈ Γ ( k = 1,2,..., N ). Defining a xk , ~ y k ) ( k = 1,2,..., N ), the approximate solution of (1)-(2) is expressed as: finite set of source points ( ~ N N u ( x, y ) ~ ¦ a (j1) u1 ( x − ~ xj, y − ~ y j ) + ¦ a (j2) u 2 ( x − ~ xj,y − ~ yj) j =1 N

v( x, y ) ~ ¦

j =1

j =1

a (j1) v1 ( x −

N ~ xj, y − ~ y j ) + ¦ a (j2) v 2 ( x − ~ xj,y − ~ yj) j =1

(3)

N N p ( x, y ) ~ ¦ a (j1) p1 ( x − ~ xj,y − ~ y j ) + ¦ a (j2) p 2 ( x − ~ xj, y − ~ yj) j =1

j =1

§p · § u u2 · ¸¸ and the vector function p := ¨¨ 1 ¸¸ is a fundamental where the pair of the matrix function G := ¨¨ 1 v v 2¹ © p2 ¹ © 1 solution of the Stokes equations (1), i.e. they satisfy the following pair of Stokes equations:


∂p1 = δ, ∂x ∂p − ∆u 2 + 2 = 0, ∂x

− ∆u1 +

∂p1 = 0, ∂y ∂p − ∆v 2 + 2 = δ, ∂y

− ∆v1 +

185

div (u1 , v1 ) = 0 (4) div (u 2 , v 2 ) = 0

Here δ denotes the Dirac distribution concentrated to the origin. The fundamental solution can be expressed in the following form (see [1,2]):

u1 ( x, y ) = −

2y2 1 §¨ log( x 2 + y 2 ) + 1 + 2 8π ¨© x + y2

· ¸, v1 ( x, y ) = 1 ⋅ 2 xy , ¸ 8π x 2 + y 2 ¹

2x 2 1 2 xy 1 § , v 2 ( x, y ) = − ¨ log( x 2 + y 2 ) + 1 + ⋅ u 2 ( x, y ) = 2 8π x 2 + y 2 8π ¨© x + y2

· ¸, ¸ ¹

p1 ( x, y ) =

1 2x ⋅ 4π x 2 + y 2

(5)

1 2y p 2 ( x, y ) = ⋅ 4π x 2 + y 2

The a priori unknown coefficients a (j1) , a (j2) can be computed by taking the boundary conditions (2) into account i.e. by solving the following linear system of equations: N (1) N x j , y k − ~y j ) + ¦ a (j2) u 2 ( x k − ~ x j , yk − ~ y j ) = u 0 ( xk ) ¦ a j u1 ( x k − ~

j =1 N

¦

j =1

j =1

a (j1) v1 ( x k

N −~ x j , yk − ~ y j ) + ¦ a (j2) v 2 ( x k − ~ x j , yk − ~ y j ) = v0 ( x k )

(6)

j =1

( k = 1,2,..., N ). Since the fundamental solution has a singularity at the origin, the source points ( ~ xk , ~ yk ) must differ from the collocation points ( x k , y k ) . The usual choice is to define the source points outside of Ω . However, the system (6) becomes extremely ill-conditioned when the distances of the source points and the boundary increase. Note that this phenomenon appears also for other partial differential equations. This remains the case if the approximate solution is expressed with nonsingular solutions of the original problem (boundary knot method, see [3]) Table 1 shows the condition numbers of the system (6) for the model problem when Ω is the unit circle; here N denotes the number of the (equally spaced) collocation points. The source points are also equally spaced along the circle centered at the origin with radius (1 + d ) . The condition number increases extremely quickly when N and especially d increase. In the next section, a regularization technique is presented which results in far less ill-conditioned systems. d \ N 0.01 0.02 0.05 0.10 0.20 0.50 1.00

8 3.1 3.7 5.3 9.9 35.6 201.0 480.4

16 7.4 10.2 23.0 131.8 355.0 1.9E+3 1.3E+5

32 20.7 36.4 363.0 1.6E+3 1.8E+3 1.3E+6 9.4E+9

64 73.8 262.0 6.4E+3 3.3E+4 6.9E+5 6.1E+11 4.5E+17

128 531.9 3.4E+4 9.1E+4 2.1E+6 8.5E+10 5.2E+16 1.1E+19

256 6.5E+4 1.8E+5 5.2E+6 4.3E+11 6.1E+17 3.1E+19 6.8E+19

Table 1. Condition numbers of the system (6). N is the number of collocation points, d is the distance of the source points from the boundary

The Regularized Method of Fundamental Solutions applied to the Stokes equations Here we use the technique proposed in [4,5]. The Laplace operators in Eq. (1) are approximated by the fourth-order differential operator ∆ ( I −

1 c2

∆ ) , where I denotes the identity operator and c is a carefully

chosen scaling constant. Then, instead of (1), the following system is to be solved:

186


∂p 1 − ∆§¨ I − 2 ∆ ·¸u + = 0, c ∂x © ¹

∂p 1 − ∆§¨ I − 2 ∆ ·¸v + = 0, c ∂y © ¹

div (u , v ) = 0

(7)

supplied with the boundary conditions (2). Of course, the solution of (7)-(2) is not expected to be unique, since it is of fourth order, i.e. and extra boundary condition can also be prescribed. However, a particular solution can easily be obtained by using the MFS. A fundamental solution of (7) can be expressed in the following form (see [6] for details):

§ ∂2E ¨ 2 ¨ § u1 u 2 · 2 ¸¸ = c ⋅ ¨ ∂y G = ¨¨ 2 v v 2¹ © 1 ¨− ∂ E ¨ ∂x∂y ©

∂ 2 E ·¸ ∂x∂y ¸ ¸, ∂2E ¸ ¸ ∂x 2 ¹

−

§ ∂Φ · ¨ ¸ § p1 · ¨ ∂x ¸ p = ¨¨ ¸¸ = ¨ ¸, © p 2 ¹ ¨ ∂Φ ¸ ¨ ∂y ¸ © ¹

(8)

where E is the fundamental solution of the sixth-order operator ∆2 (∆ − c 2 I ) , and Φ denotes the harmonic fundamental solution. In polar coordinates: E (r ) = −

2 2 § ¨ K 0 (cr ) + log(cr ) + (cr ) log(cr ) − (cr ) ¨ 4 4 2πc ©

1

4

· ¸, ¸ ¹

Φ (r ) =

1 log r 2π

(9)

(where K 0 denotes the usual modified Bessel function of the third kind). Thus, a particular solution of (7) can be obtained similarly to (3): N N u ( x, y ) ~ ¦ a (j1) u1 ( x − x j , y − y j ) + ¦ a (j2) u 2 ( x − x j , y − y j ) j =1 N

v( x, y ) ~ ¦

j =1

j =1

a (j1) v1 ( x −

N x j , y − y j ) + ¦ a (j2) v 2 ( x − x j , y − y j ) j =1

(10)

N N p ( x, y ) ~ ¦ a (j1) p1 ( x − x j , y − y j ) + ¦ a (j2) p 2 ( x − x j , y − y j ) j =1

j =1

However, now the source points ( x j , y j ) and the collocation points are allowed to coincide, since the γ − log 2 § 1 0 · ¸¸ (where γ = 0.5772... , the matrix function G is continuous at the origin, and G (0,0) = ⋅ ¨¨ 4π ©0 1¹ Euler constant). This can be deduced from standard series expansions of the Bessel functions. Thus, the a

priori unknown coefficients a (j1) , a (j2) can be computed by solving the system: N (1) N ( 2) ¦ a j u1 ( x k − x j , y k − y j ) + ¦ a j u 2 ( x k − x j , y k − y j ) = u 0 ( x k )

j =1 N

¦

j =1

j =1

a (j1) v1 ( x k

N − x j , y k − y j ) + ¦ a (j2) v 2 ( x k − x j , y k − y j ) = v 0 ( x k )

(11)

j =1

The formula (10) can be considered also a special boundary interpolation technique (cf. [4,5]). Note that, the interpolated vector field (u , v) is automatically divergence-free everywhere, which is important in a lot of applications.

In general, the system (11) is much better conditioned than (6), as can be seen in Table 2. Here Ω is again the unit circle, N denotes the (equally spaced) collocation points, c is the scaling factor appearing in (7). The crucial point is the proper choice of the scaling factor c. If it is too low, the approximate solution (10) satisfies the original Stokes equation poorly; if it is too high, (10) produces numerical singularities at the boundary source points (cf. [5]), though the solutions of (7) approximate well the solution of the original problem (1)-(2) as shown in the next section.


c \ N 4 8 16 32 64 128 256 512

8 59.5 29.8 19.0 14.7 12.6 11.3 10.6 10.0

16 442.0 175.8 80.8 48.8 36.5 30.6 27.2 25.0

32 3.4E+3 1.2E+3 465.4 202.8 118.3 86.4 71.1 62.3

64 2.7E+4 9.8E+3 3.3E+3 1.1E+3 487.0 276.9 198.5 161.0

128 2.2E+5 7.8E+4 2.5E+4 8.2E+3 2.7E+3 1.1E+3 633.8 447.6

187

256 1.7E+6 6.2E+5 2.0E+5 6.3E+4 1.9E+4 6.4E+3 2.5E+3 1.4E+3

Table 2. Condition numbers of the system (11). N is the number of collocation points, c is the applied scaling factor Error estimation

Assume for simplicity that Ω = (0,2π) × (0,+∞) and consider only the solutions of (1) which are 2π periodic with respect to x. Then the solution of (1)-(2) can be expressed in terms of the stream function ψ ∂ψ ∂ψ as u = , v=− . The stream function is biharmonic, thus, has the following Fourier series expansion: ∂x ∂y ψ ( x, y ) = ¦ (α k + β k y ) e −|k | y e ikx

(12)

k

Along the boundary i.e. at y = 0 , the velocity components u, v are prescribed: u ( x,0) = ¦ u k e ikx , k

v( x,0) = ¦ v k e ikx , where the Fourier coefficients u k , v k are assumed to be known. Using the above series k

expansions, the coefficients α k , β k in (12) can easily be computed: αk =

i vk , k

βk = uk +

i|k| vk k

(k = 1,2,..., N ) ,

(13)

whence the exact solution (u * , v * ) of (1)-(2) can be expressed as:

u * ( x, y ) = ¦ (u k − | k | u k y − ikv k y ) e −|k | y e ikx k

(14)

v * ( x, y ) = ¦ (v k − iku k y + | k | v k y ) e −|k | y e ikx k

Similarly, the solutions of (7) can be expressed with the help of the stream function again. The stream function ψ now solves the sixth-order equation ∆2 (∆ − c 2 I )ψ = 0 .

(15)

As can easily be checked, ψ has the following Fourier series expansion: 2 2 º ª ψ ( x, y ) = ¦ «(α k + β k y ) e −|k | y + δ k e − k +c y » e ikx , k ¬ ¼

(16)

Along the boundary i.e. at y = 0 , the velocity components u, v are prescribed, as earlier. However, a third boundary condition is also required, since Eq. (15) is of sixth-order. Assume for simplicity that ∆ψ ( = − rot (u , v ) ) is prescribed along the boundary: ∆ψ ( x,0) = ¦ wk e ikx . Substituting into (16), we obtain that k

the coefficients α k , β k , δ k satisfy the following system of equations for all indices k:

188


− | k | α k + βk − k 2 + c 2 δk = uk ,

− ikα k − ikδ k = v k ,

− 2 | k | β k + c 2 δ k = wk .

(17)

( k = 1,2,..., N .) The inverse of the matrix A of Eq. (17) can be calculated directly, which implies:

§αk · § uk · ¸ ¨ ¸ −1 ¨ ¨ β k ¸ = A ¨ vk ¸ , ¨δ ¸ ¨w ¸ © k¹ © k¹

(18)

where § ¨− 2 | k | ¨ 1 A −1 = ⋅ ¨¨ c 2 2k 2 + c 2 − 2 | k | k 2 + c 2 ¨ 2|k | ¨¨ ©

i k

§c2 − 2 | k | k 2 + c2 · ¨ ¸ © ¹ i 2 |k |c k 2i k

· ¸ ¸ k 2 + c 2 − | k | ¸¸ . ¸ 1 ¸¸ ¹ −1

(19)

∂ψ ∂ψ , v=− can be expressed in ∂y ∂x terms of the Fourier coefficients of the boundary conditions. After considerable long calculations, the

Now the stream function ψ as well as the velocity components u =

differences u − u * and v − v * i.e. the error of the approximation (7) can be estimated in L2 -norm in the following way: Theorem 1: There exists a constant C ≥ 0 independent of the c and the boundary conditions such that:

· C § 1 ¨¨ ¦ | k | ⋅ | u k | 2 + ¦ | k | ⋅ | v k | 2 + ¦ | wk | 2 ¸¸ = || u − u * || 2L (Ω ) ≤ 2 k k |k| c2 © k ¹ C § · 2 2 = + || v | Γ || 1 / 2 + || rot (u , v) | Γ || 2 −1 / 2 ¸ ¨ || u | Γ || 1 / 2 H H H (Γ ) (Γ ) (Γ) ¹ c2 ©

(20)

and similarly:

C § · 2 + || v | Γ || 2 1 / 2 + || rot (u , v) | Γ || 2 −1 / 2 ¸ . || v − v * || 2L (Ω ) ≤ ¨ || u | Γ || 1 / 2 2 (Γ ) (Γ ) (Γ) ¹ H H H c2 ©

(21)

From a practical point of view, the theorem says that if c is great enough, then the solutions of (7), (2) approximate well the (unique) solution of (1)-(2) independently of the boundary condition taken to rot (u , v ) . However, if (7) is solved by the MFS i.e. in the form of (10), then c should not be too great in order to avoid numerical singularities. As a rule of thumb, c should be inversely proportional to the characteristic distance of the collocation points at the boundary, as can be seen also in the simple example presented in the next section. A numerical example

As an illustrative example, consider a rotating Stokes flow in the unit circle. The stream function is as 1 2

follows: ψ (r ) = r 2 . Consequently, the velocity components are u (r , t ) = r sin t , and v(r , t ) = − r cos t (in polar coordinates). Table 3 shows the relative L2 -errors of the approximate solutions defined by the regularized MFS i.e. in the form of (7). N again denotes the number of the (equally spaced) collocation points on the boundary, and c denotes the scaling parameter appearing in (7). It can clearly be seen that the


optimal value of the scaling parameter is proportional to N i.e. inversely proportional to the distance of the neighboring collocation points. c \ N 4 8 16 32 64 128 256 512

16 12.76 8.820 4.054 1.211 4.776 8.490 11.95 15.17

32 12.89 9.248 5.238 2.243 0.419 2.367 4.384 6.319

64 12.90 9.305 5.434 2.824 1.174 0.168 1.192 2.243

128 12.91 9.312 5.461 2.919 1.462 0.598 0.060 0.596

256 12.91 9.313 5.464 2.932 1.509 0.743 0.302 0.023

Table 3. Relative L2 -errors (%) of the approximate solution of the test problem computed by the regularized MFS. N is the number of collocation points, c is the applied scaling factor Summary and conclusions

A meshless technique for solving the permanent Stokes equations has been presented. Instead of the original Stokes equations, a fourth-order approximate system of partial differential equations is to be solved. This was performed by the Method of Fundamental Solutions. However, the fundamental solution of the approximate system is continuous at the origin, which allows the source and collocation points to coincide. The use of sources located outside of the domain is thus avoided, resulting in much less illconditioned algebraic problems. The success of the technique depends on the proper choice of the scaling parameter of the method. It has been shown that the solution of the applied fourth-order system approximates well the solution of the original Stokes problem (independently of the boundary condition taken to rotation of the velocity field along the boundary), provided that the scaling parameter is great enough. However, to avoid the appearance of numerical singularities at the source points, a limitation of the scaling parameter is necessary. To achieve an optimal compromise, the value of the scaling parameter should be inversely proportional to the characteristic distance of the boundary source points. Acknowledgement: The research was partly supported by the European Union (co-financed by the European Regional Development Fund) under the project TÁMOP-4.2.2-08/1-2008-0021. References

[1] C.J.S.Alves, A.L.Silvestre Density Results Using Stokeslets and a Method of Fundamental Solutions for the Stokes Equations. Engineering Analysis with Boundary Elements, 28, 1245-1252 (2004). [2] D.L.Young, S.J.Jane, C.M.Fan, K.Murugesan, C.C.Tsai The Method of Fundamental Solutions for 2D and 3D Stokes Problems. Journal of Computational Physics, 211/1, 1-8 (2006). [3] W.Chen Symmetric boundary knot method. Engineering Analysis with Boundary Elements, 26/6, 489494 (2002). [4] C.Gáspár A Meshless Polyharmonic-type Boundary Interpolation Method for Solving Boundary Integral Equations. Engineering Analysis with Boundary Elements, 28, 1207-1216 (2004). [5] C.Gáspár A multi-level regularized version of the method of fundamental solutions. In: The Method of Fundamental Solutions – A Meshless Method. (eds: C.S.Chen, A.Karageorghis, Y.S.Smyrlis). Dynamic Publishers, Inc., Atlanta, USA, 145-164 (2008). [6] C.Gáspár Several Meshless Solution Techniques for the Stokes Flow Equations. In: Progress on Meshless Methods (eds: A.J.M.Ferreira, E.J.Kansa, G.E.Fasshauer), Computational Methods in Applied Sciences, Vol. 11, Springer, 141-158 (2009).

189

190


A Boundary Element Formulation based on the Convolution Quadrature Method for the Quasi-static Behaviour Analysis of the Unsaturated Soils Pooneh Maghoul 1, Behrouz Gatmiri 2, 3, Denis Duhamel 1 1

Université Paris-Est, UR Navier, LAMI, Ecole des Ponts, Champs sur Marne, France E-mail : [email protected]; E-mail : [email protected] 2

3

University of Tehran, Departments of Civil Engineering, Tehran, Iran Université Paris-Est, UR Navier, CERMES, Ecole des Ponts, Champs sur Marne, France E-mail : [email protected]

Keywords: Boundary element method; Boundary integral equations; Fundamental solution; Convolution quadrature method; Unsaturated soil; Porous media; Consolidation analysis

Abstract. This paper aims at obtaining an advanced formulation of the time-domain Boundary Element Method (BEM) for two-dimensional consolidation analysis of unsaturated soil. Unlike the usual timedomain BEM the present formulation applies a Convolution Quadrature developed by Lubich [1,2] which requires only the Laplace-domain instead of the time-domain fundamental solutions. Introduction In compacted fills or in arid climate areas where soils are submitted to wetting-drying cycles such as groundwater recharge, surface runoff and evapo-transpiration, fine-grained soils are not saturated with water, and contain some air. Due to capillary effects and soil-clay adsorption, the pore water is no more positive, and is submitted to suction. Prediction and simulation of unsaturated soil behaviour are of great importance in making critical decisions that affect many facets of engineering design and construction and, therefore, have been the issue of growing concern for several decades. From the mechanical point of view, an unsaturated porous medium can be represented as a three-phase (gas, liquid, and solid), or three-component (water, dry air, and solid) system in which two phases can be classified as fluids (i.e. liquid and gas). The liquid phase is considered to be pure water containing dissolved air and the gas phase is assumed to be a binary mixture of water vapor and ‘dry’ air in a non-isothermal case. In this paper first of all, the set of fully coupled governing differential equations of a porous medium saturated by two compressible fluids (water and air) subjected to quasi-static loadings is obtained. These phenomenal formulations are presented based on the experimental observations and with respect to the poromechanics theory within the framework of the suction-based mathematical model presented by [3,4]. In this model, the effect of deformations on the suction distribution in the soil skeleton and the inverse effect are included in the formulation via a suction-dependent formulation of state surfaces of void ratio and degree of saturation. The linear constitutive law is assumed. The mechanical and hydraulic properties of porous media are assumed to be suction dependent. In this formulation, the solid skeleton displacements ui , water pressure pw and air pressure pa are presumed to be independent variables. Secondly, the Boundary Integral Equation (BIE) is developed directly from those equations via the use of the weighted residuals method for the first time in a way that permits an easy discretization and implementation in a numerical code. The associated fundamental solution obtained by [5] (in both Laplace transform and time domains) is used in the BIE. Since the corresponding time-domain fundamental solution do not have a simple mathematical structure, the convolution integral appeared in the BIE seems too difficult, even impossible, to be performed analytically. For this reason, a new approach so-called “Operational Quadrature Methods” developed by [1,2] can be used to evaluate the convolution integrals. In this formulation, the convolution integral is numerically approximated by a quadrature formula whose weights are determined by the Laplace transform of the fundamental solution and a linear multistep method


191

[6]. The resulting BEM time domain formulation represents the first of its kind for two-dimensional consolidation problems. Governing Equations Governing differential equations consist of mass conservation equations of liquid and gaseous phases, the equilibrium equation of skeleton associated with water and air flow equations and constitutive relation. The assumption of infinitesimal transformation and incompressibility of solid matrix is considered. Solid Skeleton. The equilibrium equation and the constitutive law for the soil’s solid skeleton including the effects of suction are written [3]: V ij G ij pa , j pa , i f i 0 (1) V ij G ij pa

OG ij H kk 2PH ij Fijs pa pw

where P , O are Lame coefficients, pD the suction modulus matrix

w, a

(2) is the water or air pressure, G ij is the Kronecker delta and F is s ij

D.D suc 1 In which D suc is a vector obtained from the state surface of void ratio independent variables of V pa and pa pw . Fs

D suc 1

e

(3) which is a function of the

we / 1 e w pa pw

(4)

The elasticity matrix D can be presented by using the bulk modulus and the tangent modulus D D K 0 , Et D V pa , pa pw Where Et is tangent elastic modulus which can be evaluated as El Es

Et

(5) (6)

El is the elastic modulus in absence of suction and ms pa pw

Es

(7)

ms being a constant, Es represents the effect of suction on the elastic modulus. K 0 is the bulk modulus of an open system and evaluated from the surface state of void ratio 1 e we / w V pa (8) Continuity and transfer equations for water. A combination of generalized Darcy’s law for water transfer and conservation law for water mass, leads to the general equation for water transfer. The water velocity, uw , is defined as u w K w pw / J w z (9)

K 01

where J w is water unit weight. K w aw 10D w e > S r S ru / 1 S ru @ is the water permeability in which S ru is residual degree of saturation. The mass conservation law for water unit volume is written as w U w nS r / wt div U w uw 0 (10) 3.5

Continuity and transfer equations for air. Considering the generalized Darcy’s law, the air flow equation can be given as: ua K a pa / J a z (11) where K a Da J a > e 1 S ru @ a / P a is the air permeability in which Pa and e are air dynamic viscosity and void ration, respectively. Applying the mass conservation law for air, the air transfer equation will be w U a n 1 S r / wt div U a ua 0 (12) in which U a is air density and n stands for porosity. Summery of the field equations in Laplace transformed domain. By introducing (2) into (1), (9) into (10), (11) into (12), and by applying the Laplace transformation in order to eliminate the time variable of the partial differential equations, we can write compactly the transformed coupled differential equation system into the following matrix form: T T B >u p p @ > f 0 0@ 0 (13) E

i

w

a

i

with the not self-adjoint operator B :

192


1 F s w i F s wi ª P ' G i j O P w i w j º « » ˆ ˆ c U w k w ' s U w n g1 s U w Sr w j s U w nˆ g1 « » «¬ U a kac ' s U a nˆ g1 »¼ s U a 1 Sˆr w j s U a nˆ g1 where s is the Laplace parameter, kwc kw / J w and kac ka / J a .

B

(14)

In equation (14), i and j vary from one to four. The partial derivative

,i

is denoted by w i and ' w i i is

the Laplacian operator. Note the operators B in (14) are not self adjoint. Therefore, for the deduction of fundamental solutions the adjoint operator to B has to be used: s U w Sˆr w j s U a 1 Sˆr w j º ª P ' G i j O P w i w j » s U a nˆ g1 F s wi U w k wc ' s U w nˆ g1 (15) B « « » ˆ 1 s U w ng 1 F s w i U a kac ' s U a nˆ g1 ¼» ¬« Fundamental Solutions Here, the fundamental solution associated with the operator (15) is derived in the Laplace transform domain. Mathematically spoken a fundamental solution is a solution of the equation BG IG x y G t W 0 where the matrix of fundamental solutions is denoted by G , the identity matrix by I and the matrix differential operator by B . These solutions can be used in a time-dependent convolution quadrature-based BE formulation which needs only Laplace transformed fundamental solutions. In this study, because the operator type of the governing equations is an elliptical operator, the explicit 2D Laplace transform domain fundamental solution can be derived by using the method of Kupradze et al. [7] or Hörmander [8]. An overview of this method is found in the original work by [8] and more exemplary in References [5,9,10,11,12,13]. The components of fundamental solution tensor are obtained as follows:

Gij

ª 2r,i r, j Gij O K O r O K O r r r O 2 K O r O 2 K O r º ½ 2 1 2 1 1 1 ,i , j 2 0 2 1 0 1 ° ¬« ° ¼» r ¾ 2 2 ® g nkc kc G r r 2 2S O 2 P O2 O1 ° 1 a w .s ª ,i , j ij K1 O2 r K1 O1r r r K O r K O r º ° ,i , j 0 2 0 1 ¯ ¬« ¼» ¿ O2 O1 kwc kac r

G i 3

r,i § g1n .s 2 K1 O2 r K1 O1r S .s O K O r O K O r · ¨ ¸ r 2 1 2 1 1 1 ¹ 2S O 2 P kwc O22 O12 © kac O2 O1

G i 4

r,i § g1n .s 2 K1 O2 r K1 O1r 1 S .s O K O r O K O r · ¨ ¸ 2 1 2 1 1 1 r ¹ O2 O1 2S O 2P kac O22 O12 © kwc

G 3 j

r,i § g1n .s K1 O2 r K1 O1r F s O K O r O K O r · ¨ ¸ 2 1 2 1 1 1 ¹ 2S O 2 P U w kwc O22 O12 © kac O2 O1

G 33

G 34

F s 1 S r 1 g1n .s K 0 O2 r K 0 O1r 2S U w kwc kac O22 O12 O 2 P

G 4 j

r,i § 1 F s O K O r O K O r g1n .s K1 O2 r K1 O1r · ¨ ¸ 2 1 2 1 1 1 ¹ 2S O 2P U a kac O22 O12 © kwc O2 O1

G 43

1 F s S r 1 1 g1 .n s K 0 O2 r K 0 O1r O22 O12 2S U a kwc kac O 2P

G 44

1

(16b)

(16c)

(16d)

s 1 § O 2 K O r O 2 K O r § 1 F 1 S r g1n · .s K O r K O r · ¨ ¸ ¸ 2 0 2 1 0 1 0 2 0 1 2 2 ¨ c © O 2 P kac ¹ 2S U w k w O2 O1 © kac ¹

(16e) (16f)

(16a)

F s Sr gn 1 § O 2 K O r O 2 K O r § 1 ·¸ .s K 0 O2 r K 0 O1r ·¸ ¨ 2 0 2 1 0 1 2 2 ¨ © O 2P kwc ¹ k wc ¹ 2SU a kac O2 O1 ©

(16g) (16h) (16i)


193

in which K 0 Oi r is the modified Bessel function of the second kind of order zero with the argument r x [ which denoted the distance between a load point and an observation point. In the above Laplace transform domain fundamental solutions, i.e. eqs (27), Gij is the displacement of the solid skeleton in the ith direction due to unit force in the jth direction. Whereas G i 3 and G i 4 are the displacement of the solid skeleton in the ith direction due to a unit rate of water and air injection, respectively. Also, G 3 j and G 4 j are, respectively, the water pressure and air pressure due to the unit force applied in the jth direction. G 33 and G 34 are the water pressure due to a unit rate of water and air injection, respectively. Also, G 43 and G 44 are the air pressure due to a unit rate of water and air injection, respectively. Boundary Element Formulation To the authors’ knowledge, the boundary integral equations for quasi-static unsaturated poroelasticity have not yet been obtained. The boundary integral equations for this problem will be derived by taking the fundamental solution as the weighted function and using the method of weighted residuals, which is essentially an integration by parts technique. In this method, the integral equation is derived directly by equating the inner product of eq (13) and the matrix of the adjoint fundamental solutions G implying that IG x [ 0 B G (17) to a null vector, i.e. ª uD º ³: B « p w » G d: ¬ pa ¼

with

0

G

ªGDE GD 3 GD 4 º «G G G » 34 « 3 E 33 »

¬«G4 E G43 G 44 »¼

s w a ªU DE U D U D º « P s P w P a » « wsE ww wa » Pa ¼ ¬ Pa E Pa

(18)

where the integration is performed over a domain : with boundary * and vanishing body forces and sources are assumed. After integrating by parts twice over the domain according to the theory of Green’s formula and using partial integration, the operator B is transformed from acting on the vector of unknowns % . This yields the following system of integral > uD p w p a @T to the matrix of fundamental solutions G equations in index notation as

³* > Ouk ,k Fs p a p w p a nE GDE P uE ,D uD ,E nE @ GD j d* ³ uD > O G kj , k nE GDE P GD j , E G E j ,D nE @ d* * U k c p G p G d*

³

a a *

a,n

4j

a

(19)

4 j,n

U w kwc ³ p wG 3 j , n p w, n G 3 j d* s U a 1 Sˆr ³ G 4 j uD nE GDE d* s U w Sˆr ³ G 3 j uD nE GDE d* * * *

G mj ³ ui Bim d: 0 :

By substituting Eq (17) into (19) and using the property of Dirac’s delta function G x [ , we reach the transformed quasi-static unsaturated poroelastic boundary integral representation for the transformed internal displacements and pressures given in matrix form, i.e., ª uD º cI « pw » «¬ p »¼ a

S wS aS ªU DE PD PD º ª tD º «U W P wW P aW » « q » d * ³* « E »« w» «¬ U EA P wA P gA »¼ ¬ qa ¼

S ªTDE «T W ³* « E «¬ T A

E

QDwS QDaS º u ª Dº Q wW Q aW »» « p w » d * « » Q wA Q aA »¼ ¬ p a ¼

where the traction vector, the normal water flux and the normal gas flux are respectively tD V DE nE O uk , k Fs p a p w p a nE GDE P uE ,D uD , E nE qw U w k wc p w, n qa U a kac p a , n

(20)

(21a) (21b) (21c)

Also the T S , Q wS and Q aS in eq (20) can be interpreted as the adjoint terms to the traction vector tD , the water flux qw and the air flux qa as follows TDES W

TD

> OU kSE ,k s U w Sˆr PEwS s U a 1 Sˆr PEaS GD l P UDES ,l U lSE ,D @ nl > OU kW, k s U w Sˆr P wW s Ua 1 Sˆr P aW GD l P UDW,l U lW,D @ nl

(22a) (22b)

194

TDA Q wS


> OU kA, k s U w Sˆr P wA s U a 1 Sˆr P aA GD l P UDA,l U lA,D @ nl

D

Q wW Q wA

(22c)

U w kwc PDwS, n

(22d)

U w kwc PDwW ,n

(22e)

U w kwc PDwA ,n

(22f)

QDaS Q aW

U a kac PDaS, n

(22g)

U a kac P, naW

(22h)

Q aA

U a kac P, naA

(22i)

The coefficient cij in eq (20) has a value G ij for points inside : and zero outside : . The value of cij for points on the boundary * is determined from the Cauchy principal value of the integrals. It is equal to 0.5G ij for points on * where the boundary is smooth. Eq (20) can be compacted in index notation for the 2-D case as follows c [ I u j [ ; s ³ G ij x, [ ; s ti x; s Fij x, [ ; s ui x; s d * (23) * where ti

>tD

G i j

q w

ªU DE PD «U W P wW « E A wA ¬« U E P S

wS

qa @ , ui PDaS º P aW »» ; Fij P aA ¼» T

>uD

ªTDE «T W « E A ¬« T S

E

T p w p a @ and also wS aS QD QD º Q wW Q aW »» Q wA Q aA ¼»

(24)

The time dependent boundary integral equation for the unsaturated soil is obtained by a transformation to time domain. t

³ ³ G

c [ I u j [ ; t

0

ij

*

t

W ; x, [ , ti W ; x Fij t W ; x, [ ui W ; x d *

(25)

Equation (25) is an exact represent of the quasi-static response of a multiphase porous medium, involving integrations over the surface as well as the time history. For the practical problem, suitable approximations are needed for both the spatial and temporal variations of field variables. As will be shown, temporal integrations of the time functions involved will be performed numerically using an operational convolution quadrature method (CQM), as like as the spatial integration which be evaluated using numerical techniques. The salient features of the temporal and spatial integrations are outlined below. Temporal integration. Because of the complexity of the time-dependent fundamental solution presented by [5], the convolution quadrature method (CQM) (see References [1,2]) is used. In this formulation, the convolution integral is numerically approximated by a quadrature formula whose weights are determined by the Laplace transform of the fundamental solution and a linear multistep method [6]. By applying this method, the convolution integrals between the fundamental solutions and the nodal values in eq (25) are approximated by t

³G

Gij ti

ij

0

Fij ui

³

t

0

t

W ; x, [ .t j W ; x dW

N

¦ Z

G ij

N n 1

.t nj

(26a)

n 1

Fij t W ; x, [ .u j W ; x dW

N

¦ Z

F ij

N n 1

.u nj

(26b)

n 1

within each time step, the field variable (displacement/pressure and traction/flux) is assumed to remain N n 1 N n 1 constant. In eqs (26) G and F are the influence function which are defined by ZijG ZijF

m

m

x, [ x, [

m L

¦ G

m L

¦ F

L 1

2S i . ml / L

(27a)

2S i .ml / L

(27b)

ij

x, [ ; sl e

ij

x, [ ; sl e

l 0

L 1 l 0

2S il / L

/ 't . In eqs. (27) sl is given by sl J e By substituting eqs (27) into Equation (25), the time-convoluted boundary element equation is:

ci j uiN ([ )

N

¦³ n 1

*

(ZiGj ) N n 1 ( x, [ ).tin ( x)d * ³ (ZiFj ) N n 1 ( x, [ ).uin ( x)d *

*

(28)


195

Spatial integration. Using isoparametric quadratic elements and assuming a quadratic variation over both geometry and field variables, the functions (displacements and tractions) at any point over an element can be expressed in terms of the nodal values as ui ND K U ieD , ti ND K Ti De , pw ND K PweD , qw K

ND K Qwe D , pa

ND K QaeD where i 1, 2 for 2D and D

ND K PaeD and qa K

1, 2,3 for a quadratic element and ND (K ) are the shape functions in the local intrinsic coordinates (K ) of the element. Once the spatial discretization process described above has been accomplished, the nodal quantities can be brought outside the surface integrals of Eq (28), since now the integrands contain only known functions. Therefore, the discretized BE equation corresponding to point [ can be written as ci j uiN ([ )

^³ ¦¦ «¬ª¦ D N

E

n 1 e 1

3

*e

1

ZiGj

N n 1

`

3

ND (K )d * e (K ) . T jDe ¦ n

D 1

^³

*e

ZiFj

N n 1

`

n ND (K )d * e (K ) . U ejD º »¼

(29) n

where *e is the surface of the eth boundary element, E is the total number of boundary elements, UDe and e n

TD represent the nodal values at the node D of element e at the moment tn n't of U and T . The integrals which have to be evaluated over the isoparametric element, can be written in intrinsic coordinates. Then

ci j .uiN ([ )

N

E

3

^³ ¦¦ ¬«ª¦ D n 1 e 1

1

1

1

ZiGj

N n 1

`

3

ND (K ) J K dK . T jeD ¦ n

D 1

^³

1

1

ZiFj

N n 1

`

n ND (K ) J K dK . U ej D º ¼»

(30)

where J K is the Jacobian of transformation. The usual point collocations scheme, i.e. by allowing point [ to coincide sequentially with all the nodal points of the boundary, is used to establish a set of integral equations in order to obtain unknown boundary values. Also, writing systematically at each global boundary node produces a system of algebraic equations containing the generalized displacement and traction at all collocation points at time steps N of the form: c .u N ([ )

E

3

TD ¦¦ D e

e 1

where 'GieD

N

1

N n 1

³

1

1

N

'G De 1 UDe 'FDe ZiGj

N n 1

1

N 1 E

3

¦¦¦ TDe n 1 e 1D 1

ND (K ) J K dK and ' Fi De

n

'GDe

N n 1

³

N n 1

1

1

n

UDe 'FDe

ZiFj

N n 1

N n 1

(31)

ND (K ) J K dK .

Then, for each time steps it is sufficient to obtain 'GD and 'FD only for the current time steps. e

e

Conclusion In this paper, an advanced formulation of the time-domain Boundary Element Method (BEM) for twodimensional consolidation analysis of unsaturated soil is obtained. Unlike the usual time-domain BEM the present formulation applies a Convolution Quadrature developed by [1,2] which requires only the Laplacedomain instead of the time-domain fundamental solutions. References C. Lubich I. Numerische Mathematik 52, 129–145 (1988a). C. Lubich II. Numerische Mathematik 52, 413–425 (1988b). B. Gatmiri Final report of CERMES-EDF (1997). B. Gatmiri, P. Delage, M. Cerrolaza Adv. Eng. Software 29(1), 29–43 (1998). B. Gatmiri, E. Jabbari, Int. J. Sol. Struct. 42, 5971-5990 (2005a). M. Schanz and H. Antes Comput. Mech. 20, 452–459 (1997). V.D. Kupradze, T.G. Gegelia, M.O. Basheleishvili, T.V. Burchuladze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity (1979). [8] L. Hörmander Linear Partial Differential Operators (1963). [9] P.Maghoul, B.Gatmiri, D.Duhamel Int. J. Numer. Anal. Meth. Geomech. 34, 297–329 (2010). [10] B.Gatmiri, P.Maghoul, D.Duhamel Int. J. Solid. Struct. 47, 595-610 (2010). [11] B.Gatmiri, M.Kamalian Int. J. Geomech. 2(4), 381–398 (2002). [12] M.Kamalian, B.Gatmiri, M.J.Sharahi Commun. Numer. Meth. Eng. 24(9), 749-759 (2008). [13] E.Jabbari,B.Gatmiri Comput. Model. Eng. Sci. 18(1), 31-43 (2007). [1] [2] [3] [4] [5] [6] [7]

196


Elastodynamic laminate element method for lengthy structures E.V. Glushkov, N.V. Glushkova and A.A. Eremin Institute for Mathematics, Mechanics and Informatics Kuban State University, 350040, Krasnodar, Russia [email protected], [email protected] and [email protected] Keywords: laminate structures, path integral representations, wave propagation and diffraction

Abstract. Similarly as BEM, the laminate element method (LEM) is a special case of the method of fundamental solutions (MFS) in which the basis functions are constructed to meet identically all governing equations in sublayers and boundary conditions at the plane-parallel surfaces and interfaces of a layered sample considered. The paper gives a brief description of the method and a few numerical examples illustrating its advantages in the computer simulation of elastodynamic behavior of lengthy laminate structures. Introduction Many engineering structures are composed of large areas of flat or curved multilayered plates, e.g. aircraft wings and fuselages, multiple-glass panes, etc. In order to simulate their dynamic behavior conventional FE technique or finite difference discretization can be used. However, sufficient length (compared to thickness) and possible sharp difference among the elastic properties of sublayers are severe obstacles to the use of these methods. Sharp solution gradients require considerable mesh refinement leading to increased computational expenses and/or to the loss of numerical stability. Additional computational costs are also connected with a complex wave structure resulting from repeated reflections and refractions at the interfaces and defects. To overcome those difficulties various hybrid approaches such as a coupling of FEM results for areas adjacent to edges or inhomogeneities with the Lamb wave asymptotics are developed. However, there exists a way to avoid such a coupling using modified boundary elements aimed at lengthy elastic multilayered structures [1, 2]. In fact, it is a special case of meshless expansions in terms of fundamental solutions (MFS). Its main idea and difference from conventional MFS approximations is in the use of basis functions (boundary elements) in the form of fundamental solutions for the layered structure as a whole. Such basis functions called laminate elements (LEs) satisfy identically the governing equations in the sublayers and all interface and boundary conditions on the plane-parallel surfaces. Therefore, only conditions on the remaining part of the domain’s boundary are to be approximated by LEs. Owing to the semi-analytical form of LE representation and identical accounting for the interface and boundary conditions, this approach is especially convenient for elastodynamic simulation of long structures with highly contrast mechanical properties. The method has been implemented for both isotropic [1, 2] and anisotropic [3] laminates. Moreover, basing on the long-term experience in developing fast and stable algorithms of Green’s matrix calculation for stratified media that comes back to the 1980s [4], we can use LE approximations with functionally graded and fluid-filled porous-elastic materials, as well. Recent benchmark comparisons carried out with multiple-glass panes as an example has shown that a conventional FEM approach could not catch the peculiarities of stress concentration at contrast interlayers up to the absolute fail with too long samples. At the same time, the LEM results are stable disregard to the structure length and layer properties. As an illustration of LEM application, the dynamic deflections of loaded 2D and 3D lengthy glass-polymer plates as well as Lamb wave propagation and diffraction by inner inclusions are considered. The accounting for anisotropy is demonstrated by examples of wave energy spatial diagrams for wave fields generated by circle piezoelectric patch actuators in fiber reinforces laminate composite plates.


197

Laminate element method Let us consider time-harmonic oscillations ue−iωt of laminate structures; below the factor e−iωt is conventionally omitted. Classical boundary elements (BEs) intended for elastodynamic problems are constructed basing on the matrix of fundamental solutions (Green’s matrix) for an infinite homogeneous elastic space [5, 6]. Just as the BEs, the LEs are based on the fundamental matrices l(x, ξ), but derived for an infinite laminate plate or half-space as a whole. The column of the matrix are displacement vectors uj associated with point sources δ(x − ξ)ij directed along the coordinate unit vectors ij , j = 1, 2, 3. Here x = (x1 , x2 , x3 ) = (x, y, z) is a vector of spatial variables in the Cartesian coordinate system assigned by the vectors ij ; ξ = (ξ1 , ξ2 , ξ3 ) is a point of source location (LE center). The basis vectors i1 and i2 are taken to be parallel to the laminate’s boundary planes, while i3 is orthogonal to its surface. −1 With such a choice l(x, ξ) can be represented in terms of the inverse Fourier transform Fxy with respect to the horizontal coordinates x, y via the Fourier symbol L(α1 , α2 , z) = F[l(x)]: 1 −1 [L] ≡ 2 L(α1 , α2 , z)e−i(α1 x+α2 y) dα1 dα2 (1) l(x) = Fxy 4π Γ1 Γ2

The integration contours Γ1 and Γ2 go along the real axes α1 and α2 deviating conventionally into the complex planes just for rounding real poles ζk of the integrand. Eq. (1) is a start point for the development of fast and stable algorithms for the matrix l(x) calculation, which are of decisive importance for the LEM practical implementation. Here one encounters two rather independent groups of complexities: a) L calculation and b) numerical integration. To overcome the first of them we rely on recursive matrix algorithms coming back to the pioneering Haskell-Thomson-Petrashen’ solutions for elastic layered media. Their numerical stability is assured by excluding exponential factors from the diagonal matrix sub-blocks of a sparse block system to which the problem of L computation is reduced. In that way only nondegenerating exponentially free small blocks are to be inverted in the course of double-sweep recursive system solution. In more detail the algorithms are described in [2, 3]. A fast numerical integration of integrals (1) is a much more challenging problem. Even with contemporary powerful computers a straight numerical integration could hardly yield proper results. On the other hand, the level of computer expenses is radically reduced after certain analytical preprocessing using the complex variable theory, residual technique, special functions and asymptotics. Finally the integral representations for matrices l(x) are brought to close analytical formulas involving cylindrical Bessel functions (for 3D cases) and residuals of the matrix L elements at the real and complex poles ζk . The latter have to be found and calculated beforehand. Thus the crucial points allowing one to handle efficiently the LEs are the implementation of algorithms for matrix L computation together with fast and reliable methods of searching for complex poles and calculating residuals for functions given numerically. As soon as LEs obtained (i.e. there exists a code for l(x, ξ) efficient calculation) the displacement field u(x) caused in a laminate sample with possible local inhomogeneities (defects) by a given load q may be sought for in the form u = u0 + usc ,

usc ≈

N

lj (x)cj ,

(2)

j=1

where u0 is an incident field generated by q in the infinite laminate plate without defects, and usc is an additional field arising due to u0 scattering from the plate’s edges (for finite samples) and/or local defects (cracks, void, inclusions, surface irregularities, etc.). Whereas a close analytical representation for u0 is easily derived using the integral transform Fxy applied to the equations and boundary conditions for the infinite defectless structure, usc is approximated by a sum of LEs consisting of the fundamental matrices lj (x) = l(x, ξj ) and unknown vector coefficients cj . In accordance with the MFS general scheme the latter have to be chosen to minimize the

198

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz discrepancy of the boundary conditions on the surface of scatterers S. For that the LE centers ξj are allocated along S at a certain distance d from this surface. Since x = ξ j are singular points of the matrices lj (x), one must put ξ j outside the sample domain D, in which the field usc is approximated by LE sum (2). It is worthy to note that matrix l may be written as the sum l = l0 + l1 , where l0 (x, ξ) is the fundamental matrix for the homogeneous elastic space with the same properties as those of the sublayer that contains the source point ξ, while l1 (x) is composed of fields reflected from the interfaces and external sides of a laminate structure considered. Hence, the elements of matrix l0 bear the same singularity at x = ξ as that of the matrix of fundamental solutions used for classical BEs, while l1 (x) has no singularities. In that way, the most of methods derived for operating with singular BEs may be used for LEM computations as well. In particular, l(x, ξ) may be used as a singular matrix kernel of the boundary integral representation usc (x) = l(x, ξ)c(s)ds, (3) S

in which ξ = ξ(s) ran over the surface S. Its discretization, for example, in line with the s−s expansion of the unknown vector-function c(s) in terms of localized hat-splines ψj (s) = ψ( ∆sj ), ψ(t) = 1 − |t|, |t| < 1: N c(s) ≈ cj ψj (s), cj = c(sj ), j

leads to the same representation (2) in which lj (x) =

|s−sj |> 1,

−H

where er is the radial component of the Umov-Poynting power vector for a time-averaged wave energy flux in a harmonic field ue−iωt . In other words, these plots show the directivity of Lamb-wave energy radiation from the source to infinity versus the polar angle ϕ (x = r cos ϕ, y = r sin ϕ). Whereas with an isotropic plate the diagrams would be simple circles, the anisotropy results in preferable directions of energy outflow along the fiber stacking. Moreover, while at the frequency ω = 0.25 the energy is preferably radiated along the fibers of the upper ply, at the frequency ω = 1 the most part of energy flows within the interior sublayers that may not be visible from the measurements of surface displacements uz (x, y, 0) by a laser vibrometer. Acknowledgement The authors acknowledge the financial support from the Russian Ministry of Education and Science (project No 2.1.1/1231) and from the Russian Foundation for Fundamental Research (RFBR). References [1] E.V.Glushkov, N.V.Glushkova and D.V.Timofeev, A layered element method for simulation elastodynamic behaviour of laminate structures with defects. In monograph: Advances in Meshless Methods /edited by J. Sladek & V. Sladek, Tech Science Press, Forsyth, GA, USA, 17-36 (2006). [2] Ye.V. Glushkov, N.V. Glushkova, A.A. Yeremin and V.V. Mikhas’kiv, The layered element method in the dynamic theory of elasticity. Journal of Applied Mathematics and Mechanics, 73, 449–456 (2009). [3] Ye.V. Glushkov, N.V. Glushkova, A.S. Krivonos, Excitation and propagation of elastic waves in multilayered anisotropic composites. Journal of Applied Mathematics and Mechanics 74(3) (2010). [4] V.A. Babeshko, E.V. Glushkov, N.V. Glushkova, Methods of Green’s matrix calculation for a stratified elastic half-space. Zhurnal Vychislitelnoy Matematiki i Matematicheskoy Fiziki, 27(1), 93-101 (1987) (in Russian). [5] C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel, Boundary Element Techniques. SpringerVerlag (1984). [6] M. H. Aliabadi, The Boundary Element method – Applications in Solids and Structures, John Willy & Sons LTD., Vol. 2 (2002).

202


Three-dimensional eigenstrain formulation of boundary integral equation method for spheroidal particle-reinforced materials Hang Ma1, Qin-Hua Qin2 1

Department of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, P R China, [email protected] 2 Department of Engineering, Australian National University, ACT 0200, Canberra, Australia, [email protected] Keywords: eigen-strain, Eshelby tensor, spheroidal particle, boundary integral equation, iteration

Abstract. A computational model is presented using the proposed three-dimensional eigenstrain formulation for modeling the spheroidal particle-reinforced materials. The model and its solution procedure is based on the concept of equivalent inclusion of Eshelby while the eigenstrains in each inhomogeneity embedded in the matrix are determined using an iterative scheme. With the proposed model, the solution scale of the inhomogeneity problem can be significantly reduced as the unknowns are on the boundary of the solution domain only. Using the algorithm, the stress distributions and the overall elastic properties are identified using the boundary element method (BEM) for spheroidal particle-reinforced inhomogeneous materials over a representative volume element. The performance and efficiency of the proposed computational model are assessed through several examples. 1. Introduction The determination of elastic behaviour of an embedded inclusion is of considerable importance in a wide variety of physical and engineering problems. Since the pioneer work of Eshelby [1], inclusion and inhomogeneity problems have been a focus of solid mechanics for several decades. Following Eshelby’s idea of equivalent inclusion and eigenstrain solution, quite a diverse set of research work has been reported analytically [2-5] and/or numerically [6-13] in the literature. The eigenstrain solution can represent various physical problems where eigenstrain may correspond to thermal strain mismatches, strains due to phase transformation, plastic strains or fictitious strains arising in the equivalent inclusion problems, overall or effective elastic, plastic properties of composites, quantum dots, microstructural evolution, as well as the intrinsic strains in the residual stress problems [14]. The analytical models available in the literature can be the basis for understandings to predict the stress/strain distribution either within or outside the inhomogeneity and for further research of the inclusion problem. However, these analytical solutions were obtained generally for problems with simple geometries only, such as single ellipsoidal, cylindrical and spheroidal inclusion in an infinite domain. Therefore, numerical simulations with finite element methods (FEM), volume integral methods (VIM) or BEM have been used in the analysis of inhomogeneity problems with various shapes and materials. The FEM may produce results for the whole composite materials, including results inside the inhomogeneity [7], but the solution scale would be large since both the matrix and every inhomogeneity should be discretized. The VIM and the BEM seem to be more suitable for the solution of the inhomogeneity problems in comparison with the FEM. In the VIM [8-10], the zones of inhomogeneity are represented by the volume integrals, which will essentially simplify the construction of the final matrix of the linear algebraic system to which the problem is reduced to some extent after the discretization. However, as the interfaces need to be discretized in the VIM, it is suitable only for small scale problems with a few inhomogeneities. The situation in the application of BEM to inclusion problems, often coupled with the VIM [11,12], is much the same with that of the VIM in


203

which the problems of simple arrays of inclusions were solved in small scales owing to the reason similar to that in the VIM, i.e., the unknown appearing in the interfaces. For large scale problems of inmohogeneity with BEM [13], special techniques of the fast multipole expansions [15] should be employed, which leads to complexity of the solution procedure. To the authors’ knowledge, Eshelby’s idea of equivalent inclusion and eigenstrain solution has not yet been fully utilized in the area of numerical study of the inhomogeneity problems [16]. Based on the Eshelby’s idea, the authors recently proposed the eigenstrain formulation of the BIE for modeling particle-reinforced materials in two-dimensional elasticity [17]. In the present work, the computational model is extended to the three-dimensional case for modeling spheroidal particle-reinforced materials at this initial stage and solved by the BEM [18]. 2. Eigen-strain formulation of BIE In the present model, The particle and matrix are assumed to be isotropic and bonded perfectly so that the displacement continuity and the traction equilibrium remain along their interfaces. The problem domain considered is a bounded region filled with the matrix and the inclusions surrounded by the outer boundary . The inhomogeneous zones in the domain are denoted by ,with the boundary ,(, =,). The displacement and the stress fields of the problem can be expressed by the eigenstrain formulations of the BIE [17] as follows: C p ui p

³W

j

*

C p Vij p

³W *

where O*ijkl ( p, q)

q uij* p, q d* q ³ u j q Wij* p, q d* q ¦ ³ H 0jk q Vijk* p, q d: q

k

(1)

:I

*

* * q uijk* p, q d* q ³ uk q Wijk* p, q d* q ¦ ³ H kl0 q V ijkl p, q d : q Hkl0 p Oijkl *

(2)

:I

* lim ³ xlW ijk p, q d * q . In eqs (1) and (2), p and q represent the source and field points,

:H o0 *H

respectively. u*ij, *ij and *ij stand for the Kelvin’s fundamental solutions for displacements, tractions and stresses, respectively. u*ijk, *ijk and *ijk are correspondingly the derived fundamental solutions [18]. with its boundary represents an infinitesimal zone around p within ,. In eqs (1) and (2), 0ij represent the eigenstrains of particles, which are determined in an iterative manner to be described in the next section. Obviously, the eigenstrains in each particle depend on the applied stresses or strains, the geometries as well as the material constants of the particle and matrix. Following the idea of Eshelby [1], the eigenstrains, or the stress-free strains, in a particle with material being identical to matrix, or the so called equivalent inclusion, without applied stress correlate the constrained strains Cij by the Eshelby tensor Sijkl as follows if the deformed particle has been placed back into the matrix: H ijC

Sijkl H kl0

(3)

The Eshelby tensor is only geometry dependent and generally takes the form of integrals. For simple geometries, the components of Sijkl can be given explicitly and are available in literatures [3,19]. For inhomogeneity problems, by defining the Young’s modulus ratio =EI/EM, where the subscripts I and M represent the inhomogeneity and matrix, respectively, a particle under applied strains ij to be replaced by an equivalent inclusion without altering its stress state, the following relation should hold true according to Hooke’s law:

1 E1 H ijC E 2G ij H kkC H ij0

vM 0 G ij H kk 1 2vM

1 E1 H ij E 2G ij H kk

(4)

204


where 1=(1+vM)/(1+vI), 2=vM/(1-2vM)-vI/(1-2vI), and v is Poisson’s ratio. Combining eqs (3) and (4), the eigenstrains in each particle can be predicted from the given applied strains. 3. Solution procedures The present computational model for spheroidal particle-reinforced materials is solved by the BEM. However, the domain integrals in eqs (1) and (2) need to be transformed into the boundary-type integrals [20] before discretization:

³V

* ijk d :

:I

³xW

* k ij d *

³V

(5a),

*I

³ xW

* * ijkl d : Oijkl

* l ijk d *

:I

(5b)

*I

In eq (5) the assumption that the eigenstrains in each particle are constant has been used. It is known that the generalized applied strain or the applied stress at each particle will be disturbed by other particles, especially those in the adjacent zone surrounding the concerned particle. As a result, the applied strains so as to the eigenstrains should be corrected in an iterative way in the solution procedure. After discretization and incorporated with the boundary conditions, eq (1) can be written in the matrix form as: Ax = b + B (6) where A is the system matrix, B the coefficient matrix for eigenstrains, b the right vector related to the known quantities on the outer boundary, x the unknown vector. is the eigenstrain vector of all the particles to be corrected in the iteration. It needs to be pointed out that the coefficients in A, B and b are all constants so that they need to be computed only once. At the starting point, the vector is assigned by initial values with the applied strains via the equations (2) at each position of the particles at the elastic state computed irrespective of particles. Then the unknown vector x can be computed by the following iterative formulae: x

k 1

k = A 1 b + B

(7)

where k is the iteration count. Define the maximum iteration error max=max̮(k+1)-(k)̮, which is the maximum difference of eigenstrain components between the two consecutive iterations. The convergent criterion in the present study is chosen as EMmax10-3. It should be addressed here that for the evaluation of applied stresses of certain particle at ,, eq (2) should be reformed by excluding the current particle as Vij p

³W *

k

q uijk* p, q d* q ³ uk q Wijk* p, q d* q *

NI

¦

H kl0 q

J 1, J z I

³ xW

* l ijk

p, q d * q ,

(8)

p:I

*J

because the stress state at the due place are generated, in addition to the applied load, by the disturbances of all other particles in the solution domain, where NI is the total number of particles. The flow chart of the algorithm is shown in Fig. 1.

x3

x2 x1 Figure 1: The flow chart of the algorithm

Figure 2: The representative volume element

4. Numerical examples A cube is chosen as the representative volume element (RVE) as shown in Fig. 2 with triply periodically


205

spaced spheroidal particles. The particle spacing is defined in Fig. 3a. The discretization is shown also in Fig. 3 for the outer boundary (b) and the interface in one octant (c), respectively. However, it needs to be pointed out that the interface discretization has no contribution to the degree of freedom of the problem for the present algorithm since the purpose of it is only for the numerical evaluation of domain integrals in eq (1) by boundary-type quadrature using eq (5) when the distances between p and q are relatively small. Otherwise, the one-point computing [21] can achieve enough accuracy as follows if the distances are relatively large: [3

[2

[1

(a) Particle spacing 2s and radius r0

(b) Outer boundary

(c) Interface in one octant

Figure 3: The discretization

³V

* ijk d :

* | V V ijk

³V

(9a),

:I

* ijkl d :

* | V V ijkl

(9b)

:I

where V stands for the volume of , and O*ijkl=0 if pę\(,Ĥ,). (a)

1.6

(b)

Triaxial tension EI/EM=0.01

0.8

Particle

Matrix

Exact VTT Vrr

0.4

1.0

1.2

1.4

1.2

Particle

Matrix Triaxial tension EI/EM=10

1.0

0.8

0.0 0.8

Exact VTT Vrr

1.4

1.2

Computed stresses

Computed stresses

1.6

1.6

0.8

1.0

1.2

r/r0

1.4

1.6

r/r0

Figure 4: Computed stresses across the interface of a soft (a) and a hard (b) spheroidal particle in triaxial tension Algorithm

Degree of freedom

CPU time (s)

Domain decomposition

1362

153̚155

Eigenstrain formulation

492

ζ10

Table 1: Comparison of the degree of freedom and CPU time for the RVE with a single particle

In order to assess the model with the eigenstrain formulation, the stresses across the interface of a single spheroidal particle in the RVE in triaxial tension are computed and compared with the exact solutions. The results are presented in Fig. 4 to show the validity and accuracy of the algorithm. It is interesting to see from Fig. 4 that the tangential stresses on the interface computed using the eigenstrain formulation take just the average values of the two sides, the particle and matrix. In the stress computation, the technique of distance transformation [22] is employed when the point p is very close to the interface. Nevertheless, the same problem with single particle can also be solved using the traditional BEM with the domain decomposition or sub-domain technique. The degree of freedom and CPU time of the two algorithms are listed in Table 1. The results are obtained by running the program on a desk-top computer (Intel Pentium Dual CPU, 1.60GNz), showing the efficiency of the eigenstrain formulation. Figs. 5a and 5b present the computed overall properties of the RVE and the CPU time, respectively, as a function of the total particle number, NI, while the relative particle sizes, r0/s (Fig. 2a) are kept constant. It is

206


seen from Fig. 5a that the computed overall properties become stable when NI reaches 103 and above. The degree of freedom for the eigenstrain formulation holds constant, say 492 in the calculation (Table 1), independent of NI. In contrast, the same problem is difficult to be solved using the domain decomposition algorithm on the desk-top computer.

2.5

Spheroidal particle Relative size, r0/s=0.7

2.0

(a)

E/EM (EI/EM=0.01) P/EM (EI/EM=0.01) E/EM (EI/EM=10) P/EM (EI/EM=10)

4

10

CPU Time (s)

Overall Properties

3.0

1.5 1.0

Spheroidal particles Modulus ratio, EI/EM =10

(b)

3

10

2

10

r0/s=0.4 r0/s=0.7

0.5 0.0

1

1

2

10

3

10

4

10 10 Total particle number, NI

1

10

2

10

3

4

10 10 Total particle number, NI

10

Figure 5: Overall properties of the RVE (a) and the CPU time (b) as a function of total particle number, NI

E/EM (EI/EM=10) E/EM (EI/EM=0.01) P/EM (EI/EM=10) P/EM (EI/EM=0.01)

1.6 Overall Properties

1.4 1.2 1.0 0.8 0.6

E/EM Poisson's ratio P/EM

1.2

(a) Overall Properties

1.8

Spheroidal particle Total number NI=1000

0.4 0.2

1.0

Spheroidal particle Total number NI=1000 Relative size r0/s=0.6

0.8 0.6

(b)

0.4 0.2

0.2

0.3

0.4 0.5 0.6 Relative particle size, r0/s

0.7

10

-3

10

-2

-1

0

10 10 10 Modulus ratio, EI/EM

1

10

2

10

3

Figure 6: Overall properties as a function of relative particle size, r0/s (a) and the modulus ratio, EI/EM (b)

The effect of the relative particle size, r0/s, and the modulus ratio, EI/EM, on the overall properties of the RVE are presented in Figs. 6a and 6b, respectively. It can be seen from Fig. 6a that the moduli increase monotonically with r0/s for hard particles but decrease for soft particles as expected. The elastic behavior of the overall properties with the variation of EI/EM in Fig. 6b are similar to those in the two-dimensional case [17] that the most effective range of EI/EM to the overall properties is between 0.1 and 10 while the stagnancy of properties are observed in ranges when EI/EM is very small or very large.

Iteration times

6

5

4

4

2

Spheroidal particle Total number NI=1000

0 0.2

0.3

(b)

(a) Iteration Times

E/EM (EI/EM=10) E/EM (EI/EM=0.01) P/EM (EI/EM=10) P/EM (EI/EM=0.01)

8

0.4 0.5 0.6 Relative particle size, r0/s

0.7

3 Spheroidal particle Total number NI=1000 Relative size r0/s=0.6

2

1

-3

10

10

-2

-1

Tension Shear 0

1

10 10 10 Modulus ratio, EI/EM

2

10

10

3

Figure 7: The convergence behaviors of the algorithm with respect to relative particle size, r0/s (a) and the modulus ratio, EI/EM (b)

The convergence behavior of the algorithm are presented in Fig. 7, showing that the iteration times varies with a number of factors such as the size of particles or volume fractions, the ratio of modulus as well as the loading manners, etc., which is considered to reflect the effects on the stress states at locations among


207

particles. However, the convergence can be generally achieved with a few iterations. The two principal factors need to be considered which influence the convergence behavior in the present algorithm. The first would be the mutual disturbances of the stress states among particles while the second would be the mismatches between particles and the matrix. 5. Conclusion The novel computational model and solution procedure are presented for particle-reinforced composites using the proposed three-dimensional eigenstrain formulation of the BIE and solved by the BEM in the present study. As the unknowns appear only on the boundary of the solution domain, the solution scale of the problem with the present model remains fairly small in comparison with the traditional algorithm using FEM or BEM. The tangential stresses on the interface computed using the eigenstrain formulation take just the average values of the two sides, the particle and matrix. The effectiveness and efficiency of the proposed model as well as the convergent performance of the solution procedure are assessed by several numerical examples. ___________________________________________________________________________ Acknowledgement: The work was supported by the National Natural Science Foundation of China (Grant No. 10972131).

References [1] J.D.Eshelby Proceedings of the Royal Society of London A241,376-396(1957). [2] T. Mura, H.M.Shodja, Y.Hirose Applied Mechanics Review 49(10),S118-S127(1996). [3] S.Federico, A.Grilloc, W.Herzog J. Mech. Phys. Solids 52,2309-2327(2004). [4] I.Cohen J. Mech. Phys. Solids 52,2167-2183(2004). [5] L.X.Shen, S.Yi. Inter. J. Solids Struct. 38,5789-5805(2001). [6] I.Doghri, L.Tinel Comp. Meth. Appl. Mech. Eng. 195,1387-1406(2006). [7] P.A.Kakavas, D.N.Kontoni Inter. J. Num. Meth. Engng. 65,1145-1164(2006). [8] S.K.Kanaun, S.B.Kochekseraii J. Comput. Physics 192,471-493(2003). [9] J.Lee, S.Choi, A.Mal. Inter. J. Solids Struct. 38,2789-2802(2001). [10] C.Y.Dong, Y.K.Cheung, S.H.Lo A regularized domain integral formulation for inclusion problems of various shapes by equivalent inclusion method. Comp. Meth. Appl. Mech. Eng. 191,3411-3421(2002). [11] C.Y.Dong, K.Y.Lee Boundary element analysis of infinite anisotropic elastic medium containing inclusions and cracks. Eng. Anal. Bound. Elem. 29,562-569(2005). [12] C.Y.Dong, K.Y.Lee Effective elastic properties of doubly periodic array of inclusions of various shapes by the boundary element method. Inter. J. Solids Struct. 43,7919-7938(2006). [13] Y.J.Liu, N.Nishimura, T.Tanahashi, X.L.Chen, H.Munakata ASME J.Appl. Mech. 72,115-128(2005). [14] H.Ma, H.L.Deng Adv. Eng. Software 29,89-95(1998). [15] L.F.Greengard, V.Rokhlin Journal of Computational Physics 73,325-48(1987). [16] Y.Nakasone, H.Nishiyama, T.Nojiri Mater. Sci. Eng. A285,229-238(2000). [17] H.Ma, C.Yan, Q.H.Qin Eng. Anal. Bound. Elem.33,410-419(2009). [18] C.A.Brebbia, J.C.F. Telles and L.C. Wrobel Boundary element techniques—theory and applications in engineering, Springer (1984). [19] M.Rahman ASME J.Appl. Mech.69,593-601(2002). [20] H.Ma, N.Kamiya JSCE J. Appl. Mech. 1,355-364(1998). [21] H.Ma, J.Zhou, Q.H.Qin Adv. Eng. Software 41,480-488(2010). [22] H.Ma, N.Kamiya Comput. Mech. 29,277-288(2002).

208


Green’s functions, boundary elements and finite elements Friedel Hartmann University of Kassel, Kurt-Wolters-Str. 3, D-34225 [email protected] Keywords: Green’s function, boundary elements, finite elements, stiffness matrices, adaptive refinement Introduction The Green’s function G(y, x) of the Poisson equation −∆u = p

in Ω

u=0

on Γ

(1)

allows to write the solution u(x) in terms of an influence function G(y, x) p(y) dΩy . u(x) =

(2)

Ω

In boundary element analysis we replace the Green’s function by a fundamental solution g(y, x) ∂u ∂g u(x) = [g(y, x) (y) − (y, x) u(y)] dsy + g(y, x) p(y) dΩy (3) ∂n ∂n Ω Γ and we determine the unknown boundary values by solving an integral equation approximately and so construct the BE-solution ∂uh ∂g uh (x) = [g(y, x) g(y, x) p(y) dΩy . (4) (y) − (y, x) uh (y)] dsy + ∂n ∂n Γ Ω Solving the problem (1) with finite elements means that we project the exact solution u(x) onto the solution and trial space Vh = {vh ∈ H1 (Ω)|vh (x) = 0, x ∈ Γ}

(5)

that is the FE-solution uh is the best approximation in the sense of the strain energy metric a(u − uh , u − uh ) ≤ a(u − vh , u − vh ) where

v h ∈ Vh

(6)

a(u, v) :=

Ω

(∇u · ∇v) dΩ .

(7)

Tottenham’s equation It seems that these two methods follow two totally different lines of reasoning but the surprising result is that also the FE-method is a Green’s function method. This result goes back to Tottenham who in a very early paper, [1], mentioned that the FE-solution uh has the form Gh (y, x) p(y) dΩy (8) uh (x) = Ω

where Gh (y, x) is the FE-Green’s function that is the projection of the exact Green’s function −∆y G(y, x) = δ(y − x)

G(y, x) = 0

y∈Γ

(9)


209

onto the space Vh . If the nodal shape functions ϕi (x) form a basis of Vh then Gh (y, x) is the function ui (x) ϕi (y) (10) Gh (y, x) = i

where the nodal values ui = ui (x) are the solution of the system K u(x) = f

kij = a(ϕi , ϕj )

fi = ϕi (x) .

The proof of Tottenham’s equation (8) is easy and can be done on one line. Namely we have δ(y − x) uh (y) dΩy = a(Gh , uh ) = Gh (y, x) p(y) dΩy . uh (x) = Ω

(11)

(12)

Ω

First uh is considered to be the FE solution of the right-hand side p and Gh ∈ Vh assumes the role of a virtual displacement Gh (y, x) p(y) dΩy (13) a(Gh , uh ) = Ω

next Gh is considered to be the FE solution of the right-hand side δ, and uh assumes the role of a virtual displacement uh (x) = δ(y − x) uh (y) dΩy = a(Gh , uh ) (14) Ω

which explains the left-hand side. The symmetric strain energy a(Gh , uh ) plays the role of a turnstile. The inverse stiffness matrix The approximate Green’s function for the displacement u(x) at a node xk has the form uGi (xk ) ϕi (y) . Gh (y, xk ) =

(15)

i

The vector uG = {uG1 , uG2 , . . . , uGn } is the solution of the n × n system K uG = e k

(unit vector ek ) ,

(16)

which means that the columns ck of the inverse stiffness matrix K −1 uG = K −1 ek = ck

(17)

are the nodal displacements which belong to the n Green’s functions Gh (y, xk ) of the n nodes xk . cki ϕi (y) = cTk ϕi (y) . (18) Gh (y, xk ) = i

This explains why the inverse of a tridiagonal matrix is fully populated. Even if only one node xk carries a point load P = 1 the whole structure deforms. An example provides the bar in Fig. 1 a, which consists of five linear elements. The stiffness matrix and its inverse are, see Fig. 1 b ⎡ ⎡ ⎤ ⎤ 2 −1 0 0 4 3 2 1 ⎢ EA ⎢ 2 −1 0⎥ 6 4 2⎥ ⎢ −1 ⎥ , ⇒ K −1 = l ⎢ 3 ⎥. K= (19) 2 −1 ⎦ 4 6 3⎦ l ⎣ 0 −1 5 EA ⎣ 2 0 0 −1 2 1 2 3 4 And evidently are the columns of the inverse the Green’s functions of the nodes.

210


Figure 1: a) Elastic bar subdivided into five linear elements b-e) the displacements are the columns of the inverse stiffness matrix (all values times l/(5 EA)). Boundary elements When the BE solution (4) is compared with the FE solution (8) there seems not to be much agreement between the two solutions. But the BE solution (4) is just the same formula Gh (y, x) p(y) dΩy (20) uh (x) = Ω

in disguise, of course with a different Gh . To see this note that the Green’s function G can be split into a fundamental solution G and a regular part uR , u(x) = G(y, x) p(y) dΩy = g(y, x) p(y) dΩy + uR (y, x) p(y) dΩy . (21) Ω

Ω

Ω

The regular part is the solution of the problem −∆uR = 0

in Ω

uR (y, x) = −g(y, x)

y ∈ Γ.

(22)

This splitting implies that the boundary integrals in the influence function (4) are just an equivalent expression for the work done by the distributed load p acting through the regular part uR : ∂G ∂u uR (y, x) p(y) dΩy = G(y, x) (y) dsy − (y, x) u(y) dsy . (23) ∂n Ω Γ Γ ∂ν Hence it can be assumed that the boundary integrals in the BE solution (4) play the same role, ∂uh ∂G uhR (y, x) p(y) dΩy : = G(y, x) (24) (y) dsy − (y, x) uh (y) dsy , ∂n Ω Γ Γ ∂ν


211

and so we arrive at (20), which makes the two methods look alike. Though there is a difference in the Green’s function: the FE Green’s function is only mesh dependent, while the BE Green’s function is also load-case dependent. Note that the BE method uses an ingenious approach: it does not approximate uR (y, x)—this would be simply too laborious, because at every point x it would have to approximate a new function uR (y, x)—but instead substitutes for the work integral (uR [x], p) the work done by the Cauchy data u on ΓN and t = ∂u/∂n on ΓD via the conjugate terms of the fundamental solution G. That is, the program knows that it suffices to approximate the “static” data u and t leaving the effects caused by a change in the observation point x to the fundamental solution G(y, x). This is the essence of (24). Accuracy The FE method has a handicap because it must approximate both parts of the Green’s function G = g + uR while in the BE method the fundamental solution is built into the code. This handicap can be overcome if the FE-method approximates only the regular part u(x) − uh (x) = (G(y, x) − Gh (y, x)) p(x) dΩy Ω = (g(y, x) + uR (x) − g(y, x) + uhR (y, x)) p(y) dΩy Ω (25) = (uR (x) − uhR (y, x)) p(y) dΩy . Ω

Of course if the source point x comes too close to the boundary then the regular part will also become singular. Everything is a functional The importance of Tottenham’s insight lies in the fact that in linear analysis every value, every displacement, every stress, every potential value can be considered a functional J(u). For example the following expressions J(u) = u(x)

J(u) = σxx (x)

J(u) =

l 0

σxy ds

(26)

are all functionals. And so as we associate with the classical functional, J(u) = u(x), a Dirac delta δ(y − x) u(y) dΩy (27) J(u) = Ω

so we can associate with every other linear functional J(u) a certain Dirac delta. Let the functional J(u) be for example the average value of the gradient of u over a small circular region Ωc (x) with radius c centered at a point x ∈ Ω 1 (u,x (x) + u,y (x)) dΩ . (28) J(u) = π c2 Ωc (x) We postulate that this functional can be expressed as δΣ (y − x) u(y) dΩy . J(u) =

(29)

Ω

We do not say what the Dirac delta δΣ (y − x) looks like, what it is. We only say that the scalar product (= integral) of δΣ with the function u provides the value J(u). In more complicated cases it

212


is even questionable whether the Dirac delta is an integrable function, whether the expressions make sense. We simply assume they do. Next we only need to find the function G which has the delta function as its right-hand side, that is which solves the equation −∆ G(y, x) = δΣ (y − x)

G=0

on Γ

because Green’s second identity then implies δΣ (y − x) u(y) dΩy − G(x, y) p(y) dΩy = 0 B(G, u) = Ω Ω

(30)

(31)

J(u)

or

J(u) =

Ω

G(x, y) p(y) dΩy .

(32)

This is the ’trick’. The abstract Dirac delta has got a shape, G, and the scalar product of this G with the right hand side p renders the value J(u). Now we do finite elements. We approximate the Green’s function G with the n shape functions in Vh Gh (y, x) =

n

ui (x) ϕi (y)

(33)

i=1

and so the variational problem for the weak solution becomes on Vh a system of n equations a(Gh , ϕi ) = (δΣ (y − x), ϕi ) = J(ϕi )

i = 1, 2, . . . n

(34)

fi = J(ϕi ) .

(35)

or in matrix notation kij = a(ϕi , ϕj )

K u(x) = f (x)

So the extension of the FE method to Green’s functions is simple and straightforward. Infinite energy Strictly speaking the extension is not so simple because most Green’s functions have infinite energy. Hence approximating a Green’s function with finite elements is an ill-posed problem. The surprising aspect is that the FE-method does not care. All the input in an FE-program is processed by these approximate Green’s function which theoretically do not lie in the solution space. But it is remarkable how ’old-fashioned’ the ansatz of the FE-method is. The FE-method is a numerical implementation of Green’s function. And this is the same idea behind the boundary element method. So in this sense the two methods are identical twins. Goal oriented refinement The error of an FE-solution u(x) − uh (x) =

Ω

(G(y, x) − Gh (y, x)) p(x) dΩy

(36)

is simply proportional to the error in the approximate Green’s function. So given that we are interested in the value u(x) of the solution at a particular point x we must refine the mesh in such a way that the error in the Green’s function for the point value u(x) is minimized. This technique is called goal oriented adaptive refinement.


213

By making use of the Galerkin orthogonality and Schwarz’ inequality we can formulate the following estimate for the error of an FE-solution |u(x) − uh (x)| ≤ || G[x] − Gh [x]||E ||u − uh ||E

(37)

where ||.||E is the standard energy norm, ||u||2E := a(u, u) and G[x] := G(y, x) is the Green’s function. Again we face the same problem as before: theoretically such an estimate requires the Green’s function to have bounded energy what normally is not the case. But if we read u(x) as the average value of the potential over a small disk centered at x the corresponding Green’s function G(y, x) would have finite energy and then (37) would make sense. This result (37) is motivation to minimize the error in the Green’s function and the error in the solution e for the Green’s function and uh simultaneously. That is, at each refinement step an error indicator ηG an error indicator ηpe for the original problem is calculated on each element and the combined error e · η e so that the sum of the local errors η e = η e · η e provides indicator on each element is then η e = ηG p p G an upper bound for the error e |u(x) − uh (x)| ≤ ηG · ηpe . (38) e

The energy norms of the errors eG = G−Gh and eu = u−uh are calculated by measuring the eigenwork done by the element residual forces and jump terms on the element edges, so that for example 1/2

ηp = a(eu , eu )Ωe = r(eu )1/2 .

(39)

This technique was applied to the plate in Fig. 2. The first mesh, the mesh in Fig. 2 a, is the result of a standard adaptive refinement ηpe ≤ εT OL .

(40)

To push the error below the preset error margin the program has to refine all those elements—in practice only the first, say, 30%—where the error ηpe of the original (or primal) problem exceeds this margin. The mesh in Fig. 2 b is based on weighting the primal error ηp with the error ηG of the Green’s function elementwise e · ηpe ≤ εT OL . ηG

(41)

e of the numerical Green’s function is so low that this inequality In most parts of the mesh the error ηG is automatically satisfied. That is many of the refinements in Fig. 2 a are not necessary if we are only interested in the stress σyy at x.

References [1] Tottenham H (1970) “Basic Principles”, in: Finite Element Techniques in Structural Mechanics. (Eds. Tottenham H, Brebbia C), Southampton University Press, Southampton 1970 [2] Hartman F, Katz C (2008) Structural Analysis with Finite Elements, 2nd ed., Springer-Verlag Berlin Heidelberg New York

214


Figure 2: Adaptive refinement a) standard refinement ηp ≤ εT OL b) goal oriented refinement ηp ×ηG ≤ εT OL


215

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k

i

j

/ 0 " " 11

vj1

qi1

vj2

qi2

vjn

qin

f (vj1 )

yj1

f (vj2 )

yj2

f (vjn )

yjn

vk1

f (vk1 ) yk1

vk2

f (vk2 ) yk2

vkn

f (vkn ) ykn

/ 0) 2 " "

• qj = [q1 , q2 , . . . , ql ])

3 " 4

• wji = [w1 , w2 , . . . , wl ]) i j 4

wji


• vj

• f (vj ) • yj

217

!

w

! "

#

$!

" "

! " % ! &'(

1 2 2 ej (n)

$ "

N

)

(n) =

M

N M 1 2 ej (n) 2N

)

n=1 j=1

%

Vf = 0.5

Lx = Ly = 8

θ

BaT iO3 − CoF eO4

"

&*(

+"

xc

yc

"

L

" , *

"

+

-

!

xc −2.5 2.5

yc −2.5 2.5

L 0.25 2

θ 0 165

σ

2a θ

φ ϕ

y x

σ , *

218


x = 8

pmap = (p − Pmean )

Pstd pstd

p pstd p Pmean Pstd ! " Pmean = 0

Pstd = 1 # $ Ni 1 noisei = ξi u2IJ α Ni

$

i=1

Ni = 12 % % ξi % α & % '() * + " % # , -% ,. ')

" % / #

0

1 " " " $ " % 2 % " 3 3 4 " " , % 25 ,

− φ φ φ ϕ ϕ ϕ φ+ϕ φ+ϕ φ+ϕ − φ φ φ ϕ ϕ ϕ φ+ϕ φ+ϕ φ+ϕ

− 0% 2% 5% 0% 2% 5% 0% 2% 5% − 0% 2% 5% 0% 2% 5% 0% 2% 5%

xc 2.3929 3.8556 4.003 3.5975 3.3571 4.1208 3.7372 4.3766 3.8418 3.6578 4.2019 3.5711 3.9752 4.0502 4.4259 3.5463 3.3444 4.2344 3.6202 2.1881

yc 1.8715 3.8556 3.3068 4.5943 3.963 3.3647 4.0152 2.939 3.9729 3.9154 3.6343 3.2589 3.5278 4.1471 3.9713 3.568 3.8043 4.1841 3.8229 4.3293

L 0.56626 0.70548 0.64632 0.79468 0.62785 0.87338 0.70335 0.7681 0.73402 0.77309 1.3615 1.1541 1.0673 0.99984 1.3506 1.1476 1.1332 1.1819 1.1067 1.3687

θ 133.5 93.541 95.959 59.551 74.783 84.616 79.348 113.31 88.671 93.557 57.045 48.863 48.568 40.225 56.959 50.117 39.147 36.307 44.738 35.366

" % " % .* % 6 " # %


219

8

7

6

5

4

3

2

1

0

0

1

2

3

4

5

6

7

8

− φ φ φ ϕ ϕ ϕ φ+ϕ φ+ϕ φ+ϕ − φ φ φ ϕ ϕ ϕ φ+ϕ φ+ϕ φ+ϕ

− 0% 2% 5% 0% 2% 5% 0% 2% 5% − 0% 2% 5% 0% 2% 5% 0% 2% 5%

xc 2.3929 2.4951 2.803 2.3857 2.7296 2.77 2.428 2.0552 2.2272 2.3976 4.2019 3.9661 3.9858 3.9705 4.0558 4.046 4.6654 4.2846 4.0117 3.5944

yc 1.8715 2.2285 2.0634 2.2329 2.1501 2.3469 2.6912 2.567 2.4524 2.7865 3.6343 3.6471 4.0219 4.2997 3.5987 3.7323 4.363 3.677 3.5426 3.1878

L 0.56626 0.55257 0.72627 1.0578 0.7909 1.2093 0.4961 0.80822 1.1241 1.0207 1.3615 1.5021 1.0673 1.0001 1.2674 1.3263 1.6431 1.2855 1.6863 1.2313

θ 133.5 123.57 120.64 98.406 111.31 111.74 108.65 100.91 114.57 34.819 57.045 37.582 57.261 63.061 62.778 44.625 42.62 61.264 62.645 45.439

! " #

$

%% &

'%

$

& (

220


ϕ 8

6

8

Crack NN (No Noise) NN (2% Noise) NN (5% Noise)

xc = 4.20 yc = 3.63 L = 1.36 θ = 57

7

6

5

5

4

4

3

3

2

2

1

0


xc = 2.39 yc = 1.87 L = 0.56 θ = 133.5

7

1

0

1

2

3

4

5

6

7

8

0

0

1

2

3

4

5

6

7

8

8

6

8


xc = 4.20 yc = 3.63 L = 1.36 θ = 57

7

6

5

5

4

4

3

3

2

2

1

1

0

0

1

2

3

4

5

6

7

8

0


xc = 2.39 yc = 1.87 L = 0.56 θ = 133.5

7

0

1

2

3

4

5

6

7

8

**(+&)'&)'&

! " #$%&''()

,-. / 0 )! 1 2 !31 4 #0 1 2 1 5 676'86&' &'' ,&. / 0 )! 1 9 9" )#0 1 2 !31 :

;5 1 when the considered main crack-tip approaches the microcrack pair, and then a shielding effect occurs as K Imax / K 0max

@

1 wr ½ (1 2v)(r,i n j r, j ni )¾ . ® (1 2v)G ij 3r,i r, j wn 8S (1 v)r 2 ¯ ¿ Considering the constitutive relations and eq (2), the corresponding stresses are expressed as V ( p) S u~ ds S u~ ds , Tij

³

ij

kij 1k

S1

³

S2

(3)

(4)

kij 2 k

where the integral kernels are given as follows wr ½ G °3 wn (1 2v)G ij r,k v( r, jG ik r,iG jk ) 5r,i r, j r,k 3v(ni r, j r,k n j r,i r,k )° S kij ¾, 3 ® 4S (1 v)r ° ° ¯ (1 2v)(3nk r,i n j n jG ik niG jk ) (1 4v)nk G ij ¿

>

@

r

(5)

( x1 [1 ) 2 ( x2 [ 2 ) 2 ( x3 [3 ) 2 .

(6)

3ˊHypersingular integral equations of two intersecting cracks p1 j ( P ) on the crack-surface S1 and T j

Using the traction boundary condition V 3j crack-surface

S 2

where

³

p2 j ( P) on the

, the hypersingular integral equations for the unknown functions u~ij can be obtained as

³

S1

³

S1

( n2 K k 2 i

K k 3i u~1k ds

³

S 2

K k 3i u~2 k ds

n3 K k 3i )u~1k ds

³

S 2

( n2 K k 2 i

p1i ( P) ,

(7)

n3 K k 3i )u~2 k ds

p2i ( P ) ,

(8)

is the symbol of the finite-part integral , p1i ( P ) and p2i ( P) denote the tractions in the ith

direction and at the source point P on S1 and S 2 that could be obtained by considering an infinite elastic solid without cracks, and G n2 3r, 2 r,3 (v 5r,12 ) n3 (3vr, 22 15r,12 r,32 1 v) , (9a) K131 4S (1 v)r 3 G K 231 n2 3r,1r,3 (5r, 22 1 2v) n3 3r,1r, 2 (v 5r,32 ) , (9b) 4S (1 v) r 3 G n2 3r,1r, 2 (v 5r,32 ) n3 3r,1r,3 (1 5r,32 ) , (9c) K 331 4S (1 v)r 3 G K132 n2 3r,1r,3 (v 5r, 22 ) n3 3r,1r, 2 (v 5r,32 ) , (9d) 4S (1 v)r 3 G n2 3r, 2 r,3 (1 5r, 22 ) n3 ( 3vr,12 15r, 22 r,32 1 v) , (9e) K 232 4S (1 v)r 3 G K 332 n2 (3vr,12 15r, 22 r,32 1 v) n3 3r, 2 r,3 (1 5r,32 ) , (9f) 4S (1 v)r 3 G n2 3r,1r, 2 (1 2v 5r,32 ) n3 3r,1r,3 (1 5r,32 ) , (9g) K133 4S (1 v)r 3 G K 233 n2 3(1 2v)r,12 ) 15r, 22 r,32 2(1 v) n3 3r, 2 r,3 (1 5r,32 ) , (9h) 4S (1 v)r 3 G n2 3r, 2 r,3 (1 5r,32 ) n3 3r,32 (2 5r,32 ) 1 , (9i) K 332 4S (1 v)r 3

>

@

>

@

>

@

>

@

>

@

>

@

>

@

^ >

^

@

>

`

@`

268


K121

K 221 K 321

K122 K 222

K 322 K123

K 223 K 323

G 4S (1 v)r 3 G 4S (1 v) r 3 G 4S (1 v)r 3 G 4S (1 v)r 3 G 4S (1 v)r 3 G 4S (1 v)r 3 G 4S (1 v)r 3 G 4S (1 v)r 3 G 4S (1 v)r 3

>n (3vr

2 ,3

2

>n

2

>n >n

@

15r,12 r, 22 1 v) n3 3r, 2 r,3 (v 5r,12 ) ,

@

3r,1r, 2 (1 5r, 22 ) n3 3r,1r,3 (v 5r, 22 ) ,

(9k)

@

(9l)

@

(9m)

2

3r,1r,3 (v 5r, 22 ) n3 3r,1r, 2 (1 2v 5r,32 ) ,

2

3r,1r, 2 (1 5r, 22 ) n3 3r,1r,3 (1 2v 5r, 22 ) ,

^n >3r 2

^n

>n

(9j)

2 , 2 (2

@

`

5r, 22 ) 1 n3 3r, 2 r,3 (1 5r, 22 ) ,

(9n)

>

@`

2

3r, 2 r,3 (1 5r, 22 ) n3 3(1 2v) r,12 15r, 22 r,32 2(1 v) ,

2

3r,1r,3 (v 5r, 22 ) n3 3r,1r, 2 (v 5r,32 ) ,

>n

2

@

2

(9p)

@

(9q)

@

(9r)

3r, 2 r,3 (1 5r, 22 ) n3 (3vr,12 15r, 22 r,32 1 v) ,

>n (3vr

2 ,1

(9o)

15r, 22 r,32 1 v) n3 3r, 2 r,3 (1 5r,32 ) .

4ˊSingularity indexes at the front of two intersecting cracks In order to investigate the singularity at the crack-front, a local coordinate system is defined as x1 - x2 - x3 . The x1 -axis is the tangential line of the crack-front at the point Q0, x2 -axis is the internal normal line in the crack plane, and x3 is the normal of the crack. The displacement discontinuities of the crack-surface near a crack-front point Q0 can be assumed as

°u1k (Q) ® °¯u2 k (Q)

g k (Q0 )[ 2Ok , hk (Q0 )[ 2cOk ,

k

1,2,3,

(10)

where g k (Q0 ) and hk (Q0 ) are non-zero constants related to the point Q0, Ok is the singularity index (order of the singularity) at the crack-front. Consider a small semi-circle domain SH on the crack-surface including the point Q0 and using the main-part analysis method [5, 11], the following relations can be derived

³

SH ki

1 k1 j uij ds # 0, r3

i 1, 2 ; j

2 ,3; k

1, 2 ,

1 5 (3vk2 k132 1 v)u11 ds # > 4v 2(1 v)@ SO1 g1 (Q0 ) x2O1 1 cot(O1S ) , 3 r3 S g 4 (Q0 ) x2O4 1 1 ³SH 12 r 3 k23 (v 5k1 )u21ds # 2(1 v) sin(O4S ) O4 sin >O4 (D S ) D @ ,

³

SH 11

S g (Q ) x O4 1 1 5 (3vk2 k132 1 v)u21 ds # 2(v 1) 4 0 2 O4 cos > O4 (D S ) D @ , 3 3 sin(O4S ) r 1 O 1 ³SH 11 r 3 u13 ds # 2SO3 g3 (Q0 ) x2 3 cot(O3S ) , 1 5 2 O 1 ³SH 22 r 3 (3vk3 3 k12 1 v)u21ds # 2(1 v)S g4 (Q0 ) x2c 4 O4 cot(O4S ) cos(2D ) , 1 O 1 ³SH 22 r 3 k23 (v 5k1 )u21ds # 2(1 v)S g4 (Q0 ) x2c 4 O4 cot(O4S ) sin(2D ) ,

³

SH 12

(11) (12) (13) (14) (15) (16) (17)


³

SH 22

269

1 ª 10 º (3v 1)k1 k232 3 v » u23 ds # 4S g 6 (Q0 ) x2cO6 1O6 cot(O6S )(2sin 2 (2D ) 1) , 3 r 3 «¬ ¼ 1 O 1 2 2 ³SH 22 r 3 (6k3 15k3 1)u23 ds # 2S g6 (Q0 ) x2c 6 O6 cot(O6S )(4sin D cos(2D ) 1) ,

where SH 1i and SH 2i indicate that the source point P is on

S1

and

S 2

(18) (19)

, respectively. The subscript i =1

denotes that the integration point Q is on the crack-surface S1 , and i =2 indicates that Q on S 2 . Using the above relations, eqs (7) and (8), and considering O1 O2 and O2 O3 O4 O5 O , the following characteristic equations for the singularity indexes or eigenvalues could be obtained cos 2 (O1S ) cos 2 > O1 (D S ) @ 0, (20)

^cos (OS ) cos >O (D S )@` 2

2

2

O 2 sin 2 D Ô 2 sin 2 D 4 cos 2 (OS ) sin 2 > O (D S )@`

0.

(21)

The solutions of eqs (20) and (21) are presented in Fig. 2, which shows the dependence of the singularity indexes on the angle between the two crack-surfaces. 1.0

O O

Singularity indexes

0.9

0.8

0.7

0.6

0.5 0

15

30

45

60

75

90

105

120

135

150

165

180

Angle of two crack-surfaces Fig. 2 The dependence of the singularity indexes on the angle of two crack-surfaces

5ˊSingular stresses at the front of two intersecting cracks The stress intensity factors on the intersecting line of the two crack-surfaces are defined by

K lim V (r , M ) M 0 (2r )1O , ° I r o0 33 ° (22) V 32 (r , M ) M 0 (2r )1O , ® K II lim r o0 ° 1 O1 V 31 (r , M ) M 0 (2r ) , °¯ K III lim r o0 where r is the distance from P to Q0 as shown in Fig. 1. Based on the relation (10), the following integrals could be obtained by the main-part analysis method.

³

SH 1i

³

SH i1

1 k1 j uij ds # 0, r3

i 1, 2 ; j

2,3,

S g (Q ) U O1 1 1 5 (3vk2 k132 1 v)u11 ds # 2(1 v) 1 0 O1 cos > O1 (M S ) M @ , 3 3 sin(O1S ) r

S g 4 (Q0 ) U O4 1 1 k v k u ds v ( 5 ) 2(1 ) O4 sin(D SO4 DO4 M O4M ), # 23 1 21 ³SH i2 r 3 sin(O4S )

(23)

i 1,2, i 1,2,

(24) (25)

270


S g3 (Q0 ) U O3 1 1 k k u ds (5 1) 2 O3 (O3 1) sin M cos > O2 (S M ) 2M @ , i 1,2, # (26) 23 3 13 3 ³SH i1 r sin(O3S ) S g3 (Q0 ) U O 1 1 2 ³SH r 3 (15k3 6k3 1)u13 ds # 2 sin(O3S ) O3 ^cos >O3 (S M ) M @ (1 O2 )sin M sin >O3 (S M ) 2M @` , (27) 3

11

S g (Q ) U O1 1 1 k23 (5k1 v)u11 ds # 2(1 v) 1 0 O1 sin > O1 (S M ) M @ , 3 SH 21 r sin(O1S )

³

(28)

S g (Q ) U O2 1 1 k23 (5k2 1)u12 ds # 2 2 0 O2 ^2sin > O2 (S M ) M @ (O2 1) sin M cos > O2 (S M ) 2M @` . (29) 3 SH 21 r sin(O2S ) Using the above relations and considering O1 O2 and O2 O3 O4 O5 O , the singular stresses can be expressed by

³

V 31

2(1 v)

V 32

V 33

SO1 U O 1 ^ g1 (Q0 ) cos >O1 (S M ) M @ g4 (Q0 ) cos >O1 (S D M ) M @` , sin(O1S ) 1

SOU O 1 > 2 g 2 (Q0 ) f 21 (M ) 2 g3 (Q0 ) f1 (M ) g5 (Q0 ) f 23 (M , D ) g6 (Q0 ) f 2 (M , D )@ , sin(OS ) SOU O 1 > 2 g 2 (Q0 ) f1 (M ) 2 g3 (Q0 ) f32 (M ) g5 (Q0 ) f 2 (M , D ) g6 (Q0 ) f34 (M , D )@ , sin(OS )

(30) (31) (32)

where

f 21 (D ) cos M cos > O (S D ) 2D @ O sin M sin > O (S D ) 2D @ , ° ° f1 (M ) (O 1) sin M cos > O (S D ) 2D @ , ° ° f 23 (M , T ) cos > O (S D M ) M @ cos > OS D (2 O ) M OM @ ° (1 O ) cos > OS D (4 O ) M OM @ (O 1) cos > OS D (2 O ) 3M OM @ , ° ° ® f 2 (M ,T ) sin > O (S D M ) M @ (O 1) sin > O (S D M ) 3M @ ° O sin > OS D (2 O ) 3M OM @ , ° ° f (M ) cos > O (S M ) M @ (1 O ) sin M sin > O (S M ) 2M @ , ° 32 ° f34 (M , T ) 3cos > O (S D M ) M @ (O 1) cos > O (S D M ) 3M @ ° O cos > OS D (2 O ) 3M OM @ . °¯

(33)

6ˊConclusions The hypersingular boundary integral equations of two intersecting three-dimensional cracks are presented in this paper. The characteristic equations for computing the singularity indexes are obtained, from which the dependence of the singularity indexes on the angle of the two crack-surfaces is analyzed, which partially verifies the correctness of the hypersingular boundary integral equations. The singular stresses at the crackfront are obtained and presented by the main-part analysis method.

Acknowledgments The project is supported by the National Natural Science Foundation of China (No. 10872213) and the personnel exchange program of China Scholarship Council (CSC) and German Academic Exchange Service (DAAD).


References [1 ] Suo Z., Kuo C.M., International Journal of Fracture, 1992, 54: 79-100. [2 ] Shang F.L., Kuna M., Engineering Fracture Mechanics, 2003, 70: 143-160. [3 ] Qin T.Y., Tang R.J., International Journal of Fracture, 1993, 60: 373-381. [4 ] Dunn M.L., Wienecke H.A., Int. J. Solids Struct., 1996, 30: 4571-4781. [5 ] Qin T.Y., Noda A.K., ASME J. of Appl. Mech., 2002, 69: 626-631. [6 ] Kuna M., Computat. Mater. Sci., 1998, 13: 6780. [7 ] Qin T.Y., Noda N.A., JSME International Journal, 2004, 47: 173-180. [8 ] Wang Z.K., Huang S.H., Engineering Fracture Mechanics, 1995, 51: 447-456. [9 ] Chen M.C., Fracture and Damage of Advanced Materials, 2004, 125-130. [10 ] Hill L.R., Farris T.N., AIAA Journal, 1998, 36: 102-108. [11 ] Tang R.J., Qin T.Y., Acta Mechanica Sinica, 1993, 25: 665-675.

271

272


Reconstruction of elasticity fields in isotropic materials via a relaxation of the alternating procedure Liviu Marin1 and B. Tomas Johansson2 1 Institute of Solid Mechanics, Romanian Academy, 15 Constantin Mille, P.O. Box 1-863, 010141 Bucharest, Romania 2 School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Keywords: Linear elasticity; Inverse problem; Cauchy problem; Alternating iterative algorithm; Relaxation procedures; Boundary element method (BEM). Abstract. We propose two algorithms involving the relaxation of either the given boundary displacements or the prescribed boundary tractions on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [3] applied to Cauchy problems in linear elasticity. Mathematical Formulation Consider a homogeneous linear elastic material which occupies a bounded Lipschitz domain Ω ⊂ Rd , where usually d ∈ {1, 2, 3}. Let Γ0 = ∅ be an arc of ∂Ω such that meas(Γ0 ) = 0 and set Γ1 = ∂Ω \ Γ0 . In the absence of body forces, the equilibrium equations are given by [1] Lu(x) ≡ −∇ · σ (u(x)) = 0,

x ∈ Ω.

(1)

Here L is the Lamé (Navier) differential operator, σ (u(x)) = [σij (u(x))]1≤i,j,≤d is the stress tensor

associated with the displacement vector u(x) = (u1 (x), . . . , ud (x))T , while on assuming small deformations, the corresponding strain tensor ε (u(x)) = [εij (u(x))]1≤i,j,≤d is given by the kinematic relations: 1 ε (u(x)) = ∇u(x) + ∇u(x)T , x ∈ Ω = Ω ∪ ∂Ω. (2) 2 These tensors are related by the constitutive law, namely σ (u(x)) = C ε (u(x)) ,

x ∈ Ω,

(3)

where C = [Cijkl ]1≤i,j,k,l≤d is the fourth-order elasticity tensor which is symmetric and positive definite. Let n(x) = (n1 (x), . . . , nd (x))T be the outward unit normal vector at x ∈ Γ and N u(x) ≡ t(x) = (t1 (x), . . . , td (x))T be the traction vector at a point x ∈ Γ defined by N u(x) ≡ t(x) = σ (u(x)) · n(x),

x ∈ ∂Ω,

(4)

where N is the boundary-differential operator associated with the Lamé (Navier) differential operator, L, and Neumann boundary conditions on Γ. If it is possible to measure both the displacement and traction vectors on a part of the boundary Γ, say Γ0 , then this leads to the mathematical formulation of the Cauchy problem consisting of the partial differential equations (1) and the boundary conditions u(x) = ϕ(x),

N u(x) ≡ t(x) = ψ(x),

x ∈ Γ0 ,

(5)

∗ 1/2 where ϕ ∈ H1/2 (Γ0 )d and ψ ∈ H00 (Γ0 )d are prescribed vector-valued functions. Here, the space 1/2 H (Γi ), i = 0, 1 is a subset (restrictions to the boundary Γi ) of the trace space H1/2 (∂Ω) (of the 1/2 Sobolev space H 1 (Ω)). The space H00 (Γi ), i = 0, 1, consists of functions from H1/2 (∂Ω) vanishing on ∗ 1 1/2 d is the dual space of H00 (Γi )d , i = 0, 1, with the usual norms. Γ1−i , i = 0, 1, and H (Ω) This problem consisting of (1) and (5), termed the Cauchy problem, is much more difficult to solve both analytically and numerically than direct problems, since the solution does not satisfy the general conditions of well-posedness [2].


273

Alternating Iterative Algorithms with Relaxation In this section we propose two alternating iterative algorithms with relaxation which aim to reduce the computational time of the alternating iterative algorithm introduced by Kozlov et al. [3], and at the same time maintaining the accuracy of the numerical results obtained with the latter, see also Marin and Johansson [4]. Alternating iterative algorithm with relaxation I:

∗ 1/2 Step 1.1. If k = 1 then choose an arbitrary function ξ(1) ∈ H00 (Γ1 )d . Step 1.2. If k ≥ 2 then solve the direct problem ⎧ Lu(2k−2) (x) = 0, ⎪ ⎪ ⎨ t(2k−2) (x) ≡ σ u(2k−2) (x) · n(x) = ψ(x), ⎪ ⎪ ⎩ (2k−2) (x) = η(k−1) (x), u

x ∈ Ω, x ∈ Γ0 ,

(6)

x ∈ Γ1 ,

where η(k−1) (x) = u(2k−3) (x), x ∈ Γ1 to obtain u(2k−2) (x), x ∈ Ω and t(2k−2) (x) ≡ σ u(2k−2) (x) , x ∈ Γ1 . Step 2. Provided that k ≥ 2 update the unknown Neumann data on Γ1 as: ξ(k) (x) = θ t(2k−2) (x) + (1 − θ) ξ(k−1) (x),

x ∈ Γ1 ,

where the relaxation factor, 0 ≤ θ ≤ 2, is fixed. For k ≥ 1 solve the direct problem ⎧ x ∈ Ω, Lu(2k−1) (x) = 0, ⎪ ⎪ ⎨ (2k−1) u (x) = ϕ(x), x ∈ Γ0 , ⎪ ⎪ ⎩ (2k−1) t (x) ≡ σ u(2k−1) (x) · n(x) = ξ(k) (x), x ∈ Γ1 ,

(7)

(8)

to determine u(2k−1) (x), x ∈ Ω and u(2k−1) (x), x ∈ Γ1 . Step 3. Set k = k + 1 and repeat Steps 1 and 2 until a prescribed stopping criterion is satisfied. Remark 1 The value θ = 1 in eqn. (7) corresponds to the alternating iterative algorithm introduced by Kozlov et al. [3] with an initial guess for the Neumann data, whilst values θ ∈ (0, 1) and θ ∈ (1, 2) in eqn. (7) correspond to the alternating iterative algorithm introduced by Kozlov et al. [3] with an initial guess for the Neumann data and a constant under- and over-relaxation factor, respectively. Alternating iterative algorithm with relaxation II: Step 1.1. If k = 1 then choose an arbitrary function η(1) ∈ H1/2 (Γ0 )d . Step 1.2. If k ≥ 2 then solve the direct problem ⎧ Lu(2k−2) (x) = 0, ⎪ ⎪ ⎨ u(2k−2) (x) = ϕ(x), ⎪ ⎪ ⎩ (2k−2) t (x) ≡ σ u(2k−2) (x) · n(x) = ξ(k−1) (x),

x ∈ Ω, x ∈ Γ0 ,

(9)

x ∈ Γ1 ,

where ξ(k−1) (x) = t(2k−3) (x), x ∈ Γ1 , to obtain u(2k−2) (x), x ∈ Ω, and u(2k−2) (x), x ∈ Γ1 . Step 2. Provided that k ≥ 2 update the unknown Dirichlet data on Γ1 as: η(k) (x) = θ u(2k−2) (x) + (1 − θ) η(k−1) (x),

x ∈ Γ1 ,

(10)

274


where the relaxation factor, 0 ≤ θ ≤ 2, is fixed. For k ≥ 1 solve the ⎧ Lu(2k−1) (x) = 0, ⎪ ⎪ ⎨ t(2k−1) (x) ≡ σ u(2k−1) (x) · n(x) = ψ(x), ⎪ ⎪ ⎩ (2k−1) u (x) = η(k) (x), to determine

u(2k−1) (x),

x ∈ Ω, and

t(2k−1) (x),

direct problem x ∈ Ω, x ∈ Γ0 ,

(11)

x ∈ Γ1 ,

x ∈ Γ1 .

Step 3. Set k = k + 1 and repeat Steps 1 and 2 until a prescribed stopping criterion is satisfied. Remark 2 The value θ = 1 in eqn. (10) corresponds to the alternating iterative algorithm introduced by Kozlov et al. [3] with an initial guess for the Dirichlet data, whilst values θ ∈ (0, 1) and θ ∈ (1, 2) in eqn. (10) correspond to the alternating iterative algorithm introduced by Kozlov et al. [3] with an initial guess for the Dirichlet data and a constant under- and over-relaxation factor, respectively. Following the ideas of Kozlov et al. [3] and Jourhmane and Nachaoui [5], one can rewrite the 1/2 ∗ Cauchy problem (1) and (5) as finding ξ ∈ H00 (Γ1 )d such that Bθ ξ + Gθ = ξ for a certain operator Bθ and element Gθ , see Marin and Johansson [4] for a definition of them. One can prove that Bθ is self-adjoint, non-expansive, positive and one is not an eigenvalue, for θ in a certain interval. Moreover, the iterative schemes are fixed point iterations for the above operator equation; from these observations the following convergence result is obtained, see Marin and Johansson [4]: 1/2 ∗ Theorem 1 Let ϕ ∈ H1/2 (Γ0 )d and ψ ∈ H00 (Γ0 )d . Assume that the Cauchy problem (1) and (5) has a solution u ∈ H1 (Ω)d . Let u(k) be the k-th approximate solution in the alternating procedure I described above. Then there exists a number 1 < b ≤ 2 such that when the relaxation parameter θ is chosen with 1 ≤ θ ≤ b, then lim u − u(k) H1 (Ω)d = 0 (12) for any initial data element ξ

(1)

∈

k→∞

∗ 1/2 H00 (Γ1 )d .

Remark 3 Let u0 be the initial guess of the displacement and choose the traction ξ = σ D−1 u0 Γ1 , where D−1 gives the solution to the Dirichlet problem with u = u0 on Γ1 and u = ϕ on Γ0 . Starting the alternating iterative algorithm with relaxation I with this traction as guess, the second approximation will be precisely our first initial approximation. Thus, from the above theorem, convergence is settled also for the alternating iterative algorithm II. Remark 4 To present a stopping rule for the case of relaxation I with noisy data, let w(2) be the element obtained from the second approximation in the proposed alternating procedure, with the initial guess ϕ = 0, and define the element E(ϕ, ψ) by E(ϕ, ψ) = t w(2) Γ1 . (13) Then, with noisy data ϕδ and ψδ , where δ > 0, and

E(ϕδ , ψδ ) − E(ϕ, ψ) ≤ δ,

(14)

the discrepancy principle can be employed as a stopping rule, see Vainikko and Veretennikov [6]. This implies in particular that if k = k(δ) is the smallest integer with

(2k+2) (2k)

t u

≤ bδ − t uδ (15) δ (k(δ))

for given b > 1, then uδ

converges to the exact solution of (1) and (5) when δ → 0.


275

The Boundary Element Method The Lamé system (1) for two-dimensional homogeneous isotropic linear elastic materials, i.e. d = 2, can be formulated in integral form with the aid of the Second Theorem of Betti, see e.g. [1], namely cij (x)uj (x) + − Tij (x, y) uj (y) dΓ(y) = Uij (x, y) tj (y) dΓ(y), (16) ∂Ω

∂Ω

for i, j = 1, 2, x ∈ Ω, and y ∈ ∂Ω, where the first integral is taken in the sense of the Cauchy principal value, cij (x) = 1 for x ∈ Ω and cij (x) = 1/2 for x ∈ ∂Ω (smooth), and Uij and Tij are the fundamental displacements and tractions for the two-dimensional isotropic linear elasticity given by

∂r(x, y) ∂r(x, y) Uij (x, y) = C1 C2 ln r(x, y) δij − (17) ∂yi ∂yj and

∂r(x, y) ∂r(x, y) ∂r(x, y) ∂r(x, y) ∂r(x, y) − C4 nj (y) − ni (y) , C4 δij + 2 ∂yi ∂yj ∂n(y) ∂yi ∂yj (18) respectively. Here r(x, y) represents the distance between the node/collocation point x and the field point y, whilst the constants C1 , C2 , C3 and C4 are given by

Tij (x, y) =

C3 r(x, y)

C1 = −1/[8πG(1 − ν)],

C2 = 3 − 4ν,

C3 = −1/[4π(1 − ν)],

C4 = 1 − 2ν,

(19)

where ν = ν for the plane strain state and ν = ν/(1 + ν) for the plane stress state. A BEM with continuous linear boundary elements, see e.g. Brebbia et al. [7], is employed in order to discretise the integral equation (16). If the boundaries Γ0 and Γ1 are discretised into N0 and N1 continuous linear boundary elements, respectively, such that N = N0 + N1 , then on applying the boundary integral equation (16) at each node/collocation point, we arrive at the following system of linear algebraic equations A U = B T. (20) Here A and B are matrices which depend solely on the geometry of the boundary ∂Ω and material properties, i.e. the Poisson ratio, ν, and the shear modulus, G, and can be calculated analytically, and the vectors U and T consist of the discretised values of the boundary displacements and tractions, respectively. Numerical Results Example. In the following, we solve the Cauchy problem (1) and (5) for a two-dimensional isotropic linear elastic medium characterised by the material constants G = 3.35 × 1010 N/m2 and ν = 0.34 (copper alloy) in a smooth, doubly connected geometry. The following analytical solution for the displacements are assumed:

(an) 1 ui (x1 , x2 ) = V(1 − ν)xi − W(1 + ν) 2 xi 2 xi , i = 1, 2, (21) 2G(1 + ν) x1 + x2 with (σ − σ )r2 r2 σo ro2 − σi ri2 (22) , W = o 2 i 2 o i , σi = 1.0 × 1010 N/m2 , σo = 2.0 × 1010 N/m2 , 2 2 ro − ri ro − ri in the annular domain Ω = x ∈ R2 |ri < ρ(x) < ro , where ρ(x) = x21 + x22 is the radial polar coordinate of x, ri = 1 and ro = 4, which corresponds to constant internal and external pressures σi and σo , respectively, for which the stress tensor is given by V=−

(an)

σ11 (x1 , x2 ) = V + W (an)

(an)

x21 − x22 2, (x21 + x22 )

σ12 (x1 , x2 ) = σ21 (x1 , x2 ) = 2W

(an)

σ22 (x1 , x2 ) = V − W x1 x2 . 2 (x21 + x22 )

x21 − x22 2, (x21 + x22 )

(23)

276


Here Γ0 = Γo = x ∈ ∂Ω ρ(x) = ro and Γ1 = Γi = x ∈ ∂Ω ρ(x) = ri . For the inverse problem considered in this study, the BEM system of linear algebraic equations (20) has been solved using N0 = 40 and N1 = 120 continuous linear boundary elements used for discretising the boundaries Γ0 and Γ1 , respectively, while the initial guesses ξ(1) and η(1) for the traction t Γ1 and displacement u Γ1 vectors were taken to be ξ(1) (x) = 0,

x ∈ Γ1 ,

and

η(1) (x) = 0,

x ∈ Γ1 ,

(24)

respectively. Convergence of the Algorithms. In order to analyse the accuracy, convergence and stability of the proposed alternating iterative algorithms with relaxation, for k ≥ 1 we introduce the errors ⎧

⎨ u(2k−1) − u(an) L2 (Γ )d for the alternating iterative algorithm I 1 (25.1) eu (k) =

⎩ u(2k) − u(an) 2 for the alternating iterative algorithm II L (Γ1 )d and

⎧

⎨ t(2k) − t(an) L2 (Γ )d 1 et (k) =

⎩ t(2k−1) − t(an) 2 L (Γ1 )d

for the alternating iterative algorithm I for the alternating iterative algorithm II.

(25.2)

Here u(2k−1) (u(2k) ) and t(2k) (t(2k−1) ) are the displacement and traction vectors retrieved on Γ1 after k iterations using the alternating iterative algorithm with relaxation I (II), respectively. Although not presented, it is reported that, for all values of the relaxation parameter θ, both accuracy errors eu and et keep decreasing until a specific number of iterations, after which the convergence rate of the aforementioned errors becomes very slow so that they reach a plateau. As expected, for each value of θ, eu (k) < et (k) for all k ≥ 1; also, the larger the parameter θ, the lower the number of iterations and, consequently, computational time are required for obtaining accurate numerical results for both the displacement and the traction vectors on Γ1 . Similar results have been obtained for the alternating iterative algorithm with relaxation II. Regularizing Stopping Criterion. To simulate the inherent inaccuracies in the measured data on Γ0 , we assume that random noise, pu and pt , have been added into the various levels of Gaussian exact displacement u Γ0 = ϕ and traction t Γ0 = ψ data, respectively, so that the following perturbed displacements and tractions are available:

ϕδ ∈ L2 (Γ0 )d : u(an) Γ0 − ϕδ L2 (Γ0 )d = δ, and ψδ ∈ L2 (Γ0 )d : t(an) Γ0 − ψδ L2 (Γ0 )d = δ. (26) Fig. 1(a) presents, on a logarithmic scale, the accuracy error eu as a function of the number of iterations, k, obtained using the alternating iterative algorithm I, θ = 1.50 and pu ∈ {1%, 2%, 3%}. From this figure it can be seen that, for each fixed value of pu , the errors in predicting the displacement vector on the under-specified boundary Γ1 decrease up to a certain iteration number and after that they start increasing. Although not illustrated, it is important to mention that the accuracy error et has a similar behaviour. If the iterative process is continued beyond this point then the numerical solutions lose their smoothness and become highly oscillatory and unbounded, i.e. unstable. Therefore, a regularizing stopping criterion must be used in order to cease the iterative process at the point where the errors in the numerical solutions start increasing. To define the stopping criterion required for regularizing/stabilizing the iterative methods analysed in this paper, for k ≥ 1, the following convergence error is introduced:

E(k) = A U(k) − B T(k) , (27) where A and B are the BEM matrices. In the case of the alternating iterative algorithm with relaxation I, the vectors U(k) and T(k) are given by

T T Φ U(k) = , Φδ = ϕ(δ;1) , . . . , ϕ(δ;N0 ) , U(2k−1) = u(2k−1;1) , . . . , u(2k−1;N0 ) , (28.1) U(2k−1)


277

10.0

0.3

0.1 0.06

Convergence error, E

Accuracy error, eu

5.0 pu = 1% pu = 2% pu = 3%

0.03

0.01

pu = 1% pu = 2% pu = 3%

1.0 0.5

0.1 0.05

0.006 0.01 1

5

10

50 100

500

1

Number of iterations, k

5

10

50 100

500

Number of iterations, k

(a) Accuracy error eu : pu ∈ {1%, 2%, 3%}, algorithm I

(b) Convergence error E: pu ∈ {1%, 2%, 3%}, algorithm I

Figure 1: (a) The accuracy error eu , and the convergence error (c) E, as functions of the number of iterations, k, obtained using the alternating iterative algorithm I, θ = 1.50 and various amounts of noise added into u Γ0 . T(k) =

T T Ψ , Ψδ = ψ(δ;1) , . . . , ψ(δ;N0 ) , T(2k) = t(2k;N0 +1) , . . . , t(2k;N0 +N1 ) ; T(2k)

(28.2)

The alternating iterative algorithms I and II are ceased according to the discrepancy principle of Morozov, see also [4], namely at the optimal iteration number, kopt , which is the smallest integer with E(k) ≈ O(δ).

(29)

Fig. 1(b) presents the evolution of the convergence error E with respect to the number of iterations performed, k, using the alternating iterative algorithm I, θ = 1.50 and pu ∈ {1%, 2%, 3%}. By comparing Figs. 1(a) and 1(b), it can be seen that selecting the optimal iteration number, kopt , according to the stopping rule (29) captures very well the minimum values for the accuracy errors eu and et . Hence eqn. (29) represents a stabilizing stopping criterion for the alternating iterative algorithms. Stability of the Algorithms. Based on the stopping criterion (29), the numerical results for the x1 -component of the displacement and the x2 -component of the traction, obtained on the underspecified boundary Γ1 using the alternating iterative algorithm I, θ = 1.50 and pu ∈ {1%, 2%, 3%}, and their corresponding analytical values are presented in Fig. 2(a) and 2(b), respectively. It can be seen from these figures that the numerical solutions for both the displacement and traction vectors are stable approximation to their corresponding exact solutions, free of unbounded and rapid oscillations, and they converge to the exact solutions as pu −→ 0. The values of the optimal iteration number, kopt , the corresponding accuracy errors, eu (kopt ) and et (kopt ), and the CPU time, obtained using the alternating iterative algorithm I, the stopping criterion (29), various levels of noise added into the Dirichlet and Neumann data on Γ0 and various values of the relaxation parameter, θ ∈ (0, 2), are presented in Table 1. In order to assess the performance of the alternating iterative algorithm I with under-, no and over-relaxation, we exemplify by considering pu = 1% and pt = 0%: The CPU times needed for the alternating iterative algorithm I with θ = 0.50 (under-relaxation), θ = 1.00 (no relaxation) and θ = 1.50 (over-relaxation) to reach the numerical solutions for the displacement and traction vectors on Γ1 were found to be 76.28, 40.39 and 27.67 s, respectively, while the corresponding values for the optimal iteration number required, kopt , were found to be 40, 20 and 13, respectively. This means that, to attain the numerical solutions for the unknown data on Γ1 , the alternating iterative algorithm I with over-relaxation requires a reduction in the number of iterations performed and CPU time by

278


θ 0.10

0.50

1.00

1.50

1.80

0.10

0.50

1.00

1.50

1.80

pu 1% 2% 3% 1% 2% 3% 1% 2% 3% 1% 2% 3% 1% 2% 3% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

pt 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 1% 2% 3% 1% 2% 3% 1% 2% 3% 1% 2% 3% 1% 2% 3%

kopt 203 175 158 40 35 32 20 17 16 13 12 11 11 10 9 152 123 106 30 25 22 15 13 11 10 9 8 9 7 7

eu (kopt ) 0.49476 × 10−2 0.93362 × 10−2 0.13543 × 10−1 0.47990 × 10−2 0.89878 × 10−2 0.12955 × 10−1 0.45304 × 10−2 0.87296 × 10−2 0.12370 × 10−1 0.43344 × 10−2 0.80237 × 10−2 0.11618 × 10−1 0.41556 × 10−2 0.77977 × 10−2 0.11346 × 10−1 0.92226 × 10−2 0.18373 × 10−1 0.27449 × 10−1 0.89794 × 10−2 0.17090 × 10−1 0.25018 × 10−1 0.81957 × 10−2 0.14745 × 10−1 0.24051 × 10−1 0.73335 × 10−2 0.12527 × 10−1 0.19075 × 10−1 0.54442 × 10−2 0.13645 × 10−1 0.16103 × 10−1

et (kopt ) 0.47071 × 10−1 0.86381 × 10−1 0.12277 × 100 0.45935 × 10−1 0.84196 × 10−1 0.11952 × 100 0.44108 × 10−1 0.82284 × 10−1 0.11567 × 100 0.42491 × 10−1 0.77212 × 10−1 0.11010 × 100 0.40854 × 10−1 0.74962 × 10−1 0.10779 × 100 0.73617 × 10−1 0.14605 × 100 0.21775 × 100 0.76813 × 10−1 0.14596 × 100 0.21342 × 100 0.77429 × 10−1 0.13865 × 100 0.22624 × 100 0.77341 × 10−1 0.12976 × 100 0.19747 × 100 0.60551 × 10−1 0.15775 × 100 0.18018 × 100

CPU time [sec] 397.82 350.18 303.75 76.28 67.59 63.53 40.39 35.51 36.45 27.67 26.51 24.79 24.62 23.10 21.60 283.12 224.78 194.01 58.14 49.82 44.50 31.68 28.01 24.84 22.67 21.01 19.62 20.07 17.31 17.32

Table 1: The values of the optimal iteration number, kopt , the corresponding accuracy errors, eu (kopt ) and et (kopt ), and the computational time, obtained using the alternating iterative algorithm I, the discrepancy principle, various amounts of noise added into u Γ0 or t Γ0 , and various values for the relaxation parameter, θ.


279

1.0

0.3 0.2

0.5

t2/10

10

0.1 u1 0.0

0.0

-0.1 Analytical

-0.3 -1.0

-0.5

pu = 1% pu = 2% pu = 3%

-0.2

-0.8

-0.6

-0.4 /2

-0.2

0.0

(a) Displacement u1 Γ : pu ∈ {1%, 2%, 3%}, algorithm I 1

-1.0 -1.0

Analytical pu = 1% pu = 2% pu = 3%

-0.8

-0.6

-0.4

-0.2

0.0

/2

(b) Traction t2 Γ : pu ∈ {1%, 2%, 3%}, algorithm I 1

Figure 2: The analytical and numerical (a) displacement u1 Γ1 , and (b) traction t2 Γ1 , obtained using the alternating iterative algorithm I, the discrepancy principle, θ = 1.50 and various amounts of noise added into u Γ0 . approximately 35% and 67% with respect to those corresponding to the standard iterative algorithm I, see e.g. [3], and the alternating iterative algorithm I with under-relaxation, respectively. Conclusions In this paper, we proposed two algorithms involving the relaxation of either the given Dirichlet or Neumann data on the over-specified boundary for the alternating iterative algorithm of Kozlov et al. [3] applied to Cauchy problems in linear elasticity. A convergence theorem for these relaxation methods, as well as a regularizing stopping criterion, were presented. The aforementioned algorithms with relaxation were implemented, for two-dimensional isotropic linear elastic materials, by employing continuous linear boundary elements. The numerical results obtained showed the numerical stability, convergence, accuracy, consistency and computational efficiency of the proposed relaxation procedures. Acknowledgement. The financial support received by L. Marin from the Romanian Ministry of Education, Research and Innovation through IDEI Programme, Exploratory Research Projects, Grant PN II–ID–PCE–1248/2008, is gratefully acknowledged. References [1] L.D. Landau and E.M. Lifshits, Theory of Elasticity, Pergamon Press (1986). [2] J. Hadamard, Lectures on Cauchy Problem in Linear Partial Differential Equations, Yale University Press (1923). [3] V.A. Kozlov, V.G. Maz ya and A.V. Fomin, U.S.S.R. Computational Mathematics and Mathematical Physics 18, 817–825 (1991). [4] L. Marin and B.T. Johansson, Computer Methods in Applied Mechanics and Engineering, accepted (2010). [5] M. Jourhmane and A. Nachaoui, Applicable Analysis 81, 1065–1083 (2002). [6] G.M. Vainikko and A.Y. Veretennikov, Iteration Procedures in Ill-Posed Problems, Nauka Publications (1986). [7] C.A. Brebbia, J.F.C. Telles and L.C. Wrobel, Boundary Element Techniques, Springer-Verlag (1984).

280


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286


Analysis of the dynamic response of deep foundations with inclined piles by a BEM-FEM model Luis A. Padrón, Juan J. Aznárez, Orlando Maeso and Ariel Santana University Institute SIANI, Universidad de Las Palmas de Gran Canaria, Spain e-mail: {lpadron, jaznarez, omaeso, asantana}@iusiani.ulpgc.es Keywords: Boundary element – finite element coupling, soil-structure interaction, inclined piles.

Abstract. A boundary element - finite element coupling formulation is used to address the dynamic behavior of deep foundations with inclined piles, modeling the soil by boundary elements as a three-dimensional zonedhomogeneous isotropic unbounded viscoelastic medium, and the piles by monodimensional finite elements as compressible Euler-Bernoulli beams. The problem is solved in the frequency domain. The formulation is briefly presented at the beginning of the paper. Then, validation results are presented for different foundation configurations and pile-soil stiffness ratios. Introduction This work addresses the dynamic response of inclined piles and pile groups with inclined members, which are widely used in Civil Engineering but whose response under dynamic events has not been sufficiently analyzed up to date. In fact, their use in seismically active regions is discouraged by several building codes, even though there are some published results [1-3] suggesting that inclined piles may have beneficial rather than detrimental effects on the seismic response of foundation and superstructure. A number of papers provide dynamic stiffness and damping functions of vertical piles but, up to the authors knowledge, impedance functions of inclined piles have been presented only by Giannakou et al. [4] and by Mamoon et al. [5], but for very few and specific cases. For this reason, and in order to contribute in this area, the methodology presented herein has been developed. Numerical Model The boundary element method is used herein to model the dynamic response of the soil region taking into account the internal loads arising from the pile-soil interaction. These loads are modeled as distributions of tractions applied on a line dened by the pile axis, and are named ’load-lines’. On the other hand, the piles rigidity is introduced later into the system by using nite elements. The whole approach, together with the denition of the geometrical parameters of the problem, is depicted in Fig. 1. Let the soil be considered as a linear, homogeneous, isotropic, viscoelastic, unbounded region with boundary . The boundary integral equation for a time-harmonic elastodynamic state dened in the domain can be written in a condensed and general form as

where ck is the local free term matrix at collocation point ‘k’, u and p represent the displacement and traction elds in the three directions of space, u* and p* are the elastodynamic fundamental solution tensors on the boundary due to a time-harmonic concentrated load at point ‘k’, np is the number of piles, and pj represents the pile-soil interface along the load-line j . In eq (1), the two terms between brackets represent the contribution of the internal loads, being qsj and fsj vectors containing the tractions (acting within the soil) along the pile-soil interface. More precisely, fsj represents a point load placed at the tip of the pile, while qsj is the distribution of interaction loads, along the pile shaft, applied on a line dened by the pile axis, both forces fsj and qsj coming from the pilesoil interaction along the different interfaces. On the other hand, jk represents the corresponding u* tensor computed at the tip of the pile. However, a singularity arises when the collocation point ‘k’ coincides with the tip node of the pile. In such a case, this term can be computed by considering the force at the tip of the pile as a vector of uniformly distributed tractions over a circular surface with radius Rp2 = A/ , which yields a regular integral [6,7], being A the area of the pile section.


287

Figure 1. Pile foundation geometry and modeling through BEM-FEM coupling formulation

Following the usual procedure in the boundary element method, the numerical solution of eq (1) requires the discretization of the boundary surface. In this case, quadratic elements of six and nine nodes have been used. Then, over each boundary element, displacement and traction elds are approximated in terms of their values at nodal points ( u and p ) making use of a set of polynomial interpolation functions [8]. The treatment of singularities can be found in [6,7]. Now, eq (1) can be written for all boundary nodes in and all internal nodes in pi , yielding, respectively, the following two matrix equations

where H and G are coefcient matrices obtained by integration over the elements of the fundamental solution times the corresponding shape functions, u and p are the vectors of nodal displacements and tractions of the boundary elements, and pi is the vector of nodal displacements along load-line i. On the other hand, piles are discretized using three-node beam elements with 13 degrees of freedom: three displacements on each node and two rotations at each of the ends. Linear axial deformation is allowed and pile exural behavior is modeled according to the Euler-Bernoulli beam theory, but torsional response is not included in the model. After the discretization process, the dynamic behavior of pile j can be represented, in the nite-element sense, by a matrix equation where is the circular frequency of excitation, u is the vector of nodal translation and rotation amplitudes along the pile, fjext includes the external forces acting at the top and the tip of the pile, Kj and Mj are the stiffness and mass matrices of the pile, and Qj is the matrix that transforms these nodal traction components to equivalent nodal forces. As usual, matrices Kj, Mj and Qj are expressed herein as global matrices, obtained following the general assembly process of the nite element method from the elemental matrices dened for a general vertical element [6,7] and after pre and post multiplying by the corresponding rotation matrices in order to adapt to the pile inclination. Now, imposing equilibrium and compatibility conditions along the load lines, and prescribing boundary conditions, eqs (2), (3) and (4) can be rearranged in a system of equations of the type

288


Figure 2. Multi-region boundary-element model definitions and BEM mesh details.

Comparison Results The formulation presented above was implemented in a previously existent multi-region BEM FORTRAN code [9, 10]. The aim of this section is the validation of this formulation (and its implementation) through a set of comparison results corresponding to dynamic stiffness and damping functions for 2×2 pile groups. To this end, the computation of the dynamic stiffness and damping functions of pile groups is going to be validated against results from an advanced 3-D multi-region boundary element code for time-harmonic elastodynamic problems as presented in [9,10]. 3-D boundary element formulation. In this multi-region boundary element formulation, both soil and piles are modeled as continuum isotropic homogeneous linear viscoelastic regions with their actual geometries. The boundary integral representation of the displacements in each domain (soil and each pile) corresponds to eq (1) but leaving the right hand side only with the rst term. For the specic case of a single oating pile embedded in a viscoelastic half-space, the boundary element equations for each region (pile and soil) in partitioned form are:

corresponding eqs (8) and (9) to pile and soil regions, respectively. According to Fig. 2, sub-indexes 1 to 3 of the above equations correspond, respectively, to the pile connection with the rigid cap where nodal displacements are known (1), to the pile-soil interface (2), and to the soil free-traction ground surface (3). Imposing, on the above expressions, external boundary conditions together with compatibility and equilibrium along the pile-soil interfaces, the combined equations for the coupled impedance problem can be written as

All boundaries (pile-soil interfaces, pile-cap interfaces and ground surface) are discretized into a nite number of quadratic nine-node and six-node boundary elements. Details of one of the used meshes in this work are shown in Fig. 2. Note that, due to the problem symmetries, only one quarter of the geometry needs to be discretized. This multi-region BEM code, being more rigorous and versatile than the simplied BEM-FEM coupling scheme presented before, presents clear disadvantages when it comes to performing parametric studies. Such disadvantages are all related to the relatively high number of degrees of freedom involved in a boundary-element


289

model and also to the amount of work needed to produce the mesh corresponding to each one of the congurations to analyze. Both negative aspects are clearly improved by the coupling formulation, where the number of degrees of freedom is radically reduced, and the pile discretization, much more simple to dene, is independent of the soil mesh.

Table 1. Configurations used for validation of dynamic stiffness and damping functions.

Validation results. In this section, comparison results are shown for several congurations of 2×2 inclined pile groups. Three different rake angles have been considered for these plots: = 10°, 20° and 30°. The rst case corresponds always to a pile separation ratio of s/d = 5, while the other two correspond to s/d = 10. Also, for = 10° and 30°, piles are always inclined parallel or perpendicular to the direction of excitation. On the other hand, the conguration used for = 20° corresponds to a case in which the piles are inclined, symmetrically, along the cap diagonals (see Table 1). Results are shown for the two different pile-soil stiffness ratios considered in this paper. Figs. 3 to 6 present horizontal, vertical, rocking and horizontal-rocking crossed dynamic stiffness and damping functions for the congurations described above. Results corresponding to the boundary element – nite element coupling formulation presented in this work are labeled as “BEM-FEM” and plotted using points, while those obtained from the multi-region boundary element code are labeled as “BEM-BEM” and plotted using dashed lines. In the horizontal and rocking cases, results are provided for excitation modes along both horizontal axes. It can be seen that a very good agreement exists between the BEM-FEM coupling scheme used in the following section and the more rigorous boundary element code.

Figure 3. Comparison between horizontal impedances obtained by BEM-BEM and BEM-FEM models.

290


Figure 4. Comparison between vertical impedances obtained by BEM-BEM and BEM-FEM models.

Figure 5. Comparison between rocking impedances obtained by BEM-BEM and BEM-FEM models.

Acknowledgments This work was supported by the Ministerio de Ciencia e Innovación of Spain (BIA2007-67612-C02-01). The financial support is gratefully acknowledged.


291

Figure 6. Comparison between horizontal-rocking crossed impedances obtained by BEM-BEM and BEM-FEM models.

References [1] G. Gazetas and G. Mylonakis. Seismic soil-structure interaction: new evidence and emerging issues. In Geotechnical Earthquake Engineering and Soil Dynamics III ASCE, Geotechnical Special Publication II, 11191174 (1998). [2] J. Guin. Advances in soil-pile-structure interaction and non-linear pile behavior, Ph.D. Thesis, State University of New York at Buffalo (1997) [3] N. Gerolymos, A. Giannakou, I. Anastasopoulous and G. Gazetas. Evidence of beneficial role of inclined piles: observations and summary of numerical analyses. Bulletin of Earthquake Engineering, 6(4),705-722 (2008) [4] A. Giannakou, N. Gerolymos, G. Gazetas, T. Tazoh and I. Anastosopoulos. Seismic behavior of batter piles – I. Elastic Response. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, In Press (2010) [5] S.M. Mamoon, A.M. Kaynia and P.K. Banerjee. Frequency domain dynamic analysis of piles and pile groups. J Eng Mech ASCE, 116, 2237-2257 (1990) [6] L.A. Padrón, J.J. Aznárez and O. Maeso. BEM-FEM coupling model for the dynamic analysis of piles and pile groups. Engineering Analysis with Boundary Elements, 31, 473-484 (2007) [7] L. A. Padrón. Numerical model for the dynamic analysis of pile foundations, Ph. D. Thesis, University of Las Palmas de Gran Canaria (2009) (available for download at http://hdl.handle.net/10553/2841) [8] J. Domínguez. Boundary Elements in Dynamics. Computational Mechanics Publications & Elsevier Applied Science, Southampton, NY (1993) [9] O. Maeso, J.J. Aznárez and F. García. Dynamic impedances of piles and groups of piles in saturated soils. Computers and Structures, 83, 769-782 (2005) [10] F. Vinciprova, O. Maeso, JJ. Aznárez and G. Oliveto. Interaction of BEM analysis and experimental testing on pile-soil systems. In Problems in structural identification and diagnostic: General aspects and applications. Springer-Verlag, 195-227 (2003)

292


Fast Multipole Boundary Element Method (FMBEM) for acoustic scattering in coupled fluid-fluidlike problems 1

2

V. Mallardo1 , C. Alessandri1 , M. H. Aliabadi2

Department of Architecture, University of Ferrara, Italy, [email protected], [email protected] Department of Aeronautics, Imperial College London, UK, [email protected]

Keywords: Fast multipole method, Helmholtz equation, Coupled problem. Abstract. The present paper intends to couple the Fast Multipole Method (FMM) with the Boundary Element Method (BEM) in the analysis of the scattering of plane sound waves from an infinite cylinder in a fluid. The cylinder is modeled as fluid-like, i.e. it is not capable to support shear waves. A governing integral equation is derived from the Helmholtz differential equation for both the fluid and the scatterer. Such an equation is discretised and solved by using the fast multipole approach in order to speed up the CPU time. The final procedure turns out to be much faster if compared to the conventional approach. An iterative solver is adopted in order to improve the overall computational efficiency. Introduction Scattering of acoustic and electromagnetic waves are important in many engineering fields. It can give important information on the internal composition of solids and fluids and represents a useful tool in non-destructive testing: internal inhomogeneities, flaws and inclusions in an object which may be detected by investing it with ultrasound waves. There are several applications of the Finite Element Method (FEM) to scattering problems, but its inability to deal with infinite domains has forced the FEM to be coupled with analytic or other numerical methods, for instance, with the bymoment method, modal expansion, absorbing boundary condition, infinite elements. The Boundary Element Method has been successfully applied in many problems involving infinite domains (see [1-2] for a review). The one dimension lower solution space and the implicit satisfaction of the Sommerfield radiation condition are the main advantages. On the other hand, the final system of equations gives rise to dense, non-symmetric and sometimes illconditioned coefficient matrix. In [3] the cost of obtaining solutions to problems governed by the Helmholtz equation in both interior and exterior domains by means of the BEM and the FEM is studied and compared. The Authors intend to demonstrate that FEM is economically competitive with BEM but the above comparison does not take into account the development of new fast iterative procedures coupled with BEM, such as the Fast Multipole Method (FMM) (see [4] for instance) and the Hierarchical Matrix approach [5]. Solving the system of equations by the conventional BEM becomes prohibitively expensive when applied to high frequency range or cases which require iterative solutions, i.e. optimisation and identification problems. In fact, the computation of the coefficients of the matrices governing the discrete problem require O(N 2 ) operations, where N is the number of degrees of freedom (DOFs), and other O(N 3 ) operations are necessary to solve the system by using any direct solver. In 1983 Rokhlin [6] proposed an algorithm for rapid solution of classical boundary value problems for the Laplace equation based on iteratively solving integral equations of potential theory. The CPU time requirement obtained was proportional to N . The starting point was the harmonic expansion of the kernel. The algorithm appeared to be the most efficient, at that time, for the solution of large scale boundary value problems whenever the solution needed to be evaluated at a limited number of points. The procedure was then extended, a few years later, to two dimensional acoustic scattering in [7] where the Author described a similar procedure capable to reduce the CPU time requirements of the algorithm to N 4/3 . In both papers no connection with the BEM was introduced. The coupling between the BEM and the FMM occurs years later. A comprehensive review of the fundamentals of FMM to accelerate the Boundary Integral Equation Method (BIEM) with reference to 1


293

the Laplace and Helmholtz fields is surveyed in [8]. The huge improvement in the coupling technique is clear if the number of unknowns which can be handled by a common laptop is considered: with conventional BEM it is not possible to solve up to a few thousands of unknowns whereas with FMBEM problems of the size with 105 unknowns can be easily threaten. This paper intends to present a FMBEM approach for solving the scattering from an infinite cylinder immersed in a unviscous fluid and illuminated by an incident plane wave. The cylinder may contain a cavity. First the governing integral equations along with the necessary multipole expansions are exposed. Then the coupling between FMM and BEM is detailed and the advantages are underlined. A numerical example closes the paper. The proposed approach can be particularly useful for developing a fast numerical/experimental procedure aimed at identifying internal cavities by using non-destructive ultrasound waves. The interaction model The scattering of time-harmonic acoustic waves in a homogeneous isotropic unviscous acoustic medium (either finite or infinite) from a fluid-like infinite cylinder is described by the Helmholtz equation: ∇2 p(x) + k 2 p(x) = 0

(1)

which is to be satisfied in both the external medium Ωe carrying the incident wave pinc (x) and in the internal domain Ωi (see Fig. 1). The wave number k depends on the position of the point x, i.e.: k = ke = ω/ce with ce = ext. wave speed

in Ωe

(2a)

k = ki = ω/ci with ci = int. wave speed

in Ωi

(2b)

Figure 1: Infinite cylinder with cavity immersed in a fluid The total pressure in Ωe is taken as the sum of the incident wave and the scattered wave psc , i.e.: pe (x) = pinc (x) + psc (x)

(3)

The scattered pressure satisfies the Sommerfield radiation condition at infinity; scattered, incident and total pressure in Ωe and pressure in Ωi satisfy all the Helmholtz wave equation with different wave number. Continuity of the normal components of velocities and equilibrium of the pressures are enforced along the interface Γe : 1 1 qe (x) = − qi (x) ρe ρi

pe (x) = pi (x) 2

(4)

294


where ρe and ρi are external and internal densities respectively. The external and internal fluxes can be defined as ∂pe (x) ∂pi (x) qi (x) = (5) qe (x) = ∂ne (x) ∂ni (x) whereas the internal boundary Γi can be either soft or hard, i.e.: p(x) = 0

or

q(x) = 0

(6)

The flux is expressed in terms of the normal derivative of the pressure where n(x) is the outward normal to the boundary under analysis and it can assume the value of either ne or ni (see Fig. 1). The boundary integral equations corresponding to the field equation Eq. (1) can be written as: q ∗ (ξ, x)pe (x)dΓe (x) = p∗ (ξ, x)qe (x)dΓe (x) + pinc (ξ) (7) c(ξ)pe (ξ) + Γe

Γe

where ξ and x can move on the interface Γe , and c(ξ)pi (ξ) + q ∗ (ξ, x)pi (x)dΓ(x) = p∗ (ξ, x)qi (x)dΓ(x) Γ

(8)

Γ

with ξ and x collocating on both Γe and Γi . The points ξ and x are usually referred to as the source and field points, respectively; the fundamental solutions p∗ and q ∗ furnish the solution corresponding to a point disturbance in an infinite 2-D domain: i (1) p∗ (ξ, x) = H0 (kr) 4 ik (1) q ∗ (ξ, x) = − H1 (kr)r,n 4 (1)

(9a) (9b)

(1)

where H0 and H1 are the Hankel function of the first kind, 0th and 1st order respectively, r = x−ξ is the distance between the collocation point ξ and the field point x. The FMBEM coupling procedure For the above problem the conventional BEM numerical procedure is based on three steps: first, the discretisation of the boundary Γ = Γe ∪ Γi , second, the collocation of the Eq. (7) in every node of the interface Γe and of the Eq. (8) in each node (both on Γe and on Γi ) along with the boundary conditions Eqs. (4, third, the resolution of the system of equations of the type: ⎡

Hee

−Gee

0

0

⎢ i i ⎢ Hee − ρi Giee Hei −Giei ρe ⎢ ⎣ i ρi i Hie − ρe Gie Hii −Gii

⎤⎧ p ⎪ ⎪ ⎪ e ⎥⎪ ⎨ qe ⎥ ⎥ p Γi ⎦⎪ ⎪ ⎪ ⎪ ⎩ qΓ i

⎧ ⎪ pinc ⎪ ⎪ ⎪ ⎨ 0 = ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ⎭ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

(10)

after imposing the boundary conditions on the cavity, i.e. Eq. (6), in order to obtain a final square system of equations. In the present contribution constant elements are adopted: with such a choice most integrals involved are performed analytically. The discretised equation collocated at the generical node ξ i can be written as: n n c(ξi )p(ξ i ) + pj q ∗ (ξ i , x)dΓ(x) = qj p∗ (ξ i , x)dΓ(x) (11) j=1

Γj

j=1

The procedure requires the evaluation of either the integral of element. 3

p∗

Γj

or the integral of q ∗ on each boundary


295

For convenience, the complex notation is introduced, i.e. the collocation and field points are replaced by their complex representation: ξ = z0 = ξ1 + i ξ2

x = z = x1 + i x 2

(12)

The FMM relations intervene in the evaluation of integrals involved in the Eq. (11). The multipole expansion is the key point in reducing the CPU time which is necessary to evaluate each integral. If F (z0 , z)f indicates either p∗ (z0 , z)q or q ∗ (z0 , z)p, the following local expansion can be obtained: Γj

where:

F (z0 , z) f dΓ(z) =

∞ i (−1)p L−p (zL )Ip (z0 − zL ) 4 p=−∞

(13)

Ip (z) = (−i)p Jp (kr)eipθ

r, θ are the polar coordinates of z and Jp stands for the Bessel function of the The coefficients L−p are given by the following M2L translation: Ll (zL ) =

∞

Ok+l (zL − zC )P−k (zC )

(14) pth

order.

(15)

k=−∞

where z0 − zL 4( P, t n 1 ) 4( P, t n )@ 2 and substituting the value of heat source function g at the midpoint of the time step, one can rewrite Eq. (3) in the following form:

2C ( P )4( P, t n 0.5 ) 1 4S

³

*

1 2S

³

*

§ r 2 · wr 4(Q, t n 0.5 ) ¸¸ d* exp¨¨ r © 4D't ¹ wn

§ r2 · § r2 · 1 ¸¸d* ¸¸d: q (Q, t n 0.5 ) Ei¨¨ g (Q, t n 0.5 ) Ei¨¨ ³ : 4Sk © 4D't ¹ © 4D't ¹

(6)

ª r2 º 4 Q t ( , ) exp n « ' » d: C ( P )4( P, t n ) 4SD't ³ : ¬ 4D t ¼ 1

The displacement integral equation corresponding to Eq. (2) with b

C ij ( P )u j ( P )

³ U *

* ij

( P, Q )t j (Q ) T ( P, Q )u j * ij

0 can be written as follows [8]:

Eh (Q ) d* 1 2Q ³

: :H

4(Q )U ik* ,k ( P, Q )d:

(7)

where E is the Young modulus, Q is the Poisson ratio and h is the coefficient of linear thermal expansion. Cij represents the coefficient matrix of the free term, u j and t j are components of the displacement and traction vectors, respectively, and U ij* and Tij* are the fundamental solutions for the displacement and traction, respectively [8]. U ik* ,k in the domain integral in Eq. (7) can be expressed as follows:

336


(1 2Q ) r,i 4SG (1 Q ) r

U ik* ,k

(8)

There are two domain integrals in Eq. (3) and one in Eq. (7). The method used for evaluating these domain integrals is presented in next section. Evaluation of the domain integrals using the CTM The CTM is a general method for the evaluation of domain integrals with boundary-only discretization [3-5]. Consider the following 2D regular domain integral: (9) I p( x1 , x2 )d:

³

:

where : is the domain of the 2D region. Using Green's theorem, one can write [3]:

I

³ §¨© ³ *

p( x1c , x2 )dx1c ·¸dx2 ¹

x1 a

(10)

where * represents the boundary and a is an arbitrary constant. The integral in Eq. (10) can be computed as follows: K

I

¦³ ³ k 1

x1

*k a

p( x1c , x2 )dx1cdx2

(11)

where K is the number of boundary elements. The outer integral in Eq. (11) is evaluated by the Gaussian quadrature method, while the inner integral is evaluated by the composite Gaussian quadrature method. For these computations, the integrand p must be evaluated at some internal integration points. When the function p is defined by irregularly spaced data over a grid within the domain and on the boundary then this function can be evaluated by a meshfree interpolation method, such as the RPIM [6]. The interpolation approximation of the function p at an integration point can be expressed in the following general form [6]: M

p ( x1 , x2 )

¦I ( x , x ) p i

1

2

i

T P

(12)

i 1

where M is the total number of boundary nodes and internal grid points, pi is the value of the function p at the point i and Ii is its shape function. By using Eqs. (11) and (12), the following equation is obtained: M

I

¦J

q

pq

T p

(13)

q 1

where is the weight vector which depends on the geometry and location of grid points only, and the vector p contains the values of the function p at the boundary nodes and internal grid points. An important advantage of the present method, in contrast with the DRM, is represented by the fact that the existence of a particular solution for the interpolating shape functions is not required. Since the locations of the internal grid points are set, the weight vector is computed only once and used for the fast computation of all domain integrals. The first domain integral occurring in the temperature integral equation (6) is given by:

§ r2 · ¸¸d: g (Q, t n0.5 )Ei¨¨ : © 4D't ¹ The exponential integral function Ei( ) in Eq. (14) is weakly singular and is expressed as follows [7]: I1

³

Ei( x) Ei( x) ln( x)

(14)

(15)

where f

Ei ( x) 0.57721566 ¦ 1 n 1

n 1

xn n.n!

After some computations, the domain integral in Eq. (14) can be expressed as follows:

(16)


I1

ª § r2 · º ¸¸ ln(4D't )»d: g Q t ( , ) «Ei¨¨ 0 . 5 n ³: ' D t 4 ¹ ¬ © ¼ §1· §1· 2 ³ >g (Q, t n0.5 ) g ( P, t n0.5 )@ ln¨ ¸d: 2 g ( P, t n0.5 ) ³ ln¨ ¸d: : : r © ¹ ©r¹

337

(17)

The first and second integrals in Eq. (17) are regular and can be evaluated by Eq. (13). The third integral can be exactly transformed into a boundary integral using the CTM [4]. The second domain integral in Eq. (6) is:

I2

³

:

ª r2 º 4(Q, t n ) exp « »d: ¬ 4D't ¼

(18)

The integrand of this integral is not singular, but it becomes less smooth near the collocation point as 't o 0 , and therefore the integral must be carefully computed. Consequently, the integral given by Eq. (18) is regularized as follows:

I2

³ >4(Q, t :

n

ª r2 º ª r2 º ) 4( P, t n )@ exp « »d: 4( P, t n ) ³: exp « »d: ¬ 4D't ¼ ¬ 4D't ¼

(19)

The integrand of the first integral in Eq. (19) is now smoother near the collocation point and can be evaluated by Eq. (13). The second integral in Eq. (19) can be exactly transformed to the boundary using the CTM [4]. The domain integral appearing in the displacement integral equation (7) can be expressed as follows:

I3

³

:

r 4(Q) ,i d: r

(20)

This integral is weakly singular and can be regularized as follows:

I3

r,i

³ >4(Q) 4( P)@ r d: 4( P)³ :

:

r,i d: r

(21)

The first integral in Eq. (21) is regular and can be evaluated by Eq. (13), while the second integral in Eq. (21) can be exactly transformed into a boundary integral using the CTM [4]. An important point worth mentioning here is that the boundary nodes and internal grid points are constant in all time steps and therefore, the weight vector of integration, i.e. in Eq. (13), is constant in all time steps and does not change for the evaluation of all domain integrals. Evaluation of the stress in the domain The components of the stress tensor at an arbitrary point in the domain can be evaluated using the thermoelastic stress-displacement equation. For plane strain problems, this relation is given by:

V ij (x)

§ wu wu · Eh 2GQ wu k G ij G¨¨ i j ¸¸ 4(x)G ij w w x x 1 2Q wxk 1 2Q j i © ¹

(22)

The derivatives of the displacement at an arbitrary point can be evaluated using the RPIM. The components of the displacement vector at a point can be expressed by the RPIM as follows: ui ( x) T u i (23) where u i is a vector containing the values of ui at the grid points and is the vector of the RPIM shape functions at these points. Now the derivatives of the displacement components can be evaluated as [6]: ui , j (x) T, j u i (24) After the evaluation of the derivatives of the displacement components at a point, one can evaluate the stress components at that point using Eq. (22)

338


Example In this example, the domain shown in Fig. 1(a), under the following time-dependent and non-uniform heat source is considered:

g ( x, y )

(2 u 10 6 ) exp( x y ) f (t )

(W m3 )

where

t 10 f (t ) ® ¯1

t d 10 t t 10

The initial temperature is considered uniform and equal to 0.0qC. The material constants are given as: k 64 W/(m$ K ) , c 434 J/(kg $ K ) , U 7850 kg/m 3 , h 11.7 u 10 6 1/ $ K , E 210 GPa . For the BEM analysis of the problem, the boundary has been discretized using two different meshes, with 40 and 96 linear boundary elements, as shown in Fig. 1(b), while the time step size was set to 10 sec. The BEMbased results obtained have been compared with their corresponding FEM solutions. In the FEM analysis of the problem, the commercial software ANSYS has been employed, the domain has been discretized using a fine mesh with 5467 quadrilateral finite elements and 5658 nodes, see Fig. 2, and the time step size was set to 0.5 sec. Figs. 3a and b present the numerical results, obtained for the temperature and stress variation with respect to time, respectively, at the internal point A (0,0.15) and the boundary point B (0.1,0.1) . The results for the displacement variation with respect to time at the boundary point B in comparison with the corresponding FEM solution are shown in Fig. 4. As can be seen from these figures, the results obtained using the proposed BEM are in very good agreement with the FEM solution.

Figure 1: Configuration of the problem: a) the geometry and boundary conditions, b) boundary nodes and internal points

Figure 2: The FEM (ANSYS) discretization of the domain


339

Figure 3: Time-dependent variation of a) temperature, b) stress in x-direction

Figure 4: Time-dependent variation of displacement in y-direction at the point B Conclusions A BEM formulation for the analysis of uncoupled transient thermo-elasticity involving space- and timedependent heat sources was developed. In the proposed method, the domain integrals are evaluated without internal cells using the CTM and RPIM. There is no need to find particular solutions for the shape functions used in the interpolation computations in the meshfree RPIM, while the form of the shape functions can be arbitrary and sufficiently complicated. The time-dependent fundamental solution of the diffusion equation can also be efficiently employed for the thermal analysis. Since most of the generated matrices are constant in all time steps, the CPU time can be significantly reduced. References [1] T. Matsumoto, A. Guzika, M. Tanaka Int. J. Numer. Meth. Engng, 39, 1432–1458, (2005). [2] J. Chatterjee, F. Ma, D.P. Henry, P.K. Banerjee Comput. Meth. Appl. Mech. Engng., 196, 2828–2838, (2007). [3] M.R. Hematiyan Compul Mech, 39, 509-520, (2007). [4] M.R. Hematiyan Comun Numer Meth Engng, 24, 1497-1521, (2008). [5] A. Khosravifard, M.R. Hematiyan Eng Anal Bound Elem, 34, 30-40, (2010). [6] G.R. Liu, Y.T. Gu, An Introduction to Meshfree Methods and Their Programming , Springer (2005). [7] L.C. Wrobel The Boundary Element Method, Vol. 1: Applications in Thermo-Fluids and Acoustics, John Wiley & Sons (2002). [8] M.H. Aliabadi The Boundary Element Method, Vol. 2: Applications in Solids and Structures, John Wiley & Sons (2002).

340


Three-Dimensional Thermo-Elastoplastic Analysis by Triple-Reciprocity Boundary Element Method Yoshihiro OCHIAI 1

1

Department of Mechanical Engineering, Kinki University, 3-4-1 Kowakae, 577-8502, Higashi-Osaka, Japan, [email protected]

Keywords: Elastoplastic problem; Initial strain method; BEM; Thermal stress

Abstract. In general, internal cells are required to solve thermo-elastoplastic problems using a conventional boundary element method (BEM). However, in this case, the merit of the BEM, which is ease of data preparation, is lost. The triple-reciprocity BEM can be used to solve two-dimensional thermoelastoplasticity problems with a small plastic deformation without using internal cells. In this study, it is shown that three-dimensional thermo-elastoplastic problems with heat generation can be solved by the triple-reciprocity BEM without the use of internal cells. Initial strain and stress formulations are adopted and the initial strain or stress distribution is interpolated using boundary integral equations. A new computer program is developed and applied to solving several problems. 1. Introduction The finite element method (FEM) requires several repetitions of remeshing for large-plastic-deformation analysis. Elastoplastic problems can be solved by a conventional boundary element method (BEM) using internal cells for domain integrals [1]. In this case, however, the merit of the BEM, which is ease of data preparation, is lost. On the other hand, several countermeasures have been considered. Ochiai and Kobayashi proposed the triple-reciprocity BEM without using internal cells for elastoplastic problems [2]. By this method, a highly accurate solution can be obtained using only fundamental solutions of low orders and by diminishing the need for data preparation. Ochiai and Kobayashi applied the triple-reciprocity BEM (improved multiple-reciprocity BEM) without using internal cells to two-dimensional elastoplastic problems using initial stress and strain formulations [3]. In this study, the triple-reciprocity BEM is applied to three-dimensional thermo-elastoplastic problems with arbitrary heat generation. Initial strain and stress formulations are adopted and the theory is expressed using a few fundamental solutions. In this method, boundary elements and arbitrary internal points are used. The arbitrary distributions of the initial strain or stess for elastoplastic analysis are interpolated using boundary integral equations and internal points. This interpolation corresponds to a thin plate spline. In this method, strong singularities in the calculation of stresses at internal points become weak. A new computer program is developed and applied to several thermo-elastoplastic problems to clearly understand the theory.

2. Theory 2.1 Heat conduction Point and line heat sources can easily be treated by a conventional BEM. In this study, an arbitrarily distributed heat source W [1] S (q ) is treated. In steady heat conduction problems, the temperature T under an arbitrarily distributed heat source W [1] S (q ) is obtained by solving the following equation: W [1] S (q ) 2T , (1) O where O is thermal conductivity. The boundary integral equation for the temperature in the case of steady heat conduction problems is given by [2] wT (Q) wT [1] ( P, Q) CT(P) ³* {T [1] (P, Q) T (Q)}d*(Q ) O1 ³: T [1] ( p, q)W [1] S (q)d: , (2) wn wn


341

where C 0.5 on the smooth boundary and C 1 in the domain. The notations * and : represent the boundary and domain, respectively. The notations p and q become P and Q on the boundary, respectively. In the case of three-dimensional problems, the fundamental solution T [1] (p, q) in Eq. (2) for steady temperature analysis and its normal derivative are given by 1 T[1] (p, q) , (3) 4S r

wT [1] ( p, Q) wn

1 wr , 4S r 2 wn

(4)

where r is the distance between the observation point p and the loading point q. As shown in Eq. (2), when the arbitrary heat generation W [1] S ( q ) occurs in the domain, the domain integral is necessary. Therefore, the

triple-reciprocity BEM [5,6] is used. The distribution of the arbitrary heat generation W1S ( q ) in the case of a three-dimensional problem is interpolated using integral equations to transform the domain integral into a boundary integral. The following equations are used for interpolation [3,4]: 2W [1]S (q )

W 2

[2]S

W [ 2]S (q) M

(q)

¦ W

,

(5)

(qm ) ,

(6)

[ 3 ] PA

m 1

where the function W [3] PA (qm ) expresses the state of a uniformly distributed polyharmonic function in a spherical region with the radius A. The function T [ f ] ( p, q ) is defined as

2T [ f 1] ( p, q ) T [ f ] ( p, q ) . The function T

[f]

(7)

( p, q ) can be expressed as r 2 f 3 . 4S (2 f 2) !

T [ f ] ( p, q )

(8)

Figure 1 shows the shape of polyharmonic functions; the biharmonic function T [ 2] is not smooth at r 0 . In a three-dimensional case, a smooth interpolation cannot be obtained using solely the biharmonic function T [ 2 ] ( p, q ) . To obtain a smooth interpolation, the polyharmonic function with the volume distribution T [ 2 ] A ( p, q ) is introduced. A polyharmonic function with the volume distribution T [ f ] A ( p, q ) , as shown in Fig. 1, is defined as [5]

T [ f ] A ( p, q )

A

2S

S

³0 [ ³0 {³0

T [ f ] ( p, q )a 2 sin T dT } dI ] da .

(9)

The function T ( p, q ) can be easily obtained using the relationships r R a 2 aR cos T and dr aR sin T dT , as shown in Fig.1. This function is written using r instead of R, similarly to Eq. (8), 2

[ f ]A

2

2

although the function in Eq. (9) is a function of R. The newly defined function T [ f ] A ( p, q ) can explicitly be shown as 1 ^(2 fA r )(r A) 2 f (2 fA r )(r A)2 f ` r ! A , T [ f ] A ( p, q ) (10) 2r 2 f 1 !

T [ f ] A ( p, q )

1 ^(2 fA r )( A r )2 f (2 fA r )( A r )2 f ` r d A . 2r 2 f 1 !

(11)

Using Eqs. (5), (6), and (7), and Green's second identity, Eq. (2) becomes wT (Q ) wT [1] ( P, Q) T (Q )}d*(Q ) wn wn (12) [f] 2 M wW (Q) wT [ f 1] ( P, Q) [ f ] O1 ¦ (1) f ³* {T [f 1] (P, Q) W (Q )}d*(Q ) O1 ¦ T [3] A ( P, q m )W [ 3] P (q m ) . wn wn m 1 f 1

CT(P)

³* {T

[1]

(P, Q)

2.2 Initial strain formulation

342


To analyze thermo-elastoplastic problems using the initial strain formulation, the following boundary integral equation must be solved [1,2]. cij ( P, )u j ( P)

³

*

[uij[1] ( P, Q) p j (Q) pij ( P, Q)u j (Q)]d*

³

:

[1] 1] V [jki ( P, q )HI jk (q )d:

wuiT (1) ( P, Q ) wT (Q ) T (1) ui ( P, Q )}d*(Q ) wn wn T [ f 1] M 2 wu ( P, Q ) [ f ] wW [ f ] (Q) O1 ¦ (1) f ³ǌ[ i W (Q) uiT [ f 1] ( P, Q) ]d* O1 ¦ uiT [3] A ( P, q m )W [3] P (q m ) w n w n f 1 m 1

³* {T (Q )

(13)

Here, HI [jk1] (q) is the initial strain rate and cij is the free coefficient. Moreover, u j (Q ) and p j (Q ) are the j-th components of the displacement and surface traction rates, respectively. On the other hand, * and : are the boundary and the domain, respectively. As shown in Eq. (13), when there is an arbitrary initial strain rate, a domain integral becomes necessary. Denoting the distance between the observation point and the loading point by r, Kelvin's solution uij[1] ( p, q) and pij ( p, q) are given by 1 {(3 4Q )G ij r ,i r , j } , (14) 16S (1 Q )Gr 1 wr pij ( p, q) {[(1 2Q )G ij 3r ,i r , j ] (1 2Q )(r ,i n j r , j ni )} , (15) 8S (1 Q )Gr 2 wn where Q is Poisson's ratio and G is the shear modulus. The i-th component of a unit normal vector is [1] ( p, q) , uiT [ f ] ( p, q), wuiT [ f ] ( p, q) / wn denoted by ni . Moreover, let us set r ,i wr / wxi . The functions V ijk uij[1] ( p, q )

and uiT [3] A ( p, q ) in Eq. (13) are given by [1, 9]

V [jki1]] ( p, q) uiT [ f ] ( p, q )

1

{(1 2Q )(G ji r , k G ki r , j G jk r , i ) 3r , i r , j r , k } ,

8S ( 1 Q )r 2

m0T ,[i f 1) ( p, q )

wuiT [ f ] ( p, q) wn

m0 (2 f 1)r ,i r 2 f 2 , 4S (2 f )!

(17)

m0 ( 2 f 1)r 2 f 3 4S ( 2 f )!

m0T ,[i f 1] A ( p, q )

uiT [3] A ( p, q )

(16)

wr º ª «ni (2 f 3)r ,i wn » , ¬ ¼ m0 A3 r ,i (105r 6 189 r 4 A2 27 r 2 A4 A6 ) , r!A 45360 r 2

m0 rr ,i (r 6 27 r 4 A2 189r 2 A4 105 A6 ) . 45360 Denoting the coefficient of linear thermal expansion as D, m0 is given by m0

(18) (19)

rdA

uiT [3] A ( p, q)

(20)

(1 Q )D /(1 Q ) .

In an initial stress formulation using initial stress increments V [Ijk1] (q) , the domain integral 3 in Eq. (13)

is replaced by 3

³

:

[1] [1] H ijk ( P, q)V Ijk (q) dǡ ,

(21)

where [1] H ijk ( p, q ) [(1 2ǵ)(G ij r , k G ik r , j ) G jk r ,i 2r ,i r , j r , k ]

1 . 8S (1 ǵ)G r

(22)

2.3 Interpolation of initial strain Interpolation using boundary integrals is introduced to avoid the domain integral in Eq. (1). The distribution [1] S of the initial strain HI jk (q) in the case of a three-dimensional problem is interpolated using the integral

equation to transform the domain integral into a boundary integral. The following equations are used for interpolation [3,4]:

2HI jk (q) HI jk (q) , [1] S

[ 2] S

(23)


343

M

[ 2]S [ 3] PA 2HI jk (q) ¦ HI jk (qm ) ,

(24)

m 1

where 2

w 2 / wx 2 w 2 / wy 2 w 2 / wz 2 . From Eqs. (23) and (24), we obtain

4HI jk (q) [1] S

M

[ 3] PA ¦ HI jk (qm )

,

(25)

m 1

[ 3] PA where the function HI jk (q) expresses the state of a uniformly distributed polyharmonic function in a

spherical region with the radius A. We must emphasize that Eqs. (23) and (24) can be used for interpolating [1] S the complex distribution of the initial strain HI jk (q) . These equations are the same as those used for generating a free-form surface using an integral equation [6]. In this method, each initial strain component [1] S HI jk (q) ( j, k=1, 2, 3) is interpolated. 2.4 Representation of initial strain by integral equations The distribution of the initial strain is represented by integral equations. Denoting the number of points [ 3] P [ 2] S HI jk (q) as M, the curvature of the initial strain rate HI jk (q) is given by Green's second identity and Eq.

(24) as [3,4]

wHI jk (Q ) [ 2]S

CHI jk ( P )

³* {T

[ 2]S

[1]

( P, Q )

wn

M w T [1] ( P, Q ) [ 2 ] S HI jk (Q )}d* ¦ T [1] A ( P,q m )HI [jk3] PA ( qm ) . wn m 1

(26)

[1] The initial strain rate HI jk ( P) is given by Green's theorem and Eqs. (23) and (24) as [3,4] 2

w HI jk (Q )

f 1

wn

[ f ]S

¦ ( 1) f ³* {T [ f ] ( P, Q )

CHI jk ( P ) [1] S

w T [ f ] ( P, Q ) [ f ] S HI jk (Q )}d* wn

M

[ 3 ] PA ¦ T [ 2 ] A ( P, qm )HI jk ( qm ) ,

(27)

m 1

where C

0.5 on the smooth boundary and C 1 in the domain. It is assumed that

HI [jk2 ] S

(Q) is zero. For

internal points, the next equation is obtained similarly to Eq. (27). [ f ]S 2 w HI jk (Q ) wT [ f ] ( p, Q ) [ f ] S [1] S HI jk (Q)}d* cHI jk ( p ) ¦ (1) f ³ {T [ f ] ( p, Q ) * wn

wn

f 1

M

¦T

[ 2] A

( p, qm )HI jk

[ 3] PA

( qm )

(28)

m 1

If the boundary is divided into N0 constant elements and N1 internal points are used, simultaneous linear algebraic equations with (2N0+N1) as unknowns must be solved. 2.5 Triple-reciprocity boundary element method The function V [jkif ] ( p, q ) is defined as

2V [jkif 1] ( p, q ) V [jkif ] ( p, q ) .

(29)

Using Eqs. (23), (24), and (29), and Green's second identity, Eq. (13) becomes cij ( P )u j ( P )

³* [uij

[1]

wHI jk (Q ) [ f ]S

V [jkif 1] ( P, Q ) O1

2

f

w V [jkif 1] ( P, Q )

f 1

wn

wn

M

}d* ¦ V [jki3] A ( P, q )HI jk ( m ) (q ) ³* {T (Q ) m 1

[ 3 ] PA

HI [jkf ] S (Q )

wu iT [1] ( p, Q ) wT (Q ) T [1] ui ( p, Q )}d* (Q ) wn wn

M wu T [ f 1] ( P, Q ) [ f ] wW [ f ] (Q ) 1 ui[3] A ( P, qm )W [3] PA (qm ) . (30) [ i ]d* O W (Q ) uiT [ f 1] ( P, Q ) * wn wn m 1

¦ (1) ³ f 1

2

( P, Q ) p j (Q ) pij ( P, Q )u j (Q )]d* ¦ (1) f ³* {

¦

344


V ijk[ f ] ( p, q ) is obtained as [4] ( 2 f 1)(2 f 3)r 2 f 4 {(2 f 1 2 fQ )(G jk r ,i G ik r , j ) (1 2 fQ )G ij r , k (2 f 5)r ,i r , j r , k } .(31) 4S (1 Q )(2 f ) !

V ijk[ f ] ( p, q)

Moreover, the normal derivatives wV ijk[ f ] ( p, q ) / wn and V ijk[ 3] A ( p, q ) are given by [f] wV ijk ( p, q )

( 2 f 1)(2 f 3) 2 f 5 r [(2 f 5){(2 f 1 2 fQ )(G jk r ,i G ik r , j ) 4S (1 Q )(2 f ) ! wr (1 2 fQ )G ij r , k (2 f 7)r , i r , j r , k } ( 2 f 5)(r , j r , k ni r ,i r , k n j r ,i r , j nk ) wn (2 f 1 2 fQ )(G jk ni G ik n j ) (1 2 fQ )G ij nk ] , (32) wn

V ijk[ 3] A ( p, q )

A3 {18QG ij r , k r 2 (35r 4 14 A2 r 2 A4 ) 15120(1 Q )r 4

(G jk r ,i G ik r , j G ij r , k )(105r 6 63 A2 r 4 9 A4 r 2 A6 ) r ,i r , j r , k (105r 6 63 A 2 r 4 27 A4 r 2 5 A6 ) 18(1 Q )(G jk r , i G ik r , j )r 2 (35r 4 14 A2 r 2 A4 )} r ! A , (33)

V ijk[ 3] A ( p, q )

r {9QG ij r , k ( r 4 14 A 2 r 2 35 A 4 ) (G jk r ,i G ik r , j G ij r , k )(r 4 18 A2 r 2 63 A4 ) 7560 (1 Q )

4r ,i r , j r , k r 2 ( r 2 9 A2 ) 9(1 Q )(G jk r ,i G ik r , j )( r 4 14 A2 r 2 35 A4 )} r d A . (34) 3. Numerical examples The stresses in a partially heated cube with 10 mm side length shown in Fig. 1 are obtained. The temperature at the upper surface ( 5 u 10 ) is 300 ͠, and the temperature at the other part of the upper surface is adiabatic. The temperature at the y=0 and 10 surfaces is adiabatic. The temperature at the other three surfaces is 0 ͠. Young’s modulus E =210 GPa, Poisson’s ratio Q 0.3 , thermal expansion D 0.000011 , yield stress V 0 250 MPa and strain hardening H 0.05E are assumed. The cube is restricted in the y direction. Figure 2 shows the temperature distribution. Figure 3 shows the plastic zone obtained for different temperature. To ensure the accuracy of the present method, the thermal stresses in a short solid circular cylinder with 5 mm height, the radius of which is A as shown in Fig. 4, are obtained under heat generation: W

W0

( A2 r 2 ) , A2

(53)

where W0 / O 10C $ / mm 2 . The temperature is 0 ͠ at outer surface, and A=10 mm is assumed. The upper and lower surfaces are adiabatic, and the solid cylinder is restricted in the z direction. Young’s modulus E =210 GPa, Poisson’s ratio Q 0.3 , thermal expansion D 0.000011 , yield stress V 0 250 MPa and strain hardening H 0.05E are assumed. Figure 5 shows the temperature distribution with the exact solution. Figure 6 shows the circumferential, radial, axial and equivalent stress distributions. Boundary element results are shown with FEM solutions in Fig. 6.

REFERENCES [1] Telles, J. C. F., The Boundary Element Method Applied to Inelastic Problems, Springer-Verlag, Berlin, (1983). [2] Ochiai, Y. and Kobayashi, T., Initial Strain Formulation without Internal Cells for Elastoplastic Analysis by Triple-Reciprocity BEM, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 1877-1892, (2001). [3] Ochiai, Y. and Sladek, V., Numerical Treatment of Domain Integrals without Internal Cells in ThreeDimensional BIEM Formulations, CMES , Vol. 6, No. 6, pp. 525-536, (2004). [4] Ochiai Y., Three-Dimensional Thermal Stress Analysis by Triple-Reciprocity Boundary Element Method, International Journal for Numerical Methods in Engineering, Vol. 63, No. 12, pp.1741-1756, (2005)


(a) Boundary elements

(b) Internal points Fig.1 Partially heated cube

Fig.2 Temperature distribution (y=5mm)

Fig.3 Plastic zone obtained for different temperature (y=5mm)

Fig.4 Cylinder with heat generation

Fig.5 Temperature distribution in cylinder (y=2.5mm) Fig.6 Stress distribution in cylinder (y=2.5mm)

345

346


ELASTOPLASTIC ANALYSIS FOR ACTIVE MACRO-ZONES VIA MULTIDOMAIN SYMMETRIC BEM Panzeca T.1,a, Parlavecchio E.1,b, Terravecchia S.1,c, Zito L.1,d 1

Disag, Viale delle Scienze, 90128 Palermo, Italy, a

[email protected], [email protected],

c

[email protected] ,[email protected]

Keywords: SBEM; multidomain; elastoplasticity; active macro-zones; return mapping algorithm. Abstract. In this paper a strategy to perform elastoplastic analysis by using the Symmetric Boundary Element Method (SBEM) for multidomain type problems is shown. This formulation uses a self-stresses equation to evaluate the trial stress in the predictor phase, and to provide the elastoplastic solution in the corrector one. Since the solution is obtained through a return mapping involving simultaneously all the plastically active bem-elements, the proposed strategy does not depend on the path of the plastic strain process and it is characterized by computational advantages due the considerable decrease of the plastic iterations number. This procedure has been developed inside Karnak.sGbem code [1] by introducing an additional module. Introduction A multi-domain SBEM strategy [2], based on an initial strain approach, is applied for the analysis of 2D structures, in the hypothesis of elastic-perfectly plastic behaviour, von Mises model, associated flow rules and strain plane state. Let us start from the discretization of the domain in substructures (in analogy with the finite elements methods), called bem-elements, where the plastic strain accumulation have to be valuated. Then let us impose the regularity conditions, in strong form on the displacements (nodal compatibility) and in weak form on the tractions (generalized equilibrium) both evaluated on the interfaces boundary, and let us effectuate a strong variable condensation. This procedure provides a self-equilibrium stresses equation governing the elatoplastic problem and connecting stresses, valuated on the each bem-e strain points, to plastic strains, treated as volumetric distortions, through an influence matrix (stiffness matrix), negative semi-definite as for the finite elements. The same equation is used both in order to valuate the predictors within the elastic phase, and to correct the elastic solution. In the first phase the use of only self-stresses equation offers the advantage to evaluate the predictor in simple way. Indeed this equation contains influence coefficients depending on both known imposed plastic strains and the external actions amplified by load multiplier. For the generic load increment, it permits to locate all the bem-elements in which the plastic admissibility condition is violated, i.e. to define the active macro-zones which require correction techniques. Then, in the second phase, the trial solution is corrected by a return mapping algorithm, which is defined in according at the extremal paths theory [3],,in this approach used within a discrete problem. The proposed algorithm permits the simultaneous correction of the elastic solution in all the plastically active bem-elements and utilizes the same self-stresses equation in a nonlinear global system of 4xa equations in 4xa unknowns, where a is the active bem-elements number. In the present approach the approximate solution is easily obtained by using the well-known standard Newton-Raphson procedure, just used in elastoplastic problems within the Bem formulations by some authors [4,5]. In order to prove the efficiency of the proposed strategy, a numeral test, performed by the Karnak.sGbem code [1], is shown at the end of this paper.


347

1. Self-stresses equation via multidomain SBEM In this section the procedure utilized to obtain, by using the SBEM for multi-domain type problems, the equation connecting the stresses to the imposed volumetric strains, through a stiffness matrix involving all the bem-elements in the discretized system, is shown. Let us consider a bi-dimensional body having domain : and boundary * , subjected to actions acting in its plane: - forces f at the portion * 2 of free boundary, - displacements u imposed at the portion *1 of constrained boundary, - body forces b and plastic strains p in : . The external actions f , u, b may increase separately or simultaneously through the multiplier E . In the hypothesis that the physical and geometrical characteristics of the body are zone-wise variables, an appropriate subdivision of the domain in bem-elements is introduced. This subdivision involves the introduction of an interface boundary * 0 between contiguous bem-elements and, as a consequence, two new unknown quantities rising in the analysis problem, i.e. the displacements u 0 and the tractions t 0 vectors, both referred to interface boundaries. The adopted strategy [2] contemplates the study of each bem-e embedded in a unlimited domain having the same physical properties and the same thickness of the examining bem-e. It is necessary to distinguish the boundary as * of : or as * of the complementary domain : f \ : . As a consequence the boundary quantities take on a different meaning: the forces acting on the boundary must be interpreted as layered force distribution, whereas the displacements must be thought as a double layered displacement one.

1.1 Characteristic equations of the bem-e Let us start by imposing for each bem-e the following Dirichlet and Neumann conditions: u1

u1

on *1 ,

t2

f2

on * 2

(1a,b)

and introducing the Somigliana Identities (S.I.) of the displacements and of the tractions in the previous eqs.(1a,b). The following integral equation system is obtained:

u1

u1[f1 , u 2 , f0 , u 0 ] E u1[ f2 , u1 , b] u1[ p ]

on *1

t2

t 2 [f1 , u 2 , f0 , u 0 ] E t 2 [ f2 , u1 , b] t 2 [ p ]

on * 2

u0

u 0 [f1 , u 2 , f0 , u 0 ] E u 0 [ f2 , u1 , b] u 0 [ p ]

on * 0

t0

t 0 [f1 , u 2 , f0 , u 0 ] E t 0 [ f2 , u1 , b] t 0 [ p ]

on * 0

(2a-d)

p

where the vector represents the inelastic strains due to thermal or plastic actions, whose presence requires domain integrals having singular kernels, suitably studied [6,7]. The eqs.(2a-c) have to be interpreted as the response of the body on the boundaries *1 , * 2 , * 0 , respectively, with the free terms opposite in sign, whereas eq.(2d) has the meaning of traction valued on the actual interface boundary * 0 . u1 [f1 , u 2 , f0 , u 0 ] E u1 [ f2 , u1 , b] u1 [ p ] 0

on *1

t 2 [f1 , u 2 , f0 , u 0 ] E t 2 [ f2 , u1 , b ] t 2 [ p ] 0

on * 2

u [f1 , u 2 , f0 , u 0 ] E u [ f2 , u1 , b] u [ ] 0

on * 0

0

t0

0

0

P

t 0 [f1 , u 2 , f0 , u 0 ] E t 0 [ f2 , u1 , b ] t 0 [ p ]

(3a-d)

on * 0

In addiction, let us introduce the stress vector: [f1 , u 2 , f0 , u 0 ] E [ f2 , u1 , b ] [ p ]

on :

(3e)

and the boundary discretization into boundary elements by making the following modelling of all the known and unknown quantities:

f1 = t F1 , f2 = t F2 , t 0 = t F0 , u 2 = u U 2 , u1 = u U1 , u0 = u U0 , p

p p

(4a-g)

348


where t and u are shape functions regarding the boundary quantities, while p are domain shape functions used to model plastic strains p connected to the Gauss points of the bem-e. Besides, the capital letters F and U indicate the nodal vectors of the forces ( F1 on *1 and F0 on * 0 ) and of the displacements ( U 2 on * 2 and U 0 on * 0 ) defined on the boundary elements. Let us perform the weighting of all the coefficients of the eqs.(3a-d). For this purpose, the same shape functions as those modelling the causes have been employed, but introduced in an energetically dual way in according to the Galerkin approach. In this way it is possible to obtain the following block system:

0 0

A u1,u1 A f 2,u1 A u 0,u1

A u1,f 2 A f 2,f 2 A u 0,f 2

A u1,u 0 A f 2,u 0 A u 0,u 0

A u 0,f 0

U 2 A f 2,V F0 A u 0,V

P0

A f 0,u1

A f 0,f 2

A f 0,u 0

A f 0,f 0

U 0

0

A u1,f 0 A f 2,f 0

F1

A u1,V p E

A f 0,V

ˆ W 1 ˆP 2

(5a-d)

ˆ W 0 ˆ L 0

where the first and second rows represent the Dirichlet and Neumann conditions (1a,b) written in weighted u 0 and ³ u t 2 0 . The third and fourth rows regard the weighting of the displacements form ³ *1 f 1 *2 T u ³*c f c 0 and of the tractions at the interface zone. In particular the vector P0 ³* u t 0 in eq.(3d) collects the generalized tractions defined on the boundary elements of * 0 . The influence matrix, containing 4x4 block matrices, is symmetric. The introduced coefficient E is the multiplier of the boundary (U1 ) , F2 and domain b actions. Eq.(3e) defines the field stress, obtained through the S.I., i.e.: 0

aV,u1 aV,f 2

a V ,u 0

a V,f 0

F1 U 2 F0 U0

aV,V

p E ˆl V

.

(5e)

The eqs.(5a-e) may be expressed in compact form in the following way: 0 AX A 0 U 0 AV p E Lˆ P0

X A U A p E Lˆ A 0 00 0 0V 0

(6a-c)

aV X aV 0 U 0 aVV p E ˆlV where the vector X collects the sub-vectors F1 , U 2 and F0 , whereas the U 0 and p vectors characterize the displacements of the nodes in the interface zone, changed in sign, and the nodal plastic strains, respectively. The vector P0 represents the generalized (or weighted) traction vector defined in the boundary elements of the interface zone, obtained as a weighted response to all the known, amplified by E , and unknown actions, regarding boundary and domain quantities. The vector represents the stress, valued at the Gauss points, due to the all the known, amplified by E , and unknown actions. By performing a variables condensation through the replacement of the X vector extracted from eq.(6a) into eqs.(6b,c), one obtains:

D00 U 0 D0V p E Pˆ 0 dV 0 U 0 dVV p E ˆ P0

(7a,b)

These latter are the equations characteristic of each bem-e. They relate the generalized (or weighted) tractions P0 defined on the interface zone * 0 and the stresses at the bem-e domain to the nodal displacements U 0 , to the plastic strains p and the two load terms Pˆ 0 and ˆ amplified by E , respectively. These latter represent the generalized tractions vector along the interface boundary and the stresses vector in the domain with reference to each bem-e, as elastic response. Moreover, D00 , D0 V , d V 0 , d VV are the stiffness matrices of the bem-e, being D00 and d VV square matrices, D0 V and DV 0 rectangular ones.


349

1.2 Assembled system and self-stresses equation Let us subdivide the body in m bem-elements and consider for each of these the eqs.(7a,b). Thus we obtain two global relations connecting all the generalized tractions and the stresses related to the bem-elements considered, formally equal to the same eqs.(7a,b), but regarding the constitutive equations of the assembled system. Let us introduce the compatibility among the nodal displacements of the adjacent bem-elements: U0

(8)

H 0

where H is a topological matrix and 0 the nodal displacements vector of the assembled system, and the equilibrium condition among generalized tractions at the interface boundaries. HT P0

(9)

0

Using the previous eqs.(8,9), the eqs.(7a,b) become: K 00 0 K 0V p E fˆ0

0

(10a,b)

k V 0 0 k VV p E ˆ

By performing a new variables condensation through the replacement of the 0 vector extracted from eq.(10a) into eq.(10b), it is obtained:

Z p E ˆ s

(11)

where Z

1 k V 0 K 00 K 0V k VV , ˆ s

1 ˆ k V 0 K 00 f0 ˆ

(12a,b)

The eq.(11) provides the stress at the strain points of each bem-e in function of the volumetric plastic strain p and of the external actions ˆ s , the latter amplified by . The matrix Z , defined self-stresses influence matrix of the assembled system, is a square matrix having 3mx3m dimensions with m bem-elements number, full, non symmetric and semi-defined negative. The evaluation of this matrix involves only the elastic characteristic of the material and the structure geometry. The matrix Z permits to evaluate the elastic response in the Gauss points of all the bem-elements due to the plastic stain vector p , whereas the vector ˆ s collects the influence coefficients, as response to the known external actions F2 , U1 , b . 2. Active macro-zones analysis

In this section the strategy to compute the plastic strains for each loading step and at every bem-e is shown. These approaches utilize eq.(11) both to evaluate the predictors and during the corrector phase, here after shown. Let us start computing the trial stresses, i.e. the purely elastic response at the instant n 1 in each m bemelements of the discretized body. For thus purpose, eq.(11) provides all the predictors * n 1 as function of the plastic strain p n , stored up at the previous step and then imposed as volumetric distortions, and of load increment E n 1 : * n 1

Zp n E n 1 ˆ s

(13)

where Z matrix is full and regards all the bem-elements, obtained by the discretization. The check of the plastic consistency condition of the stresses computed on appropriately chosen points is performed by using the yield condition expressed in this context through the von Mises law for each bem-e: F [ n 1 ]

1 2

T n 1 M n 1 2y d 0 .

(14)

In the a bem-elements (with a d m ) where this latter inequality is violated, a return mapping phase occurs to evaluate the plastic strains and the direction of the plastic flow.

350


This phase, called corrector phase, uses the same eq.(11) to obtain the elastoplastic solution at every bem-e where the plastic consistency condition is violated. In this phase the vector , representing the end step stress, as well as the volumetric plastic strain vector p are unknown quantities. This latter is the plastic strain to impose at every active plastically bem-e in order to have the stress on the yield boundary of the elastic domain, through which the direction of the plastic flow may be defined. Obviously, inside of each loading step the corrector phase has to be repeated until all the predictors do not satisfy the plastic consistency conditions. In detail eq.(11), written for every h bem-elements ( h = 1,...,a ), is utilized to perform the elastoplastic analysis at n 1 load step simultaneously in all the plastically active macro-zones individuated in the previous predictor phase, i.e.:

a *a Z aa p a

(15)

0

where the subscript n 1 has been omitted for convenience. The Z aa matrix coefficients derive from the Z matrix present in eq.(15), by extracting the blocks relative to the a plastically active bem-elements. The double index specifies the bem-elements where the plastic strains (cause) and the related stresses (effect) arise. Let us introduce the plastic admissibility conditions for the a bem-elements:

F [ a ] d 0 , a t 0 , a F [ a ] = 0

(16a-c)

In the hypothesis that, for each h-th bem-e, the shape function definite in eq.(4g) is the same of the shape function related to the plastic multiplier, i.e. Oh p h with p t 0 , the plastic strain for the h-th active bem-e is expressed as:

ph

/h

wF w h

/h M h

(17)

The solving non linear system for all the active bem-elements is the following: a * ° f Ih { h h ¦ / k Z hk M k 0 k 1 ® ° f { 1 T M 2 0 with h 1,...a h y ¯ IIh 2 h

(18a,b)

where h is the stress solution located on the yield surface of the elastic domain, *h the elastic predictor, / h Z hh M h the direct corrective components (stress in the h-th bem-e due to distortion p h applied on the ah same bem-e) and ¦ k 1 / k Z hk M k the indirect corrective components (stress in the h-th bem-e due to distortion p k applied on the k-th bem-e) respectively. The eqs.(18a,b) comprises a system of 4xa non linear equations in 4xa unknowns (three stress components h and a plastic multiplier / h for each active bem-e). The approximate solution of this nonlinear problem involving all the plastically active bem-elements is here obtained by applying the Newton-Raphson procedure as follows: I 1j Z11 M " #

aj Z a1 M ( ) M # 0 j T 1

aj Z1a M

% # " I aj Z aa M "

0

% # " ( aj )T M

Z11 M 1j #

" Z1a M aj % #

Z a1 M 1j " Z aa M aj 0

"

0

# 0

% "

# 0

1j 1 1j # .

f I 1j #

f Iaj aj 1 aj = j 1 j 1 1 f IIj1 # # aj 1 aj

(19)

f IIaj

which, written in compact form, becomes: X aj 1

X aj J a ( X aj ) 1 f ( X aj ) .

(20)


351

The Jacobian matrix J a contains the derivatives of the functions defined in eqs.(18a,b), X aj 1 is the vector of the unknowns, X aj and f ( X aj ) are the known vectors evaluated in the j-th step. The vector X a in the j+1-th step is the solution in terms of stress and plastic multipliers evaluated on the Gauss points of all the plastically active bem-elements. Since the Jacobian operator J a ( X aj ) usually has big dimensions which coefficients have different meaning, its inverse and update could require high computational cost. In order to overcome these disadvantage, the following strategy, able to reduce the computational burden in the iterative process, is been developed. Let us consider the system of eqs.(19) here rewritten in compact form: j J V/ j 1 j . j 1 0 j

j J VV j J /V

FVj F/j

(21)

and in explicit form: j j 1 j j j 1 j j °J VV ( ) J V/ ( ) FV ® j j 1 j j °¯J /V ( ) F/

(22a,b)

Let us perform a condensation of variables by extracting the vector ( j 1 j ) from eq.(22a): j j ( j 1 j ) ( J VV ) 1[ J V/ ( j 1 j ) FVj ]

(23)

and replacing it into eq.(22b). It is obtained: j J // ( j 1 j ) F /j

(24)

where: j J //

j j j J /V ( J VV ) 1 J V/ , F /j

j j J /V (J VV ) 1 FVj F/j

(25a,b)

The proposed algorithm shows high computational efficiency because the inversion is related to only two j j and J VV of reduced dimensions. blocks J // Z is a square matrix. It is written only once and its dimension depends on the bem-elements number as a result of the discretization. This matrix is used twice: as Z having n dimension in the predictor phase second its origininary form, as Z aa having a d h dimension during the return mapping phase with reference to the plastically active bem-elements, i.e. those bem-elements where the predictor do not satisfy the yield condition. The peculiarity of the shown approach is that the return mapping process is used simultaneously in the plastically active bem-elements, avoiding the return mapping strategy for single bem-e, it showing arbitrariness and very onerous computational burden. In addition the proposal to perform simultaneously on the plastically active bem-elements involves iterative return mapping process with a very high saving in the computational times. 3. Numerical results In order to show the efficiency of the proposed method, a traction test, by using the SBEM code Karnak.sGbem [1], has been performed. In the present section a square plate with circular hole is subjected to tensile load q 100000 daN / m , as shown in Fig.1. The material characteristics are the Young’s modulus E 200000 daN / cmq and the Poisson’s ratio Q 0.29 , whereas the uniaxial yield value is V y 4500 daN / cmq . The plate geometry, shown in Fig.1, has unit thickness. The load-displacement curve is shown in Fig.2 and the solution was compared to the strongly iterative solution in the sphere of SBEM [8].

352


q

200 mm

Uy

20

90 mm

90 mm

q

a)

b)

Fig. 1. Plate with circular hole: a) problem description; b) adopted mesh.

4

3

Present approach Iterative approach

2

1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Uy

Fig. 2. Load – displacement curve.

References [1] Cucco F, Panzeca T, Terravecchia S.,(2002). The program Karnak.sGbem Release 2.1. [2] Panzeca T., Cucco F., Terravecchia S., (2002). Symmetric boundary element method versus Finite element method. Comp. Meth. Appl. Mech. Engng.; 191: 3347-3367. [3] Ponter A.R.S., Martin J.B., (1972). Some extremal properties and energy theorems for inelastic materials and their relationship to the deformation theory of plasticity. J. Mech. Phys. Solids; 20: 281-300. [4] Bonnet M., Mukherjee S., (1996). Implicit BEM formulations for usual and sensitivity problems in elastoplasticity using the consistent tangent operator concept. Int. J. Solids & Struct.; 33: 4461-4480. [5] Mallardo V., Alessandri C., (2004). Arc-length procedures with BEM in physically nonlinear problems. Engng. Anal. Bound. Elem.; 28: 547-559. [6] Panzeca T., Terravecchia S., Zito L., (2010). Computational aspects in 2D SBEM analysis with domain inelastic actions. Int. J. Num. Meth. Engng.; 82: 184-204. [7] Gao X.W., Davies T.G., (2000). An effective boundary element algorithm for 2D and 3D elastoplastic problem. Int. J. Solids & Struct.; 37: 4987-5008. [8] Panzeca T., Cucco F., Parlavecchio E., Zito L. (2009). Elastoplastic analysis by the Multidomain Symmetric Boundary Element Method. Xth International Conference on Boundary Element Techniques, (Beteq 2009).


Interaction problems between in-plane and out-plane loaded plates by SBEM. Panzeca T.1, Cucco F. 2, La Mantia A. 1, Salerno M. 1 1

DISAG, Università di Palermo, 90128 Palermo, [email protected], alessandro.lamantia@ unipa.it, [email protected]

2

Università Kore di Enna, 94100 Enna, Italy, , [email protected]

Keywords: Symmetric Boundary Element Method, in-plane loaded plate, out-plane loaded plate, multidomain approach.

Abstract. In the present paper the multidomain approach is used in the analysis of spatial structural systems made by in-plane and out-plane loaded thin plates. The plates, studied under membrane and bending conditions, are considered as macro-elements connected each other along the interface boundary through appropriate unknown kinematical and mechanical quantities. In this context the use of the substructuring techniques is an original contribution of the paper. The system made by in-plane and out-plane loaded plates is solved by using the displacement approach introduced inside the Symmetric Boundary Element Method, using an elastic relation connecting the kinematical and generalized mechanical quantities at the interface nodes.

1. Introduction. In this paper the results obtained by some of the present authors ([1], [2], [3]) in the separate analysis of stretching and bending plates inside the Symmetric Boundary Element Method (SBEM) are applied to more complex structural systems in which the plates are subjected to in-plane and out-plane actions ([4], [5]). These latter are defined as two-dimensional macro-elements connected each other along the interface boundary, supposing at this stage that the connection may occur only between orthogonal macro-elements. The spatial system obtained can be solved by means of an equation which keeps in consideration the different behaviour of every macro-element. For this purpose, an elastic equation connecting the generalized tractions and moments, defined on the interface boundaries, to the displacements and rotations of the nodes of the same boundaries is introduced. The used strategy, already employed in [2] for the in-plane loaded plates, contemplates the introduction of a characteristic matrix for every two-dimensional macro-element of the system. The definition of this matrix and the computation of its coefficients do not contemplate any distinction between free Γ 2 , constrained Γ1 or interface boundaries and, consequently, between known and unknown nodal quantities. This strategy is introduced to simplify the writing of the elastic relation valid for every macro-element. Indeed, once the characteristic matrix associated to every macro-element is written, it is possible to introduce the actual boundary conditions, by specifying the type of constraint and the imposed displacements on Γ1 and the distributed loads on Γ 2 . At the same time the unknowns of the elastic problem are characterized by the reactions at the nodes of Γ1 and the displacements at the nodes of Γ 2 . Finally, the regularity conditions between the interface boundaries of the macro-elements are used, by imposing that the generalized tractions and moments among connecting boundaries are equal and opposite in sign, as well as the nodal displacements and rotations are equal. These latter conditions lead to an equations system in which the unknowns are the displacements of the interface through which it is possible to compute all the boundary and domain quantities of each macro-element.

353

354


2. The multidomain formulation of the stretching and bending plate system. Let us consider the structural system made by thin, homogeneous, isotropic and elastic twodimensional solids J (with J=1,...N) of domain Ω , subjected to external actions and connected each other at the interface boundaries (Fig.1). The generic solid J is a plate under membrane and bending conditions. The plate boundary is distinguished into constrained Γ1 , free Γ 2 and interface Γ 0 sides. f2 , u 2 c2n,j2n

G2

G0

3 1

G1

b)

u0 ,t0

x z y

G0

2

a)

b dW

J

j0n,m0n

j 1n , c1n

u1 , f1

Fig.1: a) Interaction between two-dimensional solids, b) Geometry, constraint and load conditions for the in-plane and out-plane loaded plate. In the plate the physical and geometric properties may be different in every zone. The solid J is subjected to the following external actions: T - in-plane and out-plane body forces b = ª¬ b x b y b z º¼ ; T - in-plane and out-plane displacements and rotations u1 = ª¬ u1x u1 y u1z ϕ1n ϕ1t º¼ imposed on the constrained boundary Γ1 ; T - in-plane and out-plane forces and couples f2 = ª¬ f2x f2y f2z c2n c2t º¼ given on the free boundary Γ 2 . We want to obtain by the displacement formulation inside of SBEM the elastic response to the known external actions in terms of boundary quantities, defined along the boundary elements, each characterized by the outward normal n : T - mechanical reactions (in-plane forces, shear forces and couples) f1 = ª¬f1x f1y f1z c1n c1t º¼ on the constrained boundary Γ1 ; - kinematical quantities (in-plane and out-plane displacements and rotations) T u 2 = ª¬ u 2x u 2y u 2z ϕ2n ϕ2t º¼ on the free boundary Γ 2 ; In order to analyze the stretching and bending plate system we operate an appropriate subdivision of the solid in macro-elements. To study only a plate, a general strategy is used, based on the introduction of a matrix, called characteristic. To derive the characteristic matrix no distinction is made between the constraint, free and interface boundaries. It involves that the whole boundary is subjected to a distribution of layered mechanical actions f and to a distribution of double layered kinematical discontinuities ∆u = −u . Using this strategy, described in [2] for the in-plane loaded plate and in [6] for the out-plane loaded plate, it possible to obtain the relation

B X + Lˆ = 0 where the following compact positions are made:

(1)


ªB B = « uu ¬ B tu

B ut º ; B tt »¼

ªF º X =« »; ¬ −U ¼

ˆ º ªW Lˆ = « » ˆ ¬« P ¼»

355

(2a-c)

Each matrix B hk ( h, k = u, t) is diagonal: ª Bin B hk = « hk ¬ 0

0 º » B out hk ¼

(3)

where the superscripts in and out denote the in-plane and out-plane loaded plate. All the coefficients of the matrix B are made by double integrals and have valuated by imposing in a sequential way a distribution of causes on the boundary elements, and by computing the effects at the same boundaries through a weighting process of the response. They are computed in closed form, using linear and quadratic shape functions. B uu and B tt are diagonal and symmetric matrices, whereas B ut and B tu are in general rectangular and contain some coefficients whose integrals are considered as Cauchy Principal Values, to which the free terms are associated. The advantages of the use of the characteristic matrix B are mainly related to the generalization of the coefficients computation. Furthermore it is possible to derive from it the algebraic operators of the solving equations for any constraint and load conditions. In order to get this aim, a rearrangement of the rows and columns of the characteristic matrix is performed, as consequence of the subdivision into the constrained Γ1 , free Γ 2 and interface Γ 0 boundaries. The vectors F and U are redefined in the following way:

F = ª¬ F2 T

F1 T

T

F0 T º¼ ,

(− U ) = ª¬ −U T2

−U 0T

− U1 T º¼

T

(4a,b)

where a distinction is made among known and unknown nodal quantities and among different boundaries. In order to obtain the algebraic operators for the solution of the problem, we introduce the generalized Dirichlet and Neumann conditions on the Γ1+ and Γ +2 , written in weighted form on the boundary elements:

W1+ = 0 on Γ1+ ,

P2+ = 0 on Γ +2

(5a-b)

and the generalized kinematical condition written at the interface boundary

W0+ = 0 on Γ 0+ .

(5c)

By performing a variables condensation in eq.(1), we obtain the generalized stresses - nodal displacements equation:

P0 = D00 U 0 + Pˆ 0 .

(6)

Eq.(6) contains interface quantities and relates the generalized mechanical quantities P0 associated to the nodes of Γ 0 to the kinematical quantities U 0 of the same boundary nodes and to the load vector Pˆ 0 . The algebraic operator D00 is the stiffness matrix of the macro-element and it is symmetric and singular. The matrix D00 is diagonal and has the following form

356


ª Din D00 = « 00 ¬ 0

0 º » . Dout 00 ¼

(7)

Hereafter for notation simplicity the subscript 00 is suppressed. Eq.(7) is rewritten in the following form: PJ = D J U J + Pˆ J

(8)

where the subscript j denotes the generic macro-element. The stiffness matrix D J , associated to the interfaces nodes of the in-plane and out-plane loaded plate, is employed for the analysis of the plate interaction. We consider N plates, with j = 1, 2,..., N . Let us denote by i the interface side, being i = a, b,..., m

b

b

3 1

a

2

a)

ub = U b1 = U b 3

b)

Fig.2: The assembly process: a) system of in-plane and out-plane loaded plates, b) indication of the interface quantities. We introduce a global relation connecting the weighted mechanical quantities to the kinematical ones at the interface nodes, i.e: P = D U + Pˆ

(9)

with P = ª¬ P1Τ U = ª¬ U1Τ

Τ P2Τ ... PNΤ º¼ , Pˆ = ª¬ Pˆ1Τ Τ

Τ

Pˆ 2Τ ... Pˆ NΤ º¼ ,

U Τ2 ... U ΤN º¼ , D = diag[D1

D2

(10 a-d)

... Di ]

where the vectors P and Pˆ collect the interface mechanical quantities (tractions and moments) and the load terms of all the macro-elements, whereas the vector U collects the corresponding kinematical quantities (displacements and rotations) at the same plate interfaces. Let us introduce the displacements vector of the interface nodes: u = ª¬u Τa

ubΤ ... u Τm º¼

Τ

(11)

The regularity condition of the kinematical quantities have to be imposed, and as a consequence U=Zu

(12)

where Z is an appropriate compatibility matrix. In Fig.3 the compatibility condition for the interface b of the macro-elements J = 1 and J = 3 is shown. The equilibrium condition of the weighted mechanical quantities takes on the form


ZT P = 0

357

(13)

The use of the eqs.(9), (12) and (13) leads to the following solving system:

K u = fˆ

(14)

where one has set

fˆ = Z T P

K = ZT D Z ,

(15 a,b)

The latter matrix K is symmetric and definite. Likewise to the finite element method, the solving equations are obtained in terms of the nodal displacements.

3. Numerical test. In the following test a system of three square plates of dimensions 2x2, with physical constants E = 1 ⋅106 and ν = 0.3 and thickness s = 0.01 , is analyzed (Fig.3a). The plates are fully constrained at the lower sides and are connected each other by rollers. m y ,jy x 2 (3)

G2

G2

G2

(2)

G2

qz =1

10 1

(1)

5

(2)

J=2

6

G1(1)

13

8 7

G2

j1x =1 j8x =1

b)

(2)

G0 G0

J=1

a)

z

(3)

J=3 (1)

j7x =1

u12x=2 u 13x=1

c) f z=1

1

G1(2) yx

f z=1

m x ,jx d)

f z=-1

Fig.3: a) System of in-plane and out-plane loaded plates, b) Load condition 1, c) Load condition 2, d) Load condition 3. Three different load conditions are considered: 1. a set of kinematical quantities (Fig.3b) imposed on the constrained boundary Γ1 , to perform a rigid rotation ϕ x = 1 around the clamped side of the plate J = 1 and a rigid rotation and vertical translation of the plates J = 2 and J = 3 . Obviously, the kinematical response of the system reproduces the assigned rotation, whereas the mechanical response is null everywhere; 2. a uniformly distributed domain load q z = 1 , normally acting to the middle surface of the plate J = 1 (Fig.3c); 3. two linear distributions of boundary forces fz , given at two free sides of the plates J = 2,3 (Fig.3d).

358


The boundary of the plates is discretized into 8 elements, in each of which the modelling of the boundary quantities and the weighting of the response are performed by linear and quadratic shape functions. In the Table 1 the nodal displacements and rotations ( u xi , u zi , ϕxi , ϕyi ) and so also the nodal resultant reactive forces and moments ( Fx (i − j) , Fz(i − j) and M x (i − j) ) are shown in all the load conditions. In the latter rows of the Table 1 the errors in the global equilibrium conditions in x, z directions ( ∆Fx , ∆Fz ) and around the clamped side of plate 1 ( ∆M x ) are shown.

u x5 u z5 ϕx5 ϕ y5 u x6 u z6 ϕx 6 ϕ y6 u x10 u z10 Fz(7−8) M x(7−8) Fx(7−13) Fz(7−13) ∆Fx ∆Fz ∆M x

Load condition 1 1.3 × 10 −9 -2.000000 0.999999 1.5 × 10−9 1.2 × 10−9 -0.999999 1.000000 −7.3 × 10−9 1.999999 -2.000000 −2.3 × 10−8 −2.1 × 10−8 −2.8 × 10−9 1.3 × 10 −9 2.2 × 10−9 3.2 × 10−9 1.5 × 10−9

Load condition 2 0.990809 2.536825 -0.762869 -1.407730 0.757765 0.908269 5.654555 0.181118 -0.678301 1.834956 -3.358452 0.851098 -0.404577 -0.173775 0.013 0.722 -0.377

Load condition 3 0.645140 1.529249 -0.655026 -0.832129 0.577442 0.494687 -0.528024 -0.181885 -0.974296 2.334935 -0.460933 0.172506 -0.345508 -0.060844 0.048 -0.411 0.487

Tab.1: System of three plates: kinematical quantities, resultant reactive forces and moments, and errors in the equilibrium relations for the three different load conditions.

References. [1] Panzeca, T., Salerno, M, 2000. Macro-elements in the mixed boundary value problems. Comp. Mech. 26, 437-446. [2] Panzeca, T., Cucco, F., Terravecchia, S., 2002. Symmetric boundary element method versus finite element method. Comp. Meth. Appl. Mech. Engng. 191, 3347-3367 [3] Panzeca, T., Milana, V., Salerno, M., 2009. A symmetric Galerkin BEM for plate bending analysis. Eur. J. Mech. A/Solids, 28, 1, 62-74. [4] De Paiva, J.B, Aliabadi M.H., 2000. Boundary element analysis of zoned plates in bending. Comp. Mech, 25, 560-566. [5] Waideman, L., Venturini, W.S., 2009. An extended BEM formulation for plates reinforced by rectangular beams. Engng. Anal. Bound. Elem. 33, 983–992. [6] Panzeca T., La Mantia A., Salerno M., Terravecchia S., A multidomain approach of the SBEM in the plate bending analysis, BeTeq X., Athens July 22-24, 2009, Greece, in: Advances in Boundary Element Tecniques X, pp. 225-232, Sapountzakis E.J. and Aliabadi M.H. (Eds.), Eastleigh, United Kingdom, 2009.


359

GENETIC ALGORITHM WITH BOUNDARY ELEMENTS FOR SIMULTANEOS SOLUTION OF MINIMUM WEIGHT AND SHAPE OPTIMIZATION PROBLEMS Li Chong Lee Bacelar de Castro1 Paul William Partridge2 1,2

Department of Civil and Environmental Engineering, University of Brasília - UnB Campus Universitário Darcy Ribeiro – 70910-900 - Brasília, DF, Brazil,1 [email protected] 2 [email protected] Keywords: Genetic Algorithm, Boundary Elements, Minimum weight, Shape Optimization.

Abstract. Computational Intelligence is changing the way in which some problems are solved because it provides many powerful solution methods for researchers in all areas including for cases considered to be difficult or nearly impossible. Here Genetic Algorithms (GA) are used for optimization in 2D. In using the GA one solves a specific problem using a data structure based on chromosomes, each of which contains one possible solution to the problem. The chromosomes are structured in a binary chain and genetic operators are applied to generate new sampling points in the search space which recombines these structures preserving and improving upon critical information. The Boundary Element Method, (BEM) is combined with the GA in order to evaluate each possible solution produced, in this case for the minimum weight and shape optimization problems combined. The BEM has a significant advantage for this type of problem: as modification is carried out only on the boundary the confection of new descretizations is straightforward. If domain type methods are used a new mesh is needed for each chromosome generated. The objective function is responsible for classifying the individuals in accordance with their degree of adaptation. The shape and weight which are the project variables should be within certain limits in order to obtain useful solutions. The control of these limits – by means of constraints- is done considering penalties on the objective function. Constraints are applied to stresses, to nodal displacements and to the weight. A classical 2D structure is used for the optimization. In general the GA have a higher computational cost because of the number of evaluations, however the GA are more robust and effective in the search for initial optimized solutions, taking more time in the refinement of these solutions. 1

Introduction

In structural optimization one tries to find the best way of arranging the project variables, with or without constraints, related to structural behaviour, geometry or other factors. The goal for optimization is defined by the objective function (OF), which defines some significant quantity (cost, weight, dimensions, loading, etc.). According to Castro [1], from a physical point of view, the project variables may represent the following information about a structure:

360


x x

Mechanical or physical properties of the material; The topology of the structure, the way the elements are connected and the number of elements in a structure;

x

Geometrical shape or configuration of a structure;

x

Dimension of the transversal sections or the length of elements.

2. Optimization A problem with constraints can be transformed into a problem without constraints by association with a penalty function. The use of the penalty function, modified in this way is an efficient means of dealing with constraints ([2],[3],[4]). A modified objective function can be put in the following way [5]

F ( x)

f ( x) penal ( x )

(1)

The term penal (x), which is the penalty function, can differ in accordance with the method and the way in which it is applied to the non feasible solutions.

Since the 1990´s the following classification of these techniques has been frequently used for optimization problems with constraints [6],[7]. 2.1 Static Penalty Function. The penalty function parameters are determined by the user which might take time according to the complexity of the problem being analyzed. Ref. [5] suggests multiplying the residual of each constraint violated Si, where i = 1, 2,..., n. by a constant chosen by the user, C. The modified objective function is then given by: n

F ( x)

f ( x) C.¦ S i

(2)

i 1

2.2 Dynamic Penalty Function. For this type of penalty function, the parameters depend on the iteration (t) which is being executed. Ref. [8] puts the OF in the following way:

F ( x)

m

f ( x) C.t .¦ S Ej ( x) D

(3)

j 1

C, D and E, are constants and have the suggested values of 0.5, 2 and 2 respectively. Note that the term reaches its maximum value at the last iteration.

C .t D

2.3 Adaptive Penalty Functions. In accordance with the information obtained during the iterations the penalty parameters can be defined adaptively. Ref. [9] proposed a penalty function with logarithmic behaviour. Using this, gradually at each iteration, the OF is penalized in the proximity of non feasible solutions. The function has the following form: L

F ( x)

P

f ( x) Ri ¦¦ j 1 i 1

1 S j ( xi )

ª º½ ° « f ( x ) »° ° i » °¾ Ri | 10Ei, the last variable being defined as Ei | Int ®log10 « L P 1 « »° ° ¦ «¦ » j 1i 1 S j ( xi ) ° °¯ ¬ ¼¿

(4)


361

When Sj d 1 the OF is penalized and when Sj t 1 the function is maintained. L is the number of constraints and P is the number of project variables 3. Genetic Algorithms Genetic Algorithms can be considered to be a family of computational models inspired on evolution. These algorithms use as a starting point a set or population of chromosomes in which each one contains a possible solution for the specific problem. Genetic operators are applied which recombine the information contained in the chromosomes, preserving and improving critical information. In a wider use of the term a Genetic Algorithm is any model based on a population which uses genetic operators, selection, crossover and mutation, to generate new sampling points in the search space. Optimization is obtained by the minimization of the objective function, which for the examples considered is written in terms of the weight and the stresses. The process is terminated by a test which finishes the evolutionary process if the GA has reached some predetermined stopping point. Here that point is the largest value of the objective function found in a given number of generations, as will be considered in the example in section 5. The genetic operators are applied after the selection of the individuals by means of the roulette wheel method [10]. Crossover using from three to six points was used between the pairs of chromosomes selected, being one or two per allele. All of the bits of all of the chromosomes with the exception of the first and second were submitted to mutation with a probability of 2%. 4. The Boundary Element Method The boundary element method is a numerical method, the principal characteristic of which is the reduction of the dimension of the problem by one, [11], [12],[13]. There are many problems in engineering that can be characterized mathematically by differential equations with the appropriate boundary conditions, for example the solid mechanics problem considered here. A major difficulty with these differential equations is that most of them cannot be solved analytically. In order to obtain a solution to these problems the use of a numerical method is necessary. BEM can be easily adjusted to complex geometrical regions, in addition to which, in the absence of domain integrals all of the approximations are restricted to the boundary and regions with high gradients can be modelled with a better precision than with finite elements,[14]. The Boundary Element Method, (BEM) is used here in order to evaluate each possible solution produced, in this case for the minimum weight and shape optimization problems combined. The BEM has a significant advantage for this type of problem: as modification is carried out only on the boundary the confection of new descretizations is straightforward. If domain type methods are used a new mesh is needed for each chromosome generated. 4.1 Formulation for 2D elasticity Consider the equation of linear elasticity in :

(5)

With the boundary conditions 1st Condition of the type u i 2nd Condition of the type pi

u i on boundary *1 ;

V ij n j

pi on boundary *2 ;

The treatment for the displacement conditions and the boundary tractions are considered for example in [11].

u and p will assume approximate values and are weighted. The weighted residual expression is defined as

362


³

:

(V ij , j b j )u j * d: 0

(6)

Integrating twice by parts equation (6) one obtains

³

:

(V ij , j b j )u * d:

³

*2

( p j p j )u * d* ³ (u j u j ) p * d*

(7)

*1

This equation is the starting point for the application of the boundary element method In order to transform Eq. (7) in boundary integral equation it is necessary to apply the fundamental solution u * . For reasons of space we will consider that the details of the application of the method are well known. 5. Numerical Example A cantilever beam subject to a uniform distributed loading as shown in fig. 1. is considered. The example is a 2D linear elasticity problem without body forces. The beam is considered to be in plain stress with the following parameters qy = 300 N/mm, t = 1mm, E = 207000 MPa, Q = 0.3 and M = 1257.6 x 106 kg. It is intended to optimize the shape of the structure in order to obtain a reduction in weight not less than the material limit which is 1/4 M, or 314.4 x 10-6 kg and Vmáx= 10 Kpa. Linear boundary elements were used. Different numbers of control points and elements are considered as shown below.

Fig 1 – Cantilever Beam subjected to uniform distributed loading. A fitness function incorporating a penalty function was employed transforming the problem with constraints in one without constraints (Eq. 1). The use of the penalty function is an efficient way of treating constraints as well as permitting

that nearly feasible solutions be considered in the process of selection, in such a way that part of their genetic material be used for the next generation. The penalty term (Mass) is defined for the problem considered by Eq. (8). F x f x penal ( x) Mass penal ( M l ) (8) F (M i )

º ª Mass i º ª Vi Mass i D ¦ « 1» ¦ « 1» V i ¬ Mass max i max ¼ ¬ ¼

D

10000

(9)

A search space was chosen in the interval [0,16;0,10] which are the dimensions of the structure and using a precision of 4 decimal places in such a way that " x ! 7,51 and " y ! 7,25 With these definitions "t = 16. Table1 shows the parameters used for this problem. Population (npop)

50

Probability of mutation (pm)

0.02

Stopping criterion

95% of chromosomes equal Table 1 – Parameters for the beam.


363

Here the boundary element method (BEM) was used in the evaluation of the structure. The governing equation is considered above. Linear boundary elements are used. The fixing of the boundary is done at each generation in a random but controlled way giving a set of chromosomes which form a possible solution. Control points are predefined on the structure in the form of a grid. Initially each grid point is occupied by a boundary element, being arranged in such a way that they do not intersect. If the inclusion of an element would lead to some non-feasible configuration that element is discarded, in particular, if there is intersection the element is discarded. At each generation the new chromosomes are imposed in a new configuration with a tendency to a reduction in size because of the objective function.

N.º of control points 12 21 29 35 51 69 81 117

Precision Generation 0.0001 100 0.0001 100 0.0001 100 0.0001 100 0.0001 100 0.0001 100 0.0001 100 0.0001 100

M (kg) 614.32 x10-6 413.8 9x10-6 374.42 x10-6 315.67 x10-6 315.85 x10-6 315.26 x10-6 315.26 x10-6 315.26 x10-6

uy (mm) 0.375 0.348 0.244 0.322 0.359 0.159 0.159 0.159

Table 2 – Solution for different control points (Static Penalty Function). Considering the same problem, analysis is now done functions, the results being given in tables 3 and 4. N.º of control points Precision Generation 12 0.0001 100 21 0.0001 100 29 0.0001 100 35 0.0001 100 51 0.0001 100 69 0.0001 100 81 0.0001 100 117 0.0001 100

considering dynamic and adaptive penalty M (kg) 514.32 x10-6 433.89 x10-6 324.47 x10-6 315.67 x10-6 315.85 x10-6 315.26 x10-6 315.26 x10-6 315.26 x10-6

uy (mm) 0.245 0.328 0.244 0.322 0.259 0.159 0.159 0.159

Table 3 – Solution for different control points (Dynamic Penalty Function). N.º of control points 12 21 29 35 51 69 81 117

Precision Generation 0.0001 100 0.0001 100 0.0001 100 0.0001 100 0.0001 100 0.0001 100 0.0001 100 0.0001 100

M (kg) 393.8 9x10-6 374.12 x10-6 315.16 x10-6 315.16 x10-6 315.16 x10-6 315.16 x10-6 315.26 x10-6 315.16 x10-6

uy (mm) 0.218 0.239 0.190 0.189 0.189 0.189 0.189 0.189

Table 4 – Solution for different control points (Adaptive Penalty Function).

It should be noted that in the example the quantity of control points and the type of penalty function influence directly in obtaining new configurations.

364


Figure 2 Generation 1 with 12control points, static penalization

A

B

C

Figure 3 – A (Generation 55 with 12 control points, static penalization); B(Generation 100, with 12 control points, static penalization) and C(Generation 100, with 35, 51, 69, 81 and 117control points).

6. Conclusions

The optimization process uses a Genetic Algorithm of the generational type with binary coding and elitist strategy associated with the Boundary Element Method (BEM) applied to an optimization problem in linear elasticity. The constraints, inherent to project variables and to the structural synthesis, were considered by means of a modified objective function including a penalty term. In this way, a problem with a constraint is treated as a problem without constraints. The methodology does not present deficiencies at any point in the optimization process and the expected result is obtained. The shape optimization and minimum weight problem was solved within the restraints specified.


365

Acknowledgements The authors acknowledge the support of the Brazilian agencies CAPES and CNPq.

References [1] Erbatur, F., Hasançebi, I. T. and Kiliç H. Computers and Structures 75, 209-224 (2000) [2] Fox, R.L., Optimization Methods for Engineering Design. 2nd Edition, Addison-Wesley, (1973) [3] Haftka, R. T. and Gurdal, Z., Elements of Structural Optimization, 3th edition, Kluwer Academic Publishers, (1992). [4] Shoenauer, M. and Xanthakis, S., Proceedings of The 4th International Conference on Evolutionary Programming, 573-580. (1993). [5] Castro, L. C. L. Computational Intellegence applied to Structural Engineering, PhD Thesis, University of Brasília, (In Portuguese) (2009). [6] Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs. Third Edition, SpringerVerlag, (1996). [7] Lagaros, N. D., Papadrakakis, M. and Kokossalakis, G. Computers and Structures, 80 571-589, (2002). [8] Joines, J. and Houck, C. Proceedings of the First IEEE international Conference on Evolutionary Computation, 579-584, (1994). [9] Bezerra, L. M. Inverse Elastostatics Solutions with Boundary Elements, PhD Thesis, Carnegie Institute of Technology, Pittsburgh, USA, (1993). [10] Goldberg, D. E. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, (1989). [11] Brebbia, C. A. and Dominguez, J., Boundary Elements an Introductory Course, Computacional Mechanics Publications, and McGraw-Hill Book Company, (1989). [12] Brebbia, C.A., Telles J. C. F. and Wrobel L. C., Boundary Elements Techniques – Theory and Applications in Engineering, Springer-Verlag, (1984) [13] Wrobel, L.C and Aliabadi, M. H. The Boundary Element Method, Wiley, (2002). [14] Amano, K Journal of Computational and Applied Mathematics 53, 353-370, (1994)

366


New boundary integral equations for evaluating the static and dynamic T -stresses A.-V. Phan Department of Mechanical Engineering University of South Alabama Mobile, AL 36688, USA [email protected]

Keywords: T -stress, dynamic T -stress, symmetric-Galerkin boundary element method, elastodynamics, frequency analysis, transient responses.

Abstract 2-D boundary integral equations (BIEs) for determining the static T -stress (STS) and dynamic T stress (DTS) for cracks of arbitrary geometries are introduced in this paper. The BIE for the STS is regular while that for the DTS is only weakly singular. These BIEs can be used in the post-processing stage of a boundary element (dynamic) analysis of cracks. In this work, the proposed BIE for the DTS is formulated in the frequency domain so it can be used within a boundary element analysis in that domain. By applying the inverse fast Fourier transform to the frequency response of the DTS, its time history can be obtained. Test examples are given to validate the proposed technique and assess its accuracy.

1. Introduction In classical theories of fracture mechanics, the stress and displacement fields in the vicinity of a crack tip are characterized by a single parameter called the stress intensity factor (SIF). However, numerous experimental work (e.g., [1, 2]) have shown that a second fracture parameter, known as the elastic T stress (the first non-singular term in the series expansion of the stress component parallel to the crack and ahead of a crack tip) plays an important role in the linear elastic fracture mechanics of brittle materials under mixed-mode loading conditions. As a result, there has been increasing attempts to describe the crack tip behavior in terms of both the SIF and T -stress, and a larger and larger amount of investigation has been devoted to the numerical evaluation of the T -stress. For example, finite element method (FEM, e.g., [3, 4]), boundary element method (BEM, e.g., [5]), symmetric-Galerkin BEM (SGBEM, e.g., [6]), complex variable function method [7], etc., have been employed to calculate the static T -stress (STS). However, while there is a large number of numerical investigations for the STS, the situation is less encouraging for the dynamic T -stress (DTS). Only a few studies on the DTS can be found in the literature such as those using BEM by Sladek et al. [8], FEM by Jayadevan et al. [9, 10], and scaled boundary FEM (SBFEM) by Song and Vrcelj [11]. It is important to note that most of these DTS calculations [8–10] are based upon the interaction integral (or M -integral) method. Although the STS and DTS can be computed directly from the asymptotic expansion [12] for the crack-tip stress field, there was a concern over the fact that the numerical result was sensitive to the distance from the crack tip to a point selected for calculating the STS or DTS [8]. However, as demonstrated herein, a non-singular boundary integral equation (BIE) can be derived using the asymptotic stress expansion and thus, the STS can be directly computed at the crack-tip location. A similar technique can be used to derive a weakly singular 2-D BIE in the Fourier-space frequency domain for determining the DTS. These BIEs are general and they can be employed as a postprocessing step in any version of boundary element (dynamic) analysis of cracks. In this work, the SGBEM for elastostatics and Fourier SGBEM for elastodynamics are the versions of choice. Two test examples are presented to demonstrate the accuracy and effectiveness of using the proposed BIEs in evaluating the STS and DTS. 1


367

2. Boundary Integral Formula for the STS X2

x2 ϕ

r X1

P

x1 θ

P’

Γc+ Γc−

Figure 1: Global coordinate system X1 X2 and crack tip coordinate system x1 x2 . An exact boundary integral formula for the STS is derived in this section for cracks of arbitrary geometry. The formula is based upon the series expansion for the stress field in the vicinity of a crack tip [12] (Fig. 1) KI I KII II (1) σij (r, φ) = √ fij (φ) + √ fij (φ) + T δ1i δ1j + O(r1/2 ) 2πr 2πr where KI and KII are the mode-I and mode-II SIFs, and fijI and fijII are universal functions of angle φ. For φ = 0, one gets

σ11 (r, 0) σ22 (r, 0)

=

√

1 2πr

KI KI

+

T 0

+ O(r1/2 )

(2)

which results in the following expression for evaluating the STS:

T = lim σ11 (r, 0) − σ22 (r, 0)

(3)

r→0

Without loss of generality, consider a finite domain containing a crack composed of two surfaces Γ+ c and Γ− c symmetrically loaded. Let Γ and Γc be the boundary of the domain and the crack, respectively − and Γb = Γ \ Γc where Γc = Γ+ c ∪ Γc . In this work, Eq. (3) is evaluated using the following BIE for the stress at an interior point P as shown in Fig. 1 and a limit process is carried out as P tends to the crack tip P (r → 0):

σk (P ) =

Γb

s s Dkj (P, Q) tj (Q) − Skj (P, Q) uj (Q) dQ − lim

P →P

Γ+ c

s Skj (P , Q) ∆uj (Q) dQ = 0 (4)

where Q is a field point, tj and uj are the traction and displacement vectors on Γb , respectively, − s s ∆uj = u+ j − uj is the displacement jump across the crack surfaces, and Dkj and Skj are the − elastostatic kernel tensors. Note that, as the crack surfaces are symmetrically loaded, Σtj = t+ j +tj = 0 s and thus the integrand term −Dkj (P , Q) Σtj (Q) does not appear in the second integral. Use of Eq. (4) in Eq. (3) results in

T =

Γb

∆Dks (P, Q) tk (Q) + ∆Sks (P, Q) uk (Q) dQ + lim

P →P

Γ+ c

∆Sks (P , Q) ∆uk (Q) dQ = 0

(5)

s − Ds s s s where ∆Dks = D11k 22k and ∆Sk = S22k − S11k . While the second integral in Eq. (4) tends to infinity as P → P (the stress field σk is known to be singular at crack tips), the second integral in Eq. (5) is finite as P → P . This is expected as T -stress is a non-singular term. If Eq. (5) is discretized, the limit is only associated with the integral over the boundary Γ+ ct of the crack-tip element. Let this limit be Tct , one gets

Tct = lim

P →P

Γ+ ct

∆Sk (P , Q) ∆uk (Q) dQ

2

(6)

368

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz x2 x1

Γc+

3 (s=1)

P

P’

1 (s=0)

2 (s=0.5)

Figure 2: Crack tip element When a numerical implementation is carried out with the standard quadratic shape functions, it can be shown, as P ≡ P , that (see Fig. 2) J ∆S1 (s) = −

8µ b2 (a2 s + a1 )(a2 c22 s2 + 2a1 c22 s + a21 a2 ) π(1 − ν)(c22 s2 + 2a1 a2 s + a21 )3

J ∆S2 (s) = −

4µ b22 (2a2 c22 s3 + 3a1 c22 s2 − a31 ) π(1 − ν)(c22 s2 + 2a1 a2 s + a21 )3

(7)

where J, µ, ν are the Jacobian, shear modulus and Poisson’s ratio, respectively, and a1 = 4x1 (2) − x1 (3)

a2 = 2 x1 (3) − 2x1 (2) b2 = 2 x2 (3) − 2x2 (2) c22 = a22 + b22

(8) P

It means that the products J∆Sk (s) are not singular at the crack tip P (s = 0) as ≡ P . Thus, the limit process in Eq. (6) is not necessary and the integral can be directly evaluated at the crack tip as Tct =

1

0

∆Sk (s) ∆uk (s) J ds

(9)

The proposed technique is general and it can be implemented as a post-processing step of any version of the BEM with fracture analysis capability. In this work, the SGBEM is the version of choice.

3. Boundary Integral Formula in the frequency domain for the DTS In the Fourier-space frequency domain, the elastodynamic version of Eqs. (5) and (6) takes the following forms:

T (ω)= +

Γb

P →P

lim

P →P

and

Tct = lim

∆Dk (P, Q, ω) tk (Q, ω) + ∆Sk (P, Q, ω) uk (Q, ω) dQ + lim

P →P

Γ+ ct

Γ+ c

Γ+ c

∆Sks (P , Q) ∆uk (Q) dQ

∆Sk (P , Q, ω) − ∆Sks (P , Q) ∆uk (Q, ω) dQ

∆Sks (P , Q) ∆uk (Q) dQ + lim

P →P

Γ+ ct

(10)

∆Sk (P , Q, ω) − ∆Sks (P , Q) ∆uk (Q, ω) dQ

(11) where ∆Dk and ∆Sk are the dynamic counterparts of the static kernels ∆Dks and ∆Sks . By numerically implementing this equation using a standard quadratic element, as mentioned in the previous section, the first integral on the right hand side is non-singular as P ≡ P . The second 3


369

limit in Eq. (11) should also be bounded as its integrand is only weakly singular as P ≡ P (see [13]). In other words, the limit processes in Eq. (11) are not necessary and these integrals can also be directly evaluated at the crack tip. In this work, Eq. (10) is implemented in the post-processing stage of the Fourier SGBEM for elastodynamics.

4. Test Examples 4.1. T -stress for a circular arc crack in an infinite plate σY

8

x1 A Y

8

σX

x2 θ X θ

B

Figure 3: A circular arc crack in an infinite plate. Consider a circular arc crack in an infinite plate subjected to a remote biaxial tension as shown in ∞ = σ ∞ = σ, and θ varies from 15◦ to 165◦ with an increment of 15◦ . The analytical Fig. 3 where σX Y solution for the STS at crack tips A or B for this problem is given by [7] as 2(1 − cos θ) T = σ 3 − cos θ 0.6

0.55

% Error

(12)

1

0.4

0.8

1 T / σ (SGBEM, N=20) T / σ (Analytical)

0.8

0.545

0

0.4

0.4 % Error

-0.2

0.535 0.2

-0.4

0.53

-0.6 100 200 300 400 500 600 700 800 900 1000 N

0 0

Figure 4: Solution convergence for θ = 75◦

0.2

30

60

90 120 θ , degrees

150

0 180

Figure 5: SGBEM vs. analytical solution

4

% Error

T/σ

0.6

T/σ

0.54

0.6

% Error

T/σ

0.2

370


First, a convergence test for the numerical results of the normalized T -stress T /σ is carried out. Figure 4 shows that the numerical results for θ = 75◦ converge as the number of elements N used to mesh the crack increases. Even with a very coarse mesh, the absolute value of the percentage error is still less than 0.45%. Finally, an excellent agreement between the numerical solutions (with N =20) and the analytical solution for all the values of θ considered can be seen in Fig. 5. The maximum percentage error is less than 0.7%. 4.2. Dynamic T -stress for a mixed-mode crack in a finite plate x2

2a

σ(t)

σ(t)

H

θ

x1 H B

B

Figure 6: A plate with a centrally located mixed-mode crack The last example deals with a mixed-mode crack in a finite plate under a uniaxial tension σ(t) = σ0 H(t) (see Fig. √6). The crack has an orientation θ = 45◦ relative to the direction of the load and a length 2a = 10 10 mm. The size of the plate is (2H × 2B) = (30 mm× 60 mm). The material properties of the plate are: shear modulus µ = 76.923 GPa, Poisson’s ratio ν = 0.3, mass density ρ = 5, 000 kg/m3 , and damping ratio ζ = 1%. 50

10

40

8 30

6

real imaginary

20

4 T / T0

T1(ω)

10 0

2

-10

0

-20 -30

Fourier SGBEM (ζ = 1%) Laplace BEM [8] SBFEM [11]

-2

-40

-4 -50

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

f = ω /2π, MHz

Figure 7: Frequency response T1 (ω)

2

4

6

8

10 12 Time (µs)

14

16

18

20

Figure 8: Numerical results for T /T0

A total of 20 elements is employed on the boundary and 10 uniform elements are used to discretize the crack. For the frequency response analysis of the DTS, a frequency step ∆f = 500 Hz and a number of samplings N = 212 = 4, 096 need to be selected. These data were based on convergence studies. Figure 7 shows the conjugate symmetry of the frequency response T1 (ω) (due to a unit 5


371

Heaviside load σ(t) = H(t)) while Fig. 8 depicts the time histories of the normalized T /T0 where T0 is adopted from Reference [8] (T0 = −1.058 σ0 ) for the purpose of comparison. It can be seen from Fig. 8 that while the SGBEM solution does not agree with the BEM [8] one, there is a very good agreement between the SGBEM and SBFEM [11] curves.

5. Conclusion In this paper, 2-D BIEs were proposed for numerically calculating the STS and frequency response of the DTS. These BIEs can be implemented in the post-processing stage of a boundary element analysis of cracks. The transient response of the DTS can easily be obtained by applying the inverse fast Fourier transform to its frequency response. These BIEs are simpler than those used to determine the stress components at an interior point to a domain. They can be employed directly at the crack tip to accurately evaluate the STS and DTS without any concern about the numerical sensitivity of the results. The technique is much more computationally effective than the M -integral method as the latter requires evaluating both the stress and displacement at several interior points used as Gauss integration points. For the test examples considered in this work, there is an excellent agreement between the numerical results for the STS and the analytical solution. The transient response of the DTS agrees very well with one of only a few referenced solutions available in the literature.

Acknowledgments This research was supported in part by the NSF Grant CMMI-0653796 and NASA Grant NNM07AA09A03.

References [1] Williams, J.G., and Ewing, P.D., “Fracture under complex stress – The angled crack problem”, Int. J. Fract., 8, 441–446, 1972. [2] Ueda, Y., Ikeda, K., Yao, T., and Aoki, M., “Characteristics of brittle fracture under general combined modes including those under bi-axial tensile loads”, Engng. Fract. Mech., 18, 1131– 1158, 1983. [3] Ayatollahi, M.R., Pavier, M.J., and Smith, D.J., “Determination of T -stress from finite element analysis for mode I and mixed mode I/II loading”, Int. J. Fract., 91, 283–298, 1998. [4] Paulino, G.H., and Kim, J.-H.,“A new approach to compute T -stress in functionally graded materials by means of the interaction integral method”, Engng. Fract. Mech., 71, 1907–1950, 2004. [5] Yang, B., and Ravi-Chandar, K., “Evaluation of elastic T -stress by the stress difference method”, Engng. Fract. Mech., 64, 589–605, 1999. [6] Sutradhar, A., and Paulino, G.H., “Symmetric Galerkin boundary element computation of T stress and stress intensity factors for mixed-mode cracks by the interaction integral method”, Engng Analysis with Boundary Elements, 28, 1335–1350, 2004. [7] Chen, Y.Z., “Closed form solutions of T-stress in plane elasticity crack problems”, Int. J. Solids Struct., 37, 1629–1637, 2000. [8] Sladek, J., Sladek, V., and Fedelinski, P., “Computation of the second fracture parameter in elastodynamics by the boundary element method”, Advances in Engineering Software, 30, 725– 734, 1999.

6

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[9] Jayadevan, K.R., Narasimhan, R., Ramamurthy, T.S., and Dattaguru, B., “A numerical study of T -stress in dynamically loaded fracture specimens”, Int. J. Solids Struct., 38, 4987–5005, 2001. [10] Shin, D.K., and Lee, J.J., “Numerical analysis of dynamic T stress of moving interfacial crack”, Int. J. Fract., 119, 223–245, 2003. [11] Song, C., and Vrcelj, Z., “Evaluation of dynamic stress intensity factors and T -stress using the scaled boundary finite-element method”, Engng. Fract. Mech., 75, 1960–1980, 2008. [12] Williams, M.L., “On the stress distribution at the base of a stationary crack”, J. Appl. Mech., 24, 109–114, 1957. [13] Phan, A.-V., Gray, L.J., and Salvadori, A., “Symmetric-Galerkin boundary element analysis of the dynamic stress intensity factors in the frequency domain,” Mechanics Research Communications, 37, 177-183, 2010.

7


373

The Boundary Element Method Applied To Visco-Plastic Analysis E. Pineda¹, M.H. Aliabadi2, and J. Zapata1 1 2

Instituto Politécnico Nacional, ESIA-UZ, México, [email protected]

Department of Aeronautical Engineering, Imperial College London, South Kensington campus, London SW7 2AZ, [email protected]

Key Words: Boundary Element, Fracture Mechanics, Visco-plasticity.

Abstract: This paper presents a new formulation of the Dual Boundary Element Method to visco-plastic problems in a two-dimensional analysis. Visco-plastic stresses and strains around the crack tip are obtained until the visco-plastic strain rate reaches the steady state condition. A perfect visco-plastic analysis is also carried out in linear strain hardening (H’=0) materials. Part of the domain, the part that is susceptible to yield is discretized into quadratic, quadrilateral continuous cells. The loads are used to demonstrate time effects in the analysis carried out. Numerical results are compared with solution obtained from the Finite Element Method (FEM).

Introduction In the case of inelastic fracture mechanic problems and in problems with high temperature gradients where inelastic strain rates are proportional to high power of stress, regions with strain rate concentration provide nearly all the inelastic contribution to the stress rates [1]. The main reason for the success of the BEM (boundary element method) in fracture mechanics applications is the ability to model high stress concentration fields accurately and efficiently. A comprehensive review of the historic development of the BEM for fracture mechanics can be found in the work of Aliabadi [2]. One of the early efforts in solving non-elastic fracture mechanics problems by using BEM was made by Morjaria and Mukherjee [3] and Banthia and Mukherjee [4] where they used a crack Green's function to model the crack and obtain solution for the time-dependent stress field which was developed near the crack tip in finite plates. An alternative methodology based on the use of the Kelvin fundamental solutions was presented in [5], [6] and [7]. Recently, the DBEM (dual boundary element method) has been developed as a very effective numerical tool to model general fracture problems with numerous applications to linear elastic and non-elastic fracture problems [8]. BEM has been applied to elastoplastic problems since the early seventies with the work of Swedlow and Cruse [9] and Richardella [10] who implemented the von Mises criterion for 2D problems using piecewise constant interpolation for the plastic strains. Later, Telles and Brebbia [11] and others had, by the beginning of the eighties, developed and implemented BEM formulations for 2D and 3D inelastic, viscoplastic and elastoplastic problems (see [12] for further details). In recent years, Aliabadi and co-workers [13] have introduced a new generation of boundary element method for solution of fracture mechanics problems. The method which was originally proposed for linear elastic problems [14],[15] and [16] has since been extended to many other fields including problems involving nonlinear material and geometric behaviour [17]. In the present paper applications of the DBEM to visco-plasticity are presented. The specimens analyzed are a square plate and a plate with a hole, both of them with different crack length. The boundary was discretized with quadratic continuous and semi-discontinuous elements, but the domain with nine nodes internal cells. In visco-plasticity only the part susceptible to yielding was discretized. The von Mises yield criterion was applied so the material used for these sort of analysis were metals.

374


Boundary Integral Formulation for Visco-plasticity In the visco-plastic analysis like plasticity, the initial strain approach will be applied and the integral equation to calculate the displacement on the boundary is basically the same, the only difference is that the plastic strain is replaced with the visco-plastic strain rate. So the displacement equation can be rewritten as:

cij ( x' )u j ( x' ) ³ t 'ij ( x' , x)u i ( x' )d* *

³ u' *

ij

( x' , x)t j ( x' )d* ³ V 'ij x' , z H vp ij z d:

(1)

:

Where ui , ti and H vp ij are the displacement, traction and visco-plastic strain rates respectively. t 'ij , u 'ij and

V 'ij are the displacement, traction and third order fundamental solutions, respectively, which are functions of the positions of the collocation point x and the field point x which belong to the boundary, or the internal point z and the material properties.

The Dual Boundary Integral Equations The Displacement Equation If the displacement integral equation (1) is collocated on the upper crack surface and the free term for a smooth boundary C ij 1 / 2 is considered, the displacement integral equation can be rewritten as

1 1 u j ( x' ) u j ( x' ) ³ t ij ( x' , x)u i ( x' )d* * 2 2 vp ³ V 'ijk x' , z H jk z d:

³u *

ij

( x' , x)tj ( x)d* (2)

:

where x ' and x ' belong to the upper and the lower crack surfaces respectively. The Traction Equation From the definition of tractions followed by the application of Hooke's law and through the differentiation of the displacement boundary integral equation it is possible to define the time-dependent traction equation as

ti

V ij n j

(3)

By substituting the terms in (3) we obtain

³ S x , x u x d*

1 1 c p j x c p j x c ni x ³ Dijk x c , x t j x d* ni x * 2 2 1 ª º « ³ 6 cijk x c , z H vp f ij (H vp jk x d: jk z ) » ni z : 2 ¬ ¼

*

c ijk

c

j

where @ ij denotes the components of the derivative of the stresses with respect to the time, t i the traction rate and n j the components of the normal vector to the boundary. Equation (4) represents the traction boundary equation for a lower crack surface see [22]. Crack Modeling

(4)


375

Discontinuous Boundary Elements and Internal Cells To keep the simplicity of the boundary element and for the sake of efficiency discontinuous quadratic elements are used for the crack modeling and the general modeling strategy can be summarized as follows: x The displacement equation (2) is applied for collocation on the upper crack surface; x The traction equation (4) is applied for collocation on the lower crack surface. Basic Concepts of Visco-Plasticity In order to explain the theory of visco-plasticity it is convenient to analyze the one-dimensional rheological model see [22] for more details. A uniaxial yield stress governs the onset of the visco-plastic deformation. Once visco-plasticty begins the stress level for continuing visco-plastic flow depends on the strain hardening characteristics of the material ( ). After applying Hook’s law and boundary conditions, see Pineda [22] for further details, it is possible to obtain: (5) Expression (5) is the visco-plastic strain rate in terms of the stresses for the uniaxial case in which (.) denotes the derivative with respect to the time, . From the visco-plastic model see [22] the strain response with time can be represented by two cases. The first case is the perfectly visco-plastic material in which . In this case the visco-plastic deformation continues at a constant strain rate. The second case is the linear hardening case ( ), where after the initial elastic response, the viscoplastic strain rate is exponential and reaches the steady state condition when this value becomes zero. On the other hand, for a perfectly visco-plastic material there is always an imbalance of stress in the system which does not reduce and consequently the steady state condition can not be achieved. Benchmark Problems Centre Cracked Plate A centre cracked tension standard specimen is considered in this example. The boundary of the problem is discretized with 44 quadratic boundary elements and the domain with 84 interior quadratic cells. Viscoplastic behaviour and plane stress analysis was assumed and a constant load of 250.09 MPa. The geometry , and and boundary conditions are shown in figure 1, where . Because of the symmetry it is not necessary to model the whole plate so half of the model is used in the analysis since we can obtain the same results.

376


Va

y

o

H

o

Crack Tip

Va W

a

Figure 1. Geometry and boundary conditions for a centre cracked plate.

The material properties are x Young's modulus, U

;

x

Poisson

x

@ y 895. 9 MPa. ; Applied stress @ a 250. 29 MPa. Viscoplastic coefficient + p 0. 045

x x

s ratio,

;

Yield stress ,

Figure 2. Von Misses Stresses for a centre cracked plate.


377

The stress profiles shown in figure 2 shows a plot of the equivalent stress or von Mises stress against the distance from the crack tip. It is clear to see from this figure that the maximum stresses are located close to the crack tip where a visco-plastic zone is developed here. The tolerance applied in this case was 0.001 and the initial time step was t 0. 01 , The results were compared to the FEM calculated in ANSYS and it was found the biggest percentage relative of error is 3%.

Figure 3. Displacements for a centre cracked plate ahead the crack tip. Figure 3 shows the comparison of the displacements for a centre cracked plate by using: Discontinuous boundary elements on the crack, the rest continuous and semi discontinuous boundary elements. It is clear to see the converged solution for the different sort of elements. Plate with a hole A perforated tension specimen with geometry and boundary conditions as illustrated in figure 4, was analyzed in this example. Plane stress condition was assumed here. The plate has these dimensions: x x x

Wide w = 10 mm Hight H=36 mm Crack length a = 0.3 mm

The material selected is an aluminum alloy with the following properties: N mm 2

x

Young's modulus, E=7000

x

Poisson's ratio, 7 0. 2 ;

x

Hardening coefficient HU 0 (perfectly visco-plastic)

x

Yield stress ,

x x

Applied stress Visco-plastic coefficient

@ y 24. 3

;

N mm 2

;

378


y

x

Crack Tip

Figure 4.-Geometry and boundary conditions for a plate with hole and centred cracket

Figure 5. Von Misses stresses for a plate with a hole and centre cracked. Figure 5 shows the results for a von Misses stress distribution ahead the crack tip against the distance from the crack tip r/a. Around the crack tip the stresses start to converge to the yield stress which is logical for a perfectly viscoplastic case. It is clear that the highest stresses are around the crack tip and that the stresses start to decrease in proportion to the distance from the crack tip.


379

Figure 6.- Stresses in y direction for a half and full plate. Results for stresses ahead the crack tip are shown in figure 6. The geometry ,boundary conditions and material properties are the same as above but in this case the applied stress is a= 9.5 Mpa. . It is clear the stresses near to the crack tip are high and then decrease. Conclusions In this paper the DBEM was applied to the analysis of non-elastic crack problems in fracture mechanics. It has been demonstrated here that this method is an accurate and efficient method for modeling visco-plastic crack problems. The analysis is general and can be applied to mixed mode cracks in non-linear fracture mechanics problems. The displacement and traction boundary integral equation used are independent. In the case of the traction equation continuity of strains is required at the collocation node to guarantee the existence of the finite part integrals. Since this discontinuous boundary elements have to be used on the crack faces. The visco-plastic behaviour is represented by a plastic strain field over a region, susceptible to yield, discretized with quadrilateral quadratic continuous and discontinuous internal cells.

References [1] Providakis, C.P., Viscoplastic BEM Fracture Analysis of Creeping Metallic Cracked Structures in Plane Stress using Complex Variable Technique. Engineering Fracture Mechanics, 70, 707-720, (2003). [2] Aliabadi, M.H., Boundary Element Methods in Fracture Mechanics. Appl Mech Rev, 50, 83-96, (1997). [3] Morjaria, M., and Mukherjee, S., Numerical Analysis of Planar, Time-Dependent inelastic Deformation of Plates with Cracks by the Boundary Element Method, Int. J. Solids Structures, 17, 127-143, (1981). [4] Banthia V., and Mukherjee, S., On Stress and Line Integrals in the Presence of Cracks, Research Mechanica, 15, 151-158, (1982). [5] Tan, C.L., Lee, K.H. Elastic-Plastic Stress Analysis of a Cracked thick-walled Cylinder, Journal of Strain Analysis, 50-57, (1983). [6] Yong, L., and Guo, W.G., The calculation of J

I

Integrals of Thick-Walled Tubes with One and

Two Symmetrick Cracks by Elastoplastic BEM, Int. J. Pres. Ves. & Pipping, 51, 143-154, (1992). [7] Hantschel T., Busch, M., Kuna, M., and Maschke, H.G., Solution of Elastic-Plastic Crack Problems by an Advanced Boundary Element Method, in Proceedings of the 5th. International Conference on

380


Numerical Methods in Fracture Mechanics, A.R.. Luxmoore and D.R.J. Owen, Eds., pp. 29-40, (1990). [8] Leitão, V., An Improved Bounadry Element Formulation for Nonlinear Fracture Mechanics I, PhD Thesis, Wessex Institute of Technology, University of Porsmouth, (1993). [9] Swedlow, J. L. and Cruse, T. A. Formulation of the boundary integral equation for threedimensional elastoplastic flow, International Journal of Solids and Structures, 7, 1673-1681 (1971). [10] Riccardella, P. An Implementation of the Boundary Integral Technique for plane problems of Elasticity and Elastoplasticity, PhD Thesis, Carnegie Mellon University, Pitsburg, PA (1973). [11] Telles, J. C. F., and Brebbia,C.A. Elastic/viscoplastic Problems Elements,International Journal of Mechanical Sciences, 24, 605-618, (1982).

using

Boundary

[12] Aliabadi, M.H., The Boundary Element Method. Applications in Solids and Structures. Vol. 2. John Wiley & Sons, Ltd, West Sussex, England (2002). [13] Aliabadi,M.H. A new generation of boundary element methods in fracture mechanics,International Journal of Fracture, 86, 91-125, (1997). [14] Aliabadi,M.H. and Portela,A. Dual boundary element incremental analysis of crack growth in rotating disc. Boundary Element Technology VII, Computational Mechanics Publica-tions, Southampton, 607-616, (1992). [15] Portela, A. Aliabadi, M.H., and Rooke, D.P., Dual Boundary Element Incremental Analysis of Crack propagation. Computers & Structures, 46, 237-247, (1993). [16] Mi, Y., Aliabadi, M.H., Dual Boundary Element Method for Three-Dimensional Fracture Mechanics Analysis, Engng. Anal. with Bound. Elem., 10, 161-171, (1992). [17] Chao Y.J., Zhu, X.K., and Zhang, L., Higher-Order Asymptotic Crack-Tip .elds in a powerlawCreeping material, International Journal of Solids and Structures, 38, (2001). [18] Cisilino, A.P. and Aliabadi, M.H., Three-dimensional BEM Analysis for Fatigue Crack Growth in Welded Components, International Journal for Pressure Vessel and Piping, 70,135-144, (1997). [19] Cisilino, A.P., Aliabadi, M.H. and Otegui, J.L., A Three-dimensional Element Formulation for the Elasto-Plastic analysis of Cracked Bodies, International Journal for Numerical Methods in Engineering, 42, 237-256, (1998). [20] Hutchinson, J.W., Singular behaviour at the end of a tensile crack in a hardening material, J. Mech. Phys. Solids, 16, 13-31, (1968). [21] Zienkiewicz, O.C., and Cormeau, I.C., Visco-Plasticity and Creep in Elastic-Solids A Unified Numerical Solution Approach, International Journal for Numerical Methods in Engineering, vol. 8, 821-845, (1974). [22] Pineda, E., Dual Boundary Element Analysis for Creep Fracture, Ph. D. Thesis, Queen Mary College, University of London, (2005).


381

Optimal shape of fibers in composites with various ratios of phase stiffnesses P.P. Prochazka Czech Technical University in Prague, Civil Engineering, dept. Structural Mechanics Prague, Czech Republic [email protected]

Keywords: Composite materials, shape optimization of fibers, fiber ratio, phase stiffnesses, optimization using homogenization

Abstract. In this paper a new procedure for optimization based on homogenization of composites is proposed. This is based on boundary element method, which seems to be more efficient and more accurate in applications than finite elements. Special properties of the stress distribution (or concentration factors) on a unit cell are utilized. The optimum condition starts with minimum Lagrangian with respect to the internal parameters, identifying a position of nodes on the interfacial boundary. In order to assure solvability and uniqueness of the problem, additional conditions are postulated in terms of constraints conditions. It appears that the formulation for minimum Lagrangian leads to the optimization of both stresses and displacements in composite structures. Averaging process including integration over the unit cell follows the fact that the stresses "relax" in the larger phase and their distribution converges to constant except for a zone being in the vicinity of the fiber-matrix interfacial surface. This is another one property why boundary element method is more suitable than finite element method. The results are focused on the optimal shape of fibers in rectangular and hexagonal periodic structures for varying stiffnesses of phases. 1 Introduction Homogenization and shape optimization of fibers in a composite structure has been solved by many authors mostly by means of FEM. In this paper a new procedure for homogenization of composites is proposed similarly to [1], where a beam structure is supposed for treatment. This is based on BEM, which seems to be more efficient and more accurate in applications than the FEM solution. Special properties of the distribution of strains and stresses (or concentration factors) on a unit cell are used. Averaging process including integration over the unit cell, particularly in the case of fiber reinforced concrete (when the fiber ratio is relatively small) follows the fact that the stresses "relax" on the matrix and their distribution converges to constant except for a zone being in the vicinity of the fiber - matrix interfacial surface. This is why the boundary element method is more suitable than the finite element method, particularly in problems concerning fiber reinforced concretes. A comparison medium is employed and the jump in material properties is involved in the integral form, following the idea of Hashin-Shtrikman variational principles, [2], and the publication [3]. Then, minimum for Lagrangian is sought, including variation of the domain of fiber under subsidiary condition that the volume of the fiber is fixed and the diameter of fibers is limited), may be used and the mechanical behavior on the interface fiber - matrix is more viewable during the iteration process solving this strictly nonlinear problem. Similarly to preceding publication by Prochazka, [4], (in view of solvability) additional constrain has to be involved: the boundaries of the unit cell and of the fiber must remain disjoint. The homogenization is based on the publication [5]. It appears that in some cases (short beam, structures close to cubes, etc.) admit the removal of the restrictive condition about the limit diameter of the optimized area (of course, the constant volume condition must be fulfilled), [6]. Certain more theoretical idea can be found in [7]. It is also necessary to point out that during the iterative processes a special tricks have to be introduced, [8], as sometimes very different values of the strain energy density can occur even after small local deviation of the interfacial boundary between fiber and matrix. This is why the authors by [8] suggested a logarithmic scale which reduced the excesses in the above said values. Similar trick was used in [1] and [6]. In these papers it was shown that such an improvement basically speeds up the iteration after comparing the logarithmic scale and linear one.

382


2 Basic relations in a unit cell Denote by a bounded domain, { u p , u p

0 is its Lipschitz's boundary, both representing

the trial unit cell. Unidirectional composite is treated, so that the unit cell is two-dimensional embedded in a global (macro) coordinate system 0 x1 x 2 and local (micro) coordinate system 0 y1 y 2 . On u the displacement vector u {u1 , u 2 } be prescribed, and on p the vector of tractions p { p1 , p 2 } is given. Recall the relation stresses - tractions on the boundary p : p i

ij n j , where n { {n1 , n 2 } is the outward unit normal to

the boundary, ij , ij are components of the stress and the strain tensors, respectively. Microscopic constitutive law and micro-macro relation between stresses and strains provide w ij wy j

㺦

0 ,

,

ij =< ij > = ij d

㺦

(1)

E ij =< ij > = ij d

where ij are components of the overall stress tensor and E ij are components of the overall strain tensor, is the average or the Lebesque measure.. The boundary conditions are selected in such a way that the tractions are on the boundary on opposite sides of the same magnitude but of opposite directions and the strains, which are written as:

( u)

E ˆ(u) ,

³ ij (uˆ ) d

ˆ(u) (uˆ ),

0

(2)

where the displacement fluctuation uˆ is on the opposite sides of the same value and direction. Such boundary conditions are said periodic. Moreover, let E ij be given and create influence matrices, as described hereinafter. Hooke’s law can be written with respect to eq. (2):

ij ( y ) = Lijkl ( y ) kl ( y ) = L0ijkl kl ( y ) + ij = L0ijkl kl (uˆ ( y )) + Lijkl ( y ) E kl + îj ( y )

(3)

where the polarization tensors and ˆ is defined as ij ( y ) [ Lijkl ]( y ) kl ( y ), îj ( y ) = [ Lijkl ]( y )ˆ kl ( y ), [ Lijkl ]( y )

Lfijkl L0ijkl for y f , [ Lijkl ]( y )

m 0 Lm ijkl Lijkl for y

(3a)

and Lijkl are components of the stiffness tensor, being uniform on fiber f and matrix m. L0ijkl are the components of stiffness of the comparative medium, not yet specified. In case the stiffness is written in terms of matrix and the components of both stresses and strains are lined up as vectors it holds: f ½ 11 ° f ° ® 22 ¾ ° f ° ¯ 12 ¿

f ª L11 « f « L12 « 0 ¬

f L12 f L22 0

f ½ 0 º 11 »° f ° 0 » ® 22 ¾ , f ° G f » ° 12 ¼¯ ¿

m½ 11 ° m° ® 22 ¾ ° m ° ¯ 12 ¿

m ª L11 « m « L12 « 0 ¬

m L12 m L22 0

m½ 0 º 11 »° m ° 0 » ® 22 ¾ m° G m » ° 12 ¼¯ ¿

(4)

Substituting eq. (3) to eq. (11) gives expression, which loses meaning in the classical sense as there is a derivative of function which suffers from jump in tractions along the interface C between matrix and fiber and has to be taken in the sense of distributions: Since obviously uˆ [ H 1 ( )] 2 where H 1 is the Sobolev space of integrals of the second power of the first derivatives convergent in the generalized sense. Hence, we can construct a weak formulation responding the constitutive equations (1) and as the displacements are continuous along C we finally get that the distributed loading along C is applied as


piC p1C

m [( L11

p 2C

m [( L12

f L11 ) E11

f L12 ) E11

f ( Lm ijkl Lijkl ) E kl n j

m ( L12

( Lm 22

f L12 ) E 22 ]n1

Lf22 ) E 22 ]n 2

or

(G

m

(G

m

383

(5)

f

G ) E12 n 2 G f ) E12 n1

(5a)

and the traction p C { p1C ,p2C } are positively oriented in the direction of outward normal to the fiber. It appears that this traction is introduced independently of the external boundary conditions. The first relation can now be restored as: 㹽( L0ijkl kl (uˆ ( y )) + îj ( y )) 㹽y j

= p Cj ( y )C

(6)

where C is the Dirac function expressing that the tractions p Cj are distributed along the interface C . Taking into account the boundary conditions we obtain the relations

ijm (uˆ ( y ))

m ijkl ( y ) Ekl ,

ijf (uˆ ( y ))

f ijkl ( y ) Ekl

(7)

This process leads to a fourth-order "concentration factor tensor" Aijkl defined as

ijp ( u( y ))

p ( I ijkl ijkl ( y )) E kl

p Aijkl ( y ) E kl

(8)

where the superscript p { f for y f and p { m for y m . 3 Boundary conditions

Since symmetric unit cell in periodic structure is taken into account the assumed boundary conditions are appropriate according to Fig. 1 and Fig. 2. The boundary conditions for E 22 are analogous to that prescribed for E11 .

Figure 1: Unit cell and boundary conditions for unit shear E12

Figure 2: Boundary conditions for unit tension E11

384


4 Localization using the BEM

Now we are capable of formulating integral equations solving the problem defined by the partial differential eqs. (6) leading to obtain the "concentration factor tensor" Aijkl in eq. (8), or more precisely to get the relations eq. (7). The integral formulation can be posted as the operators L0ijkl are uniform in all domain of the unit cell. Equivalent to eq. (6) reads as: c mn ( )uˆ n ( ) =

* * u mi ( y, ) pi ( y ) d ( y ) - 㺦 p mi ( y, )uˆ i ( y ) d ( y ) + 㺦

㹽

+

㹽

㺦

㺦

* * u mi ( y , ) piC ( y ) d ( y ) + mij ( y, ) îj ( y ) d ( y )

C

(9)

where cmn are components of a tensor its components depend on the position w , if 㺃 this is the unit tensor (unit matrix), the quantities with asterisks are known kernels for the comparative medium, i.e. for example, rm ri 1 1 * u mi ( y, ) = {(3 - 4 0 ) ln( )ij + } , ri = y i - i , r = y - , . is the Euclidean norm, r r r 8G0 (1 - 0 ) etc., and G0 , 0 are elastic material constants in the comparative medium. Decoding the polarization tensor the last term in eq. (9) provides: ... + ( [ Lfijkl - L0ijkl ]

* ) mij ( y , ) ˆk ( y ) d 㺦+ [ Lmijkl - L0ijkl ] 㺦

f

(9a)

m

The overall strain E ij is assumed to be given independently of the shape of the unit cell and of the shape of the fiber. The loading of this unit cell will be given by unit impulses of E ij , i.e. we successively select E i0 j0

E j0i0

1; E ij

0 for either i 0 z i or j 0 z j .

Differentiating eq. (9) by n for 㺃 , introducing the interpolation for the arising equation and the linear interpolation in eq. (9) for 㺃㹽 and, finally, put L0ijkl = Lfijkl the first term in eq. (9a) disappears and the first relation in eq. (7) immediately follows. Putting L0ijkl = Lm ijkl the second relation in eq. (7) is obtained. f m + Aijkl = I ijkl which is unit tensor. From this one get the result that only one Recall well known identity: Aijkl

alternative for stiffness in the comparative medium need to be considered. 5 Optimization

A natural question for engineers dealing with composites could be: determine such shape of fibers that the bearing capacity of the entire composite structure increases and attains its maximum. This is a problem of optimal shape of structures and can be formulated for composites as follows: Let the uniform strain field E kl be applied to the domain (in our case, a periodic distribution of fibers is considered). This produces f m and Amnkl as described in the previous section. Let ( A f , A m , f , m ) be a real concentration factors Amnkl f m functional of Amnkl , Amnkl and f , m . The problem of optimal shape consists of finding such a domain

f from a class O 㻜{ f ; measure f = C} of admissible domains, which minimizes . It remains to state the design parameters p identifying the change of the boundary of the fibers. A natural choice is a movement of the boundary C . The Lagrangian involving the side condition using the Lagrangian multiplier is written as:


(u, f ) =

1 2

1

㺦 ij ( y) ij ( y) d + ( 㺦d f - C ) = 2 S ij Eij + ( 㺦d f - C )

f

385 (10)

f

owing to Hill's energy condition. Coefficient is the Lagrangian multiplier. Substituting eq. (8) to eq. (10) gives: f 1

(u, f ) = [ Lfijkl < A( p) fkl ( y ) > f + Lmijkl < A( p) klm ( y ) > m ]Eij E + ( 㺦 d - C ) 2 f

(11)

and only the concentration factors are dependant of the vector p . 6 Euler's equations

The stationary requirement leads to differentiation of the functional by the shape (design) parameters p s Aklf ( p) 㹽㹽Aklm ( p) 㹽 (u, ) 1 f 㹽 = [ Lijkl < > f + Lm < > m ]E ij E + ijkl 㹽p s 2 㹽p s ps 㹽㹽p s

㺦d = 0

(12)

f

which can be rewritten as:

E s + = 0,

s = 1,2,..., n

(13)

where

=

㹽Aklm ( p) 㹽Aklf ( p) 1 f > m ]E ij E > f + Lmijkl < [ Lijkl < 㹽p s 㹽p s 2 for each s =1,…,n 㹽 d 㹽p s 㺦 f

and n is the number of dof on C . If we have claimed ps, s = 1,...,n the distances of the origin from the current boundary of the fiber, Es corresponds to the strain energy density at the point of the interfacial boundary, in our case at the nodal point s . The eq. (13) requires Es to have the same value for any s. In other words, if the strain energy density E s were the same at any point on the "moving" part of the boundary, the optimal shape of the trial body would be reached. Differentiating by completes the system of Euler's equations ( Ts are triangles created by the origin and the boundary element on C ): n

¦

meas Ts = C

(14)

s 1

7 Example

The examples depicted in Fig. 3 and Fig. 4 are typical for seeking optimal shape of fibers leading to the highest bearing capacity of the composite structure. The black color describes the stiffer phase. In the first case we consider bulk modulus = 3.7 and shear modulus G = 1.11 of the softer matrix while the stiff fiber possesses the material properties as: bulk modulus = 10.4 and shear modulus G = 6.25 . The quantities are introduced in GPa. The volume ratio is divided as 50% : 50%. Note that during the iteration process tending

386


to the optimal shape of fiber the extreme distance of C from the origin located at the center of gravity of the unit cell divaricates from the origin unlimitedly. If we allow for such a phenomenon nonrealistic state can be attained and this is why we have to restrict the extension of the fibers cutting the largest ray p s for all admissible s . For this reason a reasonable iterative procedure has to substitute the standard one and not only eq. (14) has to be ensured but also the restriction of the ray limit needs to be checked. Then the realistic results depicted in Figs. 3 and 4 are reached. In the first case of the soft matrix the homogenized values of material constants are as: = 6.09 and G = 2.61 . The second case leads to the overall properties: = 6.15 and G = 2.45 .

Figure 3:

Soft matrix, stiff fiber

Figure 4:

Soft fiber, stiff matrix

8 Conclusions

In this paper a procedure leading to shape optimization of bearing capacity of a composite structure for minimum Lagrangian is proposed. Unidirectional fibers are studied although the extension of the approach to 3D case theoretically easy. Only the consumption of the computer time would increase basically. In some previous papers of the author it was proved that the optimization based on the selected functional results in both minimum displacements and overall stresses. Very interesting side result appears that the overall properties do not differ too much and classical methods (Mori-Tanaka, self consistent) are relatively accurate, although the impacts on the bearing capacity and deformation can be great. Acknowledgment: This paper was prepared under financial support of GAR, project No. P105/10/0266. Financial support of Ministry of Education and Sport of the Czech Republic, project numbers MSM 6840770001 is also acknowledged. 9 References

[1] P.P. Prochazka, V. Dolezel, and T.S. Lok Optimal shape design for minimum Lagrangian. Engineering Analysis with Boundary Elements 33, (2009), p. 447–455 [2] Z. Hashin, and S. Shtrikman On some variational principles in anisotropic and non homogeneous elasticity. J. Mech. Phys. Solids 10, (1962), p. 335–42. [3] P. Prochazka, and J. Sejnoha Behavior of composites on bounded domain. BE Communications 1996;7(1):6-8. [4] P. Procházka Homogenization of linear and of debonding composites using the BEM. Engineering Analysis with Boundary Elements Volume 25, Issue 9 , (2001), p. 753-769 [5] P.M. Suquet Homogenization techniques for composite media, Lecture Notes in Physics 272, Berlin, Springer , (1985), p. 194-278. [6] P.P. Prochazka Shape optimal design using inverse variational principles. Transaction of WIT: The Built Environment. In: Brebbia CA, Hernandez S, Kassab AJ editors. 1999, p. 40-49. [7] J. Haslinger, and J. Dvorak Optimum composite material design. M2AN 1995;29(6):657-86. [8] Y. Tada, Y. Seguchi, and T. Soh Shape determination problems of structures by the inverse variational principle, feasibility study about application to actual structures. July 1986. Bulletin of JSME 29(253): p. 253.


387

Extended stress intensity factors of a three-dimensional crack in electromagnetothermoelastic solid T. Y. Qin 1 X. J. Li1 L. N. Zhang1 1

College of Science, China Agricultural University, Beijing 100083, P. R. China, email: [email protected]

Keywords: Electromagnetothermoelastic composites, crack, hypersingular integral equation, boundary element method, extended stress intensity factors. Abstract. Using the Green’s functions, the extended general displacement solutions are obtained for a planar crack in a three-dimensional electromagnetothermoelastic solid by boundary integral equation method. Then the crack problem is reduced to solving a set of hypersingular integral equations, in which the unknown functions are the extended displacement discontinuities of the crack surface. Based on the analytical behavior of the unknown functions near the crack front, a numerical method for some typical cracks subjected to extended loads is proposed, in which the extended displacement discontinuities are approximated by the products of basic density functions and polynomials. Finally, the distributions of the extended intensity factors varying with the shape of the crack are presented. The results show that the present method yields smooth variations of extended intensity factors along the crack front accurately. 1. Introduction The electromagnetothermoelastic materials have the electromagnetic coupling effect, and have become of major interest as the functional materials such as actuators and sensors. The reliability of these structures depends on the knowledge of applied mechanical, electric, magnetic, and thermal disturbances. In the field of magneto-electro-elastic materials, an anti-plane crack was considered by Wang and Mai [1], and it is found that the SIFs in the static case were independent of electric and magnetic loadings. Sih et al [2] studied mode I and II crack initiation behavior problems. Tian and Gabbert [3] considered some multiple crack interaction problems in magnetoelectroelastic solids and evaluated the extended SIFs. Based on the integral transforms and singular integral equation method, a dynamic anti-plane impermeable internal crack perpendicular to the boundary was considered by Feng and Su [4]. A penny shaped crack under uniform heat flow in an electromagnetothermoelastic solid was considered by Niraula and Wang [5]. The main difficulties in the field are related to the mathematical complexities and efficiency of the numerical methods. In this paper, the general solutions of extended displacements for a three-dimensional crack in electromagnetothermoelastic materials subjected to the extended loads are obtained by boundary integral method. Based on the exact analytical solution of the singular stresses and extended displacements near the crack front, a numerical method for the crack problems was proposed by the finite-part integral method. 2. Hypersingular integral equations Consider a planar crack S in an infinite electromagnetothermoelastic materials. A fixed rectangular Cartesian system xi (i=1,2,3) is used. The crack is assumed to be in the x1x2 plane, and normal to the x3 axis. Using the magneto-electro-elastic form of the Somigliana identity, the extended displacements at an interior point p are expressed as I ,J =1,2,3,4,5 (1) U I ( p ) ³ TIJ ( p, Q )U J (Q )ds (Q ) S

where TIJ is the fundamental solution of the magneto-electro-elastic material and related Green’s function[6,7], U J is the extended displacement discontinuity, and can be written as

U J

u j ° ®u4 °u ¯ 5

u j u j I I I

M M M

J J J

j 1, 2,3 4 5

(2)

388


Using the boundary conditions of the crack surface, the hypersingular integral equations can be obtained as

³ (r

3

S

³ (r

2

³ (r

2

S

S

2 (c44 D0 s02 (GDE 3r,D r, E ) (GDE 3r,D r, E )¦ i 1 Ui2ti2 )uE (q ) 3r 4 r,D ¦ i 1 O33 si2ti1u6 (q ))ds (q ) pD ( p) (3) 5

r,D ¦ i 1 Aib ti2 Ui1uD (q ) r 3 ¦ n 5

5

3

¦

5 i 1

5

Uim tit un (q ) 3r 4 O3D r,D ¦ i 1 vi2 Oi- Uimu6 (q ))ds (q ) pm ( p) 5

(GDE 3r,D r, E )¦ i 1 Aib O3E ti2uE (q) 3r 4 O3D r,D ¦ i 1 Aib vi2 Oi- O33 Ui6u6 (q))ds (q) p6 ( p)

where D m=3,4,5,

5

³

5

(4) (5)

means that the integral must be interpreted as a finite-part integral, and the

coefficients related to material constants are given [8]. It can be shown that the singularities near the crack front in electromagnetothermoelastic materials are the same as that in a general homogenous material. The extended SIFs are defined as K lim 2rV (r ,T ) , K lim 2rV ( r ,T ) , K lim 2rV (r ,T ) 33 2 32 3 31 r o0 r o0 ° 1 r o0 T 0 T 0 T 0 (8) ® ° K 4 lim 2r D3 (r ,T ) , K 5 lim 2r B3 (r ,T ) , K 6 lim 2r-3 (r ,T ) r o0 r o0 r o0 T 0 T 0 T 0 ¯ where r is the distance from point p to the crack front point q0 . 3. Numerical procedure

A method proposed by Qin and Noda [9] can be generalized to solve the hypersingular integral equations (3~5) for several typical shape cracks, such as rectangular crack, elliptical crack and semi-elliptical crack. Using the behaviour near the crack front, the extended displacement discontinuities unknown functions can be written as U J ([1 , [2 ) FJ ([1 , [ 2 )W ([1 , [ 2 ), J 1,",6 (9) where WJ ([1 , [ 2 ) is the fundamental density function, and can be expressed as (a 2 [ 2 )(b 2 [ 2 ) for rectangular crack 1 2 ° ° 2 2 2 2 W ([1 , [ 2 ) ® 1 [1 a [ 2 b for elliptical crack (10) ° 2 2 2 2 for semi-elliptical crack °¯ [ 2 (1 [1 a [ 2 b ) and FJ ([1 , [ 2 ) is the unknown function, which can be approximately expressed as the polynomials

¦ ¦

FJ ([1 , [ 2 )

S

H

s 0

h 0

aJsh[1s[ 2h

(11)

here a Jmn are unknown constants. Substituting eq (9) into eq (3), eq (4) and eq (5), a set of algebraic linear equations for the determination of the unknown coefficients a Jmn can be obtained by finite-part integral method as follows, which can be determined by selecting a set of collocation points. M N (12) ¦ m 0 ¦ n 0 aImn I IJmn ( x1 , x2 ) pJ ( x1 , x2 ) . 4. Numerical solutions

In order to verify above methods and illustrate its application, some typical crack problems in a threedimensional infinite transversely isotropic magneto-electro-elastic solid subjected to normal mechanical load f V 33 , electrical load D 3f and magnetic loads B3f in infinite are calculated numerically. Here the independent

material constants for Eshelby’s tensors given by Li [10] are used. In demonstrating the numerical results, the following dimensionless intensity factors will be used F1 °° ® F4 ° °¯ F5

K1 / V 33f b , F2 f 33

b

f 33

b

K4 / D K5 / B

K 2 / V 31f b , F3

K 3 / V 32f b

(13)


389

4.1 Rectangular crack. Consider a rectangular crack in infinite electromagnetothermoelastic materials. In the case of the load is V 33f ˈ the numerical results of dimensionless SIFs with increasing the polynomial exponents are given in tables 1 for 20×20 collocation points. It shows that the results are convergent to the forth digit when polynomial exponents M=N9.Table 2 shows the variation of the maximal extended stress intensity factors when a/b=1,2,3,5,7, . The results show that the present method has convergence to the fourth digit when a/b7. Table 1 Convergence of dimensionless extended SIF F1 along with increasing the polynomial exponents x1 / a 0/11 1/11 2/11 3/11 4/11 5/11 6/11 7/11 8/11 9/11 10/11 .7536 .7536 .7536 .7529 .7534

M= 13 M= 11 M= 9 M= 7 Qin[9]

.7518 .7518 .7518 .7514 .7517

.7464 .7464 .7464 .7466 .7465

.7374 .7374 .7374 .7383 .7375

.7244 .7245 .7244 .7246 .7245

.7067 .7067 .7067 .7069 .7065

.6829 .6829 .6829 .6827 .6826

.6512 .6512 .6512 .6513 .6510

.6092 .6089 .6092 .6071 .6086

.5503 .5501 .5503 .5505 .5494

.4493 .4520 .4493 .4433 .4543

Table 2 Dimensionless maximal extended stress intensity factors a/b F1 F4 F5

1 .7536

2 .9183

Present 3 5 .9279 .9379

7 1.002

1.002

Hu[11] 1.000

Tian [3] 1.0053

F2

.8781

.9904

1.019

1.032

1.033

1.033

——

——

F3

.9903

1.282

1.408

1.498

1.524

1.537

——

——

4.2 Elliptical crack. Consider an elliptical crack in an infinite electromagnetothermoelastic solid ender the extended loads at the infinite. The numerical results of dimensionless extended stress intensity

factors for different crack shape listed in tables 3 for 13×13 collocation points. It shows that the results are agreement with those given by Ding [6] and Qin [12]. Fig.1 gives the dimensionless extended stress intensity factors for different crack shape. It is shown that results show that the maximum extended stress intensity factors vary more gently when a/b7, and are close to those for two dimensional cases. Table 3 Maximum dimensionless extended stress intensity factors F1, F4, F5 a/b=1

a/b=4/3

a/b=1.5

a/b=2

Present

0.6366

0.7230

0.7536

0.8257

Ding [6]

0.6366

ˉ

ˉ

ˉ

Qin [12]

0.6366

0.7239

0.7564

0.8267

390


Figure 1 Dimensionless extended stress intensity factors F1, F4, F5 for different a/b 4.3 Semi-ellptical crack. Consider a semi-elliptical crack in an infinite electromagnetothermoelastic solid ender the extended loads at the infinite as shown in Fig.2. The numerical results of dimensionless stress intensity factorsF1 along the curved edge for different crack shape are in Fig.3.

Figure 2 An semi-elliptical crack

Figure3 Dimensionless stress intensity factorF1 along the curved edge for different a/b 5. Conclusions


391

In this paper, a planar crack problem in electromagnetothermoelastic materials is investigated. The problem is reduced to solve a set of hypersingular integral equations by boundary integral equation method. Based on the behaviour of the extended displacement discontinuities near the crack front, a high precision numerical technique is proposed for the hypersingular integral equations, and the numerical results of the extended stress intensity factors are given for several typical crack problems. From the numerical solutions, it is shown that the maxinum extended stress intensity factors vary more gently when a b t 8 , and are closed to that of two-dimensional case. Acknowledgments

The project supported by the National Natural Science Foundation of China (No. 10872213) and the personnel exchange program with China Scholarship Council(CSC) and German Academic Exchange Service(DAAD). References

[1] B.L.Wang and Y. W. Ma Mechanics Research Communications, 31, 65-73(2004). [2] G. C. Sih, R. Jones and Z. F. Song Theoretical and Applied Fracture Mechanics, 40, 161-186(2003). [3] W.Y. Tian and U. Gabbert European Journal of Mechanics a-Solids 23, 599-614(2004). [4] W. J. Feng and R. K. L. Su International Journal of Solids and Structures, 43, 5196-5216(2006). [5] O.P. Niraula and B.L.Wang Journal of thermal Stresses, 29, 423-437(2006). [6] H. J. Ding, A.M. Jiang and P. F. Hou, W. Q. Chen Engineering Analysis with Boundary Elements, 29, 551-561(2005). [7] E. Pan Z Angew Math Phys, 53(5):815-38(2002). [8] B.J. Zhu and T.Y. Qin International J Solids and Structures, 44, 5994-6012(2007). [9] T.Y. Qin and N. A. Noda International Journal of Solids and Structures, 40, 2473-2486(2003). [10] J.Y. Li. International Journal of Engineering Science, 38, 1993-2011(2000). [11] K.Q. Hu, G.Q. Li and Z. Zhong. Mechanics Research Communications, 33, 482-492(2006). [12] Q.H. Qin and Q.S.Yang Macro-Micro Theory on Multifield Coupling Behavior of

Heterogeneous Materials, Higher Education Press, 54-155(2008).

392


Adaptive Cross Approximation and its applications R. Grzhibovskis1 and S. Rjasanow1 1

Department for Mathematics and Computer Science, University of Saarland, D-66123 Saarbrücken, Germany

Keywords: Adaptive Cross Approximation, Boundary Element Method, radial basis functions, composite materials

Abstract. The main ideas of the Adaptive Cross Approximation (ACA) for Boundary Element Method (BEM) will be presented. Some recent applications of the ACA will be reviewed. The first one is devoted to the reconstruction of the three-dimensional metal sheet surfaces obtained via incremental forming techniques by the use of the radial basis functions. In the second application, a calculation of effective elastic moduli in three-dimensional linear elasticity for highly anisotropic composite material is considered. 1. Adaptive Cross Approximation The Adaptive Cross Approximation (ACA) for dense matrices arising in BEM was first introduced in [4], [7] in mathematical and in [9] in engineering literature. Now it is a well established technique for approximation of dense matrices of different origin. The main applications lie, however, still by three-dimensional BEM. In the last years a significant progress is achieved in both theoretical foundation of the ACA [6], [5] and in the practical use of this technique in engineering and industry [10]. We refer also to the monograph [12], where a number of careful academic and more complicated numerical examples is presented. Due to an asymptotically quadratic memory requirement for the classical boundary element realisations, this method is applicable only for a rather moderate number N of boundary elements. Fortunately, all boundary element matrices can be decomposed into a hierarchical system of blocks which can be approximated by the use of low rank matrices. This approximation can be computed by the use of the ACA algorithm. A typical element of a Galerkin-BEM matrix A ∈ RN×M is of the form Z Z

ak, =

K(x, y)ψk (x)ϕ (y) dsy dsx ,

k = 1, . . . , N , = 1, . . . , M ,

(1)

Γ Γ

where K : R3 × R3 → R is either the fundamental solution of the underlining differential operator or one of its derivatives. The trial functions ϕ as well as the test functions ψk are usually compactly supported. Let us denote the centers of these supports by N M X = xk k=1 and Y = y =1 . If the kernel K is smooth then the singular values of the matrix A are exponentially decaying and the matrix can be approximated by a low rank matrix A˜ as follows ˜ F ≤ εAF , A − A

rank A˜ = r(ε) ,

where the rank r = r(ε) is a logarithmic function of the required accuracy but does not depend on the dimensions N and M. Therefore, a linear memory requirement for the matrix A˜ is achieved ˜ = O r(ε)(M + N) . Mem(A) The best possible approximation in any unitary invariant norm is given due to the Mirsky theorem [11] by the truncated Singular Value Decomposition (SVD) of the original matrix A. In BEM, however, the kernel function, as we already have mentioned above, is the fundamental solution of a differential operator, and, therefore, exposes an algebraic singularity for x → y and is infinitely smooth for x = y. Such functions are asymptotically smooth, i.e. it holds α β ∂x ∂y K(x, y) ≤ c p p! |x − y|−p K(x, y) , for all α, β ∈ N30 , (2)


393

0.05 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

Fig. 1: An admissible cluster pair for N = 2264

Fig. 2: Partitioning of the BEM matrix for N = 2264 where p = |α + β|. For asymptotically smooth functions, the admissibility condition min diam X , diamY ≤ η dist(X,Y) , 0 < η < 1

(3)

guarantees existence of a degenerate approximation r(ε)

˜ y) = K(x, y) ≈ K(x,

∑ uk (x) vk (y) ,

for all x ∈ X , y ∈ Y ,

k=1

where the rank r = r(ε) depends only on the required accuracy ε. Thus, the Galerkin matrix (1) constructed on an admissible pair (X,Y ) will be of the low rank r(ε) independent of its dimension. However, the original sets X and Y are, of course, not admissible. The usual way to obtain an approximation is an hierarchical decomposition of the matrix in a system of blocks which corresponds to an hierarchical decomposition of the sets X and Y in two systems of clusters and determination of the admissible cluster pairs corresponding to the condition (3). Instead of going into details, we refer to the monograph [12] and demonstrate an admissible cluster pair in Figure 1. The final matrix is then so called hierarchical matrix [8] an we illustrate its structure in Figure 2. In the practice, the optimal approximation of the single blocks of a hierarchical matrix can not be computed with SVD due to its high computational costs. The ACA algorithm instead delivers a good quality approximation in almost linear complexity. In this paper, we present a version of the ACA, so called ”Fully pivoted ACA” which is also not asymptotically optimal but very clear and easy to understand. Let A ∈ RN×M be a block of a hierarchical matrix.

394


Algorithm 1 1. Initialisation R0 = A , S0 = 0 . 2. For i = 0, 1, 2, . . . compute 2.1. pivot element (ki+1 , i+1 ) = ArgMax |(Ri )k | , 2.2. normalising constant 2.3. new vectors

−1 γi+1 = (Ri )ki+1 i+1 , ui+1 = γi+1 Ri ei+1 , vi+1 = R i eki+1 ,

2.4. new residual 2.5. new approximation

Ri+1 = Ri − ui+1 v i+1 , Si+1 = Si + ui+1 v i+1 .

In Algorithm 1, e j denotes the jth column of the identity matrix I. The whole residual matrix Ri is inspected in Step 2.1 of Algorithm 1 for its maximal entry. Thus, its Frobenius norm can easily be computed in this step, and the appropriate stopping criterion for a given ε > 0 at step r would be Rr F ≤ εAF . We refer again to [12] for more complicated but asymptotically optimal partially pivoted ACA algorithm.

2. Applications Interpolation with Radial basis functions In [3], a smooth reconstruction of three-dimensional metal sheet surfaces obtained via incremental forming techniques with the help of radial basis functions is considered. Radial basis functions (RBFs) have become increasingly popular for the construction of smooth interpolant s : Rd → R through a set of N scattered, pairwise N distinct data points xi , fi i=1 , xi ∈ Rd . The interpolant is introduced as N

s(x) = ∑ ai φ(|x − xi |) + Pm (x),

(4)

i=1

where ai are unknown coefficients, Pm is an mth-degree polynomial, and φ is a basis function. In addition to the interpolation condition s(xk ) = fk at each data point, we require N

∑ ak q j (xk ) = 0

k=1

for each basis function q j (x), j = 1, . . . , l(m), in the l(m)-dimensional space of mth-degree polynomials in Rd . This yields a system of linear algebraic equations, Φ Q a f = , (5) 0 c Q 0 where Φi j = φ(|x j − xi |), f is the vector of data values fi , a is the vector containing the unknown coefficients l(m) ai , c is the vector of coefficients of Pm with respect to the basis {qk }k=1 , and Qi j = q j (xi ). In our application, the data comes from optical measurements of sheet metal parts. The top and the bottom surfaces of the part are


395

measured in a fixed frame of reference, and of thickness along the part is sought. We, therefore, a distribution restrict ourselves to data in R2 , i.e. xi = xi,1 , xi,2 , and make use of the thin plate spline (TPS) basis function φ(r) = r2 log(r)

(6)

which is conditionally positive-definite of order two. Thus, a first-degree polynomial, P1 (x) = c1 + (c2 , c3 ) , x must be inserted into (4) to guarantee the invertibility of system (5). The form of (6) implies, that the matrix Φ is fully populated. It is easy to check, however, that the function (6) satisfies (2) i.e. is asymptotically smooth. The performance of the resulting interpolation procedure on synthetic data sets is summarized in Table 1. The memory requirement ("Mem") is given along with the compression rate ("rat") for the approximation as a measure of the data reduction achieved through ACA. The number of GMRES iterations (#) and CPU time ("sol. t.") are also given. We observe the almost linear storage and computation time increase. N [103 ] 1 5 10 25 50 100 500

Mem [MB] 2.5 28.1 69.2 199.5 466.4 1122.4 7540.8

rat. [%] 33.80 14.73 9.07 4.18 2.44 1.47 0.39

compr. t. [s] Nwcc U wk z w@

(20c,d)

U k w ' Nwc cc GA1 EI wccc GA wc Nvc cc U k v ' GA1 EI vccc GA vc w z

y

z

(20e)

z

y

(20f)

z

v y

y

Finally, D j , E j , E j , J j , J j ( j 1, 2,3 ) are functions specified at the beam-column ends x 0, l . Eqns. (17)(19) describe the most general boundary conditions associated with the problem at hand and can include elastic support or restraint. It is apparent that all types of the conventional boundary conditions (clamped, simply supported, free or guided edge) can be derived from these equations by specifying appropriately these functions (e.g. for a clamped edge it is D1 E1 J 1 1 , E1 J 1 1 ,

D 2 D3

E 2 E3 J 2 J 3 E 2 E 3 J 2 J 3 0 ). The solution of the boundary value problem given from eqns. (16), subjected to the boundary conditions (17)-(19), which represents the nonlinear flexural analysis of a Timoshenko beam-column, partially supported on a tensionless Winkler foundation, presumes the evaluation of the shear deformation coefficients a y , az , corresponding to the principal coordinate system Cyz . These coefficients are established equating the approximate formula of the shear strain energy per unit length

U appr .

a y Q y2

a Q2 z z 2 AG 2 AG

(21)

with the exact one given from

³:

U exact

W xz 2 W xy 2G

2

d:

(22)

and are obtained as [2] ay

1

Ny

A ³ ª¬ 4 e º¼ ª¬ 4 e º¼ d : '2 :

az

1

Nz

A ³ ª¬ ) d º¼ ª¬ ) d º¼ d : '2 :

(23a,b)

where W xz j , W xy are the transverse (direct) shear stress components, { i y w wy iz w wz is a j symbolic vector with i y , iz the unit vectors along y and z axes, respectively, ' is given from '

2 1 Q y z

(24)

Q is the Poisson ratio of the cross section material, e and d are vectors defined as § y2 z2 · e ¨Q I y ¸ i y Q I y yziz ¨ 2 ¸¹ ©

§ y2 z2 · d Q I z yzi y ¨Q I z ¸ iz ¨ 2 ¸¹ ©

(25a,b)


403

and 4 y, z , ) y, z are stress functions, which are evaluated from the solution of the following Neumann type boundary value problems [2]

24 2 I y y in 2)

2 I z z

:

in :

w4 wn

ne

w) wn

nd

on

K 1

*

*j

(26a,b)

j 1

on

K 1

*

*j

(27a,b)

j 1

where n is the outward normal vector to the boundary * . In the case of negligible shear deformations a y a z 0 . It is also worth here noting that the boundary conditions (26b), (27b) have been derived from the physical consideration that the traction vector in the direction of the normal vector n vanishes on the free surface of the beam. 3. Integral Representations Numerical Solution

According to the precedent analysis, the nonlinear flexural analysis of a Timoshenko beam-column, partially supported on a tensionless Winkler foundation, undergoing moderate large deflections reduces in establishing the displacement components u x and v x , w x having continuous derivatives up to the second and up to the fourth order with respect to x , respectively. Moreover, these displacement components must satisfy the coupled governing differential equations (16) inside the beam and the boundary conditions (17)-(19) at the beam ends x 0, l . Eqns. (16) are solved using the Analog Equation Method [1] as it is developed for hyperbolic differential equations [3]. 4. Numerical examples

In order to illustrate the importance of the nonlinear analysis and the influence of the shear deformation effect, a clamped beam-column of length l m , having a hollow rectangular cross section ( E 210GPa , v 0.3 ) and resting on a homogeneous (either bilateral or unilateral) elastic foundation of stiffness k z , as this is shown in Fig.2, is examined. 0

0.02

pz=100kN/m

z

w (m)

x y

kz

0.04

az= 3.664 ay= 1.766

h=14cm

l=5m t=4mm

0.06

y z b=23cm

0.08

0

1

2

3

4

5

x (m) Nonlinear Analysis without Shear Deformation Nonlinear Analysis with Shear Deformation Linear Analysis without Shear Deformation Linear Analysis with Shear Deformation e

Fig. 2. Clamped beam-column of hollow Fig. 3. Deflection w along the beam-column (case i) for rectangular cross section subjected to the soil stiffness k 50kN / m 2 . z uniformly distributed load (case i).

404


In Fig.3 the deflection w along the beam-column resting on a tensionless foundation with

k z 50kN / m2 and subjected to a uniformly distributed load p z 100kN / m (case i) are presented performing either a linear or a nonlinear analysis and taking into account or ignoring shear deformation effect. From this figure, the influence of the nonlinearity to the performed analysis is remarked, while the discrepancy of the obtained results due to the shear deformation effect justifies its inclusion even in thin walled sections. Moreover, in Table 1 the deflection and the bending moment at the midpoint of the beamcolumn are presented performing either a linear or a nonlinear analysis and taking into account or ignoring shear deformation effect. Finally, in Fig.4 the deflection curves of the beam-column resting on a tensionless foundation are presented for various values of the modulus k z of the subgrade reaction, performing a nonlinear analysis, taking into account shear deformation effect and demonstrating the importance of the soil stiffness in the obtained results. To illustrate the importance of the tensionless character of the subgrade reaction, the same beam-column subjected to a concentrated moment M y 100kNm at its midpoint (case ii) is also studied. In Fig.5 the deflection curves of the beam-column resting on a tensionless foundation are presented for various values of the modulus k z of the subgrade reaction, performing a nonlinear analysis and taking into account shear deformation effect. Additionally, in Table 2 the extreme values of the displacements and the soil reaction are presented for both cases of bilateral and unilateral soil reaction for various values of the modulus k z performing a geometrical nonlinear analysis and taking into account shear deformation effect. From the aforementioned figure and table, it is concluded that the unilateral character of the foundation is of paramount importance and the error occurred from the ignorance of this behavior is considerable. Without Shear Deformation Linear Analysis

Nonlinear Analysis

With Shear Deformation Linear Analysis

Nonlinear Analysis

w l /2

7.49

6.93

7.95

7.28

M y l /2

188.23

178.13

189.57

176.89

Table 1. Deflection (cm) and moment (kNm) at the midpoint of the clamped beam-column (case i), for kz

kz (kN/m2)

50kN / m 2 .

Bilateral Winkler Min w (mm)

Max w (mm)

5.59

-5.67

5.53

Max psz (kN/m) 0.276

5.39 4.01 1.45

2.694 20.007 72.675

-6.10 -7.84 -9.21

4.92 2.56 0.57

2.459 12.822 28.327

Max w (mm)

5·101

-5.59

2

-5.39 -4.01 -1.45

5·10 5·103 5·104

Unilateral Winkler Max psz (kN/m) 0.279

Min w (mm)

Table 2. Extreme values of the displacements and the foundation reaction of the beam-column (case ii). 5. Concluding remarks

The main conclusions that can be drawn from this investigation are a. The numerical technique presented in this investigation is well suited for computer aided analysis for beams of arbitrary simply or multiply connected doubly symmetric cross section.


405

b. In some cases, the effect of shear deformation is significant, especially for low beam slenderness values. c. The discrepancy of the obtained results performing a linear or a nonlinear analysis is remarkable. d. The significant influence of the unilateral character of the foundation in both the deflections and the soil reaction, especially in the case of a stiff soil is demonstrated. e. The importance of the soil stiffness to the response of the beam – column is verified. f. The developed procedure retains most of the advantages of a BEM solution over a FEM approach, although it requires domain discretization. 0

kz5 = 50000kN/m2

-0.008

kz4 = 5000kN/m2

-0.004

w (m)

w (m)

0.02

0

0.04

kz1 = 0kN/m2 kz2 = 50kN/m2 kz3 = 500kN/m2

kz3 = 500kN/m2

0.004 0.06

kz4 = 5000kN/m2

kz2 = 50kN/m2

kz5 = 50000kN/m2 0.008 0

kz1 = 0kN/m2

0.08 0

1

2

3

1

2

3

4

5

x (m) 4

5

x (m)

Fig. 4. Deflection curves of the beam-column (case i) Fig. 5. Deflection curves of the beam-column (case for various values of the modulus k z of the subgrade ii) for various values of the modulus k z of the reaction. tensionless subgrade reaction. Acknowledgments

The work of this paper was conducted from the “DARE” project, financially supported by a European Research Council (ERC) Advanced Grant under the “Ideas” Programme in Support of Frontier Research [Grant Agreement 228254]. References [1] J.T.Katsikadelis The Analog Equation Method. A Boundary-Only Integral Equation Method for Nonlinear Static and Dynamic Problems in General Bodies, Theoretical and Applied Mechanics, 27, 13-38 (2002). [2] E.J.Sapountzakis and V.G. Mokos A BEM Solution to Transverse Shear Loading of Beams, Computational Mechanics, 36, 384-397 (2005). [3] E.J.Sapountzakis and J.T. Katsikadelis Analysis of Plates Reinforced with Beams, Computational Mechanics, 26, 66-74 (2000).

406


Solution of Hot Shape Rolling by the Local Radial Basis Function Collocation Method Božidar Šarler1, Siraj-ul-Islam2 and Umut Hanoglu3 1

Laboratory for Multiphase Processes, University of Nova Gorica, Slovenia, e-mail: [email protected] 2 Laboratory for Multiphase Processes, University of Nova Gorica, Slovenia, e-mail: [email protected] 3 Laboratory for Multiphase Processes, University of Nova Gorica, Slovenia, e-mail: [email protected] Keywords: Hot shape rolling, Thermo-mechanical model, Strong formulation, Meshless methods, Radial basis functions, Collocation.

Abstract. This paper formulates the problem of hot shape rolling of steel as a sequence of multi-pass deformation steps that lead from the initial shape to the final deformed shape. The thermal and mechanical models are considered. The problem is formulated as a sequence of plane-strain problems with heat conduction in the direction perpendicular to rolling only, and uniform displacement in the direction of rolling. The material is assumed to behave elasto-plastic. Strain hardening is assumed to depend on temperature, effective strain and effective strain rate. The related thermal and mechanical models are solved in their strong forms by using the Local Radial Basis Function Collocation Method (LRBFCM) with multiquadrics shape functions and explicit time stepping. The spatial discretization of the initial and deformed cross section of the rolled product is performed through Transfinite Interpolation (TFI) and Elliptic Node Generation (ENG). 1. Introduction The purpose of this paper is the formulation of the meshless numerical solution [1, 3] of a Thermo-Mechanical Model (TMM) for hot shape rolling [2]. Hot shape rolling is a process in which the rolled product, initially usually in the shape of a rectangular cross-section obtains a highly deformed shape like for example a railway track. Numerical solution is obtained by using the recently developed LRBFCM [3]. This is the first attempt to use this meshless method in this type TMM. Majority of metal forming analysis of large deflection problems are based on finite element method (FEM). Although FEM is well established procedure, it might run into difficulty due to mesh distortion issues. Because of the truly meshless character, LRBFCM offers a viable alternative to avoid such situation. LRBFCM has recently been used in highly sophisticated simulations like multi-scale solidification modeling [4], convection driven melting of anisotropic metals [5], continuous casting of steel [6]. The TMM model is in this paper based on the Glowacki’s assumptions of the travelling slice (see Fig.1 left) with a sequence of plane deformations and an elasto-plastic constitutive rule [7, 11, 12, 13]. LRBFCM is simple expilicit meshless procedure based on multi-quadic radial basis functions. Meshless LRBFCM is developed for a sequence of multipass deformation steps from the initial shape to the final shape. Open literature on application of meshless methods to TMM is scarce. The point interpolation method based on the global weak form has been succesfully used in the field of metal forming [8]. Such approach however introduces some sort of integration that requires a background mesh and the method thus deviates from the trully meshless sense. In the present paper, the strong form approach is preserved in TMM, such as demonstrated in [9]. The paper is organised as follows. In section 2, the formulation of the thermal model is discussed, followed by the mechanical model in section 3. In section 4, the geometry confirming node generation procedure is described. Section 5 represents conclusions. The thermal model is interacting with the mechanical model through stress-strain constitutive relations. The mechanical model is interacting with the thermal model through internal heat generation due to plastic deformation. A basic scheme of the calculation procedure is shown in Fig. 1 right. 2. The Thermal Model The thermal model is designed to be able to calculate the steady state temperature distribution in the rolled product as a function of the following process parameters: steel grade, initial product dimension, initial temperature, rolling velocity, cooling of the strand and the rolls, rolls diameter, shape and position, and rolls


407

lubrication. The thermal model is defined on the complex shaped three dimensional domain : with boundary * . The steady state temperature distribution in the rolled product is defined through equation (1).

Fig. 1 Left: schematics diagram of travelling slice. Right: scheme of the thermo-mechanical coupling and discretization manipulations.

U c p vT k T S , p :3D

(1)

wT k T S ; p : 2D (2) wt with p , U , c p , T , k , and S standing for position vector, density, specific heat, temperature, thermal

Ucp

conductivity, and heat source due to plastic deformation. It is assumed that the heat transport takes place only in the direction perpendicular to the rolling direction and that the homogenous deformation takes place. This implies that the velocity in the direction of the rolling is constant over the perpendicular cross-section of the rolled product. It however changes (increases) in the rolling direction due to the reduction of the cross-section. The steady state Eq. (1) can under the posed assumptions be transformed into transient Eq. (2) with t standing for time. Eq. (2) represents a slice of the rolled product which travels through the rolling strand with the velocity vz ( z ) where the coordinate z points into the rolling direction. The time coordinate can be calculated from the rolling coordinate through relation z (t )

³

t

t0

vz ( z ) dt z0 , i.e. t ( z )

focuses on the solution of the heat equation at time t

f > z0 , t0 , vz ( z ) @ . The thermal model thus

t0 't , i.e. at z t 't on a 2D domain of arbitrary t0 i.e. z t0 to be known, as well as the boundary

cross-section : 2D . We assume the thermal field at the time t

conditions (of the Dirichlet, Neumann, or Robin type) at the time (t0 , t0 't ] . We structure the numerical method by the local explicit radial basis function collocation method. In this method, the domain and the boundary of the domain are discretized with N

N * N : points p n

^p , p `

tr

x

y

( tr is transpose) of which N : are in the

domain and N * on the boundary. Each of the points has a defined neighborhood with NZ points. On these points a collocation with the scaled (with the maximum distances between points in the subdomain xmax , ymax ) multiquadrics radial basis functions \ i

px pxn

2

/ xmax p y p yn / ymax c 2 is made. This means that 2

NZ

the temperature T and its derivatives can be calculated as T

¦\ D n

n 1

w 2T w px2

NZ

¦ w \ n 1

2

n

n

, wT w px

NZ

¦ w\

n

/ wpx D n ,

n 1

/ wpx2 D n with the collocation coefficients D n . The details of the method can be found in

[5]. This method is a truly meshless method since there is no need to define any connectivity or polygonisation between the points p n . After the calculation of the temperature field, the mechanical deformation at time

408

t


t0 't , i.e. z t 't has to be calculated. The mechanical deformation is characterized by the shape of the

rolls at position z t 't . The calculation of mechanical deformation p n t0 't

û

displacement vector u n

p n t0 u n t0 't ; with

, u yn ` is described as follows. tr

xn

3. The Mechanical Model During the rolling process multidirectional forces with varying magnitudes makes the material deform in a very short interval of time. In a real working system, the physical properties of the entry and also the exit product are usually known but it is very crucial to understand the deformation process where many different variants are combined and the material structure has changed. It not only helps us to understand the process, it also gives us a chance to make future predictions. The mechanical model of the hot shape rolling is actually the sum of forces per unit volume on the work piece. The main goal of the model is to find the displacements of virtual points in the slab. In order to understand the details of the deformation process of the slab under the deformation zone, the slab is divided into thin slices to be analyzed individually. Strong formulation of metal deformation for static problems is the following: LT + b 0 , (3) where L is the 3 u 2 derivative matrix with elements L11 w wx , L12 0 , L 21 0 , L22 w wy , L31 w wy

and L32

w wx ,

^V

, V y , V xy ` is the stress vector, and b tr

x

^b , b `

tr

x

y

the body force, that are assumed to

be present only at the boundaries (surface traction at the contact with the roll) of the system. The boundary conditions at the contact can be written as: essential boundary conditions: u uˆ on *u and natural boundary conditions:

n on *t , where uˆ is the prescribed displacement, is the surface traction

^W

,W y ` , n is tr

x

the 3 u 2 matrix of direction cosines of the normal direction at the boundary which can be defined as n11 n 32 nx , n12 n 21 0 , n 31 n 22 n y ( nx , n y represents direction cosines along the co-ordinate axis). The boundary conditions at the inlet are: u x

0 , uy

the rolling direction. The boundary conditions at the outlet are; u x

0 and u z

0 , uy

ventry , where ventry is a known value and z is

0 and v z is unknown because the final dimension of

the exiting slab is not known yet. However, in the case of plain strain model the exit speed can be calculated with conservation of mass. In a 2D system, the equilibrium equations can be written as shown below

wV x wV xy bx wpx wp y

0,

wV y wp y

wV xy wpx

by

0.

The stress-strain relation can be written as

(4)

Cep where Cep is the elastic-plastic stiffness matrix defined for

plane stress case as

ª1 Q eff 0 º » §Q Eeff « 1 · 1 1 1 «Q eff »; Q eff Eeff ¨ , , (5) Cep 1 0 2 ¨ E 2 E p ¸¸ 1 Q eff « Eeff E E p » © ¹ 0 1 Q eff 2 »¼ «¬ 0 with Eeff standing for the effective Young's modulus and Q eff for the effective Poisson's ratio, with elastic Young’s modulus E and plastic modulus E p and

^H

, H y , H xy ` stands for the strain vector. Alternatively if tr

x

the stress vector is replaced by its strain relation shown above and strain vector is replaced by its displacement relation which is = Lu , then the deformation problem can be expressed as


§ Eeff ¨¨ 2 © 1 Q eff

· ª w § wu x wu y · w Q eff ¸¸ « ¨¨ ¸ w w wp y ¸¹ wp y p p « ¹¬ x © x

§ Eeff ¨¨ 2 © 1 Q eff

·ª w ¸¸ « ¹ «¬ wp y

§ wu wu y ¨¨Q eff x wpx wp y ©

409

§ 1 Q eff ¨ ¨ 2 ©

§ wu wu y ¨¨ x © wpx wp y

· ·º ¸¸ ¸ » bx ¸ ¹ ¹ »¼

0

(6)

· w § 1 Q eff ¨ ¸¸ ¨ ¹ wpx © 2

§ wu x wu y ¨¨ © wp y wpx

· ·º ¸¸ ¸ » by ¸ ¹ ¹ »¼

0.

(7)

The displacement is in the mechanical model approximated similarly as the temperature in the thermal model by using the locally supported multiquadrics

ux p

NZ

¦\

n

(p )D xn ,

(8)

n

(p )D yn

(9)

n 1

u y p

NZ

¦\ n 1

With D nx and D ny standing for the constants to be determined from the collocation. Let us define the displacement of all nodes in the subdomain in a column vector

U = >u1 v1 u2 v2 . . . un vn @ . T

(10)

The discretization of strain can be rewritten with using its relation to displacement as

Hx Hy H xy

wu x wpx wu y wp y

NZ

w\ n

n 1

x

¦ wp NZ

w\ n

¦ wp n 1

wu x wu y wp y wpx

D xn ,

(11)

D yn ,

(12)

y NZ

w\ n

n 1

y

¦ wp

NZ

D xn ¦ n 1

w\ n D yn wpx

(13)

Using Eqs. (11) - (13) in the the Eqs. (6) - (7) we get the following equation;

1 Q eff w 2\ n D xn 2 2 1 wp x

¦ wp wp

1 Q eff w 2\ n D yn 2 2 1 wp x

¦ wp wp

NZ

¦ n

NZ

¦ n

NZ

w 2\ n

n 1

y

NZ

w 2\ n

n 1

y

D yn

1 Q eff 2

x

D xn

1 Q eff 2

x

w 2\ n D xn 2 1 wp y

0

(14)

w 2\ n D xn 2 1 wp y

0.

(15)

NZ

¦ n

NZ

¦ n

For points on the essential boundary *u the discretized form of the equations is

Eeff ª NZ w 2\ n 1 Q eff D xn «¦ 1 Q eff ¬« n 1 wpx2 2

¦ wp wp

Eeff ª NZ w 2\ n 1 Q eff « ¦ 2 D yn 1 Q eff ¬« n 1 wpx 2

¦ wp wp

Eeff ª NZ w 2\ n 1 Q eff D xn «¦ 1 Q eff ¬« n 1 wpx2 2

¦ wp wp

Eeff ª NZ w 2\ n 1 Q eff D yn «¦ 1 Q eff ¬« n 1 wpx2 2

¦ wp wp

NZ

w 2\ n

n 1

y

NZ

w 2\ n

n 1

y

D yn x

D xn x

1 Q eff 2 1 Q eff 2

º w 2\ n D xn » 2 p w 1 y ¼»

uˆ x

(16)

º w 2\ n D xn » 2 1 wp y ¼»

uˆ y .

(17)

NZ

¦ n

NZ

¦ n

For points on the natural boundary *t the discretized form of the equations is NZ

w 2\ n

n 1

y

NZ

w 2\ n

n 1

y

D yn x

D xn x

1 Q eff 2 1 Q eff 2

º w 2\ n D xn » W x 2 p w 1 y ¼»

(18)

º w 2\ n D xn » W y . 2 p w 1 y ¼»

(19)

NZ

¦ n

NZ

¦ n

3.1 Remarks on the TMM model: In the mechanical model of hot steel rolling, the strong formulation is used and all the unknowns of the system are reduced to one unknown which is the displacement vector u to be

410


calculated. The definition of stress is made in terms of strain. Many different relations can be used between stress and strain such as elastic, elastic-plastic, plane-strain, plane-stress and so on. However in realty, especially for hot rolling, it is known that the stress is not only strain dependent but also depends on other variables such as temperature and strain rate. This stress-strain relation will also be implemented in the mechanical model to clearly see the effects of the temperature and also TMM to cooperate. A sparse matrix can be created for the solution of the mechanical model. The rows of the matrix for each node need to be chosen from above Eqs. (15) - (20) for the interior as well as boundary nodes. The internal heat generation rate per unit volume S can be expressed as S : . 4. Node Generation

In numerical simulation of the hot shape rolling one has to encounter very complicated final shapes. However the initial shape during hot rolling is nearly rectangular. In order to get accurate solution, the physical domain must be covered with nodes, so that it can capture as accurately as possible the given geometry. Physically confirming nodes greatly improve quality of the solution, whereas on the other hand the non-confirming nodes are the major contributors towards the loss of accuracy and convergence of the underlying numerical procedure. Keeping in view importance of node generation, the procedure is accomplished in two stages. 4.1 Transfinite Interpolation (TFI). Through this technique we can generate initial grid which is confirming to the geometry encountered in different stages of plate and shape rolling. Suppose that there exists a transformation

r ( p[ , pK )

^ p ( p , p ), p ( p , p )`

tr

x

[

K

y

[

K

which maps the unit square, 0 p[ 1, 0 pK 1 in the computational

domain onto the interior of the region ABCD in the physical domain such that the edges p[ boundaries AB, CD and the edges pK

0, 1 map to the

0,1 are mapped to the boundaries AC, BD. The transformation is used for

this purpose is defined as r ( p[ , pK ) = (1- p[ )rl ( pK ) + [ rr ( pK ) + (1- pK )rb ( p[ ) + pK rt ( p[ ) - (1- p[ )(1- pK )rb (0) - (1- p[ ) pK rt (0)

(20)

(1- pK ) p[ rb (1) - p[ pK rt 1 where rb , rt , rl , rr represent the values at the bottom, top, left and right edges respectively.

p p ,p

4.2 Elliptic Node Generation (ENG). The mapping procedure form the physical domain, px , p y to the

computational domain p[ , pK

pK

can be described by the following functions

p[

[

x

y

and

pK px , p y having continuous derivatives all order. The nodes generated through TFI are refined through a

procedure known as EGG which uses a pair of Laplace’s equations 2 p[

0, 2 pK

0 . The partial differential

equations for the corresponding inverse problem are [10]

g 22

g 22

w 2 px w 2 px w 2 px 2 g12 g11 2 wp[ wp[ wpK wpK2

0, g 22

2 2 1 §¨ § wpx · § wpK · ·¸ ¨ ¸ ¨ ¸ , g12 J 2 ¨ ¨© wpK ¸¹ ¨© wpK ¸¹ ¸ © ¹

w 2 py wp[

2

2 g12

w2 py wp[ wpK

1 § wpx wpx wp y wp y ¨ J 2 ¨© wpK wp[ wpK wp[

g11

· ,g ¸¸ 11 ¹

w2 py wpK2

0

, where

(21)

2 2 1 §¨ § wpx · § wpx · ·¸ ¨ ¸ ¨ ¸ , J 2 ¨ ¨© wp[ ¸¹ ¨© wp[ ¸¹ ¸ © ¹

and J is Jacobian of the transformations. These equations are then discretized using the central difference formulas. We denote the value of px and p y at point i, j by xi , j and yi , j . Here, all xi , j and yi , j are unknown except those along the boundaries that are prescribed. To solve this set of equations, one may use the methods like Gauss Seidel, conjugate gradient, successive-over-relaxation (SOR) scheme or multigrid methods. In the present case we use SOR which gives fast convergence of the initial grid to the enhanced grid. Using SOR the Eq. (21) can be written as

] i,kj

Z ª b º a ] ik1, j ] ik1,1j ] ik1, j 1 ] ik1, j 1 ] ik1,1j 1 ] ik1,1j 1 c ] i,kj 1 ] i,kj 11 » 1 Z ] i,kj 1 2 a c «¬ 2 ¼


411

(22) where ]

px , p y , t is the iteration number and 1 Z 2 is the successive-over-relaxation parameter. Fig. 2

shows initial node generation through TFI and its correlation with ENG. rt ( p[ ) rr ( pK )

rl ( pK )

pK p[

rb ( p[ )

Fig 2. Transformation from computational domain to physical domain (left), TFI and nodes displacment through ENG (right). The collocation points are put on the intersection of grid lines. Summary This paper for the first time shows application of a very simple LRBFCM meshless approach in TMM of hot shape rolling. The strong formulation of the governing equations is preserved in the solution procedure. Time stepping is performed in a simple explicit way. No polygonisation and integrations are needed. The developed method is almost independent of the problem dimension. The complicated geometry can easily be coped with. The method appears efficient, because it requires solution of a set of small systems of equations of the size of the support in the thermal model and a solution of a large sparse system of equations in the mechanical model. The method is simple to learn and simple to code. The method can cope with very large problems since the computational effort grows approximately linear with the number of the nodes. Our ongoing research is focused on the numerical implementation. The numerical examples will be shown at the conference. Acknowledgement The present research has been funded by the Slovenian Research Agency, under grants P2-0379 (BŠ), Lv-441 and (SI), young researcher grant (UH), and Štore Steel Company. References [1] G. R. Liu, Mesh Free Methods, CRC Press, (2003). [2] J. G. Lenard, M. Piertrzyk, and L. Cser, Mathematical and Physical Simulation of the Properties of Hot Rolled Products, Elsevier, (1999). [3] B. Šarler and R. Vertnik Computers and Mathematics with Applications, 51, 1269-1282 (2006). [4] B. Šarler, G. Kosec, A. Lorbiecka and R. Vertnik Mater. Sci. Forum, 649, 211-216 (2010). [5] G. Kosec and B. Šarler Int. J. Cast. Met. Res., 22, 279-282 (2009). [6] R. Vertnik and B. Šarler Int. J. Cast. Met. Res., 22, 311-313 (2009). [7] M. Glowacki, Journal of Materials Processing Technology 168, 336-343 (2005). [8] W. Hu, L.G.Yao and Z.Z. Hua, Engineering Analysis with Boundary Elements, 31, 326-342 (2007). [9] R.C. Batra and G.M. Zhang, Computational Mechanics, 41, 527-545 (2008). [10] J. F. Thompson, B.K. Soni and N. P. Weatherill, Handbook of grid generation, CRC Pres (1999). [11] W.F. Chen and D.J. Han, Plasticity for Structural Engineers, Springer-Verlag, (1988). [12] Su-Hai Hsiang, Sheng-Li Lin, International Journal of Mechanical Sciences 43, 1155-1177 (2001). [13] L. Perez Pozo, F. Perazzo, A. Angulo, Advances in Engineering Software, 40, 1148-1154, (2009).

412


Regularization for a Poroelastodynamic Collocation BEM Michael Messner1,a , Martin Schanz1,b 1 Institute

for Applied Mechanics, Technikerstraße 4/II, 8010 Graz, Austria a [email protected] , b [email protected]

Keywords: time domain collocation BEM, regularization, linear poroelasticity, wave propagation, CQM

Abstract. Boundary element methods are especially suitable to study wave propagation in linear poroelasticity, where some of the most interesting applications lie in infinite or at least semi-infinite domains. The proposed collocation boundary element method is based on Biot’s theory for a linear poroelastic continuum. It uses the first boundary integral equation with only weakly singular kernels. This is possible due to a regularization of the strongly singular double layer, performed by partial integration. At the end some numerical results are presented. Introduction One of the major problems when dealing with boundary element methods is the evaluation of singular integral operators. The singularities one has to deal with differ between first and second boundary integral equation. In the engineering community, the most common approach is to use a collocation scheme only involving the first boundary integral equation. In such methods, the evaluation of the strongly singular double layer is still not straightforward. It can be shown that linear poroelasticity inherits the singular behavior from linear elasticity [11]. Thus, solutions known from there are likely to apply for the regularization in linear poroelasticity, too. Apart from analytical integration, which is limited to some special element types, a frequently used technique for the evaluation of the double layer is the numerical integration technique proposed by Guiggiani and Gigante [4]. This method subtracts the singular part from the integral’s kernel, making the evaluation of the remainder with standard techniques possible. However, due to such differences, this method tends to show numerical instability. Another common approach are analytic regularization techniques. Applying integration by parts to the whole kernel, these procedures allow to reduce the order of the singularity by moving it to the unknown quantity. Such techniques have been introduced in [5, 9, 10] in different fashions. Han’s [5] regularization for linear elastostatics was extended by Kielhorn [6] to the field of linear elastodynamics and viscoelasticity. It is found that this approach can be adapted to the regularization of the poroelastic double layer. As a result a weakly singular representation of the first boundary integral equation is obtained. Advantages of this approach are the numerical stability and the suitability for generic integration routines, since all operators of the boundary integral equation show the same singular behavior. Biot’s Theory Biot’s theory [2] is used to describe the state of a saturated linear poroelastic continuum in some domain Ω ⊂ R3 . It provides a set of constitutive, kinematic, and kinetic equations for the two constituents of a poroelastic material, i.e., the solid and the fluid phase. These equations may be combined in different ways to result in a set of equations of motion. However, as shown by Bonnet [3] it is sufficient to chose the solid displacements us and the pore pressure p to fully describe a


413

poroelastic continuum. In general this is only possible in Laplace domain, from now on denoted by L [ f (x)] = fˆ(s) with s ∈ C. Assuming homogeneous initial conditions, a homogeneous mixed boundary value problem in terms of the displacement field uˆ = [uˆ s , p] is stated by ˆ x) = 0 x˜ ∈Ω Bˆ x˜p u(˜ ˆ u(x) = gˆ D x ∈ΓD ˆt(x) = gˆ N x ∈ΓN ,

(1)

with the differential operator and the traction field e 2 Bx˜ + s (ρ − β ρ f )I (α − β )∇x˜ p ˆ 2 Bx˜ = s(α − β )∇ − sρβ f ∆x˜ + sφR x˜

ˆt(x) =

Txe sβ n x

−αnx β n ∇ sρ f x x

uˆ s (x) p(x)

.

(2)

Bxe˜ is the differential and Txe the traction operator from linear elastostatics, where differentiation takes place with respect to the respective subscripts. I is the identity, the bulk density is denoted by ρ, and the fluid density by ρ f . Biot’s effective stress coefficient is given by α = φ (1 + Q/R), where φ is the porosity and the two parameters Q and R are used to describe the coupling of the two constituents. The abbreviation β = β (κ, s, ρa ) depends on the dynamic permeability κ, the Laplace parameter s, and the apparent mass density ρa , which was introduced by Biot. For more details on these concepts the reader is referred to [2]. Boundary Integral Equation Detailed information on the derivation of the boundary integral equation (BIE) in Laplace domain can be found in [11, 14]. Here, the BIE is assumed to be given as ˆ (3) C (x)u(x) = Vˆ ˆt (x) − Kˆ uˆ (x) with x, y ∈ Γ , where the jump term is found on the left hand side. The first operator on the right hand side is commonly referred to as single layer and the second one as double layer. Their definition is given by ˆ (x − y)ˆt(y) dsy Vˆ ˆt (x) = U (4) Γ ˆ ˆ τ) dsy Kˆ uˆ (x) = lim (x − y)u(y, (5) Tˆy U ε→0 |x−y|≥ε

C (x) = lim

ε→0 ∂ Bε

ˆ Tˆy U

(x − y) dsy .

(6)

ˆ − y) is the fundamental solution of the adjoint problem and Tˆy the related In these definitions, U(x traction operator Tye sαny ˆ Ty = . (7) β −β n n ∇ y sρ f y The appearance of the latter in the double layer (5) leads to the strong singularity to be dealt with in the subsequent section. Finally, a formal inverse Laplace transformation of (3) yields the time domain BIE, where the assumption of homogeneous initial conditions still holds C (x)u(x,t) = (V ∗ t) (x,t) − (K ∗ u) (x,t) with

x, y ∈ Γ and t ∈ (0, ∞) .

(8)

The operator ∗ denotes the convolution in time. Furthermore, the spatial definition of the operators in (8) is analogous to their counterparts in Laplace domain. In the next section, this equation is used to derive a time domain collocation boundary element method.

414


Boundary Element Formulation Equation (8) is discretized in time via the convolution quadrature method (CQM) by Lubich [7] in its reformulated version by Banjai and Sauter [1]. More details on this method may be found in the referred literature and in [12]. Here only the required parts are extracted and exemplary shown for the double layer. Splitting the time interval (0, T ] into N equidistant intervals ∆t, the convolution at a discrete time tn = n∆t is approximated by the quadrature rule (K ∗ u) (x,tn ) ≈

n

∆t ˆ ∆t ∑ ωn− j (K )u (x,t j )

with

n = 1, . . . N ,

(9)

j=0

and the definition of the weights ˆ ω ∆t j (K ) ≈

R− j N ˆ ∑ K (s)ζ j N + 1 =0

with

2πi

ζ = e N+1

and s =

γ(Rζ − ) . ∆t

(10)

Here, R denotes the radius of analyticity and γ the characteristic function of the underlying multi-step method. Plugging these weights into (9) and modifying the summation indices regarding causality [1] allows to exchange the summation order. This results in a decoupled set of double layers in Laplace domain N R−n N ˆ (K ∗ u) (x,tn ) ≈ (11) K uˆ (x)ζ n with uˆ = ∑ R j u(x,t j )ζ − j . ∑ N + 1 =0 j=0 Application of the CQM in this manner to the entire time domain BIE, results in a set of = 1, . . . N/2 decoupled Laplace domain BIE to be discretized in space (taking into account that the second half is complex conjugated). For the approximation of the Dirichlet data isoparametric elements with continuous ansatz functions are used, whereas for the Neumann data discontinuous ansatz functions are more suitable. Choosing the number of collocation points to be the same as the number of unknowns on each boundary yields a discrete linear system of equations for the th problem ˆD ˆ D) ˆD ˆ D (C + K ˆth −K V gˆN,h −V = . (12) ˆ N −(C + K ˆN ˆ N) ˆN uˆh gˆD,h V K −V From the solution of the = 1, . . . N Laplace domain problems one obtains the time domain results trough the representation given in (11). Regularization of the Double Layer After the collocation boundary element method was given in the previous sections, this section’s aim is to show how the weakly singular form of the double layer is obtained. Due to the spatial nature of the singularity and the use of the CQM it is sufficient to proceed in Laplace domain. The presented representation holds for closed (piecewise Lipschitz) domains, which allows an evaluation at non smooth points of the boundary, too. Let us recall the Laplace domain double layer from (5) in more detail ⎞ ⎛ f sαny Tˆye ˆs ˆ U u ⎠ dsy , ⎝ Kˆ uˆ g (x) = lim (13) β −β n ε→0 |x−y|≥ε (pˆ s ) pˆ f f n ∇y y sρ y 0

y


415

ˆ s results in a where only the application of Tˆye onto the elastic part of the fundamental solution U strong singularity [11]. Revealing its type to be the same as in linear elastostatics, motivates the ˆ s into splitting of U ˆ s (r) + O r0 := 1 ∆I − λ + µ ∇∇ ∆χ(r) ˆ s (r) = U ˆ + O r0 with r = |x − y| , (14) U ˜ µ λ + 2µ ˆ λ and µ being the Lamé parameters, and χ(r) the scalar function resulting from Hörmander’s method [11]. The decomposition in (14) makes it possible to use the same approach as done by Kielhorn [6]. Stokes theorem is taken as a starting point Γ

∇y × a, ny dsy =

∂Γ

a, ν dσy ,

(15)

where the right hand side contour integral vanishes for closed domains. Under this assumption one can deduce

Γ

My := (ny × ∇) × .

(My a) dsy = 0 with

(16)

Since (16) holds for a being a tensor of variable order, the two following integration by parts formulas are derived

Γ Γ

(My w) e dsy = −

(My e) v dsy =

Γ

w (My e) dsy

(17)

e (My v) dsy .

(18)

Γ

Together with these two formulas, using the same representation of Tˆye as in [6] and just looking at ˆ s , all requirements are met to perform the regularization. Here its application to the singular part of U the subscript y is skipped for sake of readability

lim

ε→0 |y−x|≥ε

ˆS Tˆ e U uˆ dsy ˜

= lim

ε→0 |y−x|≥ε

= lim

ε→0 |y−x|≥ε

S 2ˆ 2ˆ ˆ M ∆ χ uˆ + I n ∇ ∆ χ uˆ + 2µ M U uˆ dsy ˜

ˆ ˆ + I n ∇ ∆ χˆ uˆ + 2µ U −∆ χˆ (M u) 2

2

˜

S

(19)

ˆ dsy . (M u)

Note that the assumption of a closed boundary is clearly violated by the the double layer’s definition as a Cauchy principal value in (5). However, if the double layer and the jump term (6) are combined [13], the boundary is closed again. As a consequence of this, Stokes theorem must be applied to the same parts of the jump term as it was for the double layer. Taking (16) into account this yields ˆ S dsy M ∆2 χˆ + I n ∇ ∆2 χˆ + 2µM U C (x) =lim ˜ ε→0 ∂ Bε (20) 2ˆ =lim I n ∇ ∆ χdsy = I (x)Φ(x) . ε→0 ∂ Bε

The jump term simplifies to the evaluation of the internal solid angle Φ(x) [8].

416


Numerical Example Despite the assumption of a closed boundary, wave propagation in a three dimensional poroelastic half space is studied, using the just presented CQM based collocation boundary element method. Such a domain does not fulfil this requirement, since the discretization needs to be truncated at some point. However, this example is motivated by the observation that this formulation yields at least as good results as we know from other formulations [11], using Guiggiani and Gigante [4] for the evaluation of strongly singular integrals. As the underlying multi-step method for the CQM a BDF-2 is chosen. The discretized half space is a flat patch of 10[m] × 10[m] using a triangulation with 900 triangular elements, which outer normal points in positive z direction. For the material data, properties of soil are assumed. The marked area of 0.75[m] × 0.75[m] at the center in Figure1(b) is loaded with a traction step load (t > 0) of t = [0, 0, −1] [N/m2 ], whereas all remaining elements are traction free. Additionally, zero pressure is applied on the entire surface. For the Dirichlet data piecewise linear continuous and for the Neumann data piecewise constant ansatz functions are used.

(a) Material properties of soil

6e-11 6e-11

4e-11 4e-11

2e-11

2e-11

0

0

s

[N/m2 ] [−] [kg/m3 ] [−] [N/m2 ] [kg/m3 ] [N/m2 ] m4/Ns

uz [m]

2.54 · 108 0.298 1884 0.48 1.1 · 1010 1000 3.3 · 109 3.5 · 10−9

E ν ρ Φ Ks ρf Kf κ

-2e-11

-2e-11

-4e-11

-6e-11

-4e-11

0

z y

•

0.01

0.02

0.03

0.04

0.05

0.06

-6e-11

x (b) Half space with loaded area at the center and observation point close to the edge

0

0.05

0.1

0.15 time [sec]

0.2

0.25

(c) Solid displacement usz at the observation point

Figure 1: Wave propagation in a poroelastic half space. The observation point is located on the diagonal at 6.13[m] from the loaded area. In Figure1(c) the arrival times for the fast compression wave can be observed at 0.0037[sec] and for the surface wave at 0.0269[sec]. The oscillation after all waves have passed, results from the CQM, which can not resolve discontinuities as one would expect there. The unsteady behavior of the solution before the quasi static solution is reached might due to the violation of the assumption of a closed boundary.


417

Conclusions A regularization technique for the double layer in linear poroelasticity was given, which is based on integration by parts for closed boundaries. This regularization results in a simplification of the jump term. Since the regularization was adopted from linear elasticity, this form of the jump term also applies to collocation boundary element methods in linear elasticity, where the same regularization technique is used. The results obtained by this method are promising, not only for closed boundaries, but as shown also for open domains. Finally, it should be mentioned that despite the title of this paper, the proposed regularization also works for Galerkin schemes using the first BIE. However, if one wants to work with a symmetric formulation as in [6] the regularization has to be extended to the second BIE, too. References [1] L. Banjai and S. Sauter. Rapid solution of the wave equation in unbounded domains. Journal on Numerical Analysis, 47(1):227–249, 2008. [2] M. A. Biot. Theory of propagation of elastic waves in a fluid-saturated porous solid. I/II. Journal of the Acoustical Society of America, 28(2):168–191, 1956. [3] G. Bonnet. Basic singular solutions for a poroelastic medium in the dynamic range. Journal of the Acoustical Society of America, 82(5):1758–1763, 1987. [4] M. Guiggiani and A. Gigante. A general algorithm for multidimensional cauchy principal value integrals in the boundary element method. Journal of Applied Mechanics, 57:906–915, 1990. [5] H. Han. The boundary integro-differential equations of three-dimensional neumann problem in linear elasticity. Numerische Mathematik, 68:268–281, 1994. [6] L. Kielhorn. A Time-Domain Symmetric Galerkin BEM for Viscoelastodynamics., volume 5 of Computation in Engineering an Science. Verlag der Technischen Universtiät Graz, 2009. [7] C. Lubich. Convolution quadrature and discretized operational calculus I/II. Numerische Mathematik, 52: 129–145/413–425, 1988. [8] V. Mantic. A new formula for the c-matrix in the somigliana identity. Journal of Elasticity, 1993. [9] J. Nedelec. Integral equations with nonintegrable kernels. Integral Equations and Operator Theory, 5: 563–672, 1982. [10] N. Nishimura and S. Kobayashi. A regularized boundary intgral equation method for elastodynamic crack problems. Computational Mechanics, 4:319–328, 1989. [11] M. Schanz. Wave Propagation in Viscoelastic and Poroelastic Continua. Springer, 2001. [12] M. Schanz. On a reformulated convolution quadrature based boundary element method. Computer Modelling in Engineering and Sciences, 2010. accepted. [13] O. Steinbach. Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer, 2008. [14] P. Urthaler and O. Steinbach. On the unique solveability of boundary integral formulations in poroelasticity. Technical Report 3, Institut für Numerische Mathematik, 2008. www.numerik.math.tu-graz.ac.at.

418


A Fast BEM for the dynamic analysis of plates with bonded piezoelectric patches I. Benedetti1, Z. Sharif Khodaei2 and M.H. Aliabadi3 1

DISAG – Dipartimento di Ingegneria Strutturale Aerospaziale e Geotecnica, Università di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy, [email protected]

Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW72AZ, London, UK, 2 [email protected], 3 [email protected] Keywords: Piezoelectric patches, Elastodynamics, DBEM, Laplace Transform Method, Adaptive Cross Approximation, Hierarchical Matrices, Fast BEM solvers. Abstract. In this paper a fast boundary element method for the elastodynamic analysis of 3D structures with bonded piezoelectric patches is presented. The elastodynamic analysis is performed in the Laplace domain and the time history of the relevant quantities is obtained by inverse Laplace transform. The bonded patches are modelled using a semi-analytical state-space variational approach. The computational features of the technique, in terms of required storage memory and solution time, are improved by a fast solver based on the use of hierarchical matrices. The presented numerical results show the potential of the technique in the study of structural health monitoring (SHM) systems. Introduction Structural reliability and safety are relevant issues in many engineering applications. In recent years, Structural Health Monitoring (SHM) has emerged as an important concept in the broader field of investigation of the smart structures [1]. The development of effective SHM systems is an inherently multidisciplinary task involving the in-depth knowledge of the damage mechanisms of the structure to be monitored, of sensor and actuator network technology and of signal processing and identification algorithms [2,3]. From a simple to a more advanced level, a SHM system should be able to provide qualitative information about the presence of damage, to give quantitative estimation of the position and severity of damage and to provide information about the structural safety and residual service life. SHM systems are generally based on the real time comparison of the local or global response of the damaged structure with the known response of the undamaged one. This can be accomplished by measuring, through a network of suitably arranged sensors, some physical variables and fields which may be affected by changes of material and geometrical conditions in proximity of the damaged area. Typical sensors for such tasks are strain gauges, accelerometers, fiber optics, piezoelectric films and piezoceramics. Piezoelectric materials, in particular, are among the most widely used smart materials because of their reliability and sensitivity [4]. One of the most relevant techniques for the quantitative assessment and identification of damage in complex structures is based on the use of ultrasonic waves. Rayleigh and Lamb waves in particular have shown great potential for the assessment of plate-like structures, which are very common in the automotive and aerospace industries [5-7]. In such applications, waves are introduced in the structure at a point and, after travelling a certain distance, are detected at other points through a suitable array of transducers. Both the generation and detection of waves is often accomplished by using piezoceramic devices bonded on or embedded into the structure. If, in the travelled distance, the wave interacts with a defect or damage, the response captured by the sensors array will differ for the expected one and, through suitable signal processing tools, useful information about the nature of the damage itself can be obtained. One of the factors enabling the successful design of general SHM systems, also in terms of production time and costs, is the availability of reliable analytical and numerical modeling tools. If Lamb waves are used for damage detection, the numerical tool should be able to accurately model the wave generation process, the wave propagation and its interaction with the defect, i.e. the wave scattering and, finally, the


419

sensors output. Much research has been carried out to provide a better understanding of the physics behind wave propagation and interaction with defects [7,8]. In this paper a boundary element model for the elastodynamic analysis of plates with bonded piezoelectric patches is presented. The piezoelectric patches are modelled by using a state-space based variational approach, which leads to a form of the equations for the patches particularly suitable for coupling with the boundary element model of the underlying structure. The elastodynamic BEM for the analysis of the plate is formulated in the Laplace domain. The storage memory and the solution time of the method, which limit the application of the BEM to large scale problems, are reduced by employing a previously developed fast solver based on the use of hierarchical matrices. Piezoelectric patch model Let us consider a three-dimensional piezoelectric patch with plane mid-surface, and constant thickness h, see Fig.1. The top and bottom surfaces are perpendicular to the poling direction x3, while x1x2 is the plane of transverse isotropy. The bottom surface of the patch is bounded on the surface of the host structure through a thin adhesive layer. The model for the piezoelectric patch is obtained by including the inertial terms in the state-space based approach used by Benedetti et al. [9,10]. Such approach leads, for the generic piezoelectric element, to the following relationship

ªuº «M » « » «V » « » ¬ D ¼TOP

ª Luu «L « Mu « LV u « «¬ Ldu

LuM LMM LVM LdM

LuV LMV LVV LdV

Lud º ª u º LM d »» « M » « » LV d » «V » »« » Ldd »¼ ¬ D ¼ BOTTOM

(1)

which relates the value of the electromechanical variables at the top patch surface to their value at the bottom surface. The vectors u, M, V and D represent the nodal values of the mechanical displacements, electric potential, mechanical stresses and electric displacements respectively. It is worth specifying that u collects the nodal components ui1, ui2, ui3 of the nodal displacements along the three directions (the subscript i refers to the i-th discretization node), V collects the out-of-plane components of stress Vi31, Vi32, Vi33, while D collects the out-of-plane components of the nodal values of the electric displacement Di3. The matrix blocks Lij Lij h, s depend on the patch thickness and, if the model is formulated in the Laplace

transform domain, on the complex parameter s. By applying suitable electrical and mechanical boundary conditions at the top and bottom surfaces, the equations for piezoelectric sensors and actuators are obtained. Piezoelectric transducers are usually realized by coating with a conductive layer, serving as electrode, both the bottom and top surface, which are then electrically equipotential. On the other end, the patch top surface is generally mechanically free. These general considerations are sufficient to write the actuator equations, while the sensor equations are obtained by taking also into account that no free electric charge can be present on the electrode surfaces. By considering that the patch equations have to be coupled with the BEM model of the host structure, it is convenient to write them in the following form t0 Vout

'u2 (l ,W ) / 'u1 (l ,W ) @ t1 > 'u2 (l ,W ) / 'u1 (l ,W ) @ t2

where t21 t2 'u1 (l ,W ) t1'u2 (l ,W ) , d 21 d1'u2 (l ,W ) d 2 'u1 (l ,W ) . The constants dD , tD and the bi-material constant H are defined in Appendix [25].

1.4

Normalized SIF

1.2 1 0.8 0.6 0.4

homog.-uncoupled

0.2

dissimilar

coupled

0 0

2000

4000

6000

8000

10000 12000

time [sec] Fig. 3 Time variations of the SIF

The stress intensity factor is normalized by the static value with homogeneous material properties. One can see in Fig. 3, that the maximum stress intensity factor is reached in infinite time for a homogeneous plate, when the maximum temperature gradient occurs. However, for an interface crack between two dissimilar materials the maximum temperature gradient is reached in a finite instant and therefore the maximum stress intensity factor is observed just at that time. Then, the stress intensity factor is slowly decreasing to the static value K Istat (interface) 2.12 105 Pa m1/ 2 . The coupling effect is again very weak on the SIF.

Acknowledgements This work is supported by the Slovak Science and Technology Assistance Agency registered under number registered under number APVV-0427-07 and the Slovak Grant Agency VEGA-2/0039/09, which is gratefully acknowledged.

References

440


[1] M. Comninou, D. Schmeser ASME Journal of Applied Mechanic, 46, 345-348 (1979). [2] C. Atkinson International Journal of Fracture 13, 807-820 (1977). [3]

G.B. Sinclair International Journal of Fracture 16, 111-119 (1980).

[4] A.F. Mak, L.M. Keer, S.H. Chen, J.L. Lewis ASME Journal of Applied Mechanics 47, 347-350 (1980). [5] R. Yuuki, S.B. Cho Engineering Fracture Mechanics 34, 179-188 (1989). [6] Y.C. Shiah, C.L. Tan Computational Mechanics 23, 87-96 (1999). [7] K.H. Park, P.K. Banerjee International Journal of Solids and Structures 39, 2871-2892 (2002). [8] V. Sladek, J. Sladek Applied Mathematical Modelling 7, 241-253 (1984). [9] G.F. Dargush, P.K. Banerjee ASME Journal of Applied Mechanic 58, 28-36 (1991). [10] I.G. Suh, N. Tosaka Theoretical and Applied Mechanics 38, 169-175 (1989). [11] P. Hosseini-Tehrani, M.R. Eslami Engineering Analysis with Boundary Elements 24, 249-257 (2000). [12] L. Gaul, M. Kögl, M. Wagner Boundary Element Methods for Engineers and Scientists, SpringerVerlag, Berlin (2003). [13] Q. Li, S. Shen, Z.D. Han, S.N. Atluri CMES: Computer Modeling in Engineering & Sciences 4, 571585 (2003). [14] R.C. Batra, M. Porfiri, D. Spinello International Journal for Numerical Methods in Engineering 61, 2461-2479 (2004). [15] J. Sladek, V. Sladek, S.N. Atluri CMES: Computer Modeling in Engn. & Sciences 6, 309-318 (2004). [16] V. Sladek, J. Sladek, M. Tanaka,, Ch. Zhang Engineering Analysis with Boundary Elements 29, 10471065 (2005). [17] J. Sladek, V. Sladek, Ch. Zhang, C.L. Tan CMES: Computer Modeling in Engineering & Sciences 16, 57-68 (2006). [18] F. Bobaru, S. Mukherjee International Journal for Numerical Methods in Engineering 53, 765-796 (2003). [19] S.N. Atluri The Meshless Method, (MLPG) For Domain & BIE Discretizations, Tech Science Press (2004). [20] J.C. Houbolt Journal of Aeronautical Sciences 17, 371-376 (1950). [21] W. Nowacki Thermoelasticity, Pergamon, Oxford (1986). [22] S.G. Lekhnitskii Theory of Elasticity of an Anisotropic Body, Holden Day, San Francisco (1963). [23] Ch. Zhang Crack Dynamics Ivankovic A, Aliabadi MH (eds.) WIT Press, Southampton 71-120 (2005). [24] S.B. Cho, K.B. Le, Y.S. Choy, R. Yuuki Engn. Fracture Mechanics 43, 603-614 (1992). [25] M. Wünsche, Ch. Zhang, J. Sladek, V. Sladek, S. Hirose, M. Kuna Int. Journal of Fracture, 157: 131147 (2009).


441

Local integral equations combined with mesh free implementations and time stepping techniques for diffusion problems V. Sladek1, J. Sladek1 and Ch. Zhang2 1

2

Institute of Construction and Architecture, Slovak Academy of Sciences, 845 03 Bratislava, Slovakia ([email protected], [email protected]) Department of Civil Engineering, University of Siegen, Paul-Bonatz-Str. 9-11, D-57076 Siegen, Germany ([email protected])

Keywords: transient heat conduction, strong formulation, weak formulation, Laplace transform, time stepping, accuracy, convergence, computational efficiency

Abstract. The paper deals with transient heat conduction in functionally gradient materials. The spatial variation of the temperature field is approximated by using a mesh free approximation, while the time dependence is treated either by the Laplace transform method and/or by the polynomial interpolation in the time stepping method. The accuracy and convergence of the numerical results as well as the computational efficiency of various approaches are compared in numerical test example.

1. Governing equations. Differential and integral formulations. The equations for transient heat conduction in isotropic and continuously non-homogeneous media is given by the partial differential equation of parabolic type with variable coefficients [1]

O (x)u,k (x, t ) ,k U (x)c(x) wu (wxt , t )

w(x, t ) , in : u [0, T ]

(1)

where u (x, t ) is the temperature field, w(x, t ) is the volume density of heat sources (for diffusion problems w 0 ), U (x) is the mass density, c( x) is the volume density of the specific heat per unit mass, and O (x) is the thermal conductivity coefficient. The first term on the left-hand side of Eq.(1) is the divergence of the heat flux vector

qk ( x, t ) O (x)u,k (x, t )

(2)

and the second term is the rate of the temporal change of the volumetric density of heat. The physically reasonable boundary conditions of the problem can be of the following types: (i) Dirichlet b.c.: u ( , t ) u ( , t ) at w: D , t [0, T ] , (ii) Neumann b.c.: ni ( )qi ( , t ) q ( , t ) at w: N , t [0, T ] ,

(3)

(iii) Robin b.c.: D u ( , t ) E ni ( )qi ( , t ) 0 at w: R , t [0, T ] , D , E \ ,

where w: w: D w: N w: R , ni ( ) is the unit outward normal vector to the boundary, D and E are real constants, and a tilde over a quantity denotes the prescribed value. The boundary conditions are to be supplemented by the initial condition, which in the present parabolic problem is the initial value of the temperature u (x,0) v( x) in : w: .

(4)

The governing equation in differential form (1) is derived form the physical balance principles which take an integral form in a continuum theory. Let us consider an arbitrary piece of continuum contained in a domain :c bounded with the boundary w:c . Then, the energy balance for a considered continuum is expressed as

442

³


ni ( ) qi ( , t )d *( )

w:c

w ³ U (x)c(x) wt u (x, t )d :(x)

:c

³

w(x, t )d :(x)

(5)

:c

In view of the Gauss divergence theorem, one can see that Eq. (5) is an equivalent of Eq. (1) since each of them can be derived from the other one under the assumption of arbitrary choice of the sub-domain :c : . Sometimes, the time evolution is investigated by using the Laplace transformation. Then, the time variable is eliminated temporarily and replaced by the Laplace transform parameter p by using the integral transformation f

³ u ( x, t ) e

u ( x, p ) :

pt

f

dt , hence

0

wu (x, t ) pt e dt wt 0

³

pu (x, p ) v(x) .

(6)

Now, in view of Eq. (6), the governing equations (1) and/or (5) can be rewritten for the Laplace transform of the temperature as

O (x)u,k (x, p) ,k pU (x)c(x)u (x, p) ³

ni ( )O ( )u,i ( , p )d *( ) p

w:c

w( x, p ) U (x)c(x)v(x)

³ U (x)c(x)u (x, p)d :(x)

:c

(7)

³ > w(x, p) U (x)c(x)v(x)@ d :(x) .

(8)

:c

The boundary conditions for the Laplace transform of the temperature can be obtained directly from Eq. (3) by performing the transformation according to Eq. (6). In the LT-approach, the numerical inversion of the LT is a key issue, since it is an ill-posed problem. Various Laplace-inversion algorithms are available in literature. Regarding good experience with the Stehfest’s algorithm [2], we shall use this algorithm in the present analysis. If g ( p) is the Laplace-transform of g (t ) , an approximate value of the inverse for a specific time t g (t )

ln 2 N t

t is given by § ln 2 · n¸ , © t ¹

¦ wn g ¨

n 1

wn

(1) N / 21

k N / 2 (2k )! , ( N / 2 k )! k !( k 1)!( n k )!(2k n)! k [( n 1) / 2]

min( n, N / 2)

¦

(9)

with [(n 1) / 2] being the integer part of ( n 1) / 2 . In the present analysis, N 10 was adopted.

2. Moving Least Sqaures (MLS)-approximation for spatial variations. Recall that the MLS approximation belongs to mesh free approximations since no predefined connectivity among nodal points is required. In this paper, we shall consider the Central Approximation Node (CAN) concept of the MLS-approximation [3, 4]. Let x q be the CAN for the approximation at a point x . Then, the amount of nodes involved into the approximation at x is reduced a-priori from Nt (total number of nodes) to N q , where N q is the number of nodes supporting the approximation at the CAN

%q

x q , i.e. the amount of nodes in the set

^xa ; wa (xq ) ! 0à 1 , where wa (x) is the weight function associated with the node x Nt

a

at the field

point x . In this paper, we employ the Gaussian weights [3-5]. The MLS-CAN approximation for spatial variation of the field variable f (x) {u (x, t ), u ( x, p )} is given by f ( x) |

Nq

¦

fˆ n( q,a )I n( q,a ) ( x) ,

(10)

a 1

where n(q, a ) is the global number of the a -th local node from % q , fˆ n ( q ,a ) {uˆ n( q,a ) (t ), uˆ n ( q ,a ) ( p )} , with fˆ n( q ,a ) being the nodal unknowns, which are different from the nodal values of physical quantities, in


443

general. In this paper, we shall specify the CAN x q as the nearest node to the approximation point x . Recall that the shape functions I m (x) are not known in closed form and a computational procedure must run for evaluation at each approximation point x . This is the main handicap of mesh-free approximations as compared with mesh-based approximations utilizing mostly polynomial interpolations. Besides the approximation of field variables, we need also their gradients which can be approximated as gradients of approximated fields (10) f, j ( x) |

Nq

¦

fˆ n( q,a )I,nj( q,a ) ( x)

(11)

a 1

and similarly, one can approximate also higher-order derivatives. Recall that the evaluation of the derivatives of the shape functions is still more complicated than the evaluation of shape functions and also the accuracy of such approximations is worse. Substituting these approximations into the governing equations (7) and/or (8) considered at nodal points x ( :c xc is a sub-domain around the node x ), one obtains the system of algebraic equations c

c

¦ K cg pM cg uˆ g ( pn )

R c ( pn ) ,

( c 1, 2,..., Nt ), ( n 1, 2,..., N )

(12)

M cg

(13)

g

where the matrix elements are given as K cg

O (xc )I,gkk (xc ) O,k ( xc )I,gk (xc ) ,

U (xc )c(xc )I g (xc ) ,

R ( pn ) w(x , pn ) U (x )c(x )v(x ) , g n(c, a ) with a % for the strong formulation (i.e. discretization of the governing PDE) which will be referred as the CPDE approach (collocation of PDE), while in the case of weak formulation starting from Eq. (8), the matrix elements are given as c

c

K cg

³

w:

c

c

n ( qK ,a )

ni ( )O ( )I,i

c

c

( )d *( ) ,

M cg

n ( q ,a ) ³ U (x)c(x)I x (x)d :(x)

,

(14)

:c

c

³ > w(x, pn ) U (x)c(x)v(x)@ d :(x) ,

R c ( pn )

:c

with g being global numbers of nodes generated by n(qK , a) and/or n(q x , a) , where q x is the nearest nodal point to the integration point x . Similarly, substituting the approximations for the temperature and its gradients into the governing equations (1) and/or (5), we obtain the system of the ordinary differential equations

§

¦ ¨¨ K cg uˆ g (t ) M cg g

©

with R c (t )

wuˆ g (t ) · ¸ R c (t ) , ( c 1, 2,..., Nt ), wt ¸¹

w( xc , t ) in the strong formulation, while R c (t )

(15)

³

w(x, t )d :(x) in the weak formulation.

:c

In order to solve the ODE (15), we employ a polynomial interpolation for the time variation of the nodal unknowns.

3. Time interpolations. Let us split the time interval [0, T ] by discrete time instants ti into a finite number subintervals m 1

[0, T ]

* [ti , ti 1 ] ,

i 0

'ti

ti 1 ti

Making use n 1 nodes, one can define element obeying interpolation of order n .

444


Linear Lagrange interpolation. The element Ti is defined as the interval Ti 't ¦ ti 1 a N (W ) ti 2i (1 W ) , a 1 2

tT

i

a

[ti , ti 1 ] with the interior points being parametrized as

W [1, 1]

(16)

since N 1 (W ) (1 W ) / 2 , N 2 (W ) (1 W ) / 2 . The time dependence of a physical variable u (t ) is approximated on Ti by the interpolation 2

u (t ) T

i

¦ ui 1 a N a (W )

a 1

1 W ui 1 ui ui 1 ui , 2 2

uk

u (tk ) .

(17)

Then, the time derivative u (t ) du (t ) / dt is approximated by the constant u (t ) T

i

1 du J (W ) dW T

i

1 ui 1 ui , 'ti

(18)

since the Jacobian of the transformation (16) is given as J (W ) dt / dW T

i

Making use a different parametrization T

'ti / 2 .

(1 W ) / 2 with T [0,1] , we obtain from (17) and (18)

u (ti T'ti ) ui T ui 1 ui T ui 1 (1 T )ui 1 u (ti T'ti ) ui 1 ui . 'ti Considering the system of the ODE (15) at t ti T'ti , we obtain §

1

¦ ¨ K cg T't ©

g

i

· M cg ¸ uîg1 ¹

(19) (20)

§ · 1 1 1 R c (ti T'ti ) ¦ ¨ (1 ) K cg M cg ¸ uîg , ( i 0,1, 2,... ) T T T'ti ¹ g ©

(21)

which is the well known T -method used in time stepping approaches for solution of the ODE.

Quadratic Lagrange interpolation.

If we choose the time instants obeying 'ti D i 't with D i {1, 2,3,...} , then it is easy to create also the higher order elements. Thus, for 3-node (quadratic) element Ti , the third node is ti 1 , the second node (mid-node) is ti , and the first node is ti k , where k is found from the condition k

¦ Di j Di . Then, the interior points on

Ti are parametrized as

j 1

ti k N 1 (W ) ti N 2 (W ) ti 1 N 3 (W ) ti WD i 't , W [1, 1] ,

tT

i

1

2

2

(22)

3

since N (W ) W (W 1) / 2 , N (W ) 1 W , N (W ) W (W 1) / 2 . A physical variable and its time derivative are approximated on Ti as u (t ) T

u (ti W'ti )

u (t ) T

u (ti W'ti )

i

i

W

W

(W 1)ui k (1 W 2 )ui (W 1)ui 1 , 2 1 du 2W 1 2W 2W 1 ui k u u , D i 't i 2D i 't i 1 J (W ) dW T 2D i 't

(23)

2

(24)

i

since the Jacobian of the transformation (22) is given by J (W ) dt / dW T

i

D i 't .

In the T -method used in time stepping approaches, the ODE (15) are considered at the time instant t ti T'ti . Both u (ti T'ti ) and u (ti T'ti ) are expressed in terms of instant values u j by Eqs. (23), and (24). Thus, in view of (23) and (24), the system of the ODE (15) can be solved subsequently for time instants unknowns by


§

· g

2T 1

1

¦ ¨ K cg TD 't T 1 M cg ¸ uî 1 i ¹ g ©

2

T (T 1)

R c (ti T'ti )

445

1T 1 g· § g ¦ K cg ¨ uî k 2(1 T )uî ¸ 1T g © ¹

1 g g ¦ M cg (2T 1)uî k 4T uî , ( i 1, 2,... ) . (T 1)TD i 't g

(25)

This sequence of equation systems should be supplemented with equations for unknowns uˆ1g . Eq. (25) is valid also for i 0 , but it involves also the nodal unknown at fictitious time instant. For this purpose, we assume that the first node on T1 is take at fictitious time instant t1 t0 't0 . Then, from (24) at t0 (i.e., i 0 and W 0 ), we have uˆ g uˆ g 2't uˆ g (t ) . (26) 1

0

1

0

Furthermore, from (15), we have Nt

c

uˆ0g

S g ¦ D gg cuˆ0g , D gg c

¦ ( M 1 ) ga K ag

gc

c

, Sg

a 1

Nt

¦ (M 1 ) ga R a (t0 )

(27)

a 1

Hence after substituting (26) and (27) into (25) at i 0 , we obtain §

· M cg ¸ uˆ1g 0 ¹

¦ (1 T )'t0 K

cg

2

¦ ¨ K cg T't g

©

1

T

1

T2

g , gc

R c (t0 T't0 )

(2 1/ T ) M

cg

1

T

¦ (1 T )'t0 K cg (2 1/ T ) M cg S g g

c D gg cuˆ0g

1

T't0

¦ (1/ T T )'t0 K cg 2M cg uˆ0 . g

(28)

g

Certain simplification of Eq. (28) takes place, if the initial temperature is uniformly distributed in the analyzed domain : . Then,

c gc

¦ D gg uˆ0

0 , since v(x) const . The other simplifications occur in the right-

gc

hand sides of Eqs. (25) and (28), when the parameter T

1 and/or w(x, t ) 0 .

Note that the matrix elements K cg and M cg are independent of the time variable as well as the Laplace transform parameter. Nevertheless, their calculation in the weak formulation is time consuming because of lengthy evaluation of shape functions and their derivatives at integration points. Without going into details [5], recall that this handicap can be removed by using the Taylor series expansions of the shape functions and the material parameters around the centre of the sub-domain :c since the number of evaluations is reduced drastically. Moreover, the integrations can be performed analytically in this approach. For comparison of efficiency of the LT and the time stepping approaches recall that the former approach requires to compute the set of nodal unknowns for several values of the Laplace parameter for each considered time instant, while the later approach needs to compute the set of nodal unknowns at each time instant before the considered time instant.

4. Numerical tests. In order to study the accuracy and convergence of numerical results, we shall consider the example for which exact solution is available. In this paper, we consider a square domain L u L occupied by medium with exponentially graded heat conduction as well as specific heat while constant mass density: U const .,

O (x) O0e2G x2 L c(x)O0 / c0 . If constant values of the temperature are prescribed on the bottom u0 and top u L of the square, while the lateral sides are thermally insulated and constant initial value of temperature v(x) const v is assumed, the exact solution is given as [6] u ex (x, t ) u L

1 e 2G x2 / L 1 e

2G

u0

e2G x2 / L e 2G 1 e2G

2

O0 f kS § kS · N [( kS / L )2 (G / L )2 ]t x e sin ¦ c O (x) k 1 k (kS )2 G 2 ¨© L 2 ¸¹

446


where ck

v u0 (1) k (u L v)eG , N

O0 . In numerical computations, we have used O0 1 c0 U c0

U,

G 1 , u0 1 v , uL 20 , w 0 . The uniform distribution of nodal points is employed with h being the distance of two neighbour nodes. The % error is evaluated as the average of % errors at all nodal points. Figure 1 illustrates the failure of the accuracy by the approach based on the weak formulation and numerical integration (LIE(ni)) for small and intermediate time instants. The linear Lagrange interpolation (LLI) has been used for time approximation. The accuracy is not improved even by increasing the total number of nodes ( Nt nnod ) and/or using quadratic Lagrange interpolation (QLI).

Fig. 1 Accuracy of temperature computed at midpoint by LIE(ni)-LLI( T

1)

Much better results are received by the LIE(ni) combined with LT approach used for treatment of the time dependence as can be seen from Fig.2

Fig. 2 Time variation of the temperature computed at midpoint by LIE(ni)-LLI and LIE(ni)-LT


447

It can be explained by the fact that the results by LIE(ni)-LLI are strongly sensitive on the accuracy of evaluation of the domain integral because the transient term in the ODE is proportional to ( 't ) 1 and the inaccuracy at early time instants affects also the accuracy at further time instants. Although large values of the Laplace parameter correspond to early time instants (which contribute to importance of domain integral too), the inaccuracy of results at early time instants has no influence on the accuracy at later time instants. Remarkably better accuracy as well as convergence of numerical results is achieved by both the CPDE and LIE(ai) approaches as shown in Figs. 3 and 4

Fig. 3 Time variation of the temperature computed at midpoint by LIE(ai) and CPDE with ( T

1)

Fig. 4 Convergence of numerical results by CPDE and LIE(ai) Finally, Fig. 5 shows that for low densities of nodes (large values of h ) the total computational time by LIE(ni) is almost one order higher than that by LIE(ai) and/or CPDE. With increasing the density of nodes

448


these differences are vanishing because the time needed for solution of for large systems of algebraic equations is becoming dominant.

Fig. 5 Comparison of total computational times by CPDE, LIE(ai) and LIE(ni)

Summary Several computational techniques are discussed for solution of transient heat conduction in FGM. For the spatial variation of temperature, the MLS-approximation is used while the time dependence is treated either by the Laplace transform or by the time stepping method. From the numerical results achieved in the test example, the following conclusions can be drawn: i The weak formulation based on the LIE with numerical integration fails for early and intermediate time instants when combined with time stepping techniques, while with the LT gives much better results. i The CPDE as well as the weak formulation based on the LIE+analytical integration yield acceptable results in combination with both the LT and time stepping techniques. The linear Lagrange interpolation gives better accuracy than quadratic LI and the best accuracy is achieved by the LT method. i The computational efficiency by the LIE(ai) is very close to that by CPDE and both are remarkably better than the efficiency by LIE(ni) especially for low and intermediate densities of spatial nodes.

References [1] L.C.Wrobel The Boundary Element Method, Vol1: Applications in Thermo-Fluids and Acoustics, Wiley (2002). [2] H. Stehfest Communication of the Association for Computing Machinery, 13, 47-49; 624 (1970). [3] V. Sladek, J. Sladek, Ch. Zhang Computational Mechanics, 41, 827-845 (2008). [4] V. Sladek, J. Sladek, Ch. Zhang Engineering Analysis with Boundary Elements, 32, 1012-1024 (2008). [5] V. Sladek, J. Sladek Local Integral Equations implemented by MLS-approximation and analytical integrations, Engineering Analysis with Boundary Elements (accepted for publication).


449

Computation of Moments in thin Plates of Composite Materials under Dynamic Load using the Boundary Element Method K. R. Sousaa , A. P. Santanaa , E. L. Albuquerqueb , and P. Solleroc a Federal Institute of Maranhao ˜ Department of Mechanical and Materials ˜ Luis, MA, Brazil 65025-000, Sao

{kerlles,andre}@ifma.edu.br b University

of Brazilia - UNB Faculty of Technology 70910-900, Brasilia, Bsb, Brazil [email protected] c University

of Campinas - UNICAMP Faculty of Mechanical Engineering 13083-970, Campinas, SP, Brazil [email protected]

Keywords: Boundary element method, radial integration method, dual reciprocity boundary element method, plates, composite materials, dynamic of plate, and stress analyses.

Abstract. This work presents a dynamic formulation of the boundary element method for moments of anisotropic thin plates. The elastostatic fundamental solution for anisotropic thin plates is used and inertia terms are treated as body forces. Domain integrals that come from body forces are transformed into boundary integrals using the radial integration method (RIM). In this method, the inertia term is approximated as a sum of approximation functions times coefficients to be determined. In this work, the augmented thin plate spline is used as the approximation function. The time integration is carried out using the Houbolt method. Only the boundary is discretized. Numerical results show good agreement with results available in literature. Introduction. Nowadays, BEM is a well-established numerical technique to deal with an enormous number of engineering complex problems. Analysis of plate bending problems using the BEM has attracted the attention of many researchers during the past years, proving to be a particularly adequate field of applications for that technique. In recent years, the boundary element formulation for plate bending has included the analysis of anisotropic problems. Shi and Bezine [10] presented a boundary element analysis of plate bending problems using fundamental solutions proposed by [16] based on Kirchhoff plate bending assumptions. Rajamohan and Raamachandran [6] proposed a formulation where singularities were avoided by placing source points outside the domain. Albuquerque et al [2] presented a method to transform domain integrals into boundary integrals in the classical plate theory for composite laminate materials. The transformation follows the radial integration method, as proposed by Gao [3]. In [1], this formulation was extended for dynamic problems. Shear deformable shells have been analyzed using the boundary element method by [13] with the analytical fundamental solution proposed by [14]. Wang and Huang [12] presented a boundary element formulation

450


for orthotropic shear deformable plates. Later, in Wang and Schweizerhof [15], the previous formulation was extended to laminate composite plates. Stress and moment computation by the BEM has been addressed by some works in literature. For example, Zao [18] and Zao and Lan [17] have discussed the computation of stresses in plane elastic problems, Knopke [4] presented and discussed the integral formulation for computation of stresses in isotropic thin plate, Rashed et al [7] presented an stress integral formulation in the BEM fo Reissner plate bending problems. To the best of authors knowledge, the computation of moments by the BEM in anisotropic plates have still not been addressed in literature. This paper proposes a numerical procedure to compute moments at internal points and at the boundary of composite laminated plates using a dynamic boundary element plate formulation that follows the Kirchhoff hypotheses. Boundary integral equations. The boundary integral formulation for anisotropic thin plate problems uses two integral equations, for displacement and rotation (see [2]). The transversal displacement equation is given by:

Kw(Q) +

Γ

Vn∗ (Q, P)w(P) − m∗n (Q, P)

Nc

∂ w(P) dΓ(P) + ∂n

Nc

∑ R∗c (Q, P)wc (P) = ∑ Rc (P)w∗c (Q, P) + i

i

i

i=1

i=1

Γ

Vn (P)w∗ (Q, P) − mn (P)

i

Ωg

b(P)w∗ (Q, P)dΩ +

∂ w∗ (Q, P) dΓ(P), ∂n

(1)

where P is the field point; Q is the source point; Γ is the boundary of the domain Ω of the plate; Ωg is the part of the domain Ω where the body force b is applied; the constant K is introduced in order to consider that the source point Q can be placed in the domain, on the boundary, or outside the domain (if the point Q is on a smooth boundary, then K = 1/2); ∂∂()n is the derivative to the outward unity vector n that is normal to the boundary Γ at the field point P; mn and Vn are, respectively, the normal bending moment and the Kirchhoff’s equivalent shear force on the boundary Γ; Rc is the thin plate reaction of corners; wc is the transversal displacement of corners; Nc is the number of corners; and the symbol * stands for fundamental solutions. The rotation equation is given by: 1 ∂ w(Q) + 2 ∂ n1 Nc

∂ R∗ci

Γ

∂V ∗ ∂ m∗n ∂w (P) dΓ(P) + (Q, P)w(P) − (Q, P) ∂ n1 ∂ n1 ∂n

∂ w∗i

i

i=1

∂ w∗ (Q, P)dΩ + ∂ n1 Ωg i=1 ∂ w∗ ∂ ∂ w∗ Vn (P) (Q, P) dΓ(P), (Q, P) − mn (P) ∂ n1 ∂ n1 ∂ n Γ Nc

∑ ∂ n1 (Q, P)wc (P) = ∑ Rc (P) ∂ n1c (Q, P) + i

b(P)

(2)

where ∂∂ n()1 is the derivative to the outward unity vector n1 that is normal to the boundary Γ at the source point Q.


451

As can be seen, domain integrals arise in the formulation owing to the presence of the body force b. In order to transform these integrals into boundary integrals, consider, as in the DRM, that the body force b is approximated over the domain Ωg as a sum of M products between approximation functions fm and unknown coefficients γm , that is: M

b(P) =

∑ γm fm + ax + by + c

(3)

m=1

with M

M

M

m=1

m=1

m=1

∑ γm xm = ∑ γm ym = ∑ γm = 0

(4)

The approximation functions used in this work is the well known thin plate spline given by: fm3 = R2 log(R),

(5)

used with the augmentation function given by equations (3) and (4) . It has been shown in some works from literature that this approximation function can give excellent results for many different formulations (see Partridge [5]). Equations (3) and (4) can be written in a matrix form, considering all source points, as: b = Fγ

(6)

γ = F−1 b

(7)

Thus, γ can be evaluated as:

For transient analysis, the body force vector is given by: ¨ b = ρ hw.

(8)

¨ is the acceleration (double dots stands for second where ρ is the material density, h is the plate thickness, w time derivative). To carry out the time integration during a interval T , this interval is divided into N equal intervals (time steps) of size ∆τ (T = N∆τ ). The acceleration for the time step τ + ∆τ is given by: ¨ τ +∆τ = w

1 (2wτ +∆τ − 5wτ + 4wτ −∆τ − wτ −2∆τ ) . ∆τ 2

(9)

Provided that wτ , wτ −∆τ , and wτ −2∆τ are known, we can compute wτ +∆τ by doing: Axτ +∆τ = yτ +∆τ

(10)

where xτ +∆τ is the vector of unknown variables and yτ +∆τ is the vector of known variables in which the elements are computed taking into account boundary conditions and computed values for prior time steps. Moments are written in terms of transversal displacement as:

452


∂ 2w ∂ 2w ∂ 2w mx = − D11 2 + D12 2 + 2D16 , ∂x ∂y ∂ x∂ y ∂ 2w ∂ 2w ∂ 2w , my = − D12 2 + D22 2 + 2D26 ∂x ∂y ∂ x∂ y ∂ 2w ∂ 2w ∂ 2w mxy = − D16 2 + D26 2 + 2D66 , ∂x ∂y ∂ x∂ y

(11)

So, in order to compute moments, it is necessary to calculate second derivatives of transverse displacement w. These derivatives are given by (see [9]):

∂ 2 w(Q) ∂ x2

=

2 ∗ ∂ V n ∂ x2

Γ

Vn (P)

Γ

Ω

∂ 2 w(Q) ∂ y2

=

b(P)

n ∂ y2

Vn (P)

Γ

Ω

∂ 2 w(Q) ∂ x∂ y

=

b(P)

n

∂ x∂ y

Γ

Vn (P)

Ω

b(P)

Nc ∂ 2 w∗ci ∂ 2 w∗ ∂ 3 w∗ (Q, P) − mn (P) (Q, P) dΓ(P) + ∑ Rci (P) (Q, P) + 2 2 ∂x ∂ n∂ x ∂ x2 i=1 (12)

(Q, P)w(P) −

Nc ∂ 2 R∗ ∂ 2 m∗n ∂ w(P) ci dΓ(P) + ∑ (Q, P) (Q, P)wci (P) − 2 2 ∂y ∂n i=1 ∂ y

Nc ∂ 2 w∗ci ∂ 2 w∗ ∂ 3 w∗ (Q, P) − m (P) (Q, P) dΓ(P) + ∑ Rci (P) (Q, P) + n ∂ y2 ∂ n∂ y2 ∂ y2 i=1

∂ 2 w∗ (Q, P)dΩ ∂ y2

2 ∗ ∂ V Γ

Nc ∂ 2 R∗ ∂ 2 m∗n ∂ w(P) ci (Q, P) (Q, P)wci (P) − dΓ(P) + ∑ 2 2 ∂x ∂n i=1 ∂ x

∂ 2 w∗ (Q, P)dΩ ∂ x2

2 ∗ ∂ V Γ

(Q, P)w(P) −

(Q, P)w(P) −

(13) Nc ∂ 2 R∗ ∂ 2 m∗n ∂ w(P) ci (Q, P) dΓ(P) + ∑ (Q, P)wci (P) − ∂ x∂ y ∂n i=1 ∂ x∂ y

Nc ∂ 2 w∗ci ∂ 2 w∗ ∂ 3 w∗ (Q, P) − mn (P) (Q, P) dΓ(P) + ∑ Rci (P) (Q, P) + ∂ x∂ y ∂ n∂ x∂ y ∂ x∂ y i=1

∂ 2 w∗ (Q, P)dΩ ∂ x∂ y

(14)

Numerical results. Consider a square clamped-plate under a uniformely distributed step load applied at time τ0 = 0 with amplitude q = 2, 07.106 N/m2 .The plate is orthotropic with the following material properties: E2 = 6895 MPa, E1 = 2E2 , G12 = 2651.9 MPa, ν12 = 0.3, ρ = 7166 kg/m3 . The edges of the plate is a = 254 mm and thickness h = 12.7 mm. This problem is equivalent to problem proposed by Sladek et al. (2007) which was analyzed using the MPLG. The static moment of the central node of the plate is given 3 2 by mstat x = 9, 54 × 10 N.m and the normalization factor of time by to = a /(4 ρ h/D). Twelve quadratic


453

discontinuous boundary elements (three per edge) with equal length and time steps ∆τ = 3.9447.10−5 s are used in the discretization of space and time, respectively. Results are obtained using 1, 9, and 25 internal points. They are shown in figurue 1. These moments are compared with a meshless Petrov Galerkin formulation ([8]) and the finite element method ([11]). Figure 1 shows moments mx at the central node of the plate as a function of time. 2.5

2

m x /m est. x

1.5

1 1 internal point 9 internal points 25 internal points FEM Sladek et al. (2006)

0.5

0

−0.5 0

0.1

0.2

t/to

0.3

0.4

0.5

Figure 1: Moment at the centre of the plate using different internal points. Results obtained with 25 internal points were closer to the solution of the finite element method and meshless than results with 1 and 9 internal points. However, results with 9 and 25 points are very close, indicating convergency. Results with one internal point are very smooth. So, the use of internal points are needed to obtain better precision. Conclusions. This paper analysed the use the radial integration method applied to transient analysis of anisotropic plates. From results, we can conclude that Acknowledgment. The authors would like to thank the State of Maranhão Research Foundation (FAPEMA) and the National Council for Scientific and Technological Development (CNPq) for the financial support of this work.

References [1] E. L. Albuquerque, P. Sollero, and W. P. Paiva. The radial integration method applied to dynamic problems of anisotropic plates. Communications in Numerical Methods in Engineering, 23:805–818, 2007. [2] E. L. Albuquerque, P. Sollero, W. Venturini and M. H. Aliabadi. Boundary element analysis of anisotropic Kirchhoff plates. International Journal of Solids and Structures, 43:4029–4046, 2006.

454


[3] X.W.Gao. The radial integration method for evaluation of domain integrals with boundary only discretization, Engineering Analysis with Boundary Elements, Vol. 26, pp. 905–916, (2002). [4] B. Knopke. The hypersingular integral equation for the bending moments mxx, mxy, and myy of Kirchhoff plates, Computational Mechanics, Vol. 15, pp. 19-30, (1994). [5] P. W. Partridge. Towards criteria for selection approximation functions in the dual reciprocity method. Engineering Analysis with Boundary Elements, 24:519–529, 2000. [6] C. Rajamohan and J. Raamachandran. Bending of anisotropic plates charge simulation method, Advances in Engineering Software, Vol. 30, pp. 369–373, (1999). [7] Y. F. Rashed, M. H, Aliabadi, C. A. Brebbia. On the evaluation of stress in the BEM for Reissner plate bending problems, Applied Mathematica Modeling, Vol. 21, pp. 155-163, (1997). [8] J. Sladek, V. Sladek, Ch. Zhang, J. Krivacek and P.H. Wen. Analysis of orthotropic thick plates by meshless local PetrovGalerkin (MLPG) method. International Journal for Numerical Methods in Engineering, v. 67, p. 1830–1850, 2006. [9] K. R. P. Sousa. Analysis of Stress in thin Plates of Composite Materials under Dynamic Load using the Boundary Element Method. Master Thesis, Faculty of Mechanical Engineering, University of Campinas., 2009. [10] G. Shi and G. Bezine. A general boundary integral formulation for the anisotropic plate bending problems, Journal of Composite Material, Vol. 22, pp. 694–716, (1988). [11] J. Useche. Shellcomp v3.4: Finite Element Analysis Program for Linear Static and Dynamic Analysis of Composite Shell Structures. Universidade Tecnolgica de Bolivar, Cartagena, Colmbia, 2008. [12] J. Wang and M. Huang. Boundary element method for ortotropic thick plates, Acta Mechanica Sinica, Vol. 7 (3), pp. 258–266,(1991). [13] J. Wang and K. Schweizerhof. Free vibration of laminated anisotropic shallow shells including transverse shear deformation by the boundary-domain element method, Computers and Structures, Vol. 62, pp. 151–156, (1997). [14] J. Wang and K. Schweizerhof. The fundamental solution of moderately thick laminated anisotropic shallow shells, International Journal of Engineering Science, Vol. 33, pp. 995–1004, (1995). [15] J. Wang and K. Schweizerhof. Fundamental solutions and boundary integral equations of moderately thick symmetrically laminated anisotropic plates, Communications in Numerical Methods in Engineering, Vol. 12, pp. 383–394, (1996). [16] B.C. Wu and N.J. Altiero. A new numerical method for the analysis of anisotropic thin plate bending problems,Computer Methods in Applied Mechanics and Engineering, Vol. 25, pp. 343–353, (1981). [17] Z. Zhao and S. Lan. Boundary stress calculation - a comparison study, Computers & Structures, Vol. 71, pp. 77-85, (1999). [18] Z. Zhao. On the calculation of boundary stress in boundary elements, Engineering Analysis with Boundary Elements, Vol. 16, pp. 317-322, (1995).


455

Meshless boundary element methods for exterior problems on spheroids E.P.Stephan1, A.Costea1 ,Q.T.Le Gia2 and T. Tran2 1

Institute for Applied Mathematics and QUEST (Center for Quantum Engineering and Space-Time Research), Leibniz University of Hanover, Am Welfengarten 1, 30167 Hanover, Germany. E-mail: [email protected] 2 School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia. Email: [email protected], [email protected], Keywords: Boundary integral equations, meshless methods, preconditioning, domain decomposition

Abstract. In geophysical applications one is interested in the Neumann problem exterior to a spheroid where the orbits of satellites are located. The satellites create data which amount to boundary conditions in scattered points. As a model problem, we consider the exterior Neumann problem of the Laplacian with boundary condition on an oblate or prolate spheroid. We propose to use spherical radial basis functions in the solution of the boundary integral equation arising from the Dirichlet-to-Neumann map. Our meshless approach with radial basis functions is particularly suitable for handling scattered satellite data. We also propose a preconditioning technique based on an overlapping domain decomposition method to deal with ill-conditioned matrices arising from the approximation problem. Here we report from [14, 11] on a meshless method with radial basis functions for the Neumann problem for the Laplacian exterior to an oblate or a prolate spheroid. In geophysical applications [1, 2] one is interested in such exterior Neumann problems where the orbits of satellites are located on spheroids. A key tool of our approach is the use of the Dirichlet-to-Neumann map which directly converts the bounday value problem into a pseudodifferential equation on the spheroid. This integral equation is then handled with Fourier techniques by expansion into appropriate spherical harmonics. Huang and Yu [6] solved this pseudodifferential equation numerically with standard boundary elements on a regular grid on the angular domain of the spherical coordinates. Our approach uses spherical radial basis functions (SRBF's) instead, allowing for better handling of scattered data. Originally we introduced the meshless boundary element method for integral equations on the sphere [4] and we used it to solve the linearized Molodensky problem [3]. The error analysis for the meshless method with radial basis functions on the sphere as done in [4] has been extended to spheroids in [14, 11]. Again the smooth solution of the pseudodifferential equation can be approximated with high convergence rates by the Galerkin solution consisting of radial basis functions. The numerical solution is obtained by an appropiate implementation [9] of the prolate and oblate spheroidal harmonics and by truncating the resulting infinite dimensional discrete Galerkin system. We present numerical results of our meshless boundary element method for equidistributed points, so-called Saff points, and for scattered data points from satellite observations. Furthermore we present iteration numbers for the conjugate gradient method applied to solve the discrete systems in case of scattered data. We list the iteration numbers for the preconditioned conjugate gradient method when an overlapping additive Schwarz method is used as preconditioner. For both oblate and prolate spheroids we obtain only mildly growing iteration numbers for the overlapping additive Schwarz method. This technique was firstly analysed for pseudodifferential equations on the sphere in [5]. ‫ʹݔ‬

‫ʹݔ‬

‫ʹݔ‬

Let ΓͲ ൌ ሼሺ‫ ͳݔ‬ǡ ‫ ʹݔ‬ǡ ‫ ͵ݔ‬ሻǣ ͳʹ ൅ ʹʹ ൅ ͵ʹ ൌ ͳǡ ܽ ൐ ܾ ൐ Ͳሽ be an oblate spheroid and Ωܿ be the ܽ ܽ ܾ unbounded domain outside the boundary ΓͲ . We consider the exterior Neumann problem: Given ݃ ‫ ʹܮ א‬ሺΓͲ ሻ, find ܷ ‫ א‬Ωܿ satisfying Δܷ ൌ Ͳ ݅݊Ωܿ μߥ ܷ ൌ ݃ ‫݊݋‬ΓͲ ܷሺ‫ܠ‬ሻ ൌ ܱሺȁȁ‫ܠ‬ȁȁെͳ ሻ ܽ‫ݏ‬ȁȁ‫ܠ‬ȁȁ ՜ ∞

(1)

456


where ȁȁ‫ܠ‬ȁȁ denotes the Euclidean norm of ‫ ܠ‬and ߥ denotes the unit outward normal vector on ΓͲ . The solution is given by the series ݊ ܷሺߤǡ ߠǡ ߮ሻ ൌ σ∞ ݊ൌͲ σ݉ ൌെ݊

ܶ݊݉ ሺ ߤ ሻ ‫ݑ‬ො ܻ ሺߠǡ ߮ሻǡߤ ܶ݊݉ ሺ ߤ Ͳ ሻ ݊݉ ݊݉

൒ ߤͲ ൐ Ͳǡ

where (ߤǡ ߠǡ ߮) denote the oblate spheroidal coordinates [6]. Here ݅ߨ݊ ݉ ܶ݊݉ ൌ ݅ሺ ሻܳ݊ ሺ݅‫ݔ‬ሻǡ݅ ʹ ൌ െͳǡ ʹ and ܳ݊݉ are the associated Legendre functions of second kind and ܻ݊݉ the spherical harmonics of degree n and ‫ݑ‬ො݊݉ the expansion coefficients. In [6] it is shown that eq (1) is equivalent to ࣥ‫ ݑ‬ൌ ݃‫݊݋‬ΓͲ

(2)

with the Dirichlet-to-Neumann map (Steklov-Poincaré operator) ࣥ. Its weak formulation is: Find ‫ͳܪ א ݑ‬Ȁʹ ሺΓͲ ሻ satisfying ‫ܦ‬ሺ‫ݑ‬ǡ ‫ݒ‬ሻǣ ൌ ‫׬‬Γ ሺࣥ‫ݑ‬ሻ‫ ݏ݀ݒ‬ൌ ‫׬‬Γ ݃‫ͳܪ א ݒ׊ݏ݀ݒ‬Ȁʹ ሺΓͲ ሻ Ͳ

Ͳ

݉ ‫כ‬ ݊ ‫ܦ݄ݐ݅ݓ‬ሺ‫ݑ‬ǡ ‫ݒ‬ሻ ൌ ݂Ͳ σ∞ ො ݊݉ ‫ݒ‬ො݊݉ ݊ൌͲ σ݉ ൌെ݊ ‫ ݊ܩ‬ሺߤͲ ሻ‫ݑ‬

‫ ݉݊ܩ݁ݎ݄݁ݓ‬ሺ‫ݔ‬ሻǣ ൌ െ

ሺͳ൅‫ ʹ ݔ‬ሻ

݀ ݉ ܶ ሺ‫ݔ‬ሻ ݀‫݊ ݔ‬

ܶ݊݉ ሺ‫ݔ‬ሻ

Ǥ

(3) (4) (5)

Proposition: There exists a unique solution for the variational problem (3). This is due to the Lax-Milgram theorem since ‫ܦ‬ሺ‫ڄ‬ǡ‫ڄ‬ሻ is continuous and coercive on the Sobolev space ‫ͳܪ‬Ȁʹ ሺΓͲ ሻ and the right hand side in eq (3) defines a continuous linear functional on ‫ͳܪ‬Ȁʹ ሺΓͲ ሻ. Here ‫ ݏ ܪ א ݑ‬ሺΓͲ ሻ, ‫ א ݏ‬Թ if and only if ʹ ∞ ݊ ʹ ‫ݏ‬ ‫ ݏ‬ሺΓ ሻ ൌ σ݊ൌͲ σ݉ ൌെ݊ ሺͳ ൅ ݊ ሻ ȁ‫ݑ‬ ො ݊݉ ȁʹ ൏ ∞ ‫ܪצ ݑ צ‬ Ͳ

The approximate solution to eq (3) is sought in a finite dimensional subspace of ‫ͳܪ‬Ȁʹ ሺΓͲ ሻ. In order to use SRBFs we take the bijection ߱ǣ ΓͲ ՜ ॺʹ ߱ሺ‫ܠ‬ሻ ൌ ሺߠ߮ǡ ߠ߮ǡ ߠሻǡ

(6)

and we define a reproducing kernel on ΓͲ as Ψሺ‫ܠ‬ǡ ‫ܠ‬′ሻ ൌ Φሺ߱ሺ‫ܠ‬ሻǡ ߱ሺ‫ܠ‬′ሻሻǡ‫ܠ‬ǡ ‫ܠ‬′ ‫ א‬ΓͲ

(7)

where Φ is defined via a univariate function ϕǣ ሾെͳǡͳሿ ՜ Թ by (see [7, 8]) Φሺ‫ܡ‬ǡ ‫ܢ‬ሻ ൌ ϕሺ‫ܢ ڄ ܡ‬ሻ‫ܡ׊‬ǡ ‫ ʹॺ א ܢ‬Ǥ

(8)

Here ϕ has a series expansion in terms of Legendre polynomials ܲ݊ of degree ݊, as ͳ ෠ ϕሺ‫ݐ‬ሻ ൌ Ͷߨ σ∞ ݊ൌͲ ሺʹ݊ ൅ ͳሻϕሺ݊ሻܲ݊ ሺ‫ݐ‬ሻ

(9)

with ൅ͳ ϕ෠ ሺ݊ሻ ൌ ʹߨ ‫׬‬െͳ ϕሺ‫ݐ‬ሻܲ݊ ሺ‫ݐ‬ሻ݀‫ݐ‬Ǥ

(10)

The kernel Ψ can be expanded into a series of spherical harmonics as ݊ ‫כ‬ ෠ Ψሺ‫ܠ‬ǡ ‫ܠ‬′ሻ ൌ σ∞ ݊ൌͲ σ݉ ൌെ݊ ϕሺ݊ሻܻ݊݉ ሺ߱ሺ‫ܠ‬ሻሻܻ݊݉ ሺ߱ሺ‫ܠ‬′ሻሻ

where we choose

(11)


ϕ෠ ሺ݊ሻ ؆ ሺͳ ൅ ݊ʹ ሻെ߬ Ǥ

457

(12)

Given a set of scattered data points ܺ ൌ ሼ‫ ͳܠ‬ǡ Ǥ Ǥ Ǥ ǡ ‫ ܯܠ‬ሽ ‫ ؿ‬ΓͲ , we take ܸ ߬ ǣ ൌ ‫݊ܽ݌ݏ‬ሼΨ݆ ǣ ൌ Ψሺ‫ ݆ܠ‬ǡ‫ڄ‬ሻǣ ‫ܺ א ݆ܠ‬ሽǤ Now, the solution of eq (3) is approximated by ‫ ߬ ܸ א ܺݑ‬satisfying the Galerkin method ‫ܦ‬ሺ‫ ܺݑ‬ǡ ‫ݒ‬ሻ ൌ ‫׬‬Γ ݃‫ ߬ ܸ א ݒ׊ݏ݀ݒ‬Ǥ

(13)

Ͳ

To this end we have to solve a linear system ‫ ܋ܣ‬ൌ ܏

(14)

where A is a matrix with entries ‫݅ܣ‬ǡ݆ ൌ ‫ܦ‬ሺΨ݅ ǡ Ψ݆ ሻǡ݅ǡ ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ‫ ܯ‬and ܏ is a vector with entries ݆݃ ൌ න ݃Ψ݆ ݀‫ݏ‬ǡ݆ ൌ ͳǡ Ǥ Ǥ ǡ ‫ܯ‬Ǥ ΓͲ

Using eq (4) and eq (11) we can write ݊ ʹ ݉ ‫כ‬ ෠ ‫݅ܣ‬ǡ݆ ൌ ݂Ͳ σ∞ ݊ൌͲ σ݉ ൌെ݊ ሾϕሺ݊ሻሿ ‫ ݊ܩ‬ሺߤͲ ሻܻ݊݉ ሺ߱ሺ‫ ݅ ܠ‬ሻሻܻ݊݉ ሺ߱ሺ‫ ݆ܠ‬ሻሻǤ

(15)

‫ܰܣ‬ ݅ǡ݆ . Then ܰ

we compute Further let N denote the number of the series terms of the truncated matrix element the actual Galerkin approximation ‫ ܺݑ‬by solving eq (14) with the Galerkin matrix ‫ ܣ‬obtained via the truncated entries ‫ܰܣ‬ ݅ǡ݆ . Let ܻ ൌ ‫ ͳܡ‬ǡ Ǥ Ǥ Ǥ ǡ ‫ ܯܡ‬be the image of ܺ under the map ߱, i.e. ‫ ݆ܡ‬ൌ ߱ሺ‫ ݆ܠ‬ሻ for ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ‫ܯ‬. As ܻ is a set of scattered points on ॺʹ , we define the mesh norm ݄ܻ of ܻ as usual ݄ܻ ൌ െͳ ሺ‫ ݆ܡ ڄ ܡ‬ሻ. ‫ܻא ݆ܡ ʹॺאܡ‬

Proposition [11]: For an oblate spheroid and truncation number ܰ chosen sufficiently large there holds the quasioptimal error estimate for the difference between the exact solution ‫ ݏ ܪ א ݑ‬ሺ߁Ͳ ሻ of eq (3) and the Galerkin solution ‫ ߬ ܸ א ܺݑ‬,ʹ߬ ൒ ‫ݏ‬ ‫ݏ‬െͳȀʹ

‫ ݑ צ‬െ ‫ͳ ܪצ ܺݑ‬Ȁʹ ሺ߁Ͳ ሻ ൑ ‫ܻ݄ܥ‬

ǡ

where the constant ‫ ܥ‬is independent from ܰ and the set ܺ used to define ߖ݆ . (The corresponding result for the prolate spheroid see [14].) Our numerical experiments plotted in Fig. 2 for the equidistant mesh of Saff points (see Fig. 1) show these predicted convergence rates ߙ , namely ߙ ൌ ʹǤͷ for ܱ݉ ൌ Ͳ ( ߬ ൌ ͳǤͷ ), ߙ ൌ ͶǤͷ for ܱ݉ ൌ ͳ ( ߬ ൌ ʹǤͷ),ߙ ൌ ͸Ǥͷ for ܱ݉ ൌ ʹ(߬ ൌ ͵Ǥͷ), cf. Table 1. To compute the entries ‫ ݆݅ܣ‬of the stiffness matrix given in eq (15), we need to compute the spherical harmonics ܻ݊݉ and the functions ‫ ݉݊ܩ‬. The functions ܶ݊݉ are calculated using the algorithm for oblate spheroidal harmonics presented in (see [9]) and we use ‫ ݉݊ܩ‬ሺ‫ݔ‬ሻ ൌ ‫݊ܩ‬െ݉ ሺ‫ݔ‬ሻ ‫ ݉݊ܩ‬ሺ‫ݔ‬ሻ ൌ ሺ݊ ൅ ͳሻ‫ ݔ‬൅ ሺ݊ െ ݉ ൅ ͳሻ

ܶ݊݉൅ͳ ሺ‫ݔ‬ሻ ǡ݉ ܶ݊݉ ሺ‫ݔ‬ሻ

ൌ Ͳǡͳǡ ǥ ǡ ݊Ǣ ݊ ൌ Ͳǡͳǡʹǡ ǥ

The right hand side terms ݆݃ are computed by using the Fourrier coefficients of ݃ and Ψ݆ and Parseval's identity. In our numerical experiments, we use Ψሺ‫ܠ‬ǡ ‫ܠ‬′ሻ ൌ Φሺ߱ሺ‫ܠ‬ሻǡ ߱ሺ‫ܠ‬′ሻሻ for arbitrary ‫ܡ‬ǡ ‫ ʹॺ א ܢ‬and Φሺ‫ܡ‬ǡ ‫ܢ‬ሻ ൌ ߩ݉ ሺඥʹ െ ʹ‫ܢ ڄ ܡ‬ሻ with ߩ݉ being locally supported radial basis functions defined by Wendland [10]; see Table 1. In this case eq (12) holds for ߬ ൌ ݉ ൅ ͵Ȁʹ see[12].

458


݉

ߩ݉ ሺ‫ݎ‬ሻ

߬

0

ሺͳ െ ‫ݎ‬ሻʹ൅

1.5

1

ሺͳ െ ‫ݎ‬ሻͶ൅ ሺͶ‫ ݎ‬൅ ͳሻ

2.5

2

‫ݎ‬ሻ͸൅ ሺ͵ͷ‫ʹ ݎ‬

3.5

ሺͳ െ

൅ ͳͺ‫ ݎ‬൅ ͵ሻ

Table 1: Wendland's RBFs Example 1: Problem (1) with oblate spheroid ΓͲ (݂Ͳ ൌ Ͷ, ߤͲ ൌ ͳ), Neumann condition and exact solution

݃ൌെ ܷൌ

ߤ ߠ ߮ሺʹ ʹ ߤ ൅ ʹ ߠሻ ͷ

݂Ͳʹ ሺ ʹ ߤ െ ʹ ߠሻʹ ߤ ߠ ߮

͵ ݂Ͳʹ ሺ ʹ ߤ െ ʹ ߠሻʹ

Let ݁ǣ ൌ ‫ ݑ‬െ ‫ ܺݑ‬where ‫ݑ‬ሺߠǡ ߮ሻ ൌ ܷሺߤͲ ǡ ߠǡ ߮ሻ, solves eq (3) and ‫ ܺݑ‬solves eq (13) with truncation number ܰ‫ ܿ݊ݑݎݐ‬ൌ ͳͲͲ. We compute ‫ ʹܮצ ݁ צ‬ሺΓͲ ሻ and ‫ͳ ܪצ ݁ צ‬Ȁʹ ሺΓͲ ሻ approximately by ܰ݉ܽ‫ ݔ‬ൌ ͳʹͲ ܰ݉ܽ‫ݔ‬ σ݊݉ ൌെ݊ ȁ‫ݑ‬ෞܺ ݊݉ െ ‫ݑ‬ො݊݉ ȁʹ ሻͳȀʹ ‫ ʹܮצ ݁ צ‬ሺΓͲ ሻ ൎ ሺσ݊ൌͲ

and

ܰ

݉ܽ‫ݔ‬ ‫ͳ ܪצ ݁ צ‬Ȁʹ ሺΓͲ ሻ ൎ ሺσ݊ൌͲ ሺͳ ൅ ݊ʹ ሻͳȀʹ σ݊݉ ൌെ݊ ȁ‫ݑ‬ෞܺ ݊݉ െ ‫ݑ‬ො݊݉ ȁʹ ሻͳȀʹ

෠ ‫ݑ݄ݐ݅ݓ‬ෞܺ ݊݉ ൌ σ‫ܯ‬ ݅ൌͳ ϕሺ݊ሻܿ݅ ܻ݊݉ ሺ߱ሺ‫ ݅ ܠ‬ሻሻ and ‫ݑ‬ො݊݉ is computed by an appropriate quadrature formula [13].

Figure 1: Image of Saff points on oblate spheroid

(16)


459

Figure 2: Log-log plot for ‫ͳܪ‬Ȁʹ ሺΓͲ ሻ errors using Wendland RBFs ߩ݉ ሺ‫ݎ‬ሻ (Saff points) for the oblate (Om) and prolate spheroid (Pm) Table 2 gives the errors in the ‫ ʹܮ‬ሺΓͲ ሻ and ‫ͳܪ‬Ȁʹ ሺΓͲ ሻ norms for scattered points. The matrix is ill conditioned and a preconditioner is required. Table 3 shows the corresponding numbers of iteration of the preconditioned conjugate gradient method (PCG) with an overlapping additive Schwarz preconditioner [mref]; stopping criteria in both cases is relative tolerance ൑ ͳͲെͳͲ . Errors of the same order as in the non-preconditioned case are obtained but PCG needs much smaller iteration numbers.(cpu: computational times in seconds, iter: numbers of iterations)

‫ܯ‬ 3470 7763 10443

‫ܻݍ‬ ߨ/140 ߨ/200 ߨ/240

‫ ʹܮצ ݁ צ‬ሺΓͲ ሻ 6.25503E-006 2.41695E-006 1.87142E-006

‫ͳ ܪצ ݁ צ‬Ȁʹ ሺΓͲ ሻ 5.01114E-005 2.13257E-005 1.62031E-005

ITER 2809 27064 30931

CPU 234.6 13323.7 17361.9

Table 2: Errors with scattered points from MAGSAT, using CG, ߩͲ ሺ‫ݎ‬ሻ, Ex.1 ‫ܯ‬ 3470 7763 10443

ߙ 0.9 0.97 0.98

ߚ -0.15500000 0.99999996 0.99999996

‫ ʹܮצ ݁ צ‬ሺΓͲ ሻ 6.24283E-006 2.41695E-006 1.87142E-006

‫ͳ ܪצ ݁ צ‬Ȁʹ ሺΓͲ ሻ 5.02580E-005 2.13257E-005 1.62031E-005

ITER 73 939 1602

Table 3: Errors with scattered points from MAGSAT, for PCG, ߩͲ ሺ‫ݎ‬ሻ, Ex.1

CPU 4.3 294.1 1085.8

460


In [14] we have analysed the foregoing meshless boundary element method for a prolate spheroid. We obtain similar results both with respect to convergence of the meshless Galerkin solution as well as the behaviour of the iteration number of PCG. For brevity we present only some numerical results. Example 2: Problem (1) with

prolate spheroid ΓͲ ([14]), Neumann condition and exact

solution

݃ሺߤǡ ߠǡ ߮ሻ ൌ െ

ξʹ ʹߠ ߮ሺ͹െ͵ Ͷߤ ൅Ͷ ʹߤ ʹߠሻ Ͷ݂ͲͶ ඥ ʹ ߤ െ ʹ ߠሺ ʹߤ ൅ ʹߠሻ͹Ȁʹ

ܷሺߤǡ ߠǡ ߮ሻ ൌ

ξʹ ʹߤ ʹߠ ߮ Ǥ ʹ݂Ͳ͵ ሺ ʹߤ ൅ ʹߠሻͷȀʹ

(17)

stopping criteria in both cases is relative tolerance ൑ ͳͲെͳͲ . Tables 4 and 5 give the errors in the ‫ ʹܮ‬ሺΓͲ ሻ and ‫ͳܪ‬Ȁʹ ሺΓͲ ሻ norms and the iteration numbers for CG and PCG for scattered points for Example 2 (stopping criteria in both cases is relative tolerance ൑ ͳͲെͺ ). We observe for the prolate spheroid that the errors of the meshless boundary element method and the performance of the PCG with overlapping Schwarz preconditioner behave as in the case of the oblate spheroid. ݉ 0 1

‫ܯ‬ 2133 7763 2133 7763

‫ܻݍ‬ ߨ/100 ߨ/200 ߨ/100 ߨ/200

‫ ʹܮצ ݁ צ‬ሺΓͲ ሻ 1.08971E-6 2.15172E-7 3.76100E-8 1.03461E-8

‫ͳ ܪצ ݁ צ‬Ȁʹ ሺΓͲ ሻ 7.79793E-6 1.81600E-6 2.54359E-7 6.99365E-8

cpu 19.2 1592.3 102 620.6

iter 1124 7518 7737 4059

Table 4: Errors with scattered points from MAGSAT, using CG, ߩ݉ ሺ‫ݎ‬ሻ, Ex.2

݉ 0

‫ܯ‬ 2133 2133 2133 2133 7763 7763 7763 7763

ߙ 0.90 0.80 0.70 0.60 0.97 0.90 0.80 0.70

ߚ -0.01 -0.66 -0.78 -0.69 0.63 -0.46 -0.76 -0.84

‫ܬ‬ 42 22 17 11 140 48 23 17

cpu 4.4 4.6 5.6 4.1 176.1 133.6 162.1 238.2

iter 68 45 36 22 226 85 59 51

‫ ʹܮצ ݁ צ‬ሺΓͲ ሻ 0.11045E-5 0.11045E-5 0.11045E-5 0.11045E-5 0.24240E-6 0.24247E-6 0.24239E-6 0.24239E-6

1

2133 2133 2133 2133 7763 7763 7763 7763

0.90 0.80 0.70 0.60 0.97 0.90 0.80 0.70

-0.01 -0.66 -0.78 -0.69 0.63 -0.46 -0.76 -0.84

42 22 17 11 140 48 23 17

10.9 6.1 7.9 2.8 323.3 104.5 55.8 154.8

251 95 75 22 738 147 40 60

0.36969E-7 0.36969E-7 0.36969E-7 0.36969E-7 0.36654E-8 0.42327E-8 0.47790E-8 0.35058E-8

Table 5: Errors with scattered points from MAGSAT, for PCG, ߩ݉ ሺ‫ݎ‬ሻ, Ex.2


References [1] Freeden, W. and Gervens, T. and Schreiner, M. Constructive approximation on the sphere of Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York, 1998. With applications to geomathematics. [2] E. W. Grafarend, F. W. Krumm and V.S. Schwarze. Geodesy: the Challenge of the 3rd Millennium. Springer, Berlin, 2003. [3] Stephan, E.P. and Tran, T. and Costea, A. A boundary integral equation on the sphere for high-precision geodesy. Computer Methods in Mechanics: Lectures of the CMM 2009, :99--110, 2010. [4] Tran, T. and Le Gia, Q. T. and Sloan, I. H. and Stephan, E. P. Boundary integral equations on the sphere with radial basis functions: error analysis. Appl. Numer. Math., 59(11):2857--2871, 2009. [5] Tran, T. and Le Gia, Q. T. and Sloan, I. H. and Stephan, E. P. Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Technical report, School of Mathematics and Statistics, The University of New South Wales., 2009, submitted. [6] Huang, Hongying and Yu, Dehao. The ellipsoid artificial boundary method for three-dimensional unbounded domains. J. Comput. Math., 27(2-3):196--214, 2009. [7] Schoenberg, I. J. Positive definite functions on spheres. Duke Math. J., 9:96--108, 1942. [8] Xu, Yuan and Cheney, E. W. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 116(4):977--981, 1992. [9] Gil, A. and Segura, J. A code to evaluate prolate and oblate spheroidal harmonics. Computer Physics Communications., 108:267--278, 1998. [10] Wendland, Holger. Scattered data approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2005. [11] Costea, A. and Le Gia, Q. T. and Tran, T. and Stephan, E. P. Solution to the Neumann problem exterior to an oblate spheroid by radial basis functions. To appear, 2010. [12] Narcowich, Francis J. and Ward, Joseph D. Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal., 33(6):1393--1410 (electronic), 2002. [13] Driscoll, James R. and Healy, Jr., Dennis M. Computing Fourier transforms and convolutions on the ʹ-sphere. Adv. in Appl. Math., 15(2):202--250, 1994. [14] Le Gia, Q. T. and Tran, T. and Stephan, E. P. Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions. Advances in Computational Mathematics, 2010.

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462


3-D Green element method for potential flows Edwin Nyirenda and Akpofure Taigbenu School of Civil and Environmental Engineering, University of the Witwatersrand. P. Bag 3, Johannesburg, WITS 2050. South Africa. Email: [email protected] Keywords: Green element method; 3-D spatial dimension; steady and unsteady potential flow problems.

Abstract Earlier formulations of the Green element method (GEM) had addressed flow problems in 1-D and 2-D spatial domains. This paper describes a 3-D formulation that is used to solve steady and unsteady potential flow problems in homogeneous media. In 2-D, the internal normal directional fluxes are either expressed in terms of the potential by a finite difference approximation that introduces some errors or retained with the introduction of an additional compatibility equation to address the closure problem (flux-based GEM). In this 3-D formulation, the former approach is followed because of the substantial increase in the level of complexity associated with retaining those internal fluxes. The advantage of GEM in implementing the singular integral equations is retained because the surface and volume integrations are always carried out in the element in which the source or collocation node is located. The current formulation is applied to steady and unsteady potential flow examples and found to exhibit satisfactory accuracy.

Introduction Previous works of numerical calculations with the Green element method (GEM) for linear and nonlinear transport phenomena in homogeneous and heterogeneous media have been in 1-D and 2-D domains [1 - 5]. They exploited the second order accuracy of the theory of the boundary element method (BEM) and the versatility of the finite element methodology in order to be able to address a wide range of transport processes. Could these computational gains of GEM be retained in 3-D spatial domains? This provides the motivation for the GEM simulations of steady and

unsteady potential flow problems in 3-D homogeneous media in an attempt similar to those of [6] and [7], albeit under the name of “multidomain BEM”. In the current work, the unique advantage in facilitating the calculations of the surface and domain integrations by the Green element approach is explored because the collation node always resides in the element over which integration is implemented. In rectangular hexagonal elements, that allows the integrations to be done analytically. Unlike in 1-D and 2-D where the internal fluxes were retained and directly calculated, here they are expressed in terms the potentials by finite differencing. Three examples are used to demonstrate the performance of the current 3-D Green element formulation, and it gives more accurate solutions than FEM.

GEM formulation for 3D Laplace equation We seek the solution to the time-dependent differential equation in a 3D domain.

wu (1) f ( x, y, z , t ) in : wt w 2 / wx 2 w 2 / wy 2 w 2 / wz 2 , u is the potential, K and S are medium parameters and

K 2u where 2

S

f ( x, y , z , t ) is the forcing term. Accompanying the differential equation are the essential conditions of the type u u on *1 and natural conditions of the type q wu / wn q on *2 , where n is the normal to the boundary * *1 *2 and the dashes indicate that those values are known. The integral representation of


463

eq (1) via the singular boundary integral theory using Green’s second identity and the fundamental solution to the Laplacian operator is

wu · § kui ³³ uq * d* ³³ qu * d* ³³³ ¨ f S ¸ u * dA 0 wt ¹ * * : © The coefficient k takes on different values depending on the included angle at the collocation point, ri .

u * 1 / 4Sr is the fundamental solution of the Laplacian operator and q* wu * / wn is its normal directional derivative. The integral equation applies to the entire domain and equally to a part of it (an element or cell). When expressed for an element, the dependent variables u and q are approximated by interpolation functions in terms of their nodal values. To obtain the global solution (solutions at all nodes in the computational domain), the element equations are aggregated in such a way that the continuity of the potential u and the compatibility of the flux are maintained across inter-element boundaries. In 1-D and 2-D, this is easy to achieve but in 3-D there are some challenges which are overcome in the coding of the formulation by adopting a convention in the labelling of the flux and potential. Using linear interpolation for the dependent variables over regular (rectangular) hexagonal elements, (2) u | I ju j Where I j is the interpolation function with respect to the j th node of the element. Introducing the expression for the interpolation into the integral eq (2) results in the discretized element equations, namely du j (3) H ij u j Gij q j Wij Fi 0 dt where * (4a) H ij kG ij I j qi dS

³³

Gij

³³ I u

j i

*e *

dS

(4b)

*e

Wij

S ³³³I j ui dA *

:

(4c)

e

* e and :e respectively represent the surface and domain of the e th element, the index i refers to the collation node and j a field node. The computation of the vector of the forcing term Fi depends on whether it is from a distributed loading in which case it is approximated by the interpolation functions similar to that for the dependent variables or from a point loading in which case it is represented by the Dirac delta function. Numerical implementation Whatever the elements that are used to discretize the domain - tetrahedral (with triangular plane surfaces) or hexahedral (with quadrilateral plane surfaces) - a local coordinate system is defined using the collocation point as origin, with the first two coordinates in the plane of the element surface and the third coordinate normal to the surface. The Green element formulation ensures that the collation point is always in the same element in which surface and domain integrations are carried out. This enormously simplifies the evaluation of the integrals, most of which are implemented analytically. The time derivative in eq (3) is simplified by the generalized finite difference approximation in time with a weighting factor D : [0,1] to become

DH where T

ij

Wij / 't u 2j DGij q 2j

TH ij Wij / 't u1j TGij q1j DFi 2 TFi1

(5)

1 D , and the indices 2 and 1 , respectively, represent the current time t 2 t1 't and previous

time t1 . The discretized equations can now be aggregated for all the elements. To retain the calculations of

464


the fluxes at all nodes as in 1-D and 2-D GEM is computationally onerous and not followed in 3-D. With the use of rectangular hexahedron solid elements, the internal fluxes are approximated by a finite differencing expression in terms of the potential along the lines of previous works in Taigbenu[1,2]. Some comments are appropriate here for the labelling of the fluxes and the compatibility conditions that should exist across inter-element faces. Although grid generation software GID generates 3-D elements with defined surfaces, hence duplication of surfaces that belong to two elements, a code was then developed not only to eliminate duplicate surfaces but also label them with respect to each element. Using Figure 1, the flux q2 6 is referenced to element 1 and is located at node 6 in a normal direction to surface 2.

Figure 1: Flux labelling convention for the inter-element compatibility relations.

Results and discussion Three numerical examples are used to demonstrate the accuracy of the foregoing 3-D GEM. The examples have been constructed so they have analytical solutions with which accuracy comparison is made. These examples are in domains that have regular geometries. Additional benchmarking is made against FEMWATER finite element package on the basis of solution accuracy, but no attempt is made to compare the computational efficiencies of FEMWATER and our GEM because of their different coding platforms.

Example 1 – 2D Poisson equation example A 2-D potential flow problem that is governed by the Poisson equation is used to validate the 3-D GEM model. The flow differential equation is w 2u w 2u wx 2 wy 2

4 in x : [0,1];

y : [0,2]

(6)

With the boundary conditions

u ( x , 0) u (0, y )

x 2 , u ( x , 2) 2

y , u (1, y )

( x 2) 2 , 0 d x d 1 ( y 1) 2 , 0 d y d 2

(7)


The exact solution is known [8]: u ( x, y )

465

( x y ) 2 . The example is numerically solved in 3-D by

specifying a z -dimension of unit length. The z 0 and z 1 planes have zero flux prescribed on them, while the specified boundary conditions are appropriately specified on other external planes. The domain is uniformly discretized into 64 elements each of size 0.25x0.50x0.25 as shown in Figure 2. The simulated potentials at some internal nodes from the current GEM and FEMWATER are compared in Table 1. It is observed that at all nodes the Green element method performed better than the finite element method.

Figure 2: Domain discretization for Example 1.

Table 1. Comparison of the potential from exact solution, GEM and FEMWATER at some selected internal nodes. GEM FEMWATER y u u Node ID x Exact Rel. error (%) Rel. error % z 12 16 27 38 49

0.25 0.50 0.75 0.25 0.50

1.50 1.50 1.50 1.00 1.00

0.25 0.25 0.25 0.25 0.25

1.5625 1 0.5625 0.5625 0.25

1.744653 1.239443 0.692075 0.777557 0.512837

12 24 23 38 105

1.843985 1.369952 0.844555 0.895536 0.691167

18 37 50 59 176

Example 2 - Transient 1-D heat conduction with Dirichlet conditions Our second example is one of transient heat conduction in a bar of unit length in which unit temperature is maintained at one end and zero at the other. Initially the bar is kept at zero temperature. The exact solution, derived by method of Laplace transform, is presented in Carslaw and Jaeger [9]. To model the onedimensional heat conduction problem, two opposite faces of the domain (shaded faces in Figure 3) have the prescribed temperature values, while the other four faces are specified as zero heat flux boundaries. The domain is discretized into 10 rectangular hexagonal elements with GID, and the node labelling is as shown in Figure 3. The numerical simulations are carried out with a uniform time step of 0.1 and the fully implicit finite difference scheme. The exact, GEM, and FEMWATER solutions for the potential (temperature) at times of 0.2 and 0.4 are presented in Figures 4a and 4b, while the average relative error from GEM and FEM at each time step is presented in Figure 4c. Figure 4c indicates that the errors from both GEM and FEM diminish asymptotically to zero as steady state is approached, and the former gives more accurate solutions than the latter.

466


Example 3 - Transient 1-D heat conduction with Neumann conditions Our third example is similar to previous one, except that unit magnitude of heat flux in and out of the bar are maintained at the ends of the bar (Neumann boundary conditions). As in the previous example, the bar is kept at zero temperature initially.

z y

x

Figure 3: 3-D discretization of 1-D heat conduction example 2. 1

1

Time=0.4

Time=0.2 Exact GEM FEM

u

u

0.75

Exact GEM FEM

0.75

0.5

0.5 0.25

0.25

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

x

x

(a)

(b) 100

GEM FEM

Error (%)

80 60 40 20 0 0

0.2

0.4 0.6 Time

0.8

1

(c)

Figure 4: Exact and numerical solutions for temperature of example 2: (a) t 0.2 , (b) t 0.4 and (c) mean error. The exact solution is well known, derived by the method of Laplace transform, and given in Carslaw and Jaeger [9]. The same discretization for Example 2 is used in this example (Figure 3). The exact, GEM, and FEMWATER solutions for the potential (temperature) at times of 0.2 and 0.4 are presented in Figures 5a and 5b. The average relative error from GEM and FEM at each time step is presented in Figure 5c. GEM gives more accurate solutions than FEM.


467

Conclusion The extension of the GEM to 3-D spatial domains that is presented in this paper is intended to explore how the computational robustness of GEM in 1-D and 2-D presents itself in 3-D. Here we have used only rectangular hexagonal elements because of their ease in accounting for the internal normal fluxes. The formulation will in future incorporate tetrahedral and general hexahedral elements. The performances of GEM and FEM were compared on three simple steady and transient potential flow problems for which analytical solutions are available, and in all cases GEM gave more accurate solutions. The normal fluxes at internal nodes are not retained in 3-D in contrast to 2-D where they were retained by the introduction of a compatibility equation which resolved the closure problem [10]. In the current work, the internal normal fluxes were expressed in terms of the potential by finite differencing, hence the use of rectangular hexagonal elements which eased the implementation of the differencing. 0.6 Time=0.4

0.6 Time=0.2

0.4

Exact GEM FEM

0.4

Exact GEM FEM

0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

(a)

(b) 80

GEM FEM

Error (%)

60 40 20

(c)

0 0

0.2

0.4 0.6 Time

0.8

1

Figure 5: Exact and numerical solutions for temperature of example 3: (a) t 0.2 , (b) t 0.4 and (c) mean error. Acknowledgement Special thanks go to the National Research Foundation which provided the financial support for this research work. References [1] A.E. Taigbenu The Green element method, Int J Num Methods in Engrg 38, 2241-2263 (1995). [2] A.E. Taigbenu The Green Element Method, Kluwer, Boston, USA, 1999. [3] A.E. Taigbenu and O.O. Onyejekwe A Flux-correct Green Element Model of Quasi Three-Dimensional Multiaquifer Flow, Water Resources Research, 36 3631-3640 (2000). [4] P.Lorinczi, S.D. Harris and L. Elliott Modified flux-vector-based Green element method for problems in steadystate anisotropic media, Engrg Anal Boundary Elements 33, 368-387 (2009). [5] V. Popov and H. Power, The DRM-MD integral equation method: An efficient approach for the numerical solution of domain dominant problems. Int J Num Methods in Engrg 44 327-353 (1999). [6] Ramšak, M. and Škerget, L., 3D multidomain BEM for a Poisson equation, Engrg Anal Boundary Elements, 33, 689 (2009). [7] X.W. Gao and T.G. Davies 3D multi-region BEM with corners and edges, Int J of solids and structures, 37, 1549 (2000). [8] R.L. Burden and J.D. Faires Numerical Analysis. Brooks/Cole, California (2001). [9] H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids. Oxford University Press, London (1959). [10] A.E. Taigbenu, The flux-correct Green element formulation for linear, nonlinear heat transport in heterogeneous media, Engrg Anal Boundary Elements, 32, 52-63 (2008).

468


BEM Fracture Mechanics Analysis of 3D Generally Anisotropic Solids 1 2.

C.L. Tan1*, Y.C. Shiah2, J.R. Armitage1 and W.C. Hsia2 Department of Mechanical & Aerospace Engineering Carleton University, Ottawa, Canada K1S 5B6 Department of Aerospace Engineering and Systems Engineering Feng Chia University, Taichung, Taiwan, R.O.C. (* Corresponding author; Email: [email protected])

Keywords: Anisotropic elasticity, Green’s functions, fracture mechanics, stress intensity factors.

Abstract. Numerical stress intensity factor solutions for cracks in 3D generally anisotropic solids are very scarce in the literature. This is particularly true of those obtained using the BEM. A new BEM formulation recently developed by the lead authors that is based on explicit algebraic forms of the fundamental solutions for 3D anisotropic elasticity is being employed together with the use of special crack-front boundary elements to obtain the stress intensity factors in 3D cracked bodies. These fracture parameters can be directly obtained using computed values of the traction coefficients at the nodes on the crack front, similar to the developments in isotropic BEM analysis. Two numerical examples are presented to demonstrate the veracity of the approach. In the first case, the results obtained by the present BEM are compared with the exact analytical solution that is available; in the second case, they are compared with those obtained using a commercial FEM code. Introduction The boundary element method (BEM) is well established as an efficient computational tool for fracture mechanics analysis in isotropic elasticity. This is also true for anisotropic elasticity in 2D. Due to the complexities in the analytical and numerical formulations, its application to practical problems in 3D general anisotropic elasticity is not as widely reported in the literature. Its use for fracture mechanics analysis of threedimensional, cracked, generally anisotropic solids is even more limited, see, e.g. [1]-[2]. In these BEM studies, the stress intensity factors (SIFs) have generally been obtained from extrapolation or correlation of the computed crack face nodal displacement data with the classical near crack-front field solutions. Noting that these near-field variations of the displacements and the stresses for anisotropy are also O(r1/2) and O(r -1/2), respectively, Saez, et al. [3] followed the very successful developments in isotropy of using the quarter-point crack-front elements with the appropriate modification of the shape functions for the tractions, to obtain displacement and traction formulas for the evaluation of the SIFs in their BEM formulation, albeit, for 3D transversely isotropic solids. They further demonstrated, as has been widely established in BEM formulations for isotropic and 2D general anisotropic elasticity, that the traction formula is less mesh sensitive than that based on the computed crack face nodal displacements. Special crack-front elements with mid-side nodes have also been developed for 3D BEM fracture mechanics analysis of isotropic bodies [4], [5]. The corresponding traction formula obtained has also been shown to be as simple and efficient for the direct determination of the SIFs. It has, however, not been employed in BEM analysis of general anisotropic solids. The lead authors have very recently [6] implemented a BEM formulation for 3D stress analysis of generally anisotropic elastic solids based on an explicit, real variable form of the fundamental solutions derived by Ting and Lee [7] and Lee [8]. In the present study, two example crack problems in 3D anisotropic elastic bodies are analysed using this formulation. The traction formula based on the special crack-front elements with mid-side nodes [4], [5] is further demonstrated as a simple and direct means to obtain the SIFs. The first example is the problem of a penny shaped crack in a transversely isotropic infinite body for which an exact analytical solution exists. For the second problem of a penny shaped crack in a cylindrical bar, the BEM results are compared with those obtained using the FEM commercial code ANSYS. Fundamental solutions for general anisotropic elasticity The boundary integral equation (BIE) for elastostatics relating the displacements uj and tractions tj at the boundary S of the homogeneous elastic domain can be written in indicial notation as

1


469

C ij ( P) u i ( P ) + ³ u i (Q) T*ij ( P , Q) dS = ³ t i (Q) U ij ( P , Q) dS S

(1)

S

In eq. (1), the value of Cij(P) depends upon the local geometry of S at the source point P; Uij(P,Q) U(x), and T*ij(P,Q) represent the fundamental solutions (or Green’s functions) of displacements and tractions, respectively, in the xi-direction at the field point Q due to a unit load in the xj-direction at P in a homogeneous infinite body. The fundamental solution Uij(P,Q) and its derivatives, Uij,l(P,Q) employed here are those derived by Ting and Lee [7] and Lee [8]; T*ij(P,Q) can be directly calculated from the strain components. The numerical formulation of the BEM is, in the main, otherwise the same as in isotropy that is well-established in the literature; the resent work employs the quadratic isoparametric element formulation. Only the final forms of the fundamental solutions are therefore presented here; details of their numerical evaluation may be found in [6]. Consider an infinite anisotropic body in which a unit load is applied at a source point P at the local origin x = 0 and the field point Q at x=(x1, x2, x3) is at distance r away. For a unit circle n* =1 on an oblique plane normal to xQ, the unit vector n* can be written in terms of an arbitrary parameter \ as

n* =n cos\ +m sin\ ,

(2)

where the vectors n, m along with x/r form a right-handed triad >n, m, x/r @ . Ting and Lee [7] have obtained

the Green’s function for displacements, U(x), to be 1

U ( x)

4S r

H ij

1

1 4S r T

4

¦ q ˆ

(n)

n

,

(3)

n= 0

ˆ ( n ) , is the adjoint of * and where Hij is the Barnett-Lothe tensor; *ˆ , with components ( p )=Q+p (R+RT )+p 2 T .

(4)

In eq. (4),

Q { Qik =Cijks n j ns , R { Rik =Cijks n j ms , T { Tik =Cijks m j ms ,

(5)

where p = tan\ and Cijks is the material stiffness matrix. A sextic equation in p is obtained by setting the determinant, | ( p ) | = 0. The six independent roots of this equation are the Stroh’s eigenvalues; they are complex, pv =D v + i E v , E v >0 , ( =1, 2, 3), and appear as conjugate pairs. Also in eq. (3), qn is given by

-1 ° °° 2 E1E 2 E 3 qn = ® ° -1 ° 2E E E ¯° 1 2 3

ª ° 3 º ½° ptn « Re ®¦ ¾ -G n 2 » »¼ ¬« ¯° t=1 pt -pt+1 pt -pt+2 ¿°

for n= 0, 1, 2,

ª ° 3 ptn- 2 pt 1 pt 2 °½ º « Re ®¦ ¾» ¬« °¯ t=1 pt -pt+1 pt -pt+2 °¿¼»

for n= 3, 4,

(6)

where the overbar in pt denotes the complex conjugate, ij is the Kronecker delta and the subscript t follows the ˆ ( n ) in terms of Q, R and T may be found in [6]. cyclic rule t = (t-3) if t>3. More explicit expressions of The traction fundamental solution, T*(x), is obtained by differentiating the Green’s function U(x), and invoking the generalized Hooke’s law. Lee[8] has derived the displacement derivative of U(x) to be as follows:

U ij ,l where

M ijklmn

2S i T

2

3

¦ p t 1

t

>

@

1 S yl H ij C pqrs ys M lqiprj yq M sliprj 4S 2 r 2

1 2 2 pt 1 pt pt 2

2

ª ·º § 1 1 ¸¸» «)cijklmn ( pt ) 2) ijklmn ( pt ) u ¨¨ © pt pt 1 pt pt 2 ¹¼ ¬

(7) (8)

470


ijklmn ( p)

ª ni n j ni m j mi n j p mi m j p 2 º ª ˆ kl ( p ) ˆ mn ( p ) º ¬ ¼¬ ¼ ( p p1 ) 2 ( p p2 ) 2 ( p p3 ) 2

(9)

where the prime in )cijklmn ( pt ) denotes differentiation with respect to p. There is no need to re-write this

ˆ ( p ) , ( p pt ) 2 , quantity more explicitly as it is relatively straightforward to program the functions * ª ni n j ni m j mi n j p mi m j p 2 º , and their derivatives into subroutines in the computer code and then ¬ ¼ apply the chain rule in the differentiation. Determination of the stress intensity factors Special crack-front quadrilateral elements are employed to treat crack problems in this work. These elements are geometrically identical to the isoparametric elements with mid-side nodes, but the functional variations of the displacements and the tractions have been modified using the appropriate shape functions to represent the proper near-front fields. The near-field variations of the displacements and the stresses for anisotropy are also O(r1/2) and O(r -1/2), respectively, as in isotropy. These shape functions may be found in [4, 5]; they were originally developed for fracture mechanics studies of 3D isotropic bodies. By equating the functional variation of the tractions in the traction-singular crack front element to the classical field solution, simple “traction formulas” may be derived for determining the SIFs, (KI , KII, KIII), directly from the BEM computed traction coefficients, ti* , at the crack-front nodes. For a crack lying in the x1 –x2 plane, these formulas for node A on the crack-front of the traction-singular element are as follows:

( K I ) A (t3* ) A S l ( K II ) A (t1* ) A S l * 2 A

( K III ) A (t )

(10)

Sl

where ( ti* )A is the computed traction coefficient in the i-th direction at node A, and l is the width of this element (normal to the crack-front). These formulas provide a direct, efficient and accurate means of determining these fracture parameters as will be demonstrated by the examples below. Numerical Examples Example (1): The first example presented here is a transversely isotropic infinite domain with a penny shaped crack for which exact analytical solution for the SIFs under different load conditions are available [9]. For the BEM analysis, the body is modeled as a cube with side lengths ten times the diameter, 2a, of the crack which lies in the mid-plane. Using the sub-regioning technique for the fracture mechanics analysis, the BEM mesh employed is shown in Fig. 1; the indicated Cartesian coordinate system has its origin at the centre of the crack. The width of the crack-front element was set to be l/a = 0.15. A graphite-epoxy composite with the following material stiffness coefficients [3] is considered: C11 = 13.92 MPa, C12 = 6.92 MPa, C13 = 6.44 MPa, C33 = 160.7 MPa and C44 = 7.07 MPa; the material axes being coincident with the global Cartesian axes. Two load cases are treated, namely, (i) remote uniform tension of 33= o and (ii) remote shear stress 23= o, applied at the top and bottom faces of the cube modeled. The problem was also solved for isotropy in both load cases as well, with Poisson’s ratio = 0.3. For case (i), the exact solution for the normalized stress intensity factor, K I / V o S a , is the same for both isotropy and transverse isotropy, the value being 0.637. The computed results using the traction formula for the corresponding materials are 0.633 and 0.638, respectively; the errors are thus less than 1%. For load case (ii), the exact solutions for the values of the material properties used can be verified to be: [ K II / W o S a 0.7490sin T , K III / W o S a 0.5243cos T ] in the isotropic solid, and in the anisotropic

3


471

x3 x2 x1

(a) Elements at exterior surface of the cube

(b) Elements at sub-region interface

Fig. 1: BEM mesh for the problem of a penny shaped crack in an infinite domain; total 216 elements 604 nodes. graphite-epoxy, [ K II / W o S a 0.8115sin T , K III / W o S a 0.4617 cos T ] ; the angle here is angular position of the crack front measured from the x1-axis. The BEM computed values are shown in Table 1; the corresponding exact solutions are shown in parentheses where it can be seen that the errors are all less than 1.5%.

Angle, (deg)

Isotropic Medium

K

* II

K

Graphite Epoxy * III

K

* II

* K III

0

0.5190 (0.5243)

0

0.4555 (0.4617)

30

0.3734 (0.3745)

0.4494 (0.4540)

0.4053 (0.4057)

0.3943 (0.3999)

60

0.6467 (0.6486)

0.2594 (0.2621)

0.7016 (0.7028)

0.2280 (0.2309)

0.7471 (0.7490)

0

0.8102 (0.8115)

0

90

0

Table 1: Computed normalised stress intensity factors, K II* shaped crack in an infinite domain under shear load.

* K II / W o S a , K III

K III / W o S a , for a penny

Example 2: The second problem analysed is a long cylindrical bar with a penny shaped crack of radius a lying in the axial plane at the mid-section of the bar, as shown in Fig. 2 The length of the bar is taken to be four times its radius R, and R/a = 2. The bar is constrained at the lower face and subjected to a uniform axial stress o at the remote end of the top surface. The BEM mesh employed is shown in Fig. 3. It was verified first with isotropic analysis for which the computed value of K I / V o S a = 0.685 at the crack periphery; this is in excellent agreement with the corresponding solution by Benthem and Koiter [10] for a very long cylinder of 0.688. For the anisotropic analysis, zinc crystal was considered as the material for the cylindrical bar. It is also a hexagonal system and has the following material stiffness coefficients in the principal directions of the Cartesian axes [11]: C11 = 160.9 MPa, C12 = 33.5 MPa, C13 = 50.1 MPa, C33 = 61.0 MPa and C44 = 38.3 MPa. However, for the analysis, the material axes are rotated clockwise by 60o about the x1-axis; the resulting material stiffness matrix 4

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in the global Cartesian system thus has the characteristics of a more generally anisotropic solid. With the remote axial load, all three modes of crack deformation will be present. To verify SIF results obtained from the BEM analysis, the problem was also analysed using the commercial FEM code ANSYS with a very refined mesh (see Fig. 4). The SIFs from the FEM analysis are calculated using extrapolation techniques from the computed stresses near the crack front at the point of interest. The computed SIF results of the BEM and FEM analysis of the problem are shown plotted in Fig. 5; here T is angular position of the crack front measured from the x1-axis. Only the results for the range of T = 0 to 90o are presented for conciseness; it should be noted that KI is symmetric about both the x1-x3 and x2-x3 planes, KII is symmetric about the x2-x3 plane but anti-symmetric about the x1-x3 plane, and KIII is symmetric about the x1-x3 plane and anti-symmetric about the x2-x3 plane. It can be seen that the agreement between the two sets of results for the SIFs obtained from the two numerical methods is very good indeed. Of interest to note is that for the material system analysed, the magnitudes of the mode II and mode III SIFs due to the remote axial load are relatively small as compared to those for mode I; they can be neglected for most practical purposes.

(a) Fig. 2: A cylindrical bar with a penny shaped crack.

(b)

Fig. 3: BEM mesh for Example 2- total 240 elements 660 nodes; (a) elements at exterior surface; (b) elements at sub-region interface

Fig. 4: FEM mesh for Example 2-total 59136 SOLID186 elements, 122226 nodes

Fig. 5: Variation of the computed normalized stress intensity factors with angular position – Example 2

5


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It should perhaps be remarked that in both examples presented above, the entire physical problem was modeled even though this was not necessary from symmetry considerations. These meshes are being employed in ongoing studies of more general cases of these cracked geometries that require the full model. Conclusions Traction-singular crack-front elements have been employed in the elastic fracture mechanics analysis of cracked 3D anisotropic solids using the BEM. With these elements, the same simple “traction formulas” as established in isotropic elasticity can be derived and employed for the determination of the stress intensity factors in a quick and direct manner from the computed traction coefficients at the nodes on the crack front. Two numerical examples have been presented in this paper to demonstrate the validity of the approach. The results of the first example have been compared with exact analytical solutions, while for the second example, they were compared with those obtained using the FEM. There was excellent agreement of all the corresponding results. Acknowledgement The authors gratefully acknowledge the financial support from the National Science and Engineering Research Council of Canada and the National Science Council of Taiwan (NSC 96-2221-E-035-011-MY3). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

E. Pan and F.G. Yuan, Int. J. Num. Methods Engng., 48, 211-237 (2000) J. Rungamornrat, Engng. Analysis Boundary Elem., 30, 834-846 (2006) A. Saez, M.P. Ariza and J. Dominguez, Engng. Analysis Boundary Elem., 20,287-298 (1997) M.L. Luichi and S. Rizzuti, Int. J. Num. Methods Engng., 24, 2253-2271 (1987) S.B. Liu and C.L. Tan, Int. J. Fracture, 72, 39-67 (1995) C.L. Tan , Y.C. Shiah and C.W. Lin, CMES-Comp Model Eng & Sci, 41, 195-214 (2009) T.C.T. Ting and V.G. Lee, Q. J. Mech. Appl. Math., 50, 407-426 (1997) V.G. Lee, Mech. Res. Comm., 30, 241-249 (2003) M.K. Kassir and G.C. Sih, in Ch.8, Mechanics of Fracture 2, G.C. Sih (ed.), Noordoff Int. Pub. (1975) J.P. Benthem and W.T. Koiter, in Ch. 3, Mechanics of Fracture 1, G.C. Sih (ed.), Noordoff Int. Pub., (1973) H.B. Huntington, The Elastic Constants of Crystals, Academic Press, New York (1958)

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A BEM analysis of the fibre size effect on the debond growth along the fibre-matrix interface L. Távara, V. Mantiˇc, E. Graciani, F. Par´ıs Escuela Técnica Superior de Ingenieros, Universidad de Sevilla Camino de los Descubrimientos s/n, 41092 Sevilla, Spain [email protected], [email protected], [email protected], [email protected] Abstract. Damage initiation in a fibre reinforced composite lamina subjected to transverse tension is analised from the micromechanical point of view. The debond onset and growth at the fibrematrix interface is studied by means of a new linear elastic-brittle interface model. The interface is modeled by a continuous distribution of linear elastic springs which simulates the presence of a thin interphase layer. In the constitutive law for the continuous distribution of springs the normal and tangential stresses across the undamaged interface are respectively proportional to the relative normal and tangential displacements until the energy available for crack propagation equals the fracture toughness of the interface, where the breaking of the interface occurs. An important feature of this law is that the energy available for crack propagation is evaluated taking into account the variation of the fracture toughness with the fracture mode mixity of a crack growing along the interface between bonded solids. The present interface model is implemented in a 2D Boundary Element Method (BEM) code. This code is used to study the failure mechanism of an isolated fibre under transverse tension. The influence of the fibre radius on the debond onset and growth is shown. Keywords: Imperfect interface, weak interface, fibre-matrix interface, Energy Release Rate, fracture mode, composites.

Introduction Fibre reinforced composite laminas under loads transverse to the fibres usually exhibit a failure mechanism called matrix failure or interfibre failure, see Par´ıs et al. [1], Correa et al. [2, 3], and Mantiˇc et al. [4]. A comprehensive review of the problem of an elastic circular (in 2D) or cylindrical (in 3D) inclusion embedded in an elastic matrix with a partial debond at their interface (modeled like an interface crack) subjected to a remote tension can be found in Távara et al. [5]. Many authors considered that a perfect bonding condition at the undamaged fibre-matrix interface is often inadequate in describing the physical nature and mechanical behavior of this interface. See Gao [6] and Hashin [7] for a review of related works modeling the fibre-matrix interface as an elastic layer with vanishing thickness. A practical way to describe the behavior of bonded solids is to model an elastic layer, sometimes called interphase, as a continuous distribution of linear elastic springs with appropriate stiffness parameters. In the present work the interface model developed in T´ avara et al. [5] is used. This model, governed by a linear elastic-brittle constitutive law, takes into account the variation of the fracture toughness with the fracture mode mixity. This interface model has been implemented in a 2D collocational BEM code and used to solve several problems of damage initiation and growth between composite laminas [8]. The present work is organized as follows. The interface failure criterion is described first. Then, the incremental BEM formulation used is presented. It is noteworthy that the equilibrium and compatibility conditions along the interfaces are imposed in a weak form allowing the use of non-conforming meshes. In the following section, the problem of a cylindrical inclusion under a remote transverse tension is described. Finally, BEM numerical solutions for the above mentioned problem of interface crack onset and growth are presented. The influence of the fibre radius on the debond onset and growth is determined and some interesting conclusions are obtained.


475

Interface failure criterion The linear elastic interface model implies the presence of a stress concentration instead of a stress singularity at the interface crack tip. In the present work, damage of a portion of an interphase layer (i.e., a thin adhesive layer) is modeled as an abrupt decrease to zero of stiffness parameters of this layer accompanied with vanishing of stresses in this portion of the layer, subsequently leading to a free separation/sliding of both surfaces. This damage occurs when a point on the fracture locus (in (𝜎, 𝜏 ) plane, as will be shown later) is achieved in a portion of the layer. The threshold (critical) normal and shear stresses (𝜎𝑐 and 𝜏𝑐 ) depend on the fracture mode mixity, which can be characterized by (fracture) energy based angle 𝜓𝐺 defined in this section. The continuous spring distribution that models the elastic layer (interphase) is governed by the following simple linear elastic-brittle law written at an interface point 𝑥: { Linear elastic 𝜎(𝑥) = 𝑘𝑛 𝛿𝑛 (𝑥) 𝛿𝑛∗ (𝑥) ≤ 𝛿𝑛𝑐 (𝜓𝐺 (𝑥)) and 𝛿𝑡∗ (𝑥) ≤ 𝛿𝑡𝑐 (𝜓𝐺 (𝑥)), interface 𝜏 (𝑥) = 𝑘𝑡 𝛿𝑡 (𝑥) (1) { Broken 𝜎(𝑥) = 0 ∗ ∗ 𝛿𝑛 (𝑥) > 𝛿𝑛𝑐 (𝜓𝐺 (𝑥)) and 𝛿𝑡 (𝑥) > 𝛿𝑡𝑐 (𝜓𝐺 (𝑥)), interface 𝜏 (𝑥) = 0 where 𝜎(𝑥) and 𝜏 (𝑥) are, respectively, the normal and tangential stresses in the elastic layer, 𝛿𝑛 (𝑥) and 𝛿𝑡 (𝑥) are, respectively, the normal and tangential relative displacements between opposite interface points. 𝛿𝑛∗ (𝑥) and 𝛿𝑡∗ (𝑥) are the maximum normal and tangential relative displacements achieved at each interface point up to the considered instant of the problem evolution. 𝛿𝑛 (𝑥) and 𝛿𝑡 (𝑥) are sometimes referred to as the value of the opening and sliding between the interface sides. 𝑘𝑛 and 𝑘𝑡 denote the normal and tangential stiffnesses of the spring distribution. Notice that the critical variables 𝛿𝑛𝑐 (𝜓𝐺 ) and 𝛿𝑡𝑐 (𝜓𝐺 ) are functions of the fracture mode mixity angle 𝜓𝐺 (𝑥) at an interface point. Thus, different values of these critical variables may be obtained at different interface points. We can define that the “spring” at a point breaks when either 𝛿𝑛 or 𝛿𝑡 reaches its critical value, 𝛿𝑛𝑐 (𝜓𝐺 ) or 𝛿𝑡𝑐 (𝜓𝐺 ). At this moment both normal and tangential stiffness are set to zero at this point. The interface failure criterion is based on the Energy Release Rate (ERR) concept. The ERR in a linear interface model is defined as the stored elastic strain energy per unit length in the unbroken “crack-tip interface spring” as shown in [9] and recently independently also in [10]. Thus, the ERR of a mixed mode crack in a linear elastic interface is defined as: 𝐺 = 𝐺𝐼 + 𝐺𝐼𝐼 =

𝜎𝛿𝑛 𝜏 𝛿𝑡 + . 2 2

(2)

It can also be defined in terms of crack tip stresses 𝜎 and 𝜏 or relative displacements 𝛿𝑛 and 𝛿𝑡 : 𝐺=

𝜏2 𝛿2 𝑘𝑛 𝛿2 𝑘𝑡 𝜎2 + = 𝑛 + 𝑡 . 2𝑘𝑛 2𝑘𝑡 2 2

(3)

The fracture mode mixity angle can be defined, rewriting equation (2) as: 𝐺 = 𝐺𝐼 + 𝐺𝐼𝐼 = 𝐺𝐼 (1 + tan2 𝜓𝐺 ), where

(4)

𝐺𝐼𝐼 . (5) 𝐺𝐼 It is assumed that a crack propagates when the ERR, 𝐺, reaches the fracture energy 𝐺𝑐 , that is: tan2 𝜓𝐺 =

𝐺 = 𝐺𝑐 .

(6)

A strong dependence of 𝐺𝑐 on the mode mixity has been observed in extensive experiments by Evans et. al. [11] and Banks-Sills and Askhenazi [12] among others. Thus, 𝜓𝐺 is an important parameter governing the interface crack growth. From several phenomenological laws for 𝐺𝑐 suggested

476


in the past [13], the following family of expressions of the fracture energy (representing the fracture toughness), is considered to be representative of a large number of bimaterial systems: 𝐺𝑐 = 𝐺𝐼𝑐 [1 + tan2 ((1 − 𝜆)𝜓𝐺 )],

(7)

where

2 𝜎 ¯2 𝜎 ¯𝑐 𝛿¯𝑛𝑐 𝑘𝑛 𝛿¯𝑛𝑐 = 𝑐 = (8) 2 2𝑘𝑛 2 corresponds to fracture energy in pure opening mode I. 𝜆 is a fracture mode-sensitivity parameter, where the typical range 0.2 ≤ 𝜆 ≤ 0.3 characterizes interfaces with moderately strong fracture mode dependence. 𝜎 ¯𝑐 and 𝛿¯𝑛𝑐 are the critical normal stress and opening displacement reached when the spring breaks in mode I.

𝐺𝐼𝑐 =

(a)

(b)

Figure 1: (a) Fracture energy 𝐺𝑐 as a function of 𝜓𝐺 for different values of 𝜆, (b) interface failure loci in (𝜎, 𝜏 ) plane for 0 ≤ 𝜓𝐺 ≤ 𝜋/2 and different values of 𝜆 with 𝑘𝑛 /𝑘𝑡 = 3 As can be seen in Figure 1(a), if 𝜆 = 0 the interface will never break in pure mode II, due to the asymptotic behavior of the function 𝐺𝑐 defined in (7). As mentioned before the use of 0 < 𝜆 < 1 in (7) yields a more realistic behavior, allowing the interface to fail in pure mode II. As shown in Figure 1 for larger values of 𝜆 the influence of the fracture mode II increases. Writing the crack propagation criteria along a linear elastic-brittle interface, making use of (4 - 8), the following general expression of the critical normal stress as a function of the angle 𝜓𝐺 is obtained: √ 𝜎𝑐 (𝜓𝐺 ) 𝜎𝑐 (𝜓𝐺 ) = 𝜎 ¯𝑐 1 + tan2 [(1 − 𝜆)𝜓𝐺 ]. cos 𝜓𝐺 and 𝛿𝑛𝑐 (𝜓𝐺 ) = . (9) 𝑘𝑛 Thus, 𝜎 ¯𝑐 = 𝜎𝑐 (0∘ ) and 𝛿¯𝑛𝑐 = 𝛿𝑛𝑐 (0∘ ). In a similar way the critical tangential stress for mixed mode can be expressed, in terms of 𝜎𝑐 (𝜓𝐺 ) and 𝜓𝐺 , as: √ √ 𝜏𝑐 (𝜓𝐺 ) 𝑘𝑡 𝜏𝑐 (𝜓𝐺 ) = 𝜎 ¯𝑐 1 + tan2 [(1 − 𝜆)𝜓𝐺 ]. sin 𝜓𝐺 and 𝛿𝑡𝑐 (𝜓𝐺 ) = . (10) 𝑘𝑛 𝑘𝑡 The plot of interface failure loci parameterized by equations (9) and (10) is shown in Figure 1(b) for 0∘ ≤ 𝜓𝐺 ≤ 90∘ , where a ratio 𝑘𝑛 /𝑘𝑡 = 3 has been considered. The normal and tangential critical stress in mixed mode were normalized with the normal critical stress in mode I, 𝜎 ¯𝑐. It should be mentioned that for 𝜆 = 0, the expression of 𝐺𝑐 in (7) becomes similar to that obtained in [12], although it was used in a different interface model. Also 𝜎𝑐 (𝜓𝐺 ) in (9) equals 𝜎 ¯𝑐 (the critical normal stress in Mode I, see Figure 1(b)) for all values of 𝜓𝐺 [8]. BEM and Linear Elastic-Brittle Interface Incremental formulation. The numerical solution of the non-linear problem formulated is based on a gradual application of the loads and displacements imposed, by means of a load factor F, 0 ≤ 𝐹 ≤ 1.


477

The solution procedure is given by a series of lineal stages, “load steps”. At the beginning of each load step the actual interface zone bonded by the adhesive layer is established, which defines the actual linear system of equations. By solving this system of equations the corresponding elastic solution is obtained. Thus, the solution of the problem will be divided into a number 𝑀 (a priori unknown) of load steps in which the values of the problem variables vary linearly: 𝜙(𝑥, 𝐹 ) = 𝐹 △𝑚 𝜙(𝑥),

(11)

with 𝐹𝑚−1 ≤ 𝐹 ≤ 𝐹𝑚 , 𝑚 = 1, ..., 𝑀 , and 𝐹0 = 0, and where 𝜙(𝑥, 𝐹 ) is the value of any problem variable at a point 𝑥 after an 𝐹 -fraction of the load is applied. △𝑚 𝜙(𝑥) is the value of the increment of the variable 𝜙(𝑥) corresponding to the unit increment of the load factor 𝐹 (or equivalently the derivative of 𝜙(𝑥) with respect to 𝐹 ), and it is obtained in the solution of the linear system of equations corresponding to the m-th load step. This solution fulfills all the conditions of the linear elastic interface formulation (and also of the frictionless contact formulation, if needed in the damaged zone) up to a certain maximum value 𝐹𝑚 of the load factor 𝐹 associated to this load step. A further increment of the load factor leads to rupture of some springs (or to a change in contact conditions). Consequently, values of variable 𝜙 at the end of each load step are defined as 𝜙(𝑥, 𝐹𝑚 ) = 𝐹𝑚 △𝑚 𝜙(𝑥) for 𝐹 = 1, ..., 𝑀 . Weak formulation of the linear elastic-brittle interface. The equilibrium and compatibility conditions along the interface zone between the two solids A and B are imposed in a weak manner, derived from the principle of virtual work. In this way, stresses along the interface zone are defined as 𝐴 the tractions along the interface part of the boundary of solid A, Γ𝐴 𝑖 ⊂ Γ . Equilibrium is imposed in 𝐵 𝐵 𝐴 a weak form at all points 𝑦 along the interface of solid B, Γ𝐵 𝑖 ⊂ Γ , which means △𝑡𝑖 (𝑦)+△𝑡𝑖 (𝑦) = 0, is guaranteed by the fulfilment of the following integral equation considered at Γ𝐵 : 𝑖 ∫ 𝐵𝜓 𝐴 𝐵 [△𝑡𝐵 𝑖 = 𝑛, 𝑡, (12) 𝑖 (𝑦) + △𝑡𝑖 (𝑦)]△𝑢𝑖 𝑑Γ𝑖 = 0 Γ𝐵 𝑖

for any field of compatible displacements △𝑢𝐵𝜓 𝑖 (𝑦). Accordingly, displacements along the interface are defined by the displacements at Γ𝐵 𝑖 . Compatibil𝐴 𝐴 𝐵 ity is imposed in a weak form at all points 𝑦 belonging to Γ𝐴 𝑖 , which means △𝛿𝑖 (𝑦) = △𝑢𝑖 (𝑦) − 𝑢𝑖 (𝑦) with △𝛿𝐴 (𝑦) being the relative displacements between opposite points of the adjacent solids, by the fulfilment of the following integral equation considered at Γ𝐴 𝑖 : ∫ 𝐴 𝐵 𝐴 𝐴 △𝑡𝐴𝜓 (13) 𝑖 [△𝑢𝑖 (𝑦) − △𝑡𝑖 (𝑦) − △𝛿𝑖 (𝑦)]𝑑Γ𝑖 = 0 Γ𝐴 𝑖

for any tractions field in equilibrium △𝑡𝐴𝜓 𝑖 (𝑦). Cylindrical inclusion under a transverse tension The problem of an elastic cylindrical inclusion inside an elastic matrix with and without a partial debond along its interface subjected to a remote uniform tension perpendicular to the debond has been studied in depth by many researchers, see references in [1, 14]. In the present study an infinitely long cylindrical inclusion is considered, with circular section of radius 𝑎, embedded in an infinite matrix, Figure 2. Both the inclusion and the matrix are considered as linear elastic isotropic materials. Let (𝑥, 𝑦, 𝑧) and (𝑟, 𝜃, 𝑧) be the cartesian and cylindrical coordinates, the 𝑧-axis being the longitudinal axis of the inclusion, and the 𝑥-axis the one parallel to the direction of the load. A uniform remote tension 𝜎𝑥∞ > 0 is applied. The semidebond angle is denoted as 𝜃𝑑 . A plane strain state is assumed in the system.

478


(b)

(a)

Figure 2: Inclusion problem configuration under remote transverse tension (a) without and (b) with a partial debond. Numerical Results Consider an inclusion bonded to the matrix along its lateral surface through a continuous distribution of springs that behave according to the linear elastic-brittle interface model introduced. It is assumed that, although strictly speaking there might be no material between the fibre and matrix, the above interface model can be used to simulate the behavior of this system. An important feature of this model is the possibility of studying not only the fibre-matrix debond (interface crack) propagation but also the debond (interface crack) onset. BEM model. A typical bi-material system among fibre reinforced composite materials is chosen for this study: glass fibre and epoxy matrix. The elastic properties of these materials, the Dundurs’ bi-material parameters, 𝛼 and 𝛽, and the harmonic mean of the effective elasticity moduli 𝐸 ∗ in plane strain, defined e.g. in [1, 4, 14], are detailed in Table 1. Tabla 1: Isotropic bi-material constants (𝑚-epoxy matrix and 𝑓 -glass fibre). Mat. 𝑚 𝑓

Poisson ratio 𝜈𝑚 = 0.33 𝜈𝑓 = 0.22

Young’s modulus 𝐸𝑚 = 2.79 GPa 𝐸𝑓 = 70.8 GPa

𝛼

𝛽

𝐸∗

0.919

0.229

6.01 GPa

The 2D BEM model represents a circular inclusion (with a radius 𝑎: 0.375 < 𝑎 < 7.5 𝜇m) inside a relatively large square matrix with a 1 mm side. 1472 continuous linear boundary elements are used: 32 elements for the external boundary of the matrix and two uniform meshes of 720 elements to model the fibre-matrix interface (therefore, the polar angle of each element is 0.5∘ ). Effect of the inclusion size. A parametric study has been carried out to determine the influence of the size of the inclusion on the onset and growth of the debond crack, as discussed in [14]. ¯𝑐 = 90MPa and 𝑘𝑛 /𝑘𝑡 = 3 are assumed. The The following interface properties 𝐺𝐼𝑐 = 2Jm−2 , 𝜎 fracture mode-sensitivity parameter is taken as 𝜆 = 0.3. The problem is solved for different inclusion radii, see A dimensionless structural parameter characterizing the interface brittle√Table 2. √ 𝐺𝐼𝑐 𝐸 ∗ 𝑎0 1 ness, 𝛾 = 𝜎¯𝑐 = 𝑎 𝑎 , introduced in [14], is also included in Table 2. The reference length ∗

𝐸 = 1.48𝜇m. 𝑎0 = 𝐺𝐼𝑐 𝜎 ¯𝑐2 In part (a) of Figure 3 the applied remote stress, 𝜎𝑥∞ , is plotted as a function of the opening, 𝛿𝑛 , obtained at point A, defined in Figure 2. In part (b) of Figure 3 the (minimum) remote stress, 𝜎𝑥∞ , needed to cause crack growth is plotted versus the semidebond angle 𝜃𝑑 . From these figures it is possible to obtain an estimation of 𝜃𝑐 and 𝜎𝑐∞ . It can be seen that after reaching 𝜎𝑐∞ , the crack growth becomes unstable. Thus, an instability phenomenon called snap-through takes place. Table 2 summarizes the values of the 𝜎𝑐∞ , 𝜎𝑐∞ /¯ 𝜎𝑐 and 𝜃𝑐 for the different values of 𝑎 studied. The results obtained are similar to those obtained in a different way in [14] in the following sense:


479

(b)

(a)

Figure 3: BEM results. (a) Applied stress with respect to normal relative displacements at point A, see figure 2, and (b) applied stress with respect to semidebond angle for different inclusion sizes. Tabla 2: Loads which produce the crack onset, 𝜎𝑐∞ , and the critical semidebond angle, 𝜃𝑐 . 𝑎 (𝜇m) 0.375 0.75 3.75 7.5

(a)

𝛾 1.99 1.4 0.62 0.44

𝜎𝑐∞ (MPa) 177.0 115.6 68.4 63.9

𝜎𝑐∞ /¯ 𝜎𝑐 1.967 1.284 0.760 0.710

𝜃𝑐 (∘ ) 40.75 48.25 69.25 > 70

(b)

Figure 4: Comparison of BEM and analytical results [5]. Inclusion size effect on the critical remote tension that produces the growth, 𝜎𝑐∞ , as function of 𝑎/𝑎0 . as the value of inclusion radius, 𝑎, becomes lower, the value of the critical stress becomes higher. It can also be seen that the value of the critical semidebond angle 𝜃𝑐 decreases when the radius of the inclusion becomes smaller. In Figure 4 the effect of inclusion size on the critical load that produces the debond onset, 𝜎𝑐∞ , is shown. The marked points on one curve represent cases solved by BEM, and the continuous lines are obtained by the analytical solution developed in [5], based on work of Gao [6]. As can be observed

480


(a)

(b)

Figure 5: Comparison of BEM and analytical results [5]. Inclusion size effect on the critical remote tension that produces the growth, 𝜎𝑐∞ , as function of 𝛾.

(a)

(b)

Figure 6: BEM results. Inclusion size effect on the critical semidebond angle 𝜃𝑐 . in Figure 4, 𝜎𝑐∞ ∼ 𝑎1 for vanishing values of 𝑎, whereas it approaches a constant value for 𝑎 → ∞, see asymptotes indicated in Figure 4(b). Notice that as the value of inclusion radius, 𝑎, becomes lower, the value of 𝜎𝑐∞ becomes higher, similarly to previous results in [14], although with different asymptotic behavior for small inclusion radii (𝜎𝑐∞ ∼ √1𝑎 in [14]). In Figure 5(a) the dependence of 𝜎𝑐∞ on the structural dimensionless parameter 𝛾 is depicted, the corresponding asymptotes being also indicated in Figure 5(b). The inclusion size effect on the critical semidebond angle, 𝜃𝑐 , is shown in Figure 6(a) and in a similar way in 6(b) this effect is shown in logarithmic scale. Conclusions The use of a new linear elastic-brittle interface model (continuous spring distribution model) to study fibre-matrix debond problems in composite materials is introduced and analyzed in the present work. This model has been proved to be an efficient way to characterize the failure of the interface of an inclusion subjected to far field transverse loads. The problem of a circular inclusion under transverse tension assuming material properties of a common composite material (glass fibre and epoxy matrix) and a linear elastic-brittle interface has been solved by the collocational BEM. The crack onset and growth along the fibre-matrix interface is modeled using the new linear elastic-brittle constitutive law


481

of the interface. The interface constitutive law included in the incremental algorithm of the BEM code has the advantage of being independent of the number of elements used in the interface. Furthermore, as the elastic-brittle interface equilibrium and compatibility equations are imposed in a weak form, the BEM code can solve problems with non-conforming meshes. From the numerical results it can be seen that after reaching 𝜎𝑐∞ , the crack growth becomes unstable. Thus, an instability phenomenon called snap-through takes place. The inclusion size effect on the debond onset is observed, as the value of inclusion radius, 𝑎, becomes lower, the value of 𝜎𝑐∞ becomes higher, similarly to previous results in [14], although with different asymptotic behavior for small inclusion radii. It can also be seen that the value of the critical semidebond angle 𝜃𝑐 decreases when the radius of the inclusion becomes smaller. Acknowledgments The work was supported by the Junta de Andaluc´ıa (Projects of Excellence TEP-1207, TEP-2045 and TEP-4051) and the Spanish Ministry of Education and Science (Project TRA2006-08077). References [1] F. Par´ıs, E. Correa, V. Mantiˇc, Kinking of transverse interface cracks between fiber and matrix, J Appl Mech, 74(2007) 703–716. [2] E. Correa, V. Mantiˇc, F. Par´ıs, Numerical characterisation of the fibre-matrix interface crack growth in composites under transverse compression, Eng Fract Mech, 75(2008) 4085-4103. [3] E. Correa, V. Mantiˇc, F. Par´ıs, A micromechanical view of inter-fibre failure of composite materials under compression transverse to the fibres, Compos Sci Technol, 68(2008) 2010–2021. [4] V. Mantiˇc, A. Blázquez, E. Correa, F. Par´ıs, Analysis of interface cracks with contact in composites by 2D BEM. in: M. Guagliano and M.H. Aliabadi (Eds.), Fracture and Damage of Composites, WIT Press, Southampton, 2006, pp.189–248. [5] L. Távara, V. Mantiˇc, E. Graciani, J. Ca˜ nas, F. Par´ıs, BEM analysis of crack onset and propagation along fibre-matrix interface under transverse tension using a linear elastic-brittle interface model. Eng Anal Bound Elem (submitted). [6] Z. Gao, A circular inclusion with imperfect interface: Eshelby’s tensor and related problems, J Appl Mech, 62(1995) 860–866. [7] Z. Hashin, Thin interphase/imperfect interface in elasticity with application to coated fiber composites, J Mech Phys Solids, 50(2002) 2509–2537. [8] L. Távara, V. Mantiˇc, E. Graciani, J. Ca˜ nas, F. Par´ıs, Analysis of a crack in a thin adhesive layer between orthotropic materials. An application to composite interlaminar fracture toughness test. CMES - Comput Model Eng Sci (in press). [9] S. Lenci, Analysis of a crack at a weak interface, Int J Fracture, 108(2001) 275–290. [10] A. Carpinteri, P. Cornetti, N. Pugno, Edge debonding in FRP stregthened beams: Stress versus energy failure criteria. Eng Struct, 21(2009) 2436–2447. [11] A.G. Evans, M. R¨ uhle, B.J. Dalgleish, P.G. Charalambides, The fracture energy of bimaterial interfaces, Metall Trans A, 21A(1990) 2419-2429. [12] L. Banks-Sills, D. Ashkenazi, A note on fracture criteria for interface fracture, Int J Fracture, 103(2000) 177-188 . [13] J.W. Hutchinson, Z. Suo, Mixed mode cracking in layered materials, in: J.W. Hutchinson, T.Y. Wu (Eds), Advances in Applied Mechanics, Academic Press, New York, 29(1992), 63-191. [14] V. Mantiˇc, Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion. Int J Solid Struct, 46(2009) 12871304.

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Nonlinear Nonuniform Torsional Vibrations of Shear Deformable Bars Application to Torsional Postbuckling Configurations E.J.Sapountzakis1 and V.J.Tsipiras2 1,2

School of Civil Engineering, National Technical University, Zografou Campus,GR-157 80 Athens, Greece

Keywords: shear deformation, secondary twisting moment deformation effect, bar, boundary element method, nonuniform torsion, nonlinear vibrations, torsional vibrations, torsional postbuckling configuration.

Abstract. In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the effects of geometrical nonlinearity (finite displacement – small strain theory) and secondary twisting moment deformation. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are subjected to the most general axial and torsional (twisting and warping) boundary conditions. The resulting coupling effect between twisting and axial displacement components is also considered and a constant along the bar compressive axial load is induced so as to numerically examine the dynamic response at the (torsional) post-buckled state. A coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an independent warping parameter is formulated. The resulting equations are further combined to obtain a single partial differential equation with respect to the angle of twist. The problem is numerically solved employing the Analog Equation Method (AEM), a BEM based method, leading to a system of nonlinear Differential – Algebraic Equations (DAE). The main purpose of the present contribution is to numerically investigate linear or nonlinear free vibrations of bars at torsional postbuckling configurations. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy. 1. Introduction When arbitrary torsional boundary conditions are applied either at the edges or at any other interior point of a bar due to construction requirements, this bar under the action of general twisting loading is leaded to nonuniform torsion. In this case, apart from the well known primary (St. Venant) shear stress distribution, normal and secondary (warping) shear stresses arise formulating the warping moment (bimoment) and secondary twisting moment (bishear), respectively [1, 2]. The secondary twisting moment deformation effect (STMDE) is associated with the inclusion of warping shear stresses in the global equilibrium of the bar and the performance of an accurate analysis of bars of closed shaped cross sections [3]. This effect generally necessitates the introduction of an independent warping parameter in the kinematical components of the bar (apart from the angle of twist), increasing the difficulty of the problem at hand. Besides, since weight saving is of paramount importance, frequently used thin-walled open sections have low torsional stiffness and their torsional deformations can be of such magnitudes that it is not adequate to treat the angles of cross section rotation as small. Thus, the study of nonlinear effects on these members becomes crucial. When finite twist rotation angles are considered, the induced geometrical nonlinearities result in effects that are not observed in linear systems. In such situations the possibility of an analytical solution method is significantly reduced and is restricted to special cases of boundary conditions or loading. During the past few years, the nonlinear nonuniform torsional dynamic analysis of bars has received a good amount of attention in the literature. Di Egidio et al. in [4-5] presented a FEM solution to the nonlinear flexural-torsional vibrations of thin-walled open beams taking into account in-plane and out-of-plane warpings and neglecting warping inertia. In these papers, the torsional-extensional coupling is taken into account but the axial boundary conditions are not general. Moreover, Simo and Vu-Quoc in [2] presented a FEM solution to a fully nonlinear (small or large strains, hyperelastic material) three dimensional rod model based on a geometrically exact description of the kinematics of deformation. Pai and Nayfeh in [6] studied a geometrically exact nonlinear curved beam model for solid composite rotor blades using the concept of local engineering stress and strain measures and taking into account the in-plane and out-of-plane warpings. In the last two research efforts, the out-of-plane buckling of a framed structure and a helical spring have been analyzed respectively, thus the extensional-torsional coupling is not discussed. Rozmarynowski and


483

Szymczak [7] and Sapountzakis and Tsipiras [8] focus to the problem of nonlinear torsional vibrations. Nevertheless, dynamic analysis of bars at a torsional post-buckled state, as this is presented for the static case in [9], is not performed. Although free or forced vibrations of bars at flexural postbuckling configurations are well studied both numerically and experimentally [10], however this is not the case for buckled bars at a torsional post-buckled state. To the authors’ knowledge, only Mohri et. al. [11] proposed a FEM solution to the linear free vibration analysis of pre- and post-buckled open thin-walled cross section beams subjected to special boundary conditions, neglecting warping inertia. In all of these research efforts the angle of twist per unit length is considered as a warping parameter with the exception of the aforementioned research effort of Simo and Vu-Quoc [2] who employed an independent one. In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the effects of geometrical nonlinearity (finite displacement – small strain theory) and secondary twisting moment deformation. The resulting coupling effect between twisting and axial displacement components is also considered and a constant along the bar compressive axial load is induced so as to numerically examine the dynamic response at the (torsional) post-buckled state. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows. i. The cross section is an arbitrarily shaped doubly symmetric thin or thick walled one. The formulation does not stand on the assumptions of a thin-walled structure and therefore the cross section’s torsional and warping rigidities are evaluated “exactly” in a numerical sense. ii. The adopted numerical procedure can efficiently analyze torsional vibrations of bars at postbuckling configurations without employing any assumptions on the form of the modeshapes of deformation. iii. A BEM approach is employed (requiring boundary discretization exclusively for the cross sectional analysis) resulting in line instead of area elements of the FEM solutions (requiring the whole cross section to be discretized), while a small number of elements are required to achieve high accuracy. 2. Statement of the problem 2.1. Displacements, strains, stresses Let us consider a prismatic bar of length l , of constant arbitrary doubly symmetric cross-section of area A . The homogeneous isotropic and linearly elastic material of the beam cross-section, with modulus of elasticity E , shear modulus G and mass density U occupies the two dimensional multiply connected region

: of the y,z plane and is bounded by the * j j 1,2,...,K boundary curves, which are piecewise smooth,

i.e. they may have a finite number of corners. In Fig. 1a Syz is the principal bending coordinate system through the cross section’s shear center. The bar is subjected to the combined action of the arbitrarily distributed or concentrated time dependent conservative axial load n x,t and twisting mt mt x,t and

warping mw mw x,t moments acting in the x direction. Under the aforementioned loading, the displacement field of the bar accounting for large twisting rotations is assumed to be given as

u x, y,z,t um x,t K x x,t ISP y,z

v x, y,z,t z sinT x x,t y 1 cos T x x,t w x, y,z,t

y sin T x x,t z 1 cos T x x,t

(1a) (1b) (1c)

where u , v , w are the axial and transverse bar displacement components with respect to the Syz system of axes; T x x,t is the angle of twist; ISP is the primary warping function with respect to the shear center S [1];

K x x,t and um x,t denote an independent warping parameter and an “average” axial displacement of the bar’s cross section, respectively, that will be later discussed. Employing the strain-displacement relations of the three - dimensional elasticity, exploiting the assumptions of moderate displacements and employing eqns (1), the nonvanishing strain resultants are obtained as

484

H xx


umc K xc ISP

1 2 y 2 z 2 T xc 2

J xy K xc

wISP T xc z wy

J xz K xc

wISP T xc y wz

(2a,b,c)

Considering strains to be small and employing the second Piola – Kirchhoff stress tensor, the work contributing stress components are defined in terms of the strain ones as S xx

S xz

GJ xz , where E is obtained from Hooke’s stress-strain law as E *

*

E H xx ,

S xy

GJ xy ,

E 1 Q ª¬ 1 Q 1 2Q º¼ . E is

frequently considered instead of E ( E | E ) in beam formulations [12] and is adopted in the present study as well. By exploiting eqns (2), the stress resultants are obtained as S xx

1 2º ª E «umc K xc ISP y 2 z 2 T xc » 2 ¬ ¼

S xy

P S S xy S xy

S xz

P S S xz S xz

(3a,b,c)

where P S xy

§ wI P · GT xc ¨ S z ¸ ¨ wy ¸ © ¹

P S xz

§ wI P · GT xc ¨ S y ¸ ¨ wz ¸ © ¹

(4a, b)

denote the well known primary (St. Venant) shear stress distribution accounting for uniform torsion [13] and S S xy

G K x T xc

wISP wy

S S xz

G K x T xc

wISP wz

(5a, b)

denote the secondary (warping) shear stress distribution accounting for nonuniform torsion. 2.2. Primary warping function ISP , “average” axial displacement um The primary warping function is evaluated independently by exploiting local equilibrium considerations along the longitudinal x axis from the solution of the following boundary value problem [1]

2ISP

0 in :

wISP wn

zn y ynz on * j

(6a,b)

where 2 w 2 / wy 2 w 2 / wz 2 is the Laplace operator and w / wn denotes the directional derivative normal to the boundary * . Since the problem at hand has Neumann type boundary condition, the evaluated warping function contains an integration constant, which is resolved by inducing the constraint

³ IS d: P

0 leading

:

to the fact that the primary warping function does not influence the axial stress resultant and that um represents an average axial displacement of the bar’s cross section 2.3 Warping shear stress distribution, independent warping parameter K x

By substituting eqns (3-5) on the differential equation describing local equilibrium along the longitudinal axis x and the associated boundary condition [8], it is easily concluded that these equations can not be satisfied. Moreover, a warping shear stress distribution including a secondary warping function ISS has been proved not to violate both the aforementioned equilibrium equation and the associated boundary condition, as proposed in [14]. Therefore, employing eqns (5) to obtain accurate values of warping shear stresses is of doubtful validity. Nevertheless, the present formulation makes it possible to accurately analyze bars of either


485

closed or open shaped cross sections and account for warping shear stresses in global equilibrium, which has not been achieved in previous research efforts [14]. 2.4. Equations of global equilibrium

To establish global equilibrium equations, the principle of virtual work

³ S xxGH xx S xyGJ xy S xzGJ xz dV ³ U uG u vG v wG w dV ³ t xG u t yG v t zG w dF

V

V

(7)

F

under a Total Lagrangian formulation, is employed. In the above equations, G denotes virtual quantities,

denotes differentiation with respect to time, V , F are the volume and the surface of the bar,

respectively, at the initial configuration and t x , t y , t z are the components of the traction vector with respect to the undeformed surface of the bar. Performing the decomposition of shear strains into primary and secondary parts, as it is described for shear stresses in eqns (3b-c, 4, 5), the contribution of shear stresses in the virtual work of internal forces can be written after some algebraic manipulations as l

I1

³

x 0

ª M tPGT xc M tS GK x GT xc º dx ¬ ¼

(8)

where M tP , M tS are the primary and secondary twisting moments, respectively [1], defined here as

M tP

ª

· § P ·º P wIS z ¸ S xz y ¸ » d: ¨¨ ¸ ¸ © wy ¹ © wz ¹ »¼ § wISP

³ «« S xy ¨¨

:

P

¬

M tS

P § S wISP S wIS ³ ¨ S xy S xz ¨ wy wz :©

· ¸¸ d: ¹

(9a,b)

Substituting eqns (4-5) into eqns (9), the above stress resultants are given (with respect to the kinematical components) as M tP

GI tT xc

M tS

GI tS K x T xc

(10a,b)

where It , I tS are the primary [1] and secondary [15] torsion constants, respectively given here as It

§

³ ¨¨ y

:©

2

z2 y

wISP wI P z S wz wy

· ¸¸ d: ¹

I tS

§ wI P wI P AT ³ ¨ y S z S ¨ wz wy :©

· ¸¸ d: ¹

(11a,b)

with AT defined as the “effective shear area due to the restrained torsional warping”. Below, it is assumed that AT 1 , which evidently leads to the relation I t I P I tS ( I P is the polar moment of inertia [8]). Substituting the stress resultants given in eqns (3), the strain ones given in eqns (2) and the displacement components given in eqns (1) to the principle of virtual work (eqn (7)), the governing partial differential equations of the initial boundary value problem of the bar are obtained after some algebra as

m EAumcc EI PT xcT xcc n x,t U Au

3 2 U I P T x G I t I tS T xcc GI tSK xc EI PP T xc T xcc EI P umc T xcc EI P umcc T xc 2 U CS K x ECSK xcc GI tS K x T xc mw x,t

(12a) mt x,t

(12b) (12c)

486


subjected to the initial conditions ( x 0,l )

um x,0 um0 x

T x x,0 T x0 x

um x,0 um0 x T x,0 T x x

x0

(13a,b)

K x x,0 K x0 x

K x x,0 K x0 x

(13c,d,e,f)

together with the boundary conditions at the bar ends x 0,l

a1 N D 2 um

D3

E1 M t E 2T x

E3

E1 M w E 2K x

E3

(14a,b,c)

where N , M t , M w are the axial force, twisting and warping moments at the bar ends, respectively given as N

EAumc

1 2 EI P T xc 2

Mt

G I t I tS T xc GI tSK x EI P umc T xc

1 3 EI PP T xc 2

Mw

ECSK xc (15a,b,c)

while ai , Ei , Ei ( i 1,2,3 ) are time dependent functions specified at the boundary of the bar, CS is the warping constant with respect to the shear center S [1] and I PP , appearing in eqns (12b), (15b), is a geometric cross sectional property given as I PP

³ y

:

2

z2

2

d : . The boundary conditions (14) are the

most general boundary conditions for the problem at hand, including also the elastic support. It is worth here noting that the expressions of the externally applied loads with respect to the components of the traction vector can be easily deduced by virtue of the right hand side of eqn (7). It is pointed out that all the relations established so far are completely analogous to those of the Timoshenko beam theory, modeling the shear - bending loading conditions of bars. A significant reduction on the set of differential equations can be achieved by neglecting the axial inertia term of eqn (12a), a common assumption among dynamic beam formulations. Ignoring this term, two partial differential equations with respect to T x x,t , K x x,t can be obtained, which are further combined (by performing similar algebraic manipulations with those presented in [16]), leading the initial boundary value problem to a single partial differential equation with respect to T x x,t . After neglecting the higher order term

U CS U I P Tx

GI , this equation is written for the case of constant along the bar axial load as S t

§ 1 EI · 2 EI I 2 U I P T x U CS ¨ PS PS N ¸ T cc U CS nS ª«3 T xc T xcc 6T xcT xccT xc 3TxcT xcT xcc 3Txc T xc º» ¨ N GI ¸ x ¬ ¼ GI A GI t t t © ¹

§1 · ECS ª I I 3 3 2 3 2 § · º 3 T xcc 9T xcT xccT xccc T xc T xcccc» ECS ¨ PS N ¸T xcccc ¨ GI t P N ¸T xcc EI n T xc T xcc EI n S « ¨ N GI A ¸ A 2 2 GI ¬ ¼ © ¹ t t © ¹ U CS ECS mt S mtcc mt mwc GI tS GI t

(16)

where I n is a nonnegative geometric cross sectional property related to the geometrical nonlinearity, given as I n I PP I P 2 A . Eqn (16) must satisfy the pertinent initial conditions (13c, d) and boundary conditions (14b, c), where the independent warping parameter K x and the twisting and warping moments M t , M w are given (at the bar ends x 0,l ) as

K x T xc

· ECS § 1 ECS EI n ª I 3 2 2 º ¨ P N ¸T xccc « 3T xc T xcc 2 T xc T xccc» GI tS ¨© N GI tS A ¸¹ GI tS GI tS ¬ ¼

(17a)


487

§1 · EI n ª IP · IP 1 3 § 3 2 2 º ¨ GI t A N ¸T xc ECS ¨¨ N S N ¸¸T xccc 2 EI n T xc ECS «3T xc T xcc 2 T xc T xccc» GI t A ¹ GI tS ¬ ¼ © ¹ ©

Mt

(17b)

§1 · EI n ª 3 I 2 º ECS ¨ PS N ¸T xcc ECS « T xc T xcc » ¨ N GI A ¸ GI tS ¬ 2 ¼ t © ¹

Mw

(17c)

In eqns (16-17) N is an auxiliary geometric constant related to the STMDE given as N

I tS

I

t

I tS .

Comparing the formulated reduced initial boundary value problem and the one presented in [8] where the STMDE is not taken into account, it is concluded that this effect alters the expressions of warping inertia, warping stiffness and external loading and induces higher order nonlinear inertia and stiffness terms in the governing partial differential equation. Some nonlinear stiffness terms are also induced in the kinematical and stress components at the bar ends.

3. Integral Representations – Numerical Solution According to the precedent analysis, the nonlinear nonuniform torsional vibration problem of shear deformable bars reduces to establishing the displacement component T x x,t having continuous partial derivatives up to the fourth order with respect to x and up to the second order with respect to t , satisfying the nonlinear initial boundary value problem described by the governing equation (16), the initial conditions (13c, d) along the bar and the boundary conditions (14b, c) at the bar ends x 0,l . This problem is solved using the Analog Equation Method [17], as this is developed for hyperbolic differential equations in [18].

4. Numerical example 2,1 u 10 8 kN / m 2 , G

An open thin-walled I-shaped cross section bar ( E

U

2

4

8,002kN sec / m ) of length l

4,0m , having flange and web width t f

tw

8,1 u 107 kN / m 2 ,

0,01m , total height and

total width h b 0,20m has been studied. The geometric constants of the bar are computed, employing A 5,800 u 10 -3 m 2 , I P

400 constant boundary elements, as -7

6

-7

4

, I tS

-4

5,434 u 10 -5 m4 , I n

4

-7

1,631 u 10 -7 m6 ,

6

I PP 6 ,722 u 10 m , I t 2,080 u 10 m 5,413 u 10 m , CS 1,200 u 10 m . The bar’s ends are simply supported according to its torsional boundary conditions, while the left end is immovable and the right end is subjected to a compressive axial load according to its axial boundary conditions. The numerical results have been obtained by employing 21 nodal points along the bar length. Load - frequency relation Present study - AEM Mohri et. al. [11] - Analytical expression

350

Angle of twist at x=l/2 linear fundamental modeshape nonlinear fundamental modeshape

1.44

300

1.43 250

angle of twist (rad)

fundam. natural frequency (sec-1)

400

200 150 100 50 0

1.42 1.41 1.4 1.39 1.38 0

0.01

1.37

0.02

0.03

0.04

0.05

0.06

0.07

0.08

t(sec)

1.36 1.35

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

(a) 1.34 (b) relation [11] along with obtained pairs of values (a) and time histories of the angle of twist at Axial load (kN)

Fig. 1. N - Z f

the midpoint of the free vibrating bar for a postbuckling ( N

5000kN ) axial loading (b).

488


In Fig. 1a, the load - frequency relations given in [11] of the bar undergoing small amplitude torsional free vibrations both in the pre- and post-buckling region are presented along with pairs of values ( Z f , N ) obtained from the proposed method by employing the linear fundamental modeshape of the angle of twist as

initial twisting rotations T x0 x (along with zero initial twisting velocities T x0 x ) [8] and by ignoring the STMDE. From Fig. 1a, the validity of the proposed method is concluded. Moreover, in Fig. 1b the obtained time history of the angle of twist T x l / 2,t at the midpoint of the bar for a postbuckling ( N 5000kN ) axial loading is presented, demonstrating that the buckled bar undergoes multifrequency vibrations. For comparison purposes, in Fig. 1b the time history of T x l / 2,t employing the nonlinear fundamental modeshape of the angle of twist as initial twisting rotations T x0 x (along with zero initial twisting

velocities T x0 x ) is also included ( N 5000kN ), showing that the initiation of free vibrations with the nonlinear modeshape does not induce higher harmonics in the response of the bar. In Fig. 2 the time histories of the angle of twist T x l / 2,t at the midpoint of the bar, for two cases of the initial midpoint angle of twist amplitude ( T x0 l / 2 2,20 rad , T x0 l / 2 2,50 rad ) are presented taking into account or ignoring the STMDE. It is concluded that the positions around which vibrations are performed depend on the initial amplitude of vibration, while for the larger value of T x0 l / 2 they

coincide with the static equilibrium position of the prebuckling configuration. The initial amplitude of vibration affects also the natural frequency the bar. Finally, from Fig. 2 it is also observed that the STMDE does not influence the kinematical components of buckled bars of open thin-walled cross sections, undergoing large amplitude vibrations. This conclusion does not depend on the magnitude of the initial midpoint angle of twist amplitude. 3 2.5 2

Angle of twist (rad)

1.5 1 0.5 0 -0.5

0

0.02

-1

0.04

0.06

t(sec)

0.08

0.1

0.12

Angle of twist at x=l/2 initial midpoint angle of twist amplitude = 2.20rad - with STMDE initial midpoint angle of twist amplitude = 2.20rad - without STMDE initial midpoint angle of twist amplitude = 2.50rad - with STMDE initial midpoint angle of twist amplitude = 2.50rad - without STMDE

-1.5 -2 -2.5 -3

Fig. 2. Time histories of the angle of twist at the midpoint of the free vibrating bar taking into account or ignoring the STMDE ( T x0 l / 2 2,20 rad , T x0 l / 2 2,50 rad ). 5. Concluding remarks

The main conclusions that can be drawn from this investigation are a. The numerical technique presented in this investigation is well suited for computer aided analysis of cylindrical bars of arbitrarily shaped doubly symmetric cross section, supported by the most general twisting and warping boundary conditions and subjected to the combined action of arbitrarily distributed or concentrated time dependent conservative axial and torsional loading. b. The geometrical nonlinearity leads to coupling between the torsional and axial equilibrium equations and alters the modeshapes of vibration. Consequently, the initiation of small amplitude free vibrations of buckled bars with the linear fundamental modeshape as initial twisting rotations induces higher harmonics in the bar’s response, but its fundamental natural frequency is only slightly affected.


489

c. Large twisting rotations have a profound effect on both the positions around which vibrations are performed and the fundamental natural frequency of buckled bars undergoing large amplitude free vibrations. For large initial amplitudes of vibration, the positions around which vibrations are performed coincide with the static equilibrium position of the prebuckling configuration. d. The secondary twisting moment deformation affects negligibly the kinematical components of buckled bars of open shaped thin-walled cross sections undergoing free vibrations of small or large amplitude. e. The developed procedure retains most of the advantages of a BEM solution over a FEM approach, although it requires longitudinal domain discretization. Acknowledgements

The authors would like to thank the Senator Committee of Basic Research of the National Technical University of Athens, Programme “PEVE-2008”, R.C. No: 65 for the financial support of this work. References

[1] E.J.Sapountzakis and V.G.Mokos, Warping shear stresses in nonuniform torsion by BEM, Computational Mechanics, 30, 131-142 (2003). [2] J.C.Simo, and L.Vu-Quoc A Geometrically-exact rod model incorporating shear and torsion-warping deformation, International Journal of Solids and Structures, 27, 371-393 (1991). [3] J.Murín and V. Kutis An effective finite element for torsion of constant cross-sections including warping with secondary torsion moment deformation effect, Engineering Structures, 30, 2716-2723 (2008). [4] A.Di Egidio, A. Luongo and F. Vestroni A non-linear model for the dynamics of open cross-section hinwalled beams--Part I: formulation, International Journal of Non-Linear Mechanics, 38, 1067-1081 (2003). [5] A.Di Egidio, A. Luongo and F. Vestroni A non-linear model for the dynamics of open cross-section thin-walled beams--Part II: forced motion, International Journal of Non-Linear Mechanics, 38, 10831094 (2003). [6] P.F. Pai and A.H. Nayfeh A fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three-dimensional stress effects, International Journal of Solids and Structures 31, 1309-1340 (1994). [7] B.Rozmarynowski and C.Szymczak Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section, Journal of Sound and Vibration, 97, 145-152 (1984). [8] E.J.Sapountzakis and V.J.Tsipiras Nonlinear nonuniform torsional vibrations of bars by the boundary element method, Journal of Sound and Vibration in press. [9] C.Szymczak Buckling and initial post-buckling behavior of thin-walled I columns, Computers & Structures, 11, 481-487 (1980). [10] S.A.Emam and A.H.Nayfeh Postbuckling and free vibrations of composite beams, Composite Structures, 88, 636-642 (2009). [11] F.Mohri, L.Azrar and M.Potier-Ferry Vibration analysis of buckled thin-walled beams with open sections, Journal of Sound and Vibration, 275, 434-446 (2004). [12] V. Vlasov Thin-walled elastic beams, Israel Program for Scientific Translations, Jerusalem, (1963). [13] E.J.Sapountzakis Torsional vibrations of composite bars by BEM, Composite Structures, 70, 229-239 (2005). [14] E.J.Sapountzakis and V.J.Tsipiras Warping shear stresses in nonlinear nonuniform torsional vibrations of bars by BEM, Engineering Structures, 32, 741-752 (2010). [15] E.J. Sapountzakis and V.G. Mokos Secondary torsional moment deformation effect by BEM, Proceedings of the 10th international conference of Advances in Boundary Element Techniques, Athens, Greece, 81-88 (2009). [16] E.J Sapountzakis and J.A. Dourakopoulos Flexural-torsional buckling analysis of composite beams by BEM including shear deformation effect, Mechanics Research Communications, 35, 497-516 (2008). [17] J.T. Katsikadelis The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies, Theoretical and Applied Mechanics, 27, 13-38 (2002). [18] E.J. Sapountzakis and J.T.Katsikadelis Elastic deformation of ribbed plates under static, transverse and inplane loading, Computers & Structures, 74, 571-581 (2000).

490


Harmonic Analysis of Spatial Assembled Plate Structures Coupled with Acoustic Fluids Using the Boundary Element Method Useche, J.a , Albuquerque, E.L.b , Shoefel, S.b a Universidad Tecnologica ´ de Bol´ıvar Cartagena de Indias, BOL, Colombia

[email protected] b University

of Campinas - UNICAMP Faculty of Mechanical Engineering 13083-970, Brasilia, SP, Brazil {eder}@unb.br

Keywords: Boundary Element Method, Acoustic Fluid, Thin-Walled Structures, Shear Deformable Plates, Dynamic response, Coupled Fluid-Structure Interaction.

Abstract In this work, a full boundary element formulation for the harmonic analysis of spatial thin-walled structures coupled to internal acoustic fluid is presented. The thin-walled structure is modeled using a boundary element approach where Reissner plate and the plane elasticity formulations are coupled. Three dimensional plate structures are modeled using a spatial multi-region formulation where compatibility equations of displacements and equilibrium equations of tractions are applied to degrees of freedom along the interface of the plates. On the other hand, the acoustic fluid is modeled using a boundary element formulation for the Helmholtz equation applied to internal acoustic fluids. Domain integrals for both, fluid and structure equations, were treated using the dual reciprocity method. Results obtained in numerical examples are in good agreement with finite element results, showing the accuracy and efficiency of the proposed boundary element formulation for dynamic analysis of spatial thin-plate structures coupled to internal acoustic fluids.

1

Introduction

Acoustic fluid-structure interaction analysis (FSI) is important in most engineering fields involving problems where dynamic analysis of structures can not be conceived without regard to its interaction with contained or external fluids. Due to the complexity of fluid-structure interaction phenomena, finding mathematical closed solutions to the governing equations represents a cumbersome job. Numerical models arises as alternative approaches to the problem solution. Traditionally, these models have been based on finite element method (FEM) [5]. In these models, the structure is modeled using plate or shell models based on the classical thin plate theory or the shear deformable thick plate theory, while fluid is modeled using the Helmholtz equation for acoustic fluids [4]. Many works and different types of formulations have been proposed and published in the literature using these approaches. Since consolidation of the boundary element method (BEM) as reliable numerical method for structural and fluid analysis, the FSI analysis of problems has been carried out using hybrid formulations based on the BEM to model the acoustic or fluid media and the FEM to model the structural response [9]. The main advantage of such formulations lies in a substantial reduction in the number of the degrees of freedom in the discretization of the fluid domain, since BEM only requires the contour domain discretization. However, in


491

this formulations is necessary to discretize the entirely structure due to the use of the FEM. The boundary element analysis of spatial plate structures using the thick plate theory was first presented by Dirgantara and Aliabadi [2]. In this work, the concept of associated plates in space was used to model spatial structures under static loads. In Palermo [6], the harmonic response of spatial structures using a frequency domain BEM for shear deformable plates, is developed. However, despite the fact that the BEM have been used for the dynamic-analysis of spatial structures and for the analysis of acoustic fluids, to the best of authors knowledge, these formulations do not have been used for the FSI analysis using a full boundary element formulation for such purpose. In this work, a new full boundary element formulation for the harmonic analysis of spatial thin-walled structures coupled to acoustic fluid is presented. The structure is modeled using a boundary element formulation where shear deformable plate and plane elasticity are coupled. Three dimensional associations are carried out using a spatial multi-region formulation where compatibility equations of displacements and equilibrium equations are applied to degrees of freedom along the interface of the plates. On the other hand, the acoustic fluid is modeled using a boundary element formulation for the acoustic wave equation. Fluidstructure coupling equations were established considering the continuity of the normal acceleration of the particles at fluid-structure interfaces. Domain integrals on both, fluid and structure equations, were treated using the dual reciprocity boundary element method - DRM. The developed formulation was used to study the linear vibration response of spatial structures coupled to internal acoustic fluids.

2

Boundary element formulation for acoustic wave equation

The dynamic pressure of an ideal inviscid fluid under small perturbations in a spatial region Ω f confined by the boundary surface Γ f , is governed by the wave equation [3] given by: p,αα + s =

1 p¨ c2

(1)

where p is the fluid pressure, c stands for wave propagation velocity, s represents a source term and p¨ = ∂ 2 p/∂ t 2 . Indicial notation is used throughout this work. Greek indices vary from 1 to 2 and Roman indices from 1 to 3. The boundary integral formulation for this equation, without considering source term, can be written as [7]: k(x )p(x ) +

Γf

Q∗ (x , x)p(x)dΓ f =

Γf

P∗ (x , x)q(x)dΓ f +

1 c2

Ωf

P∗ (x , x) p(x)dΩ ¨ f

(2)

where x and x are field and collocation points, respectively, q = ∂ p/∂ n n is the outward normal vector at boundary coefficient k is a function of the geometry at the collocation points P∗ (x , x) and Q∗ (x , x) = ∂ P∗ /∂ n are fundamentals solutions for pressure and gradient pressures for three dimensional acoustic problems. In order to threat the domain integral, the dual reciprocity boundary element method (DRM) was used. In this way, equation (2) can be re-written as: k(x )(x ) + +

Γf

Q∗ (x , x)p(x)dΓ f =

Γf

P∗ (x , x)q(x)dΓ f

1 NDRM αm (t) ci Pîm + Q∗ (x , x)Pˆm dΓ f − P∗ (x , x)Qˆ m dΓ f ∑ 2 c m=1 Γf Γ¨ f

(3)

where NDRM represents the number of total DRM collocations points; Pˆm (x , x) and Qˆ m (x , x) represent particular solutions for the Laplace equation; coefficients α¨ m are related to p¨ through: p¨ = Fim α¨ m (t). In

492


Figure 1: Fluid-structure interaction problem this work, approximation functions fm = 1 + rm were used. In order to discretize boundary surfaces of the acoustic medium, N boundary quadrilateral elements were used and p(x) and q(x) were assumed to be constant over each element and equal to their values at the mid-element node. In this way, equation (4) can be written in compact matrix form as: f Mp¨ + f Hp = f Gq (4) where f M is the fluid mass matrix, p and q are vectors of nodal pressures and normal derivative of pressure, respectively, at the field points.

3

Boundary element formulation for plate structures

In this work, spatial plate structures were modeled using the multi-region formulation proposed as presented by [2]. In this formulation, the structure is divided into several regions and equilibrium and compatibility equations along the interface edges are imposed. In each region, formulation is formed by coupling the boundary element formulation of shear deformable plate bending and the plane stress elasticity formulation.

3.1 Boundary element formulation for in-plane stress Now consider an elastic plate of thickness h occupying the area Ωs , bounded by the contour Γs , in the x1 x2 plane. Governing equations for the dynamic response considering plane stress conditions are given by [1]: Guα ,β β +

G u + bα = ρs u¨α 1 − 2ν β ,β α

(5)

where G is the shear modulus, ν is Poisson ratio and uα represents in-plane displacements, ρs is the mass density, u¨α represents in-plane accelerations, and bα are in-plane body forces. The boundary integral formulation for equation (5) is given by: cαβ u(x ) +

Γs

∗ Tαβ (x , x)uβ (x)dΓs =

+

Γs

ρs h

∗ Uαβ (x , x)tβ (x)dΓs +

Ωs

1 h

∗ Uαβ (x , x)u¨β (x)dΩs

Ωs

∗ Uαβ (x , x)bβ (x)dΩs

(6)

∗ (x , x) and U ∗ (x , x) are the fundamental solutions for plane stress elasticity, respectively [1]; where Tαβ αβ tβ is the traction vector at boundary and cαβ (x ) is a constant that depends on the geometry at collocation


493

point. In order to threat the integral domain containing inertia terms in this equation, the DRM was used. In this way, equation (7) can be written as [7]: cαβ u(x ) + +

Γp

∗ Tαβ (x , x)uβ (x)dΓs =

ρs DRMS m ∑ α¨ l (t)[cαβ Uˆ lmβ − h m=1

Γp

∗ Uαβ (x , x)tβ (x)dΓs +

∗ Uαβ (x , x)Tˆlmβ (x)dΓ p +

Γp

1 h

Γp

Γp

Zi,∗α (x , x)nα (x)bβ dΓs

∗ Tαβ (x , x)Uˆ lmβ (x)dΓ p ]

(7)

In this equation, NDRMS represents the number of total DRM collocations points used in the plate; Uˆ lmβ (x) and Tˆlmβ (x) are the particular solutions to equivalent homogeneous equation considering approximation functions fm = 1 + rm . Coefficients α¨ ml (t) are related to u¨β through: u¨β = Fil α¨ ml (t). Applying the boundary element method to this equation, considering discontinuous quadratic elements to discretize de boundary of the plate, we obtain: s Mu¨ + s Hu = s Gt + s Bb (8) 3.2 Boundary element formulation for plate bending The dynamic bending response for the plate was modeling using the classical Reissner plate theory. In this way, governing equations considering rotary inertia and distributed shear forces, f3 and moments, fα , are given by [6]: D

1−ν 2ν ρs h3 vγ ,γ δαβ ) −C(vα + v3,α ) + fα = (vα ,β β + vβ ,αβ + v¨α 2 1−ν 12 C(vα ,α + v3,αα ) + f3 = ρs hv¨3

(9) (10)

In these equations, vα and v¨α denote rotations and angular accelerations about x1 and x2 axis, respectively; v3 and v¨3 represent transverse deflection and transverse acceleration, respectively. D = Eh3 /12(1 − v2 ) and C = D(1 − ν )λ 2 are the bending and shear stiffness of the plate, respectively; λ 2 = 10/h is called the shear factor. The boundary integral formulation for equations (10) and (11) is given by: cikp vk (x ) +

Pik∗ (x , x)vk (x)dΓs =

Γs

Γs

+

Ωs

Vik∗ (x , x)sk (x)dΓs +

Ωs

Vik∗ (x , x) fk (x)dΩs

Vik∗ (x , x)Λik v¨k (x)dΩs

(11)

In this equation, Vik∗ and Pik∗ are the fundamental solutions for shear deformable plates; sα represents inplane moments and s3 is the shear force; Λik tensor is defined as: Λα ,β = 1/12ρs h3 δαβ and Λ33 = ρs h. DRM was used to transform domain integrals related to inertial terms into boundary integrals. In this work, only constant distributed pressure and moments are considered as external loads acting on the plate surface. The first domain integral at the right hand side of equation (11) was transformed into the boundary integral using the divergence theorem [8]. In this way, this equation can be re-written as: cik vk (x ) + +

Pik∗ (x , x)vk (x)dΓs =

Γs DRMS

∑

m=1

β¨lm [cikVˆlkm −

Γs

Γs

Vik∗ (x , x)sk (x)dΓs + fk

Vik∗ (x , x)Pˆlkm dΓs +

Γs

Γs

Zi,∗α (x , x)nα (x)dΓs

Pik∗ (x , x)Vˆlkm dΓs ]

(12)

In this equation Vˆlkm and Pˆlkm are the particular solutions to the equivalent homogeneous equations (11), 3 /9 for the approximation of angular velocities and the function considering the function fm = 1 − λ 2 rm

494


p(t) = P.Sin(wot) acoustic fluid

c plate structure

x3 x2

x1

b

a

rigid wall at bottom

clamped edges

Figure 2: Fluid-structure interaction problem fm = 1 + rm for the approximation of the transverse accelerations. Zi∗ are the particular solutions of the equation Zi,∗θ θ = Vi3∗ ; nα is the normal vector to boundary at field point x. Coefficients β¨lm are related ¨ k through expression: Λm ¨ k = Film β¨lm . Applying the boundary element method to equation (12), to Λm ik w ik w considering discontinuous quadratic elements to discretize de boundary of the plate, we obtain: p

M¨v + p Hv = p Gs + p Bf

(13)

Finally, the complete equations for a plate, considering in-plane and bending response is obtained through equations (8) and (13), in matrix form, as: pM pH pG pB v s f 0 0 0 0 v¨ = + (14) + u t b 0 sM 0 sH 0 sG 0 sB u¨ In order to model spatial structures, they are divide into several regions with at least one region for each plate component, as proposed by [2]. For each region, boundary element influence matrices, given by equation (14) in their local coordinate system, are assembled. Plates are connected through common boundaries and the overall system matrices are formed by enforcing along the interface boundary of the plate, compatibility equations of displacements and equilibrium equations of tractions are satisfied. In general, equations for the dynamic response of spatial structure assembled with n-plates can be written as: s

¨ + s Hw = s Gr + s Bz Mw

(15)

¨ 2 ...,w ¨ n }, w = {w1 , w2 . . . , wn }, r = {r1 , r2 . . . , rn } and z = {z1 , z2 . . . , zn }, ¨ = {w ¨ 1, w In this equation, w ¨ n = {¨v, u} ¨ n , wn = {v, u}n , rn = {s, t}n , and zn = {f, b}n where for the n-plate: w

4

Fluid-structure coupling equations

Fluid-structure coupling equations are given by compatibility considerations about normal pressure and dynamic pressure force acting at fluid-structure interface. Mathematically, these conditions can be written


495

−5

10

FEM Model BEM Model

−6

10

−7

Log10(U)

10

−8

10

−9

10

−10

10

−11

10

−12

10

0

5

10

15 20 25 Frequency [Hertz]

30

35

40

Figure 3: Amplitude of displacement at point x = 0.5, y = 1.0, z = 1.0 on structure as [5]: ¨ n.∇p = q ≡ −ρs Cw w p ≡ Cz z

(16)

That is, pressure gradient acting on the fluid-structure interface Γ f s (see figure 1) are related to normal acceleration of the plate and the acoustic pressure is equilibrated with pressure on plates in contact with the acoustic fluid. In these equations, Cw and Cz represents connectivity matrix joining fluid and structural degree of freedom at fluid-structure interfaces. Finally, replacing equations (16) into equations (4) and (15) we obtain the coupled fluid-structure equation problem (without considering external distributed forces acting on the structure): sM s H −s B C−1 sG ¨ 0 0 w w r fs z = (17) + fM fG ρ C fH 0 0 fG p q p¨ fs s w where s B f s and f G f s are sub-matrices obtained from s B and f G, respectively. In order to perform harmonic analysis, small linear and harmonic responses are considered in this work.

5

Numerical results

In order to verify the proposed formulation, the harmonic response of box-shaped structure containing an acoustic fluid is analyzed (see figure 2). Structure have dimensions a = b = 1.0 m, c = 2.0 m, thickness t = 0.015 m, Young’s modulus E = 210 GPa, Poisson’s ratio v = 0.33, and density ρs = 7850 kg/m3 . The base of the structure is campled to a rigid wall. The acoustic fluid have ρ f = 1000 kg/m3 and c = 1450 m/s. A distributed time harmonic pressure with amplitude of Pmax = 1.0 Pa was applied to the acoustic fluid at the free surface as showed in figure 2. Plates were discretized using 12 quadratic discontinuous boundary elements at each borders and 32 DRM collocation points at each plate. Similarly, 32 constant

496


2

10

FEM Model BEM Model 1

10

0

Log(P)

10

−1

10

−2

10

−3

10

−4

10

0

5

10

15 20 25 Frequency [Hertz]

30

35

40

Figure 4: Amplitude of pressure at point x = 0.5, y = 0.5, z = 1.0 on fluid

surface elements, rectangular geometry were used at each fluid boundary surfaces. Collocation points are coincident with those DRM plate collocation points. Furthermore, 64 DRM collocation points were used at the fluid domain. In order to check the validity of the formulation, a finite element analysis was carried out using the fluid-structure analysis capabilities offered by the commercial software ANSYS 10.0. The FEA model was built using the FLUID30 element defined by eight nodes and three degree of freedom at each node to model the contained fluid. Structure was modeled using the SHELL93 element. The structure was meshed into 200 shell elements and the contained fluid was discretized using 500 fluid elements. Figure 3 shows the spectral response in frequency domain for amplitude of displacement at point with coordinates x = 0.5, y = 1.0, z = 1.0 on the structure. A good correlation for the spectral response when compared with the FEM solution is found. In a similar way, figure 4 shows the spectral response in frequency domain for amplitude of pressure at a point with coordinates x = 0.5, y = 0.5, z = 1.0 on the structure. Again, good correlation with the spectral response when compared with the FEM solution is found.

6

Conclusions

A full boundary element formulation for the harmonic analysis of spatial structures coupled with acoustic fluids was presented. The boundary element formulation for the structure was based on the shear deformable plate theory using the spatial assembled plate approach. BEM formulation for fluid was developed based on the acoustic fluid equation. In both cases, static fundamental solutions were used. In order to couple the structural response with fluid response, coupling equations were proposed. DRBEM was used to threat domain integrals of the BEM raising from inertial terms. Results obtained show good correlation when compared with those obtained by the FEM.


497

References [1] C. A. Brebbia, J. Dominguez. Boundary Elements: An Introductory Course. WIT Press, Southampton, 1993. [2] T. Dirgantara, M. H. Aliabadi. Boundary element method analysis of assembled plate-structures. Communications in Numerical Methods in Engineering, 17:749–760, 2001. [3] J. Dominguez. Boundary Elements in Dynamics. Computational Mechanics, London, 1993. [4] S. Kirkup. Fluid structure interaction Integrating Sound Software, London, 2007. [5] H. P. -P Morand, R. Ohayon. Fluid structure interaction. John Wiley and Sons, New York, 1995. [6] L. Palermo. On the harmonic response of plates with the shear deformation effect using the elastodynamics in the boundary element method Engineering Analysis with Boundary Elements, 31(2):176186, 2007. [7] P. W. Partridge, C. .A. Bebbia, L. C. Wrobel. The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, London, 1992. [8] Y. F. Rashed. Boundary Element Formulations for Thick Plates Topics in Engineering Vol 35. WIT Press, Southampton, 2000. [9] A. Siamak, J. H. Paul, D. T. Wilton, Coupled Boundary and Finite Element Method for the Solution of the Dynamic Fluid-Structure Interaction Problem. Springer, New York, 1992.

498


On the Numerical Analysis of Damage Phenomena in Saturated Porous Media Eduardo T Lima Junior1, Wilson S Venturini1 and Ahmed Benallal2 1

2

São Carlos School of Engineering, University of São Paulo – São Carlos, Brazil [email protected], [email protected]

Laboratoire de Mécanique et Technologie, ENS de Cachan/CNRS – Cachan, France [email protected]

Keywords: porous media, damage mechanics, consolidation, BEM.

Abstract. This work is devoted to the numerical analysis of saturated porous media, taking into account the damage phenomena on the solid skeleton. The porous media is taken into poro-elastic framework, in full-saturated condition, based on the Biot’s Theory. A scalar damage model is assumed for this analysis. An implicit boundary element method (BEM) formulation, based on time-independent fundamental solutions, is developed and implemented to couple the fluid flow and two-dimensional elastostatic problems. The integration over boundary elements is evaluated by using a numerical Gauss procedure. A semi-analytical scheme for the case of triangular domain cells is followed to carry out the relevant domain integrals. The non-linear problem is solved by a Newton-Raphson procedure. Numerical examples are presented, in order to validate the implemented formulation and to illustrate its efficiency.

Introduction The study of porous materials is extremely relevant in several areas of knowledge, such as soil and rock mechanics, contaminant diffusion, biomechanics and petroleum engineering. The mechanics of porous media deals with materials where the mechanical behavior is significantly influenced by the presence of fluid phases. The response of the material is highly dependent on the fluids that flow through the pores. Biot [1] was the first to propose a coupled theory for three-dimensional consolidation, based on the Terzaghi’s studies on soil settlement [2]. This thermodynamically consistent theory is described in the book by Coussy [3], who improved significantly the knowledge on poromechanics. Cleary [4] presented the fundamental solutions to porous solids, representing the first contributions on integral equations dedicated to this kind of problems. Among others pioneers BEM works applied to porous media, the ones from Cheng and his collaborators [5-7] are well-known, using the direct BEM formulation. In the field of material mechanics, we note the modelling of nonlinear physical processes, as damage and fracture. Processes of energy dissipation and consequent softening have been extensively studied, so that one can count on a wide range of models already developed. Continuum Damage Mechanics (CDM) deals with the load carrying capacity of solids whose material is damaged due to the presence of micro-cracks and micro-voids. CDM was originally conceived by Kachanov [8], to analyze uniaxial creeping of metals subjected to high-order temperatures. Several authors studied and developed models related to CDM. Lemaitre and colleagues [9-10] contributed significantly to the field. In this work, we use the model of Marigo [11], who presented a scalar isotropic model for brittle and quasi-brittle materials. The first applications of BEM to damage mechanics reported in the literature are Herding & Kuhn [12] and Garcia et al [13]. Recently, we can cite the works of Sladek et al. [14], Botta et al. [15] and Benallal et al. [16]. These works include non-local formulations to treat strain localization phenomena and associated numerical instabilities. Some aspects on the numerical analysis of porous media experiencing damage are found in Cheng & Dusseault [17] and Selvadurai [18]. Due to the increasing complexity of models developed for engineering problems, robust numerical models capable to provide accurate results with the least possible computational effort are looked for. In this scenario, BEM appears as an interesting choice for obtaining numerical solutions in various engineering applications. In this paper, a non-linear set of transient BEM equations is developed, based on Betti’s reciprocity theorem, to deals with isotropic-damaged porous media. The description of porous solid is done in a Lagrangean approach. Marigo’s damage model is applied with a local evaluation of the thermodynamic force associated to damage. Regarding the BEM numerical procedure, the integration over boundary elements is evaluated by using a numerical Gauss procedure. A semi-analytical scheme for the case of triangular domain cells is followed to carry


499

out the relevant domain integrals. A Newton-Raphson procedure is applied to solve the non-linear system, with a consistent tangent operator. This is done in the light of the procedure introduced by Simo and Taylor [19] for finite elements.

Governing Equations The following free energy potential is considered,

U\( kj , D, I I0 )

2 1 1 (1 D) kjE dkjlm lm b 2 M ª¬Tr kj º¼ 2 2

1 2 M I I0 bM I I0 Tr kj 2

(1)

where the constants M and b represent the Biot modulus and Biot coefficient of effective stress, respectively. In the case of saturated media, filled by an incompressible fluid, the Biot coefficient assumes unit value. In fullsaturated conditions, the lagrangian porosity I measures the variation of fluid content per unit volume of porous material. The bulk density is described by U . E djklm represents the isotropic drained elastic tensor. jk denotes the strains in the solid skeleton. Assuming isotropic case, the damage is represented by the scalar-valued internal variable D , which defines the internal state of the material, taking values between zero (sound material) and one (complete degradation). The initial porosity field is indicated by I . The derivatives of free energy potential with respect to the internal variables lead to the associate variables, that are the total stress V jk , the pore-pressure p and the thermodynamical force Y conjugated to D .

V jk

U

w\ w jk

p p0 U Y

U

w\ wD

(2)

(3)

(1 D)E djklm lm bM ª¬bTr jk I I0 º¼ G jk

w\ w I I0

M ª¬I I0 bTr jk º¼

1 jk E djklm lm 2

(4)

Using equations (2) and (3) the total stress tensor is written as

V jk

E jklm Hlm DE jklm Hlm b p p0 G jk

(5)

from which it is seen that it includes three different contributions, being the first one the effective stress Vefjk , acting on the grains of the solid matrix, and the second one the stress due to damage Vdjk . In addition to the state laws given above, it is necessary to define a damage criterion. In Marigo's model it takes the form:

F (Y , D) Y N(D)

(6)

The term N(D) represents the maximum value of Y reached during the loading history, and is adopted here in its simple linear form N(D)

Y0 AD , where parameters Y0 and A are material dependent. The damage evolution

becomes from the consistency condition F (Y , D)

0 , resulting in:

500


Y A D

(7)

The fluid flow through the porous space can be described by Darcy's law. Assuming a laminar flow, this law considers a linear relationship between the flow rate and the pressure gradient:

k ª¬ p,k f k º¼

Qk

(8)

k the scalar permeability coefficient, defined as a P function of the intrinsic permeability k and the fluid viscosity P . The fluid body force is represented by f k .

In this simple version, it is assumed isotropic, with k

The fluid mass balance equation, assuming no external fluid sources, is written as:

d Uf I Uf Q k ,k dt

0

(9)

The following equilibrium and compatibility relations, added to appropriate boundary conditions complete the set of equations that describes the poro-elasto-damage problem, in quasi-static conditions:

V jk,k b j

0

1 u k, j u j,k 2

H jk

(10)

(11)

Integral Equations In order to couple the behaviour of the solid and fluid phases, two sets of integral equations are derived. The first one is related to the elastostatics problem, for which a pore-pressure field is distributed over the domain, while the other equation refers to the pore-pressure itself. In order to obtain the integral equations one can use Betti’s reciprocity theorem, which can only be applied to elastic fields. Thus, in the case of elasticity, assuming the effective stress definition:

³ V jk (q)Hijk (s, q)d: ³ H jk (q)Vijk (s, q)d: ef

*

*

:

(12)

:

³ V jk (q) V jk (q) bG jk p(q) Hijk (s, q)d: ³ H jk (q)Vijk (s, q)d: d

*

*

:

(13)

:

where s and q represent the source and field points, and X* is the fundamental solution for the variable X , from now on. The direction i refers to the application of the unit load on the source point into the fundamental domain. In elastostatics, one applies the well-known Kelvin fundamental solutions. By applying the divergence theorem to equation (13), and considering the transient nature of the problem, one obtains the following integral equation for displacements on the boundary points S:

Cik u k (S)

³ T k (Q)uik (S, Q)d* ³ Tik (S, Q)u k (Q)d* *

*

*

*

H*ijk (S, q)d: ³ V djk (q)H*ijk (S, q)d: ³ bG jk p(q) :

:

(14)


501

The stresses at internal points are obtained by differentiating equation (14), now written for internal points, and applying Hooke's law, which leads to

³ Sijk (s, Q)u k (Q)d* ³ Dijk (s, Q)T k (Q)d* ³ R ijkl (s, q)V dkl (q)d:

V ij (s)

*

*

:

(15)

@ TLij ª¬V dkl (s) º¼ ³ bGkl R ijkl (s, q)p(q)d : TLij >bGkl p(s) :

where Sijk , Dijk and R ijkl are the derivatives of the fundamental solutions, and TLij are the free-terms coming from differentiation. The integral equation for the pore-pressure can be obtained in a similar way, defining the proportional flow vector Q pr Q k kf k kp,k in order to apply Betti's Theorem k

³ >Qk kfk @ p,k (s, q)d: ³ Qk (s, q)p,k (q)d: *

*

:

(16)

:

from what the divergence theorem leads to write:

p(s)

³ Q*K (s, Q)p(Q)d* ³ p* (s, Q)Q K (Q)d* ³ p* (s, q)Q k,k (q)d: ³ p*,k (s, q)kf k (q)d: *

*

:

(17)

:

K indicates the outward normal direction to the boundary. Assuming Qk,k

I (see (9)) and, neglecting the

body force f k ,we get:

p(s)

³ Q*K (s, Q)p(Q)d* ³ p* (s, Q)QK (Q)d* ³ p* (s, q)I(q)d: *

*

(18)

:

For convenience, it is possible to take the derivative I(q) from (3), so that the pore-pressure is given by the following equation:

p(s)

ª1 º ³ Q*K (s, Q)p(Q)d* ³ p* (s, Q)Q K (Q)d* ³ p* (s, q) « p(q) bTr H (q) » d: M ¬ ¼ * * :

(19)

Considering a finite time step 't n = t n +1 t n and a corresponding variable increment 'X = X n+1 X n , one can integrate equations (14), (15) and (19) along the interval 't , leading to the following set of equations, in terms of the variable increments:

³ 'Tk (Q)u ik (S, Q)d* ³ Tik (S, Q)'u k (Q)d* ³ bG jk 'p(q)Hijk (S, q)d:

Cik 'u k (S)

*

*

*

*

*

:

³ 'Vdjk (q)H*ijk (S, q)d:

(20)

:

'Vij (s)

³ Sijk (s, Q)'u k (Q)d* ³ Dijk (s, Q)'Tk (Q)d* ³ R ijkl (s, q)'Vdkl (q)d: *

*

:

TLij ª¬ 'Vdkl (s) º¼ ³ bGkl R ijkl (s, q)'p(q)d: TLij >bGkl 'p(s) @ :

(21)

502


c(s)p(s)

³ Q*K (s, Q)p(Q)d* ³ p* (s, Q)Q K (Q)d* *

*

(22)

1 1 1 ³ p* (s, q)'p(q)d: ³ bp* (s, q)Tr 'H(q) d: 't : M 't : Algebraic Equations and Solution Procedure

The numerical solution of the boundary value problem requires both the time and space discretizations. It should represent the system of equations in a discrete way along the linear boundary elements and into the triangular domain cells in order to obtain the approximate values of the variables of interest. One defines the number of boundary points by N n and the number of internal nodes by N i . The appropriate discretization of the integrals on (20)-(22), followed by some algebraic manipulations inherent to BEM, leads to the following system:

> H @^'u` > G @^'T` > Q@^'Vd ` b >Q@> IK @^'p` ^'V`

> HL@^'u` > GL@^'T` > QL @^'Vd ` b > QL @> IK @^'p`

^p `

ª¬ HP(i) º¼ ^p` ª¬GP(i) º¼ ^V`

(i)

(23) (24)

1 b ªQP(i) º¼ ^'p(i) ` ª¬QP(i) º¼ > Tr @^'H` M 't ¬ 't

(25)

> @ come from the integration ^ ` are prescribed or unknown

The subscript (i) refers to internal points. The influence matrices represented by of the fundamental solutions and its derivatives. The variables represented by

variables along the boundary or over the domain. After some arrangements, the system given above is written as

> E @^'H` ^'Ns` ª¬> QS@ > I@º¼ ^'Vd ` b ª¬> QS@ > I@º¼ > IK @^'p(i) `

(26)

1 ª ª ºº «¬> I@ M 't ¬QP (i) ¼ »¼ ^'p (i) `

(27)

where ^'Ns` and

^Np` 'bt ¬ªQP

(i)

º ¼ > Tr @^'H`

^Np` are vectors containing prescribed values and >E@ the drained elastic tensor. Finally,

arranging the two equations in a single one, in terms of ^'H` only, leads to

ª E º ^'H` ¬ ¼

> 'Ns@ ^Np` ª¬QSº¼ ^'Vd `

(28)

which contains the new terms:

^ ` Np

ªE º ¬ ¼

1 ª ªQP (i) º º b ª¬QSº¼ > IK @ «> I@ ¼ »¼ M 't ¬ ¬

1

^Np`

1 ª º b2 1 ª ªQP (i) º º ªQP (i) º > Tr @» «> E @ ª¬QSº¼ > IK @ «> I@ » ¬ ¼ ¬ ¼ M 't 't ¬ ¼ «¬ »¼

(29)

(30)

Due to the presence of correction terms associated with damage, equation (28) is non-linear at each time increment, and can be written:


503

^ `

^Y ^'H ` `

ª¬ E º¼ ^'H n ` > 'Ns @ Np ª¬QSº¼ ^'Vd n ` 0

n

(31)

The solution is carried out by a Newton-Raphson’s scheme. An iterative process is required to reach equilibrium. Then, from iteration i , the next try i 1 is given by 'H in1 'H in G'H in . The correction G'Hin is

^

` ^ ` ^

`

^

`

calculated from the first term of the Taylor expansion, as follows:

^

^

w Y ^'Hin `

`

Y ^'Hin `

where the derivative

w ^'H

^

w Y ^'Hin ` w ^'H

i n

`

i n

`

`

^G'H ` i n

0

(32)

` is the consistent tangent operator.

Numerical Example To illustrate the BEM formulation applied to poro-elastic media we first analyze the consolidation of a semiinfinite plane, under a strip uniform load of width 2a . Due to the symmetry, we consider only the half-plane, as shown in Fig. 1. The load is applied instantaneously at t 0 . An analytical solution was proposed by Schiffman et al. [20], in terms of an adjusted time factor W , and the dimensionless values of pore-pressure, total and

(1 X ) 2Gk t, p (1 2X ) J f

effective stresses: W

O

x1

p(W ) , V p(0)

V11 (W ) V22 (W )) and Vef V11 (0) V22 (0))

ef V11 (W ) Vef22 (W )) . ef V11 (f) Vef22 (f))

draining surface

x2

10 a

10 a

Figure 1. Problem definition and internal cells mesh adopted From Fig. 2 one can observe the response on the point O(0,a/2) (see Fig.1). The results obtained with the proposed model and the ones presented in [20] are compared. In the early-time response, just after the loading, fluid flow is slow, inducing the highest pore-pressures. One should note the occurrence of Mandel-Cryer effect, which is characterized in this plane strain consolidation by an increase in the pore-pressure at early times compared to the initial pore-pressure. With time, the drainage process leads to an increase in effective stress field, accompanied by a proportional pore-pressure reduction, until its vanishing.

504


Figure 2. Normalized pore-pressures, total and effective stresses Poro-elastic column subject to damage. Let us consider another consolidation case. It consists on a soil layer of thickness equal to 10m, resting on a rigid impermeable base. A constant unit load is progressively applied on the top surface of the layer over 1000s, under drained conditions (Fig. 3). The response is compared for the poroelastic and elasto-damage behaviours, besides the coupled response. The results presented correspond to the bottom of the layer. The material parameters, assuming the layer made of Berea Sandstone, are defined as follows 13 (Detournay & Cheng [21]): E 14400 MPa, X 0.2 , b 0.79 , M 12250 MPa, k 1.9 x 10 m2 and 5 7 9 P MPa·s. For the damage model, we adopt the parameters Y0 10 MPa and A 2 x 10 MPa. The analysis involving damage are presented up to the limit load.

draining surface

Figure 3. Problem definition and internal cells mesh adopted The strain behaves in a similar way for the poro-elastic and the elastic materials, increasing almost linearly up to the total time (Fig. 4). The difference between the two curves results from the fluid phase flow. Taking into account the damage, the contribution of the fluid is also significant, leading to a delay in the damage process (Fig. 5b).


505

Figure 4. Strain evolution for all the considered behaviors In Fig. 5a, it can be observed that the elasto-damaged material has an intermediary behaviour between the porous media and the damaged porous media. In addition, it reaches the maximum load before the poro-elastodamaged material, with a higher deterioration level (Fig. 5b). It is interesting to note the augmentation of the pore-pressure in the presence of damage, beyond the threshold defined on the simple poro-elastic case.

Figure 5. a) Effective stress field evolution b) Damage parameter evolution On the numerical stability, the presented model shown to be almost independent of the time step adopted, having been tested values from 0,001 up to 10s, without any observable changes on the response. It should be noted that, in the presence of damage, the response is represented only up to around 300s, which corresponds to the limit load as we have a load control. Besides, strain softening in the constitutive law causes localization phenomena, which leads to physically meaningless results and imposes difficulties on the numerical solution, requiring the use of regularization techniques.

506


Figure 6. Pore-pressure evolution

Conclusions and Perspectives A BEM formulation to poro-elasto-damaged material was presented. The model has shown a reasonable level of coupling between the damage and the fluid seepage. The literature, on theoretical and experimental levels, poses several interesting questions, among which the variations that the damage state imposes on the poro-elastic parameters. Some developments in this way are being made in the presented model, in order to improve the solidfluid interaction.

Acknowledgements To CNRS, FAPESP and Île-de-France Region for the financial support.

References [1]M.A.Biot General theory of three-dimensional consolidation J. Appl. Physics 12 155-164 (1941). [2]K.Terzaghi Die berechnung der burchlassigkeitsziffer des tones aus dem verlauf der hydrodynamischen spannungserscheinungen Sitz. Akad. Wiss. Abt. IIa 132 125-138 (1923). [3]O.Coussy Poromechanics, Chichester: John Wiley & Sons (2004). [4]M.P.Cleary Fundamental solutions for a fluid-saturated porous solid International Journal of Solids and Structures 13 785-806 (1977). [5]A.H.D.Cheng and J.A.Liggett Boundary integral equation method for linear porous-elasticity with applications to fracture Int. J. Numer. Meth. Eng. 20 279-296 (1984). [6]A.H.D.Cheng and E.Detournay On singular integral equations and fundamental solutions of poroelasticity International Journal of Solids and Structures 35 4521-4555 (1998). [7]A.H.D.Cheng and M.Predeleanu Transient boundary element formulation for linear poroelasticity International Journal of Applied Mathematical Modelling 11 285-290 (1987). [8]L.M.Kachanov Time of rupture process under creep conditions Izvestia Ak. Nauk 8 26-31 (1958). [9]J.Lemaitre A course on Damage Mechanics, Berlin: Springer-Verlag (1992). [10]J.Lemaitre and J.L.Chaboche Mécanique des Matériaux Solides, Paris: Dunod (1985).


507

[11]J.J.Marigo Formulation d’une loi d’endommagement d’un materiau élastique Comptes rendus de l’académie des sciences 292 série II 1309–1312 (1981). [12]U.Herding and G.Kuhn A field boundary element formulation for damage mechanics Engineering Analysis with Boundary Elements 18 137-147 (1996). [13]R.Garcia, J.Florez-Lopez and M.Cerrolaza A boundary element formulation for a class of non-local damage models International Journal of Solids and Structures 36 3617-3638 (1999). [14]J.Sladek, V.Sladek and Z.P.Bazant Non-local boundary integral formulation for softening damage International Journal for Numerical Methods in Engineering 57 103-116 (2003). [15]A.S.Botta, W.S.Venturini and A.Benallal BEM applied to damage models emphasizing localization and associated regularization techniques Eng. Anal. Bound. Elem. 29 814-827 (2005). [16]A.Benallal, A.S.Botta and W.S.Venturini On the description of localization and failure phenomena by the boundary element method Comp. Meth. Appl. Mech. Eng. 195 5833-5856 (2006) [17]H.Cheng and M.B.Dusseault Deformation and diffusion behaviour in a solid experiencing damage: a continous damage model and its numerical implementation International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts 30 1323-1331 (1993). [18]A.P.S.Selvadurai On the mechanics of damage-susceptible poroelastic media Key engineering Materials 251252 363-374 (2003). [19]J.C.Simo and R.L.Taylor Consistent tangent operators for rate-independent elastoplasticity Computer Methods in Applied Mechanics and Engineering 48 101-118 (1985). [20]R.L.Schiffman, A.T-F.Chen and J.C.Jordan An analysis of consolidation theories Journal of the Soil Mechanics and Foundations Division SM 1 285-312 (1969). [21]E.Detournay and A.H.D.Cheng Fundamentals of poroelasticity Comprehensive Rock Engineering: Principles, Practice and Projects v. II, Great Britain: Pergamon Press (1993).

508


Efficient Solution of Acoustic Radiation Problems by Boundary Elements and Interpolated Transfer Functions Otto von Estorff1 and Olgierd Zaleski2 1

Hamburg University of Technology, D-21071 Hamburg, Germany, Estorff@ tu-harburg.de 2

Kasernenstr. 12, D-21073 Hamburg, Germany, [email protected]

Keywords: Acoustics, boundary element method, transfer functions, frequency interpolation, efficient solution.

Abstract. An efficient technique for solving acoustic radiation problems defined by known Neumann boundary condition is presented. It is based on an indirect boundary element formulation which is combined with a special source simulation technique. This combination allows an accurate and systematic computation of acoustical quantities in wide ranges of the considered frequency range. Since the new methodology is using a truncated series approach, the present contribution also provides a suitable estimation of the approximation error. Thus the presented procedure can be used for a reliable design of acoustical systems.

Introduction Acoustics is one of a small number of engineering fields where the boundary element method (BEM) is a widely used numerical method [1], [2], [5], even in commercial software tools. In the majority of cases, it is employed to predict sound radiated by vibrating structures. In classical BE formulations, the acoustic response is calculated by solving the system of equations for each load condition, typically defined by surface vibrations, i.e. normal velocities, of the structure. Because of the disadvantageous properties of BEM system matrices [2] this direct solving procedure is known to be rather computer time consuming. Consequently, there is still a strong need for fast and accurate numerical approaches, which increase the efficiency of the classical BEM. Interesting approaches, different from the ones suggested hereafter, can be found, for instance, in [6]. If for a given geometry multi-frequency simulations need to be performed, the calculation of so-called Acoustic Transfer Functions (ATF) turns out to be very efficient [9]. With known ATFs, the calculation of sound pressure values caused by different structural vibrations is a task for which the computation time is nearly negligible. The computational effort for an acoustic analysis involving ATFs is caused by the calculation of the ATFs themselves. A quite efficient solution to determine the ATFs is proposed in this paper. An additional step towards a reduction of the computational effort must be seen in a combination of the ATFs with a new approximation technique: Using a special source technique the ATFs are interpolated between only a few ATF values computed in a smart way by the BEM. Finally an error bound can be defined in order to control the accuracy of the overall calculation of the ATFs. In order demonstrate the accuracy and applicability of the new approach, a numerical example is discussed in detail. The procedure turns out to be computationally very powerful and it seems to be a promising step towards a more efficient calculation of complex sound radiation problems. Boundary Element Methods in Acoustics Most numerical methods in acoustics have in common that they seek to find a solution for the acoustic wave equation which describes the propagation of waves in a fluid assuming that all amplitudes are small and that the fluid is inviscid. In the majority of the formulations it is assumed that the acoustic fluid is subjected to a harmonic excitation with frequency . Then the wave propagation can be described by the well-known Helmholtz equation

2 p k 2 p

J,

(1)

where p is the acoustic pressure and k Z / c the wave number. c denotes the sound wave velocity and J the amplitude of a harmonic excitation in the interior of the fluid. The linearized equation


p

iUZv

509

(2)

yields a relation between the pressure gradient p and the particle velocity v, where is the density of the fluid. Considering the general case of a mixed boundary value problem, the boundary conditions for the solution of eq (1) can be written as

p

p on S1 and

wp wn

ipZvn on S2 ,

(3)

where n is the normal vector of the structural surface S=S1 U S2, and over bar “-“ marks the known boundary values on S1 and S2, respectively. Employing Green’s identity, the three-dimensional fundamental solution G ( X , Y ) , and the relation defined in eq (2), eq (1) can be transformed to the Helmholtz boundary integral equation [2]

ª

C X p X

³ «¬ p Y S

º wG X ,Y iU Z v n Y G X ,Y » dS Y wnY ¼

(4)

where p is the acoustic pressure at an arbitrary point X inside the fluid. Y is a point on the boundary S and C(X) a factor depending on the location of X. It should be noted that eq (4) is known as the direct integral representation of eq (1). Alternatively, an indirect form can be found. In this case, using a variational approach, as suggested by Hamdi [6] or Coyette [1], the differential eq (1) can be turned into an integral equation of the form

ª

p X

³ «¬P Y S

º wG X ,Y V Y G X ,Y » dS Y wnY ¼

(5)

where P ( Y ) p p and V ( Y ) v n v n are the double and single layer potential, respectively. The superscripts “+” indicates the positive side of the boundary S (positive direction of the normal vector), while “–” is marking the negative side. G represents, as before, the three-dimensional fundamental solution. More details can be found, e.g., in [2]. Note that only in the case of very simple geometries and boundary conditions, the eqs (4) and (5) can be solved analytically. For more complex systems numerical formulations, as discussed in [2], have to be used. Acoustic Transfer Functions Basic Idea. The Acoustic Transfer Functions are frequency dependent functions which represent a relation between the surface velocities of a sound radiating structure and the sound pressure level at a given field point (microphone position) [5]. A single ATF between the ith node of a discretized vibrating surface and one arbitrary field point is equal to the sound pressure induced at this field point, when the normal velocity vn is set to 1 at the ith node only, and set to 0 everywhere else at the surface of a vibrating structure. It should be pointed out that an ATF only depends on the configuration of the acoustic domain, i.e., on the geometry, on the fluid properties (sound velocity and mass density), on the acoustic surface treatment (local impedance or admittance), on the excitation frequency Z, and, of course, on the location of the field point. However, an ATF is independent of the loading condition. With known ATFs the acoustic response p can simply be found as the matrix product of the ATFs with the operational structural response vn as shown by the equations {p} = [ATF] {vn}

(6)

The rows of the acoustic transfer matrix [ATF] are equal to single ATFs of the model. The high efficiency of the acoustic simulations is achieved with the concept of the ATFs since it allows the same set of ATFs to be re-used to compute and compare sound fields in many different applications [5], [9]. Due to the definition of the ATFs it is easy to see that their direct calculation might become very expensive in means of the CPU time. It requires the evaluation of the field point pressure for as many excitation vectors as there are nodes in the boundary element model. Efficient Calculation. In order to reduce the computational effort when calculating ATFs a new efficient procedure is proposed in this chapter. It is based on the computation of two particular sound fields. The first one

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occurs when a monopole source is placed at the position X0 of the field point for which the ATFs are to be found. In such a case the Helmholtz eq (1) for the sound pressure pr can be written as

2 pr k 2 pr

G X X 0 ,

(7)

where the Dirac function G is used to define the source. The second sound field appears under the condition of a completely sound reflecting surface S B of the considered structure. For its sound pressure pz the Helmholtz eq (1) leads to

2 pz k 2 pz

0.

(8)

After a few steps the equation

G X X 0

p z 2 pr k 2 pr p z pr 2 p z k 2 pr p z

(9)

can be obtained from (7) and (8). Together with the third Greens identity [2] eq (9) leads to

S

wp · § wp r p z p r z ¸dS ¨ wn wn ¹ S B Sf ©

³

pr X 0

(10)

where S denotes the whole boundary of the acoustic domain, i.e. its boundary at infinity S f and the surface of the considered structure S B . For the evaluation of the integral (10) over S f , the Sommerfeld condition for the sound radiation at infinity [2] is taken into account, such that

§ wp · lim r ¨ ikp ¸ r of © wr ¹

0.

(11)

It can be seen directly that after applying the condition (11) first for pz and then for pr , two additional equations can be found that lead, in combination, to

§ wp r

³ ¨© wn

p z pr

Sf

wp z · ¸dS wn ¹

0

(12)

For two further terms in the remaining integral over S B one can write

wpz wn

i Z U vnz

0

(13)

since the normal velocity vnz vanishes at the sound reflecting surface S B and

wp r wn

i Z U v nr .

(14)

In order to obtain a single ATF between the ith node and the field point at X0 , the integral (10) can finally be reconstructed in the vicinity S Bi of this node which results in

³ iZ U v

nri

p zi dS

p ri X 0 .

(15)

S Bi

According to the definition of the ATFs, the normal velocity vnri is set to 1. The pressure pzi is given by a definition of a point source [2]. Taking into account these two conditions pri from eq (15) is equal to the wanted ATF.

Frequency Interpolation and Error Bounds A further speed-up of the acoustic calculation can be obtained if the detailed BEM calculation of the ATF is reduced to a limited number of master frequencies, while the remaining values are interpolated. Such an


511

interpolation can be done using a special source simulation technique [8], which can be adapted to the specific characteristic of the ATF algorithm. Using global spherical coordinates r , I , 4 , where r is the distance from the origin and I and 4 are the angles in meridional and azimuthal directions, respectively, one may approximate an ATF by the following summation [3][8]:

¦ ¦ P cos4 e N

ATF | F

l

m

l

l m ! h2 kr c l m ! l j l , m

imM j

j

l 0 m l

where

Pl

m

cos 4 j

are Legendre polynomials [3] and the ratio

1 m l

2 l!

1 cos

2

4j

m 2

d

cos 4 1 d cos 4

l m

2

(16)

l

j

l m

(17)

j

l m ! l m !

is used for normalization. The functions hl2 are

spherical Hankel functions of the second kind [3]. In practical calculations, only a finite number N of source functions can be used. Therefore, eq (16) is only an approximation of the Acoustic Transfer Functions, whose accuracy mainly depends on the choice of N and on the unknown factors cl,m. Details may be found in [10]. The usage of the proposed ATF approximation depends on the availability of reliable error bounds. The series (16) is truncated at a sufficiently high number N and the knowledge of the error caused by this action is essential for a proper design of the ATF approximation. Following [10], the truncation error of eq (16) can be simplified to

Rd

f

¦

l N 1

hl2 kr

l m ! ¦ l m ! c l

l ,m

,

(18)

m l

and an error bound H ATF for the ATF approximation can be derived. One obtains the relation

H ATF

2

kr 4

tR

(19)

which holds for kr t 1,22 . More details are given in [10].

Numerical Example In order to discuss the suggested procedure and to show its applicability and accuracy, a sound radiating structure as given in Fig. 1, namely an exhaust manifold, shall be considered. In particular, the interpolation of a representative ATF between a node and a single field point (see Fig. 1) will be discussed in detail. The distance from the chosen node to the considered field point is 1 m. In all calculations the acoustic fluid is assumed to be air (c = 340 m/s, U = 1.225 kg/m3).

node

field point

Figure 1. BE acoustic mesh of the exhaust manifold with a node and a field point for the ATF calculation.

512


6,0E03

6,0E03

4,0E03

4,0E03

2,0E03

imag (ATF)

real (ATF)

First a comparison of the reference ATF with the ATF computed using the fast BEM technique according to eq (15) is shown in Fig. 2. Both ATFs are given in the frequency range between 10 Hz and 500 Hz with 'f = 2 Hz. The proposed method leads to very accurate results for the real as well as for the imaginary part of the considered ATF. The evaluation of the relative error, which is lower than 3% in this case, confirms the good quality of the achieved results.

0,0E+00 2,0E03

0

100

200

300

400

500

600

2,0E03 0,0E+00 2,0E03

0

100

200

300

400

500

600

4,0E03

4,0E03

6,0E03

6,0E03

Frequency [Hz]

Frequency [Hz]

Figure 2. The real (left) and the imaginary (right) parts of the considered ATF computed directly with the BEM ( ) and the proposed fast technique ( x x ).

1,0E+01 1,0E+00

Error

1,0E01 1,0E02

Theor

1,0E03

N=5

1,0E04

N=3

1,0E05

N=1

1,0E06 1,0E07 1,0E08 1,0E09 0

10

20

30

40

50

1,0E+00

1,0E+00

1,0E01

1,0E01

1,0E02

1,0E02

1,0E03

1,0E03

1,0E04

1,0E04

Error

Error

kr

1,0E05

1,0E05

1,0E06

1,0E06

1,0E07

1,0E07

1,0E08

1,0E08

1,0E09

1,0E09

0

10

20

30

kr

40

50

0

10

20

30

40

50

kr

Figure 3. Error comparison for ATF approximations with 'fMaster = 100Hz (above), 'fMaster = 50Hz (bottom left) and with 'fMaster = 20Hz (bottom right).


513

In the next step the approximation of the ATF is used in order to speed-up the overall calculation. The computation frequencies are expended to 2500 Hz with 'f = 5 Hz. The source necessary for the approximation given by eq (16) is submerged directly underneath the considered node at a distance of 10 mm. For the reference solution the standard indirect BE calculation is used. During the course of this study, the number of source functions N is chosen first to N = 1 then to N = 3, and finally to N = 6. It should be noted that the second boundary condition ( vn = 0) has been prescribed only at 6 nodes. For the considered frequencies a series of ATF approximations with three different distances between the master frequencies, 'fMaster = 100 Hz, 50 Hz, 20 Hz, is discussed for each chosen N. To judge about the accuracy of the ATF approximation and the validity of the error bound H ATF , the evaluation of the absolute error is shown in Fig. 3. Values smaller then 10-20 are not plotted because of the logarithmic scale of the y axis. It is seen that error curves obtained in computations behave as the theoretical error H ATF . The biggest differences between the reference ATF and its approximation occur always at the mid frequencies between two master frequencies. In practical computations the relation between the error in these mid frequencies and its theoretical estimation H ATF can be checked before the whole approximation is conducted. Obviously a very good accuracy of the approximated ATF can be expected, when an error of an arbitrary approximation does not exceed the values of H ATF . This can be achieved even for small parameters N by an appropriate choice of 'fMaster. Conclusions The usage of the standard BEM for the investigation of sound radiation problems is rather popular. In practice however, the necessary computation time often leads to the decision not to employ the methodology. For this reason, more efficient methodologies, special algorithms and reliable approximate formulations have been discussed in this contribution. In some cases the new approach may lead to a reduction (up to 90 percent) of the overall simulation cost. In particular, the usage of Acoustic Transfer Functions, their computation by a reciprocal formulation, and a new frequency interpolation scheme turned out to be very effective and promising for future applications. References [1] J.P. Coyette An advanced boundary method for exterior acoustic problems, Proceedings of StruCoMe, Paris (1988). [2] O. von Estorff (Ed) Boundary Elements in Acoustics – Advances and Applications, WIT Press, Southampton (2000). [3] O. von Estorff and O. Zaleski Efficient Acoustic Calculations by the BEM and Frequency Interpolated Transfer Functions, Eng. Analysis with Boundary Elements, 27, 683-694 (2003). [4] O. von Estorff, S. Rjasanow, M. Stolper and O. Zaleski Two efficient methods for a multi-frequency solution of the Helmholtz equation, Computing and Visualisation in Science, 159-167, Springer, Heidelberg (2005). [5] O. von Estorff, M. Markiewicz and O. Zaleski Validation of Numerical Methods in Acoustics: What can we expect?, Computational Methods for Acoustics Problems, Saxe-Coburg Publications, Stirling (2008). [6] M. Fischer, U. Gauger and L. Gaul A multipole Galerkin boundary element method for acoustics, Eng. Analysis with Boundary Elements 28, 155-162 (2004) [7] M.A. Hamdi (Ed) A variational formulation by integral equations for the solution of the Helmholtz equation with mixed boundary conditions (in French), Compterendus Académie des Sciences 292, Paris (1981). [8] M. Ochmann and A. Homm The Source Simulation Technique for Acoustic Scattering, Acta Acustica 82, 159, (1996). [9] M. Tournour, J.-P. Rossion, L. Bricteux and C.F. McCulloch Getting useful FEM and BEM Vibro-acoustic solutions faster, using new solution methodologies, Proceedings of ISMA 2002, Leuven, (2002). [10] O. Zaleski and O. von Estorff A Fast Solution of the 3D Helmholtz Equation by BEM and Interpolated Transfer Functions. The Sixteenth International Congress on Sound and Vibration (ICSV16), Krakow (2009).

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A FAST SOLVER FOR BOUNDARY ELEMENT ELASTOSTATIC ANALYSIS J O Watson School of Mining Engineering University of New South Wales Sydney 2052, Australia [email protected] Keywords: boundary elements, elastostatics, equation solver

Abstract A method of acceleration of the solution of simultaneous equations approximating to boundary integral equations of elastostatics is proposed. Surface nodes are assigned to cells of an octree clustering scheme. For submatrices of small coefficients, approximations are made whereby for the purpose of Gaussian elimination small numbers of row and column vectors may represent the fully populated matrix. The method is demonstrated by analysis of stress at the Kiruna iron ore mine, Sweden.

Introduction In the analysis by boundary element methods of large problems of elastostatics, the solution of simultaneous equations approximating to boundary integral equations takes more computer time than matrix and load vector construction, and possibly more than equation solution in an equivalent finite element analysis. It has been clear for some time that a faster method than Gaussian elimination on a fully populated matrix is required. Progress has been made in the application of adaptive cross approximation (ACA) and heirarchical matrices [1]. However, the effectiveness of those methods has generally been demonstrated for problem geometries such as cuboids with very fine mesh that are quite unlike the convoluted domains for which real engineering problems must be solved, usually with coarser mesh. Furthermore, in those methods no direct reference is made to physical characteristics such as the decay of kernels to zero at infinity which might be expected to be the key to fast solution: if they indeed take advantage of such factors, they do so through the intermediary of linear algebra theory. The method proposed here is applicable to systems of equations obtained by discretisation of the integral equations [2] lim ³ [Tij(x,y) uj(y) ε → 0 S – S(x,ε) + s(x,ε)

– Uij(x,y) tj(y)] dSy = 0

(1)

lim ³ [Vij(x,y) uj(y) ε → 0 S – S(x,ε) + s(x,ε)

– Wij(x,y) tj(y)] dSy = 0

(2)

in which Uij(x,y) is the fundamental solution and for example Wij(x,y) = ns(x) ∂Uij(x,y)/∂xs

(3)

where ns(x) is the unit outward normal to S at x. The boundary S – S(x,ε) + s(x,ε) is as shown in Figure 1. Equations (1) and (2) are conventionally written with free terms: the forms shown above avoid the need for consideration of Cauchy principal values and Hadamard finite parts, and are consistent with the method of simple solutions by which computations are carried out [2] and which in particular permits equation (2) to be taken not only at x on a smooth part of S but also on an edge or at a corner.


Figure 1

515

Boundary for which integral equations are taken

Overview of the method The system of simultaneous equations is constructed by nodal collocation. In a mixed boundary value problem, equation (1) is taken in a direction in which displacement is unknown, equation (2) in one in which traction is unknown. Therefore, all matrix coefficients are integrals of kernels T and W multiplied by shape functions. Since both T and W are of order 1/r2 where r is distance from x to y, it follows that in the system of equations Au = f (4) where u are unknown surface nodal displacements and tractions, all matrix coefficients are bounded in absolute value by K/r2, where r is distance between the pair of nodes to which the 3x3 submatrix containing a coefficient corresponds. It is upon this result, together with the existence of Taylor series expansions for kernels and unknown displacements and tractions, that the validity of the proposed method rests. Economy of solution is achieved by lumping of submatrices of small coefficients into row and column vectors, resulting in similar sparsity to that of ACA. Nodes are assigned to cells of an octree node clustering scheme, and equation (4) rewritten as n

Σ Apquq = fq for p = 1, 2, ... ,n

(5)

q=1

Clusters are chosen such that each lies on a smooth contiguous part of S.

Row lumping In the approximate system n

Σ Bpquq = fq for p = 1, 2, ... ,n

(6)

q=1

a submatrix Bpq is taken to equal Apq, unless the diameter Dp of row cluster p divided by distance Rpq between that cluster and column cluster q is less than a threshold θr, in which case 3x3 submatrices of Bpq are taken to vary linearly with respect to local coordinates (ξp1, ξp2) of nodes of cluster p (see Figure 2). Specifically, column c of 3x3 submatrices is given by Bpqrc = β pq0c + ξp1rβ pq1c + ξp2rβ pq2c where β pq0c etc are average and weighted average values of 3x3 submatrices Apqrc:

(7)

516


Figure 2

Row and column clusters l

l β pq0c = Σ Apqrc

(8)

r=1 l

Ip1 β pq1c = Σ ξp1r Apqrc

(9)

r=1 l

Ip2 β pq2c = Σ ξp2r Apqrc

(10)

r=1

In equations (7) to (10), (ξp1r, ξp2r) are local coordinates of node r of cluster p, and Ips are moments of inertia of the cluster. In the solution of equation (6), for Dp/Rpq < θr the nine average and weighted average equations formed by submatrices β pq0c, β pq1c and βpq2c act as proxies for the equations defined by equation (7), and only those nine equations are stored and operated upon during forward reduction.

Column lumping A further reduction of equation solution time is achieved by taking in equation (6) unknown displacement or traction over a cluster to vary linearly with respect to (ξq1, ξq2) if Dq/Rpq < θc, to yield the system n

Σ Cpqvq = gq for p = 1, 2, ... ,n

(11)

q=1

where the vector vq consists of uq followed by three auxiliary triples aqs which are displacement or traction at the origin of coordinates (ξq1, ξq2), and derivatives of displacement or traction in the directions ξq1 and ξq2. The vector gq consists of fq followed by nine zeroes. An offdiagonal submatrix for which Dq/Rpq < θc is of the form

ª0 R pq º C pq = « » ¬0 0 ¼

(12)

where

R pq

ª R pq 01 «R pq 02 =« « « ¬« R pq 0l

R pq11 R pq12 R pq1l

R pq 21 º R pq 22 »» » » R pq 2l ¼»

(13)


in which

517

m

Rpq0r = Σ Bpqrc

(14)

c=1 m

Rpq1r = Σ ξq1c Bpqrc

(15)

c=1 m

Rpq2r = Σ ξq2c Bpqrc

(16)

c=1

A leading diagonal submatrix is given by

ªB pp C pp = « ¬ Ep

0º D p »¼

(17)

where

ª −I −I −I º « » E p = « − ξ p11I − ξ p12I − ξ p1mI » «− ξ p 21I − ξ p 22I − ξ p 2 mI » ¬ ¼

(18)

and

ªI 0 « D p = «0 I p1I «0 0 ¬

0 º » 0 » I p 2 I »¼

(19)

In equation (17), Ep and Dp are coefficients of constraint equations which define the auxiliary triples in terms of nodal values of displacement and traction. In the solution of equation (11), for cases in which both Dq/Rpq < θc and Dp/Rpq < θr which predominate as problem size increases, only 81 coefficients are stored and operated on during forward reduction. The exact solution of equation (6) is obtained by bifactorisation of the matrix C formed by submatrices Cpq, followed by an iteration in which reference is made to the matrix B formed by submatrices Bpq: v = v1 + v2 + v3 + ... where

Cvk = gk-1,

f0 = f

and

f k = f k-1 - Buk

for k = 1, 2, 3, ...

(20)

The only error in satisfaction of equation (4) is that incurred by row lumping.

Implementation The boundary elements are quadratic isoparametric triangles and Lagrangian quadrilaterals, and infinite elements with with asymptotic and plane strain functional variation [3]. The maximum number of nodes per cluster is currently 30, and average cluster size for the largest problems to date approaches 20 nodes. This indicates an attenuation factor (by which the number of nonzero matrix coefficients is reduced by lumping) tending to about 40 with increasing problem size. Default values of the thresholds θr and θc currently are both 0.5, and for θc equal to 0.5 error in satisfaction of equation (6) obtained by lumping of submatrix rows generally is negligible after five iterations as defined by equation (20). The matrices B and C are held on disc files. To minimise the input-output overhead during forward reduction of C, computations are blocked so that at least 500 unknowns are eliminated each time the matrix C is read from and rewritten to disc. The current problem size limit is 15000 degrees of freedom and byte addressable memory requirement 50Mb independently of problem size.

518


A difficulty has been encountered in the solution of problems in which there are insufficient constraints on displacement of elements to prevent rigid body motion. For this important class of problems the approach has been to set up to six arbitrarily chosen nodal displacements to zero, by setting offdiagonal coefficients of the corresponding equations to zero, and the leading diagonal coefficient to 1.0. This works for column lumping alone, but with both row and column lumping the iterative solution defined by equation (20) does not converge. The reason is believed to be that when weighted average equations act as proxies for submatrices, an inconsistency arises during forward reduction whereby in the elimination of an unknown from an equation, the value of the pivot is not the same for all coefficients of the equation. The consequence is that partially reduced equations no longer admit rigid body displacements without surface traction (i.e. with zero load vector coefficient) which in turn results in the arbitrarily chosen constrained nodes exerting nonzero support reactions. In a modification intended to resolve the difficulty now in a final stage of development, small adjustments are made during forward reduction to coefficients of submatrices for which Dp/Rpq > θr, to restore the admissibility of the rigid body displacements referred to above. Example: Kiruna iron ore mine, Sweden The mine at Kiruna extends 4km along strike (N-S) as shown in Figure 3. Today it is an underground operation beneath the former open cut, which is partially filled by fragmented waste rock. It would be impossible by any method to carry out a three dimensional analysis of the entire mine, so a representative E-W slice is so modelled and variations of stress and displacement to north and south taken to be two dimensional by the use of plane strain infinite boundary elements [3]. Within the E-W slice there is an extreme range of element size, from one kilometre length to only two metres square. In Figure 7 is shown mesh in the neighbourhood of the feature of primary interest, a drawpoint at which rock failure has been observed. The total number of finite boundary elements is 324, and of nodal degrees of freedom 3789. Run statistics, including computed stress (compression positive sign convention) at the point A on the brow of the drawpoint shown in Figure 7, are shown in Table 1 for various values of θr and θc.

Figure 3

Extent of three dimensional model

Figure 5

Figure 4

E-W cross section of mine

General view of finite boundary elements of the mesh


Figure 6 Detail of mesh viewed from below

519

Figure 7 Boundary element mesh near drawpoint

Run times are for an Intel 1.86MHz Core Duo processor. Problem size is at the bottom of the range for which the techniques described here are useful: with no lumping equation solution takes half the run time, and equation solution time is approximately halved by lumping.

θr

θc

0.0 0.2 0.4 0.6

0.0 0.2 0.4 0.6 Table 1

total run time (sec) 100 87 78 71

solver run time (sec) 49 37 28 21

stress at A (MPa) -10.3 -9.9 -8.1 -20.2

Run statistics for analysis of Kiruna mine

Conclusion The method is demonstrated to perform as intended for the example of the preceding Section. On the basis of limited evidence to date, the reduction in solver run time is least for such a problem with extreme mesh gradation, probably because gradation to some extent performs the same function as lumping of matrix coefficients. The linear rather than constant approximations of kernels and displacement or traction, and relatively small maximum number of nodes per cluster, were chosen largely to promote effective treatment of smaller problems and that objective does indeed appear to have been achieved. Without the ability to accelerate solution of minimally constrained problems the method is of limited value, but the difficulty is now well understood and a remedy in sight. It is arguable that the method will prove superior to ACA or hierarchical matrices since the attainment of sparsity is due partly to approximations of the matrix and partly to approximations of displacement or traction which incur no error in the solution, rather than wholly due to approximations of the matrix. Finally, there is the prospect of an improvement of accuracy of row lumping, obtained by taking the variation of kernels with respect to first argument to be a linear function divided by r2, rather than just a linear function as at present.

References [1] S Rjasanow and O Steinbach The Fast Solution of Boundary Integral Equations, Springer (2007). [2] J O Watson Int. J. Numer. Meth. Engng, 65, 1419-1443 (2006). [3] G Beer and J O Watson Int. J. Numer. Meth. Engng, 28, 1233-1247 (1989).

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Stress analysis of cracked structures considering crack surface contact by the boundary element method Wilhelm Weber, Kai Willner, Paul Steinmann and Günther Kuhn University of Erlangen-Nuremberg, Chair of Applied Mechanics Egerlandstr. 5, 91058 Erlangen, Germany [email protected], [email protected], [email protected], [email protected] Keywords: Dual BEM, Dual Discontinuity Method, crack surface contact, 3D crack propagation

Abstract. The stress analysis of cracked structures by the boundary element method (BEM) under the consideration of crack surface interaction in order to simulate fatigue crack propagation is presented. The coincident crack surfaces are separated by the utilization of the Dual BEM which deals with the displacement and the traction boundary integral equation (BIE). By the application of the dual discontinuity method – a special formulation of the Dual BEM – the discontinuities of the displacements and the tractions at the crack are utilized. For the crack surface contact a unilateral contact is assumed in the normal direction and the tangential behavior is described by Coulomb’s frictional law. In order to regularize the hard contact formulation the penalty method is utilized. An incremental iterative procedure based on a radial return mapping algorithm is applied for the solution of the non-linear problem. For the simulation of crack propagation an implicit time integration scheme in terms of a predictor-corrector procedure is applied. Numerical examples are presented to analyze the influence of friction on the deformation of the structures and on the behavior of cracks. Introduction A fracture mechanical analysis for the assessment of structural integrity of components includes the simulation of three dimensional crack growth. Due to the non-linear behaviour of crack growth an incremental procedure is needed for the simulation of fatigue crack propagation, cf. Fig. 1. Each increment starts with a stress analysis of the current crack configuration including the calculation of the fracture mechanical parameters. Next, the 3D crack growth criterion based on linear-elastic fracture mechanics is evaluated for the determination of the new position of the crack front. Finally, the mesh of the numerical model is adapted to consider the new crack geometry in the next increment.

Fig. 1: Three steps of an increment.


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Due to its nature the boundary element method (BEM) in terms of the dual BEM [1, 2] is especially suited for stress concentration problems. Here, a special formulation for cracked structures of the BEM – the dual discontinuity method (DDM) [3, 4, 5, 6] – is utilized. This method offers two advantages. On the one hand, the numerical complexity is reduced. On the other hand crack surface interaction can be easily considered by this method. Since the crack growth criterion is mainly based on the fracture mechanical parameters these values are determined from the stress field in front of the crack front by a sophisticated extrapolation method [6, 7]. The 3D crack growth criterion is evaluated for the determination of the new crack front geometry [8, 9]. Since the crack surface interaction causes non-proportional mixed mode a complete characteristic load cycle has to be evaluated for the calculation of the cyclic stress intensity factors (SIFs). If only the state of stress of the current crack geometry is taken into account a linear prediction of the new crack front is obtained. In order to consider the changing stress field between two discrete crack fronts corrector steps are required. This leads directly to an implicit time integration scheme realized in the framework of a predictor-corrector procedure which additionally yields an optimization of the new crack front with respect to its shape and location [8, 9]. Finally, the numerical model has to be adjusted to the new crack geometry. Since the BEM is utilized this task is less complicated compared to volume orientated methods, see Fig. 1c. For surface breaking crack fronts the discretization of the outer boundary around the surface breaking points has to be adjusted. It is done by a local re-meshing procedure [7] using a direct paving algorithm [10]. The interaction of the crack surfaces causes a non-linear reaction of the structure on the applied load. In order to consider this behaviour an incremental procedure has to be applied in the stress analysis. In principle the state of contact is not a-priori known and has to be determined iteratively within each increment. For an efficient determination of the state of contact the hard contact formulation has to be softened. In this work the well-known penalty method is utilized. Within this method the contact tractions are defined via a constitutive law with respect to penetrations of the hard contact formulation. Therewith, the frictional contact problem is solved by a radial return mapping scheme [11, 12, 13]. Stress analysis The elastic boundary value problem is described by the Lamé-Navier equation (1) as well as by prescribed Neumann and Dirichlet boundary conditions. The tractions at the boundary are defined via the Cauchy formula with respect to the stress tensor and the outward orientated normal vector. Since the solution of the contact problem requires integration in time the rate formulation is utilized. The dots on the field variables denote the total time derivative of the quantities. By the introduction of the discontinuities of the displacements and the tractions ,

(2) (3)

at the crack surfaces as new variables in the framework of the dual discontinuity method (DDM) the strongly singular displacement boundary integral equation (BIE) is given by

522


(4)

for source points at the outer boundary

and by

(5)

for source points at the crack surfaces and . The free term depends on the geometry at the source point. and denote the known Kelvin fundamental solutions. Usually, the displacement and the displacement discontinuity at coincident points at the crack are unknown. However, the evaluation of eq (5) is identical for the coincident points and the problem is not solvable. To overcome this problem, the hypersingular singular traction BIE

(6)

is additionally applied. The superscripts n, c and c at the field variables denote the part of the whole surface at which the corresponding point is placed. For the application of the BEM only the surface has to be discretized with boundary elements. The normal surface is meshed with continuous elements. For the evaluation of the hypersingular traction BIE (6) on the crack -continuity for the tractions and -continuity for the displacements at the collocation points are required and a smooth boundary is assumed [14]. Therefore, discontinuous elements are utilized at the crack. In case of surface breaking cracks, the transition to the crack surface is meshed with edge- and node-discontinuous elements. The relevant BIEs are evaluated within the framework of a collocation procedure. Relevant in this context means that the displacement BIE (4) is applied for source points at the outer boundary. If Dirichlet boundary conditions are prescribed at the source point on the displacement BIE (5) is evaluated. Otherwise, the traction BIE (6) is applied. After a rearrangement according to the boundary conditions the linear system of equations (7) is obtained [4, 5, 6]. The third part of the equations in (7) results from the evaluation of the remaining BIE for the source points at the crack surface . Here, denotes the identity matrix. The vector contains all unknown boundary values of the normal boundary and the unknown discontinuities at the crack respectively the unknown boundary values at . Obviously, only the reduced system of equations (8) has to be solved. The remaining unknowns are directly calculated in a post processing step via


523

(9) to have all field quantities available e.g. for the visualization. Crack surface interaction Since the BEM in terms of the DDM deals directly with the relative displacements (displacement discontinuities) and the discontinuities of the tractions it is especially suited for the consideration of interaction effects of the crack surfaces. The boundary values at the crack are defined with respect to a local orthogonal cartesian coordinate system at the collocation points that is orientated in the normal and tangential directions in order to to distinguish between the different behaviour in these directions. In the present context it is assumed that the crack surfaces belong initially to the Neumann boundary and no prescribed traction acts on them. Therefore, only tractions due to the crack surface contact occur at the crack. Frictional contact. For the behaviour of the crack surfaces in the normal direction a unilateral contact is assumed. That means the crack surfaces can not penetrate each other and the gap must fulfil the condition (10) Due to the definition of the displacement discontinuities (2) condition (10) is written as (11) for the normal displacement discontinuity the normal direction are possible at :

. Furthermore, only compressive contact tractions in (12)

The behavior in the tangential direction is described by a frictional law. In the present context Coulombs frictional law is applied. Within this criterion the effective tangential traction (13) is limited with respect to the compressive normal traction

and the frictional coefficient : (14)

In case of stick is smaller than . Otherwise, in case of slip is equal to and the direction of the tangential traction is opposite to the relative sliding direction such that energy is dissipated: (15) Boundary integral equations for crack surface contact. Contact tractions in normal and tangential direction as a result of the crack surface interaction have to be considered. Due to the principle of action and reaction the contact tractions are equal according to the amount but with opposite sign (equilibrium condition). Therewith, the discontinuities of the contact tractions vanish: respectively

(16)

By the consideration of this condition within the BIEs (4) - (6) the integrals concerning the traction discontinuities vanishes similarly as initial traction free crack surfaces are assumed.

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Therefore, the contact tractions remain only in the traction BIE (12) in an integral free way. In case of stick these tractions can be directly calculated in a post-processing step. In the slip mode the tangential tractions are known from the frictional law. Penalty method. For an efficient determination of the state of contact during the simulation the hard contact formulation is soften in terms of the penalty method. By the consideration of eq (12) a linear law given by (17) with a constant normal contact stiffness is assumed for the normal contact. In case of contact – stick or slip – the total tangential relative displacements are composed by a reversible elastic part and the slip : (18) Between the reversible displacements and tangential tractions also a linear constitutive law with the constant tangential contact stiffness is assumed: (19) Within the traction BIE (6) the total time derivative of the traction vector is required. These values are determined by a linearization procedure of the constitutive equations (17) and (19) under consideration of the frictional law (14) resulting in

(20)

The matrix depends on the state of contact and is determined during the solution procedure [11, 12]. Solution of the contact problem. Due to the non-linear behaviour of the contact problem a time integration of the rate formulation in terms of an incremental iterative procedure has to be applied. Therefore the rate values marked by a dot become incremental values , . Along this the linear system of equations (14) is written as (21) Since the system matrix of the increment consists of the matrix it is not constant during the simulation. It depends on the state of contact of the collocation points at the crack which has to be iteratively determined within each increment by a radial return mapping algorithm [11, 12, 13].The total boundary values of the increment are calculated by the accumulation of the incremental values respectively

(22)

Example A single edge crack (SEC) specimen of the material steel as sketched in Fig. 2 is investigated. It is loaded by a constant compressive force of . This force ensures that the crack surfaces are in contact. Beside this, the specimen is loaded with a torsional moment that increases from to . Then, this moment is decreased to . Overall, this procedure is successively applied four times.


525

Fig. 2: Geometry and loading of SEC-specimen Exemplarily for all points of the crack front the state of stress at the point which is located at the mid of the crack front is analyzed. Due to the symmetry of the problem only has to be considered at this point. Fig. 3 shows the -value versus the acting torsional moment for different frictional coefficients. For the -value is directly linked to the torsional moment. Otherwise, hysteresis curves are observed that exhibit counter-clockwise curves in this diagram.

Fig. 3:

at

for different frictional coefficients.

References [1] A. Portela, M.H. Aliabadi and D.P. Rooke Int. J. Num. Meth. Eng., 33, 1269-1287 (1992). [2] Y. Mi and M.H. Aliabadi Eng. Anal. Bound. Elem., 10, 161-171 (1992). [3] T.A. Cruse Boundary element analysis in computational fracture mechanics, Kluwer Academic Publishers (1988). [4] A. Cisilino and M.H. Aliabadi Int. J. Pres. Piping, 70, 135-144 (1997). [5] A. Cisilino and M.H. Aliabadi Int. J. Num. Meth. Eng., 42, 237-256 (1998). [6] P. Partheymüller, M. Haas and G. Kuhn Eng. Anal. Bound. Elem., 24, 777-788 (2000). [7] K. Kolk Automatische 2D-Rissfortschrittssimulation unter Berüsichtigung von 3D-Effekten und Anwendung schneller Randelementformulierungen, VDI Fortschritt-Berichte (2005). [8] W. Weber, P. Steinmann and G. Kuhn Int. J. Fract., 149, 175- (2008). [9] W. Weber, K. Willner and G. Kuhn Eng. Frac. Mech., to appear (2010). [10] T.D. Blacker and M.B. Stephenson Int. J. Num. Meth. Eng., 32, 811-847 (1991). [11] T.A. Laursen: Computational Contact and Impact Mechanics, Springer (2003). [12] P. Wriggers: Computational Contact Mechanics, John Wiley & Sons Ltd (2002). [13] K. Willner: Kontinuums- und Kontaktmechanik: Synthetische und Analytische Darstellung, Springer (2003). [14] M. Guiggiani, G. Krishnasamy, F.J. Rizzo and T.J. Rudolphi J. Appl. Mech., 59, 604-614 (1992).

526


Fatigue Crack Growth in Functional Graded Materials by Meshless Method P.H. Wen1a and M.H. Aliabadi2b 1

Department of Engineering, Queen Mary, University of London, London, UK, E1 4NS 2

F Department of Aeronautics, Imperial College, London, UK, SW7 2BY a

[email protected], [email protected]

Keywords: Meshless method, stress intensity factors, enriched radial basis functions, fatigue crack growth, Paris law, functionally graded materials. Abstract. This paper presents the investigation for fatigue crack growth path prediction using the meshless method. Based on the variational principle of the potential energy, meshless method has been implemented with enriched radial basis interpolation functions to evaluate mixed-mode stress intensity factors, which are introduced to capture the singularity of stress at crack tip. Paris law and the maximum principle stress criterion are adopted for defining the growth rate and direction of the fatigue crack growth respectively. Constant crack length increment and tangential approach are utilized in numerical examples. Validations are demonstrated by two numerical examples of plates with an edge crack.

Introduction Recently, meshless approximations have received much interest since Nayroles et al [1] proposed the diffuse element method. Later, Belyschko et al [2] and Liu et al [3] developed the element-free Galerkin method (EFGM) and reproducing kernel particle methods, respectively. A key feature of these methods is that they do not require a structured grid and are hence called meshless. Recently, Atluri and co-workers presented a family of Meshless methods, based on the Local weak Petrov-Galerkin formulation (MLPGs) for arbitrary partial differential equations by Atluri [4] with moving least-square (MLS) approximation. MLPG is reported to provide a rational basis for constructing meshless methods with a greater degree of flexibility. Local Boundary Integral Equation (LBIE) with moving least square and polynomial radial basis function (RBF) has been developed by Sladek et al [5, 6] for the boundary value problems in anisotropic non-homogeneous media. Both methods (MLPGs and LBIE) are meshless, as no domain/boundary meshes are required in these two approaches. A number of studies have investigated crack propagations using meshless method. The application of meshless method to linear elastostatic fracture mechanics, i.e. evaluation of stress intensity factors and analysis of crack growth, were demonstrated by Fleming et al [7], Rao et al [8] and Duflot and NguyenDang [9] using enriched basis function in the moving least square interpolation. Wen and Aliabadi [10] developed a meshless method using enriched radial basis functions and satisfactory accuracy has shown to be obtained for both static and dynamic problems. This paper aims to address the applications of meshless method in fracture mechanics with enriched radial basis function and develop this method to crack growth processes. The formulation and numerical implementation of meshless method for crack growth process are presented. An enriched radial basis function has been introduced to catch up the singularity of stresses near crack tip. The crack growth is simulated simply by adding two new nodes, i.e. one node indicates new crack tip and one is split from previous crack tip. Compared with classic finite element method, meshing and re-meshing for crack propagation problems are not necessary in this approach. The incremental crack length is obtained by integrating the Paris law for a given increment of load cycles. Two numerical examples are presented to demonstrate the validity of the proposed method.

Variational principle of potential energy Consider homogeneous anisotropic and linear elasticity, the relationships between stresses and strains by Hooke’s law are given by


527

V ij (x) C ijkl (x)H kl (x) C ijkl (x)u k ,l (x), (1) where H kl u k ,l u l ,k / 2, and C ijkl denotes the elasticity tensor which has the following symmetries C ijkl

C jikl

C klij .

(2)

For a homogeneous isotropic solid, we have

O (x)G ijG kl P (x)G ikG jl G ilG jk

Cijkl (x)

(3)

where and µ are the Lame’s constants, which are functions of coordinates x in general case. For the orthotropic plane-strain state, Hooke’s law can also be written, in matrix form, as

V 11 ½ ° ° ®V 11 ¾ °V ° ¯ 11 ¿

H 11 ½ ° ° D®H 22 ¾ °H ° ¯ 12 ¿

D

(4)

where

Q (y ) 1 Q (y )

ª « 1 « E (y )[1 Q (y )] « Q (y ) [1 Q (y )][1 2Q (y )] «1 Q (y ) « « 0 «¬

D(y )

1 0

º » » » 0 » 1 2Q (y ) » » 2[1 Q (y )] »¼ 0

(5)

in which, E(y) is the Young’s modulus and (y) the Poisson’s ratio. Both them are functions of the position of analysis point. With the shape functions, the displacements u(y) at the point y can be approximated in terms of the nodal values in a local domain, called as support domain, as n

¦I

u i (y )

k

(y, x k )uˆ ik

(y, x)uˆ i

(6)

k 1

where

Î1 (y, x1 ), I 2 (y, x 2 ),..., I n (y, x n )`

( y , x)

^

(7)

^

`

n T i

`

uˆ , uˆ ,..., uˆ , i 1,2 at point x k x , x , where k 1,2,..., n(y ), I k the and the nodal value uˆ i shape function and n(y) the number of nodes in the local supported domain. For the two dimensional elasticity, we can rearrange the above relation in a matrix form as

û1 , u 2 `T

u(y )

ª « ¬0

( y , x)

uˆ

ûˆ , uˆ , uˆ 1 1

1 2

2 1

1 i

2 i

(k ) 1

(k ) 2

(y, x)uˆ 0º » ¼

(8)

ªI1 0 I 2 «0 I 0 1 ¬

, uˆ 22 ..., uˆ1n , uˆ 2n

... I n

0

I 2 ...

`

0

0º I n »¼

(9)

T

(10)

Therefore, the relationship between strains and displacements is given by

(y )

ª wI1 « « wy1 « 0 « « wI « 1 ¬« wy 2

0 wI1 wy 2 wI1 wy1

wI 2 wy1 0 wI 2 wy 2

0 wI 2 wy 2 wI 2 wy1

...

wI n wy1

...

0

...

wI n wy 2

º 0 » » wI n » uˆ wy 2 » wI n »» wy1 ¼»

B(y, x)uˆ .

(11)

Considering the variation of the total potential energy, with respect to each nodal displacement yields 2×N a linear algebraic equation system in a matrix form as >K @2 N u2 N uˆ 2 N f 2 N (12) where N is the number of node in the domain and on the boundary. The stiffness and mass matrices can be written as

528


³B

K

T

(y , x)D(y )B(y , x)d:(y )

(13)

:

x i i 1,2,...N , and nodal force vector is defined by

for nodes x

³

f

T

:

(y , x)b(y )d:(y ) ³ T (y , x)t (y )d*(y )

^b1 ,b2 `T

where b

(14)

*V

is the vector of body force, vector of traction t

^t1 , t 2 `T ,

in which t i

V ij n j , ni

denotes a unit outward normal vector, denotes the boundary on which the traction is given. For a concentrated force acting at node i, we may determine the nodal force vector directly by

^F , F `

i T 2

i 1

fi

(15)

The approximation scheme A sub-domain : y as shown in Figure 1 is introduced in a neighbourhood of point y and is defined as local support domain. In order to guarantee unique solution of the interpolation problem, The distribution of function u in the sub-domain : y over a number of randomly distributed notes ^x i `, i 1,2,..., n(y ) can be interpolated by t

n

¦R

u (y )

k

k 1

(y,x)a k ¦ Pj (x)b j

R (y , x)a P (y )b

(16)

j 1

node xi

sub-domain y

y2

*

y1

field point y

O Figure 1. Support domain y for RBF interpolation of the field point y in a cracked body. along with the constraints n

¦ P (x k

j

)a j

0,

1d k d t

(17)

j 1

where ^Pk `k t

t

1

is a basis for Pm 1 , the set of d-variate polynomials of degree d m 1 , and

§ m d 1· ¨¨ ¸¸ is the dimension of Pm 1 . A set of linear equations can be written, in the matrix form, as d © ¹ R 0 a Pb

where matrix

uˆ ,

P0T a

0

(18)


ª R ( x1 , x1 ) R ( x1 , x 2 ) « R(x , x ) R(x , x ) 2 1 2 2 « « . . « . . « « . . « ¬« R(x n , x1 ) R(x n , x 2 )

R 0 (x, x)

... R(x1 , x n ) º ... R(x 2 , x n ) »» » ... . » P0 (x) ... . » » ... . » ... R(x n , x n )¼»

Solving these equations in (27) gives

P

T 0

R 01 P0

1

P0T R 01uˆ ,

>

R 01 I P0 P0T R 01 P0

1

ª P1 (x1 ) P2 (x1 ) « P (x ) P (x ) 2 2 « 1 2 « . . « . « . « . . « ¬« P1 (x n ) P2 (x n )

529

... Pt (x1 ) º ... Pt (x 2 ) »» ... . » ». ... . » ... . » » ... Pt (x n )¼»

@

P0T R 01 uˆ

(19) where I denotes the diagonal unit matrix. To capture singular stresses at crack tip, enriched radial basis function has been selected as following

b

Rk ( y , x) where r

a

c2 y xk

R ( y, x)

r e D r

2

(20)

y y c ; c and are free parameters; y c ( y , y ) denotes the location of the crack tip. (c) 1

(c) 2

Substituting the coefficients a and b from (29) into (24), we can obtain the approximation of the field function in terms of the nodal values (22)

u i (y )

>

R (y, x)R 01 I P P T R 01 P

1

@

P T R 01 P(y ) P T R 01 P

1

P T R 01 uˆ

^

n

¦I k 1

k

(y )uˆ ik (21)

`

where R (y , x) ^R1 (y , x1 ), R2 (y , x 2 ),..., Rn (y , x n )` and P (y ) 1, y1 , y 2 , y12 , y1 y 2 , y 22 ... are the set of radial basis functions centred around the point y and the set of d-variate polynomials respective. It is worth noting that this shape function depends uniquely on the distribution of scattered nodes within the support domain and it has the property of the Kronecker Delta function. In addition, the inverse matrix of coefficient R 01 (x) depend on the distribution of the node xi located in the support domain only, therefore it is easy to evaluate the first and higher order derivatives of shape function from Equation (8). From Equation (21), we have n

u i , j (y )

¦I

k, j

uˆ ik

i 1,2

(22)

k 1

From Equation (20), we have

Rk , j ( y , x k ) P,1 (y ) P, 2 (y )

y j x (jk ) c2 y xk

2

^0, 1, 0,2 y1 , y 2 , 0,...` ^0, 0, 1, 0, y1 , 2 y 2 ,...`

y j y (jc ) § 1 · D r ¸e D r j 1,2 . ¨ r ©2 r ¹

(23) (24) (25)

Therefore, the first order derivative for displacements is of 1 / r singularities at crack tip. In addition, the introduction of the polynomial terms has very slight effect on the accuracy (see Wen and Aliabadi [11]), therefore we have, a R 01uˆ and b 0 simply.

Fatigue crack growth analysis To evaluate stress intensity factors, both J-integral and crack opening displacement (COD) are used in this paper. For a mixed-mode fracture problem and constant material properties, the stress intensity factors are related with the J-integral for plane-strain as following

1 Q 2 (y c ) 2 K I K II2 J E (y c )

(26)

530


³ Wn

where J

1

*'

t E u E ,1 d*' and *' is an arbitrary closed contour, oriented in the anti-clockwise

direction, starting from the lower crack surface to the upper one and incorporating the crack tip and W V DE H DE / 2 is strain energy density in the field. Alternatively SIFs, for plane strain problem, can be obtained

KI

E (y c ) 2S 'u 2 , K II 8[1 Q 2 (y c )] r0

E (y c ) 2S 'u1 8[1 Q 2 (y c )] r0

(27)

where r0 indicates the gap between calculation point on the crack surface to the crack tip. Obviously for the functionally graded materials, the arbitrary closed contour or the distance r0 should be small enough in order that the material properties are considered as constants. It is worth to point out that in this calculation of J-integral, the coordinate system should be a local system, i.e. direction 1 is along the direction of crack growth. As there are two variables in Equation (26), we need to introduce another equation. The simplest and most direct way is to introduce the ratio of the crack opening displacements (COD), i.e. \ (r0 ) 'u 2 / 'u1 , therefore the mixed-mode stress intensity factors can be evaluated by

E (y c ) J , K II [1 Q 2 (y c )](1 \ 2 )

KI

\K I

(28)

where a circle of radius r0 centred at the crack tip is selected to be a J-integral contour. In general case, the crack path is a curved smooth path. However, crack propagation is simulated by successive linear increments in this approach, which direction needs to be determined. Several criteria have been proposed to describe the local direction of mixed-mode crack growth. The maximum principal stress criterion is adopted in this paper. This criterion postulates that the growth of crack will occur in a direction perpendicular to the maximum principal stress (the minor principal direction at the crack tip). Thus, at each crack tip, the local direction of crack growth is determined by the condition that the shear stress is zero, that is K I sin T c K II (3 cos T c 1) 0 (29)

Tc

>

@

2 arctan \ r \ 2 8 / 4

(30) Obviously the direction of crack growth depends on the ration \ only near the crack tip. The equivalent stress intensity factor is defined for the mixed-mode fracture as follows

K eq (a )

K I cos 3

Tc 2

3K II cos 2

Tc 2

sin

Tc

(31)

2

and the growth rate of crack are computed by the generalized Paris’ law

da dN

C'K eqm (a)

where 'K eq

(1 R) K eq ,max , R

(32)

K eq ,mix / K eq ,max

V min / V max , C and m are empirical constants

characteristic of the material. The constant m, Paris constant, is typically in the range 3-4 for common steal and aluminum alloys. At the end of each incremental analysis, the equivalent stress intensity factors are calculated and the number of cycles 'N i necessary to growth an arbitrary crack-extension increment size

'a is obtained by a 'a 1 i 'N i 'K eq m (a)da . C a³i

(33)

Numerical examples A square plate under a mode II fatigue load A square plate (20cm×20cm) was investigated for a mode II fatigue load by Kim and Lee [12]. Constant cyclic mode II traction in the range from 0 to 165MPa is applied on the upper and lower left edge of the plate in the opposite direction. In this case, 41×41 notes uniformly distribute in the domain and 17


531

properties are: E 210GPa , Q 0.3 , 200MN/m1.5 . The fatigue crack growth is simulated by the tangential technique with crack increment length of 2.5 mm ( 'a ). The initial crack

extra

nodes

around

the

crack

1.886 u 10 12 m/(MN/m1.5 ) m , m

C

tip.

The

material

3.0 and K c

lengths are specified as 1cm, 2cm, 4cm and 8cm respectively with the same incremental crack growth. The cycling number against the crack length a (mm) is presented with different free parameter in Figure 2. The stress intensity factors K I ( a ), K II ( a ) in unit of MN/m1.5 and crack growth paths are shown in Figure 3. Two different crack increments were used by Kim and Lee [12] in terms of the increment of cycling number, i.e. 'N 200 and 'N 4000 respectively. Figure 3 shows the growth paths for these four different initial cracks when free parameter is taken to 1. In addition, the selection of free parameter has limited effects on the results of SIFs and the cycling number. Obviously the growth path of the 1cm initial crack forms a sharp curve with a small radius of curvature due to the effect of boundary. In this case, the bending effect dominates the crack propagation direction. Very good agreement has been achieved both for the number of load cycles and crack growth paths with the results given by Kim and Lee [12] using the boundary element method. 6

165MN/m 2

=0.0 Series1 5

=0.1 Series2

10cm a (cm)

=1.0 Series3

a0

4

Series4 Kim and Lee (N=200) [12] Series5 Kim and Lee (N=4000) [12]

3

10cm 2

1 0.E+00

20cm

1.E+04

2.E+04

3.E+04

4.E+04

5.E+04

6.E+04

Number of load cycles N

Figure 2. Geometry of a edge cracked square plate under a mode II load and cycling number vis crack length. 400

10

Series2

=1 Series3

Series4

=10 Series5

Series6

Kc Series7

a0 a0 5

KI

y2 (cm)

KI, KII (MN/m1.5)

300

=0 Series1

200

1 cm 2 cm

a0

4 cm

a0

8 cm

0 0

5

10

15

20

Kim and Lee [12]

100

-5

KII 0 1

2

3

4

a (cm)

5

6

-10

y1 (cm)

Figure 3. Stress intensity factors vs crack length with different free parameters and growth paths with different initial crack lengths.

532


Functionally graded material cracked square plate under a mode II fatigue load The plate of same geometry and boundary conditions in above is investigated for a mode II fatigue load with considering functionally graded materials. Constant cyclic mode II traction with ratio R 0 is applied on the upper and lower left edge of the plate in the opposite direction. It is assumed that Young’s modulus E is function of coordinate except Poisson’s ratio Q ( 0.3) . In this example, Young’s modulus E has an exponential variation in y1 as

E 0 exp( E y1 / b)

E (y )

(34)

where E ln( E1 / E 0 ), with E 0 and E1 corresponding to the E-value at y1 0 and y1 b (=20cm) respectively. The nodes are uniformly distributed (41×41) in the domain and 17 extra nodes around the crack tip are used. The fatigue crack growth is simulated using the tangential technique with crack increment length of 2.5 mm ( 'a ). The initial crack lengths are selected as 1cm and 4cm respectively with the same incremental crack growth. We considered five ratios of K E1 / E 0 for each initial crack length, i.e. K equals 0.1, 0.5, 1, 5 and 10 respectively. Crack growth paths for the initial crack length of 1cm are shown in Figures 4 for each ratio of K . When ratio E1 / E 0 ! 1 , the crack propagation becomes unstable for a large crack length. It is because that the crack tip is too close to the boundary and therefore a fine increment of crack length needs to be introduced. Results by Kim and Lee for constant Young’s modulus are presented in the same figure for comparison. In general case, SIF of mode I increases when ratio E1 / E 0 decreases. Similar conclusion can be observed for the initial crack length of 4cm. Growth paths are presented in Figure 4 and all of them are stable for different ratios of young’s modulus. In the case of small ratio K , mode I stress intensity factors are increased significantly. Therefore, we can conclude that the functionally graded material has large influence both on crack growth path and stress intensity factors. y1 (cm)

y1 (cm)

0

0 0

2

4

6

8

10

0

a0

-2

2

4

6

8

-2

a0 E1 / E0

-4

5

Kim and Lee [12]

-6

y2 (cm)

y2 (cm)

10

0.1

-4

-6

Kim and Lee [12]

-8

-8

E1 / E0 E1 / E0 -10

10

E1 / E0

0.1

1

E1 / E0

0.5 -10

10

5

E1 / E0

0.5

1

Figure 4. Fatigue crack growth paths for functionally graded materials, left: initial crack length 1cm; right: initial crack length 4cm.

Conclusion This paper has presented the meshless method applied to two-dimensional fracture mechanics and fatigue crack growth. Apparently this method does not require any discretizations of the domain including the boundary. By introducing the enriched RBF interpolation, the singularities of stress in the order of

1/ r can be captured at the crack tip. Functionally graded materials are introduced in the numerical modelling and their effects both on the stress intensity factors and crack growth path are significant. Computational procedures are demonstrated by three examples. Satisfactory accuracy is shown to be achieved compared with the results by BEM et al. Conclusions can be summarised as following: 1. The meshless method is efficient for mixed-mode crack propagation problem analysis;


533

2.

The enriched RBFs are flexible and simple to program with little increase in computational efforts; 3. The system stiffness matrix is symmetric and sparse; 4. The accuracy of the number of load cycles dependents on the crack increment. Although this paper analyses the fatigue crack growth for two dimensional edge crack problems, the proposed method can be easily extended to the embedded crack problems and the crack closure problems with friction between two crack surfaces under compressive load. References [1] B. Nayroles, G. Touzot and P. Villon Computational Mechanics, 10, 307-318 (1992). [2] T. Belytschko, Y.Y. Lu and L. Gu Int. J. Numerical Methods in Engineering, 37, 229-256 (1994). [3] W.K. Liu, S. Jun and Y. Zhang Int. J. Numerical Methods in Engineering, 20, 1081-1106 (1995). [4] S.N. Atluri The Meshless Method (MLPG) for Domain and BIE Discretizations, Forsyth, GA, USA, Tech Science Press (2004). [5] V. Sladek, J. Sladek, M. Tanaka and Ch Zhang Engng Analy with Boundary Elements, 29, 829-843 (2005). [6] J. Sladek, V. Sladek, P.H. Wen and M.H. Aliabadi CMES: Computer Modeling in Engineering & Sciences, 13, 103-117 (2006). [7] M. Fleming, Y.A. Chu, B. Moran, T. Belytschko, Y.Y. Lu and L. Gu Int. J. Numerical Methods in Engineering, 40, 1483-1504 (1997). [8] B.N. Rao and S.A. Rahman Int. J. Pressure Vessels and Piping, 78, 647-657 (2001). [9] M. Duflot and H. Nguyen-Dang Int. J. Numerical Methods in Engineering, 59, 1945-1961 (2004). [10] P.H. Wen and M.H. Aliabadi Durability of Structures and Health Monitoring, 3 (2), 107-119, 2007. [11] P.H. Wen and M.H. Aliabadi Communications in Numerical Methods in Engineering, 24 (8), 635-651 (2008). [12] K. Kim and H. Lee Int. J. Numerical Methods in Engineering, 72, 697-721 (2007).

534


An Analysis of Elastic plates under Concentrated loads by NonSingular Boundary Integral Equations Kuang-Chong Wu 1 and Ze-Ming Chang 2 1

2

Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan, [email protected]

Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan

Keywords: concentrated loads, elastic plate, boundary integral equation.

Abstract. The objective of this paper is to discuss the stress distributions in an infinite elastic plate subjected to concentrated loads. The loading cases considered include: (a.) a horizontal concentrated force at the center of the plate; (b.) a vertical concentrated force on the plate surface; previous studies on the subject were mostly for isotropic materials using Fourier series expansion. In this study the elasticity solutions are separated into two parts. The first part is the solution obtained by the mechanics-of-materials approach, while the second part is a correction term, which is calculated using nonsingular boundary integral equations. The method is applied to calculate the stress distributions for isotropic and orthotropic plates. The numerical results agree well with the existing results.

Introduction With respect to a rectangular coordinate system (xΌ,x΍), let an infinite plate occupy the region -Ќ < xΌ < Ќ,

0

x΍ h. The paper is concerned with the solution for the plate in the presence of a concentrated load as shown in Fig. 1.

x2

F

h

x1 Figure 1: An infinite plate under a concentrated loadʳ

Filon [1] used Fourier series to obtain an approximate solution for a finite strip, -L xΌ L, 0 x΍ h, under concentrated loads. In the case of an infinite strip, for which L becomes infinite, Filon's solution is expressed in terms of Fourier integrals. A more detailed study of the stress distribution near the point of a transverse concentrated load on the surface of the plate was made by von Karman and Seewald [2]. A similar analysis for a longitudinal concentrated force in the middle of the plate was made by Howland [3]. The aforementioned works are all for isotropic materials. Filon's approach was extended to orthotropic materials by Yu [4] and to general


535

anisotropic materials by Wu and Chiu [5]. Here an alternative formulation is developed. The elasticity solutions are separated into two parts. The first part is the solution obtained by the mechanics-of-materials approach, while the second part is a correction term, which is calculated using nonsingular boundary integral equations developed by Wu [6]. The method is applied to calculate the stress distributions for isotropic and orthotropic plates subjected to a longitudinal force in the middle or a transverse load on the surface. Numerical examples are given to validate the proposed method.

Integral Equations For a two-dimensional anisotropic elastic body D bounded by a contour C, Wu et. al. [7] derived the following dual set of boundary integral equations:

E d ( x, m ) E t ( x, n )

ˆ (x ) Ut ˆ (x )]ds [ W ˆ (x )]dA, ³ [ Wd ³ ˆ (x ) Uf ˆ (x ) W ˆ t (x )]ds [V ³ [Vd ³ ˆ (x ) Wˆ f (x )]dA, *

'

*

'

'

C

'

D

*

'

T *

'

'

C

T

'

(1)

D

where d is the gradient of the displacement u at x = (xΌ, x΍) in the direction of m = (mΌ, m΍), t is the traction on the plane normal to n obtained by rotating m by 90 degrees counterclockwise, E =1, if x D , E = 1/2, if x C , s is the arc length of C, d* is the tangential gradient of the displacement and t* is the traction at x' on C, is the dislocation density and f is the body force in D, ˆ (x, m, x ' ) 1 Im[ A m1 p*m2 BT ], W z* z*' ( s ) S ˆ ( x, m , x ' ) U

1

S

Im[ A

m1 p*m2 AT ], z* z*' ( s )

(2)

m1 p*m2 T B ]. z* z*' ( s ) The direction of increasing s is such that when describing a circuit around the contour the domain of ˆ ( x, m , x ' ) V

1

S

Im[B

interest is to the left. In eq (2) Im denotes the imaginary part, A = (aΌ, a΍, aΎ), B = (bΌ, b΍, bΎ), and m1 p*m2 z* z*' ( s )

represents a diagonal matrix with the diagonal elements given by

m1 pk m2 , k =1, ,2 3. zk zk' ( s )

Here zk x1 pk x2 and Im[ pk ] > 0 and ak, bk and pk are determined by the following eigenvalue problem: § N1 N 2 · § a · §a· (3) p¨ ¸, ¨ T ¸¨ ¸ N N b ©b¹ 1 ¹© ¹ © 3 where N1

T1R T , N 2

T1 , N 3

RT1R T Q,

Qik =Ci1k 1 , Rik =Ci1k 2 , Tik =Ci 2 k 2 ,

(4)

536


and C is the elasticity tensor. For x C , either set of equations in eq (1) can be used to solve for either the unknown boundary displacement gradients or the tractions from the specified boundary data. For the problem of interest it is more convenient to use an alternative form of eq (1) given by [6] for x C * ' *T ' ˆ *d* (x ' ) W ˆ *T t * (x ' ))dsc (V (5) d* (x) L1ST t * (x) ³ (V ³ ˆ (x ) Wˆ f (x ))dA, C

D

where 1

ˆ * (x, m, x ' ) V

S 1

ˆ * ( x, m , x ' ) W

S

Im[ BT

1

Im[ BT

m1 p*m2 BT ], z* ( s ) z*' ( s ' ) m1 p*m2 AT ], z* ( s ) z*' ( s ' )

1

(6)

1 Re ª« BT AT º» ¬ ¼ ˆ * is non-singular as x coincides with x' [6]. Note that V

L1ST

A Concentrated Force in A Plate Consider first the case where a central longitudinal force F

F G x1 G x2 h / 2 e1 is applied, where

G is the Dirac delta function and eD is the base vector along the xD direction. Let the resulting stress be expressed as (7) V ij V ijc V ij s , where V ij s is the stress given by

V 11 s

FH ( x1 ) / h, V 12 s

V 2 2s

(8)

0.

The corresponding displacement gradients along the x2 -axis is: wu1 / wx2

0, wu2 / wx2

s

s

S12c FH x1 / h,

(9) s

where SD 1c is the reduced compliance and H is the unit step function. In view of eq (8) and (9), V ij may also be regarded as the stress due to the body force f s and the dislocation density s in the plate given by s s f G x1 Fe1 / h, S12c FG x1 e 2 / h. (10) Thus the stress V ijc can be computed from the self-equilibrium force system F f s and the dislocation density s using eq (5). Let CΌ: -Ќ < xΌ >

@ @

where both opposite crack-faces are considered as a set of corresponding parallel capacitors. If both crackfaces are considered as electrically impermeable, eq. (10) simplifies to Di (x *c , t ) Di (x *c , t ) 0 . (11) In contrast, if both crack-faces are treated as electrical permeable, eq. (10) leads to Di (x *c , t ) Di (x *c , t ) , M( x *c , t ) M(x *c , t ) 0 . (12) In the sense of the weighted residual, the time-domain displacement Galerkin-BIEs for a cracked solid can be written as

³ \(x) u (x, t ) d* ³ \(x) ³ >u J

x

*b

*b

³

G IJ ( x, y , t ) t I ( y , t )

³

\ (x) t GIJ (x, y , t ) 'u I (y , t ) d*y d*x , *b

*

c

@

t GIJ (x, y , t ) u I (y , t ) d*y d*x

*b

(13)


543

where uIJG(x,y,t) und tIJG(x,y,t) are the dynamic displacement and traction fundamental solutions, (x) is the test function, b=u+t,, and an asterisk denotes the Riemann convolution defined by t

³ g(x, t W)h (x, W) dW .

g ( x, t ) h ( x, t )

(14)

0

The time-domain traction Galerkin-BIEs are obtained by substituting eq. (13) into the constitutive equations (2) and taking the limit process xc±

³ \(x) t (x, t ) d* ³ \(x) ³ >v J

x

*

*

c

c

³ \ ( x) ³ w

* c

G IJ ( x, y , t ) t I ( y , t )

@

w GIJ (x, y , t ) u I (y , t ) d*yd*x

*b

G IJ ( x, y , t ) 'u I ( y , t )

d*yd*x ,

(15)

* c

where vIJG(x,y,t) and wIJG(x,y,t) are the traction and the higher-order traction fundamental solutions, which are defined by v GIJ (x, y , t ) C pIKse p (x)u GKJ , s (x, y , t ) t GJI (x, y , t ) , (16) (17) w G ( x, y , t ) C e ( x ) C e ( y ) u G ( x, y , t ) . IJ

pIKs p

qJLr q

KL , sr

The displacement BIEs (13) are strongly singular, while the traction BIEs (15) are hypersingular. The dynamic fundamental solutions for homogenous and linear piezoelectric solids derived by Wang and Zhang [5] are implemented in the present TDBEM. It should be mentioned that the dynamic fundamental solutions cannot be expressed in closed-form. In 2D case, they can be represented by a line-integral over a unit circle. Integrating by parts and applying the properties of convolution integrals, the dynamic fundamental solutions can be divided into a singular static and a dynamic part. The singular static part can be reduced to an explicit expression while the dynamic part can be given only as a line-integral over a unit circle.

Numerical solution algorithm To solve the strongly singular displacement BIEs (13) and the hypersingular traction BIEs (15), a Galekinmethod is applied for the spatial discretization [6]. The external boundary and the crack are discretized by linear elements. At the crack-tips, square-root shape-functions are used to describe the local behavior of the CODs properly. This ensures an accurate and direct calculation of the dynamic intensity factors from the numerically computed CODs. Strongly singular and hypersingular boundary integrals are computed analytically by special techniques [2]. The temporal discretization is performed by a collocation method. By using linear temporal shape-functions, time integrations can be performed analytically. The line-integrals over the unit circle arising in the dynamic fundamental solutions are computed numerically by using standard Gaussian quadrature formula. After temporal and spatial discretizations and invoking the initial conditions and boundary conditions, an explicit time-stepping scheme can be obtained as K 1 º ª x K (C1 ) 1 «D1 y K (B K k 1 t k A K k 1 u k )» . (18) k 1 ¼ ¬ In eq. (18) yK is the vector of the prescribed boundary data, while xK represents the vector of the unknown boundary data, which can be computed time-step by time-step. The investigated initial-boundary value problem involves two different non-linear crack-face boundary conditions. An iterative crack-face contact analysis [1,4] at the time-step, where a crack-face intersection occurs, is performed to take eq. (8) into account in the numerical algorithm. Furthermore, at the time-step, where the crack is open, an additional iterative solution procedure is developed to solve the non-linear semi-permeable electrical crack-face boundary condition (10). It should be mentioned that the solution of non-linear crack-face boundary conditions is an application field where the BEM is very attractive and efficient, since the generalized tractions and displacements are primary variables in the BIEs.

¦

Numerical results To show the effects of the different crack-face boundary conditions on the dynamic intensity factors (IFs), several numerical examples are investigated. The following loading parameter is introduced in order to measure the intensity of the electrical impact

544


e 22 D 0 , (19) H 22 V0 with 0 and D0 being the loading amplitudes. For convenience, the dynamic intensity factors are normalized by using K I (t) K II ( t ) e 22 K IV ( t ) K *I ( t ) , K *II ( t ) , K *IV ( t ) , K 0 fV0 Sa , (20) K0 K0 H 22 K 0 F

where f=1 if a mechanical loading is applied and f=F in the case of a pure electrical loading. In the first example, we consider a rectangular plate with a central crack of length 2a as shown in Fig. 1. The plate is subjected to a pure electrical impact loading D(t)=D0H(t), where D0 is the loading amplitude and H(t) is the Heaviside step function.

x 1 D( t )

D( t )

x2

Poling direction 2a

w h

Fig. 1: A rectangular plate with a central crack under an electrical impact loading The geometry of the cracked plate is determined by h=20mm, 2w=h and 2a=4.8mm. The material PZT-5H with the mass density =7500kg/m3 and the elastic constants C11 126.0 GPa , C12 84.1GPa , C22 117.0 GPa , C66 23.0 GPa ,

(21) 6.5 C / m 2 , e22 23.3 C / m 2 , e16 17.0 C / m 2 , H11 15.04 C /(GVm) , H 22 13.0 C /(GVm) is considered. The spatial discretization of the external boundary is done by an element-length of 1mm and the crack is divided into 12 elements. A time-step of cLt/h=0.04 is used in all computations. The normalized dynamic intensity factors obtained by the present time-domain BEM for D0=1 and D0=-1 using impermeable (11) and permeable (12) electrical crack-face boundary conditions and plane strain assumption are presented in Fig. 2. e21

Fig. 2: Normalized dynamic intensity factors for D0=-1 and D0=1


545

The change of the direction in the applied electrical impact loading expressed in an opposite sign in the loading amplitude D0 leads only to a change of the sign in the normalized dynamic intensity factors and does not change their amplitudes. The mode-II dynamic intensity factor vanishes, since no shear stress components are induced in the case of the investigated loading, poling direction normal to the crack-face and transversally isotropic material behaviour. Since an electrical impact is applied the normalized mode-I dynamic stress intensity factor starts from a non-zero value due to the quasi-electrostatic assumption for the electrical field, which implies that the cracked piezoelectric plate is immediately subjected to the electrical impact and therefore the crack opens at t=0. In contrast, the elastic waves also induced by the electrical impact loading need some time to reach and excite the crack. It can be clearly seen, that both investigated pure electrical impact loadings lead to physically meaningless material interpenetrations in various time intervals which requires an advanced crack-face contact analysis. According to the geometry of the cracked plate and the applied loading, the dynamic intensity factors of both crack-tips are identically. The normalized dynamic intensity factors of the present time-domain BEM obtained for semi-permeable electrical crack-face boundary condition (10) and contact conditions (8) are shown in Fig. 3 for D0=1 and in Fig. 4 for D0=-1.

Fig. 3: Normalized dynamic intensity factors for D0=1 and non-linear crack-face conditions

Fig. 4: Normalized dynamic intensity factors for D0=-1 and non-linear crack-face conditions In contrast to the results given in Fig. 2, the applications of the contact conditions (8) and different electrical crack-face condition (10)-(12) lead to quite different curves of the mode-I and mode-IV intensity factors for opposite loading directions D0=-1 and D0=1. The normalized mode-I dynamic intensity factors are very similar without a significant difference between the impermeable, the permeable and the semi-permeable

546


electrical crack-face conditions using different permittivity c. In contrast, the normalized mode-IV dynamic intensity factors show a significant influence of the electrical permittivity c. The impermeable crack-face boundary condition (11) leads to the strongest electrical field near the crack-tips. As a consequence the behaviour of the mode-I and mode-IV intensity factors is quite different and the mode-IV intensity factors show only weak time dependence. For the permeable crack-face boundary condition (12), as though the crack does not exist for the electrical field and the curves of mode-I and mode-IV intensity factors have an identical behaviour. The results of the semi-permeable cracks-face boundary condition (10) are between the bounds given by the impermeable and permeable electrical crack-face boundary conditions, which can be clearly recognized for both considered electrical impact loadings. Further it can be observed that due to the coupling effects between the mechanical and electrical fields a negative mode-I intensity factor can be obtained even in the case of a small crack-opening-displacement u2(x,t)>0. In the second example as shown in Fig. 5, let us consider a rectangular piezoelectric plate with a central crack of length 2a subjected to a combined tensile impact loading (t)=0H(t) and an electrical impact loading D(t)=D0H(t) using F=-1.

Fig. 5: A rectangular plate with a central crack subjected to impact loading The numerical calculations are carried out for the geometrical parameters and the material properties given in the previous example. The external boundary of the piezoelectric plate is divided into elements with a length 1.0mm, the crack is discretized by 12 elements, and a time-step of cLt/h=0.04 is chosen. Plain strain condition is assumed. The normalized dynamic intensity factors of the present time-domain BEM are shown in Fig. 6.

Fig. 6: Normalized dynamic intensity factors for F=-1 and non-linear crack-face boundary conditions


547

In order to point out the influence of the scattered wave fields on the dynamic intensity factors, the corresponding static results are given in table 1. Table 1: Normalized static intensity factors for different electrical permittivity c Nc K I K II K IV 1 . 05 0 . 00 1.03 0.0 (imperm.) 0.1 1.05 0.00 0.66 0.5 1.05 0.00 0.13 2.0 1.05 0.00 0.37 5.0 1.05 0.00 0.59 1.05 0.00 0.81 f (perm.) Here again, the normalized mode-I dynamic intensity factors are very similar without a significant difference for all investigated electrical crack-face boundary conditions and permittivity c while the normalized mode-IV dynamic intensity factors are quite different which can be clearly seen in Fig. 6. Since no shear stress components are induced by the investigated loading the mode-II dynamic intensity factor vanishes. Due to the applied loading direction of the electrical impact loading a material interpenetration does not occur in the investigated time interval and therefore a contact analysis is not required. The mode-I dynamic stress intensity factor starts from a non-zero value with different amplitudes depending on the electrical permittivity c between the bounds of the permeable and the impermeable solutions. The difference between the mode-IV intensity factors for the impermeable, permeable and semi-permeable crack-face boundary conditions is more pronounced for the dynamic loading, which is induced by the scattered wave fields.

Summary This paper presents a hypersingular time-domain BEM for dynamic crack analysis in homogeneous and linear piezoelectric solids using non-linear mechanical and electrical crack-face boundary conditions. Cracks in finite piezoelectric solids subjected to different impact loadings are considered. A combination of the classical displacement BIEs and the hypersingular traction BIEs is applied in the time-domain BEM. A Galerkin-method is used for the spatial discretization, while a collocation method is implemented for the temporal discretization. The dynamic time-domain fundamental solutions for homogenous, anisotropic and linear piezoelectric solids are used. An iterative solution algorithm is developed to handle the non-linear semi-permeable electrical crack-face boundary condition. In order to avoid the possible crack-face contact a second iterative solution procedure is implemented. Numerical results are presented and discussed to investigate the influences of the different non-linear electrical and mechanical crack-face boundary conditions on the dynamic intensity factors.

Acknowledgement

This work is supported by the German Research Foundation (DFG, project-no. ZH 15/6-1 and ZH 15/6-3), by the Spanish Ministry of Science and Innovation (project DPI2007-66792-C02-02) and by the Council of Innovation, Science and Business of Junta de Andalucía (project P06-TEP-02355). The financial support is gratefully acknowledged.

References [1] Aliabadi M.H., The Boundary Element Method Volume 2, Applications in Solids and Structures. Computational Mechanics Publications, John Wiley & Sons, 2002. [2] Gray L.J., Evaluation of singular and hypersingular Galerkin boundary integrals: direct limits and symbolic computation. In: Advances in Boundary Elements, ed. V. Sladek, J. Sladek, Computational Mechanics Publishers: Southampton, UK, pp. 33-84, 1998.

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[3] Hao T.H., Shen Z.Y., A new electric boundary condition of electric fracture mechanics and its applications. Engineering Fracture Mechanics, 1994; 47: 793-802. [4] Phan A.-V., Napier J.A.L., Gray L.J., Kaplan T. Symmetric-Galerkin BEM simulation of fracture with frictional contact. International Journal for Numerical Methods in Engineering, 2003; 57: 835-851. [5] Wang C.-Y. and Zhang Ch., 3-D and 2-D dynamic Green’s functions and time-domain BIEs for piezoelectric solids. Engineering Analysis with Boundary Elements, 2005; 29: 454-465. [6] Wünsche M., García-Sánchez F., Sáez A., Zhang Ch., A 2D time-domain collocation-Galerkin BEM for dynamic crack analysis in piezoelectric solids. Engineering Analysis with Boundary Elements, 2010; 34: 377-387.


549

Fast BEM for 3-D Elastodynamics based on pFFT Acceleration Technique Z. Yan1, J. Zhang2 and W. Ye3 1

Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong, SAR

Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, P.R. China [email protected] 2


3


[email protected]

[email protected] Keywords: elastodynamics, accelerated BEM, pFFT acceleration technique, porous solids

Abstract. An efficient boundary element method accelerated by the precorrected-FFT technique for the modeling of 3-D elastodynamic problems is developed and described in this paper. A frequency-domain approach is employed in which the dynamic problem is transferred in its frequency domain and is solved using the developed method. Validation of the method is conducted via an example with analytical solution and the application of the developed method to the modeling of a porous solid under an oscillatory loading is also presented. Introduction Elastodynamics is a classical subject which has been encountered in a variety of engineering fields. For many practical problems particularly those of three dimensions, numerical simulation is the only viable approach for obtaining accurate solutions. For linear problems with complex 3-D domains, the boundary element method (BEM) provides an attractive approach since only the boundary of the problem domain needs to be discretized. Since its invention, the BEM has become one of the most powerful numerical methods for solving engineering problems and applications of the BEM can be found in many fields. One disadvantage of the classical BEM, as compared to the finite element method, is the final discretized linear system which unfortunately is fully populated. Such a feature, to a certain extent, has limited the applications of the BEM to large-scale problems. The celebrated acceleration techniques, such as the fast multipole method (FMM) [1], the precorrected Fast Fourier Transformation (pFFT) technique [2], and the combined fast Fourier transform and multipole method [3], when combined with the iterative linear system solvers, have greatly reduced the computational time as well as the memory required in solving the discretized system, making the BEM suitable for large-scale problems. Successful applications of the accelerated BEM can be found in the areas of microelectromechanical systems [4-5], composite materials [6-7], elastodynamics [8-10], etc. In this paper, the development of a fast BEM approach based on the pFFT acceleration technique for frequency-domain 3-D elastodynamic problems is described. The motivation for such a development is due to the fact that the pFFT acceleration technique is relatively kernel independent and is very easy to implement. The targeted applications of the present work are the dynamic response of porous solids under an oscillatory loading. The precorrected-FFT Boundary Element Method for Elastodynamics Boundary Integral Formulation

550

2


Consider a problem that is harmonic in time with an angular frequency of Z , the governing equation for the elastodynamics in the absence of body force is (1) ª¬ P 2 O P º¼ u x, Z UZ 2 u x, Z 0 , where O , P are the two Lame’s constants, U is the density and u

ª¬u1 ,u2 ,u3 º¼ is the displacement. Based

on the fundamental solution of eq. (1) corresponding to a unit harmonic body force in an infinite elastic space, an equivalent boundary integral formulation of eq. (1) can be derived which reads (2 ) cij [ ui [ , Z ³ ª¬Gij x, [ , Z W i x, Z Fij x, [ , Z ui x, Z º¼dS x ,

where W i x,Z is the traction at the field point x and cij [ is the solid angle at the evaluation point [ . On smooth boundaries, the coefficient cij [

0.5G ij . It is G ij when the evaluation point is inside the domain

and 0 when it is outside the domain. The subscripts i and j in the above equations denote the index of the degrees of freedom and the Einstein summation convention is implied. The fundamental solutions Gij and Fij are

1 ª aG br,i r, j º , ¼ 4SUcs2 ¬ ij

1 4S

Gij x, [ ,Z

Fij x, [ ,Z

(3)

· 2b § § b · § wr wr · wr · °§ r n n r 2r,i r, j ¸ 2b,r ¨ r,i r, j ¸ ®¨ a,r ¸ ¨ G ij r ¹ © wn , j i ¸¹ r ¨© j ,i wn ¹ wn ¹ © °¯© , 2 ½° · § 2b · § c p ¨© a,r b,r r ¸¹ ¨ c 2 2¸ r,i n j ¾ © s ¹ ¿°

(4)

with a and b

1 ° jks r § 1 1 · jk r § c 2 2 2 ¸ e p ¨ s2 ®e ¨ 1 j ¨c r ¯° k r k s sr ¹ © © p

·§ 1 1 · ½° j ¸¨ ¸¨ k r k 2 r 2 ¸¸ ¾ p ¹© p ¹ ¿°

1 ° jks r § 3 3 · jk r § c 2 ·§ 3 3 · °½ 2 2 ¸ e p ¨ s2 ¸¨1 j 2 2 ¸ ¾ . In eqs. (3) and (4), the symbol j denotes the ®e ¨ 1 j ¨ c ¸¨ r °¯ ks r ks r ¹ k r k r ¹¸ °¿ p p p © © ¹©

imaginary unit, n

> n1 n2 n3 @

T

represents the outward unit normal vector, r is the Euclidean distance

between [ and x, G ij is the Kronecker delta function, c p

O 2P U and

cs

P U denote the

velocities of the elastic dilatational wave and the shear wave respectively and k p , ks are the corresponding

wave numbers. One can also express kernel Fij as Fij x, [ ,Z Tikj x, [ ,Z nk . This form is particularly useful

in the implementation of the precorrected-FFT technique. For unsteady problems, by employing Fourier transformation, a time-domain problem can be transferred into a frequency-domain problem in which the transferred displacement satisfies the same governing equation as that for oscillatory elastodynamics, i.e., eq. (1). Hence the boundary integral formulation (2) and the developed method are equally suitable for solving the unsteady problem in its frequency domain. Precorrected-FFT Technique The precorrected-FFT technique is one of the popular acceleration schemes. This scheme is easy to implement and has the advantage of being relatively kernel independent. Similar to the other acceleration schemes such as the fast multipole method, the main idea of the precorrected-FFT acceleration scheme is that instead of computing the influence matrix entries corresponding to the far-field interaction explicitly and then multiplying them with the sources, the far-field interactions are computed via an approximate method. More specifically, in the precorrected-FFT (pFFT) technique, a parallelepiped is constructed to enclose a three-dimensional problem after it has been discretized into n surface panels. This parallelepiped is then subdivided into an array of small cubes so that each small cube contains only a few panels. It should be pointed out that the surface panels and the pFFT cubes can intersect with each other. There is no need to


551

maintain any consistency between the surface panels and the cubes. The acceleration of surface integration is achieved by exploiting the fact that the integrands in the surface integrals such as those in eqs. (3) and (4) have piecewise-smooth convolution form. Thus with the aid of the uniform grid formed by the cubes in the parallelepiped, these integrals can be computed approximately using the Fast Fourier Transform technique. To ensure accuracy, such an approximation is only employed for far-field interactions, that is, integrals in which the evaluation point is far away from the field panel. For nearby interactions, direct evaluation is required. For a detailed description of this technique, readers are referred to [2] and [11]. Numerical Implementation and Examples For simplicity, a piecewise constant collocation scheme is employed to discretize the integral equation shown in eq. (2). The resultant system is solved using the generalized minimal residual algorithm accelerated with the precorrected-FFT technique. The direct calculation of near-field interaction requires the integration of two kernel functions shown in eqs. (3) and (4) on each panel. For non-singular cases, standard Gaussian quadrature can be employed to numerically evaluate these integrals. For singular cases, a standard degenerate mapping [12] is employed to regularize kernel G before integration. For the integration of F kernel which is strongly singular, a subtraction and addition method is applied as in eq. (5). F ij

§ s· s ¨© F F ¸¹ F , ij ij ij

(5)

where Fijs corresponds to the elastostatic case. The singular part , Fijs , is integrated analytically on a flat panel using the approach proposed by Cruse [13]. To validate the developed method and algorithm, a spherical cavity with a radius of a 5.4 m embedded in an infinite elastic medium is modelled first. The Young’s modulus of the elastic medium is E 62 GPa . Its Poisson ratio is Q 0.25 and density is U 2.67 u 103 kg/m 3 . The cavity is subject to a sudden and maintained constant pressure of p0 displacement is,

ur

1 1 df 1 f c p r dt r 2

where f

p0 a 2 2 UJ c p

6.9 MPa. The analytical solution for the radial

t t0,

ª 1 ·º J t § «1 e ¨ cos J st s sin J st ¸ » and s © ¹¼ ¬

(6)

1 ,J 1 2Q

1 2Q c p . 1 Q a

The frequency-domain solution can be readily obtained based on eq. (6). The frequency-domain elastodynamic equation (eq. (2)) was solved numerically with a traction boundary condition of p0 jZ . Only non-zero frequencies are considered in the validation. Several surface meshes have been employed. The smallest number of nodes per S-wavelength is 6.7 which corresponds to the case with 192 elements and f 325Hz . The radial displacement as a function of frequency f is plotted in Fig. 1 together with results obtained from the conventional BEM and the analytical frequency-domain solution. The almost identical curves obtained from the pFFT accelerated BEM and the conventional BEM show that the errors produced by the acceleration are negligible, similar to that in the steady case ([11]). Such a fact indicates that the precorrected-FFT acceleration technique works well for rapidly oscillating kernels. The convergence of the accelerated boundary element method can also be observed from the figure as the results obtained from the finer mesh are much closer to the analytical solution than those from the coarse mesh. It is also observed that the convergence rate and the accuracy are independent of the oscillating frequency, which to a certain extent illustrates the relatively kernelindependent feature of the pFFT technique. To examine the efficiency of the method, the consumed computational time for the case with a frequency of 100 Hz is plotted in Fig. 2. The number of cubes used varies with the number of surface elements. For 192

3

552

4


panels, a total of 81 cubes are used. The asymptotic lines correspond to O n , O n log n and O n

2

are also plotted for comparison. As expected, the CPU time of the pFFT accelerated BEM is O n log n

.

while the CPU time of the BEM without acceleration, i.e., the direct BEM, is O n -8

2

-6

5.0x10

4.5x10

-6

conventional BEM with 192 elements pFFT BEM with 192 elements pFFT BEM with 19200 elements analytical

0.0 -8

-5.0x10

conventional BEM with 192 elements pFFT BEM with 192 elements pFFT BEM with 19200 elements analytical

4.0x10

-6

3.5x10

-6

3.0x10

-7

-1.0x10

-6

2.5x10

-6

ur

ur

-7

-1.5x10

2.0x10

-6

1.5x10

-7

-2.0x10

-6

1.0x10 -7

-2.5x10

-7

5.0x10 -7

-3.0x10

0.0

-7

-7

-3.5x10

-5.0x10 0

50

100

150

200

250

300

350

0

frequency Hz

50

100

150

200

250

300

350

frequency Hz

(a) real part

(b) imaginary part

Fig. 1 Radial displacement as a function of frequency f

1000000

100000

pFFT BEM Direct BEM 2 O(n ) O(n*log(n)) O(n)

time (s)

10000

1000

100

10

100

1000

10000

number of panels

Fig. 2. Computational time as a function of the number of panels To test the code for problems with relatively complex domains, a unit cube with eight unconnected, randomly distributed spherical voids shown in Fig. 3(a) is simulated. For simplicity, the radius of each void is set to be the same and is 0.14 m . The material properties of the cube are set to be E 70 GPa, v 0.25, U 2.7 u 103 kg/m 3 . The cube is subject to an oscillatory pressure with an amplitude of 100 MPa on its top surface and is fixed at its bottom surface (ABDE). All other surfaces are free of traction. The displacements of the cube at different frequencies are calculated by discretizing the domain with constant triangular elements as shown in Fig. 3(b). For convergence study, several meshes at different levels of discretization are generated. The smallest number of nodes per S-wavelength is 214. Due to the lack of analytical solution of this problem, ANSYS, a commercial FEM package, is employed to obtain the reference solution for comparison. To ensure sufficiently accurate result, a FEM model with 57193 tetrahedral elements was employed. It is worth mentioning that for this problem, the BEM meshes for the


553

cube and each void are generated separately while the FEM mesh must be generated as a whole. The meshing advantage of the BEM is clearly illustrated in this example. Figure 4 plots the calculated vertical displacements at the center point C on the top surface of the cube as functions of the frequency. Several curves correspond to different discretization levels are displayed. The FEM results are also plotted for comparison. It is evident that the BEM results converge uniformly to the reference results at all frequencies.

(a) Schematics

(b) BEM mesh Fig. 3: A cube with eight spherical voids

−1.8

x 10

−3

−1.85

Magnitude of displacement/m

pFFT,Element number=2516 pFFT,Element number=14448 −1.9

pFFT,Element number=21754 FEM,Element number=57193

−1.95

−2

−2.05

−2.1 50

100

150 Frequency/Hz

200

250

Fig. 4: Comparison of displacement magnitude of a point in the vertical direction between pFFT and FEM

5

554


6

Summary A 3D accelerated BEM approach based on the precorrected-FFT technique is developed for oscillatory elastodynamic problems. The accuracy and efficiency of the developed method are evaluated using examples in which either the analytical solutions or the FEM solution are available for comparison. The performance of the pFFT accelerated BEM for solving elastodynamic problems is similar to that of

elastostatic problems, i.e., both the computational time and memory are of O n log n , where n is the size of the final discretized linear system. It has also been found that the performance of the method does not depend on the oscillatory frequencies that have been tested, which to a certain extent, reflects the kernelindependent nature of the method.

References [1] Greengard, L., and Rokhlin, V., 1997, "A new version of the fast multipole method for the Laplace equation in three dimensions," Acta Numer., 6, pp. 229-269. [2] Phillips, J. R., and White, J. K., 1997, "A Precorrected-FFT method for electrostatic analysis of complicated 3-d structures," IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 16(10), pp. 1059-1072. [3] Ong, E. T., Lim, K. M., Lee, K. H., and Lee, H. P., 2003, "A fast algorithm for three-dimensional potential fields calculation: Fast Fourier Transform on Multipoles," Journal of Computational Physics, 192(1), pp. 244-261. [4] Ding, J., and Ye, W., 2004, "A fast integral approach for drag force calculation due to oscillatory slip Stokes flows," International Journal for Numerical Methods in Engineering, 60(9), pp. 1535-1567. [5] Frangi, A., and Gioia, A. D., 2005, "Multipole BEM for the evaluation of damping forces on MEMS," Computational Mechanics, 37(1), pp. 24-31. [6] Liu, Y., Nishimura, N., and Otani, Y., 2005, "Large-scale modeling of carbon-nanotube composites by a fast multipole boundary element method," Computational Materials Science, 34(2), pp. 173-187. [7] Liu, Y. J., Nishimura, N., Otani, Y., Takahashi, T., Chen, X. L., and Munakata, H., 2005, "A fast boundary element method for the analysis of fiber-reinforced composites based on a rigid-inclusion model," Journal of Applied Mechanics, Transactions ASME, 72(1), pp. 115-128. [8] Sanz J.A., Bonnet M. , Dominguez J., 2008, "Fast multipole method applied to 3-D frequency domain elastodynamics," Engineering Analysis with Boundary Elements 32, pp 787– 795. [9] Chaillat Stéphanie, Bonnet M., Semblat J. F., 2008, "A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain," Comput. Methods Appl. Mech. Engrg. 197, pp 4233–4249. [10] Takahashi T., Nishimura N., Kobayashi S., 2003, "A fast BIEM for three-dimensional elastodynamics in time domain," Engng. Anal. Bound. Elem. 27, pp 491–506. [11] Masters, N., and Ye, W., 2004, "Fast BEM solution for coupled 3D electrostatic and linear elastic problems," Engineering Analysis with Boundary Elements, 28(9), pp. 1175-1186. [12] Nagarajan A and Mukherjee S, 1993, "Mapping Method for Numerical Evaluation of TwoDimensional Integrals with 1/r Singularity," Computational Mechanics, 12, pp. 19-26. [13] Cruse, T. A., 1969, "Numerical solutions in three dimensional elastostatics," International Journal of Solids and Structures, 5(12), pp. 1259-1274.


555

A New Time Domain Boundary Integral Equation of Elastodynamics Z. H. Yao Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China [email protected] Keywords: Boundary integral equation; time domain boundary integral equation; boundary element method; elastodynamics Abstract. The traditional time domain boundary integral equation (TDBIE) of elastodynamics is formulated based on the time dependent fundamental solution and the reciprocal theorem of elastodynamics. The time dependent fundamental solution of the elastodynamics is the response of the infinite elastic medium under a unit concentrate impulsive force subjected at a point and at an instant, including not only the pressure wave and shear wave, but also the Laplace wave with speed between that of P and S waves. In this paper, a new TDBIE is derived directly from the initial boundary value problem of the partial differential equation of elastodynamics, and using the integral equation in weighted residual format. In the new TDBIE the D’Alembert solution of the elastodynamics, namely the spherical convergent pressure wave and shear wave are applied as the kernel functions respectively. In this way, the system of TDBIE obtained is much simpler than the traditional one.

Introduction The traditional TDBIE of elastodynamics is briefly introduced in this paper, which is based on the reciprocal theorem of elastodynamics and using the time dependent fundamental solution of the elastodynamics equations. This fundamental solution is the response of the infinite elastic medium under a unit concentrated impulsive force subjected at a point and at an instant, which includes not only the pressure wave and shear wave, but also the Laplace wave with speed between that of P and S waves. And then a new TDBIE is derived directly from the initial boundary value problem of the partial differential equation of elastodynamics, and using the integral equation in weighted residual format. In the new TDBIE the D’Alembert solution of the elastodynamics, namely the spherical convergent pressure wave and shear wave are applied as the kernel functions respectively. In this way, the system of TDBIE obtained is much simpler than the traditional one.

Traditional TDBIE of elastodynamics The formulation of the traditional TDBIE is based on the reciprocal theorem of elastodynamics, and using the time dependent fundamental solution.

The time dependent fundamental solution of elastodynamics. The time dependent fundamental solution of elastodynamics satisfies the governing equation U (c12 c22 ) ukjs ,ij + U c22 ukis , jj U ukis = G ki ' (P, Q) ' (W , t ) (1) where the physical meaning of ukjs (P, W ; Q, t ) is the displacement component in x j direction of a field point Q of the infinite elastic medium at instant t resulted by the unit concentrate force in the direction of xk subjected at the source point P and instant W ; c1 , c2 are the wave speed of the pressure wave and shear wave respectively, U the mass density of the elastic medium, G ki the Kronecker G , and ' (W , t ) , ' (P, Q) the Dirac Delta function. The displacement fundamental solution can be written as ukis P, W Q, t

ª § § 1 ° t c r· r ® 3r ,k r ,i G ki « H ¨ t c ¸ H ¨ t c 4SU r °¯ r 2 c1 ¹ c2 «¬ © ©

·º ¸» ¹ ¼»

ª1 § r· 1 § r · º G ki § r · ½° + r ,k r ,i « 2 ' ¨ t c, ¸ 2 ' ¨ t c, ¸ » 2 ' ¨ t c, ¸¾ c c c c c c » 1 ¹ 2 2 ¹¼ 2 2 ¹° © © ¬« 1 © ¿

where t = t W , H is the Heaviside function. The corresponding traction fundamental solution is

(2)

556

tkiS P, W q, t


1 4S

ª§ w\ F · § wr F§ wr · · «¨ wr r ¸ ¨ wn G ki r ,i nk ¸ 2 r ¨ r ,k ni 2r ,k r ,i wn ¸ ¹© ¹ © ¹ ¬©

2

º · § w\ wF wF wr § c12 F· ¨ 2¸¨ 2 ¸ r ,k ni » r , k r ,i wr wn © c22 r¹ »¼ ¹ © wr wr

(3)

where \

§ c22 ª § r · r ·º 1 § r · t c « H ¨ t c ¸ H ¨ t c ¸ » ' ¨ t c, ¸ c2 ¹ c c2 ¹ r 3 «¬ © 1 ¹» © ¼ r ©

F

r · c22 1 § r· 2 § 3\ ' ¨ t c, ' ¨ t c, ¸ ¸ r © c2 ¹ c12 r © c1 ¹

1 ª § « ' ¨ t c, r 2 «¬ © 3F 1 ª § « ' ¨ t c, r r 2 ¬« ©

w\ wr

F

wF wr

r

r · r § r ·º ¸ ' ¨ t c, ¸» c2 ¹ c2 © c2 ¹ »¼ r · r § r · º c22 1 ª § r · r § r ·º « ' ¨ t c, ¸ ' ¨ t c, ¸» ¸ ' ¨ t c, ¸» c2 ¹ c2 © c2 ¹ ¼» c12 r 2 «¬ © c1 ¹ c1 © c1 ¹ »¼

This time dependent fundamental solution includes not only pressure wave and shear wave in the infinite elastic medium, but also the Laplace wave, which has a speed within that of pressure and shear waves.

The traditional TDBIE of elastodynamics. The Betti reciprocal theorem in elasticity has been generalized to the elastodynamic case. For the two independent dynamic state of the same elastic solid in same time space domain, ui1 , ti1 , f i 1 , ui01 , ui01 , f i 1 and ui 2 , ti 2 , fi 2 , ui0 2 , ui0 2 , f i 2 , the reciprocal theorem can be written as

ti( ) (q, t ) * ui( ) (q, t ) dS (q ) + fi( ) (Q, t ) * ui( ) (Q, t ) dV (Q) U ui( ) (Q, t ) * ui( ) (Q, t ) dV (Q) 1

S

2

1

2

1

V

2

V

= ti( ) (q, t ) * ui( ) (q, t ) dS (q ) + fi( ) (Q, t ) * ui( ) (Q, t ) dV (Q) U ui( ) (Q, t ) * ui( ) (Q, t ) dV (Q) 2

1

2

S

1

2

V

(4)

1

V

where * denotes the convolution integral. If the dynamic state to be solved is taken as the state (1), and the state corresponding to the fundamental solution is taken as state (2), then ui( ) (q, t ) = ukis (p, W ; q, t ) ukis (p, q; t W ) 2

( 2)

ti (q, t ) = t (p, W ; q, t ) t (p, q; t W ) s ki

s ki

ui( ) (Q, t ) = ukis (p, W ; Q, t ) ukis (p, Q; t W ) 2

( 2)

f i (Q, t ) = G ki ' (p, Q) ' (W , t )

2 ui( ) (Q, t ) = u kis (p, W ; Q, t ) u kis (p, Q; t W )

2 ui( ) (Q, t ) = uksi (p, W ; Q, t ) ukis (p, Q; t W )

The convolution integral in Eq. (4) should be expressed as t

ti( ) (q, t ) * ui( ) (q, t ) ukis (p, q; t W ) ti (q, W ) dW , ...... 1

2

t0

Therefore, the Eq. (4) can be rewritten as Cki (p) ui (p, t ) + =

V

V

t t0 t

S

t t0

tkis (p, q; t W ) ui (q, W ) dW dS (q )

V

ukis (p, Q; t W ) f i (Q, W ) dW dV (Q) +

S

t t0

t t0

U ukis (p, Q; t W ) ui (Q, W ) dW dV (Q)

ukis ( p, q; t W ) ti (q, W ) dW dS (q )

(5)

U ukis (p, Q; t W ) ui (Q, W ) dW dV (Q)

t0

Integrating the last integrals in both sides by parts, the time domain displacement boundary integral equation can be finally obtained, Cki (p) ui (p, t ) + =

V

+U

V

t t0

S

t t0

tkis (p, q; t W ) ui (q, W ) dW dS (q ) + U u kis (p, Q; t t0 ) ui (Q, t0 ) dV (Q)

ukis (p, Q; t W ) f i (Q, W ) dW dV (Q) +

V

S

t t0

ukis ( p, q; t W ) ti (q, W ) dW dS (q )

ukis (p, Q; t t0 ) ui (Q, t0 ) dV (Q)

This boundary integral equation can be solved by TDBEM.

(6)


557

Derivation of a new TDBIE of elastodynamics The weighted residual integration form of the elastodynamics. The governing equation of the elastodynamics is well-known, U c12 c22 u j ,ij Q, t U c22 ui , jj Q, t f i Q, t U ui Q, t 0

(7)

which is a vector field equation; therefore the divergence and the curl of this equation should also be zero, namely 2 2 2 ª º °¬ U c1 c2 u j ,ij Q, t U c2 ui , jj Q, t fi Q, t U ui Q, t ¼ ,i 0 ® 2 2 2 °¯ekmi ¬ª U c1 c2 u j ,ij Q, t U c2 ui , jj Q, t fi Q, t U ui Q, t ¼º ,m

(8) 0

The weighted residual integral equations corresponding to Eq. (7) can be written as t1 U (c12 c22 ) u j ,ij (Q, t ) + U c22 ui , jj (Q, t ) + f i (Q, t ) U ui (Q, t ) ,i w(1) (Q, t ) dtdV = 0 V t0

t1 U (c 2 c 2 ) u , (Q, t ) + U c 2 u , (Q, t ) + f (Q, t ) U u (Q, t ) , w(2) (Q, t ) dtdV = 0 e 2 i jj i i kmi 1 2 j ij m k V t0

(9)

As w(1) (Q, t ) , wk(2) (Q, t ) are arbitrary weighted functions, these equations are also identical with Eq. (8). These integral equations can be transformed using the generalized Gauss identities. For example, the first equation of Eq. (9) can be finally converted to

V

+

S

S

+

S

V

(

t1 t0

)

1 1 1 (1) ,i (Q, t ) ui (Q, t ) dtdV U (c12 2c22 ) w( ) , jji (Q, t ) + U c22 w( ) , jij (Q, t ) + w( ) ,ijj (Q, t ) U w

t1 t0

t1 t0

t1 t0

U (c 2c ) u , (q, t ) + U c (u , (q, t ) + u , (q, t )) + f (q, t ) U u (q, t )n w(1) (q, t ) dtdS j ji j ij i jj i i i 2 1

2 2

2 2

U (c 2 2c 2 ) u , (q, t )G + U c 2 (u , (q, t ) + u , (q, t )) w(1) , (q, t ) n dtdS 1 2 2 k k ij j i i j i j

(

(10)

)

U c 2 2c 2 w(1) , (q, t )G + U c 2 w(1) , (q, t ) + w(1) , (q, t ) u (q, t ) n dtdS 2) 2 kk ij ji ij j ( 1

i t =t1

t =t

1 1 fi (Q, t ) w ,i (Q, t ) dtdV + U ui (Q, t ) w ,i (Q, t ) dV U ui (Q, t ) w ( ) ,i (Q, t ) dV = 0 t = t0 t =t0 V V

t1

(1)

t0

(1)

In the first integral term, the integrand related to the weighted function is

1 ,i Q, t U c12 2c22 w1 , jji Q, t U c22 w1 , jij Q, t w1 ,ijj Q, t U w

1 ,i P, W ; Q, t (11) U c12 w1 ,i , jj P, W ; Q, t U w

It can be seen that w is just the scalar potential function of the elastic pressure wave, and w(1) ,i is the displacement component of the pressure wave. In order to eliminate the time end terms related to instant t = t1 in the last two integration terms of Eq. (9), it should take the spherical convergent pressure wave as the weighted function, which is convergent at the source point p at the instant W . (1)

Derivation of a TDBIE related to spherical convergent pressure wave. To derive the boundary integral equation directly, let the spherical convergent pressure wave is convergent to a boundary point p at the instant W , and w(1) ,i is rewritten as u1is . The direction of x1 of the Cartesian coordinate system is taken the outward normal direction at boundary point p. To extract the singular point from the integral domain, a small spherical surface with the center at point p and a radius of G is applied (Fig. 1). In this way, Eq. (9) should be rewritten as lim G 0

S

G

lim G 0

W

G c1

t0

W

S S

V V G

G 0

lim

+ lim G 0

S S + S G

+ lim G 0

U (c 2 2c 2 ) u s , G + U c 2 (u s , +u s , ) (p, W ; qG , t ) n (q ) u (qG , t ) dtdS j i 1 2 1k k ij 2 1j i 1i j

V V G

W t0

G c1

t0

W t0

G c1

G c1

u1si (p, W ; q, t ) U (c12 2c22 ) uk ,k G ij + U c22 (u j ,i +ui , j ) (q, t ) n j (q ) dtdS

U (c 2 2c 2 ) u s , G + U c 2 (u s , +u s , ) (p, W ; q, t ) n (q ) u (q, t ) dtdS j i 1 2 1k k ij 2 1j i 1i j

u1si (p, W ; Q, t ) f i (Q, t ) dtdV lim

U u1si (p, W ; Q, t0 ) ui (Q, t0 )dV = 0

G 0

V V G

u1si ( p, W ; Q, t0 ) U ui (Q, t0 )dV

(12)

558


Fig. 1 A small spherical surface to extract the singular point p It should be noted here that in this equation instant t1 is taken as W G c1 to avoid the singularity, the first integral is removed because the kernel function of spherical pressure wave satisfies the homogeneous wave equation of pressure wave, and the second integral is removed because the dynamic response to be solved satisfies the equations of motion. In this formulation, the equation of motion has been once more differentiated in Eq. (7); therefore the requirement of its differentiability is one order higher than conventional case, both for the domain points and boundary points. The limit of the integrals in Eq. (12) except the first one is quite simple; therefore Eq. (12) can be rewritten as lim G 0

SG

S

V

W t0

W t0

G

t (p, W ; qG , t ) n j (qG ) ui (qG , t ) dtdS

c1 s 1i

W

u1si (p, W ; q, t ) ti (q, t ) n j (q ) dtdS + t1si (p, W ; q, t ) n j (q )ui (q, t ) dtdS S

W t0

t0

(13)

u1si (p, W ; Q, t ) fi (Q, t ) dtdV u1si (p, W ; Q, t0 ) U ui (Q, t0 )dV + U u1si (p, W ; Q, t0 ) ui (Q, t0 )dV = 0 V

V

where the third integral term is a Cauchy principal value integral term. In the spherical coordinate system shown in Fig. 1, where the point p is taken as the origin, the scalar potential of the spherical convergent wave can be expressed as a H r c1 W t r

w(1)

a H r c1t c r

(14)

which satisfies the wave equation 1 p, W ; Q, t 0 U c12 w1 , jj p, W ; Q, t U w

(15)

In the Cartesian coordinate system, the displacement can be written as § H r c1t c · ¨ ' r , c1t c ¸ r © ¹

ar ,i r

u1si

(16)

and the corresponding traction is ani ª 1 §1 ·º U c12 2c22 ' c r , c1t c 2 U c22 ¨ ' r , c1t c 2 H r c1t c ¸ » r «¬ r ©r ¹¼

t1si

n j r , j r ,i

aª 1 §1 ·º 2 U c22 'c r , c1t c 6 U c22 ¨ ' r , c1t c 2 H r c1t c ¸ » r «¬ r ©r ¹¼

(17)

On the small spherical surface S G , there is only uniformly distributed normal traction applied. For a boundary point p on the smooth part of boundary, the resultant of the traction should be in the normal direction, lim G 0

SG

= lim G 0

W t0

SG

G

t (p, W ; qG , t ) ui (qG , t ) dtdS

c1 s 1i

W t0

G c1

1 ar ,i 2 1 U c ' (G , c1 (W t )) 4 U c22 ' (G , c1 (W t )) 2 H (G c1 (W t )) ui (qG , t ) dtdS

G G 1 G

(18)


559

Finally it is obtained lim G 0

SG

W t0

G

t (p, W ; qG , t ) ui (qG , t ) dtdS = 4S a U

c1 s 1i

If we take the constant a u1 (p, W ) =

V

W t0

S

W t0

c22 u1 (p, W ) c1

(19)

c1 , Eq. (13) can be rewritten as 4SU c22

u1si (p, W ; q, t ) ti (q, t ) n j (q ) dtdS +

S

W t0

t1si (p, W ; q, t ) n j (q ) ui (q, t ) dtdS

u1si (p, W ; Q, t ) fi (Q, t ) dtdV u1si (p, W ; Q, t0 ) U ui (Q, t0 )dV

(20)

V

+ U u1si (p, W ; Q, t0 ) ui (Q, t0 )dV V

This is the first one of the new TDBIE, where the spherical convergent pressure wave is applied as the kernel function, H r c1t c · c1r ,i § ¨ ' r , c1t c ¸ 4SU c22 r © r ¹

u1si t1si

(21)

· c1ni ª§ c12 1 §1 ·º «¨ 2 ¸ ' c r , c1t c 2 ¨ ' r , c1t c 2 H r c1t c ¸ » r r 4S r «¬© c22 © ¹ »¼ ¹ c1n j r , j r ,i ª º 1 1 § · 'c r , c1t c 3 ¨ ' r , c1t c 2 H r c1t c ¸ » 2S r «¬ r ©r ¹¼

(22)

For the simpler cases with zero initial conditions and without body force, Eq. (27) can be simplified as u1 (p, W ) =

S

W t0

u1si (p, W ; q, t ) ti (q, t ) n j (q ) dtdS +

S

W t0

t1si (p, W ; q, t ) n j (q ) ui (q, t ) dtdS

(23)

TDBIE related to spherical convergent shear waves. The TDBIE related to the spherical convergent shear waves can be derived from the second equation of Eq. (9). The vectorial potential of the spherical convergent shear waves can be expressed as b H r c2 t c r

w2(2)

w3(2)

b H r c2 t c r

(24)

The corresponding displacements can be written as e ji 3

u2s j

H r c2 t c · br ,i § ¨ ' r , c2 t c ¸ r © r ¹

u3s j

e ji 2

H r c2 t c · br ,i § ¨ ' r , c2 t c ¸ r © r ¹

(25)

The equations derived from the second equation of Eq. (9) are similar to Eq. (13), namely lim G 0

S

G

t0

W

S

V

W

t0

W t0

G c2

tkis (p, W ; qG , t ) n j (qG ) ui (qG , t ) dtdS

ukis (p, W ; q, t ) ti (q, t ) n j (q ) dtdS +

S

W t0

tkis ( p, W ; q, t ) n j (q ) ui (q, t ) dtdS

ukis (p, W ; Q, t ) f i (Q, t ) dtdV ukis ( p, W ; Q, t0 ) U ui (Q, t0 )dV

(26)

V

+ U u (p, W ; Q, t0 ) ui (Q, t0 )dV = 0 s ki

V

k = 2, 3

Because of the length limitation, skipped the lengthy derivation, the first TDBIE related to spherical convergent shear waves can finally be written as u2 (p, W ) =

V

W t0

S

W t0

u2s i (p, W ; q, t ) ti (q, t ) n j (q ) dtdS +

W t0

t2s i ( p, W ; q, t ) n j (q) ui (q, t ) dtdS

u2s i (p, W ; Q, t ) fi (Q, t ) dtdV u2s i (p, W ; Q, t0 ) U ui (Q, t0 )dV

+ U u 2s i (p, W ; Q, t0 ) ui (Q, t0 )dV V

S

V

(27)

560


The equation for the case of k 3 can be derived similarly, and finally the TDBIEs can be summarized as uk p, W ³

V

³

W

t0

u

³ U u V

³

S

s ki

s ki

³

W

t0

ukis p, W ; q, t ti q, t n j q dtdS ³

S

W

³ t p, W ; q, t n q u q, t dtdS s t0 ki

j

i

p, W ; Q, t fi Q, t dtdV ³V u p, W ; Q, t0 U ui Q, t0 dV s ki

(28)

p, W ; Q, t0 ui Q, t0 dV k

1, 2, 3

and for the simpler cases with zero initial conditions and without body force, this equation can be simplified as uk p, W

³

S

³

W

t0

ukis p, W ; q, t ti q, t n j q dtdS ³

S

W

³ t p, W ; q, t n q, t u q dtdS s t0 ki

j

k

i

(29)

1, 2, 3

Summary Based on the general method for the derivation of boundary integral equation, a system of new TDBIE has been derived directly from the partial differential equations of elastodynamics. The spherical convergent pressure and shear waves are taken as the kernel functions in the derived TDBIE respectively. In comparison with the traditional TDBIE, the new derived TDBIE is not only much simpler, but also with clear physical meaning. For the solution of the new TDBIE, the traditional TDBEM can be applied easily, and the computational efficiency could be enhanced, because the kernel functions are much simpler than traditional one. The resulted linear algebraic equation system can be solved time step by step. During the first several steps, the matrices are quite sparse. The numerical implementation will be done in near future.

References [1] M. H.Aliabadi The Boundary Element Method, Vol2: Applications in Solids and Structures, Wiley (2002). [2] H. Antes Finite Elements Analysis and Design, 1, 313-322 (1985). [3] T. A. Cruse and F. J. Rizzo Journal of Math. Anal. Appl., 22, 244- 259 (1968). [4] D.M. Cole, D.D. Kosloff and J.B. Minster Bulletin of Seismological Society of America, 68, 1331-1357 (1978). [5] T. A. Cruse Journal of Math. Anal. Appl., 22, 341- 355 (1968). [6] Q. H. Du and Z. H. Yao Acta Mechanica Solida Sinica, No. 1, 1-22 (1982 in Chinese). [7] K. F. Graff Wave Motion in Elastic Solids. Columbus, Univ. of Ohio Press, USA (1975). [8] D. L. Karabalis and D. E. Beskos Earthquake Engineering Structures and Dynamics, 12, 73-93 (1984). [9] M. Kogl and L. Gaul CMES-Computer Modeling in Engineering & Sciences, 1, 27-43 (2000). [10] G.D.Manolis and D.E.Beskos Boundary Element Methods in Elastodynamics. Unwin Hyman, London (1988). [11] W. J. Mansur A time stepping technique to solve wave propagation problems using the boundary element method. PhD Thesis, University of Southampton, U. K.(1983). [12] D. Nardini and C. A. Brebbia Boundary Element Methods in Engineering, Springer-Verlag, Berlin, Germany, pp. 312-326 (1982). [13] Y.Niwa, S. Kobayashi, and M. Kitahara: Developments in Boundary Element Methods, Vol. 2, Appl. Sci. Publications, London, U.K., pp. 143-176 (1980). [14] D. Soares and W. J. Mansur CMES-Computer Modeling in Engineering & Sciences, 8, 153-164 (2005). [15] Z. H. Yao CMES-Computer Modeling in Engineering & Sciences, 50, 21-45 (2009).


561

Regularization of the divergent integrals in boundary integral equations V.V. Zozulya Centro de Investigacion Cientifica de Yucatan A.C., Calle 43, No 130, Colonia Chuburná de Hidalgo, C.P. 97200, Mérida, Yucatán, México. E-mail: [email protected] Keywords: weakly singular, singular, hypersingular integrals, regularization, boundary integral equations. Abstract. This paper considers divergent integrals with different type of singularities, which arise when the boundary integral equation (BIE) method is used to solve boundary value problems in the theory of potentials. The main equations related to formulation of the boundary integral equation and boundary element methods in 2-D and 3-D cases are discussed in details. For their regularization an approach based on the theory of distribution and application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular and hypersingular integrals in 2-D case, have been constructed. 1. Introduction. In recent years, more and more of publications is devoted to the boundary integral equation methods (BEM) and its application science and engineering. It is because the BIE is a very powerful tool for solution of the mathematical problems science and engineering [1]. When the BIE are solved numerically divergent integrals have to be calculated. Numerical methods developed for regular integrals calculation can not be used for their calculation. There are many methods for calculation of the divergent integrals, for references see review articles [2-4] and references there. We will consider here in more details method of the divergent integrals regularization developed in [5-14] and it application in the 2-D and 3-D BIE. The method is based on the theory of distributions and idea of finite part integrals according to Hadamard. In our previous publications approach based on the theory of distributions has been developed for regularization of the divergent integrals with different singularities. We apply the approach based on the theory of distributions and finite part integrals for the problems of fracture mechanics firstly in [5]. Then it was further developed for regularization of the hypersingular integrals in static and dynamic problems of fracture mechanics in [13, 14] respectively. Further development of this approach and application of the Green’s theorems in the sense of theory of distribution has bean done in [5, 6] for piecewise constant and in [7, 8] piecewise linear approximation respectively. The equations presented in [12] permit transforms divergent hypersingular integrals into the regular ones. The developed approach can be applied not only for hypersingular integrals regularization but also for a wide class of divergent integral regularizations and any polynomial approximation. In this paper, the approach for the divergent integral regularization which is based on the theory of distributions and Green’s theorems is further developed and applied for the potential theory problems. The divergent integral regularization have been done for 2-D and 3-D the weakly singular and hypersingular integrals and regular formulas for their calculation have been obtained. The weakly singular and hypersingular integrals piecewise constant approximation have been considered for arbitrary convex polygon. It is important to mention that in presented equations all calculations can be done analytically, no numerical integration is needed. 2. Statement of problem and BIE. Let consider a homogeneous region, which in 2-D or 3-D Euclidean space occupies volume V with smooth boundary wV . The region V is an open bounded subset of the Euclidean space with a C 0,1 Lipschitzian regular boundary wV . In the region V we consider scalar function u (x) that subjected to Poisson equation 'u (x) f (x) 0, x V (1)

Here '

n

¦w

i

w i is the Laplace operator, w i

w wx i denotes the partial derivatives with respect to space,

i 1

f (x) is given in the region function. Throughout this paper we use the Einstein summation convention. If the eq (1) is defined in an infinite region, then its solution must satisfy additional conditions at the infinity in the form

562

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz in 2-D case u ( x)

O(ln(r 1 )) , w n u (x)

O (r 1 ) for r o f

p ( x)

1

(2) in 3-D case u (x) O(r ) , w n u (x) p (x) O(r 2 ) for r o f Here ni are components of the outward normal vector, w n ni w i is a derivative in direction of the vector n(x) normal to the surface wV , r is the distance in the Euclidian space. If the body occupied a finite region V with the boundary wV , it is necessary to establish boundary conditions. We consider the mixed boundary conditions in the form u (x) M (x) , x wVu , p(x) \ (x) , x wV p The boundary contain two parts wVu and wV p such that wVu wV p

(3)

and wVu wV p

wV . On the part

wVu is prescribed unknown function u (x) and on the part wV p is prescribed it normal derivative

p (x) respectively. In order to establish integral representations for the function u (x) and it normal derivative p (x) we start from second Green theorem in the form * * * * (4) ³ (u 'u u 'u )dV ³ (uw n u u w n u ])dS wV

V

which take place for any two functions u (x) and u * ( x) with continuous first and second derivatives within the region V . Let us consider solution of the elliptic partial differential eq (1) in an infinite space for the function f * ( x) G ( x y ) 'U (x y ) G (x y ) 0, x, y 3 Solution of this equation is called the fundamental solutions. In 2-D and 3-D cases it has the form 1 1 1 U (x y ) ln , U (x y ) 2S r 4S r ( x1 y1 ) 2 ( x 2 y 2 ) 2 Here r respectively. Now considering that

and r

( x1 y1 ) 2 ( x 2 y 2 ) 2 ( x 3 y 3 ) 2

u * (x) U (x y ) and p i* (x)

(5)

(6)

for 2-D and 3-D case

w n u * ( x ) W ( x, y )

(7)

from eq (4) we obtain the integral representation for the unknown function u (x) u (y )

³ ( p(x)U (x y ) u (x) W (x, y ))dS ³ b(x)U (x y )dV

wV

(8)

V

The kernels U (x y ) and W (x, y ) are called fundamental solutions for Laplace equation. After some transformations and simplifications the expression for the kernel W (x, y ) will has the following form n (x)( x i y i ) W ( x, y ) i (9) DS r E Applying to eq (8) the differential operator of normal derivative w n ni (x)w i we will find integral representation for the unknown function p (x) in the form p(y )

³ ( p(x) K (x, y ) u (x) F (x, y ))dS ³ b(x) K (x, y )dV

wV

(10)

V

The kernels K (x, y ) and F (x, y ) may be obtained applying the differential operator w n ni (y )w i to the kernels U (x y ) and W (x, y ) respectively. After some transformations and simplifications the expression for the kernels K ij (x, y ) and Fij (x, y ) have the form · n i (y )( x i y i ) 1 § n i ( x)n j (y )( x i y i )( x j y j ) Ë (11) , F ( x, y ) n i (x)n i ( y ) ¸¸ E E ¨ 2 r DS r DS r © ¹ In the eq (9) and (11) D 4 , 2 and E 3 , 2 in 3-D and 2-D cases respectively. The kernels U (x y ) , W (x, y ) , K ij (x, y ) and Fij (x, y ) contain different kind singularities, therefore K ( x, y )

corresponding integrals are divergent. Simple observation shows that the kernels in the integral representations (8) and (10) tend to infinity when r o 0 .


563

In the 3-D case with x o y U (x y ) o r -1 , W ( x, y ) o r -2 , K (x, y ) o r -2 , F (x, y ) o r 3

(12)

U ( x y ) o ln(r 1 ), W (x, y ) o r -1 , K (x, y ) o r -1 , F (x, y ) o r 2

(13)

In the 2-D case with x o y

Tending y in eq (8) and (10) to the boundary wV and taking into consideration boundary properties of the kernels (9), (11) we obtain representation of the functions u ( x) and p (x) on the smooth parts of boundary surface wV in the following form r

1 u (y ) 2

³ ( p(x)U (x y )u (x) W (x, y ))dS ³ p(x)U (x y )dV ,

wV

V

(14)

1 # p(y ) ³ ( p (x) K ( x, y )u (x) F (x, y ))dS ³ p (x) K (x, y )dV 2 wV V The plus and minus signs in these equations are used for the interior and exterior problems, respectively. To transform the BIE into the finite dimensional BEM equations we have to split the boundary wV into a collection of finite boundary elements (BE) wV

N

wV n , wV n wV k

, if n z k .

n 1

(15)

On each BE we shall choose Q nodes of interpolation and the shape functions M q (x) . Then the displacement and traction on each BE wVn will be approximately represented in the form u x | ¦ u n x q M q (x), x wVn , p x | ¦ p n x q M q (x) x wVn Q

Q

q 1

q 1

(16)

and on the whole crack surface wV in the form u x | ¦ ¦ u n x q M q (x), x wVn , px | ¦¦ p n x q M q (x), x wVn Q

N

N

n 1

n 1 q 1

N

Q

N

(17)

n 1

n 1 q 1

Substitution of the expressions (17) in eq (14), gives us the finite-dimensional representations for the vectors of displacements and traction on the boundary in the form

where

1 m u y r 2

¦ ¦ >U y

1 m p y r 2

¦ ¦ >K y

N

Q

nq

r

n 1 q 1 N

Q

nq

r

@

, x m p nj x m W nq x r , x m u n x U f , y , Vn ,

@

, x m p n y m F nq y r , x m u n x m K f , y , Vn ,

n 1 q 1

U nq y r , x m

³ U y

r

, x M q x dS , W nq y r , x m

wVn

K nq y r , x m

³ K y

³ W y

r

, x M q x dS ,

wVn r

, x M q x dS , F nq y r , x m

wVn

(19)

³ F y , x M x dS . r

(18)

q

wVn

In the case of piecewise constant approximation the finite-dimensional representations for the vectors of displacements and traction on the boundary take the form 1 m u y r 2 1 m p y r 2 where

¦ >U y N

n

r

n 1

¦ >K y

@

, x m p nj x m W n x r , x m u n x U f , y , Vn ,

N

nq

n 1

r

@

, x m p n y m F nq y r , x m u n x m K f , y , Vn ,

(20)

564

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz U n y r , x m

³ U y

r

, x dS , W n y r , x m

³ W y

wVn

K n y r , x m

³ K y

r

, x dS ,

wVn r

, x dS , F n y r , x m

³ F y

wVn

r

, x dS .

(21)

wVn

2. Divergent integrals and distributions. Let us consider function f (x) that contain singular points in the region x V in n-D space and definite integral I 0 ³ f ( x)dx (22) V

The classical approach can not provide the meaning of the integral I 0 . The integrals with singularities can not be considered in usual (Riemann or Lebegue) sense. In order to such integrals have sense it is necessary special consideration of them. Following [5] we consider here the above divergent integrals in the sense of distribution. To do that we introduce function g (x) , such that the function f (x) can be presented in the form 'k g (x) ,

f ( x)

(23)

where ' w w , which is called the k – dimensional Laplace's operator. This representation of the function f (x) has to be considered in the sense of distribution in the region V . k

2k 1

2k 2

To do that we introduce test function M (x) C f ( R n ) . Then the eq (22) can be presented in the form of distributions ( f ,M)

³ f (x)M (x)dx ³ M (x)'

k

V

H

g ( x ) dx .

(24)

VH

Application of the Green theorem gives the following identity, which also take place only in the sense of distributions

³ [M (x)' g (x) (1) k

k

k 1

¦ (1) ³ [M (x)w

g (x)'k M ( x)]dV

i 1

n

'k i 1 g (x) g (x)w n 'k i 1M (x)]dS .

(25)

wV

i 0

V

Here, w n ni w i is the normal derivative on the surface with respect to x and ni (x) is a unit normal to the surface. Taking into account eq (24) we obtain equality k 1

¦ (1) ³ [M (x)w

F .P.³ f (x)M (x)dV

i 1

wV

i 0

V

n

'k i 1 g (x) g (x)w n 'k i 1M ( x)]dS (1) k ³ g (x)'k M (x)]dV

(26)

V

which can be consider as definition of the finite part (F.P.) of the divergent integral according to Hadamard in the sense of distribution in n-D case. This equation can be used for the divergent integrals calculation. For 1 the singular function f (x) of the form f (x) we have rm k 1 P P M ( x) 1 F .P.³ m dV ¦ ( 1) i 1 ³ ['k i 1M (x)w n m i 2i m i 2i w n 'k i 1M (x)]dS (1) k ³ m 2 k 'k 1M ( x)]dV , (27) r r i 0 V r wV V r 1 k 1 where Pk (1) k i 0 for k , m ! 1 . (m 2i ) 2 In the case M (x) 1 the above equations are significantly simplified. The eq (26) has the form I0

F .P.³ f (x)dV V

³w

n

'k 1 g (x)dS , for k 1 I 0

wV

F .P.³ f (x)dV V

³w

n

g (x)dS ,

(28)

wV

From eqs (27) and (28) follows Jk

F .P. ³

Sn

dS rk

1 ( k 2) 2

³w

wS n

n

1 dl r k 2

rn 1 dl (k 2) w³S n r k

Here rn xD nD and D 1,2 . For a circular area integral (29) can be easy calculated and the following result be obtained 2S r 1 2S 2S , J 1 2SR , J 3 J k ³ nk dl dM k 2 ³ k 2 R r ( k 2 ) R ( k 2 ) R 0 wS n

(29)

(30)


565

Here polar coordinates are used, were R and M are the circle radius and polar angle respectively. In the 1-D case singular function of one variable f (x) is defined in the region x V [a, b] and can be represented in the form d k g ( x) f ( x) , (31) dx k which also has to be considered in the sense of distributions as it was shown in the eq (24). In the same way as in 2-D case integrating by path we obtain x b

b d k g ( x) d k 1 g ( x) d k M ( x) (1) k ³ M ( x) dx , dx (32) k 1 k dx k dx dx a a x a This equality can be consider as definition of the F.P. of the divergent integral according to Hadamard in the sense of distribution in 1-D case and can be used for the divergent integrals calculation in 1-D case. For the 1 we have function f (x) of the form f ( x) rm b

F .P.³ g ( x)

b

F .P.³

M ( x) r

a

m

k 1

¦ (1)

dx

i 0

i 1

x b

d i Pi d k 1i M ( x) dx i r m k dx k 1i

x a

b

(1) k ³ a

Pk d k M ( x) , r m k dx k

(33)

1 for k , m ! 1 . (m i) In the case M (x) 1 the above equations are significantly simplified. The eq (32) has the form

where Pk

(1) k i

k 1 0

x b

b d k 1 g ( x) x , for k 1 I 0 F .P.³ f ( x)dx g ( x) x k 1 dx a a x a which is the well known Leibniz's formula for the definite integral. Examples of the divergent integrals calculation in the 1-D case are presented below b 1 1 1 (a y ) ln F .P.³ ln dx (b y ) ln x y b y a y a b

F .P.³ f ( x)dx

I0

b

F .P.³ a

dx x y

ln

b b y dx , F .P.³ 2 ay a ( x y)

b a

(34)

(35)

1 1 b y a y

4.2. Piecewise constant approximation in the 1-D case.

Let us consider a straight BE of the length ' n . The piecewise constant approximation is the simplest one. We transform global coordinates in the way that they are related to the local coordinate [ [1, 1] by the equations x1 ([ ) r ' n [ , x 2 ([ ) 0 , n1 ([ ) 0 , n 2 ([ ) 1 . (36) Interpolation function has the form 1 [ [' n , ' n ] ¯0 [ [' n , ' n ].

M 0 ([ ) ®

(37)

Fundamental solutions (6), (9), (11) have the following simple form U (x y )

1 1 ln , W ( x, y ) 2S ' n[

0 , K ( x, y )

0 , F ( x, y )

1

1

S ' n[ 2

.

(38)

Applying corresponding formulas from [9] and considering divergent integrals as it was shown above we get 1

J0

1

³ ln ' [ '

1

n

n

d[

§ § 1 2' n ¨1 ln¨¨ ¨ © 'n ©

·· ¸¸ ¸ ¸ ¹¹

1

,

J2

1

³' [

1

n

2

d[

2 'n

(39)

566


5. Piecewise constant approximation in the 2-D case. The piecewise constant approximation is the simplest one. Interpolation functions in this case do not depend on the FE form and dimension of the domain. In 2-D case they have the form 1 x S n , ¯0 x S n .

M q ( x) ®

(40)

In order to simplify situation we transform global system of coordinates such that the origin is located at the nodal point, where y 0 0 , the coordinate axes x1 and x 2 are located in the plane of the element, while the axis x3 is perpendicular to that plane. In this case x 3 0 and n1 0 , n 2 0 , n 3 1 and fundamental solutions have the following simple form U (x y )

1 , W ( x, y ) 4S r

0 , F ( x, y )

0 , K ( x, y )

1 . 4S r 3

(41)

Regular representations for integrals with these kernels can be easy calculated using above approach. From the eq (29) follows rn rn dS dS J 1 ´F .P. ³ (42) ³ dl , J 3 F .P.S³ r 3 w³S r 3 dl . Sn r wS n r n n Calculations of the integrals (42) will be done using the local rectangular coordinate system with its origin located in the point y q , the x1 and x 2 axis located in the plane of the polygon and the x3 axis perpendicular to this plane as it is shown on Fig. 1. x2

nˆ 2 (k )

nˆ (k )

nˆ1 (k ) ( x1 (k ), x2 (k ))

x1

( y1q , y 2q )

Sn

1

2'1

Fig. 1. Global coordinates of the vertexes are ( x1k , x 2k ) . They can be calculate though the nodal points and unit vector normal to the contour x ik 1 x ik x 2k 1 x 2k x k 1 x1k x i (k ) , nˆ1 (k ) , nˆ 2 (k ) 1 . (43) 2 2' k 2' k The coordinates of an arbitrary point on the contour wVn may be represented in the form

Adv. Bound. Elem. Tech. Eds: Ch Zhang, M H Aliabadi, M Schanz x1 ([ )

x1 (k ) [' k nˆ 2 (k ) and x 2 ([ )

567

x 2 (k ) [' k nˆ1 (k )

(44)

where x1 (k ) and x 2 (k ) are the coordinates of the k-th side of the contour, nˆ (nˆ1 , nˆ 2 ) is a unit vector normal to the contour and [ [1,1] is a parameter of integration along the k-th side, 2' k is the length of a k-th side. These are some useful notations r ([ ) '2k [ 2 2[' k r (k ) r 2 (k ) , r (k ) x12 (k ) x22 (k ) , rn (k ) xD (k )nˆ D (k ) , r (k ) x1 (k )nˆ 2 (k ) x 2 (k )nˆ1 (k ) , rn ([ ) rn (k ) , 2' k ( x1k 1 x1k ) 2 ( x 2k 1 x 2k ) 2 . (45) Using these notations the integrals under consideration may be represented in a convenient form for the calculation. U n y r , x m

³ U (y

K

r

¦ ³ U y

, x)dS

F n y r , x m

r

, x dl ,

k 1 lk

Sn

K

¦ ³ F y

³ F (y r , x)dS

(46)

r , x dl .

k 1 lk

Sn

Here indexes r and m indicate number of nodes. Substituting eqs (43)-(45) into eqs (42) we obtain formulas for calculation of the corresponding integrals over each side of polygon in the form 1 1 r (k ) r (k ) J 1 (k ) ³ n ' k d[ , J 3 (k ) ' k ³ n3 d[ (47) 1 r ([ ) 1 r ([ ) Now these integrals can be calculated over polygon using the formulas K

¦ r (k ) I

J1

n

K

¦ rn (k ) I 3, 0

, J3

1, 0

k 1

(48)

k 1

Here we use the following notation for the integrals 1

( ' k ) l 1 ³

I p ,l

[l

p 1 r ([ )

d[

(49)

These integrals may be calculated analytically 1

I 1, 0

1 d[ 1 r ([ )

'k ³

ln r (k ) ' k [ r ([ )

1

1

, 1

I 3, 0

'k ³

1

1 r ([ ) 3

d[

' k [ r (k ) (r 2 (k ) r2 (k ))r ([ )

1

(50) 1

Integrals (47) were calculated for the cases of triangular and quadrangular domain of integration. For a regular triangle with unit side we obtain J 1 2,281 , J 3 18 , and for a square with a unit side we obtain J 1 3,525 , J 3 11.31 respectively. Then integrals in eqs (46), over any convex polygon may be represented in the form

U n y r , xq

1 J1 , W n y r , x q 4S

K n y r , xq

0 , F n y r , xq

1 J3 4S

(51)

It is important to mention that all calculations here can be done analytically, no numerical integration is needed. Conclusions. Based on the theory of distribution and Green theorems the approach for the divergent hypersingular integrals regularization is developed here and applied for the BIE methods. We consider the 1D and 2-D divergent integrals over arbitrary convex polygon for piecewise constant approximation.. The divergent integrals over the BE have been transformed to regular ones over contour of the BE. Convenient for their calculation regular formulae have been obtained. In the presented equations all calculations can be done analytically, no numerical integration is needed. It is important to mention that proposed methodology easy can be applied for regularization of the divergent integrals in elastostatics and elastodynamics and for calculation regular integrals when collocation point situated outside BE. Also developed here methodology can applied to regularization of the divergent integrals in the case of quadratic and higher BE.

568


References [1]

Brebbia C.A., Dominguez J. Boundary elements. An introductory course, WIT Press, Southampton, 1998.

[2]

Chen J.T. and Hong H.-K. Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Applied Mechanics Review, 52(1), pp. 17-33, 1999. Sladek V. and Sladek J. (eds.) Singular Integrals in Boundary Element Methods, WIT Press, Southampton, 1998. Tanaka M., Sladek V. and Sladek J., Regularization techniques applied to boundary element methods. Applied Mechanics Review, 47, 457-499, 1994. Zozulya V.V., Integrals of Hadamard type in dynamic problem of the crack theory. Doklady Academii Nauk. UkrSSR, Ser. A. Physical Mathematical & Technical Sciences, 2, 19-22, 1991, (in Russian). Zozulya V.V. Regularization of the divergent integrals. I. General consideration. Electronic Journal of Boundary Elements, 4(2), 49-57, 2006. Zozulya V.V. Regularization of the divergent integrals. II. Application in Fracture Mechanics. Electronic Journal of Boundary Elements, 4(2), 58-56, 2006. Zozulya V.V. Regularization of the hypersingular integrals in 3-D problems of fracture mechanics. In: Boundary Elements and Other Mesh Reduction Methods XXX. (Eds. P.Skerget and C.A. Brebbia), WIT Press, Southampton, Boston, 2008, 219-228. Zozulya V.V. The Regularization of the Divergent Integrals in 2-D Elastostatics. Electronic Journal of Boundary Elements, Vol. 7, No. 2, 2009, pp.50-88. Zozulya V.V. Regularization of hypersingular integrals in 3-D fracture mechanics: Triangular BE, and piecewise-constant and piecewise-linear approximations, Engineering Analysis with Boundary Elements, 34(2), 2010. 105-113. Zozulya V.V. Regularization of the Divergent Integrals in Boundary Integral Equations for Elastostatics. In: Integral Methods in Science and Engineering. Vol. 1. Analytic Methods. (Eds. C.Constanda and M.E.Perez), Birkhäuser, 2010, pp. 333-347. Zozulya V.V. and Gonzalez-Chi P.I. Weakly singular, singular and hypersingular integrals in elasticity and fracture mechanics, Journal of the Chinese Institute of Engineers, 22(6), pp. 763-775, 1999. Zozulya V.V. and Lukin A.N. Solution of three-dimensional problems of fracture mechanics by the method of integral boundary equations. International Applied Mechanics, 34(6), 544-551, 1998. Zozulya V.V. and Men’shikov V.A. Solution of tree dimensional problems of the dynamic theory of elasticity for bodies with cracks using hypersingular integrals, International Applied Mechanics, 36(1), 74-81, 2000.

[3] [4] [5] [6] [7] [8]

[9] [10]

[11]

[12] [13] [14]


569

On Levi functions Wolfgang L. Wendland Universit¨ at Stuttgart Abstract: Fundamental solutions to elliptic partial differential operators are explicitly known only in particular cases whereas Levi functions can always be constructed. In this lecture, the simple case of a second order operator with variable coeffients will be considered and with Levi Functions a system of domain– boundary integral equations for the Dirichlet problem will be obtained. The mapping properties of the corresponding operators will provide the opportunity of employing efficient solution techniques. E.E. Levi: Sulle equazioni lineari totalmente ellittiche alle derivati parziali. Rend. Circ. Math. Palermo 24 (1907) 275-317. D. Hilbert: Grundz¨ uge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig 1912. A. Pomp: The Boundary–Domain Integral Method for Elliptic Systems. Springer–Lecture Notes 1683, 1998. G.C. Hsiao, W.L. Wendland: Boundary Integral Equations, Springer–Verlag Berlin 2008.

1

570


Domain integrals in a boundary element algorithm S. Nintcheu Fata1 and L. J. Gray2 Computer Science and Mathematics Division, Oak Ridge National Laboratory Oak Ridge, TN 37831-6367, USA 1 [email protected] 2 [email protected]

Keywords: Domain integrals, Newton potential, boundary integrals, Fundamental theorem of calculus, Divergence theorem.

Abstract. A rigorous and efficient algorithm which employs only the underlying boundary discretization to compute domain integrals appearing in a boundary element formulation has been developed. In the proposed approach, a three-dimensional domain integral with a continuous or weakly-singular integrand is first transformed into an integral over the boundary of the designated domain. The resulting surface integral is then carried out via standard quadrature rules commonly used for boundary elements. This transformation of a domain integral into an equivalent boundary integral is achieved via the use of straight-line integrals that traverse the domain of interest. Moreover, it is established that this domain-to-boundary integral transformation is derived from an extension of the first fundamental theorem of calculus to higher dimension, and the divergence theorem. Employed together with the singular treatment of surface integrals that is well studied in the literature [1, 2], the proposed method can be utilized to effectively solve the three-dimensional Poisson equation in the context of a collocation or a Galerkin boundary element method. Several key features of this study, including the extension of the fundamental theorem of calculus to higher dimension, are highlighted. In addition, numerical examples dealing with mixed boundary-value problems for the Poisson equation on representative test geometries are carried out successfully to validate the proposed method.

References [1] S. Nintcheu Fata, Explicit expressions for 3D boundary integrals in potential theory, Int. J. Num. Meth. Eng., 78(1), pp. 32–47, 2009. [2] S. Nintcheu Fata, L. J. Gray, Semi-Analytic Integration of Hypersingular Galerkin BIEs for threedimensional Potential Problems, J. Comput. Appl. Math., 231(2), pp. 561–576, 2009.

The submitted manuscript has been authored by a contractor of the U.S. Government under Contract No. DE-AC05-00OR22725. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.


MESHFREE MICRO-SCALE MODELING AND STRESS ANALYSES OF 3D ORTHOGONAL WOVEN COMPOSITES L. Y. Li Department of Aeronautics, Imperial College, London SW7 2AZ, UK

P. H. Wen School of Engineering and Materials Science, Queen Mary University of London, London E1 $NS, UK

M. H. Aliabadi Department of Aeronautics, Imperial College, London SW7 2AZ, UK

Abstract In this paper, evaluation of 3D orthogonal woven fabric composite elastic moduli is achieved by applying Meshfree methods on the micromechanical model of the woven composites. A new, realistic and smooth fabric unit cell model of 3D orthogonal woven composite is presented. As an alternative to Finite Element Method, Meshfree Methods show a notable advantage, which is the simplicity in meshing while modelling the matrix and different yarns. Radial basis function and moving kriging interpolation are used as the shape function construction respectively. The Galerkin method is employed in formulating the discretized system equations. The numerical results are compared with the Finite Element results and the experimental ones. Keywords: Meshfree method; 3D orthogonal woven fabric composites; elastic moduli; Galerkin; radial basis; moving kriging

571

Advances in Boundary Element Techniques XI

Advances in Boundary Element Techniques XI

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