Advances in Boundary Element Techniques XII

Advances in Boundary Element Techniques VII

ISBN 978-0-9547783-8-5

EC ltd

Advances in Boundary Element Techniques XII

Edited by E. L. Albuquerque M H Aliabadi

Advances In Boundary Element and Meshless Techniques XII

Advances In Boundary Element and Meshless Techniques XII

Edited by E. L. Albuquerque M H Aliabadi

EC

ltd

Published by EC, Ltd, UK Copyright © 2011, Published by EC Ltd, P.O.Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

All Rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, except under terms of the Copyright, Designs and Patents Act 1988. Requests to the Publishers should be addressed to the Permission Department, EC, Ltd Publications, P.O.Box 425, Eastleigh, Hampshire, SO53 4YQ, England.

ISBN: 978-0-9547783-8-5

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 and Meshless Techniques XII 13-15 July 2011, Brasilia, Brazil Steering Committee:

Gatmiri,B (France)

M.H. Aliabadi, Imperial College London E.L. Albuquerque, University of Brasilia P. Sollero, University of Campinas P. Partridge, University of Brasilia E. Mesquita, University of Campinas L. Palermo Jr., University of Campinas L.M. Bezerra, University of Brasilia C.T.M. Anflor, University of Brasilia

Gray,L (USA)

Organising Steering Committee:

Marin, L (Romania))

Hematiyan,M.R. (Iran) Liu,G-R (Singapore) Mallardo,V (Italy) Mansur, W. J (Brazil) Mantic,V (Spain) Matsumoto, T (Japan)

Professor Eder Lima de Albuquerque

Mattheij, R.M.M (The Netherlands)

Department of Mechanical Engineering Faculty

Mesquita,E (Brazil)

of Technology, University of Brasilia

Millazo, A (Italy)

Brasilia, DF, Brazil

Minutolo,V (Italy)

[email protected]

Ochiai,Y (Japan)

Professor Ferri M H Aliabadi

Panzeca,T (Italy)

Department of Aeronautics

Partridge,P (Brazil)

Imperial College, South Kensington Campus

Perez Gavilan, J J (Mexico)

London SW7 2AZ

Pineda,E (Mexico)

[email protected]

Prochazka,P (Czech Republic)

International Scientific Advisory

Qin,Q (Australia)

Committee

Saez,A (Spain)

Abascal,R (Spain)

Sapountzakis E.J. (Greece)

Abe,K (Japan)

Sellier, A (France)

Baker,G (USA)

Seok Soon Lee (Korea)

Blasquez,A (Spain)

Shiah,Y (Taiwan)

Chen, Weiqiu (China)

Sladek,J (Slovakia)

Chen, Wen (China)

Saldek, V (Slovakia)

Cisilino,A (Argentina)

Sollero.P. (Brazil)

Davies,A (UK)

Tan,C.L (Canada)

Denda,M (USA)

Telles,J.C.F. (Brazil)

Dong,C (China)

Wen,P.H. (UK)

Dumont,N (Brazil)

Watson,J (Australia)

Estorff, O.v (Germany)

Wrobel,L.C. (UK)

Gao,X.W. (China)

Yao,Z (China)

Garcia-Sanchez,F (Spain)

Zhang, Ch (Germany)

PREFACE The Conferences on Boundary Element and Meshless 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), Athens, Greece (2009) and Berlin, Germany (2010). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the FINATEC, University of Brasilia during 13th-15th July 2011. 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. 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. We would like to dedicate the conference to the memory of our friend and college Prof Wilson Venturini.

Editors July 2011

Contents Topological optimization for potential problems using Bezier smoothed boundary C T M. Anflor, R J Marczak Numerical aspects of the use of the radial integration method in an anisotropic shallowshell boundary element formulation L J M Jesus, E L Albuquerque, P Sollero Implementation of viscoplastic analysis for multi-region problems using the boundary element method F E S Anacleto, G O Ribeiro, T S A Ribeiro Some comments about the performance of the hyper-singular boundary element formulation in potential problems C F Loeffler, R G Peixoto, E R Lovatte A combined SGBEM and conic quadratic optimization approach for limit analysis T Panzeca, F Cucco, E Parlavecchio, L Zito A fully kinematic solution of mixed boundary value problems of elastostatics J O Watson A meshless BEM for transient thermoelastic crack analysis in functionally graded materials A Ekhlakov, O Khay, Ch Zhang, J Sladek, V Sladek Preconditioning of BEM systems of equations by using a generic subregioning algorithm F C de Araújo Alternative derivations of fundamental solutions for anisotropic heat transfer R J Marczak, M Denda Decompositions of Cijkl aiming the inversion of acoustic tensors for fundamental solution derivations R J Marczak Computation of the effective thermal conductivity of functionally-graded random micro-heterogeneous materials via the fast multipole BEM M Dondero, A Cisilino Shape sensitivity analysis of 3D acoustic problems based on BEM and its application to topology optimization T Matsumoto, T Yamada, T Takahashi, C J Zheng, S Harada Computation of displacements in anisotropic plates by the boundary element method A Reis, E L Albuquerque J F Useche, H. Alvarez Numerical analysis of failure in laminate composites using phenomenological based criteria D I G Costa, E L Albuquerque, A Reis, G Panosso, P Sollero Investigation of test function in meshless local integral equation method P H Wen, M H Aliabadi An overview of boundary element formulations for micro and nano fluids L C Wrobel A domain integral equation formulation of an inhomogeneous bending thin plate Y S Yang and C Y Dong Fundamental solution based FEM for nonlinear thermal radiation problem Qing-Hua Qin and Hui Wang

1 7

13

19 25 33 38

44 50 57

66

72

78

84 90 98 104 113

Time integrations in solution of diffusion problems by Local integral equations and Point interpolation method V. Sladek, J. Sladek and Ch. Zhang Large deflection analysis of plates stiffened by parallel beams E.J. Sapountzakis and I.C. Dikaros Strategies for the implementation of 2D quadratic boundary elements on graphics hardware – GPU Josué Labaki, Euclides Mesquita and Luiz Otávio Saraiva Ferreira A fast hierarchical BEM for 3-D anisotropic elastodynamics A. Milazzo, I. Benedetti, M. H. Aliabadi Coupled evolution of damage and fluid flow in a mandel-type problem Eduardo T Lima Junior, Wilson S Venturini, Ahmed Benallal Stability analysis of composite laminate plates under non-uniform stress fields by the boundary element method P. C. M. Doval, E. L. Albuquerque, and P. Sollero Expedite implementation of the boundary element method Ney Augusto Dumont and Carlos Andrés Aguilar Marón Use of generalized Westergaard stress functions as fundamental Solutions N A Dumont, E Y Mamani Vargas Direct solution of differential equations using a wavelet-based multiresolution method Rodrigo Bird Burgos, Raul Rosas e Silva and Marco Antonio Cetale Santos A study of dual reciprocity for three dimensional models applied to the solution of Pennes Bioheat equation F R Bueno, P W Partridge Stress analysis of thin plate composite materials under dynamic loads using the boundary element L. S. Campos, K. R. P. Sousa, A. P. Santana, A. dos Reis, E. L. Albuquerque5 and P.Sollero Deformation analysis of thin plate with distributed load by triple-reciprocity boundary element method Y Ochia, T Shimizu An active noise control sensitivity formulation for three dimensional BEM A Brancati, M H Aliabadi and V Mallardo Transient heat conduction by the boundary element method: an alternative D-BEM approach J A M Carrer, M F Oliveira, A L Ferreira Building three-dimensional analysis considering bending plates by a BEM/FEM coupling L. de Oliveira Neto Damage detection in beams using boundary elements and wavelet transform R S Y C Silva, L M Bezerra, L A. P. Peña Response of isosceles triangular canyon to harmonic Rayleigh eave A Eslami Haghighat, A Parvaneh A novel semi-analytical method with diagonal coefficient matrices for the analysis of elastostatic problems M.I. Khodakarami, N. Khaji Elastic property prediction of 3D braided composites by meshfree methods L. Li and M. H. Aliabadi

119 125 133 139 145 156 162 170 176

182

190

196 202 209

217 224 230 236 242

BEM study of a fibre-matrix interface crack under biaxial transverse loads L Tavara, V Mantic, E Graciani, F Paris Three-Dimensional fundamental solution for unsaturated poroelastic media under dynamic loadings P Maghoul, B Gatmiri, D Duhamel BEM implementation of energetic solutions for quasistatic delamination problems C. G. Panagiotopoulos, V. Mantic, T. Rouıcek The dual-boundary-element formulation using the tangential differential operator and incorporating a cohesive zone model for elastostatic cracks P. C. Gonçalves, L. G. Figueiredo, L. Palermo Jr., S. P. B. Proença Slow viscous migration of a solid particle near a plane wall with a slip condition N Ghalia, A Sellier, L Elasmi, F Feuillebois Green tensor for a general non-isotropic slip condition A Sellier and N Ghalia Application of the OMLS interpolation to evaluate volume integrals arising in static and time–dependent elastoplastic analysis via D–BEM K . da Silva, J C F Telles and F C de Araújo A dual reciprocity boundary element formulation for transient dynamic analysis of shallow shells J F Useche A coupled boundary and finite element methodology for solving fluid-structure interaction problems M Barcelos, C T M. Anflor, É L. Albuquerque Analysis of acoustic wave Propagation in shallow water using the method of fundamental solutions J. A. F. Santiago, E. G. A. Costa, L. Godinho and A. Pereira Modal analysis of thick plates using boundary elements W P Paiva, P Sollero, E L Albuquerque Analytical expressions for radial integration BEM X-W Gao, K Yang, J Liu Drilling rotations and partition of unity in composite plates by the boundary element method P M Baiz, E L Albuquerque, P Sollero The analysis of single lap joint stress distributions by the BEM R Q Rodríguez, C A O Souza, P Sollero, E L Albuquerque Higher-order Green’s function derivatives and BEM evaluation of stresses at interior points in a 3D generally anisotropic solid Y C Shiah and C L Tan Numerical investigation of the electromagnetic crack-face boundary conditions in fracture analysis of magnetoelectroelastic materials J Sladek, V Sladek, Ch Zhang, M Wünsche Variable temperature in creeping materials with the boundary element method E Avalos Gauna, E Pineda León, D Samayoa Ochoa, A. Rodríguez-Castellanos Shape optimization of fibers with given properties of phases by boundary elements P P Prochazka nsBETI method using localized Lagrange multipliers L Rodriguez-Tembleque, J A Gonzalez, R Abascal, K C Park

249 255 261 269 275 281 287

293

299

305

312 318 324 330 336

343 349 354 360

Analysis of the formulation D/DR-BEM for problems of anisotropic elastodynamics S Santos, R Dias, J A M Carrer, E L Albuquerque, L A de Lacerda MFS iteration algorithms with relaxation for the stable temperature reconstruction in two-dimensional steady-state isotropic and anisotropic heat conduction problems from incomplete boundary data L Marin A boundary element model for piezoelectric dynamic strain sensing of cracked structures A Alaimo, A Milazzo, C Orlando A spectrally accurate quadrature for 3-d boundary integrals G Baker and H Zhang Inverse analysis of solidification problems using the mesh-free radial point interpolation method A. Khosravifard, M. R. Hematiyan

366

375

383 391 398

Advances in Boundary Element and Meshless Techniques XII

1

TOPOLOGICAL OPTIMIZATION FOR POTENTIAL PROBLEMS USING BEZIERSMOOTHED BOUNDARY REPRESENTATION Carla T. M. Anflor¹ and Rogério J. Marczak2 1

Faculdade do Gama, Universidade de Brasília

PO Box 8114, 72405-610, Gama – DF – Brasil, [email protected] 2

Departamento de Engenharia Mecânica, Universidade Federal do Rio Grande do Sul

Rua Sarmento Leite 425, 90050-170, Porto Alegre – RS – Brasil, [email protected] Keywords: Shape optimization, boundary elements, topological derivative, Bezier, Orthotropic Materials. Abstract. This paper aims to demonstrate the final result of an optimization process when a smooth technique is introduced between intermediary iterations of a topological optimization. In a topological optimization process is usual irregular boundary results as the final shape. This boundary irregularity occurs when the way of the material is punched out is not very suitable. Avoiding an optimization post-processing procedure some techniques of smooth are implemented in the original optimization code. In order to attain a regular boundary a smoothness technique is employed, which is, Bezier curves. An algorithm was also developed to detect during the optimization process which curve of the intermediary topology must be smoothed. For the purpose of dealing with non-isotropic materials a linear coordinate transformation was implemented. Afterwards, the smoothed and nonsmoothed topologies are compared and discussed. Introduction Classical topological optimization, based on the homogenization or the density approach [1,2], often used for elasticity problems presents a several drawbacks. The final and intermediary topology results an appearance of sawtooth shape boundaries, which requires a post-processing. This final shape in a sensibility analysis using topological derivative also results in an irregular shape due to the way of the material was punched-out. An irregularity in the boundary, such as sized rectangular cells, is not suitable because it causes a field concentration around sharp corners. To overcome these defects, a new methodology for topology optimization is implemented in the original code. The idea consists in smooth the boundary during the process optimization, where shape and topology optimization are simultaneously performed in the design process. A smoothness technique is implemented and after its final shapes is compared with those obtained from the classical optimization solution. Figure 1 illustrates a BEM mesh smoothed by Bézier interpolation.

(a)

(b)

Figure 1. Example of boundary smoothing. (a) Original result. (b) Beziér-smoothed result. Attaining this objective, the concepts of a topological derivative (DT) and a selection of smooth method are introduced and successfully combined with the classical shape optimization. To demonstrate the feasibility of the method, a simple problem in heat transfer is tested and compared with the traditional optimization methods.

2

Eds: E L Albuquerque & M H Aliabadi

Optimisation Method The main idea of an optimization method is to identify unnecessary portion of a structure and then to eliminate this portions progressively using a determined numerical method model. After a numerical heat analysis, heat flux densities in the domain are determined. It often happens that the flux levels on some parts are lower than those in other. This implies that material of different points may have different performance and contribution to the structural behavior. In this topic two methods will be briefly described, which are ESO and the DT+BEM, being the last one used in the present work. Evolutionary structural optimization using a finite element method. A method based on the main idea of elemental removal was named as Evolutionary Structural Optimizazion (ESO) and was developed by [3]. This method was also implemented by [4] and [5]. Figure 2 depicts the results obtained by the authors method, being the initial design composed by all FEM mesh elements and after the final shape resulted from the removal of a determined elements. This method is fundamented in FEM and the function objective is explicit.

Figure 2. Shape optimization using ESO: (a) initial boundary condition and (b) final shape obtained by Li et al.(1999). Topological derivative optimization using Boundary element method A topological derivative formulation for Poisson Equation is used throughout this work [6,7,8]. The total potential energy is used to achieve the sensitivity cost function when the topology of the original analysis domain is changed. In this case the objective function is implicit, it means that, for another objective function different from total potential energy a new deduce for the DT formulation is require to determine the new sensitivity cost function. The DT in this work is coupled to BEM as numerical tool solution. The BEM presents some advantages, such as, no mesh dependence and the mesh is only on the boundary. This last one characteristic provided by BEM turn this method attractive when compared to FEM considering the computational cost. The present optimization scheme should be implemented with another numerical solution method, such as, FEM. Figure 3 illustrates the idea of the DT+BEM optimization method implemented in [9,10]. Step 1:

Step 2:

Initial problem Solve the problem using BEM solver Evaluate DT at internal points

Remove material

yˆ

y DT + BEM

Start

x

Figure 3. BEM iterative scheme for material removal.

xˆ


3

Bézier curves Generally in an optimizations process the final topology results in a non-smoothed geometry. It requires an employment of techniques of smoothness during the optimization process. In order to study the behavior of a final topology a technique where previously chosen to be implemented in the actual optimization code. The most popular techniques to deal with these irregular geometries are Bezier curves, Douglas-Peucker and B-Splines algorithm. This work uses the Bézier curves to smooth the topology resulted during the optimization process. The Bézier curve was pioneered used in a modeling of surface in automobile design by Renault [11].Bézier defines the curve P(u) in the terms of the location n+1 control points pi. n

P(u ) = ¦ pi Bi ,n (u )

(1)

i =0

where Bi ,n ( u ) is a blending function or polynomials of Bernstein Bi ,n (u ) = C ( n, i ) u i (1 − u )n −i

(2)

n! and C ( n, i ) = is the binomial coefficients. i !(n − i )!

The eq. (1) is a vector and could be expressed by writing equations for the x an y parametric functions separately: n

x (u ) = ¦ xi Bi ,n (u ) i =0

(3)

n

y (u ) = ¦ yi Bi ,n (u ) i =0

xi and yi are the coordinates of the control points of the curve, always n+1 points. The union of this points form the vertices of the control polygon of the Bézier curve. These points are responsible to control the shape of the curve, with the parameter u varying between 0 to 1. Further details should be attained in [12]. As exposed until here the techniques of smoothness curves are well established in the literature and there is no problem in apply them. But in an optimization problem the effort relies in identifying which portions of the intermediary topology must be smoothed. There are some parts of the topology that can not be smoothed, such as, the portion with prescribed boundary conditions or the portion which is a straight line. In order to overcoming this problem a routine is developed in the present work to identify during an iterative optimization process which curves must be or not smoothened. This routine was introduced inside the optimization algorithm after the step of removal material. Figure 4 depicts the scheme of identification and smoothness of curves resulted during the optimization process. b.c.

boundary condition (b.c.)

3

1

Identifying curves that must be smoothed

SL b.c.

SL b.c.

b.c. b.c.

b.c.

b.c.

b.c. Optimization routine BEM + DT

2

Smooth the three curves identified

Joint the geometries

SL = Straight Line

Figure 4. Scheme of identification and smoothness.

b.c.

4


Numerical example A previous example presented [10] is revisited here in order to prove the smoothness algorithm developed. This case refers to linear steady state heat transfer problems and the potential energy is used as the cost function. The material removal history is illustrated for some iterations. The iterative process is halted when a target material volume fraction ( A f / A0 , where A f and A0 are the final and initial volume, respectively) is achieved, regardless the type of material medium. This provided a simplified criterion to compare the topologies generated using Bézier curves as technique of smoothness. Linear discontinuous boundary elements integrated with 4 Gauss points are used in all cases. Unless specified otherwise, all cases used holes with λ = 0.03 . Cross heat conductor. This example refers to a square domain subjected to low and high temperature boundary conditions on the middle of opposite sides. The problem is depicted in fig.5, where TH is the high temperature (373 K) and TL is the low temperature (273 K). The remaining boundaries are insulated. All possible cases will be studied: isotropic, orthotropic and anisotropic materials and they are to be optimized until Af ≈ 0.4 A0 is achieved.

TH

TL

ȍ

TH

TL

initial

iteration 18

iteration 34

Figure 5. Evolution history for isotropic case. Initially, an isotropic case was analyzed with k11 = k22 = 1. Symmetry was not used to provide a direct comparison to the subsequent anisotropic cases (which cannot use symmetry). Figure 5 also shows the evolution of material removal for r = 0.02lref. It is important to observe that the algorithm delivered fairly symmetric solutions throughout the process. The condition Af ≈ 0.4ÂA0 was achieved after 34 iterations. The second case represents a highly orthotropic material, with the conductivities set to kxx = 5 and kyy = 1 (see fig.6). As expected, material is selectively removed so that the heat flux along the x direction is increased. The stop criterion Af ≈ 0.4 A0 was achieved after 30 iterations.

iteration 12

iteration 25

iteration 30

Figure 6. Evolution history for orthotropic case. The third case considers an anisotropic material with kxx = 1, kyy = 1 and kxy = 0.5. The evolution history is presented in fig.7, showing that the initial symmetry is lost after the first iterations, as expected. Contrary to the previous cases, the internal cavity resulted in a rhombic shape since the Cartesian axes are not parallel to the main axes of the constitutive matrix.


iteration 12

iteration 34

5

iteration 41

Figure 7. Evolution history for anisotropic case. Figure 8 shows the percentage of material removed versus the number of iterations for each case studied in example 5.4. All cases were stopped when about 40% of material was removed. These cases were analyzed without the aid of symmetry, for comparison purposes. Obviously, anisotropic cases cannot use symmetry in general, but in many practical situations it is possible (or even expected) to align the axes of the component with the principal directions of the constitutive matrix. In such cases, smoother designs can be obtained. In order to provide a further benchmark, this example is re-analyzed for the isotropic and orthotropic cases using only one quadrant of the original geometry. Figure 9 shows the final topologies obtained for both cases while fig.10 depicts the same topology after the smoothness process. The material was removed initially with r = 0.04 lref and then r = 0.02 lref. for the remaining iterations. This simple expenditure helps generate smoother boundaries in the final design. 100 anisotropic case orthotropic case isotropic case

95 90

Area Removed %

85 80 75 70 65 60 55

0

5

10

15

20

25

30

35

40

45

Iterations

Figure 8. Material removal history for example 3.

TH

TL

TL

TH

(a)

(b)

Figure 9. Final topologies for: (a) isotropic and (b) orthotropic examples.

6


(a)

(b)

Figure 10. Beziér final topologies for: (a) isotropic and (b) orthotropic examples. Conclusions The presented work uses the DT to calculate the sensibility of the domain when a small hole is introduced. Those portions with lower sensibility have material removed by an optimization code. This procedure results in an irregular shape due to the way of the material is punched out. The objective of this paper was attained with the development of an algorithm of smoothness. The case studied here had its intermediary geometries smoothed as the iterative optimization process had evolved. The smoothness algorithm allowed avoiding a post-processing procedure in order to eliminate those sawtooth shape boundaries. The particularities of the joint of DT and BEM were preserved, such as, no mesh dependence and low computational cost. This new methodology presents a suitable procedure without lost accuracy turning the present optimization code (BEM + DT + SMOOTH) attractive. References [1] D.N.Dyck, IEEE Trans. Magn., 30, 3415-3418 (1994). [2] J.K.Byun, IEEE Trans. Magn., 35, 3718–3720 (1999). [3] Y.M.Xie, G.P.Steven, Comput. Struct, 49, 885-896 (1993). [4] D.N.Chu, Y. M.Xie, A.Hira, G. P.Steven, , Finite Elements in Analysis and Design, 24, 197-212 (1997). [5] Q.Li, G.Steven, Y.Xie,, O.Querin, Int. J. Heat Mass Transfer, 47, 5071–5083 (2004). [6] J.Céa, S. Garreau, P.Guillaume and M.Masmoudi, Comput. Methods Appl. Mech. Engrg., 188, 713-726 (2000). [7] R.Feijóo, A.Novotny, C.Padra, and E.Taroco, In Idelsohn, S., Sonzogni, V., and Cardona, A., eds. Mecánica Computacional, Vol. XXI, 2687-2711, Santa-Fé-Paraná, Argentina (2002). [8] A.Novotny,, R.Feijóo, E.Taroco, E., and C. Padra, Comput. Methods Appl. Mech. Engrg., 192, 803–829 (2003). [9] R.J.Marczak, Engineering Analysis with Boundary Elements, 31, 793-802 (2007). [10] C.T.M.Anflor and R.J. Marczak, Int. Journal of Heat and Mass Transfer, 52, 4604-4611 (2009). [11] W. M.Newman, R.F.Sproull, Principles of interactive computer graphics, McGraw-Hill (1982). [12] S. Harrington, Computer Graphics: A programming approach, McGraw-Hill (1983).


Numerical aspects of the use of the radial integration method in an anisotropic shallowshell boundary element formulation L. J. M. Jesus1 , E. L. Albuquerque2 and Paulo Sollero3 1

2

Faculty of Mechanical Engineering, State University of Campinas Campinas, Brazil, [email protected]

Department of Mechanical Engineering, Faculty of Technology, University of Braslia Braslia, DF, [email protected] 3 Faculty of Mechanical Engineering, State University of Campinas Campinas, Brazil, [email protected]

Keywords: Laminated composites, shallow shells, radial integration method. Abstract. The radial integration method is a suitable technique to transform domain integrals into boundary integrals. It is quite appropriated for anisotropic materials because it is a pure numerical technique that does not require the computation of approximation functions as in dual reciprocity boundary element method. However, a special attention should be payed on the numerical integration because it has strong influence on accuracy and computation cost of the method. This paper presents an analysis of performance of the radial integration method, considering accuracy and computational cost, when it is used in a boundary element formulation of symmetric laminated composite shallow shells. Introduction This paper aims to analyze the problems of shallow shells composite laminates, using the boundary element method. The formulation for plane elasticity and the classical theory of plates (Kirchhoff plates) are coupled. Due to curvature of the shell, domain integrals arise in the formulation. These integrals are transformed into boundary integrals using the radial integration method. The formulation of the boundary element method for anisotropic plane elasticity was developed by [2] to problems of fracture mechanics and elastic-static and extended to other problems in the research of [1] and [3]. The formulation of boundary elements for the classical theory of anisotropic plates was developed by [4, 5], and extended to other problems in the work of [6, 7, 8, 9, 10]. The formulation of shallow shells was developed by [11] for isotropic shells and by [12] for anisotropic shells. These last two studies have used the radial integration method for the transformation of domain integrals remaining in the formulation into boundary integrals. The computational cost of this formulation was high however, especially for anisotropic formulation. The main contribution of this paper is to examine the sensitivity of the radial integration method in relation to the number of integration points for the formulation of anisotropic shallow shells. As quoted in the literature, in [13] and [14], the radial integration method demand few integration points to obtain a solution close to the analytical solution. In these two studies, good results are obtained with four integration points. However, this was not carried out for anisotropic shallow shells. The main focus of this paper is a study on the number of integration points that are needed to obtain results with good precision. Boundary integral equations Consider a shallow shell of an anisotropic elastic material with the mid-surface being described by z = z(x1 , x2 ). The base-plane of the shell is defined in a domain Ω in the plane x1 , x2 whose boundary is given by Γ. Using the equilibrium equation of shallow shells, the reciprocity relation, and the Green theorem,

7

8


it is possible to derive integral equations that can be divided in terms of plane elasticity and plate bending formulations as shown by Zhang and Atluri [15] for isotropic shallow shells. These equations are coupled by the domain integrals that arise in both of them. Integral equations for the plane elasticity formulation are given by:

cij uj +

+

Ω

Γ

t∗ik (Q, P )uk (P )dΓ(P ) =

κkj wu∗ik,j (Q, P )dΩ +

Ω

Γ

u∗ik (Q, P )tk (P )dΓ(P )

u∗ik (Q, P )qk (P )dΩ(P ),

(1)

where i, j, k = 1, 2; uk is the displacement in directions x1 and x2 , ti = Nij nj , Nij are membrane forces applied in the shell; u3 stands for the displacement in the normal direction of the shell surface; κ depends on the curvature radii Rij of the shallow shell; kij are the inverse of curvature radii. P is the field point; Q is the source point; and asterisks denote fundamental solutions. The anisotropic plane elasticity fundamental solutions can be found, for example, in [16]. The constant cij is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. The integral equation for the plate bending formulation is given by:

Ku3 (Q) +

=

Γ

Nc i=1

Rci (P )u∗3ci (Q, P ) +

+

+

+

Γ

Γ

Ω

Vn∗ (Q, P )w(P ) − m∗n (Q, P ) Ω

Nc ∂w(P ) dΓ(P ) + Rc∗i (Q, P )u3ci (P ) ∂n i=1

q3 (P )u∗3 (Q, P ) dΩ

Vn (P )u∗3 (Q, P ) − mn (P )

∂u∗3 (Q, P ) dΓ(P ) ∂n

Cκnj ui (P )u∗3 (Q, P ) dΓ(P ) +

Ω

C

κij ∗ u (Q, P )u3 (P ) dΩ ρij 3

[Cκij (P )u∗3 (Q, P )],j ui (P ) dΩ,

(2)

where ∂() ∂n is the derivative in the direction of the outward vector n that is normal to the boundary Γ; mn and Vn are, respectively, the normal bending moment and the Kirchhoff equivalent shear force on the boundary Γ; Rc is the thin-plate reaction of corners; u∗3ci is the transverse displacement of corners; q3 is the domain force in the x3 direction; The constant K is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. As in the previous equation, an asterisk denotes a fundamental solution. Fundamental solutions for anisotropic thin plates can be found, for example, in [5]. As can be seen, domain integrals arise in the formulation owing to the curvature of the shell. In order to transform these integrals into boundary integrals, consider that a body force b is approximated over the domain Ω as a sum of M products between approximation functions fm and unknown coefficients γm , that is: b(P ) =

M

γ m fm .

(3)

m=1

The approximation function used in this work is: fm = R2 log(R), Equation (3) can be written in a matrix form, considering all source points, as:

(4)


9

b = Fγ

(5)

γ = F−1 b

(6)

Thus, γ can be computed as:

Body forces of integral equations (1) and (2) depend on the displacements. So, using equation (6) and following the procedure presented by Albuquerque et al. [17], domain integrals that come from these body forces can be transformed into boundary integrals. Then, by discretization of these boundary integrals, a matrix equation can be obtained. Finally, after applying boundary conditions, this matrix equation is transformed in a linear system that can be solved to find the unknowns of the shell problem. Numerical results In order to assess the accuracy of the proposed formulation, consider a square spherical shallow shell of a symmetric laminated cross-ply composite material, as shown in Figure 1. The geometry and material properties of the shell are: ratio between the edge length and thickness a/h =100, ratios between elastic moduli E1 /E2 = 25, G12 /E2 = 0.5, Poisson ratio ν12 = 0.25. The laminated has four layers with orientation [0o /90o /90o /00 ]. All layers have equal thickness of h/4. A uniformly distributed load in the transversal direction (internal pressure) q3 = qo (q1 = q2 = 0) is considered.

a

x2

a

R x1 R h Figure 1: Square spherical shallow shell. This problem was analised with different ratios between edge length and curvature radius a/R1 = a/R2 = a/R, (R12 = R21 = 0). Three meshes were used. Mesh 1 has 12 quadratic discontinuous boundary elements and 9 internal points, mesh 2 has 20 quadratic discontinuous boundary elements and 25 internal points, and mesh 3 has 28 quadratic discontinuous boundary elements and 49 internal points. All meshes have elements of equal length and uniformly distributed internal points. Boundary conditions considered for the shell are t1 = u2 = w = mn = 0 at x1 = 0 and x1 = a, and u1 = t2 = w = mn = 0 at x2 = 0 and x2 = a (see Reddy [18]). Table 1 shows results for shells under uniformly distributed load. Results are compared with analytical results presented by Reddy [18]. The formulation proposed by [18] is for moderate deep shells. As can be noticed in Table 1, results converge to the analytical solution with mesh refinement. It is also possible to notice that there is a good agreement of the proposed formulation with the analytical solution of Reddy [18] even for small ratio between and curvature radius and edge length (R/a = 1, for example). Comparing the last line of Table 1 (results obtained with the most refined mesh with

10


Table 1: Transversal displacement of the centre of the cross-ply shallow shell w ˜ = 103 wE2 h3 /(qo a4 ) under uniformly distributed load. Results computed using approximation function fm2 . R/a Ref. [18] w ˜ (mesh 1) w ˜ (mesh 2) w ˜ (mesh 3)

∞ 6.8331 6.8100 6.8014 6.7991

100 6.7772 6.7489 6.7452 6.7437

50 6.6148 6.5721 6.5817 6.5824

20 5.6618 5.5495 5.6238 5.6352

10 3.7208 3.5482 3.6819 3.7044

5 1.5358 1.4238 1.5129 1.5290

2 0.2844 0.2620 0.2800 0.2833

1 0.0715 0.0649 0.0701 0.0712

the second line of these Tables (results obtained by Reddy [18]), we notice that differences are always smaller than 1 %. As this shell is very thin (a/h =100), the effect of the shear deformation is very small. Because this, results obtained by the proposed method, that uses the formulation of Kirchhoff for thin plates, has a good agreement with results obtained using a shear deformable theory. In order to study the sensitivity of the method with the number of integration points, results were obtained using mesh 3 with different number of integration points. Normalized displacements at the centre of the shell computed with different integration points are shown in Figures 2, 3, and 4. 5 Mesh 1 Mesh 2 Mesh 3 Reddy (1985)

4

3

w

2

1

0

−1

−2

2

4

6

8 10 12 14 Number of integration points

16

18

20

Figure 2: Sensitivity of transverse displacement of the shell under load a uniformly distributed load versus the variation of number of integration points for R/a = 10. As shown in Figures 2, 3 and 4, the results with few integration points (2 or 4 points) have high errors when compared with results obtained by [18]. However, increasing the number of integration points, results of boundary element method converge quickly to the result presented by [18]. For the problem analysed here, it is concluded that good results can be obtained using 6 integration points. Conclusions This paper presented an analysis of performance of the radial integration method, considering accuracy and computational cost, when it is used in a boundary element formulation of symmetric laminated composite shallow shells. It was shown that very good results can be obtained with few integration points (6 integration points). This result makes this method very suitable for the treatment of these types of problems, since the computational cost is not high and the fact that we do not need to calculate particular solutions makes the radial integration method advantageous, because of the easy implementation, when compared with the dual reciprocity boundary element method.


3000 Mesh 1 Mesh 2 Mesh 3 Reddy (1985)

2500

2000

w

1500

1000

500

0

−500

2

4

6


16

18

20

Figure 3: Sensitivity of transverse displacement of the shell under load a uniformly distributed load versus the variation of number of integration points for R/a = 20. 8.4 Mesh 1 Mesh 2 Mesh 3 Reddy (1985)

8.2 8 7.8

w

7.6 7.4 7.2 7 6.8 6.6 6.4

2

4

6


16

18

20

Figure 4: Sensitivity of transverse displacement of the shell under load a uniformly distributed load versus the variation of number of integration points for R/a = 50. Acknowledgment The first author would like to thank the CNPq (The National Council for Scientific and Technological Development, Brazil) e AFOSR (Air Force Office of Scientific Research, USA), for financial support for this work.

References [1] E. L. Albuquerque Numerical analysis of dynamic anisotropic problems using the boundary element method. Tese (Doutorado) — Unicamp, Dept. Mec. Comput., July 2001. [2] SOLLERO, P. Fracture mechanics analysis of anisotropic laminates by the boundary element method. Tese (Doutorado) — Wessex Institute of Technology, 1994. ˆ A. R. Critérios de falha e otimiza¸c ao de estruturas de materiais comp´ [3] GOUVEA, ositos usando o método dos elementos de contorno. Dissertao (Mestrado) — Unicamp, Dept. Projeto Mecˆ anico., fevereiro 2006.

11

12


[4] PAIVA, W. P. Análise de problemas est´ aticos e dinâmicos em placas anisotrópicas usando o método dos elementos de contorno. Tese (Doutorado) — Unicamp, Departamento de Projeto Mecnico, fevereiro 2005. [5] ALBUQUERQUE, E. L. et al. Boundary element analysis of anisotropic kirchhoff plates. International Journal of Solids and Structures, v. 43, p. 4029–4046, 2006. [6] TORSANI, F. L. Implementa¸cão do c´ alculo de tens˜ oes em placas laminadas de materiais comp´ ositos usando o método dos elementos de contorno. Dissertao (Mestrado) — Unicamp, Dept. Projeto Mecˆ anico., agosto 2007. [7] SANTANA, A. P. Formula¸c˜ oes dinˆ amicas do método dos elementos de contorno aplicado a an´ alise de placas finas de compósitos laminados. Dissertao (Mestrado) — Unicamp, Dept. Projeto Mecˆ anico., novembro 2008. [8] SOUZA, K. R. Análise de Tensões em Placas Finas de Materiais Compósitos sob Carregamento Dinâmico usando o Método dos Elementos de Contorno. Dissertao (Mestrado) — Unicamp, Dept. Projeto Mecˆ anico., novembro 2009. [9] REIS, A. Formula¸c˜ oes do Método dos Elementos de Contorno para Análise de Placas de Materiais Comp´ ositos Laminados. Tese (Doutorado) — Unicamp, Departamento de Projeto Mecˆ anico, 2010. [10] REIS, A. et al. Computation of moments and stresses in laminated composite plates by the boundary element method. Engineering Analysis with Boundary Elements, v. 35, p. 105–113, 2011. [11] ALBUQUERQUE, E. L.; ALIABADI, M. H. A boundary element formulation for boundary only analysis of thin shallow shells. CMES - Computer Modeling in Engineering and Sciences, v. 29, p. 63–73, 2008. [12] ALBUQUERQUE, E. L.; ALIABADI, M. H. A boundary element analysis of symmetric laminated composite shallow shells. Computational Methods in Applied Mechanics and Engineering, v. 199, p. 2663–2668, 2010. [13] GAO, X. The radial integration method for evaluation of domain integrals with boundary only discretization. Engn. Analysis with Boundary Elements, v. 26, p. 905–916, 2002. [14] JESUS, L. J. M.; ALBUQUERQUE, E. L.; SOLLERO, P. Further developments in the radial integration method. XXXI CILAMCE - Congresso Ibero Latino Americano de Mtodos Computacionais em Engenharia. Buenos Aires, Argentina: [s.n.], 2010. [15] ZHANG, J. D.; ATLURI, S. N. A boundary/interior element method for quasi-static and transient response analysis of shallow shells. Computers and Structures, v. 24, p. 213–223, 1986. [16] SOLLERO, P.; ALIABADI, M. H. Fracture mechanics analysis of anisotropic plates by the boundary element method. Int. J. of Fracture, v. 64, p. 269–284, 1993. [17] ALBUQUERQUE, E. L.; SOLLERO, P.; PAIVA, W. P. The radial integration method applied to dynamic problems of anisotropic plates. Communications in Numerical Methods in Engineering, v. 23, p. 805–818, 2007. [18] REDDY, J. N. Exact solution of thick laminated shells. ASME Journal of Engineering Mechanics, v. 323, p. 319–330, 1985.


Implementation of Viscoplastic Analysis for Multi-region Problems Using the Boundary Element Method F. E. S. Anacleto, G. O. Ribeiro e T. S. A. Ribeiro Depto. de Engenharia de Estruturas, Escola de Engenharia, Universidade Federal de Minas Gerais Avenida Antônio Carlos, 6627, CEP: 31270-901, Belo Horizonte, MG, Brasil. [email protected], [email protected]; [email protected] Keywords: viscoplasticity, time-dependent behavior, multi-region domain.

Abstract. This work deals with the implementation of the boundary element method (BEM) for multi-region domain problems taking into account that the domain is composed of elasto/viscoplastic materials. The viscoplastic responses are obtained according to Perzyna’s flow rule. In order to obtain the solution of the multiregion problem, each region of the domain is solved separately by applying the classical BEM considering the interfacial nodes restrained against displacements and the other nodes with the specified boundary conditions. By the successively application of the BEM with a unit displacement value prescribed to each degree of freedom correspondent to each interfacial node and direction, a stiffness matrix is evaluated for each region. Assembling a global stiffness matrix and enforcing the compatibility and equilibrium conditions at the interfacial nodes the displacements can be evaluated at the interfaces. From these results, the remaining unknowns can be obtained. This technique can also be used for coupling BEM and finite element method. The efficiency and accuracy of the BEM implementation are analyzed by means of numerical examples. Introduction The domain of some important engineering problems like excavations and tunnels present non-homogeneous material properties, which can be approximated by a zoned domain, i.e., the domain is supposed to be made up by distinct regions also known as sub-regions, each one with specific material properties. These multi-regions with their specific boundaries are modeled independently and then coupled along interfaces. Besides this, many materials, such as concrete, polymers and soils, exhibit a time-dependent behavior, i.e., for a constant load different deformation and stress patterns are developed within the body as time passes. The main objective of this work is the implementation of the BEM for multi-region domain problems, under the hypothesis that the domain is composed of elasto/viscoplastic materials. In order to simulate the viscoplastic flow, the approach presented by Venturini [1] is adopted, and to solve the multi-region problem, the stiffness matrix method, which is presented by Beer et al. [2], is used. Constitutive Relations The uniaxial representation of the elasto/viscoplastic mechanical model is shown in Fig. 1. This model simulates a material which behaves linearly elastic up to reach a limit value when irreversible time-dependent strains appear. In Fig. 1 ∆ is a stress increment, ∆ is an elastic strain increment, ∆ is a viscoplastic strain increment, is an elastic parameter, is a viscous parameter and is the yield stress which is an scalar function of the hardening parameter .

Fig. 1 Elasto/Viscoplastic Model

13

14


The constitutive equation for the elasto/viscoplastic model generalized for tridimensional problems can be written as [1]

∆ ∆ ∆

(1)

where ∆ , ∆ , ∆ $, and stands for the components of the total stress increment, the total strain increment, the viscoplastic strain increment and the constitutive elastic tensors, respectively. It is important to note that eq (1) is defined in an initial stress sense and is valid only for a small increment of time ∆. In a similar way as in the elastoplastic analysis, a yield function and a flow rule are needed. The von Mises yield criterion and Perzyna's viscoplastic flow rule are adopted in order to perform the analysis and can be written, respectively, as J

(2)

∆ 〈 〉 ∆

(3)

where J is the second invariant of the stress deviatoric tensor and denotes any convenient reference value of

for the dimensionless representation of ∆ . In eq (3) the 〈 〉 notation implies that if , 〈 ⁄ 〉 . Then, for an increment of time ∆, an increment of the initial stress tensor can be expressed as

∆ ∆

(4)

The time increment value cannot be chosen freely due to numerical instability. In the current work, the time step length limit proposed by Cormeau [3] for the associative von Mises flow rule is used which can be written as ∆ "

# $ % &

(5)

where % is the Poisson coefficient and & is the Young's modulus. The time step limit is evaluated for each subregion based on its specific material properties and from these time steps values, the smallest value is adopted. Boundary Integral Equation The elasto/viscoplastic displacement boundary integral equation (BIE) can be developed in a similar fashion as the classical displacement BIE, but in this case the BIE must be expressed in an incremental form taking into account the incremental constitutive equation (eq (1)). The incremental displacement BIE based on the infinitesimal strain tensor, taking into account initial stresses, can be written as [1]

+ ', 1 /Ω 1∆ '∆( ' ) * + ', -∆. - /Γ ) 0 + ', -∆( - /Γ $ ) & Γ

Γ

(6)

Ω

+ where * + , 0 + and & represents, respectively, the displacement, traction and strain Kelvin fundamental $ denote, respectively, the displacement, traction and initial stress components. solutions, and ∆( , ∆. and ∆ In eq (6) the letters ', - and 1 represents, respectively, the source point, a field point at the boundary, and a field point in the domain. This is the same BIE of the elastoplastic formulation based on initial stress, the main difference is that an elastoplastic analysis proceeds in load increments while a viscoplastic analysis proceeds in time increments [2].


15

Boundary Element Method Following the usual procedures of the BEM, the boundary Γ is discretized into continuous linear and quadratic boundary elements, over which the boundary values (displacement and tractions) are approximated based on the respective boundary nodal values. The domain Ω is discretized into internal cells also using continuous linear and quadratic approximations. Viscoplastic formulation. By means of the discretization and the respective approximations in the viscoplastic BIE (eq (6)), an incremental system of equations is obtained, as 2034∆(5 2*34∆.5 $ 2&34∆ 5

(7)

where 4∆(5, 4∆.5 and 4∆ 5 denote vectors containing, respectively, nodal incremental values of displacements, tractions and initial stresses. The matrices 2*3, 203 and 2& 3 respectively contain the coefficients generated by the : multiplied by the shape functions numerical integrations of Kelvin fundamental solutions * + , 0 + and 6789 generate. Multi-region method. Equation (7) is valid only for one homogenous domain. In order to achieve the numerical solution of a multi-region problem using the stiffness method procedure, at the beginning each subregion ; has to be solved separately considering the prescribed loads together with the restriction applied at the interfacial nodes against displacements. This will result in a vector of tractions along the interface for each sub> region 4∆.5> @=

(8)

where matrix 2C3 contains the integration coefficients correspondent to the unknown boundary values while vector 4F5 gives the influence of the prescribed boundary values. Then a stiffness matrix 2G3 is assembled for each sub-region ;. In order to do this, the problem is solved ;H times for each sub-region (see eq (9)), where ;H stands for the number of degrees of freedom of all coupled nodes. Each solution is obtained by applying a unit displacement to each degree of freedom of the interface, one at a time. In this phase the matrix 2C3 is constant and only vector 4F5 has to be revaluated for each solution. A

4∆.5> @I

(9)

After assembling the stiffness matrix, the real solution at the boundary free nodes 4∆?5> @ and the real tractions at the boundary coupled nodes 4∆.5> < can be written as a function of the real displacements at the boundary coupled nodes 4∆(5> < , as can be seen in eq (10). > > 4∆.5> 4∆.5> 4∆.5> B$M > > 4∆?5@ 4∆?5@= 4∆?5@E, 4∆?5@ , L , 4∆?5> @>NO

(10)

By assembling the stiffness matrices for all sub-regions and considering the equilibrium and compatibility relations at the interfaces (see eq (11)) it is possible to evaluate the real displacements 4∆(5> < at the interface using eq (12). 4∆.5Q< $ 4∆.5QQ < 4∆(5Q< 4∆(5QQ
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

24


,Q VLPLODU SUREOHPV JRYHUQHG E\ WKH 3RLVVRQ HTXDWLRQ VXFK DV WKH RQHGLPHQVLRQDO EDU ZLWK YDULDEOH GHQVLW\VXEPLWWHGWRXQLIRUPWUDFWLRQLQZKLFKWKHEDVLFYDULDEOHLVQRWFRPSOHWHO\QXOORQWKHERXQGDU\ )+6GLGQRWUHSHDWWKDWSHUIRUPDQFH>@7KHLUUHVXOWVZHUHZRUVHWKDQWKHFODVVLFDOIRUPXODWLRQDWWHVWLQJ WRWKHLQIOXHQFHRIWKHH[SUHVVLRQ LQWKHUHVXOWV

)LJXUH±$YHUDJH3HUFHQWDJH(UURULQVWUHVVDVDIXQFWLRQRIWKHQXPEHURIERXQGDU\HOHPHQWV

&RQFOXVLRQV

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

5HIHUHQFHV

>@&$%UHEELD-&7HOOHVDQG/&:UREHO%RXQGDU\(OHPHQW7HFKQLTXHV6SULQJHU9HUODJ >@&$%UHEELD7KH%RXQGDU\(OHPHQW0HWKRGIRU(QJLQHHUV3HQWHFK3UHVV >@ : -0DQVXU 3 )OHXU\ -U -36 $]HYHGR $ 9HFWRU $SSURDFK WR WKH +\SHUVLQJXODU %(0 )RUPXODWLRQIRU/DSODFHV(TXDWLRQLQ',QW-RXUQDORI%(0&RPPSS >@ -&) 7HOOHV $ $ 3UDGR +\SHUVLQJXODU )RUPXODWLRQ IRU ' 3RWHQWLDO 3UREOHPV LQ $GYDQFHG )RUPXODWLRQVLQ%RXQGDU\(OHPHQW0HWKRG&KDS(OVHYLHU/RQGRQ >@ ( 5 /RYDWWH &RPSDUDomR GH 'HVHPSHQKR HQWUH DV )RUPXODo}HV 6LQJXODU H +LSHUVLQJXODU GR 0pWRGRGRV(OHPHQWRVGH&RQWRUQR0DVWHU7KHVLV33*(08)(6LQSRUWXJXHVH >@ $ $ 3UDGR 8PD )RUPXODomR +LSHUVLQJXODU GR 0pWRGR GRV (OHPHQWRV GH &RQWRUQR SDUD 3UREOHPDV%LGLPHQVLRQDLV0DVWHU7KHVLV3(&&233(8)5-LQSRUWXJXHVH >@ 5* 3HL[RWR )RUPXODomR FRP 'XSOD 5HFLSURFLGDGH +LSHUVLQJXODU GR 0pWRGR GRV (OHPHQWRV GH &RQWRUQR SDUD 3UREOHPDV GH 3RWHQFLDO (VFDODU %LGLPHQVLRQDLV 0DVWHU 7KHVLV 33*(08)(6 LQ SRUWXJXHVH


25

A COMBINED SGBEM AND CONIC QUADRATIC OPTIMIZATION APPROACH FOR LIMIT ANALYSIS Panzeca T.1,a, Cucco F.2, Parlavecchio E.1,b, Zito L.1,c 1

Dicaa, Viale delle Scienze, 90128 Palermo, Italy,

a

[email protected], [email protected], [email protected]

2

Università Kore di Enna, Cittadella universitaria, 94100 Enna, Italy, [email protected]

Keywords: SGBEM; multidomain; lower bound limit analysis; nonlinear programming. Abstract. The static approach to evaluate the limit multiplier directly was rephrased using the Symmetric Galerkin Boundary Element Method (SGBEM) for multidomain type problems [1,2]. The present formulation couples SGBEM multidomain procedure with nonlinear optimization techniques, making use of the self-equilibrium stress equation [3-5]. This equation connects the stresses at the Gauss points of each substructure (bem-e) to plastic strains through a self-stress matrix computed in all the bem-elements of the discretized system. The analysis was performed by means of a conic quadratic optimization problem, in terms of discrete variables, and implemented using Karnak.sGbem code [6] coupled with MathLab. Finally, some numerical tests are shown and the limit multiplier values are compared with those available in the literature [4,8]. The applications show a very important computational advantage of this strategy which allows one to introduce a domain discretization only in the zones involved in plastic strain action and to leave the rest of the structure as elastic macroelements, therefore governed by few boundary variables. Introduction The multidomain Symmetric Galerkin Boundary Element Method (SGBEM), developed by Panzeca et al. [1], is utilized to riformulate the lower bound limit theorem [3], which represents a powerful tool for providing directly, by means of mathematical programming techniques, the safety condition of a structure. In many practical engineering problems, which do not need information about the stress and strain histories, it proves more advantageous than elastoplastic incremental analysis, because it makes it possible to obtain the limit factor directly, saving computational costs. The proposed multidomain approach, through the von Mises yield rule, leads to an optimization problem of a linear objective function subjected to quadratic constraints to be solved by nonlinear mathematical programming. The main characteristic of the SGBEM, in comparison with other approaches, is that the compatibility and equilibrium are guaranteed at all the points of the domain and only in discrete form along the boundary. The proposed strategy uses the self-equilibrium stress equation [3,5], obtained by means of a displacement approach to define the self-equilibrium stress field, and the non linear constraints of the classical lower bound limit approach. Indeed, this equation connects stresses, computed at each bem-e Gauss point, to plastic strains through an influence matrix (self-stress matrix) which is non-symmetric, negative semidefinite and fully-populated. Then the constrained optimization problem was rephrased in the canonical form of a Conic Quadratic Optimization (CQO), in terms of discrete variables, and implemented using the Karnak.sGbem code [5] coupled with an optimization toolbox of MatLab. In order to prove the efficiency of the proposed approach, the latter approach was compared with the elastoplastic iterative analysis via multidomain SGBEM, developed by the same authors [5], through some examples. 1. Self-equilibrium stress equation via multidomain SBEM The proposed strategy uses the stress equation [5], obtained by means of a displacement approach of the SGBEM, to define the self-equilibrium stress field p Z p . Indeed the following equation

Z E ˆ e

(1)

provides the stress at the strain points of each bem-e as a function of the volumetric plastic strain p and of the external actions ˆ e , the latter amplified by . The matrix Z , defined as the self-stress influence matrix of the assembled system, is a square matrix having 3mx3m dimensions, with m bem-elements, which is full-

26


populated, non-symmetric and semi-definite negative. The evaluation of this matrix only involves knowledge of the material elastic characteristics and of the structure geometry. The reader can refer to Panzeca et al. [5] for a more detailed discussion of the characteristics of this equation introduced for a multidomain SGBEM problem. 2. Lower bound limit analysis via CQO In order to evaluate the limit multiplier directly, the lower bound approach was rephrased by means of SGBEM for multidomain type problems. In the hypothesis of the von Mises yield function, which is a convex quadratic function, the static theorem leads to a numerical optimization problem of a linear objective function subjected to linear and quadratic constraints. Therefore the analysis was developed by solving a constrained nonlinear optimization problem using known mathematical programming methods. The present formulation couples a multidomain SGBEM procedure with optimization nonlinear techniques through the introduction of the self-equilibrium stress field, defined in eq.(1). According to the lower bound theorem, the safety condition for the structure is guaranteed by a stress state satisfying the yield condition, the latter rephrased in terms of discrete variables, i.e.: F [ i ] d 0

(2)

with i = 1....m bem-elements and i

ˆ ie p

representing the total stress as the sum of the elastic stress vector ˆ ie , due to external actions, and the selfequilibrium stress vector p . The classical static approach makes it possible to obtain the lower bound factor EL as the maximum of the load factors E for which the structure does not fail: max E °EL (E , p ) ® e °¯ F [E ˆ i p ] d 0

(3)

Since the self-equilibrium stress vector p is a function of the volumetric plastic strain vector p , through the following relation: p

Zp

(4)

the optimization problem can be written as follows: EL max E (E ,p ) ° ° ®s.t. : ° e °¯ F [Eˆ Zp ] d 0

(5)

or in explicit form: max E EL ( E , p1 ,"p m ) ° °s.t. : ° e ® F1 [E ˆ 1 Z11 p1 " Z1m p m ] d 0 °# ° ° Fm [E ˆ em Z m1 p1 " Z mm p m ] d 0 ¯

(6)

In the hypothesis of the von Mises yield law, the present approach allows one to write the problem through the optimization of an objective linear function subjected to quadratic constraints only:

Advances in Boundary Element and Meshless Techniques XII max E EL ( E , p1 ,.....p m ) ° ° s.t.: °1 2 e T e ® 2 (Eˆ 1 Z11 p1 " Z1m p m ) M (E ˆ 1 Z11 p1 " Z1m p m ) y d 0 ° °# ° 1 (Eˆ e Z p " Z p )T M (Eˆ e Z p " Z p ) 2 d 0 m m1 1 mm m m m1 1 mm m y ¯2

27

(7)

where M is a constants matrix and y the uniaxial yield stress. In order to solve the previous problem, it was rewritten in the general form of a Conic Quadratic Optimization problem, as follows: min y °° ( y ) ®s.t. : ° T °¯y B y d 0

(8)

where B is a symmetric positive matrix and y is the unknown quantities vector. The canonical form (8) is obtained by collecting in the B matrix the constant terms of the eq.(7), i.e. for the i-th bem-e: Fi

T M T ˆ ie Zi1"Zim E p T1 " p Tm 1 d 0 E p T1 " p Tm ˆ ie Zi1 " Zim

2V2y

y yT Bi

(9)

and in compact form: Fi

y T Bi y 1 d 0

(10)

The lower bound problem can be rewritten as follows: min cT y ° (y) ° s.t.: °° T ®y B1 y 1 d 0 °# ° °yT B m y 1 d 0 °¯

(11)

where the vector cT 1 0 " 0 has been introduced. Problem (6), in the explicit form (11), was implemented by coupling the Karnak.sGbem code with an optimization toolbox of Matlab 7.6.0. Also in this procedure using multidomain SGBEM it was possible to reduce the size of the problem. Since this method makes it possible to introduce a domain discretization exclusively in the zones of potential store of the plastic strains, the remaining part of the structure can be considered as made up of elastic macroelements, and therefore governed by few boundary variables. This aspect makes the strategy proposed extremely advantageous. 3. Incremental elastoplastic analysis A concise treatment of the utilized strategy for incremental elastoplastic analysis via multidomain SGBEM, called active macro-zone analysis, is shown in this section, the full version being found in [5]. It allows one to compute the plastic strains for each loading step and at each bem-e by using eq.(1) both to evaluate the predictors and to perform the corrector phase. Let us start by evaluating the trial stresses, i.e. the purely elastic response at the load instant n 1 in each m bem-e of the discretized body. For this purpose eq.(1) provides all the predictors * n 1 as a function of the plastic strain vector p n , stored up at the n step, and of all the increments 'p inside the n 1 step, and hence also as a function of the external load ˆ e :

28

* n 1

Eds: E L Albuquerque & M H Aliabadi Zp n 1 n 1 ˆ e

with p n 1

p n 'p,

n 1

n '

(12)

where the fully-populated Z matrix regards all the bem-elements obtained by substructuring. A check on 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 [*i n 1 ]

1 2

*i Tn 1 M *i n 1 2y d 0 with i 1...m

(13)

In the a bem-elements (with a d m ) where the latter inequality is violated, a return mapping phase is required to evaluate the plastic strains and the direction of the plastic flow. This phase, called the corrector phase, uses the first term of eq.(1) to obtain the elastoplastic solution at every plastically active bem-e. In this phase the vector , representing the end step stress, as well as the volumetric plastic strain vector 'p , are unknown quantities. The latter is the plastic strain to impose at every plastically active bem-e in order to have the stress on the yield boundary of the elastic domain, thus making it possible to know the direction of the plastic flow. Obviously, inside each loading step the corrector phase has to be repeated until all the predictors satisfy the plastic consistency conditions of all the a bem-elements. In detail, the elastoplastic algorithm allows one to write, for all the active h bem-elements ( h = 1,...,a ), a non-local system at the n 1 load step simultaneously in all the plastically active macro-zones identified in the previous predictor phase, i.e.: A

*A Z AA 'p A

(14)

F [ A ] d 0 , ' A t 0 , ' A F [ A ] = 0

(15a-c)

where eqs. (15a-c) are the plastic admissibility conditions of the a bem-elements and the vectors A and 'p A collect the related stress and plastic strain a subvectors. The subscript n 1 has been omitted for convenience. The Z AA matrix coefficients are derived from the Z matrix present in eq.(1), by extracting the blocks relating to the a plastically active bem-elements. In the hypothesis that, for each bem-e, the plastic multiplier is modelled through 'Oh p ' h with p t 0 , the plastic strain vector, relating to all the a plastically active bem-elements, is expressed as: 'p A

' A w A F [ A ]

' A M A A

(16)

The solving nonlinear system is the following: * ° A A ' A Z AA M A A ® °¯ F [ A ] 0

0

(17)

or in explicit form, using the von Mises yield law and the plastic flow role 1 *1 '/ 1 Z11 M 1 ! '/ a Z1a M a 0 ° °# ° * '/ Z M ! '/ Z M 0 a 1 a1 1 a aa a ° a ®1 T 2 0 M 1 y °2 1 °# ° °¯ 12 Ta M a 2y 0

(18)

where A is the stress solution located on the yield surface of the elastic domain of all the active bemelements, *A the elastic predictor, ' A Z AA M A A the corrective components (containing local and nonlocal contributions). The approximate solution of this nonlinear problem involving all the plastically active bem-elements, in terms of A and ' A , can be obtained by applying the Newton-Raphson procedure. The existence of the solution to the nonlinear problem (18) is guaranteed except when plastic collapse takes place.


29

4. Numerical results In order to show the efficiency of the proposed method, two numerical tests were performed under the following hypotheses: plane strain condition, elastic-perfectly plastic von Mises law and associated plastic flow. The advantage of this approach in comparison with other analysis methods lies in a better response, because it is based on satisfaction of the equilibrium and compatibility conditions inside each bem-e: this happens thanks to use of the fundamental solutions in the analysis process. 4.1 Cantilever beam The cantilever beam in Fig.1a, subjected to shear forces EQ was considered as a bi-dimensional body. Its domain was discretized by using 138 bem-elements, each having eight nodes, characterized by linear modelling of the boundary quantities, and having decreasing size (Fig.1b) near the constraint zone. The elastic characteristics of the beam are Young’s modulus E 210000 MPa and Poisson’s ratio Q 0.3 . Uniaxial yield value is V y 200 MPa . The ratio size between the length and height is H / L 0.25 . Only the potentially plastic zones were meshed ( ! L / 3 consistently with the beam theory) and the rest is studied as an elastic macro-element whose boundary is discretized by elements 12 cm long (Fig.1b). This made it possible to reduce the number of variables and the computational burdens.

Q

H

L

a)

b)

A

q

L/3

Fig. 1. Cantilever beam: a) geometric description; b) mesh adopted.

The limit load factor, computed by the lower bound analysis (CQO), was compared with the numerical solution obtained through the incremental elastoplastic strategy and using the well-known analytical solution (beam theory) as it shown in Fig.2.

30


Q Qmax 1

0.8 Lower bound limit analysis (CQO)

0.6

Elastopl. analysis (active macro-zone)

0.4

Analitical solution

0.2

0.5

1

1.5

2

2.5

3

Displacement of point A

Fig. 2. Load factor-displacement curves.

Method

E Q / Q max

Lower bound limit analysis (Zhang et al. [4])

0.9830

Elastoplastic analysis (active macro-zones)*

0.9942

Analytical solution (beam theory)

1.0000

Elastoplastic analysis (Zito et al. [8])

1.0385

Lower bound limit analysis (CQO)*

1.0395

Table 1: Collapse load factor, obtained using the present approaches (*) and compared to other formulations. The aim is to compute the collapse load factor and to compare (see Table 1) the numerical solution, obtained through the strategies proposed in this paper, with other methods present in the literature [4,8]. 4.2 Thin square plate with circular hole In the present subsection a thin square plate with a circular hole, subjected to a uniform tensile load q , was considered. The internal discretization consisted of 144 bem-elements (Fig.3), each having four-nodes, characterized by linear modelling of the boundary quantities. The material characteristics are Young’s modulus E 206700 MPa and Poisson’s ratio Q 0.29 .Uniaxial yield value is V y 450 MPa . The constrained inequalities were satisfied with tolerance Tol =10 8 . In Fig.4 the limit load factor is compared with the numerical solution obtained through the incremental elastoplastic strategy, and with other incremental and direct methods available in the literature. The lower bound approach directly provided the limit multiplier, whereas the incremental elastoplastic approach also made it possible to evaluate the load-displacement curve.


q

200

A

20

90

90

a)

b)

Fig. 3. Thin square plates with circular holes: a) geometric description; b) mesh adopted. q 439.4560

400

300

Lower bound limit analysis (CQO)

Elastopl. analysis (active macro-zone)

200

Elastopl. analysis (Zito, et al. [8])

100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Displacement of point A

Fig. 4. Load factor-displacement curves.

5 Conclusions A strategy utilizing the multidomain SGBEM for rapidly performing lower bound limit analysis as a conic quadratic optimization problem has been shown in this paper. The present multidomain approach, called the displacement method [1], has made it possible to consider step-wise physically and geometrically nonhomogeneous materials and to obtain a self-equilibrium stress equation regarding all the bem-elements of the structure. Since this equation is inserted in the plastic admissibility conditions, the related quadratic equations represent the constraints of a nonlinear optimization problem, solved as Conic Quadratic Optimization. Further, the strategy has made it possible to introduce a domain discretization exclusively of zones involved in plastic strain stores, leaving the rest of the structure as elastic macroelements, and therefore governed by few boundary variables. It considerably limits the number of variables of the problem and makes the proposed strategy extremely advantageous. The implementation of the procedure by means of the Karnak.sGbem code, coupled with an optimization toolbox of Matlab 7.6.0, made it possible to perform some numerical tests showing the high performance of the algorithm for solution accuracy and low computational cost.

31

32


References. [1]

Panzeca T., Cucco F., Terravecchia, S., (2002). “Symmetric Boundary Element Method versus Finite Element Method”. Computer Methods in Applied Mechanics and Engineering. 191, 3347-3367.

[2]

Perez-Gavilan J.J. and Aliabadi M.H., (2001). “Symmetric Galerkin BEM for Multi-connected bodies”. Commun. Numer. Meth. Engng. 17, 761-770.

[3]

Panzeca T, (1992). “Shakedown and limit analysis by the boundary integral equation method”. Eur. J. Mech., A/Solids. 11, 685-699.

[4]

Zhang, X., Liu, Y., Zhao, Y., Cen, Z., (2002). “Lower bound limit analysis by the symmetric Galerkin boundary element method and Complex method”. Computer Methods in Applied Mechanics and Engineering. 191, 1967-1982.

[5]

T. Panzeca, E. Parlavecchio, S. Terravecchia, L. Zito., (2010) “Elastoplastic analysis for active macrozones via multidomain symmetric Galerkin BEM”, International Conference on Boundary Element Techniques (Beteq 2010), Athens, 346-352.

[6]

Cucco F, Panzeca T, Terravecchia S.,(2002). The program Karnak.sGbem Release 2.1, DICA, Palermo.

[7]

Gao XW, Davies TG., (2000), “An effective boundary element algorithm for 2D and 3D elastoplastic problem”. Int. J. Solids Struct.; 37, 4987-5008.

[8]

Zito L., Parlavecchio E., Panzeca T., (2011). “On the computational aspects of a symmetric multidomain BEM approach for elastoplastic analysis”. Journal of Strain Analysis for Engineering Design, 46, 103120.


33

A Fully Kinematic Solution of Mixed Boundary Value Problems of Elastostatics J O Watson School of Mining Engineering University of New South Wales Sydney 2052, Australia [email protected] Keywords: boundary elements, elastostatics, shape functions

Abstract In the analysis of mixed boundary value problems for piecewise homogeneous domains, it is the current practice to interpolate unknown traction over an element by taking the same shape functions as for displacement. The work described here is an investigation of whether accuracy can be improved by taking the variation of traction over constrained surface elements and interfaces between homogeneous subdomains to be restricted to that which can derive from the chosen variation of displacement, and a demonstration of how that variation can be used to improve the accuracy of coupling of the boundary element and finite element methods.

Introduction Whereas in the finite element method as usually implemented the unknowns in terms of which the approximate solution is defined are nodal displacements, in the direct formulation boundary element method for elastostatics they may be nodal values of displacement or traction. Such a mixed system is obtained if part of the surface of a domain of analysis is constrained against displacement, or the domain is piecewise homogeneous in which case traction as well as displacement is unknown over boundary elements on interfaces between homogeneous subdomains. Let lαi(y) be direction cosines of three local directions α = 1, 2, 3 (not necessarily orthogonal), in which a point y on the surface S of a homogeneous domain or subdomain is fixed or free to move. Then the boundary integral equation for that region is [1]

′

′

lim [T αβ(x,y) u β(y) ε → 0 S – S(x,ε) + s(x,ε)

– U′αβ(x,y) t′β(y)] dSy = 0

(1)

The boundary S – S(x,ε) + s(x,ε) is as shown in Figure 1, and for example u′β(y) = lβj(y) uj(y) T′αβ(x,y) = λαi(x) λβj(y) Tij(x,y)

(2)

In equation (2), uj(y) is displacement and Tij(x,y) is traction on S at y due to load at x in an infinite elastic space, λαi lαj = δij and for y ∈ s(x,ε) we take λβj(y) = λβj(x). S consists of a part S1 which forms interfaces with other homogeneous subdomains (interface elements) and a part S2 on the surface of the domain of analysis (surface elements). Over S1 and parts of S2 which are not subject to any constraints on displacement, the local directions α are taken to be the global Cartesian coordinate directions. No constraint on displacement is perfectly rigid, so in reality unknown surface tractions only occur where boundary elements are defined on a plane of symmetry and an analysis performed for that part of the domain to one side of the plane [1]. Accordingly, it is supposed that over S2 there can only exist constraint in the direction α = 1 normal to that surface.

34


Figure 1

Boundary for which integral equations are taken

It is the usual practice to interpolate unknown traction over an element by taking the same shape functions as for displacement, ignoring the fact that this allows some variations of traction to which correspond variations of displacement that cannot be represented by those shape functions. The purpose of the work presented here is twofold: to determine whether accuracy can be improved by the use of shape functions taken such that the variation of traction over an element is restricted to that which can derive from the chosen variation of displacement, and to establish a fully kinematic scheme of functional interpolation which allows boundary element subdomains to be interfaced seamlessly with finite elements.

The variation of traction over an element For a surface element subject to constraint, the variation of unknown traction is defined by reference to local Cartesian directions x1α with direction cosines k1αi, the first and second of which lie in the plane tangent to the element and the third of which coincides with the outward normal to the homogeneous subdomain under consideration. Local components of displacement are given by n

u1α = Σ Mc(ξ,η) u1αc

(3)

c=1

where Mc are shape functions of the intrinsic coordinates (ξ,η), and u1αc is displacement of node c of the element. According to Hooke’s law, local components of traction are given by (t11, t12, t13) = ( µ1 [q131 + p11 ] , µ1 [q132 + p12 ] , (λ1 + 2µ1) p13 + λ1 [q111 + q122] )

(4)

where λ1 and µ1 are the Lamé constants, p1α is the value of ∂u1α /∂x13, and q1αj = ∂u1α /∂x1j for j = 1,2. The variations of p1α and q1αj are n

p1α = Σ Mc(ξ,η) p1αc

(5)

c=1

and n

q1αj = Σ ∂Mc /∂x1j u1αc

(6)

c=1

Therefore, if displacements are quadratic functions of the intrinsic coordinates, p1α and q1αj are quadratic and linear functions respectively and the variation of traction is hybrid linear/quadratic. In the system of equations to be solved, traction unknowns are replaced by normal derivatives of displacement p1α. For an interface between homogeneous subdomains, let superscripts 1 and 2 denote first (i.e. lower numbered) and second subdomain at an element. The local directions lαi of equation (2) are taken to be


35

mutually orthogonal and such that l3i coincides with the outward normal to the first subdomain. Local Cartesian directions x1α are taken coincident with lαi so k1αi = lαi, and for construction of the discrete system approximating to the boundary integral equation for that subdomain, substitutions are carried out according to equations (4) to (6). For the second subregion, the variation of traction is defined by reference to local Cartesian directions x2α with direction cosines k2αi = -k1αi. By Newton’s third law of motion the local components of traction t2α are equal to t1α, and substitutions are again carried out according to equations (4) to (6). In the system of equations to be solved, interface tractions are replaced by normal derivatives of displacement p1α. The boundary integral equations and discrete system The system of simultaneous equations is constructed by nodal collocation. At interface nodes, equation (1) is taken for α = 1, 2, 3. At a surface node, equation (1) is taken for a direction (i.e. for a value of α) in which displacement is unknown, whereas the hypersingular equation [1]

′

′

lim [V αβ(x,y) u β(y) ε → 0 S – S(x,ε) + s(x,ε)

– W′αβ(x,y) t′β(y)] dSy = 0

(7)

in which for example V′αβ(x,y) = ns(x) ∂T′αβ(x,y)/∂xs

(8)

is taken for a direction in which traction is unknown. In equation (8), ns(x) is the unit outward normal to the surface upon which the traction t′α(x) acts. All matrix coefficients of equations taken at surface nodes are integrals of the kernels T′ and W′ and are therefore of order 1/r2 where r is distance from x to y, which improves the effectiveness of algebraic clustering for simultaneous equation solution [2]. Coefficients of equations taken at interface nodes may be of order 1/r2 or 1/r.

Coupling with finite elements At the present stage of development of boundary elements, it is generally necessary to couple to finite elements to carry out nonlinear analysis such as that of elastoplastic material behaviour. Whilst efficient elastoplastic boundary elements will eventually be developed, there remains a need for finite elements in other circumstances. To date, coupling has generally been used to model domains of infinite extent as an alternative to infinite finite elements or the continuation of a finite element mesh to an artificial outer boundary [3], and the accuracy of computed results near the interface between boundary and finite elements has not been of great importance. However, if finite elements are used to model details such as thin parts of a structure or linings such as shotcrete on rock then the interface may be near a concentration of stress, and such accuracy is essential. The proposed kinematic scheme of functional interpolation over boundary elements allows pointwise satisfaction of Newton’s third law of motion over an interface between finite and boundary elements, rather than satisfaction in an average sense as achieved by computation of equivalent nodal forces from nodal tractions. For example, a nine-node quadratic displacement variation boundary element can be coupled with a modified hexahedral finite element as shown in Figure 2. Over the boundary element, displacements are interpolated by shape functions Mc(ξ,η) as in equation (3). The degrees of freedom of the finite element may be taken as displacements of nodes on face ζ = 1, displacements of nodes on ζ = -1 and derivatives of displacement with respect to ζ at nodes on ζ = -1. The shape functions by which these degrees of freedom are multiplied are of the form Mc(ξ,η)Nd(ζ) where, for displacements at ζ = 1, displacements at ζ = -1 and derivatives of displacement at ζ = -1 respectively,

36


N1(ζ) = 0.25(1 + ζ)2 N2(ζ) = 1 - N1(ζ) N3(ζ) = 0.5(1 - ζ2)

(9)

and coupling of the elements is achieved by expressing the normal derivatives of displacement p1α in terms of derivatives with respect to ξ, η and ζ of displacement of the finite element. Example: inhomogeneous cantilever beam The beam shown in Figure 3 is modelled as two subdomains, with a total of 304 nine-node quadrilateral boundary elements. Young’s modulus and Poisson’s ratio of subdomain 1 are 10000MPa and 0.0 respectively, and analyses are carried out for Young’s modulus and Poisson’s ratio of subdomain 2 equal to 10000MPa and 0.0 (homogeneous case), 5000MPa and 0.0, and 10000MPa and 0.3. It is found that

Figure 2

Figure 3

Coupling of boundary and finite element

Cantilever beam of length 1.0m, cross section 0.25m x 0.25m


z

0.2500 0.1875 0.1250 0.0625 0.0000

exact homogeneous σy τyz 192 144 96 48 0 -48 -96 -144 -192

0.0 10.5 18.0 22.5 24.0 22.5 18.0 10.5 0.0

Table 1

E2 = 10000, ν2 = 0.0 τyz σy 192 144 96 48 0 -48 -96 -144 -196

0.3 10.2 18.7 22.2 24.7 22.2 18.7 10.2 0.3

E2 = 5000, ν2 = 0.0 τyz σy 186 146 98 49 0 -49 -98 -146 -186

0.6 10.5 18.6 21.9 24.3 21.9 18.6 10.5 0.6

37

E2 = 10000, ν2 = 0.3 τyz σy 193 144 97 49 0 -49 -97 -144 -193

-15.8 8.2 14.2 23.0 25.5 23.0 14.2 8.2 -15.8

Computed bending and shear stresses on interface (in MPa)

differences between stresses computed according to the proposed fully kinematic scheme and the conventional approach are negligible. Computed direct and shear stresses at nodes equally spaced along the line AB on the interface between subdomains shown in Figure 3 are given in Table 1. The computed values of Ĳyz at the points A and B for Poisson’s ratio 0.3 are meaningless as the exact solution exhibits stress singularities at those points [4], accurate boundary element modelling of which must await generalisation of an existing analysis of cracks and notches [5] to the multi-material case.

Conclusion The fully kinematic scheme is at an early stage of development and testing. No differences between results obtained according to that scheme and by taking the same variation of tractions as that for displacements have yet been observed, but for the examples considered so far the exact variations of traction are uniform and linear, or nearly so. It may be that for other cases the proposed scheme yields results which differ significantly from those obtained by the conventional approach. Perhaps the most important potential application of the proposed coupling with finite elements is to the modelling of thin parts of a structure or mechanical component, stresses in which are difficult to compute accurately by the boundary element method as presently implemented. Possible remedies are a local adaptation of boundary elements in which account is taken of approximations for thin plates and shells, or the use of finite elements. If the interfacing of finite and boundary elements is of sufficient accuracy then the latter option, which is more straightforward, can be adopted.

References [1] [2] [3] [4] [5]

J C Lachat and J O Watson Int. J. Numer. Meth. Engng, 10, 991-1005 (1976) J O Watson Int. Conf. Boundary Element Techniques (2010) G Beer Int. J. Numer. Meth. Engng, 19, 567-586 (1983) D B Bogy J. Appl. Mech. ASME, 35, 460- (1968) J O Watson Int. J. Numer. Meth. Engng, 65, 1419-1443 (2006)

38


A meshless BEM for transient thermoelastic crack analysis in functionally graded materials 1

A. Ekhlakov , O. Khay1, Ch. Zhang1, J. Sladek2 and V. Sladek2 1

2

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

Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia, [email protected], [email protected]

Keywords: Functionally graded materials, boundary element method, Cartesian transformation method, dynamic stress intensity factors, thermal shock, Laplace-transform Abstract. A meshless boundary element method for transient thermoelastic crack analysis in two-dimensional, isotropic, continuously non-homogeneous and linear elastic functionally graded materials subjected to thermal shock is presented. Fundamental solutions for homogeneous, isotropic and linear elastic materials are used for the boundary-domain integral equation formulation. The Cartesian transformation method is applied for the evaluation of the arising domain integrals. Numerical results are presented and discussed to show the accuracy and the efficiency of the proposed method. Introduction Functionally graded materials (FGMs) are composite materials with a continuously variable composition of the constituent phases over volume [1]. Originally, the concept of FGMs has been proposed for thermal barrier materials during a space plane project. Due to their improved mechanical and thermal properties, they receive increasingly growing research interest in material and engineering sciences. A typical example of FGMs is the ceramic/metal structure that possesses the desirable properties of metals such as high toughness, high mechanical strength and bonding capability as well as high heat, wear and corrosion resistances of ceramics. Because of the inherent brittle nature of ceramics cracks or crack-like defects may develop during the fabrication or their services. Therefore, it is important for the design and optimisation of FGMs to carry out crack analysis for a better understanding of the failure and fracture properties. In this paper, a BEM for transient thermoelastic crack analysis in two-dimensional (2-D), isotropic, continuously non-homogeneous and linear elastic FGMs is developed. The equations of motion and the thermal balance equation constitute the governing equations of the transient linear thermoelasticity. The Laplace-transform technique is applied to eliminate the time-dependence in the governing equations. A boundary-domain integral equation representation is derived from the generalized Betti’s reciprocal theorem for FGMs in conjunction with the fundamental solutions for homogeneous and linear thermoelastic solids. The boundary-domain integral equations (BDIEs) are obtained for the unknown mechanical and thermal fields. The BDIEs contain domain integrals, which describe not only the dynamic effects resulting from the inertia term and the initial conditions but also the material's non-homogeneity. The radial transformation method has been developed to transform the arising domain integrals into boundary integrals [2, 3]. In this analysis, the domain integrals are computed by using the Cartesian transformation method (CTM) without domain discretization [4, 5]. A collocation method is implemented for the spatial discretization. The final time-dependent solutions are obtained by using the Stehfest’s algorithm [6]. Numerical results are presented and discussed to demonstrate the accuracy and efficiency of the proposed CTM-BEM as well as the effects of the material gradation on the dynamic stress intensity factors (SIFs). Boundary-domain integral equations Let us consider a continuously non-homogeneous, isotropic and linear elastic FGM in a 2-D domain. The material parameters such as the mass density (x) , the Young's modulus E ( x) , the thermal conductivity k (x) , the specific heat c(x) , the linear expansion coefficient D(x) , etc. are assumed to depend continuously on the Cartesian coordinates, while the Poisson’s ratio Q is a constant. In this case, the elasticity tensor is expressed as


0 cijkl (x) cijkl P (x)

with

ci0jkl

Q Gij Gkl Gki Glj G kj Gli 1 2Q

and P x

39 E x

2 1 Q

,

(1)

where P(x) is the shear modulus, Gij denotes the Kronecker symbol. In the absence of body forces and heat sources, the equations of motion and the generalized heat-conduction equation in transient coupled thermoelasticity are given by ª¬ k ( x ) T , i ( x, t ) ¼º ( x )c ( x ) T ( x, t ) K( x ) u k , k ( x, t ) 0, V ij , j ( x, t ) ( x ) ui ( x, t ) 0, (2) ,i where Vij is the stress tensor. Unless otherwise stated, the conventional summation rule over double indices is implied, a comma after a quantity indicates spatial derivatives, a dot over a quantity denotes time derivative, and Latin indices take the values of 1 and 2 . Applying the Laplace-transform to the governing equations (2) and using the Duhamel-Neumann relations [7] yields k 0 cijkl (3) Puk ,lj P, j uk ,l JT,i J ,i T p 2ui 0, T,ii k,i T,i Np T Kpuk , k 0, where J and N are the stress-temperature modulus and the thermal diffusivity, denotes the complex Laplacetransform parameter and the superimposed bar denotes the Laplace-transformed quantities. Integral representations of the displacements and the temperature at an arbitrary point are derived from the generalized Betti’s reciprocal theorem for FGMs by using the fundamental solutions of the Laplace-transformed linear coupled thermoelasticity for homogeneous materials [7]. By moving the observation point to the boundary x * or keeping it in the domain x : the following system of BDIEs for the mechanical and thermal fields at the boundary and interior points is obtained as ª º 1 u j x, p ³ «ui y , p Tij x, y , p ti y , p U ij x, y , p » d * y E y y ( ) D « *¬ ¼» ª º k (y ) u N 0 ³ « T y , p Z j x, y , p q y , p U j x, y , p » d * y Fj x, p 0, D E y y ( ) « * ¬ ¼» (4) º N 0 K0 p ª 1 ti y , p U i x, y , p » d * y T x, p «ui y , p Ti x, y , p J 0 ³* ¬« E y D (y ) ¼» ª º k (y ) T N0 ³ « T y , p F x, y , p q y , p T x, y , p » d * y F x, p 0, E y D (y ) « * ¬ ¼» where x and y represent the source and the observation points, ti (x, p ) Vij (x, p )n j (x) are the components of the traction vector, q (x, p) k (x) T,i (x, p) ni (x) is the heat flux and ni denotes the components of the outward unit normal vector. Here, a tilde denotes the ratio of the non-homogeneous quantity to the homogeneous quantity that is designated by a zero subscript. The functions F j( u ) and F ( T ) describe the material non-homogeneity, which completely vanish for a homogeneous solid, and are defined as ª 2(1 Q) 1 § E , j D , j · § · F j( u ) ( x, p) p 20 ³ ¨ 1¸U ij (x, y , p )ui (y , p )d : y P 0 ³ « ¨ ¸ U ik (x, y , p ) D ¹¸ 1 2Q D ©¨ E E D ¹ :© :« ¬ §1 ·§ 1 ·º U ij , k (x, y , p) U kj ,i (x, y , p ) ¸» ui , k ( y, p)d : y ¨ 1¸¨ © D ¹© 1 2Q ¹¼ ª § c º § E, j D , j · · ³ « p¨ 1 ¸ U j ( x, y , p ) J 0 ¨ ¸U kk (x, y , p) » T( y, p )d : y ¨ ¸ D D E E © ¹ « :¬ © ¹ ¼» ª§ k º · 1 k § E ,i D ,i · N 0 ³ «¨ 1 ¸ U j , i ( x, y , p ) ¨¨ ¸¸U j (x, y , p ) » T,i (y , p )d : y D ¹ K0 ED © E :« ¼» ¬© E D ¹

(5)

40

Eds: E L Albuquerque & M H Aliabadi N0 K0 p ° 2 § · ® p 0 ³ ¨ 1¸U i (x, y , p)ui (y , p )d : y J 0 °¯ E D ¹ :©

F ( T ) ( x, p )

ª 2(1 Q ) 1 § E , j D , j · ½° § 1 ·§ 1 ·º U i , j (x, y , p ) U j , i ( x, y, p ) ¸» ui , j (y , p )d : y ¾ P 0 ³ « ¨ ¸U i (x, y, p) ¨ 1¸¨ ¨ ¸ D ¹ © D ¹© 1 2Q ¹¼ °¿ « 1 2Q D © E :¬ (6) ª § c º § E, j D , j · · p³ « ¨ 1 ¸ T ( x, y , p ) N 0 K0 ¨ ¸ U j ( x, y , p ) » T ( y , p ) d : y ¨ E D ¸¹ »¼ :« © ¬ © ED ¹ ª § k º · 1 k § E,i D , i · N0 ³ « ¨ 1¸ T,i (x, y , p) ¨¨ ¸¸ T (x, y , p) » T,i (y , p )d : y . K0 E D © E D ¹ :« ¬ © ED ¹ ¼» The BDIEs (4) contain boundary and domain integrals with singular kernels. The strongly singular integrals are interpreted in the sense of the Cauchy principal value. Making use of the singularity subtraction technique and the variable transformation technique the strong and weak singularities in eqs. (4) can be removed [8]. Numerical solution procedure A collocation method is employed for the spatial discretization of the BDIEs (4) in the Laplace-transformed domain. The boundary * is divided into N b quadratic boundary elements and N d internal points, which do not coincide, are selected in the interior of the domain : . Thus, the total number of nodes is N N b N d . In order to avoid the domain discretization into internal cells for evaluating the domain integrals (5) and (6) the CTM is applied [4, 5]. The functions (5) and (6) can be rewritten in matrix form as F (x, p) ³ Fk (x, y , p )nk (y )u( y, p )d * y ³ G (x, y , p )u(y , p )d : y , (7) *

:

where F is the vector of functions Fi ( u ) and F ( T ) , u is the vector containing the displacements ui and the temperature T , and the 3 u 3 matrices Fk and G are obtained from the definitions (5) and (6). The vector u is approximated by a series of prescribed basis functions and a polynomial N

u ( x, p )

¦ a 1

a

( p )Ia (x) a ( p ) b i ( p) xi

N

with

¦D

N

a i

( p) 0,

a 1

¦D

a i

( p) x aj

0,

(8)

a 1

where a , a and b i are the vectors of the unknown coefficients, and the 4-th order spline-type radial basis function is used Ia ( x) 1 6 R 2 8 R 3 3R 4 , (9) where R || x x a || is the distance from the application point x a to the field point x . The unknown coefficient vectors a , a and b i can be determined by applying the application point x a in eqs. (8) to every node. Thus, a set of linear algebraic equations can be obtained as u (x, p ) x p (10) where is the vector containing the coefficients a , a and b i . The matrix is invertible, because two nodes do not coincide, and we obtain ( p ) 1 ( x) u (x, p ). (11) It is worth noting that the matrix 1 is independent of the values of the Laplace-transform parameter p and needs to be calculated once only. Substitution of eq. (5) into the domain integral in the right–hand side of eq. (7) and application of eq. (11) results in 1 (12) ³ G (x, y, p)u(y, p)d : y I (x, p) (x)u(x, p) with I (x, p) ³ G (x, y, p)(y )d : y . :

:

The domain integral in eq. (12) is expressed as Nb ª y1 ( y2 ) º I (x, p ) ¦ ³ « ³ G (x, y , p )(y ) dy1* » dy2 , (13) q 1 *q « »¼ ¬ a where a is an arbitrary constant that can be taken as the mean value of x1 over the boundary * . By using N l integration intervals along the y1 -direction, we obtain


Nb

I ( x, p )

Ni

41

Nj

Nl

¦ J ¦ w ¦ J ¦ w G (x, y (K , K ), p)(y (K , K )) , q

l

i

q 1

i 1

j

l 1

i

j

i

(14)

j

j 1

where N i and N j are the numbers of the integration points for the integration over the boundary element * q and the l -th interval of the inner integrals, Km , wm and J m are the Gaussian points, the weights and the Jacobian of the transformation, respectively. In this manner, the domain integrals can be evaluated without domain discretization. After numerical integrations, applying the prescribed boundary conditions and a rearrangement of the equations, we obtain a system of 3N linear algebraic equations that can be written in matrix form as § ª A b 0 º ª Db º · ª x b º ª y b º (15) ¨¨ « i » « i » ¸¸ « i » « i » . © ¬ A I ¼ ¬ D ¼ ¹ ¬u ¼ ¬ y ¼ Here, x b is the 3N b vector of the unknown values of the displacements ui , the tractions ti , the temperature T and the heat flux q at the boundary collocation points, ui is the 3N d vector of the unknown displacements ui and temperature T at the internal nodes, y b and y i denote the 3N b and 3N d vectors composed of the prescribed boundary conditions. The sizes of the matrices A b , A i , D b and D i are 3N b u 3 N b , 3N d u 3 N b , 3N b u 3 N and 3N d u 3N , respectively, and I is the identity matrix. The system of linear algebraic equations (15) is solved numerically for discrete values of the Laplace-transform parameter p to obtain the boundary unknowns x b and the interior primary field quantities ui . The final time-dependent solutions can be calculated by using the Stehfest’s algorithm [6]. Once the displacement components ui ( x , t ) are numerically found, dynamic SIFs can be conveniently computed by the extrapolation technique following directly from the asymptotic expansions of the displacements in the vicinity of the crack-tip [9]. For a crack located on the x1 -axis, the dynamic mode-I and mode-II SIFs are related to the crack-opening-displacements ' ui ( x , t ) by ° K I t °½ ® ¾ ¯° K II t ¿°

2S tip 1 P lim Ho0 N 1 aH

°'u2 H, t ½° ® ¾, ¯° 'u1 H, t ¿°

(16)

where N 3 4Q or N (3 Q ) / (1 Q) for plane strain or plane stress, respectively, P tip is the shear modulus at the crack-tip, and H is a small distance from the crack-tip to the considered node on the crack-faces. Numerical results In order to test the proposed CTM-BEM, we consider an edge crack of length a 0.4 w in a rectangular, isotropic, continuously non-homogeneous and linear elastic FGM plate with the width w 1 and the length 2 l 3 w as shown in Fig. 1. The cracked plate is subjected to a cooling thermal shock T( x, t ) T 0 H(t ) on the left lateral side, whereas the right one is kept zero temperature. Here, T0 1 deg is the constant amplitude and H(t ) is the Heaviside step function. The material gradation in the xi -direction is assumed to have exponential laws E ( xi )

E0 exp(D g | xi |) ,

Dg

ln( Eb / E0 ) / A , E g

Eb

E (A ) , kb

k (A ) , cb

k ( xi )

k0 exp(E g | xi |)

ln(kb / k0 ) / A and J g

and

c( xi ) c0 exp( J g | xi |)

ln(cb / c0 ) / A , where E0

with

E (0) , k 0

gradient k (0) , c0

parameters c (0) and

c ( A ) are the Young’s modulus, the thermal conductivity, and the specific heat on the

left side and the right side for the material gradation in the x1 -direction, or on the bottom and the top side for the gradation in the x2 -direction, i.e. A is equal to w or l , respectively. The mass density, the Poisson’s ratio and the linear thermal expansion coefficient are taken as constants (x ) 1 , Q 0.25 and D(x) 0.02 , respectively, and the plane strain condition is assumed in the numerical analysis. Due to the symmetry of the problem with respect to the x1 -axis, only one half of the problem domain is modelled as shown in Fig. 1b. In this case, the mode-I dynamic SIF occurs, whereas the mode-II SIF is identically zero. In the discretization, 48 boundary nodes and 125 internal nodes are used. For a better approximation of the displacement and temperature fields near the crack-tip,

42


the density of nodes in the vicinity of the crack-tip is increased. For convenience, the dynamic SIF and the time are normalized as K I K I / E0 D 0 T0 Sa and t t k0 / a 20 c0 .

Fig. 1: An edge crack in a FGM plate First, we consider the cracked FGM plate with the material gradation parallel to the crack-line ( x1 - direction). The time variations of the normalized dynamic mode-I SIF for two selected gradient parameters D g E g J g r0.7 are presented in Fig. 2. To test the accuracy of the proposed CTM-BEM, the numerical results are compared with those provided by FEMLAB code. There is a very good agreement of the present CTMBEM and the FEM results. The normalized mode-I SIF for the corresponding homogeneous material is also given in Fig. 2 as dash-dot lines. The first peak of the SIF decreases with decreasing of the gradient parameters (Fig. 2a). The velocity of wave propagation in this case is also decreasing. Therefore, the peak value of the normalized SIF is shifted to a larger time instant. The opposite tendency is observed in Fig. 2b with increase of the gradient parameters.

Fig. 2: Normalized dynamic SIF for a material gradation in the x1 -direction Finally, we consider a cracked FGM plate with the material gradation in the in the x2 -direction perpendicular to the crack-line for the same geometrical configuration and gradient parameters as analyzed above. The timedependence of the normalized dynamic mode-I SIF is presented in Fig. 3. One can see a quite good agreement of the numerical results obtained by the CTM-BEM and the FEM. A similar behavior of the SIF as in the previous example is observed. The first peak of the normalized mode-I SIF for the negative gradient parameters (Fig. 3a) is smaller than that for the material gradation in the x1 -direction (Fig. 2a). On the contrary, for the positive gradient parameters it is significantly larger and reached at smaller time instants. The numerical results imply that the maximum amplitudes of the normalized dynamic SIFs and the timeinstants, at which they occur, depend significantly on the direction of the material gradation and the values of the gradient parameters.


43

Fig. 3: Normalized dynamic SIF for a material gradation in the x2 -direction Summary A meshless BEM for 2-D transient thermoelastic crack analysis in isotropic, continuously non-homogeneous and linear elastic FGMs under thermal shock is presented in this paper. Fundamental solutions of linear coupled thermoelasticity for homogeneous, isotropic and linear thermoelastic solids in the Laplace-transformed domain are employed for the boundary-domain integral equation formulation. The material non-homogeneity is described by domain integrals, which are evaluated by using the CTM. A collocation-based BEM is developed in the Laplace-transformed domain. The numerical inversion of the Laplace-transform is performed by Stehfest’s algorithm. The temporal variations of the dynamic SIFs for an edge crack in 2-D FGM plate are presented and compared with those obtained by the FEM. Numerical results demonstrate that the material gradation may have significant influences on the dynamic SIFs. Acknowledgement This work is supported by the German Research Foundation (DFG, Project No.: ZH 15/10-2), which is gratefully acknowledged. References [1] S. Suresh and A. Mortensen Fundamentals of Functionally Graded Materials: Processing and Thermomechanical Behaviour of Graded Metals and Metal-Ceramic Composites, IOM (1998). [2] X.W. Gao Engineering Analysis with Boundary Elements, 26(10), 905-916 (2002). [3] X.W. Gao Journal of Applied Mechanics, 69(2), 154-160 (2002). [4] M.R. Hematiyan Computational Mechanics, 39(4), 509-520 (2007). [5] M.R. Hematiyan Communications in Numerical Methods in Engineering, 24(11), 1497-1521 (2008). [6] H. Stehfest Communications of the ACM, 13(1), 47-49 (1970). [7] J. Balaš, J. Sládek and V. Sládek Stress Analysis by Boundary Element Methods, Elsevier (1989). [8] L.C. Wrobel and M.H. Aliabadi The Boundary Element Method, Vol 2: Applications in Solids and Structures, Wiley (2002). [9] J. Sládek, V. Sládek and C.Z. Zhang Computational Materials Science, 32(3-4), 532-543 (2005).

44


Preconditioning of BEM systems of equations by using a generic subregioning algorithm F.C. de Araújo Dept. Civil Eng, UFOP, Ouro Preto, MG, Brazil; [email protected]

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

Abstract. The generic substructuring algorithm, developed in previous works, is directly employed to construct block-diagonal preconditioners for BEM systems of equations. As the BE matrices for each BE subregion are independently assembled, the corresponding L and U factors, needed for constructing the preconditioner, are easily calculated. The Bi-CG solver, known for having irregular convergence behavior, is considered in the analyses to highlight the relevance of the preconditioning proposed. Complex 3D representative volume elements (RVEs) of carbon-nanotube (CNT) composites are analyzed to show the performance of the SBS-based preconditioned iterative solver. The models contain up to several tens of thousands of degrees of freedom, and the importance of the preconditioning technique is also discussed in the context of developing general (parallel) BE codes. Introduction

Today's available parallel computer architectures allied with the parallelism embedded in Krylov solvers have been, in the last decades, an appealing combination for devising efficient scalable parallel codes for solving large-order engineering problems [1, 2]. In general, direct solvers present the following disadvantages: they may be exceedingly CPU time-consuming and memoryconsuming for large-order models, and their parallel implementation is awkward. However, devising reliable efficient iterative solvers for non-symmetric systems of equations, as BEM systems typically are, has been a truly tough problem. In fact, despite the number of outstanding scientific works in this area in the last six decades, iterative solvers for general nonsymmetric matrices are a still open question [3]. In general, Krylov solvers are virtually the possible alternatives for dealing with this class of systems. Among them, long-recurrence algorithms (GMRES and variants) have disadvantages as large memory requirements for large problems and non-rare convergence stagnation in practice, and should indeed be avoided (at least purely). Thus, short-recurrence ones (Bi-CG and variants) remain as the most attractive options. 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 have been derived by combining the Bi-CG solver with residualminimization methods, as the GMRES, and so solvers such as the Bi-CGSTAB(l) [4] and the GPBiCG (generalized product Bi-CG) [5] have been devised. Additionally, preconditioners may be employed to accelerate the iterative process [1]. In the technical literature [6-8], a series of preconditioners have been reported for BEM solvers. 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, 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. They are also very important to parallelize incomplete LU-based and block-diagonal-based preconditioners.

In this paper, the BE substructuring algorithm [9-10] is employed to construct a global and block-diagonal-based preconditioner for the BE model at hand. In other words, the subregion-bysubregion (SBS) technique, applied to decompose a certain problem domain into a generic number of coupled BE models, is considered to define the block matrices for the block-diagonal 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 can be highlighted. Several representative volume elements (RVEs) of carbon-nanotube (CNT) composites simulated with BE models (with tens of thousands of degrees of freedom) are considered to show the performance of the preconditioning. The relevance of the preconditioning is pointed out not only in the context of iterative solver efficiency but also concerning the 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

im

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

(2)

m i 1

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:

4 (four subregions), explicitly assembled, it

46


:

: : :

:

A 11 H12 H13 H14 G12 H21

:

G21 A 22

:

G13

H41

G14 G23

H23 H24 H32

H31

:

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

In this system, note that H ij

H ji

G ij

G ji

y1

:

y2

:

y3

:

y4

:

.

(3)

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 block-diagonal matrices 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

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 block-diagonal 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.

3. Results and discussions The performance of the block-diagonal SBS-based preconditioner is measured by analyzing the hexagonal-packed long CNT-based composites shown in Fig. 1, wherein representative volume elements (RVEs) based on 1u 1 , 2 u 2 , 3u 3 , 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


47

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 hexagonal-packed long-CNT RVEs nsub* nel** nnodes† ndof‡ sparsity (%)

model

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

6 17 34 86

138 656 1,464 4,040

856 3,456 7,800 21,720

2,568 10,368 23,400 65,160

72 86 93 97

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

Figure 1. Hexagonal-packed long-CNT-based RVEs. Table 2. Engineering constants for the hexagonal-packed long-CNT RVEs model

E1 /E m

1u1 2u 2 3u 3 5u5

1.8081 1.8074 1.8074 1.8126 1.8131 1.3255

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

3.617%

Q 12 ,Q 13

Q 23

1.0889 1.0839 1.0916 1.0813 -

0.2943 0.2936 0.2931 0.2927 -

0.5107 0.5107 0.5185 0.4997 -

-

-

-

E2 /Em , E3 /Em

48


Table 3. Performance data for the hexagonal-packed long-CNT RVEs; tol 1.0 u 10 8 n. of CPU time (s) CPU time (s) system order n. of iterations (BE SBSiterations (Jacobi) based ILU)† (Jacobi) (BE SBSbased ILU)

model

1x1 unit cell, strain state 1 1x1 unit cell, strain state 2 2x2 unit cells, strain state 1 2x2 unit cells, strain state 2 3x3 unit cells, strain state 1 3x3 unit cells, strain state 2 5x5 unit cells, strain state 1 5x5 unit cells, strain state 2 †

2,568

68

446

3

5

2,568

85

451

3

5

10,368

249

1170

29

64

10,368

296

1166

33

64

23,400

316

1696

79

211

23,400

477

2413

114

297

65,160

614

4537

490

1713

65,160

884

4058

565

1476

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 block-diagonal one is applied. The decaying of the Euclidean residual norm, į 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 block-diagonal preconditioning. 1000

100 100

10 Preconditioner Jacobi BE SBS-based block diagonal

1 0.1 0.01 0.001 0.0001 1E-005

0.1 0.01 0.001 0.0001 1E-005

1E-006

1E-006

1E-007

1E-007

1E-008

1E-008

1E-009

Preconditioner Jacobi BE SBS-based block diagonal

1

residual Euclidean norm

residual Euclidean norm

10

1E-009 0

500

1000

1500

2000

2500

iteration

3000

3500

4000

4500

0

500

1000

1500

2000

2500

3000

3500

4000

iteration

(a) Strain state 1 (b) Strain state 2 Figure 2. Residual norm vs. iteration: 5 u 5 -unit-cell, square-packed long CNT 4. Conclusions and prospects The BE SBS technique proposed in previous papers ([9], [10]) is straightforwardly used to construct

block-diagonal-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 block-diagonal preconditioning, compared to the


Jacobi (diagonal) one, is considerably more efficient. In fact, the BE-SBS-based block-diagonal 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).

49

50


Alternative Derivations of Fundamental Solutions for Anisotropic Heat Transfer R.J. Marczak1 and M. Denda2 1

Mechanical Eng. Dept. - UFRGS - Rua Sarmento Leite 425, Porto Alegre, RS 90050-170, Brazil, [email protected] 2 Mechanical & Aerospace Dept. - Rutgers University - 98 Brett Road, Piscataway, NJ 088548058, USA, [email protected]

Keywords: Heat transfer, General anisotropic solids, Fundamental solutions, Fourier and Radon transforms.

Abstract. This work presents two new methods to derive the fundamental solution for three-dimensional heat transfer problems in the general anisotropic media. Initially, the basic integral equations used in the definition of the general anisotropic fundamental solution are revisited. We show the relationship between three, two and onedimensional integral definitions, either by purely algebraic manipulation as well as through Fourier and Radon transforms. Two of these forms are used to derive the fundamental solutions for the general anisotropic media. The first method gives the solution analytically for which the solution for the orthotropic case agrees with the well known result obtained by the domain mapping, while the fundamental solution for the general anisotropic media is new. The second method expresses the solution by a line integral over a semi-circle. The advantages and disadvantages of the two methods are discussed with numerical examples. Introduction The steadily increasing use of new and non-conventional materials by the modern industry has created a demand for reliable simulation methods in the design and analysis of mechanical components. Many of these new materials exhibit non-isotropic properties, making the differential operator of the governing equations somewhat more complicated than the well known isotropic case. The numerical or analytical treatment of orthotropic potential problems is generally accomplished by coordinate transformations which map the original problem into an equivalent isotropic one [1,2,3]. This method has the clear advantage of allowing the use of analytical solutions and/or computer codes developed for the isotropic media. However, the flux boundary conditions must also be transformed before their imposition on the new domain. While this is not a problem in domains composed of a single material, the application of the compatibility conditions along the interfaces of multi-material problems [4] can be cumbersome because the transformation for each subregion is different and therefore boundary interfaces in the original domain may no longer be aligned after the transformation. This is an important issue in problems that require the knowledge of the fluxes during the solution phase, like in the iterative solution methods or the inverse problems [5] where the flux boundary conditions is measured from the actual components. Most of the published works on non-isotropic potential problems are based on the transformation of the coordinate axes into the principal ones. In this situation, the thermal diffusion tensor (K) is diagonal, but the aforementioned drawbacks persist. Furthermore, special cases such as the functionally graded materials cannot be handled because there are no guarantees that the mapping will be invertible. It is interesting to derive the fundamental solution for the heat transfer referring to the generic, nonprincipal set of coordinate axes. To the best of authors' knowledge, no closed form fundamental solution for the fully anisotropic potential problems has been derived yet without the use of coordinate transformation. This is intrinsically due to the higher complexity of the differential equation when the thermal diffusion tensor is fully populated, making the separation of variables only conditionally possible. Reference [6] showed how to derive the two-dimensional fundamental solution for the isotropic Poisson's equation using the integral definitions. Similar ideas will be used here for the three-dimensional, anisotropic case. We first review the integral representations for the fundamental solutions as generally used to derive solutions for the vector-field problems. These representations have seldom been used for the isotropic or orthotropic scalar problems since the direct derivation


51

of solutions for these cases can be accomplished with little analytical effort. In the anisotropic case, however, the non-diagonal form of the constitutive tensor makes the use of such integral representations a more straightforward way to derive the fundamental solution. One of these definitions are applied to the general anisotropic heat transfer case, and then particularized to the orthotropic one. The agreement of the later with the well known orthotropic fundamental solution for Poisson's problems validates the applicability of the procedure. Furthermore, both cases can be reduced to the isotropic case, something not always possible in crystal physics. This one-step analytical approach, however, can be applied with relative ease only in cases of point source fundamental solutions. Other types of fundamental solutions will require more intense analytical effort, and although the explicit evaluation of the line integrals is all that is required, the calculation of the derivatives of the fundamental solution will become tedious, or even impossible. In such cases, the explicit use of Radon transform [7] to derive the fundamental solutions has proven to be a very interesting alternative, because other line integral forms can be employed [6,8]. This approach has been previously applied for crystal elastodynamics such as timetransient wave scattering, and not only allowed the analytical evaluation of the surface integrals over the boundary elements, but also reduced the computational effort drastically. More important, the fundamental solutions remain expressed by line integrals over a semi-circle for potential and flux as well. This is a significant advantage over the conventional analytical procedure which, as already mentioned, may fail to deliver fully analytical, closed form expressions for the later. We derive and test fundamental solutions for three-dimensional anisotropic heat transfer problems using both approaches. General expressions for the scalar fundamental solutions The steady-state heat conduction equation in three-dimensions, defined in the coordinate system x , can be written as L(∂ x )u (x) = b(x) x ∈ R3 , (1) where u and b are the temperature and heat source fields, respectively. The differential operator, L(∂ x ) = k ij ∂ i ∂ j , (2) is the Laplace's operator defined in terms of the thermal diffusion tensor k ij . As a companion of this differential operator, define a scalar-valued quadric,

L(x) = k ij xi x j ,

(3)

in terms of k ij which is called Green-Christoffel tensor (or acoustic tensor) [9,10]. The fundamental solution for the scalar differential operator L(∂ x ) is defined by the unit source applied at y and is given by the solution of the equation [11], L(∂ x )u ∗ (x, y ) = − δ(x − y ) x, y ∈ R 3 , (4) in an infinite space, where δ is the Dirac's delta operator. It is generically called the potential fundamental solution. The dual fundamental solution, which is the heat flux in the unit normal direction n, can be determined by the Fourier law, q ∗ (x, y ) = ni (x ) k ij u,∗j (x, y ) . (5) The solution of eq.(4) can be obtained following the former works of Kröner [12] and Synge [13], where Fourier transform was applied on both sides of eq.(1), and the concentrated excitation was reproduced by applying the limit on the distributed source. The result is the fundamental solution

u ∗ ( x, y ) =

1 8π 3

³³³

S3

L−1 (ξ) e −i ξ⋅( x− y ) dS 3 (ξ ),

(6)

where the integral is carried out over the unit sphere in the transformed space, with the center located at the origin as shown in Fig.. Notice that L−1 (ξ) is the inverse of the scalar valued quadric defined by (3). We observe from eq.(6) that the fundamental solution is constructed by the superposition of the wave functions. Applying the theory of harmonic functions, it can be further reduced to an integral over an arbitrary closed surface containing y [9],

u ∗ ( x, y ) =

1 ∆ y ³ ³ 2 L−1 (ξ) ξ ⋅ (x − y ) dS 2 (ξ ), S 16π 2

(7)

52


where ∆ y is the Laplacian operator. The final result of (7) does not depend on the size and shape of S 2 , so a unity sphere is generally used. According to eq.(8), the fundamental solution u ∗ is constructed in terms of the surface wave functions. This equation can further be transformed to give the line integral representation [12,13],

u ∗ ( x, y ) =

1 8π 2 r

³L

−1

S1

(ξ) dS 1 (ξ ),

(8) 1

where the integration is performed on a circumference ( S ) formed by the intersection of the plane perpendicular to x − y and the unit sphere S 2 . This equation can be more conveniently written in the spherical coordinates as [14]

u ∗ ( x, y ) =

( )

1 2 π −1 0 L ( ξ ) dϕ ξ 0 , 8π 2 r ³0

(9)

where ξ 0 is a parametric coordinate. Equations (8-9) are often recognized as direct consequences of the application of Radon transform [7,15], although by no means that is the only way to derive it. Its two most remarkable features are the relative simplicity of the integration needed compared with other equivalent expressions, and the preservation of the distribution properties necessary for continuous differentiation, as in eq.(5). It is worth mentioning that this form of line integral representation is seldom used in the scalar potential problems. The operational forms of eqs.(6), (7) and (8) may vary depending on the analytical procedures adopted in their derivation, but the equivalence between them has been proved [13,15]. The development of the fundamental solutions for the potential problems given by (1) can use either eq.(6), (7) or (8), as they are all continuously differentiable. We consider (7) to be the best starting point for the general methodology for three reasons. First, it is easier to integrate since it incorporates a reduction in the original integration order of (6). Secondly, although a number of one-dimensional forms like (9) can be obtained from (7), the apparent simplicity of line integral does not remain so when evaluating its derivatives as in eq.(5). Finally, eq.(7) is directly related to Radon transform whose inversion process is simpler than the corresponding inversion of the Fourier transform. The fundamental solution obtained by Radon transform is given in the form,

u ∗ ( x, y ) =

1 8π 2

³³

S2

L−1 (ξ ) δ(ξ ⋅ (x − y ) ) dS 2 (ξ ).

(10)

Transformations among eqs.(8), (9) and (10) can be performed in a number of ways. For example, eq. (8) can be reduced to an infinite integral along a line belonging to the plane perpendicular to x − y [16,17] using the symmetry property of L−1 and decomposing ξ in terms of an arbitrary parameter ψ , p = tan ψ , such that ξ = n cos ψ + m sin ψ, and the (m,n) plane is perpendicular to the vector x − y :

u ∗ ( x, y ) =

π

1 +∞ 1 + 2 −1 L (ψ ) dψ = 2 ³ L−1 ( p ) dp. 2 ³− π 4π r −∞ 4π r 2

(11)

One should note that the simplicity of the Laplacian operator allows the direct integration of any of eqs.(6), (7) and (8) or the alternative expressions given by either eqs.(9), (10) or (11). The choice of one particular form depends on the complexity of the integrand in each case. For example, eq.(10) can be integrated only if a suitable expression for the Dirac's delta is used [8,18], while eqs.(8), (9) and (11) relies on one dimensional integration. Most of them have been extensively used in the past to obtain tensorial fundamental solutions in crystal physics, in which L is a matrix that requires inversion. In comparison, the present case of potential fundamental solution is much simpler as shown below. Fundamental solution for 3D anisotropic Laplacian operator Analytical form In this section, eq.(11) will be used to derive the fundamental solution of an anisotropic diffusive medium. The general anisotropic thermal diffusion tensor is given by

ª k11 k12 K = «« k 22 «¬sym.

k13 º k 23 »». k33 »¼


53

The quadric is given by

L(ξ) = ξT Kξ = ξ1 k11 + ξ 2 k 22 + ξ 3 k33 + 2 ξ1ξ 2 k12 + 2 ξ1ξ 3 k13 + 2 ξ 3ξ 2 k 23 . If we substitute ξ = n cos ψ + m sin ψ into the above quadric we get 2

2

2

L(ψ ) = ni n j k ij cos 2 ψ + (n j mi k ij + ni m j k ij )sin ψ cos ψ + mi m j k ij sin 2 ψ = Q cos 2 ψ + R sin ψ cos ψ + T sin 2 ψ.

The parameters Q, R and T are the scalar counterparts of the similar (tensorial) terms found in crystal physics [16]. Use this result in eq.(11) to get

u ∗ (x, y ) =

· 1 + π2 § 1 1 1 ¸ dψ = π ¨ 4πr ∆ 4π 2 r ³− 2 ¨© Q cos 2 ψ + R sin ψ cos ψ + T sin 2 ψ ¸¹

( (

)

(

where ∆ = QT − R 2 / 4 = ni n j m k ml k ij k kl − n j mi k ij + m j ni k ij

))

2

/ 4. After factoring ∆ and recalling that m

and n are the unit vectors perpendicular to r = x − y , one can rewrite:

r 2 ∆ = r32 k11 k 22 + r22 k11 k 33 + r12 k 22 k 33 + 2r3 r2 (k11 k 23 − k12 k13 ) + +2r3 r1 (k12 k 23 − k13 k 22 ) + 2r2 r1 (k12 k 33 − k13 k 23 ) − r32 k12 − r12 k 23 − r22 k13 . 2

If we define

2

(k k − k12 k13 ) r r r r 2∆ = 1 + 2 + 3 + 2r3 r2 11 23 + k11 k 22 k 33 k11 k 22 k 33 k11 k 22 k 33 2

r2 =

2

+ 2r3 r1

2

2

(k12 k 23 − k13 k 22 ) k11 k 22 k 33

+ 2r2 r1

(k12 k 33 − k13 k 23 ) k11 k 22 k 33

r32 k12 + r12 k 23 + r22 k13 k11 k 22 k 33 2

−

2

2

,

then we can write u ∗ as

u∗ =

1 4πk r

(12)

where

∗ = uORT

1 4πk

1 r12 k11

+

r22 k22

+

r32 k33

(13)

Evidently (13) must be used with a coordinate system aligned with the principal directions. As for the heat flux, it is found to be

q∗ =

−1 (q1n1 + q2 n2 + q3 n3 ) 2π k 3 r 2

where the terms q i are given by:

q1 = (c2233 k11 + c1233 k12 + c1223 k13 ) r1 + (c1233 k11 + c1133 k12 + c1123 k13 ) r2 +

(c1223 k11 + c1123k12 + c1122 k13 ) r3 ,

q2 = (c2233 k12 + c1233 k 22 + c1223 k 23 ) r1 + (c1233 k12 + c1133 k 22 + c1123 k 23 ) r2 +

(c1223 k12 + c1123 k 22 + c1122 k 23 ) r3 ,

q3 = (c2233 k13 + c1233 k 23 + c1223 k33 ) r1 + (c1233 k13 + c1133 k 23 + c1123 k 33 ) r2 +

(c1223 k13 + c1123 k 23 + c1122 k33 ) r3 ,

and the symbol cijkl = k ij k kl − k il k jk is used for compactness.

(14)

54


Line Integral Form Let (e1 , e 2 , e 3 ) be a set of unit orthogonal base vectors in the global coordinate system. If

S 2 by ξ = sin φ cos θ e1 + sin φ sin θ e 2 + cos φ e 3 , then the integral (10) can be

we express the unit sphere written as

u ∗ ( x, y ) =

1 4π 2

³³

1 4π 2

³³

π

π

0

0

L−1 (ξ ) δ[ξ ⋅ (x − y )] sin φ dφ dθ

(15)

where symmetry with respect to θ has been used. This equation can further be modified if we introduce s = ξ/ sin φ = cos θ e 1 + sin θ e 2 + ctanφ e 3 and λ = ctanφ . Then (15) can be written

u ∗ ( x, y ) =

π ∞

0

−∞

L−1 (s)δ[s ⋅ (x − y )] dλ dθ.

(16)

The infinite integral in (16) can be reduced to the Cauchy integral along the λ -axis, which can be evaluated by the Residue theorem to give π

u ∗ (x, y ) = −ℜ ³ u ∗ (θ) 0

sign ( x3 − x30 ) dθ, ( x − y ) ⋅ s ∗ ( θ)

(17)

where

s ∗ (θ) = cos θ e1 + sin θ e 2 + λ∗ e 3 , 1 1 , u ∗ (θ) = 2 2π ∂ λ L(s ∗ (θ)) and λ∗ is one of the two characteristic roots of L(s) with positive imaginary part. The corresponding flux across the plane with the unit normal n is given by π

q ∗ ( x, y ) = ℜ ³ q ∗ ( θ ) 0

sign ( x3 − x30 ) dθ, [(x − y ) ⋅ s ∗ (θ)]2

(18)

where q ∗ (θ) = ni s j k ij u ∗ (θ) = ni qi∗ . The integrands of the line integrals (17) and (18) are given by the product of a function of the material properties only and a simple algebraic function. Notice that the partial differentiation of the integral can be performed by changing the order of the integration and differentiation. There is no question on the advantage of the analytical form (12) of the fundamental solution. However the advantage stops here. If we want to use the analytical solution for the continuous distribution of the heat sources, then it is not straightforward to integrate the distribution analytically. Derivation of the fundamental flux solution in such cases, if not impossible, is tedious. In contrast to this, the line integral form (17) of the fundamental solution is much simpler than the analytical solution and is flexible. Any associated fundamental solutions, that can be obtained from this either by differentiation or integration, can be obtained by simply applying the operation to the integrand of (17). Since the integrand is a simple plane wave function its derivative and integration can be performed readily. The only disadvantage is the need for numerical integration, which can be performed by the standard Gauss quadrature. On the other hand, the integrand functions in eqs.(17) and (18) are very well behaved. Numerical examples In order to assert the validity of the fundamental solutions developed for the anisotropic case, eq.(12) was implemented for three types of materials with hexagonal and trigonal symmetries. They are Graphite (hexagonal), Quartz (trigonal), and Calcite (trigonal), respectively with the following thermal diffusivity written in their principal axes [10]:

0 º 0 º 0 º ª355 0 ª 6.5 0 ª4.18 0 K G = ««0 355 0 »» , K Q = ««0 6.5 0 »» , K C = ««0 4.18 0 »» [W/mD C]. «¬0 «¬0 «¬0 0 89»¼ 0 11.3»¼ 0 4.98»¼ The application of eq.(13) referenced to a coordinate system coincident with the principal axes is straightforward. The numerical test performed here was therefore limited to verifying if the fundamental solution obtained in a arbitrary coordinate system, as given by eq.(12), reproduces the same results obtained in the ∗ orthotropic case. Initially, a coordinate system ( x, y, z ) was used to evaluate u ORT for all three materials as a


55

function of a load point placed along the line (a, a, 0 ) . These fundamental solutions are plotted as solid lines in

fig.1. A new coordinate system ( x ′, y ′, z ′) was generated by rotating the original one by 30, 60, and 90 degrees

along the axes x, y, and z, respectively. In the ( x ′, y ′, z ′) system, the thermal diffusivity tensors become obviously non-diagonal, allowing the direct application of eq.(12), which are plotted as symbols in fig.1. Excellent agreement was found in all cases analyzed. As a second example, a fundamental solution for Graphite was derived for its principal coordinate axes as in eq.(13), again considering the load point acting along the line (a, a, 0 ) . The expression for u ∗ in this case is:

u∗ =

3.16567 × 10 −4 . a

(19)

(

)

The ( x, y, z ) system was then rotated along the direction 0, 2 / 2, 2 / 2 by an angle of π / 6 . Rotating the diffusivity tensor accordingly and substituting these properties in eq.(12) produced exactly the same function given by eq.(19).

0.05

u* [`]

0.04

0.03 Graphite 0.02

Quartz Calcite

0.01

0 0

2

4

6

8

10

a [m] Figure 1: Comparison of the Green’s functions obtained by eq.(13) in the principal (x, y, z) system (solid lines) and by eq.(12) in an abritrary (x’, y’, z’) system. Conclusions In this paper we have initially presented a review of several integral identities used to derive fundamental solutions for the Laplace operator, aiming its application in integral equation methods. The relationship between the one, two and three-dimensional forms, as well as their relation to Fourier and Radon transforms have been shown, whose use in potential problems has never been fully explored. The application of either eq.(6) and (7) or eq.(8) in analytical and semi-analytical derivations, respectively, to other types of scalar problems seems to be very attractive. By replacing the operator L it is possible to use the procedure outlined here to derive different types of fundamental solutions, such as time-dependent problems, problems with spatially variable constitutive parameters, and problems with heat sources distributed along lines and arcs, to name a few. The methodology was applied to the fully anisotropic heat transfer equation, generating a less known solution for the problem, which

56


exempts the need for coordinate transformations. A numerical version of the fundamental solution was also derived using the line integral method. Numerical examples show that both fundamental solutions agree. However, analytical forms like eq.(12) may be impossible to obtain in cases subjected to excitations other than a unit source. In such cases, particular forms of the line integral (10) will provide a much easier solution for both potential and flux.

References [1] Mulholland, G., and Gupta, B. Heat transfer in a three-dimensional anisotropic solid of arbitrary shape. J. Heat Transfer 99, 135-137, (1977). [2] Chang, Y., and Tsou, C.H. Heat conduction in an anisotropic medium homogeneous in cylindrical coordinates, steady states. J. Heat Transfer 99, 132 -134, (1977). [3] Poon, K., Tsou, R., and Chang, Y. Solution of anisotropic problems of first class by coordinatetransformation. J. Heat Transfer 101, 340-345, (1979). [4] Akif Atalay, M., Dilara Aydin, E., and Aydin, M. Multi-region heat conduction problems by boundary element method. Int. J. Heat Mass Transfer 47, 1549-1553, (2004). [5] Mera, N., Elliott, L., Ingham, D., and Lesnic, D. Use of the boundary element method to determine the thermal conductivity tensor of an anisotropic medium. Int. J. Heat Mass Transfer 47, 4157-4167, (2001). [6]

Deans, S.R. The Radon Transform ans Some of its Applications. John Wiley & Sons, (1983).

[7] Wang, C.Y. and Achenbach, J.D. Elostodynamic Fundamentală Solutions for Anisotropic Solids, Geophys. J. Int. 118, 384-392,(1994). [8] Wang, C. and Denda, M., 3D BEM for general anisotropic elasticity. Int. J. Solids & Structures 44, 7073-7091, (2007). [9] Mura, T. and Kinoshita, N., Green’s functions for anisotropic elasticity. Phys. Stat. Sol. (b) 47, 607618, (1971). [10]

Nye, J.F., Physical Properties of Crystals. Clarendon, Oxford (1957).

[11] Stakgold, I., Green’s functions and boundary value problems. Wiley-Interscience Publications, New York, (1979). [12] Kröner, E., Das fundamentalintegral der anisotropen elastischen differentialgleichungen. Zeitschrift für Physik 136, 402-410, (1953). [13]

Synge, J.L., The Hypercircle in Mathematical Physics. Cambridge University Press, (1957).

[14] Li, X. and Wang, M., Three-dimensional Green's functions for infinite anisotropic piezoelectric media. Int. J. Solids & Structures 44, 1680-1684, (2007). [15]

Gelf'and, I., Graev, M.I. and Vilenkin, N.Y., Generalized functions vol.5, Academic Press, (1966).

[16] Ting, T. C.T. and Lee, V.G., The three-dimensional elastostatic Green's function for general anisotropic linear elastic solids. Mech. Appl. Math. 50, 407-426, (1997). [17] Tonon, F., Pan, E., and Amadei, B., Green’s functions and boundary element method formulation for 3D anisotropic media. Computers & Structures 79, 469-482, (2001). [18] Wang, C., Denda, M., and Pan, E. Analysis of quantum-dot-induced strain and electric fields in piezoelectric semiconductors of general anisotropy. Int. J. Solids & Structures 43, 7593-7608, (2006). [19] Rucker, W.M., and Richter, K.R. Three-dimensional magnetostatic field calculation using boundary element method. IEEE Trans. on Magnetics 24, 23-26, (1988).


57

Decompositions of Cijkl Aiming the Inversion of Acoustic Tensors for Fundamental Solution Derivations R.J. Marczak Mechanical Eng. Dept. - UFRGS - Rua Sarmento Leite 425, Porto Alegre, RS 90050-170, Brazil, [email protected]

Keywords: Constitutive equations, anisotropy, fundamental solution, acoustic tensor, boundary element method.

Abstract. It is well known that the derivation of new forms of fundamental solutions can be significantly simplified by the use of area or line integral forms. However, in any case, it is necessary to invert the acoustic tensor corresponding to the media being analyzed, and this is the major drawback of the current trend for new anisotropic fundamental solutions. This work presents various forms of decomposition of the constitutive tensor, and its application in restating the integral forms used to derive fundamental solutions. Lower rank, spectral, Kelvin, Hamilton, and Gibb’s decompostions are presented, analyzed, and compared regarding possible simplifications that they can produce.

Introduction Let a general, second order linear differential equation Lij (∂ x )u j (x) = bi (x) x ∈ R3 ,

(1) ∗

where the elliptic operator L admits a self-adjoint L and is given by: L jk (∂ x ) = cijkl ∂i ∂ l .

(2)

and x defines the coordinate system to which c is referred. An important feature of eq.(2) relies in its redefinition a non-differential operator [1, pp.124], commonly referred to as Green-Christoffel tensor or acoustic tensor [2,3,4]: L jk (x) = cijkl xi xl , (3) so that, if L is to be inverted, then, by Cramer's rule one has −1

L−1 (x) = det L(x) −1 adjL(x) = D ( x ) A (x)

(4)

where det ( L ) and adj ( L ) are the determinant and the adjugate of L ( x ) , respectively. Now recall that the primal fundamental solution for L with a pole at y is the solution of the equation [5]: Lij (∂ x )U jk (x, y ) = −δik δ(r ) x, y ∈ R 3 , (5)

where δ(⋅) is the Dirac's delta operator and r = x − y , r = r . In this form, U is generally referred to as displacement fundamental solution. The dual (or traction) fundamental solution can be determined by direct differentiation of the solution of eq.(5): (6) Tij (x, y ) = nk ( x ) ckilmU lj ,m (x, y ) , where n is the unit normal vector to the plane where T is evaluated. The objetive of this note is to revisit the now popular expression used to derive fundamental solutions

58


for boundary integral equation methods, given by: 1 U ij (x, y ) = 2 v³ 1Lij −1 (ξ) dS1 ( ξ ) , (8) 8π r S where the integral is carried out over a unit sphere, with special attention to spectral forms that can help to evaluate the inverse of L. The fundamental solution (8) is commonly written using modulation functions : 1 U ij (x, y ) = 2 U ij 8π r Where U ij (x, y ) = v³ 1L−ij1 (ξ) dS 1 ( ξ ) S

(9) A(ξ) 1 dS ( ξ ) S D (ξ ) and the integrand is typically integrated using the Residue theorem. Since the theorem applies only for distinct poles, eq.(a) suffer from degeneracy when materials with lower symmetries are analyzed. Therefore, any scheme that could help to invert the operator L – hopefully without degeneracy – would simplify significantly the tremendous work of analytical inversion of L. = v³

1

As is well known, eq.(9) has been used under different disguises for several types of problems, including multiphysical coupled ones. The derivation of strictly analytical, closed form expressions for fundamental solutions is dependent on the analytical invertibility of L. Any case of material symmetry will make these calculations rather cumbersome due to the application of L'Hopital rule. In addition, the inability of L−1 , as given by eq.(9), in degenerating to higher symmetries for specific algebraic forms remains an open question.

Since there are a several ways of representing a second order tensor, it is worth to investigate alternative forms of expliciting both, C and L in order to verify whether substantial simplification in the analytic expressions for the integrands in eq.(9) can be attained. Therefore this report aims to explore alternative ways to decompose either among the possibilities published elsewhere, aiming the analytic evaluation of U and U . Unfortunately, not all of these approaches are completely developed for all types of anisotropy. Decomposition of cijkl in lower-rank tensors Cowin [6] showed how to decompose a fourth-rank tensor in two symmetric second-rank tensors and an irreducible, completely symmetric and traceless fourth-rank tensor. The two second-order tensors are defined as:

WIJ = cijkk

VIJ = cikjk ,

which are called dilatational stiffness and Voigt stiffness tensors, respectively. If Voigt notation is to be used, these tensors assume the forms:

ªC11 + C12 + C13 C16 + C26 + C36 C15 + C25 + C35 º W = «« C12 + C22 + C23 C14 + C24 + C34 »» sym C13 + C23 + C33 ¼» ¬« ªC11 + C55 + C66 V = «« «¬ sym

C16 + C26 + C45 C22 + C44 + C66

C15 + C46 + C35 º C23 + C34 + C56 »» C33 + C44 + C55 »¼

(10)

The traces of these tensors are:

tr ( W ) = C11 + C22 + C33 + 2 ( C12 + C13 + C23 )

tr ( V ) = C11 + C22 + C33 + 2 ( C44 + C55 + C66 )

and their deviatoric parts are:

(11)

Advances in Boundary Element and Meshless Techniques XII o = W − tr ( W ) I W 3

59

o = V − tr ( V ) I V 3

Now the decomposition of ^ can be accomplished by using two scalars a and b and the following identity:

cijkm = a δij δkm + b δik δ jm + δij Akm + δik B jm

} {

{

}+ z

(12)

ijkm

where ] is the aforementioned fourth-rank tensor [6] and the following notation was used:

{ Aij Bkm } = Aij Bkm + Akm Bij

Aij Bkm = Aij Bkm + Aim Bkj

In order to satisfy the original symmetries of ^ , one must have: o = 5 W − 4V o o − 2W o 7A 7B = 3V and eq.(12) assumes the form: 2tr ( W ) − tr ( V ) 3tr ( V ) − tr ( W ) δij δ km + δik δ jm + cijkm = 15 30 5 o km − 4 δ V km + 3 δ V jm − 2 δ W o jm + δij W ij ik ik 7 7 7 7

{

} {

} {

} {

}

(13)

+ zijkm

The ] tensor ( zijkm ) can be written in contracted form (Z) as done with ^ and its Voigt’s version C. Now a relationship between L, W, and V can be assembled noting that each entry in eq.(3) is a quadratic form: jk

L jk (ξ) = ξ ⋅ C ⋅ ξ

(14)

In the absence of any material symmetries, the tensors C

with C

21

12

31

=C , C

with ξ = {ξ1

ξ2

ªC11 11 C = ««C16 ¬«C15

C16 C66 C56

C15 º C56 »» C55 ¼»

C

ªC15 13 C = ««C56 «¬C55 ªC56 23 C = ««C25 «¬C45

C14 C46 C45

C13 º C36 »» C35 »¼

C

C46 C24

C36 º C23 »» C34 »¼

C

13

= C , and ªC11 « 12 C = «C « « 13 «¬C

C44 32

jk

12

22

33

are given by:

ªC16 = ««C66 ¬«C56

C12 C26 C25

C14 º C46 »» C45 ¼»

ªC66 = ««C26 «¬C46

C26 C22 C24

C46 º C24 »» C44 »¼

ªC55 = ««C45 «¬C35

C45 C44 C34

C35 º C34 »» C33 »¼

(15)

23

C = C . L is written in a single matrix equation as L = MT CM where 12 13 C C º ªξ 0 0 º » 22 23 (16) C C » M = ««0 ξ 0 »» » 23 33 » «¬0 0 ξ »¼ 9×3 C C » ¼ 9×9

ξ3 } and 0 is a 3 × 1 null column vector. Comparing the matrices in eqs.(16) with the ones in T

eqs.(10), it becomes apparent that, if C is made symmetric such as:

ªC11 C12 « 21 22 C = «C C 31 32 «C C «¬

13 C º 23 » C » 33 » C » ¼ 9×9

60

where C


21

( )

12 T

= C

31

( ) , and C = (C ) 13 T

(

23 T

32

, C = C

)

ik k

Wij = C e

, then the dilatational stiffness is (17)

j

where e k are the canonic base vectors. The Voigt stiffness becomes merely

( ) ij

Vij = tr C

(18)

and each entry of L remain as in eq.(14). To the best of the authors knowledge, eqs.(17) and (‘8) have not been reported in the literature. We believe the submatrices in C can be used to rewrite the three-dimensional Navier equations under Stroh formalism, still incomplete in the literature. Another interesting aspect found in the ik

proposed C matrices is their close relation to the ones found in Stroh formalism [7,8]. For instance, the Navier equations for two-dimensional elasticity can be written as:

(

)

Q u,11 + R + RT u,12 +Tu,22 = 0 where u is the generalized displacement vector and the matrices Q, R, and T are given by (similar, but not identical to the ones found in most references [9]):

Q = [ ci1k1 ] ,

R = [ ci1k 2 ] ,

T = [ ci 2k 2 ] .

Now, surprisingly, it happens that 11

12

Q=C , or

R=C ,

(

11

12

C u,11 + C + C

21

T=C

) u,

12

22

,

22

+C u,22 = 0

a result not found in the literature as well. Although still unproven, it is natural to expected that in threeik

dimensional elasticity the Navier equations can be written assigning the suitable C the displacements.

to each second derivative of

Spectral decomposition Sutcliffe [10] summarized the basis for writing constitutive tensors using their spectral components. Although the interest was the decompositon of the C tensor to use in the equation σ IJ = CIK ε KJ , a useful spectral decomposition of a generic acoustic tensor is also presented (eq.(29) in [10]). Here these basic concepts are revisited, and particularized for the present case. Let the eigenproblem

^ : N = λN associated to the fourth order constitutive tensor ^ as in the non-contracted equation s = ^ : e . Since ^ is real and symmetric, the eigenspaces associated to the 6 eigenvalues λ I are orthogonal. The corresponding eigentensors can be constructed respecting the orthonormality condition N I : N J = δ IJ , projection operator can be assigned to each distinct eigenvalue as:

I =

¦ NL ⊗ NL ,

I = 1,..., K

I , J = 1,..., 6 . A (19)

L∈κi

where κi contains the set of indices related to the eigenvalue λ I with multiplicity K I = dim ( κ I ) , thus satisfying ¦ K I =1K I = 6 . Being eigentensors, the projection operators I , I = 1,..., K span the whole space of symmetric fourth-order tensors and satisfy the following relations: K

¦I = , I =1

I : J = δ IJ I ,

I , J = 1,..., K

where denotes the fourth-order identity tensor. The spectral decomposition of ^ and its inverse therefore become a sum from 1 to K components:

Advances in Boundary Element and Meshless Techniques XII K

61

K

1 λ I =1 I I

^ = ¦λ I I ,

^ −1 = ¦

I =1

assuming no eigenvalue is null. The importance of spectral decompositions in the present context lies on the possibility of using the same elastic projection operators to reconstruct a tensor and its inverse as well. In this regard, eq.(3) reads, in tensorial notation L ( ξ ) = ξ ⋅ ^ ⋅ ξ and its decomposition in the eigenspace becomes K

L ( ξ ) = ¦λ I I ,

(20)

I =1

where

I =

¦ ML ⊗ ML ,

I = 1,..., K

and

ML = NL ⋅ ξ .

L∈κi

Therefore, the inverse of L is: K

L−1 ( ξ ) = ¦

1

(21)

I =1 λ I I

Equation (20) has been applied to determine wave speeds and acoustic axes in anisotropic elastodynamic problems.[11,12]. Even more interesting seems to be the extensive application of eqs.(20) and (21) to investigate strain localization in anisotropic elastoplastic materials, to define corrected orthotropic yield criteria, and to estabilish constitutive relations for non-isotropic hyperelasticity (see, for instance [13,14,15]), in most cases developed apparently without knowledge of its importance in the derivation of fundamental solutions for anisotropic media (with few exceptions such as [16,17]). The explicit forms of the eigentensors used in eq.(19) are readily available for isotropic, cubic, and tetragonal materials ([11,12,13,18]). Kelvin's decomposition Lord Kelvin (apud [19]) has proposed the representation of a general tensor as a sum of a diagonal tensor and a dyad [19,20]: L (ξ) = D + n ⊗ n (22) where

D = diag[d1 (ξ ), d 2 (ξ ), d 3 (ξ )] and

n = {n1

n2

n3 } = {β1ξ1 β 2 ξ 2 T

β3ξ3 } . (23) T

Apparently, the decomposition (22) was also developed independently in [21], but it is not applicable to all types of anisotropy, however. Substituting eqs.(15) particularized for each type of material in eq.(14) it is possible to verify the forms that L assumes for rhombic, tetragonal, hexagonal and cubic (RTHC) symmetries. Ph. Boulanager and Hayes [22] have studied this kind of decomposition for RTHC materials, showing that degenerate cases can be dealt with by appropriate factorization of both terms in eq.(22). Further details about Kelvin's decomposition can be found in refs.[20,22,23]. Equation (22) can be further reduced with the aid of symbolic algebra computer programs. The following expressions were obtained for higher symmetries: •

Cubic:

ªξ12C11 + ξ 22C 44 + ξ 32C44 « L=« « sym. ¬

(C44 + C12 ) ξ 2 ξ1 ξ12C 44 + ξ 22C11 + ξ 32C 44

(C44 + C12 ) ξ 3ξ1 (C44 + C12 ) ξ 3ξ 2

º » » 2 2 2 ξ1 C 44 + ξ 2C 44 + ξ 3 C11 »¼

with D = diag ª a ξ12 + b, a ξ22 + b, a ξ32 + b º and the dyad can be particularized as n ⊗ n = c ξ ⊗ ξ . ¬ ¼

62


•

Hexagonal ( m 5 ):

ªξ12C11 + 12 (C11 − C12 ) ξ 22 + « + ξ 32C44 « « L=« « « sym. « ¬

1 2

º

(C11 + C12 ) ξ 2ξ1

(C44 + C13 ) ξ3ξ1 »

(C11 − C12 ) ξ12 + ξ 22C11 +

(C44 + C13 ) ξ3ξ 2 »»

1 2

»

+ ξ32C44

» ξ12C44 + ξ 22C44 + » » + ξ 32C33 ¼

with D = diag ª d ξ12 + e, d ξ22 + e, f ξ32 + b º

¬

•

Tetragonal ( m 6 ):

ªξ12C11 + ξ 22C66 + ξ 32C 44 « L=« « sym. ¬

¼

(C66 + C12 ) ξ 2 ξ1 ξ12 C66 + ξ 22C11 + ξ 32C 44

(C44 + C13 ) ξ 3ξ1 (C44 + C13 ) ξ3ξ 2

º » » 2 2 2 ξ1 C 44 + ξ 2C 44 + ξ 3 C33 »¼

with D = diag ª g ξ12 + e ξ22 + b ξ32 , e ξ12 + g ξ22 + b ξ32 , b ξ12 + b ξ22 + h ξ32 º ¬ ¼ •

Orthorhombic:

º ªξ12C11 + ξ 2 2C66 + (C66 + C12 ) ξ2ξ1 (C55 + C13 ) ξ3ξ1 » « 2 C + ξ 3 55 » « 2 2 » « ξ1 C66 + ξ 2 C22 + ( ) L=« + ξ ξ C C 44 23 3 2 » 2 + ξ C 3 44 » « 2 2 « ξ1 C55 + ξ 2 C44 + » sym. » « 2 + ξ3 C33 ¼ ¬ with D = diag ª g ξ12 + e ξ22 + b ξ32 , e ξ12 + g ξ22 + b ξ32 , b ξ12 + b ξ22 + h ξ32 º ¬ ¼

Expliciting the factors used above as functions of the material constants only, one has:

a = C11 − C12 − 2C44 b = C44 c = C12 + C44 d = C44 − C66 e = C66

f = C33 − C44 −

( C13 + C44 )2 C11 − C66

g = C11 − C12 − C66 h = C33 −

( C13 + C44 )2 C12 + C66

and the dyad n ⊗ n in the last three cases remain as in eq.(23). Hamilton's decomposition Also known as Hamilton's cyclic form, the Hamilton's decomposition of L is given by:

1 L ( ξ ) = λI + λ′ ( a1 ⊗ a 2 + a 2 ⊗ a1 ) 2

(24)


63

The numbers λ, λ′ , and the vectors a1 , a 2 are uniquely defined for L [19,24]. Less restritive forms of eq.(24) exist. For example, studying strain localization in transversely isotropic materials, Zhang et al. [15] obtained the following form for the acoustic tensor: L ( ξ ) = α1I + α 2 ξ ⊗ ξ + α3 ( e3 ⊗ ξ + ξ ⊗ e3 ) + α 4e3 ⊗ e3 (25) where e3 stands for the transverse direction, and:

c2 c5 2 + ( e3 ⋅ ξ ) 2 2 c · § α3 = ¨ c3 + 5 ¸ ( e3 ⋅ ξ ) 2¹ ©

c2 2 c5 · 2 § α 4 = ¨ c4 + ¸ ( e3 ⋅ ξ ) 2¹ ©

α1 =

α 2 = c1 +

with α1 strictly positive to satisfy the positive definiteness of C. The parameters c1 to c5 , not given in the original reference, were found to be:

c1 = C12 c2 = C11 − C12 c3 = C13 − C12

c4 = C11 − 2C13 + C33 − 4C44 c5 = C12 − C11 + 2C44

The determinant D (ξ) can be factored in the compact form:

D (ξ ) =

1 ( c1 + c2 ) ( c2 + c5 ) + ( ( c3 + c5 )( c2 − c3 ) + c4 ( c1 + c2 ) ) ( e3 ⋅ ξ )2 + 2 1 4 + ( 2c3 ( c3 + 2c5 ) − c4 ( 2c1 + c2 ) + c5 ( c4 + 2c5 ) ) ( e3 ⋅ ξ ) 2

and therefore, the inverse of eq.(25) can be written as:

L−1 ( ξ ) = β1I + β2ξ ⊗ ξ + β3 ( e3 ⊗ ξ + ξ ⊗ e3 ) + β4e3 ⊗ e3 where:

β1 =

1 α1

β2 =

1 −α 2 ( α1 + α 4 ) + α32 α1D

(

)

(

(26)

(

)

β3 =

1 −α1α3 + α 2α 4 − α32 ( e3 ⋅ ξ ) α1D

β4 =

1 −α 4 ( α1 + α 2 ) + α 32 α1D

(

)

)

Whereas very promising for purely numerical applications, the authors are not aware of decompositions such as eq.(26) for materials with lower simmetries. Gibb's decomposition The Gibb's decomposition is another form for writing a tensor using its principal axes {a, b, c} . It assumes the following form for L:

L ( ξ ) = λ1a ⊗ a + λ 2b ⊗ b + λ3c ⊗ c

(27)

where the eigenvalues are ordered as λ1 ≥ λ 2 ≥ λ 3 . Some specific details about this type of decomposition can be found in[24]. More important than a possible inverse counterpart of eq.(27) is the fact that when it is combined with eq.(24) (and making use of the identity a ⊗ a + b ⊗ b + c ⊗ c = I ), the Kelvin's decomposition as given by eq.(22) can be obtained. Isotropic seed Spectral decompositions may allow an easy way to invert a tensor, but it is impossible to find all eigenprojections in the fully anisotropic case. On the other hand, expansions of that type leads one to think about more ordinary decompositions in the form (the arguments ξ are removed for short): L = A+B (28) Here it is worth to note that the Kelvin decomposition is already expressed as such sum by setting:

64

Eds: E L Albuquerque & M H Aliabadi A = D and B = n ⊗ n

Similarly, the Hamilton decomposition can be also recast as in eq.(28) by using:

1 A = λI and B = λ′ ( a1 ⊗ a 2 + a 2 ⊗ a1 ) 2 Of course one could split L in a larger number of tensors, but two terms are particularly attractive in view of some well known expressions used to invert the sum of two matrices. If the acoustic tensor can be decomposed in such a way that one of its components is a purely isotropic part, the remaining part must contain the contribution of all lower symmetries: L = L ISO + L ANI (29) where the second term eventually encompasses all non-isotropic contributions of C. Then the fundamental solution will have a similar structure:

U = U ISO + U ANI Therefore, the question that naturally arises is 'Do anisotropic fundamental solutions have an extractable isotropic nuclei?'. Apparently, there is no reason precluding one to define a suitable tensor L ISO since this is a purely algebraic procedure. The isotropic part, however, must be such that the remaining term is not singular. The author's preliminary experiments showed that it is possible to choose a L ISO in such a way that the complementary part L ANI vanishes in the isotropic case, but this is strongly dependent on the choice of L ISO . 1 would vanish. One In many cases, reduction to isotropy could not be achieved because the denominator of L−ANI could argue that this is a direct consequence of the degeneracy. Although materials with higher symmetries, in particular isotropy, are degenerate, this seems to be a mathematical degeneracy. There is no apparent reason for taking the type of material as a necessary condition for degeneracy, as fully anisotropic materials may exhibit the same degeneracy for appropriate values of their stiffness. If this assertion is correct, then eq.(29) is valid for any type of material. Assuming so, the next step would be to determine L ISO . This is not a simple task because there are infinite possibilities. One should remember, however, that the resulting isotropic contribution represents a material which does not necessarily have a direct relationship with the original one.

Decomposition of L by 21D projection operators One last intriguing question that remains in the present context is the possibility of further decomposing L ANI . Browaeys and Chevrot [25] have proposed expressing the C tensor as a vector defined over a 21-dimensional space:

(

X = C11, C22,C33 , 2C23 , 2C13 , 2C12 , 2C44 , 2C55 , 2C66 , 2C14 , 2C25 , 2C36 , 2C34 , 2C15 , 2C26 , 2C24 , 2C35 , 2C16 , 2 2C56 , 2 2C46 , 2 2C45

)

After finding projection operators for each type of symmetry, these can be cummulatively applied to X in order to extract its components. Therefore, any triclinic vector X can be decomposed by a cascade of projections into a sum of vectors belonging to each symmetry class:

X = X ISO + X HEX + XTET + XORT + X MON + XTRI Using the definition of L this decomposition could be used to restate eq.(29) as:

L = LISO + LANI = LISO + LHEX + LT ET + LORT + LMON + LT RI If such decomposition can alleviate the analytical effort to obtain L−1 or simplify the resulting expressions is something still unproven. References [1]

Gelf'and, I., Graev, M.I. and Vilenkin, N.Y., Generalized functions vol.5, Academic Press (1966).

[2]

Mura, T. and Kinoshita, N., Green’s functions for anisotropic elasticity. Phys. Stat. Sol. (b) 47, 607-

Advances in Boundary Element and Meshless Techniques XII 618 (1971). [3] Barnett, D. M., The precise evaluation of derivatives of the anisotropic elastic Green’s functions. Phys. Stat. Sol. (b) 49, 741-748 (1972) [4]

Achenbach, J. D., Wave Propagation in Elastic Solids. Elsevier, New York (1973).

[5] Stakgold, I., Green’s functions and boundary value problems. Wiley-Interscience Publications, New York (1979). [6] (1989).

Cowin, S. C., Properties of the anisotropic elasticity tensor. Q. J. Mech. App. Math. 42, 249-266

[7]

Stroh, A. N., Dislocations and cracks in anisotropic elasticity. Phil. Mag. 8, 625-646 (1958).

[8]

Ting, T. C. T., Anisotropic Elasticity - Theory and Applications. Oxford University Press (1996).

[9] Liou, J. Y., Sung, J. C., On the Barnett-Lothe tensors for anisotropic elastic materials. European Journal of Mechanics A/Solids 27, 1140-1160 (2008). [10] Sutcliffe, S., Spectral decomposition of the elasticity tensor. ASME Journal of Applied Mechanics 59, 762-773 (1992). [11] Zuo, G. H., Schreyer, H. L. A note on pure-longitudinal and pure-shear waves in cubic crystals. J. Acoust. Soc. Am. 98, 580-583 (1995). [12] Zuo, Q. H. Upper bound on wave speeds in anisotropic materials based on elastic projection operators. Int. J. Theoretical and Applied Multiscale Mechanics 1, 16-29 (2009). [13] Mahnken, R., Anisotropic creep modeling based on elastic projection operators with applications to CMSX-4 superalloy. Comput. Methods Appl. Mech. Engrg. 191, 1611-1637 (2002). [14] Mahnken, R., Anisotropy in geometrically non-linear elasticity with generalized Seth-Hill strain tensors projected to invariant subspaces. Comm. Numer. Methods Eng. 21, 405-418 (2005). [15] Zhang, Y. Q., Lu, Y., Yu, M. H., Investigation of strain localization in elastoplastic materials with transversely isotropic elasticity. Int. J. Mech. Sci. 45, 217-233 (2003). [16] Suvorov, A. P., Dvorak, G. J., Rate form of the eshelby and hill tensors. Int. J. Solids & Structures 39, 5659-5678 (2002). [17] Masson, R. New explicit expressions of the hill polarization tensor for general anisotropic elastic rods. Int. J. Solids & Structures 45, 757-769 (2008). [18] Mehrabadi, M. M., Cowin, S. C., Horgan, C. O., Strain energy density bounds for linear anisotropic elastic materials. Journal of Elasticity 30, 191-196, (1993). [19]

Fedorov, F. I., Theory of Elastic Waves in Crystals. Plenum Press, New York (1968).

[20] Hlinka, J., Klotins, E., Application of elastostatic Green function tensor technique to electrostriction in cubic, hexagonal and orthorhombic crystals. J. Phys.: Condens. Matter 15, 5755-5764 (2003). [21] Nambu, S., Sagala, D. A. Domain formation and elastic long-range interaction in ferroelectric perovskites. Physical Review B 50, 5838-5849 (1994). [22] Boulanger, P., Hayes, M., Acoustic axes for elastic waves in crystals: Theory and applications. Proc. Royal Soc. London A 454, 2323-2346 (1998). [23] Fedorov, F. I., Fedorov, A. F. On the structure of the Green-Christoffel tensor. J. Phys. A: Math. Gen. 24, 71-78 (1991). [24] Norris, A. N., Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and poroelastic solids. Proc. Royal Soc. London A 447, 175-188 (1994). [25] Browaeys, J. T., Chevrot, S. Decomposition of the elastic tensor and geophysical applications. Geophys. J. Int. 159, 667-678 (2004).

65

66


Computation of the Effective Thermal Conductivity of Functionally-Graded Random Micro-Heterogeneous Materials Via the Fast Multipole BEM M. Dondero and A. Cisilino Welding and Fracture Division, INTEMA, Faculty of Engineering, University of Mar del Plata-CONICET. Av. Juan B. Justo 4302, Mar del Plata, Argentina, [email protected] Keywords: Effective thermal conductivity, random composites, Boundary Element Homogenization, Representative Volume Element, Functionally graded materials.

Method,

Abstract. This work introduces a numerical methodology for the computation of the effective thermal conductivity (ETC) of random micro-heterogeneous materials using representative volume elements (RVE) and the Fast Multipole Boundary Element Method (FMBEM). The methodology is applied to solve a foam-like microstructures consisting of a random distribution of circular isolated holes. The FMBEM solver together with a Genetic Algorithm (GA) optimization procedure is successfully used to design the microstructures for materials with functionally graded ETCs. Numerical results are experimentally validated.

Introduction Effective thermal conductivity (ETC) of micro-heterogeneous materials is an active research field as it has been for over a century. The importance of micro-heterogeneous materials like granular metal and ceramics, or polymeric open-cell foams lies in their applications in high performance insulation, packed beds, heterogeneous catalysts, composite materials and powder metallurgy. The size, shape, physical properties and spatial distribution of the micro-structural constituents largely determine the macroscopic, overall behaviour of these multi-phase materials. From the point of view of materials design, it would be highly attractive to tailor the material microstructure in order to obtain the desired set of macroscopic properties. One remarkable example of this concept can be found in the so-called Functionally Graded Materials (FGM), where particular spatial variations of local material properties can be used to generate materials with a set of unique properties. The computational modeling of the material microstructure together with homogenization techniques are widely used to predict the macroscopic behavior of random heterogeneous materials [1]. The homogenization of microstructures with randomly distributed components uses statistically representative volume elements (RVE) technique. In order to make the material simulated data reliable, the RVE sample must be small enough to be considered as a material point with respect to the size of the domain under analysis, but large enough to be a sample statistically representative for the microstructure. Thus, a RVE usually contains a large number of heterogeneities, and therefore the computations could be expensive. The aim of this work is predicting and verifying results for heat conduction in functionally-graded microheterogeneous materials. To this end, a methodology based on the numerical modeling of RVE using the Fast Multipole Boundary Element Method (FMBEM) and Genetic Algorithms (GA) is introduced. The capabilities of the proposed methodology are accessed via the analysis of a functionally graded material. The numerical predictions are compared to analytical and experimental results.

The Fast Multipole Boundary Element Method The high computational effort involved in this work motivated the utilization of the FMBEM. The FMBEM reduces the computational cost of the direct BEM, from an order of O(N3) to a quasi-linear, where N is the number of degrees of freedom of the system. This reduction is achieved by multilevel clustering of the boundary elements into cells, the use of the multipole series expansion for the evaluation of the fundamental solutions in the far field and the use of an efficient iterative solver. Additionally, the multipole algorithm leads to a matrix-free calculation scheme. The implementation follows that proposed by Liu and Nishimura [2]. The model boundary is discretized using constant elements (see Figure 1). The evaluations of the integrals are carried out analytically. The system of equations is solved by a preconditioned GMRES algorithm from the slatec public library available at netlib (http://www.netlib.org/). The parameters of the algorithm were set as follows: 12 expansion terms for the FMBEM, and 300 elements per cell. The tolerance for the GMRES convergence was set 10-7. Computations were


67

carried using the nodes of a Debian-based GNU/Linux diskless cluster consisting of eight Intel Pentium 4 CPUs with 2 gigabytes of RAM each. A detailed description of the implementation can be found in a previous work [3].

Representative volume element size determination In this study the material microstructure is assumed to be of the hole-matrix type in two dimensions. The holes are circular, isolated and randomly distributed. The steady state heat transfer is assumed governed by the Laplace equation. A representation of the microstructure is illustrated in Figure 1. In order to size the RVE, a series of FMBEM analysis were performed over sets of samples with void volume fractions, f , in the range 0dfd0.5. Boundary conditions for the samples were specified in order to induce a global one-dimensional heat flux q in the y-coordinate direction (see Figure 1). The normalized ETC of the micro heterogeneous material, D= K / k0, (where K is the ETC of the micro heterogeneous material and k0 is the thermal conductivity of the matrix) was computed as the ratio between the heat flux through a specimen containing holes and the heat flux through a geometrically similar specimen without the holes. The following number of holes per sample sequence was used to study the dependence of the effective responses on the sample size: 10, 30, 60, 100, 150, 200 and 300. Every computation was performed 20 times using models with a different random distribution of the holes. The resulting model discretizations ranged from 504 to 13024 elements. The solution of a model like the one illustrated in Figure 1 on a laptop with a Pentium4 @ 3Ghz processor and 1Gb of RAM using the direct BEM formulation took around 53 seconds, whereas the Fast Multipole formulation needed only 13 seconds. This speed difference justified the use of FMBEM for solving the 4 u 7 u 20 = 560 models for the RVE size determination tests. The mean and the standard deviation of the ETC results were calculated for each set of samples. The results are illustrated in Figure 2. It can be seen that the dispersion of the data diminishes as the number of holes per sample increases. Also, an increase in the data dispersion with the void fraction is observed at low number of holes per sample, nevertheless this dispersion is reduced as the number of holes increases. Justified by the somewhat ad-hoc fact that for two successive enlargements of the number of holes the responses differed from one another, on average, by less than 0.5%, the 200-hole samples were selected as RVE for further tests. {GGdG{Y

BEM node BEM element

xGdW

xGdW

s

{GGdG{X s

Figure 1: FMBEM model of a representative volume element with a void volume fraction f = 0.3 and 344 holes.

Optimization Using Genetic Algorithms Genetic Algorithms (GA) simulate the natural evolution; hence their components are the chromosomes, the genetic material that dictates unique properties of the individuals. A GA emulates the phenomena that take place during reproduction of species making use of the genetic operators. The latest are natural selection, pairing and mutation. Individuals live in an environment determined by the objective (or fitness) function,

68


where they compete for survival and only the best succeed. The GA code using in thi s work is based on PIKAIA, a self-contained, genetic-algorithm-based optimization subroutine developed by Charbonneau and Knapp [4].

f=0.1

0.8

D

0.6 f=0.3

0.4 f=0.45 f=0.5

0.2

0

50

100

150

200

250

300

Number of holes

Figure 2: Effective thermal conductivity, D= K / k0, as a function of the number of holes for a set of given void volume fractions, f. Error bars indicate the dispersion of the results computed using 20 different random distributions of the holes. T=T2

y

f1

zone 1

f2

zone 2 f3

Q=0

Q=0

… fi

zone n y

fm

x

f

T=T1

Figure 3: Domain division, boundary conditions and piecewise linear void fraction distribution of the RVE sample.

The GA is used to optimize the spatial distribution of the holes in the microstructure in order to obtain a given temperature distribution in the y-direction, T(y) (the objective function). The optimization problem is solved by dividing the model domain into n zones (parallel bands in Figure 3) of equal length with linear distribution of the void volume fraction. This approach results in a piecewise linear interpolation of the void volume fraction, f(y), which defined in terms of m=n+1 discrete fi values. The fi are selected as design variables for the GA and they are codified into a chromosome. The fitness of the individuals (the fitness function) is the deviation of its temperature field from the objective temperature field, T(y). This is assessed using a least-squares scheme for the differences between the FMBEM results and T(y) for a set of p internal points evenly distributed over the complete model domain: 2

p

¦ ª¬T ( y ) t j

fitness (individual i) =

j 1

p

j

º¼

(1)


69

where tj is the temperature solution at the jth internal point. In order to make the fitness value independent of the number of evaluation points the definition of the fitness function implies an average. The GA was implemented to run in parallel by incorporating MPI routines to PIKAIA. The algorithm uses a master-slave scheme: the master node is in charge of the management of the AG (creating and populating each generation) and the slave nodes are dedicated to the evaluation of the fitness of the individuals by solving the FMBEM models. The parallel version of the GA runs on Beowulf cluster mentioned before.

Example: Material with continuous variation in the thermal conductivity It is proposed to design a material with a thermal conductivity with a continuous variation in the y-direction. The variation is chosen to have Gaussian-like shape (see Figure 4) and it is written in terms of Padé polynomials as follows: k ( y) A (2) kmin k0 ( y P )2 V 2 where the constants were set kmin = 0.2805, A = 3 2 S , μ = L/2 = 30, σ = 20. It is worth noting that the maximum and minimum conductivity values correspond to the void volume fractions f=0 and f=0.5 respectively.

Normalized conductivity k(y)/k0

1.0

proposed calculated

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.0

7.5

15.0

22.5

30.0

37.5

45.0

52.5

60.0

Coordinate y [mm]

Figure 4: Proposed and obtained conductivity variations along the sample.

The problem boundary conditions are those depicted in Figure 1 with the prescribed temperatures T1=20°C and T2=100°C. The resulting objective temperature field is T ( y)

0.3133 1.989 y 16.41 tan 1 0.0857 y 2.5690 .

(3)

Figure 5 depicts the microstructure of one the best fitted individuals together with the contour plot for the temperature field and the optimal void volume fraction (fi = 0.477, 0.383, 0.319, 0.143, 0.059, 0.079, 0.295, 0.492, 0.292). This result was achieved after approximately 20 generations. Further generations resulted only in small fluctuations due to the random nature of the microstructure. In accordance with the maximum conductivity values, the minimum void volume fraction occurs in the central part of the sample. The void volume fraction results in Figure 5 were correlated to the ETCs results in Figure 2. The result is plotted in Figure 4. It can be seen that the calculated conductivity posses the same general trend of that used for the formulation of the problem, with the maximum value in the sample central zone.

70


0.0 0.1 0.2 0.3 0.4 0.5

Void fraction f(y)

(a)

(b)

Figure 5: (a) optimized microstructure with temperature-map, and (b) void fraction solution. Heat source

Thermocouple

Metallic plate

Heater

Sink

Thermocouple

Structure

Cooling system

Figure 6: Left: Experimental setup. Right: thermal image showing the temperature map of the specimen, the heat source (red) and the heat sink (blue).

Experimental validation A macroscopic sample of functionally-graded heat-conducting material for the solution of the problem presented in the previous section was constructed by machining circular holes in a highly-conductive 1100 AA-Grade aluminum plate. This design is intended to approach the theoretical case of a highly-conductive continuous phase with non-conductive inclusions. Figure 6 illustrates the experimental setup. Temperatures labeled as “T = T 2” and “T = T 1” were imposed by means of a large-capacity heat source and a heat sink. The heat source was powered by a 400W electric resistance, while the sink consisted in a water cooling system and an electric resistance used to control its temperature. A 1/2” thick layer of alumina wool was placed beneath the bottom face and along the sides of the specimens in order to minimize convectio n and radiation heat losses. Temperature maps were measured via thermal images acquired with an infrared thermographic camera (Fluke Ti-30). To this end, the top face of the plate was painted matte black to maximize and homogenize the infrared emittance. Figure 6 shows typical infrared picture. Figure 7 depicts the experimental (gray-filled areas), FMBEM and analytical results along the sample (ydirection). Error bars for the FMBEM results indicate the dispersion of the temperatures in the x-direction. The analytical solutions (with and without heat loss) were computed using the one-dimensional Laplace equation with the thermal conductivity as a function of the position. The heat-loss solution accounts for an estimation of the convection heat transfer to the air over the specimen surface. The function relating the


71

thermal conductivity with the void volume fraction was obtained after a polynomial fit ting of the results in Figure 1 [5]. The agreement between the four sets of results is found very good. 140

120

T / qC

100

80

60

Experimental FMBEM simulation Analytical Analytical w/heat loss

40 40

60

80

100 120 y / mm

140

160

180

Figure 7: Temperature distribution along the plate for the Material B.

Conclusions It has been presented in this work an efficient numerical tool for the design of foam-like microstructures with functional-graded thermal conductivity. The devised methodology is based on a parallel Genetic Algorithm as optimization method with a Fast Multipole (FMBEM) formulation for the evaluation of the fitness function using representative volume elements. The FMBEM is specially suited for the optimization method: the boundary-only discretization strategy makes the model data generation a simple task, while the fast multipole formulation results in important savings in computing time when compared to direct BEM. The performance of the proposed methodology was demonstrated for a material with continuous variation in the thermal conductivity. Very good agreement was achieved between the computed results, analytical solutions and experimental results.

References [1] T.I. Zohdi and P. Wriggers, “Introduction to Computational Micromechanics, Lecture Notes in Applied and Computational Mechanics”, Volume 20, First edition, Springer-Verlag, Berlin, 2005, cc. 1, 4, 9. [2] Y.L. Liu and N. Nishimura, “The Fast Multipole Boundary Element Method for Potential Problems: A Tutorial, Engineering Analysis with Boundary Elements 30 (5) (2006) 371-381. [3] M. Dondero, A. Rodriguez Carranza, A. P. Cisilino, G. Stavroulakis, “Fast multipole BEM and Genetic Algorithms for the design of foams with functional-graded thermal conductivity” in Recent Advances in Boundary Element Methods, Springer, Berlin, 2009. [4] Charbonneau P. and Knapp B. "PIKAIA: Optimization (maximization) of user -supplied 'fitness' function ff over n-dimensional parameter space x using a basic genetic algorithm method." National Center for Atmospheric Research, Boulder CO 80307-3000, US. (http://download.hao.ucar.edu/archive/pikaia) [5] M. Dondero, A.P. Cisilino, J.M. Carella and J.P. Tomba J.P., “Effective thermal conductivity of functionally-graded random micro-heterogeneous materials using representative volume element and BEM ”, to appear in the International Journal of Heat and Mass Transfer.

72


Shape Sensitivity Analysis of 3D Acoustic Problems Based on BEM and Its Application to Topology Optimization

T. Matsumoto1,a , T. Yamada1,b , T. Takahashi1,c , C. J. Zheng2,d , S. Harada1,e 1 Department

of Mechanical Science and Engineering, Nagoya University Furo-cho, Chikusa-ku, Nagoya City, 464-8603, Japan

2 Department

of Modern Mechanics, University of Science and Technology of China Hefei, China

a [email protected], b [email protected], c [email protected] d [email protected], e s

[email protected]

Keywords: Acoustics sensitivity analysis, Adjoint method, 3-D problem, Topology optimization, Level-set

method Abstract This paper presents a design sensitivity formulation for acoustic problems based on the adjoint method and the boundary element method, and its application to topology optimization of acoustic field. The objective function is assumed to consists only of boundary integrals and quantities defined at certain number of discrete points. The adjoint field is defined so that the sensitivity of the objective function does not include the unknown sensitivity coefficients of the sound pressures and particle velocities on the boundary and in the domain. Since the final sensitivity expression does not have the sensitivity coefficients of the sound pressure and particle velocity on the boundary, BEM analyses only for the primary acoustic field and the adjoint field are needed to calculate the sensitivity of the objective function. The derived formulation is applied to topology optimization of a sound scatterer placed in an infinite space. The level-set method is utilized in the iterative process of obtaining the optimum shape of the scatterer. Introduction Due to the development of fast computation algorithms[1], BEM can be considered as a strong analysis tool for shape optimization problems that requires re-meshing in the shape modification process. Shape optimization problems are usually solved by minimizing an objective function. In order to calculate the objective function and its sensitivities with respect to design variables, we can use BEM based on either the direct-differentiation method[2] or the adjoint variable method[4]. Direct-differentiation method uses a boundary integral equation for the sensitivity coefficients of the boundary quantities. Hence, when the number of design variables are large, we have to repeat calculations of the sensitivities of the boundary quantities for all the design variables. On the other hand, adjoint method defines a system that can eliminate the unknown sensitivities of the quantities on the boundary and in the domain, we have to solve only the original problem and the adjoint problem when calculating the sensitivities of the objective function. In this paper, an adjoint method approach is shown for shape and topology optimization of acoustic field. The objective function is assumed to be defined with the sound pressure and the particle velocity on the boundary and with quantities at a finite number of internal points. The boundary element method is used for the analysis of the original acoustic problem and the corresponding adjoint problem. The adjoint problem and the sensitivity expression are derived for a typical form of objective function that is appropriate for BEM. Some numerical examples are shown to demonstrate the effectiveness of the present approach. Boundary integral equations The governing differential equation for the propagation of time-harmonic acoustic waves in a homogeneous and isotropic acoustic medium is the following Helmholtz equation: r 2 p.x/ C k 2 p.x/ D 0

(1)


73

where p.x/ is the sound pressure at point x, r 2 is the Laplace operator, and k D 2f =C is the wavenumber with is the frequency f and sound speed C . The boundary conditions are p.x/ D p.x/; N on p @p q.x/ D D i!v D q.x/; N @n

(2) on q

(3)

where n denotes the outward normal direction, i the imaginary unit, ! the circular frequency, the density of the medium, and v the particle velocity. An overscribed bar indicates that the value is given on the boundary. The integral representation of the solution to the Helmholtz equation is Z Z p.x/ C q .x; y/p.y/ d .y/ D p .x; y/q.y/ d .y/ (4)

where x is the collocation point, y is the source point, and p .x; y/ is the fundamental solution, for 2-D acoustic problems it is given as i .2/ p .x; y/ D H0 .kr/ 4

with r D jx yj

(5)

and for 3-D, p .x; y/ D

e ikr 4 r

(6)

.2/

where H0 is the Hankel function of the second kind of order 0. Also, q .x; y/ is the normal derivative of p .x; y/. The boundary integral equation is obtained by taking the limit of point x to the boundary , as follows: Z Z C.x/p.x/ C q .x; y/p.y/ d .y/ D p .x; y/q.y/ d .y/ (7)

where C.x/ is a constantR that becomes 1/2 when x lies on a smooth part of the boundary. The integral symbol denotes that the integral is evaluated in the sense of Cauchy’s principal value. Also, an additional boundary integral equation is combined with Eq.(7) for exterior problems to avoid computation errors occurring at fictitious eigen-frequencies. It is obtained as the normal derivative of Eq.(7) at x, as follows: Z Z C.x/q.x/ C D qQ .x; y/p.y/ d .y/ D pQ .x; y/q.y/ d .y/ (8)

R where .Q/ D @. /[email protected]/, and the integral symbol D denotes that the integral is evaluated in the sense of finite part of divergent integral. For exterior acoustic problems, a linear combination of Eqs.(7) and (8) is used [3]. Sensitivity analysis The objective functions of shape and topology optimization problems suitable for BEM acoustic analyses can be written in the following form: Z XZ I D f .p; q; N q; q/ N d C g.p; p; N vi ; vN i / ı.x z s / d ; z s 2 (9)

s

where f is a function defined with boundary sound pressure and its normal derivative, and g is a function defined with internal sound pressure and particle velocity components vi , .i D 1; 2; 3/. The overscribed bar . N / denotes a complex conjugate, and ı.x z s / is Dirac’s delta function where z s is a point in the domain. Then, the sensitivity of I with respect to an arbitrary shape parameter becomes Z Z @f @f @f @f pPN C qPN d C f dP I0 D pP C qP C @pN @q @qN @p

74

Eds: E L Albuquerque & M H Aliabadi X Z @g @g @g @g vPN i ı.x z s / d vP i C pP C pPN C @pN @vi @vNi @p s XZ XZ s P z / d C C g ı.x g ı.x z s / dP C

s

(10)

s

where a dot . P / denotes material derivative. We now consider an augmented objective function J with an equality constraints as follows: J DI CR and

Z RD

(11)

2 N r p C k 2 p C r 2 pN C k 2 pN d

(12)

where is a Lagrange multiplier. We can use J as the objective function instead of I because p is the solution of Eq.(1) and R is identically 0. Integrating by parts for R, we have the following weak form. Z Z Z Z Z Z N ;i ni d C N d C RD p N ;i p;i d pN;i ni d ;i pN;i d C k 2 p k 2 pN d (13)

where index i D 1; 2; 3 denotes the cartesian components of the corresponding vector and the index after a comma denotes the differentiation with respect to the coordinate. Also, Einstein’s summation convention is assumed for repeated indices. After some lengthy manipulation, we obtain the expression of the sensitivity of J , as follows: Z Z @f @f @f @N @ @f pP C pPN d C C C N qP C qPN d J0 D @p @n @pN @n @q @qN Z Z N i xP i d C N ;i q C N ;j p;j ni C p C N ;i C k 2 pn N i xP i d ;i qN C ;j pN;j ni C pN;i C k 2 pn ) Z ( Z X @g @g N C i!v/ ı.x z s / pP d N ;i i C k 2 N C N .dP/ C C .f C i! v @p @v ;i s ) Z ( X @g @g ;i i C k 2 C ı.x z s / pPN d C @pN @vN ;i s Z X Z N ;i xP i C .;jj C k 2 /pN;i xP i d g;i ı.x z s /xP i d (14) .N ;jj C k 2 /p

s

where

@ @n

(15)

and the following relationships are used in the derivation. .pP;i / D pP;i p;m xP m;i P D m xP m .P;i / D ;im xP m N N .;j p;i xPj /;i D ;ij p;i xPj C N ;j p;i i xPj C N ;j p;i xPj;i J0

(16) (17) (18) (19)

In the above expression, contains the unknown sensitivity coefficients of the sound pressure and its normal derivative on q and p , respectively, and the unknown sensitivity of the sound pressure in . In order to eliminate these quantities, we use the solution of the following adjoint problems: X @g @g ı.x z s / D 0; x 2 ;i i .x/ C k 2 .x/ C (20) @pN @vN ;i s

@f ; x 2 p @qN @f @ .x/ D ; .x/ D @n @pN

.x/ D

(21) x 2 q

(22)

By using the solutions of the adjoint problems for , J 0 can be simplified up to the following expression: Z Z @g @N @ @g @g N @g pPN d C pP C J0 D C C qP C qPN d @p @n @pN @n @q @qN q Z Z p 2N N N ;i q C ;j p;j ni C p N ;i C k pni xP i d C N i xP i d C ;i qN C ;j pN;j ni C pN;i C k 2 pn Z N C i!v/ C .g C i! v N .dP/ ( ) X Z @h @h @h @h p;m xP m C pN;m xP m h;i xP i ı.x z s / d C (23) @p @q ;i @pN @qN ;i s Topology optimization We use a level-set method approach[5] for controlling the shape and topology of the domain. Level-set function is a scalar function of the point in the domain. In order to obtain an optimum topology, a fixed design domain, in which the optimum domain is included, is usually defined. The level-set function .x/ is defined in the fixed-design space and takes the value as follows: .x/ > 1;

8x 2 n @

.x/ D 0; 8x 2 @ .x/ < 0;

8x 2 D n

(24) (25) (26)

By considering this level-set function as the design variable, we can control the shape and topology of the domain. In the level-set approach in [5], the objective function is again augmented by adding a regularization term, as follows: Z F DJ C jrj2 d (27) D

The variation of the level set function with respect to fictitious time t is assumed to be proportional to the gradient of the objective function, i.e., @ dF dJ (28) D K./ D K./ r 2 @t d d Once dJ =d is obtained by BEM, the distribution of can be obtained by solving the above Eq.(28) by using FEM for a fixed design domain for which the level-set function is defined [5]. The fixed design domain is usually of simple geometry like a rectangular solid domain, and can be discretized with simple structured mesh. Therefore, FEM analysis for the fixed design domain is very simple and can be done efficiently. Numerical examples 2D example. We consider a rectangular region as shown in Fig.1. We define the following objective function to make the sound pressure at the center of the cavity close to a certain value p0 . Z 1 (29) J D jp.x; y/ p0 j2 ı.x 2:5/ ı.y 0:5/ d 2 The design variable is assumed to be the width L of the rectangular cavity. We divided the boundary of the rectangular cavity into 120 quadratic elements uniformly. The exact solution of the sound pressure is given as p.x/ D tan.kL/ sin.kx/ C cos.kx/

(30)

76


Fig. 1

A rectangular cavity model.

where k is the wave number. We assume that the target sound pressure as p0 D 0. Then, the sensitivity of J with respect to L becomes k sin.kx/ dJ D tan.kL/ sin.kx/ C cos.kx/ dL cos2 .kL/

(31)

The governing equation and the boundary condition of the adjoint problem becomes as follows: ;i i .x; y/ C k 2 .x; y/ C .p.x; y/ p0 /ı.x 2:5/ı.y 0:5/ D 0;

.x; y/ 2

D 0 on p @ D D 0 on q @n

(32) (33) (34)

We show in Fig.2 the distribution of the adjoint solution obtained by BEM and in Table 1 the sensitivity values and their errors obtained for various discretizations. The sensitivity errors are found to decrease in accordance with the increase of the number of elements and become accurate.

y

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

x

Fig. 2

Distribution of the obtained adjoint variable .

Table 1 Obtained sensitivity values and their errors. Number of elements Sensitivity Error [%] 48 1:88909 101 5.677 120 1:79874 101 0.623 240 1:78997 101 0.132 480 1:78841 101 0.045 960 1:78814 101 0.030

0.060 0.054 0.048 0.042 0.036 0.030 0.024 0.018 0.012 0.006 0.000

Advances in Boundary Element and Meshless Techniques XII 3D example for topology optimization. Next we consider a sound scatterer to reduce the sound pressure at an observation point shown in Fig.3. Fast-multipole BEM is used to calculate the original acoustic problem and the adjoint problem. We show in Fig.4 the obtained geometry of the scatterer for 340Hz. We find that the present approach can be applied to such a practical problem with three-dimensional complicated geometry. 5m 1m

1m 1m

1m Observation point

Sound source Fixed design domain D

Fig. 3

A sound scatter model to reduce the sound pressure at the observation point.

Fig. 4

Obtained scatter geometry.

Concluding remarks An adjoint method approach based on BEM has been shown for shape and topology optimization of acoustic field. Because the BEM is based on boundary only discretization, the objective function is assumed to be a functional only with boundary sound pressure and particle velocity, and with quantities only at a finite number of internal points. An adjoint system is defined so that the unknown sensitivities of the sound pressure and particle velocity on the boundary and unknown quantities in the domain are eliminated from the gradient of the objective function. On each calculation step of the gradient of the objective function, BEM calculations are repeated only for the original problem and the adjoint problem. Some numerical examples have been provided, and have demonstrated the effectiveness of the present approach. References [1] V.Rokhlin J. Comput. Phys., 60, 187–207, (1985). [2] T.Matsumoto, M.Tanaka, Y.Yamada JSME International Journal, Ser.C, 38(1), 9–16 (1995). [3] A.J.Burton, G.F.Miller Proc. Roy. Soc. Lond. A.323, 201–210 (1971). [4] E.J.Haug, K.K.Choi, V.Komkov Design Sensitivity Analysis of Structural Systems, Academic Press, (1986). [5] T.Yamada, K.Izui, S.Nishiwaki, A.Takezawa Comput. Methods Appl. Mech. Engrg., 199, 2876–2891 (2010).

77

78


Computation of displacements in anisotropic plates by the boundary element method A. Reisa , E. L. Albuquerquea J. F. Usecheb and H. Alvarezb a University of Bras´ılia - UnB Faculty of Mechanical Engineering Bras´ılia, DF, Brazil

[email protected] [email protected] b University Technologic of Bol´ıvar Faculty of Mechanical Engineering Cartagena, Colombia

[email protected] [email protected]

Keywords: Mindlin theory, thick plates, Radon transform, anisotropic plates.

Abstract. This paper presents a boundary element formulation for the computation of displacements at internal points of laminated composite thick plates. Fundamental solutions for anisotropic thick plates are obtained using Hrmander operator and Radon transform. So, they do not have a closed form and numerical integration is necessary to compute fundamental solutions in each field point. Integral equations for moments are developed and all derivatives of the fundamental solution are computed. A special integration technique is used in order to improve performance of the method. The obtained results are in good agreement with literature. Introduction. It is well known from literature that the use of the classical Kirchhoff plate model is inappropriate for composite plates. Unlike the Kirchhoff theory of thin plates, Reissner theory takes into account the transverse shear deformation, which is important in many practical applications. The sixth-order plate problem, was formulated in a boundary integral equation method form for the Reissner model by [11], who employed the Hörmander method for the derivation of the fundamental solution. An integral formulation for the Mindlin model was developed by [2], whereas [13] presented a unified integral formulation for both models. After the original works of [11], many references have reported the application of boundary elements to bending analysis of thick plates, most of them using the Reissner model as, for example, [6], [5], [14], and [7]. As we can see, a large number of articles with the analysis of isotropic plates can be found in literature. However, only few works can be found with the analysis of orthotropic plates. [4] presented a boundary element method of moderately thick orthotropic plates. In [12] the previous formulation was extended to laminated composites. This work presents a boundary element formulation for anisotropic thick plates. It uses the fundamental solution proposed by [12] that takes into account the effects of shear deformation and was derived by means of Hörmander operator and the Radon transform. Domain integrals which come from transversal applied loads are exactly transformed into boundary integrals by a radial integration technique. Some numerical examples concerning orthotropic plate bending problems are analyzed with the BEM.


1

79

EQUATIONS OF EQUILIBRIUM

Consider a plate constructed of a finite number of homogeneous, uniform-thickness layers of an orthotropic material. Relations between the generalized displacement and strains are:

κ1 = ∂ ψx ∂y

κ6 = ε5 =

∂ω ∂x

∂ ψx ∂x ,

+

∂ ψy ∂x ,

κ2 = ε4 =

∂ω ∂y

∂ ψy ∂y ,

+ ψy , (1)

+ ψx ,

in which ω is the displacement along the z-direction, and ψx and ψy are the rotation of the plates along the x and y co-ordinate axes, respectively. The relations between the stress resultants and strains are: Mi = Di j κ j ,

i, j = 1, 2, 6,

hk

i, j = 1, 2, 6,

Q1 = A45 ε4 + A55 ε5 ,

Q2 = A44 ε4 + A45 ε5 ,

Di j = ∑Nk=1

Ai j = ∑Nk=1

(k) 2 hk−1 Qi j z dz

hk

(k) hk−1 Qi j Ki K j dz,

(2)

i, j = 4, 5,

in which hk is the vertical distance from the midplane, z = 0, to the upper surface of the kth lamina. K4 and K5 are the shear correction factors. Qkij are the plane stress reduced stiffness coefficients of the kth lamina. The equilibrium equations of the plates are:

∂ M1 ∂ M6 + − Q1 + mx = 0, ∂x ∂y ∂ M6 ∂ M2 + − Q2 + my = 0, ∂x ∂y ∂ Q1 ∂ Q2 + + q = 0, ∂x ∂y

(3)

Substituting Eq.(1) and Eq.(2) into Eq.(3), we obtain the following differential equations using the generalised displacement as basic unknowns: Li jU j + qi = 0, i, j = 1, 2, 3 where U j indicate ψx , ψy and ω , qi represent the generalized loads, i.e.qi indicate mx , my and q, respectively, Li j are the differential operators.

(4)

80


2 FUNDAMENTAL SOLUTIONS The fundamental solutions of the symmetric laminated anisotropic thick plate taking into account the transverse shear deformation are a set of particular solutions of the differential Eq. (4) under a unit concentrated load, i.e., the solutions satisfy the following inhomogeneous differential equations: Li∗jUk∗j (ζ , x) = −δ (ζ , x)δki ,

(5)

in which δ (ζ , x) denotes the Dirac delta function, ζ represents the source point, x is a field point, and Uk∗j represents the generalized displacement in the jth direction of the field point x of an infinite plane when a unit point load is applied in the kth direction of the source point ζ , (see [12]). Using the plane wave decomposition method, [3], we can transform Eq. (5) into a set of ordinary differential equations, δ (ζ , x) and Uk∗j (ζ , x) may be expanded into a plane wave:

δ (ζ , x) = −

1 4π 2

2π 0

| ω1 (x − ζ ) + ω2 (y − η ) |−2 d θ ,

Uk∗j (ζ , x) =

2π 0

U˜ k∗j (ρ )d θ ,

(6)

(7)

where (ω1 , ω2 ) are the co-ordinates of a point on the unit circle, i.e., ω1 = cos(θ ), ω2 = sin(θ ), (x, y) and (ζ , η ) are the co-ordinates of the field point and the source point, respectively. Substituting Eq.(6) and Eq.(7) into Eq.(5), and considering the differential relationship ∂∂xα = ωα ddρ , we obtain a set of ordinary differential equations as follows: L˜ i∗jU˜ k∗j (ρ ) =

1 4π 2

| ω1 (x − ζ ) + ω2 (y − η ) |2 δki , i, j, k = 1, 2, 3

(8)

Following Hörmander’s operator method, the solutions of Eq. (4) can be written as: j U˜ k∗j (ρ ) =co L˜ ad jk φ (ρ ),

(9)

where φ (ρ ) is a unknown scalar function and co L˜ ∗ is the cofactor matrix of the operator L˜ ∗ . Thus, the fundamental solutions of symmetrical laminated anisotropic plates including transverse shear deformation can be written as: ∗ U˜ αβ (ρ ) = aαβ D4 φ (ρ ) −Cωα ωβ D2 φ (ρ ), ∗ ˜ Uα 3 (ρ ) = −U˜ 3∗α (ρ ) = fα D3 φ (ρ ) −Cωα Dφ (ρ ), ∗ U˜ 33 (ρ ) = AD4 φ (ρ ) − BD2 φ (ρ ) +Cφ (ρ ),

(10)

where D4 φ (ρ ) = d k φ (ρ )/d ρ k , (k = 1, 2, ...) Substituting Eq. (9) into Eq. (8), we obtain the following equation: d4 dρ 4

d2 1 − p2 ϕ (ρ ) = 2 2 | ρ |−2 , 2 dρ 4π a

in which a = A11 d11 + a12 d12 , b = a11 A55 +Cd11 ω12 + a12 A45 +Cd12 ω1 ω2 + f1 d13 ,

(11)


81

p2 = b/a. The solution of Eq.(11) can be written as follows:

φ (ρ ) =

1

p2 ρ 2 log | ρ | +2 log | ρ | +3 + exp(pρ ) ρ exp(pσ ) − exp(−pρ ) dσ . σ −∞

∞ exp(−pσ )

8π 2 p4 a

ρ

σ

dσ + (12)

In the numerical calculation of the fundamental solutions, we have to deal with the following integrals: I1 =

2π 0

F1 (θ )Dk φ (ρ )d θ ,

k = 1, 2, 3, 4, 5.

(13)

where F1 (θ ) is a function depending only on θ . In the range of 0 e 2π there are two points which make ρ = 0. We first determine a value θ0 which makes ρ = 0 and split (0, 2π ) into four intervals. As the integrand is a periodic function, four intervals are: [θ0 , θ0 + π /2], [θ0 + π /2, θ0 + π ], [θ0 + π , θ0 + 3π /2], and [θ0 + 3π /2, θ0 + 2π ]. A Gaussian formula is used to evaluate the integral I1 for each interval. The value θ0 is determined by: x−ζ . θ0 = arctan − y−η

(14)

Details of the implementation the Equations (7), (11)and (12)can be found in [4] and [12].

3 BOUNDARY INTEGRAL EQUATIONS The integral equation can be derived by considering the integral representation of the governing Eq.(3) via the following integral identity: Ω

[(Mαβ ,β − Qα )Uα∗ + (Qα ,α + q)U3∗ ]dΩ = 0,

(15)

where Ui∗ (i = α , 3) are the weighting functions. Integrating by parts (applying Green’s second identity) and making use of the algebraic relationships, it gives: U j (ζ ) +

Γ

Pi∗j (ζ , x)U j (x)dΓ =

Γ

Ui∗j (ζ , x)Pj (x)dΓ +

Ω

q(x)Ui3∗ (ζ , x)dΩ.

(16)

By taking the point ζ to the boundary at the position ζ ∈ Γ, Eq. (16) can be written as:

ci j (ζ )U j (ζ ) + − Pi∗j (ζ , x)U j (x)dΓ =

Γ

Γ

Ui∗j (ζ , x)Pj (x)dΓ +

Ω

q(x)Ui3∗ (ζ , x)dΩ,

(17)

where− denotes a Cauchy Principal Value integral, ζ , x ∈ Γ are source point and field point, respectively. The value of ci j (x) is equal to δi j /2 when x is located on a smooth boundary. Eq.(17) represents three integral equations, two (i = α = 1, 2) for rotations and one (i = 3) for deflection. The last integral on the right hand side of equation (17), that is a domain integral, is transformed into a boundary integral using the procedures presented by [1].

82

4


NUMERICAL RESULTS

To validate the procedures implemented and to assess the accuracy of solutions, consider a clamped circular plate subjected to a uniform distributed load for the different values of Ex /Ey and h. The material constants are Gxy = Gxz == 0.6Ey GPa, Gyz = 0.5Ey GPa, νLT = 0.25. The laminate is discretized into 16 constant elements per side. Results are compared with [4], as shown in Table (1). As it can be seen, there is a good agreement with literature. Table 1: Center deflection of the orthotropic thick circular plate of clamped boundary.

Ex /Ey 3 10

a/(h/2) w(ref.[4]) ˆ 10 103,78 5 16,32 10 51,15 5 9,99

w(MEC) ˆ 106,48 15,90 52,56 9,83

Error[%] 2,60 2,55 2,77 1,54

Conclusions. This paper presented a boundary element formulation for anisotropic thick plates. The fundamental solution was derived by means of Hormander operator and the Radon transform. The domain integrals are transformed into boundary integrals by a radial integration technique. The agreement obtained by the proposed method and results from literature.

References [1] 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. [2] C. S. De Barcellos, and L. H. M. Silva. A boundary element formulation for the Mindlins plate model. In: C. A. Brebbia, W. S. Venturini, editors. Boundary element techniques: applications in stress analysis and heat transfer. Southampton: CMP; p. 122–130, 1989. [3] I. M. Gel’fand and G. E. Shilov. Generalized Functions, Vol.1, Academic Press, New York, 1967. [4] J. Wang and M. Huang. Boundary element method for orthotropic thick plates. Acta Mechanica Sinica, 7:258–266, 1991. [5] J. T. Katsikadelis and A. J. Yotis. A new boundary element solution of thick plates modeled by Reissner’s theory. Engineering Analysis with Boundary Elements, 12:65–74, 1993. [6] S. Y. Long, C. A. Brebbia, and J. C. F. Telles. Boundary element bending analysis of moderately thick plates. Engineering Analysis with Boundary Elements, 5:64–74, 1988. [7] Y. F. Rashed, M. H. Aliabadi, and C. A. Brebbia. On the evaluation of the stresses in the BEM for Reissner plate-bending problems. Applied Mathematical Modelling, 21:155-163, 1997. [8] E. Reissner. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics, 12:69-77, 1945.


[9] J. Wang and K. Schweizerhof. A boundary integral equation formulation for moderately thick laminated orthotropic shallow shells. Computers and Structures, 58:277–287, 1996. [10] J. C. F. Telles. A self adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. International Journal for numerical Methods in Engineering, 24:959-973, 1987. [11] V. F. Weeën. Application of the boundary integral equation method to Reissner’s plate model. International Journal for Numerical Methods in Engineering, 18:1–10, 1982. [12] J. Wang and K. Schweizerhof. Fundamental solutions and boundary integral equations of moderately thick symmetrically laminated anisotropic plates. Communications in Numerical Methods in Engineering, 12:383–394, 1996. [13] T. J. Westphal and C. S. De Barcellos. Applications of the boundary element method to Reissner’s and Mindlins plate model. In: M. Tanaka, C. A. Brebbia, T. Honma, editors. Proceedings of the 12h international conference on boundary element technology. Computational Mechanics Publications. Southampton, 1:467–477, 1990. [14] X. Yan. A new BEM approach for Reissner’s plate bending. Computers and Structures, 54:1085–1090, 1995.

83

84


Numerical Analysis of Failure in Laminate Composites Using Phenomenological Based Criteria D. I. G. Costa1 , E. L. Albuquerque2 , A. Reis2 , G. Panosso1 and P. Sollero1 1 State

2

University of Campinas - Mechanical Engineering Faculty, 13083-970, Campinas, SP, Brazil {dalmodj,gpanoso,sollero,adriana}@fem.unicamp.br

Brasilia’s University - Darcy Ribeiro Campus, North wing - Brasilia - Brasil [email protected]

Keywords: Composite Materials, Boundary Element Method, Failure Criteria.

Abstract. This paper presents a numerical analysis of laminated composite material failure under the action of quasi-static loads applied in the plane of the laminate by means of phenomenological based criteria. These criteria can not only forseen if the composite will failure but also predict the mode of failure. The stress in each laminae is computed using the boundary element method for plane elasticity. LaRC03 and Puck failure criteria are used to evaluate if there is some damage in any point of the laminate. Failure modes are observed and compared. Introduction. The use of the boundary element method has increased in many different areas. This method has some intrinsic characteristics that make it attractive in comparison to other numerical technics. For example, discretization is easier because only the boundary must be discretized. One of the disadvantages is the difficulty to find fundamental solutions in anisotropic problems, the case of most composite materials. Despite this feature, several formulations were developed for this type of material, for the case of plane elasticity it could be mentioned [14, 15], [10], [2, 3] and [4, 5]. After some stress-strain analysis, it is required to define a criterion that state if the material will fail or not. In composite material context, most of the failure criteria up to now do not indicate satisfactorily whether the material will fail, mainly in the compression domain of the matrix or fiber. Failure criteria like TsaiWu [16] , based on a polynomial interaction of the unidirectional strengths of the material, state the possibility of failure but do not give any information about the failure modes. Failure criteria for fiber composites, according to Hashin [11], should distinguish among the various different failure modes of the composite and model each of them separately. This identification is a required feature if the criterion intends to be useful for progressive damage analysis, therefore failure criterion developed in basis of phenomenological models are preferable, for example LaRC03 [8], Puck [12, 13] and Hashin’s criteria. Boundary integral equations for membranes. Using the Green’s second identity [7], the equilibrium equation can be transformed in to an integral equation given by:

c jk u j (P) +

S

T jk (P, Q)u j (Q) ds(Q) =

S

U jk (P, Q)t j (Q) ds(Q),

(1)

where u j is the displacement vector, t j is the traction vector given by: ti = Ni j n j ,

(2)

n j is a unity vector that is normal to the boundary in field point P, Ui j and Ti j are the displacement and traction fundamental solutions for an anisotropic body under plane stress. P(x1 , x2 ) e Q(x1 , x2 ) are field and load points, respectively. The c jk constant accounts for the position of load point.


85

Failure Criteria. In this work, two phenomenological based failure criteria are used to evaluate if there is some damage in any point of the laminate, Puck and LaRC03 criteria. Puck’s criteria can be divided in a fiber and a interfiber failure criteria. For fiber failure, the criterion state the following conditions: ν f 12 1 mσ f σ22 = 1 ε1 + (3) ε1T Ef1 ν f 12 1 mσ f σ22 + (10γ21 )2 = 1, ε1 + ε1C Ef1

and

(4)

where ε1T and ε1C are the longitudinal fracture strains, for tension and compression, respectively. mσ f accounts for a “stress magnification effect” and assume the value of mσ f = 1.3 for glass fiber and ν

mσ f = 1.1 for carbon fiber. Eq (3) is used when ε1 + Eff121 mσ f σ22 has a positive value, otherwise, if has a negative value, eq (4) is used. For interfiber failure, the criteria is based on Mohr-Coulomb theory and states that the fracture is created exclusively by stresses which act on the fracture plane. There are three different failure modes, called mode A, mode B and mode C. The mode A and B indicates a 0◦ fracture plane. The mode C presents different fracture plane angles that increases from 0◦ to ≈ 51◦ . For mode A, eq (5) is used and is valid for σ22 ≥ 0:

τ21 S21

2

2 σ22 2 (+) YT (+) σ22 + 1 − p⊥ + p⊥ = 1. S21 YT S21

For mode B, the condition is given by:

2 1 (−) (−) 2 τ21 + p⊥ σ22 + p⊥ σ22 = 1, S21 that is valid for σ22 < 0 and 0 ≤ στ2122 ≤

RA⊥⊥ |τ21c | .

(6)

For mode C, the condition is given by:

⎤ ⎡⎛ ⎞2 2 σ22 ⎥ YC ⎢⎝ τ21 ⎠ + = 1, ⎦ ⎣ (−) YC (−σ22 ) 2 1 + p⊥⊥ S21 valid for σ22 < 0 and 0 ≤ στ2122 ≤ RA⊥⊥

|τ21c | . RA⊥⊥

(5)

(7)

Where p are inclination parameters, obtained experimentally.

is the fracture resistance of the fracture plane, calculated from the results of an uniaxial transverse compression test. LaRC03 criterion is based in fracture plane concepts developed by [12]. This fracture plane angle is calculated using Mohr-Coloumb effective stresses. The criteria is a set of six failure indexes, three for fiber failure and three for matrix failure. For matrix cracking, the failure indexes are: a) Matrix under tensile load: FIM = (1 − g)

σ22 YisT

+g

σ22 YisT

2 +

τ12 SisL

2 ,

(8)

where g is a ratio of fracture toughnesses in mode I and II. The “is” subscript indicates local properties that are affected by the position of the lamina in the laminate. b) Matrix under compression load: FIM =

mT τeff ST

2 +

mL τeff Sis

2 ,

(9)

86


for σ11 < Y C and FIM =

T τeff ST

2 +

L τeff Sis

2 ,

(10)

T and τ L are the effective stresses acting on fracture plane. The angle of this for σ11 ≥ Y C , where τeff eff fracture plane for a generic stress state is calculated by searching the one which maximizes the failure index in eq (10), within the interval 0 < α < α0 . α0 is the angle of fracture for uniaxial compression. For fiber failure, the failure indexes are: a) Fiber under tensile load:

FIF =

σ11 , XT

(11)

b) Fiber under compressive load: FIF =

m |τ12 |m + η L σ22 L Sis

,

(12)

m < 0 and for σ22

FIF = (1 − g)

m σ22 YisT

+g

m σ22 YisT

2 +

m τ12 SisL

2 ,

(13)

m ≥ 0, where σ m and τ m are stresses in the misaligned coordinates of kinked fibers. The anfor σ22 22 12 gle of this kinked region is a function of both, applied stresses and misalignment angle of pure fiber compression.

Numerical results. In order to compare both failure criteria, consider a plate consisting of symmetrical laminate with four laminae with the lay-up [0/90]S with thickness t = 0, 002 m each, with Boron fibers in an Epoxy matrix. The laminate is subjected to a load in x direction with N = 10000 kN. The boundary is discretized in 36 quadratic elements, 28 in the external boundary and 8 in the central hole. The mechanical and strength properties are given in Table 1. Table 1: Mechanical and strength properties of Boron-Epoxy material E1 E2 G12 ν12 XT YT XC YC S (GPa) (GPa) (GPa) (MPa) (MPa) (MPa) (MPa) (MPa) 204

18,5

5,59

0,23

1260

61

2500

202

67

Stresses and strains were computed in a boundary element code developed by Albuquerque [1]. The routines that compute the indexes of Puck and LaRC03 criteria were implemented in this code, using this input data. Results for the lamina 2 (90◦ ) are shown in Fig. 1. As can be seen, the agreement in localization of failure occur but Puck’s criteria indicates lower level of stresses which would lead to failure. Fig. 2 shows the failures modes at points of the boundary and domain. The correspondence of numbers of failure modes are shown in Table 2. The criteria agree showing equivalents type of failures, 3 (Puck) and 2 (LaRC03) indicating matrix crack under tensile loading and 2 (Puck) and 4 (LaRC03), indicating fiber failure under compression.


87

Table 2: Correspondence between numbers and failures modes for the two criteria Number PUCK LaRC03 1 eq (3) (Fiber) eq (10) (Matrix) 2 eq (4) (Fiber) eq (8) (Matrix) 3 eq (5) (Matrix) eq (11) (Fiber) 4 eq (6) (Matrix) eq (12) (Fiber) 5 eq (7) (Matrix) eq (13) (Fiber) 6 eq (9) (Matrix)

(a)

(b)

Figure 1: Failure Index using a) PUCK criterion and b) LaRC03 criterion

1.2

0.8

y [m]

0.6

0.4

0.2

0

-0.2 -0.2

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

0

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 333 3 3 3 3 3 3 3 2 2 2 3 3 3 3 3 3 2 2 2 3 3 33 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

1

3

0.2

0.4

0.6 x [m]

(a)

0.8

1

0.8

0.6 y [m]

1

1.2

0.4

0.2

0

1.2

-0.2 -0.2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 222 2 2 2 2 2 2 2 4 4 4 2 2 2 2 2 2 4 4 4 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0.2

0.4

0.6

0.8

1

1.2

x [m]

(b)

Figure 2: Failure modes obtained using a) PUCK criterion and b) LaRC03 criterion, the correspondence of numbers and failure modes can be seen in Table 2

88


Conclusions. This paper compared results of two phenomenological based criteria. From the results, it can be concluded that there is a good agreement between LaRC03 and Puck’s criteria in indicate the most likely regions to occur the failure. However, in this case, Puck indicates possibility of failure in lower levels of stress mainly in the matrix tensile mode. Concerning failure modes predictions both criteria had good agreement for this laminate. Acknowledgments. The authors would like to thank CAPES for the financial support of this work.

References [1] E. L. Albuquerque Numerical analysis of dynamic anisotropic problems using the boundary element method. Thesis — Unicamp, Dept. Mec. Comput., July 2001. [2] E. L. Albuquerque, P. Sollero, and M. H. Aliabadi. The boundary element method applied to time dependent problems in anisotropic materials. International Journal of Solids and Structures, 39:1405–1422, 2002. [3] E. L. Albuquerque, P. Sollero, and M. H. Aliabadi. Dual boundary element method for anisotropic dynamic fracture mechanics. International Journal for Numerical Methods in Engineering, 59:1187–1205, 2004. [4] E. L. Albuquerque, P. Sollero, and P. Fedelinski. Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems. Computers and Structures, 81:1703–1713, 2003. [5] E. L. Albuquerque, P. Sollero, and P. Fedelinski. Free vibration analysis of anisotropic material structures using the boundary element method. Engineering Analysis with Boundary Elements, 27:977–985, 2003. [6] 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. [7] C. Brebbia, J. Dominguez. Boundary Elements: An Introductory Course. Computational Mechanics Publications, 1989. [8] C. Dávila, P. Camanho, C. Rose. Failure criteria for FRP laminates. Journal of Composite Materials, v. 39, p. 323–345, 2005. [9] I.M. Daniel and O. Ishai. Engineering Mechanics of Composite Materials. Oxford University Press, 2006. [10] A. Deb. Boundary element analysis of anisotropic bodies under thermomechanical body force loadings. Computers and Structures, v. 58, p. 715–726, 1996. [11] Z. Hashin. Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics, v. 47, p. 329–334, 1980. [12] A. Puck, H. Schurmann. Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology, v. 58, p. 1045–1067, 1998. [13] A. Puck, H. Schurmann. Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology, v. 62, p. 1633–1662, 2002.

[14] P. Sollero, M. Aliabadi. Fracture mechanics analysis of anisotropic plates by the boundary element method. International Journal of Fracture, v. 64, p. 269–284, 1993. [15] P. Sollero, M. Aliabadi. Anisotropic analysis of composite laminates using the dual boundary element method. Composite Structures, v. 31, p. 229–234, 1995. [16] S. Tsai, E. Wu. A general theory of strength test for anisotropic materials. Journal of Composite Materials, v. 5, p. 58–80, 1971.

90


Investigation of Test Function in Meshless Local Integral Equation Method P.H. Wen1a and M.H. Aliabadi2b 1

Department of Engineering, Queen Mary, University of London, London, UK, E1 4NS 2

Department of Aeronautics, Imperial College, London, UK, SW7 2BY a

[email protected], [email protected]

Keywords: Meshless method, local integral equation method, radial basis function, analytical solutions, elasticity. Abstract. In this paper, the exact forms of integrals in the meshless local boundary integral equation method are derived and applications were demonstrated for the elasticity problems. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the Radial Basis Function (RBF) is employed for the implementation of displacements. As the closed forms of the local boundary integrals are obtained, there are no any domain or boundary integrals to be calculated in this approach.

Introduction Engineering computations have made a startling progress over the past three decades and the most successful methods among them are the finite element method and boundary element method. It is wellknown that the finite element method (FEM) is the most widely used numerical method which can effectively deal with linear, nonlinear and large scale problems. The boundary element method (BEM) is now a well-established technique for the analysis of certain engineering problem with particular advantages. But many integrals over the domain occur in the boundary integral equations for most nonlinear problems. Moreover, pure boundary integral formulations are available only if the fundamental solution of the governing operator is known. 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] proposed element-free Galerkin method and reproducing kernel particle methods, respectively. One key feature of these methods is that they do not require a structured grid and are hence meshless. Recently, Atluri and his colleagues presented a family of Meshless methods, based on the Local weak Petrov-Galerkin formulation (MLPGs) for arbitrary partial differential equations [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 method (LBIE) with moving least square and polynomial radial basis function (RBF) has been developed by Sladek et al [5] for the boundary value problems in anisotropic non-homogeneous media. Both methods (MLPG and LBIE) are meshless, as no domain/boundary meshes are required in these two approaches. However, Galerkin-base meshless methods, except MLGP presented by Atluri[6] still include several awkward implementation features such as numerical integrations in the local domain. A variety of local interpolation schemes that interpolate the randomly scattered points is currently available. The moving least square and radial basis function interpolations are two popular approximation techniques recently. With comparisons of these two techniques, the moving least-square approximation is generally considered to be one of the best schemes with a reasonable accuracy, particularly for static elasticity demonstrated by Wen et al [7]. Hardy[8] and Hon et al [9] used multiquadric interpolation method for solving linear differential equation. Numerical results indicate that these two methods provide a similar optimal accuracy in solving both 2D Poisson’s and parabolic equations. In this paper, the meshless local boundary integral method is presented for the solid mechanics. With the use of radial basis functions, the analytical solutions for all domain integrals in the weak form of are derived. The weak formulation for the governing equations with a unit test function or any polynomial test functions are obtained exactly for the local domain integrals. As the closed form of the local boundary


91

integrals are obtained, the computational time are reduced significantly. Two examples are presented to demonstrate the application of this approach.

Meshless local integral equation method

Consider a linear elastic body in three dimensional domain : with boundary w: . The equations of equilibrium can be written as V ij , j f i 0 (1)

where

ij

are stresses and f i is the body force (i

1, 2 for 2D and i 1, 2, 3 for 3D) . By Hook’s law, one

has

§ wu1 wu · wu · E § wu1 wu1 · E § wu 2 ¸ (2) ¨¨ ¨ ¨ Q 2 ¸¸ , V 22 Q 1 ¸¸ , V 12 2 ¨ 2 ( 1 x x x x w w Q ) ¨© wx 2 wx 2 ¸¹ w w Q 1 2 ¹ 1 ¹ © 1 © 2 where E is Young’s modulus, Q is Possion ratio and P E / 2(1 Q ) the shear modulus. The boundary E 1 Q 2

V 11

conditions are given as

ui

ui

on *u and t i

V ij n j

ti

on *t

(3)

in which u i and t i are the prescribed displacements and tractions respectively on the displacement boundary *D and on the traction boundary *T , and ni is the unit normal outward to the boundary * . In the local boundary integral equation approache, the weak form of differential equation over a local integral domain : s can be written as

³ (V

ij , j

f i )u i* d:

0

(4)

:s

where u i* is a test function. By use of divergence theorem, Eq.(3) can be rewritten in a symmetric weak form as

³V

w: s

ij

n j u i* d* ³ (V ij u i*, j f i u i* )d:

0

(5)

:s

If there is an intersection between the local boundary and the global boundary, a local symmetric weak form in linear elasticity may be written as

³V

:s

u d: ³ tiui*d* ³ tiui*d*

* ij i , j

in which, w: s

Ls

*D

³ t u d* ³ f u d: * i i

*T

* i i

(6)

:s

*s Ls , *s is a part of the local boundary located on the global boundary and Ls is the

other part of the local boundary inside the local integral domain : s ; *D is the intersection between the local boundary w: s and the global displacement boundary; *T is a part of the traction boundary as shown in Figure 1. The local weak forms in Eq.(5) and Eq.(6) are a starting point to derive local boundary integral equations if appropriate test functions are selected. A step functions can be used as the test functions u i* in each integral domain

u i* (x)

M i (x) at x (: s w: s ) . ® at x : s ¯ 0

(7)

where M i (x) is arbitrary function. For M i (x) 1 and zero body force f i 0 , the local weak forms Eq.(5) and Eq.(6) are transformed into simple local boundary integral equations (equilibrium of local integral domain) as

³ t d* i

w: s

and

0

(8)

92


³ t d*

³ ti d*

i L s *D

(9)

*T

support domain of x

node x

Local integral domain s

Node in support domain k

Ls

*s=*D+*T

Figure 1. Arbitrary distributed node, support domain of x, local integral domain for weak formulation.

The approximation scheme

Consider a local domain w: s shown in Figure 1, which is the neighbourhood of a point x

( ^x1 , x 2 ` ) and is considered as the domain of definition of the RBF approximation for the trail function at x and also called as support domain to an arbitrary point x. Generally the support domain is chosen as a circle centred at x. To interpolate the distribution of function u in the local domain w: s over a number of randomly distributed nodes [= ^ 1 , 2 ,..., K `, k function u at the point x can be expressed by

([ k1 , [ k 2 ), k

1,2,..., K ], the approximation of

K

u ( x)

¦R

k

(x, k )a k

R (x)a(x)

(10)

k 1

^R1 (x,), R2 (x,),..., RK (x,)` is the set of radial basis functions centred around the point x K ( x1 , x 2 ) ], â k `k 1 are the unknown coefficients to be determined. The radial basis function selected

where R (x)

[ multi-quadrics [20,21] as

c 2 ( x1 [ k1 ) 2 ( x 2 [ k 2 ) 2

Rk (x, k )

(11)

where c is a free parameter. In order to guarantee unique solution of the interpolation problem, the displacement field can be interpolated by K

u i ( x)

¦R k 1

T

k

(x, )a k ¦ Pt (x)bt

R (x)a P(x)b

(12)

t 1

along with the constraints K

¦ P ( t

k

)a k

0,

1d t dT

k 1

(13)

where ^Pt `t 1 is a basis for PT 1 , the set of d-variate polynomials of degree d T 1 . In this paper, following polynomials are considered (T=6) P 1, x1 , x 2 , x12 , x1 x 2 , x 22 (14) A set of linear equations can be written, in the matrix form, as R 0 a P0 b u i , P0 a 0 (15) where matrix T

^

`

Advances in Boundary Element and Meshless Techniques XII ª R1 ( 1 ) R2 ( 1 ) « R ( ) R ( ) 2 2 « 1 2 « . . « . « . « . . « ( ) ( R R 2 K ) ¬« 1 K

R 0 ( )

93

RK ( 1 ) º R K ( 2 ) »» . » » . » . » » RK ( K )¼» K uK

... ... ... ... ... ...

(16)

and

ª P1 ( 1 ) P2 ( 1 ) « P ( ) P ( ) 2 2 « 1 2 « . . « . « . « . . « P P ( ) ( 2 K ) ¬« 1 K

P0 ( )

... ... ... ... ... ...

PT ( 1 ) º PT ( 2 ) »» . » » . . » . » » PT ( K )¼» K uT

(17)

Solving these equations in Eq.(15) gives

b

u, a

u

P

T 0

R 01 P0

1

P0 R 01 , T

>

R 01 I P0 P0 R 01P0 T

1

P0 R 01 T

@

(18)

where I denotes the diagonal unit matrix. Substituting the coefficients a and b from Eq.(18) into Eq.(12), we can obtain the approximation of the field function in terms of the nodal values u (x) (x)u, (x) R (x) P(x) . (19) in which (x) is defined as shape function, matrix R (x) and P (x) are scale (1 u K ) and (1 u 6) matrix respectively. It is worth noting that the shape function depends uniquely on the distribution of scattered nodes within the support domain and it has the Kronecker Delta property.

Analytical solutions for test functions Firstly, consider a unit test function, i.e. M i (x) 1 and the local domain is enclosed by several straight lines as shown in Figure 2, therefore, the local boundary integral equation becomes

³V

ij

n j d*

w: s

L

¦ n ³V lj

l 1

ij

d*

(20)

*l

where L is number of straight line and M : indicates the number of nodes collocated in domain. Suppose there are nodes both in the domain and on the boundary, M M : M T M D , where M T and M D are numbers of nodes on the traction/displacement boundaries and consider the radial basis function interpolation in Eq.(19) and relationship between stress and strain in Eq.(2), Eq.(4) becomes K

L

ª

K

§

E

¦¦ «¦ ¨© 1 Q k 1 l 1

¬i

1

2

T · º § E · F1il n1l PF2il n2l ¸D ik ¦ ¨ G1tl n1l PG2tl n 2l ¸E tk » u1( k ) 2 ¹ ¼ ¹ t 1 © 1 Q

T ª § EQ º (k ) § EQ l l · l l · 0 ¦¦ «¦ ¨ 1 Q 2 F2il n1 PF1il n2 ¸D ik ¦ ¨ 1 Q 2 G 2tl n1 PG1tl n 2 ¸ E tk » u 2 ¹ ¼ ¹ k 1 l 1 ¬i 1 © t 1© K L T º (k ) ª K § EQ § EQ l l · l l · ¦¦ «¦ ¨ 1 Q 2 F1il n 2 PF2il n1 ¸D ik ¦ ¨ 1 Q 2 G1tl n 2 PG2tl n1 ¸ E tk » u1 ¹ ¹ ¼ k 1 l 1 ¬i 1 © t 1© K

L

K

T º (k ) ªK § E § E l l · l l · ¦¦ «¦ ¨ 1 Q 2 F2il n 2 PF1il n1 ¸D ik ¦ ¨ 1 Q 2 G2tl n 2 PG1tl n1 ¸E tk » u 2 ¹ ¹ ¼ k 1 l 1 ¬i 1 © t 1© K

L

0

(21a)

(21b)

94


for x k , k

1,2,...M : , where the integral functions

b

L

( xbl 1 , xbl 2 ) 1

sl ( x1l , x 2l )

xk

x2

El

2

l

nl

a l a1

l a2

(x , x )

x1

Figure 2. Local integral domain with straight boundary lines. sl

wRi

³ wx

F jil

0

ds, G j1tl

j

sin E l , n2l

Consider n1l

sl

wPt

0

j

³ wx

ds

(22)

cos E l , we have solutions in closed form

F1il

(r2 r1 ) cos E l [( x al 1 [ i1 ) sin E l ( x al 2 [ i 2 ) cos E l ] sin E l ln(d1 / d 2 )

F2il

(r2 r1 ) sin E l [( x al 2 [ i 2 ) cos E l ( x al 1 [ i1 ) sin E l ] cos E l ln(d1 / d 2 )

r1

c 2 ( x al 1 [ i1 ) 2 ( x al 2 [ i 2 ) 2

r2

c 2 ( xbl 2 [ i1 ) 2 ( xbl 2 [ i 2 ) 2

(23)

( x al 1 [ i1 ) cos E l ( x al 2 [ i 2 ) sin E l r1

d1

( xbl 1 [ i1 ) cos E l ( xbl 2 [ i 2 ) sin E l r2 Next considering a linear test function M i (x)

d2

xi with a rectangular local integral domain, the weak forms of the governing equations can be derived as following K

ª

§

K

E

¦ «¦ ¨© 1 Q k 1

¬i

2

1

T · § E · º H il111 PH il221 ¸D ik ¦ ¨ I 111 PI tl221 ¸E tk » u1( k ) 2 tl ¹ ¹ ¼ t 1 © 1 Q

T ª K § EQ · 121 § EQ · 121 º ( k ) ¦ «¦ ¨ 1 Q 2 P ¸ H il D ik ¦ ¨ 1 Q 2 P ¸I tl E tk » u 2 ¹ ¹ k 1¬i 1 © t 1© ¼

(24a)

K

K

ª

K

§ EQ

¦ «¦ ¨© 1 Q k 1

¬i

2

1

0

T º · § EQ · P ¸I tl122 E tk » u1( k ) P ¸ H il122D ik ¦ ¨ 2 1 Q ¹ ¹ t 1© ¼

T º (k ) ªK § E § E 222 112 · 222 112 · ¦ «¦ ¨ 1 Q 2 H il PH il ¸D ik ¦ ¨ 1 Q 2 I tl PI tl ¸E tk » u 2 © ¹ © ¹ k 1¬i 1 t 1 ¼ for x k , k 1,2,...M : , where the integral functions

(24b)

K

xb 1 xb 2

H ilpqr

w 2 Ri x r dx1 dx 2 , I tlpqr p wx q

³ ³ wx

xa 1 xa 2

xb1 xb 2

w 2 Pt x r dx1 dx 2 p wx q

³ ³ wx

x a 1 xa 2

It is not difficult to obtain all integrals in the closed form as

0

(25)

Advances in Boundary Element and Meshless Techniques XII § ( x2 [ i 2 ) 2 c ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ¨ 2 ©

H il111

· ( x12 c 2 [ i22 ) ln[ x 2 [ i 2 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ] ¸¸ 2 ¹

xbl 1

xbl 2

xal 1

xal 2

(26a)

§ ( x1 [ i1 ) 2 c ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ¨ 2 ©

H il121

· [c 2 ( x1 [ i1 ) 2 ] ln[ x1 [ i1 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ] ¸¸ 2 ¹

xbl 1

xbl 2

xal 1

xal 2

§ ( x2 [ k 2 ) 2 c ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ¨ 2 ©

H il221

95

(26b)

(26c)

xbl 1

xbl 2

xal 1

xal 2

xbl 1

xbl 2

xal 1

xal 2

[ i1 ( x 2 [ i 2 ) ln[ x1 [ i1 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ] § ( x1 [ i1 ) 2 c ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ¨ 2 ©

H il112

(26d)

[ i 2 ( x1 [ i1 ) ln[ x 2 [ i 2 c ( x1 [ i1 ) ( x 2 [ i 2 ) ] 2

2

2

§ ( x2 [ i 2 ) 2 c ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ¨ 2 ©

H il122

· [c 2 ( x1 [ i1 ) 2 ] ln[ x 2 [ i 2 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ] ¸¸ 2 ¹

· ( x 22 c 2 [ i22 ) ln[ x1 [ i1 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ] ¸¸ 2 ¹

I I

111 4l

I

112 4l

xbl 2

xal 1

xal 2

§ ( x1 [ i1 ) 2 c ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 ¨ 2 ©

H il222

pqr tl

xbl 1

0 for t I

221 6l

I

222 6l

2I

xbl 1

xbl 2

xal 1

xal 2

(26e)

(26f)

1,2,3 121 5l

( xbl 2 x al 2 )[( xbl 1 ) 2 ( x al 1 ) 2 ]

122 5l

( xbl 1 x al 1 )[( xbl 2 ) 2 ( x al 2 ) 2 ]

2I

(27)

Finally, considering the general case, if the test functions are selected as M i (x) x1m x 2n with a rectangular local integral domain as shown in Figure 2, the following integrals can be used to derive the analytical formulations for the weak form of governing equations

³³

w 2 Rk m n x1 x 2 dx1 dx 2 wx 22

w 2 Rk

³ ³ wx wx 1

x1m x 2n dx1 dx 2 2

· § n w ¨¨ x 2 nx 2n ¸¸ ³ x1m Rk dx1 n(n 1) ³ ³ x1m x 2n 2 Rk dx1 dx 2 x w 2 ¹ ©

(28)

x1m x 2n Rk n ³ x1m x 2n 1 Rk dx1 m ³ x1m 1 x 2n Rk dx1 mn ³ ³ x1m 1 x 2n 1 Rk dx1 dx 2

96


A part from the collocation points in the domain, we need to consider the traction/displacement boundary conditions for the nodes collocated on the boundary. For the nodes on the traction boundary, Eq.(9) should be introduced

³ t d* i

* *T

³ t i d* for x k k

1,2,..., M T

(29)

*T

For the displacement boundary nodes, we can introduce the displacement equation directly, i.e. u i ( k ) u i , k 1,2,...M D . Therefore, there are total 2 u ( M : M T M D ) linear algebraic equations which are used to determine the same number unknowns of displacements either in the domain or on the traction boundary.

Numerical example

A square plate of width a subjected to a uniform shear load W 0 on the top is analysed. the Poisson’s ratio =0.3. The nodes can be arbitrary or uniformly distributed in the domain. Free parameter in the radial basis function c ' , where ' indicates the minimum distance between the nodes in the local integral domain. The bottom of the plate is fixed as shown in Figure 3. To observe the accuracy and convergence of this method, we consider four different shapes for the local integral domain, i.e. triangle, square, octagon and circle (L=3, 4, 8 and 128) as shown in Figure 4. The shear stress at the middle point of the bottom V 12 (a / 2,0) is studied and the uniformly distributed nodes 441(21×21) is selected. The normalized shear stresses are listed in Table 1, where the solution given by BEM is V 12 ( a / 2,0) / W 0

0.8667 . In addition,

the parameter D / ' min varies from 0.2 to 2, D is character parameter of integral domain as shown in Figure 4. General speaking, the number of straight line of integral domain has very little influence on the accuracy. However, the square integral domain is of slight higher accuracy than other shapes. In addition, the convergences with the node number are observed for the constant text function and linear text function. The results for the shear stress are presented in Table 2 with parameter D / ' min 1 . W0

a

x2 x1

Figure 3. Square plate subjected to uniform shear load on the top.

D

L=3

L=4

L=8

L=128

Figure 4. Different shapes of local integral domain and character parameter D.

Advances in Boundary Element and Meshless Techniques XII Table 1. Shear stress V 12 ( a / 2, 0) / W 0 with different local integral domain shape D/min 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Error

Triangle Square 0.8882 0.8880 0.8872 0.8868 0.8856 0.8848 0.8837 0.8822 0.8814 0.8793 0.8789 0.8763 0.8763 0.8734 0.8738 0.8707 0.8715 0.8685 0.8694 0.8670 1.6% 1.3%

Octagon Circle 0.8879 0.8878 0.8862 0.8859 0.8836 0.8831 0.8804 0.8796 0.8769 0.8760 0.8735 0.8725 0.8705 0.8695 0.8681 0.8674 0.8667 0.7404 0.8349 1.0540 1.5% 4.6%

Table 2. Shear stress V 12 ( a / 2, 0) / W 0 with different density of nodes M Constant text function Linear test function

11×11 0.9546 0.9287

21×21 0.8793 0.8743

31×31 0.8733 0.8720

41×41 0.8722 0.8719

Error 3.3% 2.3%

Conclusion In this paper, the analytical formulations for the meshless local boundary integral method were derived. Hence there are not any domain and boundary integrals in computation process and computation time consequently will be reduced. Many observations were carried out for several examples, such as the influence of the parameter of integral domain ( D / ' min ) and the number of distributed nodes (M) in the domain, the accuracy by the use of constant and linear test functions etc. Finally the cracked plate was calculated without special treatments for the stress singularity at crack tip. By observing several numerical examples, it can be conclude with: (1) Meshless method can be fulfilled without any integrals; (2) Computation time is reduced; (3) Numerical solutions are stable and convergent for suitable selection of free parameters; (4) It can be extended to dynamic, elastoplastic, plate/shell bending and fracture problems easily. References [1] B. Nayroles, G. Touzot & P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics, 10, 307-318, 1992. [2] T. Belytschko, Y.Y. Lu & L. Gu, Element-free Galerkin method, Int. J. Numerical Methods in Engineering, 37, 229-256, 1994. [3] W.K. Liu, S. Jun & Y. Zhang, Reproducing kernel particle methods, Int. J. Numerical Methods in Engineering, 20, 1081-1106, 1995. [4] S.N. Atluri & T. Zhu, A new meshless local Peyrov-Galerkin (MLPG) approach to nonlinear problems in computational modelling and simulation, Comput Model Simul Engng, 3, 187-196, 1998. [5] J. Sladek, V. Sladek & Ch. Zhang, Heat conduction analysis in nonhomogeneous anisotropic solid, ZH Yao, MW Yuan, WX Zhong edited, Computational Mechanics, Tsinghua University Press and Springer, 609-614, 2004. [6] S.N. Atluri, The Meshless Method (MLPG) for Domain and BIE Discretizations, Forsyth, GA, USA, Tech Science Press, 2004. [7] P.H. Wen and M.H. Aliabadi, An Improved Meshless Collocation Method for Elastostatic and Elastodynamic Problems, Communications in Numerical Methods in Engineering, 24 (8), 635-651, 2008 [8] R.L. Hardy, Multiquadric equations of topography and other irregular surface, J. Geophys. Res., 76, 1905-1915, 1971. [9] Y.C. Hon & X.Z. Mao, A multiquadric interpolation method for solving initial value problems, J. Scientific Computing, 12, 51-55, 1997.

97

98


An Overview of Boundary Element Formulations for Micro and Nano Fluids Luiz C. Wrobel1 1

School of Engineering and Design, Brunel University, Uxbridge UB8 3PH, UK E-mail: [email protected]

Keywords: Boundary element method; microfluids; nanofluids; gas bubbles; microvasculature

Abstract. This paper presents an overview of boundary element formulations for the flow of micro and nano fluids, with particular emphasis on biomedical applications. This brief review concentrates on flows at very low Reynolds numbers (Stokes flows). The biomedical applications are related to the general problem of the transport of a gas bubble through the microvasculature, dominated by surface tension as inertial effects are negligible in the microcirculation. Recently, bubbles in nanoflows and microfluidic circuits have attracted interest for a variety of applications, including the removal of bubbles from channels and inducing and controlling flow. The transport of microbubbles is also relevant to the problem of air embolism and to a novel concept of tumour treatment based on gas embolotherapy. Therefore, it becomes very important to increase our understanding of the fluid/bubble interface evolution. BEM formulations and applications to some of these biomedical problems are discussed in the paper. Introduction This paper is primarily motivated by the need to develop a better understanding of the phenomena of bubble dynamics in cardiovascular flow. Understanding the dynamics of cardiovascular bubbles is important for treating air embolism, using microbubbles as vehicles for targeted drug delivery, and the potential clinical implementation of gas embolotherapy methods [1]. Despite considerable work aimed at understanding the mechanisms involved in cardiovascular bubble dynamics and at using these mechanisms for therapeutic and diagnostic methodologies, many questions remain unanswered. Further studies are needed to provide more detailed information regarding the fundamental interfacial dynamics of microbubbles and the interaction of microbubbles with real vessels and tissue [1]. The mathematical formulations of biomedical flow problems mostly assume very low values of the Reynolds number, as viscous forces largely dominate inertia forces. They also ignore the particulate nature of blood and treat it as a continuum, and assume proportionality between the stress and rate of strain, thus modelling blood as a Newtonian fluid. In addition, the gas inside the bubble is assumed to be ideal and the expansion/contraction of the bubble is treated as isothermal. Advances in the development of microfluidic devices in general require a greater understanding of the dynamics of bubbles travelling through channels and bifurcations since bifurcations are used in microchannel mixing devices. Previous fundamental fluid mechanics studies have considered the removal of bubbles from channels [2] and inducing and controlling flow [3]. BEM formulations have been developed to study the time-dependent motion of a semi-infinite bubble in a channel [4], the steady movement of a semi-infinite bubble contacting the channel walls [5], and the transport of small droplets through bifurcations [6]. However, the BEM literature contains very little work on long bubbles moving through bifurcations. Experiments have shown that a bubble can either travel through a bifurcation without contacting the wall or move in a ‘stick and slip’ or continuous motion while contacting the wall. In cases when the contact between the bubble and the solid surface only partially wets the surface, the wetted portion of the solid surface is delimited by a certain contact line. When moving contact lines are introduced in the model, the no-slip boundary condition at the contact line introduces a stress singularity [7]. In order to remove the stress singularity at the contact line, Cox [8] postulated that slip between the liquid and the solid or some other mechanism must occur very close to the contact line. Cox [8] then developed a contact angleslip velocity relation which was implemented in a BEM formulation to study the dynamics of the movement of the contact line when one liquid displaces an immiscible second liquid where both are in contact with a solid surface,


99

while Schleizer and Bonnecaze [9] used the BEM to study the dynamic behaviour and stability of a twodimensional immiscible droplet between parallel plates. The droplet is attached to the lower plate and forms two contact lines that are either fixed or mobile. More recently, Eshpuniyani et al. [7] studied the pressure-driven channel flow with a bubble sticking and sliding along one of the walls. The moving contact lines were modelled using a Tanner law in which the contact line speed is linearly proportional to the deviation of the contact angle from its equilibrium position. Eshpuniyani et al. [7] also modelled the effect of contact angle hysteresis (the difference between the advancing and receding contact angles), allowing the prediction of the stick-slide behaviour of bubbles which affects their long-term evolution and dynamics. Calderon et al. [10] considered the transport of a pressure-driven semi-infinite bubble which contacts the parent channel walls of a bifurcation as an initial model of long microbubble transport through microvessel bifurcations. Using the BEM allows a more precise determination of the location and shape of the bubble interface, which is important in calculation of pressures, stresses, and velocities in two phase flows. It also provided valuable information about the stresses and velocities near the contact line and allowed the comparison of the computational work to previous experimental and theoretical models. Experimental [11,12], theoretical [13] and numerical [14,15] simulations at micro/nano scales have provided clear evidence that wall slip occurs at fluid-solid interfaces, and show that the degree of boundary slip is a function of the liquid viscosity and the shear rate. Luo and Pozrikidis [16] developed a BEM formulation for studying slip flow over a spherical particle in infinite fluid and near a plane wall. In the case of a wall-bounded flow, the numerical model was axisymmetric and thus reduced to a one-dimensional integral equation in cylindrical coordinates. The results demonstrated that the torque and drag forces over a sphere are reduced when slip is considered. Ding and Ye [17] solved oscillatory slip Stokes flow problems by using a system of integral equations for the surface velocity and the normal derivative of its tangential component. The resulting integral equation for the normal derivative contains singularities of the Cauchy and Hadamard (hypersingular) types. Frangi et al. [18] employed a combined velocity-surface traction integral equation to study fluid damping in micro-electromechanical systems (MEMS), which also contains Cauchy and Hadamard singularities. Recently, Nieto et al. [19] developed a BEM formulation to study slip flow in rotating mixers, based on the use of the normal and tangential projections of the velocity integral equation, resulting in a weakly-singular mixed system of integral equations for the normal and tangential components of the surface traction. The paper will now concentrate on the modelling of the slip boundary condition and the equations for simulating and tracking moving contact lines. Fundamental Equations and Boundary Conditions Fluid flow in the micro and nanoscales is characterised by very low values of the Reynolds numbers. In this case, the problem can be modelled by the Stokes system of equations [20]:

w 2ui wx j wx j wu i wxi

wp wxi

(1)

0

(2)

in non-dimensional form, where u i and p are the components of the fluid velocity and pressure, respectively. The equivalent integral representation is of the form: c( x) u i ( x) ³ K ij ( x, y ) u j ( y ) dS y ³ u ij ( x, y ) f j ( y ) dS y S

0

(3)

S

where f i are the components of the surface traction. The fundamental velocity and traction for two-dimensional problems are given by [20]:

100

Eds: E L Albuquerque & M H Aliabadi ( xi y i )( x j y j ) º 1 ª §1· «ln¨ ¸ G ij » 4S «¬ © r ¹ »¼ r2

(4)

1 ( xi y i )( x j y j )( x k y k ) n k ( y ) S r4

(5)

u ij ( x , y )

K ij ( x, y )

in which r is the distance between the source point x and the field point y , i.e. r

x y .

The Navier slip boundary condition states that the relative tangential fluid velocity, u tf , with respect to the tangential wall velocity, U tw , is directly proportional to the tangential projection of the local shear rate, J t [19]

u tf U tw

Ls J t

(6)

The coefficient Ls is called the slip length and represents the hypothetical outward distance at the wall needed to satisfy the no-slip flow condition [21]. For planar surfaces, the tangential rate is given by the normal derivative of the tangential projection of the velocity field at the boundary surface, i.e. J t wu t / wn . Nieto et al. [19] argue that this expression has been wrongly used in the past for solving nanoflow problems with curved boundaries, as it fails to account for the variation of the normal velocity in the streamwise direction at those curved boundaries. To complete the boundary conditions at the wall, the following no-flux condition across the stationary boundaries needs to be considered [19]:

u i ni

(7)

0

Slip boundary conditions for curved and moving surfaces are discussed by Nieto et al. [19], who developed a BEM formulation for slip flow in rotating mixers which was applied to the study of concentric and eccentric Couette flow between rotating cylinders, and to single rotor mixers. The BEM formulation of Eshpuniyani et al. [7] assumes that a bubble is in contact with the upper wall of a channel. A Tanner law is used to specify the contact line velocities at the bubble-wall interface, with the velocity linearly going to zero away from the contact lines to satisfy the no-slip condition. The Tanner law is expressed as follows [7]:

k (T D T S )

u cl

(8)

where u cl is the contact line velocity, T D is the dynamic contact angle between the bubble surface and channel wall at the contact line, T S is the static contact angle and k is the constant of proportionality in Tanner’s law. Thus, in this model, the contact line moves with the objective of restoring the contact angle to its ‘equilibrium’ value, at a speed that is proportional to the deviation of the contact angle from its static position [7]. If contact angle hysteresis is included in the model, the contact line motion is governed by a modified Tanner law

u cl

k (T D T A )

for

TD T A

u cl

k (T D T R )

for

TD ! TR

u cl

0

for

T A TD TR

(9)


101

where T A and T R are the advancing and receding contact angles, respectively [7]. The stress boundary condition at the bubble interface is given by

'f i

N ni

(10)

where 'f i is the nondimensional modified stresses exerted by the two fluids and N is the curvature of the interface [7]. The values of the constant k in Tanner’s law and the static contact angle T S (or the advancing and receding contact angles T A and T R when hysteresis effect is included) depend on the properties of the two fluids and the surface. Once the flowfield is solved for with the boundary conditions (8-10), the following kinematic boundary condition is used to advance the interface shape in time: u i ni

wYi ni wt

(11)

implying that, at any given point, the interface Y moves at the local velocity u . Once the interface is advanced in time in the above manner, the new bubble volume is computed and the bubble pressure is updated using the ideal gas law for isothermal conditions, i.e. bubble pressure times the bubble volume is kept constant. A new flowfield in then calculated for this new bubble shape and pressure with the same boundary conditions, and the algorithm repeats itself in this manner [7]. Calderon et al. [10] modified the formulation of Eshpuniyani et al. [7] by introducing a transitional ‘slip length’ between the contact lines and the no-slip zone, with the slip velocity within the slip length given by u

x· § u cl ¨1 ¸ l¹ ©

(12)

where x is defined as the distance along the solid boundary from the contact line. Thus, the contact line moves at speed u cl and no-slip applies at the wall at distances greater than the slip length l away from the contact line. Conclusions

Eshpuniyani et al. [7] have shown that the effect of microbubbles on the flow rate through channels is that, initially, the flow rates at the inlet and outlet are driven by a rapid contraction or expansion. At later stages, the inflow rate is always lower than for the equivalent flow without any bubble. A bubble can thus be used to drive the flow away from a given portion of the vasculature. The advancing interface of the bubble pushes the fluid along, thus causing the outflow rate to be higher than the inflow rate. When contact angle hysteresis is included, there are cases where the bubble comes to a halt. In such situations, the outflow rate and inflow rate are equal, thus signifying overall occlusion of the flow through the channel. Since contact angle hysteresis is an effect present in reality, it can thus be used to predict the range of conditions for which bubbles of different sizes stick, and to compute the percentage of occlusion for these cases [7]. The BEM model presented by Calderon et al. [10] allows the calculation of the velocity profiles, pressures and stresses associated with a bubble travelling through a bifurcation. The results from pressure and wall shear stress will give insight into the forces and stresses cells might experience, and will guide experiments to examine possible bioeffects induced by bubble motion. The finding of recirculation regions near the contact line, even in Stokes flow, is relevant to the design of microdevices in which mixing is desired [10].

102


While these studies have revealed many important features of the bubble evolution and dynamics and its influence on the surrounding flow field and the wall stresses, there are a number of other important features and parameters that need more careful studies. In addition, many of the simplifying assumptions have to be relaxed to allow for more realistic simulations that include effects of wall flexibility and roughness, non-Newtonian nature of blood, pulsatility of flow, etc., as well as three-dimensional effects, which are of importance in practical applications.

References [1] J.L. Bull, Cardiovascular bubble dynamics, Critical Reviews in Biomedical Engineering, 33, 299-346 (2005). [2] F.M. Chang, Y.J. Sheng, S.L. Cheng and H.K. Tsao, Tiny bubbles removal by gas flow through porous superhydrophobic surfaces: Ostwald ripening, Applied Physics Letters, 92, Paper 264102 (2008). [3] Z.Z. Yin and A. Prosperetti, A microfluidic ‘blinking bubble’ pump, Journal of Micromechanics and Microengineering, 15, 643-651 (2005). [4] D. Halpern and D.P. Gaver, Boundary element analysis of the time-dependent motion of a semi-infinite bubble in a channel, Journal of Computational Physics, 115, 366-375 (1994). [5] C. Huh and S.G. Mason, The steady movement of a liquid meniscus in a capillary tube, Journal of Fluid Mechanics, 81, 401-419 (1977). [6] M. Manga, Dynamics of drops in branched tubes, Journal of Fluid Mechanics, 315, 105-117 (1996). [7] B. Eshpuniyani, J.B. Fowlkes and J.L. Bull, A boundary element model of microbubble sticking and sliding in the microcirculation, International Journal of Heat and Mass Transfer, 51, 5700-5711 (2008). [8] R.G. Cox, The dynamics of the spreading of liquids on a solid surface. Part 1: Viscous flow, Journal of Fluid Mechanics, 168, 169-194 (1986). [9] A.D. Schleizer and R.T. Bonnecaze, Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows, Journal of Fluid Mechanics, 383, 29-54 (1999). [10] A.J. Calderon, B. Eshpuniyani, J.B. Fowlkes and J.L. Bull, A boundary element model of the transport of a semi-infinite bubble through a microvessel bifurcation, Physics of Fluids, 22, Paper 061902 (2010). [11] C. Neto, V. Craig and D.R.M. Williams, Evidence of shear-dependent boundary slip in Newtonian liquids, The European Physical Journal E: Soft Matter and Biological Physics, 12, 71-74 (2003). [12] C. Neto, D.R. Evans, E. Bonaccurso, H. Butt and V. Craig, Boundary slip in Newtonian liquids: A review of experimental studies, Reports on Progress in Physics, 68, 2859-2897 (2005). [13] M.T. Matthews and J.M. Hill, Newtonian flow with nonlinear Navier boundary conditions, Acta Mechanica, 191, 195-217 (2007). [14] P.A. Thompson and S.M. Troian, A general boundary condition for liquid flow at solid surfaces, Nature, 389, 360-362 (1997). [15] L.S. Kuo and P.H. Chen, A unified approach for nonslip and slip boundary conditions in the lattice Boltzmann method, Computers and Fluids, 38, 883-887 (2009). [16] H. Luo and C. Pozrikidis, Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall, Journal of Engineering Mathematics, 62, 1-21 (2008). [17] J. Ding and W. Ye, A fast integral approach for drag force calculation due to oscillatory slip Stokes flows, International Journal for Numerical Methods in Engineering, 60, 1535-1567 (2004). [18] A. Frangi, G. Spinola and B. Vigna, On the evaluation of damping in MEMS in the slip flow regime, International Journal for Numerical Methods in Engineering, 68, 1031-1051 (2006).

[19] C. Nieto, M. Giraldo and H. Power, Boundary integral equation approach for Stokes slip flow in rotating mixers, Discrete and Continuous Dynamical Systems - Series B, 15, 1019-1044 (2011). [20] H. Power and L.C. Wrobel, Boundary Integral Methods in Fluid Mechanics, Computational Mechanics Publications, Southampton, 1995. [21] N. Nguyen and S. Wereley, Fundamentals and Applications of Microfluidics, Second edition, Artech House, Norwood, 2006.

104


A domain integral equation formulation of an inhomogeneous bending thin plate Y.S. Yang and C.Y. Dong Department of Mechanics, School of Aerospace Engineering, Beijing Institute of Technology, China

Keywords Inhomogeneous bending thin plate, RIM, domain integral equation method

Abstract. One curvature domain integral equation is proposed to investigate the inhomogeneous infinite bending thin plate. The radial integral method (RIM) is used to transform the domain integrals into the inclusion-matrix interface integrals. For a single or two inclusions with circular shape, the numerical solutions are obtained which were compared with the analytical solutions available or the finite element solutions. Introduction Inhomogeneous thin plate structures exist in many engineering application. The famous examples are the plate with holes in heat transfer and honey structures in aerospace. Therefore, the research about inhomogeneous bending thin plate has the important theoretical meanings, and the important actual engineering application values. In analyzing the bending thin plates, the analytical method [1,2] is only suitable for simple geometric shape inclusion problems. Numerical methods such as the finite element method (FEM) [3] and the boundary element method (BEM) [4] can be used to study many complicated bending thin plates. In the FEM, the whole problem domain is discretizedd into the finite elements, especially more elements near the interfaces are needed to capture the local moment changes. In the BEM, the complicated fundamental solutions, e.g. orthotropic material inclusions, are needed to study the inhomogeneous bending thin plates. In order to avoid the use of the fundamental solution of the non-isotropic material inclusions, one curvature domain integral equation for an infinite inhomogeneous bending thin plate will be deduced in this paper. The obtained integral equation only contains the inclusion domain integrals and can be used to solve the problems with the isotropic, orthotropic or anisotropic inclusions. Basic formulation The generic configuration considered in this paper (Figure1) is an infinite bending thin plate containing some inclusions made of possibly anisotropic material. The given bending moments Mx and My are applied on the remote place. Perfect bonding is assumed along the interfaces among the inclusions and the matrix. y

Myy

Mxx x

Mxx

Myy

Fig.1: Inhomogeneous bending thin plate


105

The moment M DE and curvature N OP are related through the constitutive equations

M DE

DDEOP N OP

˄1˅

i M DE

ci i DDEOP N OP

˄2˅

in the matrix, and

in the i-th inclusion. Isotropic elasticity is assumed in the matrix, so that the corresponding elasticity tensor DDEOP has the form:

DDEOP

D[QGDE G OP (1 Q )GDOG EP ]

˄3˅

ci are the bending stiffness components of the matrix and the i-th In Eqns (1) to (3), DDEOP and DDEOP

inclusion, respectively. The subscripts D , E , O , P 1, 2 . GDE is the Kronecker symbol, and D

Eh3 where E is elastic modulus, Q is Poisson’s ratio and h is the plate thickness. 12(1 Q 2 )

The elastic properties of the inclusions are possibly anisotropic, and the corresponding elasticity ci i i DDEOP 'DDEOP where 'DDEOP denotes the (possibly anisotropic) tensor is expressed as DDEOP contrast of elastic properties between the matrix and the i-th inclusion. Based on the virtual work principle, one has the following identity for the i-th inclusion

³

:i

§ ww* · * Q w M ¨ ¸ d* n n ³*i © wn ¹

ci * DDEOP d: N OP NDE

˄4˅

where the symbols with * are the known fundamental solutions generated by an unit point force applied at a fixed source point P along the z direction in which is vertical to x-y plane. Eqn (4) can be further expressed as

³

:i

§ ww* · i * * Q w M ¨ ¸ d * ³:i 'DDEOP N OP NDE d : n n ³*i © wn ¹

* DDEOP N OP NDE d:

˄5˅

The left hand side integral can be given as

³

:i

* Considering M OP ,OP

ww · § * ³ ¨ Qn* w M n* ¸ d * ³:i M OP ,OP wd : *i wn ¹ ©

* M OP N OP d :

˄6˅

' P , one has

w( P)

³

*i

§ * * ww · * ¨ Qn w M n ¸ d * ³:i M OP N OP d : wn ¹ ©

˄7˅

By means of Eqn (5), Eqn (7) becomes w P

³ Q w

*

*i

n

M nT n* d * ³

*i

Q w M T d * ³ * n

* n n

:i

i * 'DDEOP N OP NDE d:

˄8˅

For the same point P being within the i-th inclusion, the integral equation for the infinite bending thin plate with the interfaces * i (i=1,2,…,NI which is the total inclusion number) is taken as

106

Eds: E L Albuquerque & M H Aliabadi NI

NI

i 1

i 1

w0 ( P) ¦ ³ ª¬Qn ( w) w* w Qn* º¼ d * ¦ ³ ª¬T ( w) M n* M n ( w) T * º¼ d * *i *i

0

˄9˅

Using the bonding condition along the interfaces among the inclusions and the matrix, and adding Eqns (8) and (9), one can obtain NI

* l w0 P ¦ ³ l 'DDEOP N OP NDE d:

w P

l 1

˄10˅

:

In order to obtain the second derivative of the transverse displacement with respect to the source point P, Eq (10) can be changed into the following form r w0 P ³ r 'DDEOP N OP P N OP Q NDE* d :

w P

:

* r 'DDEOP d: N OP P ³ r NDE :

From the definition of the curvature, i.e. N mn 0 ( P) ³ N mn ( P ) N mn

:

r

'D

r

DEOP

r 'DDEOP N OP ( P) ³

wxm wxn

:r

:l

l 1,l z r

˄11˅

* l 'DDEOP d: N OP NDE

w, mn , one gets from Eq (11)

N OP (Q) N OP ( P )

* (Q; P) w 2NDE

NI

¦³

* w 2NDE (Q; P)

wxm wxn NI

d:

¦³

l 1,l z r

:

d:

l 'DDEOP N OP (Q) l

* (Q; P) w 2NDE

wxm wxn

˄12˅

d:

The domain integrals appearing in Eq (12) can be transformed into boundary integrals using the radial integration method (RIM) proposed by Gao [2]. To do this, the curvature in Eq (12) is approximated by a series of prescribed basis functions, i.e.

N mn (q)

NA

¦a

2

A 1

NA

¦a

A mn

A 1

2

2

0 i ij I (q) cmn ¦ cmn xi ¦¦ cmn xi x j

A mn A

i 1

NA

¦a

NA

A A mn i

A 1

˄13˅

i 1 j 1

x

¦a

A A mn i

x x Aj

0

˄14˅

A 1

where NA is the total number of application points with all boundary nodes and some selected A 0 i ij internal points. The symbols amn , cmn , cmn and cmn are the coefficients to be determined. xiA is the Cartesian coordinates at the application point A. Substituting Eqs (13) and (14) into Eq (12) as well as using the RIM lead to N 0 mn ( P) N mn ( P) I A ( P ) ³

*r

ª 1 1 wr A r aOP F1 (Q)gDE mn ( x p , x Q ) d *(Q) ® 'DDEOP « ³*r p Q 4S D ¬ r ( x , x ) wn ¯

NI ½° º lD mn ( x p , xQ ) 1 wr s p Q nE d *(Q) » ¦ 'DDEOP p q ³*s r ( x p , xQ ) wn F2 (Q) gDE mn ( x , x )d *(Q)¾¿° r (x , x ) ¼ s 1, s z r

NI ½ l ( x p , xQ ) 1 0 r 1 wr s F (Q) gDE mn ( x p , xQ )d *(Q) ¾ cOP ®'DDEOP ³ D mn p Q nE d *(Q) ¦ 'DDEOP ³ * r(x , x ) * s r ( x p , x Q ) wn 3 4S D 1, s s r z ¯ ¿


º l ( x p , xQ ) wr 1 i ° r ª 1 cOP ®'DDEOP « ³ r F (Q)r,i gDE mn ( x p , xQ ) d : xi ( P) ³ D mn p Q nE d *(Q) » * r ( x p , x Q ) wn 4 * r(x , x ) 4S D ¬ ¼ ¯° NI

¦

s 1, s z r

107

s 'DDEOP ³

*s

½ 1 wr F4 (Q) gDE mn ( x p , xQ )d *(Q) ¾ r ( x p , x q ) wn ¿

º l ( x p , xQ ) 1 ij ° r ª 1 wr cOP ®'DDEOP « ³ s F5 (Q)gDE mn ( x p , x Q ) d *(Q) xi ( P ) x j ( P) ³ D mn p Q nE d *(Q) » p Q * * r(x , x ) 4S D ¬ r ( x , x ) wn ¼ ¯° NI

¦

s 1, s z r

s 'DDEOP ³

*s

˄15˅

½ 1 wr F6 (Q)gDE mn ( x p , xQ )d *(Q) ¾ r ( x p , xQ ) wn ¿

where gDE mn ( x p , x q ) 8r,D r, E r,m r, n 2r,D r,nG E m 2r, E r,nGD m 2r,m r, nGDE 2r, E r,mGD n 2r,D r, mG E n 2r,D r, E G mn GD nG E m GD mG E n G mnGDE lD mn ( x p , x q )

Fi (Q)

2r,D r, m r,n r,D G mn r,mG an r,nGD m

r ( x p , x q ) (I ( q ) I ( P )) A A dr (q) ³0 ° r ( P; q ) ° r ( x p , xq ) I (q ) ° A dr (q ) ³0 ° r ( P; q ) ° ln r ( x p , x Q ) °° ® r ( x p , xQ ) ° ° r ( x p , xq ) xi (q) x j (q) xi ( P) x j ( P) dr (q) ° ³0 r ( P; q ) ° ° r ( x p , x q ) xi ( q ) x j ( q ) dr (q) ° ³ 0 r ( P; q ) °¯

˄16˅

˄17˅

i 1 i

2

i i

3 4

i

5

i

6

˄18˅

A 0 i ij Eqs (15) and (14) can be used to determinate the coefficients, i.e. amn , cmn , cmn and cmn , by a set of algebraic equations

â11A ` ½ ° ° ° a12A ° ° ° ° a22A ° >G @ ® ¾ °c 0 ° 11 ° ° ° ° ° 22 ° ¯c22 ¿

^N110 ( P)` ½ ° ° °^N120 ( P)` ° ° ° ° 0 ° ®^N 22 ( P)`¾ °0 ° ° ° ° ° °0 ° ¯ ¿

˄19˅

where [G] is the coefficient matrix. Once Eq (19) is solved, the moment at all the points being within inclusions can be obtained using the constitutive relation. For the points being within the matrix, the corresponding curvature can be calculated by the following equation NI

°

s 1

°¯

s 0 N mn ( P) N mn ( P) ¦ 'DDEOP ³* ®³0 N OP (Q) r (q)

* w 2NDE ( P; Q) ½° ¾ d *( q ) wxm wxn °¿

˄20˅

108


By means of the constitutive relation, the moment at all the points being within the matrix can be obtained. Numerical examples 1. A bending thin plate with one inclusion One cylindrical inclusion with the radius r 1m is embedded in an infinite bending thin plate subjected to the remote bending moments M x M y 1Nm (see Figure 2). The plate thickness is

taken as h 0.01m . Young’s moduli of the plate and the inclusion are E 1011 Pa and EI 2 u1011 Pa , respectively. The corresponding Poison’s ratios are chosen to be v Q I 0.3 . y

y Myy

Mxx

U r Mxx x

x

Myy

Fig.2: One inclusion embedded in an infinite Fig.3: Distribution of boundary and internal nodal nodes thin plate In numerical implementation, the calculation model is considered (see Figure 3). The inclusion-matrix interface is discretized into 16 quadratic boundary elements. The internal source point numbers are chosen as 192. The radial basis function is taken as R. The results for M x and M x are shown in Figures 4 and 5 together with theoretical solution, respectively. The results from the present method and the analytical method are in good agreement with each other.

Fig.4: Distribution of moment M x along x

Fig. 5: Distribution of moment M y along x

2. One circular inclusion with orthotropic material One orthotropic circular inclusion with the radius r 1m is embedded into an infinite thin plate subjected to the remote moments M x M y 1Nm (see Figure 2). The plate thickness is taken as h 0.01m . Young’s modulus and Poison’s ratios of the plate are the same as the above example. The inclusion material parameters are E1 2 u1011 Pa , E2 5 u1011 Pa , G12 7 u1010 Pa , Q 12 0.1 and Q 21 0.25 . The principal axial of the material is being in the x axis.

The distribution of the boundary and internal source points is the same as the example 1 (see


109

Figure 3). The radial basis function is taken as R. The finite element model of the quarter square plate with the edge length 20r is shown in Figure 6. The related data are given in Table 1. The boundary condition is applied along the x and y axes. The commercial software AYSYS is used to carry out the finite element calculation. The good agreement between the present solution and the FEM solution is observed as shown in Figures 7 and 8. Table 1. Finite element model parameters Total Total node Inclusion element no. no. element no.

Interface element Number 20

1500

4681

Element type

300

(a)

Shell93

(b)

Fig. 6: Finite element model: (a) 1/4 plate; (b) element distribution near inclusion 1.35 1.3

RBF only

Mx(Nm)

1.25

RBF with linear

1.2

RBF with quadratic

1.15

FEM

1.1 1.05 1 0.95 0

1

2

3

4

5

x(m)

My(Nm)

Fig.7: Distribution of moment M x along x RBF only RBF with linear EBF with quadratic FEM interface

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

1

2

3

4

5

x(m)

Fig 8: Distribution of moment M y along x

110


3. Interaction between two circular inclusions Two circular inclusion with the same radius r 1m are embedded into an infinite thin plate subjected to the remote moments M x M y 1Nm . The distance between two inclusion centers is 3m. The plate thickness is taken as h 0.01m . Young’s modulus and Poison’s ratio of the plate are chosen as E 2 u1011 Pa andQ 0.3 . The distribution of the boundary and internal source points is show in Figure 9. The radial basis function is taken as R. The finite element model of the square plate with the edge length 20m is shown in Figure 10. The element and node numbers are given in Table 2. The boundary condition is applied along the x and y axes. The commercial software AYSYS is used to carry out the finite element calculation for comparison. The results from RBF with linear and quadratic polynomials are in agreement with each other. The finite element result is also shown in Figures 11 and 12. The moment distribution between the FEM solution and the present result is in good agreement, except for those near the interface. Table 2. Finite element model parameters Interface Total element Total node Inclusion Element element no. no. element no. type Number 40 1839 5708 225 Shell93

y 1. 0 0. 5

U - 2. 5 - 2. 0

- 1. 5

- 1. 0

- 0. 5

0. 5

1. 0

U 1. 5

2. 0 2. 5

- 0. 5

x

- 1. 0

Figure 9ˊNode distribution in two inclusions

Fig. 10: Finite element model: (a) 1/4 plate; (b) element distribution near inclusion


1.275 RBF only RBF with linear RBF with quadratic FEM interface interface

1.225

Mx(Nm)

1.175 1.125 1.075 1.025 0.975 0

1

2

3

4

5

x(m)

Fig. 11: Distribution of moment M x along x 1.3 RBF only RBF with linear RBF with quadratic FEM interface interface

1.2

My(Nm)

1.1 1 0.9 0.8 0.7 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x(m)

Fig. 12: Distribution of moment M y along x Conclusions One curvature domain integral equation is presented to analysis the inhomogeneous infinite thin bending plates subjected to the remote moment. The radial integral method is adopted to transform the domain integrals into the interface integrals. Numerical results from the present method are compared with the analytical solutions available and the finite element solutions. Their good agreement shows the rightness of the present method. Acknowledgements

The authors would like thank the reviewers for their constructive comments. This work was supported by the National Natural Science Foundation of China under Grant nos. 10772030 and 11072034, the Cultivation Special Purpose Project of the Science and Technology Innovation Major Program of Beijing Institute of Technology.

111

112


References 1. C.W. Bert, Mech Res Commun, 15, 55-60, (1988). 2. C.W. Bert, J Strain Anal Eng, 36, 341-345, (2001). 3. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, 5th ed., McGraw-Hill (1989). 4. C.A., Brebbia, J.C.F.Telles and L.C. Wrobel, Boundary Element Techniques – Theory and Applications in Engineering, Springer (1984). 5.

X.W. Gao, J Applied Mech, 69, 154-160, (2002).


113

Fundamental solution based FEM for nonlinear thermal radiation problem Qing-Hua Qin1 and Hui Wang1,2 1

2

Research School of Engineering, Australian National University, Canberra, ACT, 0200, Australia, email: [email protected]

College of Civil Engineering and Architecture, Henan University of Technology, Zhengzhou, China, 450052, email: [email protected]

Keywords: Hybrid finite element, fundamental solution, heat transfer, thermal radiation

Abstract. A new hybrid finite element model (FEM) with elementary-boundary integrals only is developed for the solution of multimode heat transfer problems involving heat conduction, convection and nonlinear radiation in the paper. Based on the fundamental solution (FS) for the heat transfer problems, the internal and boundary temperature fields are independently assumed and the boundary integral-based variational functional is formulated to produce the final stiffness equation and connection between the two independent fields, and then, the temperature is determined by a special iteration procedure. Two examples involving the convection and radiation boundary conditions are performed using the proposed approach and the results show a good agreement between the proposed method and ABAQUS. 1. Introduction The determination of temperature distribution in a medium (solid, liquid, gas or combination of these phases) is the main objective of the thermal conduction analysis. As an alternative to the theoretical analysis and experiment measurements, numerical simulations are playing more and more important role for the solution of thermal related problems. Among all numerical methods, the most popular ones including FEM [1], boundary element method (BEM) [2-4], meshless methods [5-10], and hybrid Trefftz (HT) FEM [11-19] have well-developed to solve heat conduction problems during the past decades. However, they seldom involve multimodal analysis related to surface convection and surface radiation. The difficulties in solving conjugate thermal problems are associated with their nonlinearity and high computing cost. This paper is devoted to the development and application of a novel hybrid finite element formulation (HFS-FEM) to analyze the conjugate heat transfer problems by constructing a new hybrid variational functional and designing a simple iteration procedure. The developed method is based on the fundamental solution of the problem of interest, rather than the T-complete functions in HT-FEM, and inherits the advantages of HT-FEM against other techniques like FEM and BEM. At the same time, some drawbacks of the HT-FEM are removed in solving many engineering problems [20,21]. 2. Basic equations Let us consider the plane heat conduction equation in Cartesian coordinates ( X 1 , X 2 )

§ w u ( P) w u ( P) · k¨ ¸ 0 2 wX 22 ¹ © wX 1 2

:

2

P:

(1)

which is complete for any problem only if appropriate boundary conditions stated below are given

u u0 ° ®q q0 °q h (u u ) HV (u 4 u 4 ) c a a ¯

on *u on * q on * c

(2)

114


where k represents the thermal conductivity of materials, u sought temperature field, q

k wu / wn

surface heat flux, n the unit normal to w: , ua environment temperature, H the emissivity of the surface, and V the Stefan Boltzmann constant, which has the value 5.669 u 10 8 W/m2K4. *

w:

*u * q *c

is the boundary surface of the domain : . Evidently, the radiation condition described by the third equation in eq (2) is highly nonlinear, an iteration strategy is, therefore, needed. To this end, we first rewrite the convection and radiation condition as (3) q hc (u ua ) HV (u 4 ua4 ) h(u ua ) in which (4) h(u ) hc HV (u ua )(u 2 ua2 ) The formulation (3) is similar to the convective condition stated by the first part of the third expression in eq (2). Then, we can construct the following iteration algorithm to remove the nonlinearity term appeared in eq (4) (5) q h(u ( n 1) )(u ( n ) ua ) where u ( n 1) and u ( n ) are the temperatures obtained in the iteration step (n-1) and (n), respectively. When the residue is less than the specified tolerance G , that is n

¦ (u

( n 1) i

ui( n ) ) 2 d G

(6)

i 1

the iteration stops and yields final convergent results. For the sake of convenience, the fundamental solution used in our algorithm is given here. If P and Q are source and field points, the fundamental solution of a plane heat conduction problem is

u * ( P, Q )

1 ln r 2 kS

(7)

where r is the distance between P and Q . The fundamental solution satisfies the governing equation (1) at any field point Q . 3. Fundamental solution based finite element formulation To deal with the convective condition, the elementary variational integral 3 me presented in [20] is modified as

3 me

1 1 2 ku,i u,i d: ³ qud* ³ q u u d* ³ h u ua d* * qe *e 2 ³:e 2 *ce

(8)

where u and u are independent fields defined in the element and on the element boundary, respectively. In the hybrid FS FE approach, temperature field within an element, element e, can be approximated by ns

ue ( P )

¦ u ( P, Q )c *

j

ej

N e ( P)ce

(9)

j 1

where cej is unknown coefficients and ns is the number of virtual source points Q j outside the element e . In order to enforce the conformity of temperature field between adjacent elements, temperature field over the element boundary is independently assumed as

ue ( P )

nd

¦ N d i

i

d N e e

(10)

i 1

where N i is the shape function defined on the element boundary, and di the nodal temperature. nd is the number of nodes on the element edge. Appling the Gauss theorem, the hybrid variational functional (8) can be converted to the following one involving boundary integrals only:


3 me

1 h 2 qud* ³ q0ud* ³ qud* ³ u ua d* * qe *e * ce 2 2 ³*e

115

(11)

Substitution of eqs (9)–(10) into eq (11) gives

1 1 c eT H ec e d Te g e cTe G ed e d Te Fed e d Te f e a e 2 2

3 me

(12)

where

He Fe

³

*e

³

*ce

Q Te N e d*

Ge

TN d* hN e e

fe

³

*e

³

*ce

d* Q Te N e

ge

³

T d* hua N e

ae

³

*eq

T q d* N e 0

hua2 d* *ce 2

(13)

The stationary value of 3 me in eq (12) with respect to ce and d e yields the following relations

K ed e = g e fe

(14)

ce = H e1G ed e

(15)

and

4. Numerical results In order to verify the proposed algorithm, two examples including convective and radiation conditions are considered in this paper. The first example described in Fig. 1 involves a rectangular plate with convection conditions. The convection coefficient hc has different values in the computation. The conventional FE solutions obtained from ABAQUS with same mesh as that in the proposed model are provided for the purpose of comparison. As was shown in Fig. 1, due to large temperature change between the upper corner point temperature ( 180 o C ) and ambient temperature ( 25 o C ), the temperature there might change dramatically, hence, more elements are placed there (see Fig. 2). Results in Fig. 3 show the temperature variation on the convection edge. It can be seen from Fig. 3 that a good agreement between results of HFSFEM and those of ABAQUS is achieved.

Fig. 1 Rectangular plate with convection edge

Fig. 2 Mesh used in HFS-FEM and ABAQUS with 25 eight-node elements

116


Ǥ͵

ǤͶǣ Ǧ

The second example involves the same plate as the one used in the first example, while a radiation condition is added to the existing convective edge. The material emissivity H is 0.725. In the computation, the same mesh as shown in Fig. 2 is employed. If the initial value of temperature is taken to be 25 o C and the iteration tolerance is chosen as 10-6, after 4 iterations, the convergent results can be obtained and the temperature distribution for two convective coefficients (hc=0.5 and 50) is plotted in Fig. 5, from which a good agreement between results of HFS-FEM and ABAQUS can be found. Additionally, the temperature at the corner point (0.4, 0) are 164.4 o C ( hc 0.5W/m 2 K ) and 39.5 o C ( hc 50W/m 2 K ), respectively, in the presence of thermal radiation, while the temperature values are 168.8 o C ( hc 0.5W/m 2 K ) and 39.5 o C ( hc 50W/m 2 K ), respectively, in the absence of radiation effect. It is clearly evident that the convection effect is greater than the radiation effect, especially for the larger convective coefficient.


Fig. 5 Temperature variation on the convection and radiation edge 5. Conclusions A fundamental solution based finite element model for the solution of plane heat conduction with nonlinear radiation condition has been formulated in the paper. Based on the fundamental solution for the potential problem, two independent internal and boundary temperature fields are assumed independently and a modified variational functional is developed. The use of fundamental solution makes all integrals beibg over the element boundary. Two examples involving the convection boundary condition and radiation boundary condition are performed using the proposed approach and results show a good agreement between the proposed method and ABAQUS.

References [1] Chandrupatla, T.R. and Belegundu, A.D., Introduction to finite elements in engineering. 3rd ed. New Jersey: Prentice Hall (2002) [2] Q.H. Qin, Nonlinear analysis of Reissner plates on an elastic foundation by the BEM, Int J Solids Structures, 30, 3101-3111 (1993) [3] R.A.Bialecki and G.Wecel, Solution of conjugate radiation convection problems by a BEM FVM technique. J Quantitative Spectroscopy and Radiative Transfer, 84, 539-550 (2004) [4] J.Blobner, R.A.Blalecki, and G.Kuhn, Transient Non-linear Heat Conduction-Radiation Problems-A Boundary Element Formulation. Int J Numer Meth Eng, 46, 1865-1882 (1999) [5] Q.H. Qin, Meshless Approach and its Application in Engineering Problems, in: Computational Mechanics Research Trends, Hans P. Berger (ed.), Chapter 6, pp 249-289, New York, Nova Science Publishers (2010) [6] H. Wang, Q.H. Qin and Y.L. Kang A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media, Archive Appl Mech, 74, 563-579 (2005) [7] H. Wang, Q.H. Qin and Y.L. Kang A meshless model for transient heat conduction in functionally graded materials, Computational Mechanics 38, 51-60 (2006) [8] H. Wang and Q.H. Qin, A meshless method for generalized linear or nonlinear Poisson-type problems, Eng Analysis with Boundary Elements, 30(6), 515-521 (2006) [9] H. Wang and Q.H. Qin, Meshless approach for thermo-mechanical analysis of functionally graded materials, Eng Analysis with Boundary Elements, 32, 704-712 (2008) [10] L.H.Liu, J.Y.Tan, and B.X.Li, Meshless approach for coupled radiative and conductive heat transfer in one-dimensional graded index medium. J Quantitative Spectroscopy & Radiative Transfer, 101, 237-248 (2006)

117

118


[11] Q.H.Qin and H.Wang, Matlab and C Programming for Trefftz Finite Element Methods. CRC Press, Taylor & Francis Group LLC (2008) [12] Q.H.Qin, The Trefftz Finite and Boundary Element Method. Southampton: WIT press (2000) [13] Q.H.Qin, Hybrid Trefftz finite element approach for plate bending on an elastic foundation, Appl. Mathe. Modelling, 18, 334-339 (1994) [14] Q.H.Qin, Hybrid Trefftz finite element method for Reissner plates on an elastic foundation, Comp. Meth. Appl. Mech. Eng., 122, 379-392 (1995) [15] Postbuckling analysis of thin plates by a hybrid Trefftz finite element method, Comp. Meth. Appl. Mech. Eng., 128, 123-136 (1995) [16] J. Jirousek and Q.H.Qin, Application of Hybrid-Trefftz element approach to transient heat conduction analysis, Compu. & Struc., 58, 195-201 (1996) [17] Q.H.Qin, Transient plate bending analysis by hybrid Trefftz element approach, Communi. Numer. Meth. Eng., 12, 609-616 (1996) [18] Q.H.Qin, Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach, Computational Mechanics, 31, 461-468, 2003 [19] Q.H.Qin, Variational Formulations for TFEM of Piezoelectricity, Int J Solids Structures, 40, 63356346 (2003) [20] H. Wang and Q.H. Qin, Hybrid FEM with fundamental solution as trial function for heat conduction simulation Acta Mechanica Solida Sinica 22, 487-498 (2009) [21] H. Wang and Q.H. Qin, Fundamental solution based finite element model for plane orthotropic elastic bodies, European Journal of Mechanics - A/Solids, 29, 801-809 (2010)


119

Time integrations in solution of diffusion problems by Local integral equations and Point interpolation method 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, weak formulation, time stepping, Laplace transform, accuracy

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 in a local weak formulation, while the Laplace transform and/or the time stepping techniques are employed for treatment of the time evolution. The accuracy of the numerical results are studied and discussed in numerical test example using the exact solution as benchmark.

1. Governing equations. Local integral equation formulations. The transient heat conduction with the absent volume density of heat sources w(x, t ) is governed by the diffusion equation [1]

O (x)u,k (x, t ) ,k U (x)c(x) wu(wxt , t )

0 , in : u [0, T ]

(1)

where u (x, t ) is the temperature field. In isotropic and continuously non-homogeneous media, the material parameters, such as the mass density U (x) , the volume density of the specific heat per unit mass c( x) , and the thermal conductivity coefficient O (x) are spatially dependent and the governing equation is the partial differential equation (PDE) of parabolic type with variable coefficients. The physically reasonable boundary conditions of the problem are of three types: (i) Dirichlet b.c., (ii) Neumann b.c., (iii) Robin b.c. 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: . According to the energy balance in an arbitrary finite part of the continuum, :c bounded with the boundary w:c , we may write the integral equivalent of the differential governing equation (1) as

³

w:

ni ( )O ( )u,i ( , t )d *( ) c

w ³ U (x)c(x) wt u (x, t )d :(x)

(2)

0

:c

Sometimes, the Laplace transform technique is efficient tool for treatment of the time evolution. Then, the time variable is eliminated temporarily and replaced by the Laplace transform parameter p . The governing equations (1) and/or (2) 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

w:c

ni ( )O ( )u,i ( , p )d *( ) p

U ( x ) c ( x )v ( x )

³ U ( x ) c ( x )u ( x , p ) d : ( x )

:c

(3)

³ U ( x) c ( x)v ( x) d : ( x) .

(4)

:c

The boundary conditions for the Laplace transform of the temperature can be obtained by direct application of the Laplace transformation to the prescribed boundary conditions. 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

120


available in literature. Regarding good experience with the Stehfest’s algorithm [2], we shall use this algorithm in the present analysis with taking 10 values of the transform parameter for each time instant. The derived local integral equations (LIE) (2) and/or (4) are the restriction relationships, which should be satisfied together with the prescribed initial and boundary conditions in solving initial-boundary value problems.

2. Point Interpolation Method (PIM) - approximation for spatial variations. The PIM belongs to mesh free approximations since no predefined connectivity among nodal points is required [3]. In a sub-domain : q surrounding the nodal point x q , the approximated field variable f (x) {u ( x, t ), u (x, p)} can be written as N

q

¦ f (x n( q ,a ) )M ( q ,a ) (x)

f (x) : q

,

( N q {1, 2, ... , N t } )

(8)

a 1

where n( q , a ) is the global number of the a -th node associated with x q , the shape functions M ( q ,a ) (x) are expressed in terms of the basis functions defined in the Cartesian coordinate space, N q is the number of nodes in the support domain of the point x q , Nt is the total number of nodes. In this paper, we shall consider the basis functions as a combination of polynomials and radial basis functions (RBF), given by multiquadrics [4]. Such a combination improves the accuracy and numerical stability of the approximation [3,4]. The gradients of the field variable f (x) can be approximated as gradients of the interpolation (8). Substituting these approximations into the prescribed boundary conditions collocated at boundary nodes and into the LIE (2) considered at interior nodes, one obtains the system of the ordinary differential equations (ODE) for nodal values of temperature § wu g (t ) · ( c 1, 2,..., Nt ), (9) ¦ ¨¨ K cg u g (t ) M cg wt ¸¸ 0 , g © ¹

where K cg

³

w:

n ( qK ,a )

ni ( )O ( )I,i

( )d *( ) ,

M cg

n ( q ,a ) ³ U (x)c(x)I x (x)d :(x) ,

:c

c

with g being the global numbers of nodes generated by n(qK , a ) and/or n(qx , a) , where qx is the nearest

nodal point to the integration point x . Similarly, in the case of the Laplace transform (LT) approach, one obtains the system of algebraic equations which should be solved for unknown nodal values of the Laplace transforms of temperature for particular values of the transform parameter pn ( n 1, 2, ... , 10 ).

3. Linear Lagrange Interpolation (LLI) for time variations. If we approximate the time variation of nodal values of temperature, the ODEs in the above derived semidiscretized formulation convert to the algebraic equations for nodal values of temperature at particular time instants in fully discretized formulation. Let us split the time interval [0, T ] by discrete time instants ti into a finite number of subintervals [ti , ti 1 ] . In the case of linear interpolation the element Ti is defined as the interval Ti [ti , ti 1 ] with the interior points being parametrized as 2

¦ ti 1 a N a (W )

tT

i

ti

a 1

'ti (1 W ) ti T'ti , 2

W [1, 1] ,

T

(1 W ) / 2 [0,1] ,

(10)

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 T ui 1 (1 T )ui 2 2

u (ti T'ti ) ,

uk

u (tk ) .

(11)


121

Then, the time derivative u (t ) du (t ) / dt on Ti is approximated by the constant ui 1 ui / 'ti , since the 'ti / 2 .

Jacobian of the transformation (10) is given as J (W ) dt / dW T

i

ti T'ti , we obtain the sequence of algebraic equations

Considering the system of the ODE (9) at t

§

1

¦ ¨ K cg T't g

©

i

· M cg ¸ uig1 ¹

§

1

1

¦ ¨ (1 T ) K cg T't g

©

i

· M cg ¸ uig , ¹

( i 0,1, 2,... )

(12)

which is the well known T -method used in time stepping approaches for solution of the ODE with T (0,1] .

4. Numerical tests. In order to study the accuracy and convergence of numerical results, we shall consider the example for which the 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 for t [0, T ] and constant initial value of temperature v(x) const v is assumed, the exact solution is available [5]. In numerical computations, we have used O0 1 c0 U , G 1 , u0 1 v , u L 20 , w 0 . The uniform distribution of nodal points is employed with h being the distance between two neighbour nodes.

122


Fig.1 Evolutions of error distributions by LLI and LT approaches using 121 nodes and various time steps It can be seen from Fig.1 that considerable errors are localized in space and time for short time steps (early time instants). The inaccuracy of the numerical results by the LLI approach is partially decreased and remarkably delocalized with increasing the time step, while in the case of LT approach the spatial delocalization at later time instants is marginal as compared with the substantial increase of accuracy.

Fig.2 Time evolutions of the computed temperature and its accuracy by the LLI and LT approaches at the selected point ( x1 , x2 ) using 1296 nodes and three various lengths of the time step

Figure 2 shows the numerical results at the point ( x1 , x2 ) which is the nearest node to the point ( L / 2, 0.9 L ) lying close to the top side of the analyzed domain, where the prescribed boundary temperature u L 20 represents a sudden change at t 0 with respect to the initial value v 1 u0 . It can be seen that better accuracy is achieved by the LT approach especially when the longer time steps are employed. Note that three characteristic lengths play a role in this fully discretized transient field problem: (i) lh h represents the distance between two neighbouring nodes (ii) lS

Ri h represents the radius of the influence domain which is used for selection of nodes contributing

to the meshless approximation at certain point (for the Ri -factor we have used the value Ri

3.001 )

(ii) lT O / U c 't is the characteristic length corresponding to the time step 't and represents the horizon reached by the heat conduction during the time step 't with respect to the certain point. In time stepping techniques, the optimal choice is lT | h , because information at previous time instant

ti 1 cannot reach the nearest neighbour node at ti , if lT h ; on the other hand, if lT ! h , information at a node from the nearest neighbour node is not fresh.

Fig.3 Spatial distributions of the temperatures and their accuracies by the LLI and LT approaches at the time instant t 't for three values of the time step 't Now, we can explain the inaccuracy of numerical results by both the LLI and LT approaches at several early time instants ti

i't ( i 1, 2, ..., 5 ) if 't

4 u 104 . In this case lT

2 u 102 , hence the choice of

124


1296 uniformly distributed nodes is almost optimal, since h 1 /( 1296 1) 0.02857 . Then, at early time instants, the prescribed boundary value u L 20 (which is remarkably different from the initial value v 1 ) affects also the approximation in the boundary layer L lS x2 L lT , i.e. at points which lie behind the horizon of the heat conduction from the top of the analyzed domain. Quite different is the situation near the bottom of the analyzed domain, where the prescribed boundary value of the temperature is the same as its initial value u0 v 1 . With increasing 't , lT is increasing and hence not only the time interval but also the boundary layer of inaccurate results by the time stepping approaches become wider. On the other hand, in the case of the LT approach the solution at a time instant is independent of the time step and therefore the accuracy at the time instants ti

i u 't2 (which appears after i steps 't2

4 u 102 ) is the same as the

accuracy after (10 u i ) steps 't3 4 u 103 and/or after (100 u i ) steps 't4 4 u 104 . These conclusions are confirmed by the numerical results presented in Fig.2 and also in Fig.3. Thus, short time steps in time stepping approach are required not only for getting time evolution at early time instants, but also for acceptable accuracy at later time instants. On the other hand, the LT approach is indifferent to the time steps and accurate results can be obtained also at later time instants. The only disadvantages of the LT approach is lower computational efficiency when the solution is required at many time instants, since the set of algebraic equations is to be solved for several values of the Laplace transform parameter for each time instant.

Summary The local weak formulation is proposed for solution of transient heat conduction in FGM. The spatial variation of temperature is approximated by using the PIM, while the time dependence is treated either by the Laplace transform or by the time stepping method. The time evolution as well as the spatial distribution of the accuracy of numerical results has been studied and discussed.

Acknowledgements This article has been produced with the financial assistance of the European Regional Development Fund (ERDF) under the Operational Programme Research and Development/Measure 4.1 Support of networks of excellence in research and development as the pillars of regional development and support to international cooperation in Bratislava region/Project No. 26240120020 Building the centre of excellence for research and development of structural composite materials – 2nd stage. This work has been partially supported by the Slovak Science and Technology Assistance Agency registered under number APVV-0032-10, the Slovak Grant Agency VEGA-2/0039/09 and the German Research Foundation (DFG, ZH 15/14-1), which are gratefully acknowledged.

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] G.R. Liu Mesh free methods, moving beyond the finite element method, CRC Press, Boca Raton (2003). [4] V. Sladek, J. Sladek, Ch. Zhang Computational Mechanics, 41, 827-845 (2008).

[5] V. Sladek, J. Sladek, M. Tanaka, Ch. Zhang Engineering Analysis with Boundary Elements, 29, 1047-1065 (2005).

Large Deflection Analysis of Plates Stiffened by Parallel Beams E.J. Sapountzakis1 and I.C. Dikaros2 1 School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece, [email protected] 2 School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece, [email protected]

Keywords: Elastic stiffened plate, plate reinforced with beams, nonlinear analysis, bending, nonuniform torsion, warping, ribbed plate, boundary element method, slab-and-beam structure

Abstract. A boundary element method for the nonlinear analysis of plates stiffened by arbitrarily placed parallel beams of arbitrary doubly symmetric cross section subjected to arbitrary loading is developed. The plate-beam structure is assumed to undergo moderately large deflections and the nonlinear analysis is carried out by retaining nonlinear terms in the kinematical relations. Introduction Structural plate systems stiffened by beams are widely used in buildings, bridges, ships, aircrafts and machines. Stiffening of the plate is used to increase its load carrying capacity and to prevent buckling especially in case of in-plane loading. Moreover, composite reinforced concrete slabs stiffened by steel or prestressed concrete beams have been widespread in recent years due to the economic and structural advantages of such systems. However, these structures are prone to failure of the bond between the plate and the beams. Besides, having in mind the importance of weight saving in engineering structures, the study of nonlinear effects on the analysis (large deflection analysis) of stiffened plates becomes essential. This non-linearity results from retaining the squares of the slopes in the strain–displacement relations (intermediate non-linear theory). In this paper a general solution for the nonlinear analysis of plates stiffened by arbitrarily placed parallel beams of arbitrary doubly symmetric cross section subjected to arbitrary loading based on the structural model of Sapountzakis and Mokos, proposed in [1] is presented. According to this model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting in two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. The utilization of two interface lines for each beam enables the nonuniform torsional response of the beams to be taken into account. The unknown distribution of the aforementioned integrated tractions is established by applying continuity conditions in all directions at the two interface lines. Six boundary value problems are formulated and solved using the Analog Equation Method (AEM) [2], a BEM based method. The solution of the aforementioned plate and beam problems, which are nonlinearly coupled is achieved using iterative numerical methods. Statement of the problem Consider a thin plate of homogeneous, isotropic and linearly elastic material with modulus of elasticity E , shear modulus G and Poisson’s ratio Q , having constant thickness h p and occupying the two-dimensional multiply connected region : of the x, y plane bounded by the * j j

0,1, 2,..., K boundary curves, which are

piecewise smooth, i.e. they may have a finite number of corners. The plate is stiffened by a set of i 1, 2,..., I arbitrarily placed parallel beams of arbitrary doubly symmetric cross section of area Abi , which may have either internal or boundary point supports. The material of the beams is considered to be homogeneous, isotropic and linearly elastic with modulus of elasticity Ebi , shear modulus Gbi and Poisson’s ratio Q bi . For the sake of convenience the x axis is taken parallel to the beams. The stiffened plate is subjected to the lateral load g g x , x : ^ x, y` . For the analysis of the aforementioned problem a global coordinate system Oxy for the

126


analysis of the plate and local coordinate ones Oi xi y i corresponding to the centroid axes of each beam are employed (Fig.1a). The solution of the problem at hand is approached by the improved model proposed by Sapountzakis and Mokos in [1]. According to this model, the stiffening beams are isolated from the plate by sections in its lower outer surface, taking into account the arising tractions at the fictitious interfaces. Integration of these tractions along each half of the width of the i-th beam results in line forces per unit length in all directions in two interface lines, which are denoted by qixj , qiyj and qizj ( j 1, 2 ) encountering in this way the nonuniform distribution of the interface transverse shear forces qiy . The aforementioned integrated tractions result in the loading of the i-th beam as well as the additional loading of the plate. Their distribution is unknown and can be established by imposing displacement continuity conditions in all directions along the two interface lines, enabling in this way the nonuniform torsional response of the beams to be taken into account. The arising additional loading at the middle surface of the plate and the loading along the centroid axis (coinciding with the shear center axis) of each beam can be summarized as follows n

t

Middle Surface of the Plate

( 0)

s Hole K

( K )

H ole 1

LI

( 1 )

q

i

Beam i

bf

x, up

b 2f

L1

f ji

m ipy1

qxi 1

x, u p

*

K * j 0 M

f ji

i mbzj

i mbyj

2 xi

hbi

f ji : Interface line (j=1,2)

f ji

1

z, wp

f ji 1

mipx2

qxi 2 m py2

mipx1 2

xi

y, vp

Beam 1

y, vp

qzi i

i z1

q iy1

Li

( )

q iy 2

b f

Beam I

Oi { C i

i

i mbxj

q

q xji

i zj

q iyj

yi

yi

z

i

zi

(a) (b) Figure 1. Two dimensional region occupied by the plate (a) and structural model and directions of the additional loading of the plate and the i-th beam (b). a. In the plate (at the traces of the two interface lines j=1,2 of the i-th plate-beam interface) (i) A lateral line load qizj . (ii) A lateral line load wmipyj wx due to the eccentricity of the component qixj from the middle surface of the plate. mipyj

qixj h p 2 is the bending moment..

(iii) A lateral line load wmipxj wy due to the eccentricity of the component qiyj from the middle surface of the plate. mipxj

qiyj h p 2 is the bending moment..

(iv) An inplane line body force qixj at the middle surface of the plate. (v) An inplane line body force qiyj at the middle surface of the plate.

b. In each (i-th) beam ( O i x i y i z i system of axes) (i) A perpendicularly distributed line load qizj along the beam centroid axis Oi xi . (ii) A transversely distributed line load qiyj along the beam centroid axis Oi xi . (iii) An axially distributed line load qixj along the beam centroid axis Oi xi . i (iv) A distributed bending moment mbyj

qixj eizj along Oi y i local beam centroid axis due to the eccentricities eizj

of the components qixj from the beam centroid axis. eiz1

eiz 2

hbi 2 are the eccentricities.

Advances in Boundary Element and Meshless Techniques XII i (v) A distributed bending moment mbzj

127

qixj eiyj along Oi z i local beam centroid axis due to the eccentricities

eiyj of the components qixj from the beam centroid axis. eiy1 i (vi) A distributed twisting moment mbxj

bif 4 , eiy 2

bif 4 are the eccentricities.

qizj eiyj qiyj eizj along Oi xi local beam shear center axis due to the

eccentricities eizj , eiyj of the components qiyj , qizj from the beam shear center axis, respectively.

eiz1

eiz 2

hbi 2 and eiy1 bif 4 , eiy 2 bif 4 are the eccentricities. The structural models and the aforementioned additional loading of the plate and the beams are shown in Fig.1b. On the base of the above considerations the response of the plate and of the beams may be described by the following boundary value problems.

a. For the plate. Taking into account geometrical nonlinearity, the Von Kármán plate theory assumption is adopted, according to which the deflection of the plate is no longer small as compared to the plate thickness, while it remains small in comparison with the rest dimensions of the plate. Within the context of this assumption, the displacement field of an arbitrary point of the plate, as implied by the Kirchhoff hypothesis, is given as u p x, y , z u p x, y z

ww p

v p x, y , z v p x , y z

wx

ww p

w p x, y , z

wy

w p x, y

(1a,b,c)

where u p , v p , w p are the inplane and transverse displacement components of an arbitrary point of the plate and u p x, y , v p v p x, y and w p w p x, y are the corresponding components of a point at its middle surface. Applying the principle of virtual work, the system of partial differential equations of equilibrium of the plate in terms of the displacement components, is obtained as up

· ww ¸ p 1 Q ¸ wx 1 Q ¹ 2 2 1 Q w § wu p wv p · ¨§ 2 w w p w w p ¸· ww p 1 Q 2v p ¨¨ ¸¸ 2 1 Q wy © wx wy ¹ ¨ 1 Q wy wx 2 ¹¸ wy 1 Q © 2u p

2 2 1 Q w § wu p wv p · ¨§ 2 w w p w w p ¨¨ ¸ ¸ 2 wy ¹ ¨ 1 Q wx 1 Q wx © wx wy 2 ©

ª§ ° wu p 1 § ww p ¨¨ D 4 w p C ® «¨ ° «¨ wx 2 © wx ¯ ¬«©

· ¸¸ ¹

2·

§ wv ww ¸ Q ¨ p 1 ¨§ p ¸ ¨ wy 2 ¨© wy ¹ ©

· ¸¸ ¹

w 2 w p ww p wxwy wy

w 2 w p ww p wxwy wx

Eh p

1 Q 2 ,

D

Eh3p 12 1 Q 2

¸¸

1 I § 2 i ¦ ¨ ¦ q yjG y y j Gh p i 1¨ j 1 ©

¸¸

·

(2a)

¹

·

(2b)

¹

2 ·º 2

w wp ¸» 1 Q ¸ » wx 2 ¹ ¼»

2 2 ª § wu § wu p wv p ww p ww p · w w p «¨§ wv p 1 § ww p · ¸· ¨ p ¨¨ ¨¨ ¸¸ ¸ ¸ ¸ Q ¨ wx « ¨ y x x y x y y y 2 w w w w w w w w © ¹ © ¹ «¬© ¹ © 2 i i i ª ½ § I 2 w wp ° wm pxj wm pyj ww pj wwipj g ¦ « ¦ ¨ qizj q ixj qiyj 2 ¾ wy wx wx wy wy °¿ i 1 « j 1©¨ ¬

where C

1 I § 2 i ¦ ¨ ¦ qxjG y y j Gh p i 1¨ j 1 ©

2 ·º

1 § ww p ¨¨ 2 © wx

· ¸¸ ¹

· ¸G y y j ¸ ¹

»»

¸» ¸» ¹ ¼»

º

(2c)

¼

are the membrane and the bending rigidities of the plate,

respectively and G y yi is the Dirac’s delta function in the y direction. The corresponding boundary conditions can be written as a p1u pn a p 2 N pn

J p1w p J p 2 R pn

a p3

J p3

E p1u pt E p 2 N pt

G p1

ww p wn

G p 2 M pn

E p3

G p3

(3a,b)

H1k w p H 2 k Tw p

k

H 3k ,H 2k z (3c,d,e)

128


where a pl , E pl , J pl , G pl ( l 1, 2,3 ) are functions specified at the boundary and H lk ( l 1, 2,3 ) are functions specified at the k corners of the plate; u pn , u pt and N pn , N pt are the boundary membrane displacements and forces in the normal and tangential directions to the boundary, respectively. R pn and M pn are the effective reaction along the boundary and the bending moment normal to it, respectively, which, employing intrinsic coordinates (i.e. the distance along the outward normal n to the boundary and the arc length s ) are written as 2 ª ww p w w § w wp D « 2 w p Q 1 ¨ N « wn ws ¨ wswn ws © ¬ ª § w2w ww p · º p ¸» M pn D « 2 w p Q 1 ¨ N « wn ¸ » ¨ ws 2 © ¹ ¬ ¼

R pn

·º ww p ww p ¸ » N pn N pt wn ws ¸» ¹¼

(4a)

(4b)

in which N s is the curvature of the boundary. Finally, Tw p

k

is the discontinuity jump of the twisting moment

Tw p at the corner k of the plate, while Tw p is given along the boundary as § w2w ww p · p ¸ N D Q 1 ¨ ws ¸ ¨ wswn © ¹

Tw p

(5)

The boundary conditions (3a-3d) are the most general boundary conditions for the plate problem including also the elastic support, while the corner condition (3e) holds for free or transversely elastically restrained edges k. All types of the conventional boundary conditions can be derived form these equations by specifying appropriately the functions a pl , E pl , J pl and G pl ( l 1, 2,3 ) (e.g. for a clamped edge it is a p1 E p1 J p1 G p1 1 , a p2

E p2

a p3

E p3 J p 2 J p3 G p 2 G p 3 0 ).

b. For the (i-th) beam. Each beam undergoes transverse deflection with respect to z i and y i axes, as well as axial deformation and nonuniform angle of twist along xi axis. The displacement field of an arbitrary point of a cross section (taking into account moderate large displacements and rotations) can be derived with respect to those of its centroid as

ubi xi yiTbzi xi ziTbyi xi ddxTbx MSP yi , zi i vbi xi , y i , z i vbi xi z i sin Tbx xi yi ª¬«1 cos Tbxi xi º¼» i wbi xi , y i , z i wbi xi y i sin Tbx xi zi ª«¬1 cos Tbxi xi º»¼ i

ubi x i , y i , z i

dvbi

i Tby xi

dwdx cos T x i b i

i sin Tbx xi dxi

i bx

i

i Tbz xi

(6a) (6b) (6c)

dvbi

dwdx sin T x

i cos Tbx xi dxi

i b i

i bx

i

(6d,e)

where ubi , vbi , wbi are the axial and transverse displacement components with respect to the Oi y i z i system of axes;

ubi

ubi xi , vbi

vbi xi

and wbi

wbi xi

are the corresponding components of the centroid Oi ;

are the angles of rotation of the cross section due to bending, with respect to its i dx denotes the rate of change of the angle of twist Tbx xi regarded as the torsional curvature

i i i i Tby Tby xi , Tbz Tbz xi

i centroid; dTbx

and M SP is the primary warping function with respect to the cross section’s shear center (coinciding with its


129

i centroid). Employing again the principle of virtual work and considering the angle of twisting rotation Tbx and i the transverse displacement vb of the i-th beam to have relatively small values, the governing differential equations in terms of the displacement components can be written as

§ d 2u i dwi d 2 wi · b b b¸ Ebi Abi ¨ i i2 ¸ ¨ dxi 2 dx dx © ¹ Ebi I zi Ebi I iy

Ebi CSi

d 4vbi

Nbi

dxi 4 d 4 wbi dxi 4

Nbi

d 4Tbx dx

(7a)

j 1

3 i 2 i 2 i i § d 4 wi i b T 2 d wb dTbx d wb d Tbx Ebi I zi Ebi I iy ¨ i2 i 4 bx i3 i ¨ dx dx dx dxi 2 dxi 2 © dx

d 2 vbi

d 2 wbi

2

j 1©

d 2Tbx dx

§

¦ ¨¨ qizj qixj

dxi 2

Gbi I ti

i4

2

¦ qixj

i2

Nbi

dwbi dxi

2

§

¦ ¨¨ qiyj qixj j 1©

dvbi dxi

i · dmbzj ¸ (7b) dxi ¸¹

i · dmbyj ¸ dxi ¸¹

(7c)

ª 2 i 2 i § 2 i d vb d wb d wb Ebi I zi Ebi I iy « ¨ « dxi 2 dxi 2 ¨ dxi 2 Abi dxi 2 © ¬«

I ip d 2Tbx

· ¸ ¸ ¹

2 · i º ¸ Tbx » » ¸ ¹ »¼

2

§

i qixj ¦ ¨¨ mbxj j 1©

i · I ip dTbx ¸ i Ab dxi ¸¹

(7d) where I iy , I zi are the principal moments of inertia; I ip is the polar moment of inertia, while I ti and CSi are the torsion and warping constants of the i-th beam with respect to the cross section’s shear center (coinciding with its centroid). The expression of the axial stress resultant Nbi is written as

Nbi

ª i du 1 § dwi Ebi Abi « b ¨ b i « dx 2 ¨© dxi «¬

· ¸ ¸ ¹

2º

» » »¼

(8)

Moreover, the corresponding boundary conditions of the i-th beam at its ends x 0, l are given as

abi 1ubi D bi 2 Nbi D bi 3 i Ebi1vbi Ebi 2 Rby Ebi 3 i J bi 1wbi J bi 2 Rbz i i G bi1Tbx G bi 2 M bt

(9) i Ebi1Tbz i i J b1Tby

J bi 3

G bi1

G bi 3

i Ebi 2 M bz i i J b 2 M by

Ebi 3 J bi 3

i dTbx i G bi2 M bw dx

G bi3

(10a,b) (11a,b) (12a,b)

i i where the angles of rotation of the cross section due to bending Tby , Tbz given from eqns. (6d), (6e) are

simplified to i Tby

dwbi dx

i Tbz

i

dvbi dx

i

dwbi i Tbx dxi

(13a,b)

i i i i Rby , Rbz and M bz , M by are the reactions and bending moments with respect to y i , z i axes, respectively, given

as i Rby

Nbi

3 i 3 i § d 2 wi dT i b bx d wb T i d vb Ebi I zi ¨ bx 2 3 i i i ¨ dx dx dx dx dxi 3 ©

dvbi i

2 i i · · § d 3 wi i b T d wb dTbx ¸ ¸ Ebi I iy ¨ bx 3 2 i i ¸ ¨ dx dx dxi ¸¹ ¹ ©

(14a)

130

i Rbz

Nbi

dwbi


Ebi I iy

d 3 wbi

(14b)

dxi dxi3 2 i 2 i §d w i b T d vb Ebi I zi ¨ bx i 2 ¨ dx dxi 2 ©

i M bz

· d 2 wbi i ¸ Ebi I iy Tbx ¸ dxi 2 ¹

i M by

Ebi IYi

d 2 wbi

(14c,d)

dxi 2

i i M bt , M bw are the torsional and warping moments at the boundaries of the beam, respectively, given as i M bt

Gbi I ti

i dTbx

dxi

Ebi CSi

i d 3Tbx

dxi3

Nbi

i I ip dTbx Abi dxi

i M bw

Ebi CSi

i d 2Tbx

(15a,b)

dx 2

i i i i i i i Finally, D bk ,Ebk ,Ebk ,J bk ,J bk ,G bk ,G bk ( k 1, 2,3 ) are functions specified at the boundaries of the i-th beam ( x 0, l ). The boundary conditions (9-12) are the most general boundary conditions for the beam problem including also the elastic support. All types of the conventional boundary conditions (clamped, simply supported, free or guided edge) can be derived form these equations by specifying appropriately the aforementioned coefficients. Eqns. (2), (7) constitute a set of seven coupled and nonlinear partial differential equations including thirteen i unknowns, namely u p , v p , w p , ubi , vbi , wbi , Tbx , qix1 , qiy1 , qiz1 , qix 2 , qiy 2 , qiz 2 . Six additional equations are

required, which result from the displacement continuity conditions in the direction of xi , y i and z i local axes along the two interface lines of each (i-th) plate – beam interface. These conditions can be expressed as In the direction of xi local axis: i h p wwip1 hbi dwbi bif § dvbi dwbi i · dTbx u ip1 ubi ¨ along interface line 1 ( f ji 1 ) (16a) Tbx ¸ + i ISiP i i i ¸ dx f1 2 wx 2 dx 4 ¨© dx dx ¹ i h p wwip 2 hbi dwbi bif § dvbi dwbi i · dTbx u ip 2 ubi T I iP along interface line 2 ( f ji 2 ) ¨ ¸+ (16b) bx ¸ dxi S f 2 2 wx 2 dxi 4 ¨© dxi dxi ¹

In the direction of y i local axis:

vip1 vbi

h p wwip1 2

hi b T ibx along line 1 ( f ji 1 ) 2 wy

vip 2 vbi

h p wwip 2 2

hi i b T bx along line 2 ( f ji 2 ) 2 wy

Lx 18 m

FF A

g

a x

0, 2m CC

CC

hb

Ly

9m

(17a,b)

3m B FF

y

1m Ly

5m 9m

a

(a) Figure 2. Plan view (a) and section a-a (b) of the studied stiffened plate.

hb m

It m4

CS m 6

0,50 1,00 1,50 2,00

2,85863 E–02 1,40579 E–01 2,93644 E–01 4,57382 E–01

3,17495 E–04 1,34406 E–04 3,78965 E–03 2,03227 E–02

Table 1: Torsion and warping constants of the studied stiffening beam for various beam heights hb .

(b)

Nonlinear analysis hb m

Linear analysis

0,50 1,00 1,50 2,00

0,2674 0,0796 0,0450 0,0336

0,50 1,00 1,50 2,00

0,7231 0,5401 0,5018 0,4878

0,50 1,00 1,50 2,00

0,2670 0,1120 0,0891 0,0833

FEM FEM Solid model Shell-shell model [3] [3] Center of the plate 0,2093 0,1980 0,2100 0,0707 0,0778 0,0885 0,0396 0,0498 0,0622 0,0292 0,0352 0,0479 Middle of the free edge A of the plate 0,4477 0,4325 0,4470 0,4085 0,3990 0,4310 0,3946 0,3875 0,4251 0,3890 0,3810 0,4215 Middle of the free edge B of the plate 0,2626 0,2210 0,2680 0,1231 0,1230 0,1790 0,0952 0,1030 0,1570 0,0877 0,0918 0,1450

AEM Present study

FEM Shell-beam model [3] 0,2110 0,0581 0,0286 0,0189 0,4529 0,4333 0,4228 0,4187 0,2570 0,1550 0,1400 0,1360

Table 2: Displacement w p m of the studied stiffened plate for various beam heights hb .

8

6

4

2

0 0

2

4

6

8

w p max

10

12

14

0, 4477m

16

18

0.46 0.44 0.42 0.4 0.38 0.36 0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02

w p max

(a)

w p max

0, 4470m (c)

0, 4325m (b)

w p max

0, 4529m (d)

Figure 3. Contour lines of w p of the studied stiffened plate for hb 0,5m , employing the present study (a) and a FEM solution using solid elements (b), shell elements (c) or shell and beam elements (d).

132


In the direction of z i local axis: bif i along interface line 1 ( f ji 1 ) wip1 wbi Tbx 4

wip 2 wbi

bif 4

i Tbx along interface line 2 ( f ji

2)

(18a,b)

fj is the value of the primary warping function with respect to the shear center of the beam cross

where ISiP

section (coinciding with its centroid) at the point of the j-th interface line of the i-th plate – beam interface f ji . In all the aforementioned equations the values of the primary warping function M SiP ( y i , z i ) should be set having the appropriate algebraic sign corresponding to the local beam axes. Numerical solution

According to the precedent analysis, the nonlinear analysis of plates stiffened by arbitrarily placed parallel beams subjected to an arbitrary loading reduces in establishing the displacement and force components u p , v p , i w p , ubi , vbi , wbi , Tbx , qix1 , qiy1 , qiz1 , qix 2 , qiy 2 , qiz 2 . These components must satisfy the coupled boundary value problems described by eqns. (2), (3), (7), (9)-(12) subjected to the continuity conditions (16)-(18). The numerical solution of the aforementioned problems is accomplished employing the Analog Equation Method [2].

Numerical example

A rectangular plate ( a u b 18 u 9m2 , h p 0, 2m , E Ebi 3 u 107 kN m 2 , Q Q bi 0, 2 ) stiffened by a rectangular beam of 1 m width eccentrically placed with respect to the plate center line (Fig. 2) subjected to a uniformly distributed load g 160kN m 2 is examined. The plate is clamped along its small edges, while the rest of the edges are free according to the transverse and inplane boundary conditions. The stiffening beam is also clamped at its ends according to the transverse, axial and torsional boundary conditions. In Table 1 the torsion It and warping CS constants of the beam cross section for various beam heights hb are presented. In Table 2 the obtained displacements w p of the stiffened plate at its center and at the middle of the free edges A and B (Fig. 2a) for various beam heights are shown as compared with those obtained from a FEM solution [3,4]. More specifically a shell-beam model using rigid offsets for the plate-beam connection, a shell-shell model and a solid one are employed. For the aforementioned finite element models Euler-Bernoulli beam elements, 4-noded quadrilateral shell elements or 8-noded hexahedral solid elements have been used. It is worth here noting that in the shell-beam model, due to the coupling of various structural elements with rigid offsets, kinematic or static assumptions cannot be always valid at the interface of the plate and the beams. Also, the shell-shell model can be applied only for thin walled rectangular beams, while in the solid model a large number of solid elements in the plate must be used in order to avoid shear or membrane locking effects. In Fig. 3 the contour lines of displacement w p of the stiffened plate for hb 0,5m are presented as compared with those from the aforementioned FEM solution [3,4]. From the aforementioned figure and table, the accuracy of the proposed method is demonstrated.

References [1] [2] [3] [4]

E.J. Sapountzakis and V.G. Mokos, An Improved Model for the Analysis of Plates Stiffened by Parallel Beams with Deformable Connection, Computers and Structures 86, 2166-2181 (2008). 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). FEMAP for Windows. Finite element modeling and post-processing software. Help System Index, Version 10, (2008). Siemens PLM Software Inc., NX Nastran User’s Guide, (2008).


133

Strategies for the Implementation of 2D Quadratic Boundary Elements on Graphics Hardware – GPU Josué Labaki1, Euclides Mesquita2 and Luiz Otávio Saraiva Ferreira3 1,2,3

Department of Computational Mechanics, School of Mechanical Engineering, State University of Campinas – Unicamp. 200 Mendeleyev St., 13083-970. Campinas, SP, Brazil. 1

[email protected]; [email protected]; [email protected]

Keywords: High Performance Computing, Graphics Hardware, Boundary Element Method.

Abstract. In the BEM formulation using continuous elements, some of the terms of the influence matrices [H] and [G] may be given as the sum of two or more coefficients. In parallel implementations such as on graphics hardware (GPU), these two terms can be calculated simultaneously, but they cannot be written simultaneously to the same address of the influence matrices. This paper presents two strategies to overcome this restriction. An elementary but representative two-dimensional potential problem discretized by quadratic boundary elements is solved. The code was developed on an NVidia CUDA programming environment and executed on a GeForce GTX 280 graphics card. 1. Introduction

Graphics hardware (GPU) is currently an important trend of research in high performance computing. It has shown astonishing performances on dealing with problems from many branches of computational mechanics. In previous works, the authors presented implementations of the Boundary Element Method (BEM) on GPU [5, 6, 7]. The performance of the GPU over the CPU was investigated on dealing with the assembly of the influence matrices and other important steps of the BEM. Formulations of the BEM for potential and elastostatic problems with constant elements were treated. In all the cases, the GPU presented an outperformance ranging from one to two orders of magnitude, which increased with the number of elements of the problem. This paper examines the GPU implementation of a particular case of a BEM formulation in which a classical formulation of continuous quadratic elements is adopted. The remaining steps of the method are roughly the same as of other formulations with lower order or discontinuous elements, and were already investigated in previous works of the authors [5, 6, 7]. Araujo and Gray [10] have also presented a substructuring BEM formulation in which problems can be solved by the BEM without the need of assembling influence matrices. The first section of the paper states the problem addressed by this research. This section describes two different strategies to overcome the restriction of two computing units trying to write in a memory address simultaneously. Finally, the presented implementations are used to determine the matrices of influence of a simple potential problem. Its performance is compared with an ordinary CPU serial code. 2. Implementation on the GPGPU

The authors have already investigated the implementation of discontinuous boundary elements in previous papers [5, 6, 7]. It was shown that in that case, every term Hij of the matrix influence matrix of stresses [H] was determined independently of any other term. That formulation is the easiest case to be coded in a parallel algorithm. In the present formulation using continuous elements, a given term Hij can be composed by the sum of two terms of influence [8]. If two different units of calculation – threads – are assigned to

134


calculate these terms of influence, they will not be able to write their results in the address Hij of [H] at the same time, and this issue ought to be worked out carefully. In the present implementation, however, every term of the influence matrix of displacements [G] is independent of each other. This work investigates two strategies to treat the problem of two threads concurring to write simultaneously in the same address of matrix [H]. 2.1 First strategy - GPUA Consider two terms of influence him and hin, which are supposed to be written in the address Hij of matrix [H]. The index i (i=1, 2N) represents the i-th line of [H], and j is the index of a column of [H]. As Hij = him + hin applies for any index i, it can be seen that this summation is independent of the index i, and therefore this operation can be performed in parallel for different values of i. In terms of matrix notation, this means that the calculation of an entire column of [H] (several different lines with different indices i) can be performed in parallel. Consider the reduced example shown in Eqs. (1) and (2). In this example, N = 2, and therefore [H] and [G] are of size 2N u 2N = 4 u 4 and 2N u 3N = 4 u 6.

>H@

ª h11 h13 «h h 23 « 21 « h 31 h 33 « ¬ h 41 h 43

h12 h 22 h 32 h 42

h14 h16 h 24 h 26 h 34 h 36 h 44 h 46

h15 º h 25 »» h 35 » » h 45 ¼ 2 Nu2 N

(1)

>G @

ª g11 g12 «g « 21 g 22 « g 31 g 32 « ¬ g 41 g 42

g13 g 23 g 33 g 43

g14 g 24 g 34 g 44

g16 º g 26 »» g 36 » » g 46 ¼ 2 Nu3N

(2)

g15 g 25 g 35 g 45

Notice that the indices m and n of him and hin go from 1 to 3N = 6, while i=1, 2N. This matrix, in particular, could be calculated in a loop of 3N = 6 indices. In each step, all the terms of a column can be calculated simultaneously. Figure 1 depicts this example. Darkened cells represent the terms that are being added to [H] and [G] in the present iteration. Notice that, in this approach, more than one value is added to the same address of [H], but not concurrently. For instance, in the first iteration, the term h11 is added to H11, and in the third iteration, the term h13 is added to the same address.

Figure 1. Reduced example of the execution of the first algorithm. A number of threads is chosen in order to perform the calculations. As the level of parallelization is one-dimensional (i. e., only the terms of a single column are to be calculated simultaneously), these


135

threads are distributed among one-dimensional thread blocks of 16 threads. The size of a onedimensional grid is calculated so as to contain as many blocks as needed to accommodate all the 2N terms that a column of [H] or [G] presents. In the results section, this first approach is referred to as GPUA. 2.2 Second strategy - GPUB In the second strategy, two independent kernels are created. All the terms of [H] and [G] that must be written in the same address are separated in these two distinct kernels. Because the two kernels do not run simultaneously, there is no concurrent memory access to a same address of the matrices. Figure 2 shows the execution of this second algorithm on the same reduced example of N=2 elements. This time, only two serial steps are performed, which corresponds to the execution of each kernel. In the first step, all the terms of [H] are calculated by the first kernel, as well as 2N u 2N terms of [G]. In the second step, the second kernel calculates the remaining terms of [H] and adds them to the previously calculated values. It also determines the remaining terms of [G].

Figure 2. Reduced example of the execution of the second algorithm. Two-dimensional thread blocks of 4 u 4 threads are created to perform the calculations. Notice that in the execution of the first kernel, 2N u 2N threads are required to perform the calculations. On the other hand, only 2N u N threads are used by the second kernel. The size of a two-dimensional grid is calculated so as to contain as many blocks as needed to accommodate all the 2N u 2N threads for the execution of the first kernel, and 2N u N threads for the second kernel. In the Results section, this second approach is referred to as GPUB. In both strategies GPUA and GPUB, the matrices [H] and [G] are allocated as vectors of size 2N u 2N and 2N u 3N in the global memory of the GPU and passed as argument to the kernel(s) that will perform the calculations of their terms Hij and Gij. The data of the problem, such as the coordinates of the nodes and the incidence of the elements are allocated in the global memory and passed as arguments as well. Because all the data of the problem stay in the global memory throughout all the calculations, they are accessible to all the active threads of the GPU. In parallel execution, each thread of the grid will have its own index i (i=1, 2N). Based on this index, the threads will be able to univocally determine, from the data of the problem (node coordinates, element incidence, etc.) the parameters needed to perform the integration shown by Eqs. (6) and (7). In this paper, ten-node Gaussian Quadrature is adopted to perform these integrations. The ten-terms loop referring to the Gaussian Quadrature is performed sequentially by each thread. When quadratic elements are adopted, part of these integrals is singular, what requires special nodes and weights for Gaussian integration. The discussion of singular integration, however, is out of the scope of this paper. References on this subject are available at Kane [8] and many others. Single-precision floating-point operations were used throughout the present implementation, although the present graphics card (an NVidiaTM GeForce GTX 280) is capable of dealing with doubleprecision floating-point operations. The GPU still underperforms the CPU in double-precision arithmetics, as it was reported by Mesquita et al. [15].

136


The present implementations were applied to determine the matrices of influence of an elementary potential problem with quadratic boundary elements, and the results are reported in the next section. The other steps of the solution of a problem by the BEM involve essentially matrix-vector multiplication and the solution of a dense linear system of equations which are independent on the type (continuous or discontinuous) and order (constant, linear, quadratic, etc.) of the boundary element. The performance of the GPU on dealing with these other steps of the BEM was already investigated by the authors [5]. 3. Results

The present implementations are capable of dealing with two-dimensional problems, discretized by N quadratic boundary elements. As input data, it must be provided a vector containing the coordinates (xi, yi) of the vertices of the 2N nodes and a vector containing the relationship of incidence of the elements (which nodes belong in each elements). The thermal heat conduction problem depicted by Fig. 3 was treated. The problem refers to a square plate of unitary edge. Each edge is discretized by N/4 elements of same length. In the example of Fig. 3, N=8 elements are depicted, in a total of 2N=16 nodes. The red dots denote nodes that are shared by two elements. The black dots denote nodes at the center of their elements.

Figure 3. Two-dimensional square plate of unitary edge.

Figure 4. Time spent by the GPU and the CPU to calculate [H] and [G].


137

The time consumed to fill the matrices [H] and [G] was measured for several numbers of elements N. In the GPU, this time corresponds to the time spent by the specific kernel(s) that calculates these matrices, summed with the time to complete all memory copies necessary for the kernel(s) to be executed. These times are compared to a serial code written in pure C language. In the CPU, this time corresponds to the time spent by the specific function that performs these calculations. Figure 4 shows the elapsed times for values of N between 4 and 1,000 elements. At the beginning of the graphic, it can be observed that there is a number of elements before which the use of CPU is more advantageous than both GPU implementations. The reason to that is that, in order to execute the kernel that calculates [H] and [G] on the GPGPU, a few allocations and copies of memory are needed, which are not necessary in the CPU. As this allocation time is rather short and depends little on the number of elements N, the increase of N causes it to dissolve in the total execution time of the kernel(s). Beyond this point, the superiority of performance of both GPU implementations is observed. In the final experiment, in which a problem of 1,000 elements (2,000 nodes) was considered, the implementation GPUA obtained the matrices [H] and [G] in a time 46.2 times shorter than the CPU. The second approach (GPUB), however, presents a peak of outperformance of 32.1 times when N=100 elements are used, but starts to be comparatively less efficient for larger number of elements (see Figure 5).

Figure 5. Outperformance of GPU implementations over the CPU. 4. Concluding remarks

This paper has described the implementation of continuous quadratic boundary elements for twodimensional potential problems on graphics processing devices. A classical serial implementation was rewritten under the SIMD parallel programming paradigm. Two different approaches are presented to deal with the problem of simultaneous memory access to the terms of the influence matrix [H]. In the first one, concurring access is avoided by dealing with each of the columns of [H] in serial execution. In the second approach, this issue is addressed by using two different kernels to access the memory serially. The paper reports the performances of CPU and these two different parallel GPU algorithms on determining the influence matrices of a simple potential problem.

138


The results for the first approach is the most efficient one for a number of elements larger than N | 280. For the largest case reported, N=1000 elements, the first approach outperformed the CPU by 46.2 times. Numerical results show that the second approach is the most efficient technique up to N | 280 elements, presenting a performance peak at N=100 elements. But at the larger example, N=1000, the second GPU approach still outperforms the CPU by 32.1 times. The examples shown in this article indicate that GPGPUs are able to build Boundary Elements coefficient matrices in a very efficient way, compared to the classical CPU implementation. 5. Acknowledgments

The research leading to this article has been funded by Capes, CNPq, Fapesp and Faepex/Unicamp. This is gratefully acknowledged. 6. References

[1]

Shreiner, D.; Woo, M.; Neider, J.; Davies, T. (eds.), 2005. OpenGL Programming Guide, 5th edition Addison-Wesley. [2] Jones, W., 2004. Beginning DirectX9. Premior Press. [3] CUDA, 2008. Developer’s Zone. http://www.nvidia.com/ object/cuda home.html. [4] Owens, J. D; Luebke, D.; Govindaraju, N.; Harris, M.; Krüger, J.; Lefohn, A. E.; Purcell, T. J., 2007. A survey of general-purpose computation on graphics hardware. Computer Graphics Forum; 26 : 80–113. [5] Labaki, J., Ferreira, L. O. S, Mesquita, E., 2009. Implementation of Constant Boundary Elements for 2D Potential Problems on Graphics Hardware – GPU. 20th International Congress of Mechanical Engineering (COBEM 2009), Gramado. [6] Labaki, J., Ferreira, L. O. S, Mesquita, E., 2009. Implementation of Quadratic Boundary Elements for 2D Potential Problems on Graphics Hardware – GPU. 30th Iberia-Latin-American Congress on Computational Methods in Engineering (CILAMCE 2009), Armacao de Buzios. [7] Labaki, J., Ferreira, L. O. S, Mesquita, E., 2010. Implementation of the Boundary Elements Method for 2D Elastostatics on Graphics Hardware – GPU. 11th Pan-American Congress of Applied Mechanics (PACAM 2010), Foz do Iguaçu. [8] Kane, J. H., 1994. Boundary Element Analysis in Engineering Continuum Mechanics. Prentice Hall Englewood Cliffs. [9] Wloka, M.; Zeller, C.; Fernando, R.; Harris, M., 2004. Programming Graphics Hardware. http://http.download. nvidia. com/ developer/presentations/2004/ Eurographics/ EG_04_TutorialNotes.pdf [10] Araújo, F. C.; Gray, L. J., 2007. On the Efficiency of Generic BE Substructuring Algorithms Based on Krylov Solvers. ICCES – International Conference on Computational and Experimental Engineering and Sciences. Vol.2, no.2, pp.41-46.

Advances in Boundary Element and Meshless Techniques XII A Fast Hierarchical BEM for 3-D Anisotropic Elastodynamics A. Milazzo1 , I. Benedetti1 , M. H. Aliabadi2 1

Dip. di Ingegneria Civile, Ambientale e Aerospaziale, Viale delle Scienze, I90128 Palermo, Italy.

[email protected], [email protected] 2

Dept. of Aeronautics, Imperial College London,South Kensington Campus, London, SW7 2AZ, UK.

[email protected] Keywords: Anisotropic Elastodynamics, Hierarchical Matrices, Fast BEMs

Abstract. In the present paper a fast hierarchical boundary element method for three-dimensional anisotropic elasticity time-harmonic problems is presented. The approach is based on the use of hierarchical matrices for the representation of the collocation matrix, whose admissible low rank blocks are generated by the Adaptive Cross Approximation algorithm (ACA). The performances of ACA are investigated for this kind of problem. The system solution is computed by using the iterative GMRES solver which takes full advantage of the hierarchical format. The accuracy of the proposed technique for anisotropic time-harmonic elasticity is assessed together with its effectiveness in terms of the reduction in memory storage and analysis time. Introduction The inherent anisotropic behavior of composite materials and the complexity arising in their modelling make the use of numerical methods necessary for the dynamic analysis. Thanks to its features, the Boundary Element Method (BEM) has proven effective in the analysis of elastodynamic and wave propagation problems, for which different techniques have been proposed and established as accurate and efficient analysis tools [1, 2]. In the field of BEM for elastodynamics, much effort has been devoted to the analysis of isotropic solids, both in 2-D and in 3-D, whereas not much attention has been dedicated to anisotropic materials, especially for 3-D problems [3–6]. Actually, the lack of explicit closed-form fundamental solutions makes the formulation and implementation of BEM difficult for anisotropic materials. Additionally, it generally results quite timeconsuming because of the requested numerical computation of the fundamental solution kernels [7]. Many investigations have been carried out to improve the computational efficiency of the BEM and different techniques have been proposed, such as the fast multipole method, the methods based on the hierarchical matrices and other approaches. All of the above techniques originate from the idea of reducing the computational complexity of the matrix-vector multiplication, which is the recurrent operation in iterative solvers for linear systems, so as to gain relevant computational advantages in the solution of the unsymmetric and fully-populated BEM system. In this framework, hierarchical matrices provide a purely algebraic tool for the approximation of boundary element matrices, thus resulting appealing for problems in which the analytical closed-form expression of the kernels are not available, like in the anisotropic elastodynamics case. Their application to 3-D anisotropic elastostatic problems has proven effective, allowing remarkable reduction in the required computational time for both assembly and solution, together with memory storage savings [8]. In the present paper, a Fast BEM based on the use of the hierarchical matrices is presented for the solution of anisotropic time-harmonic elastodynamic applications. The core of the proposed technique is the construction of the approximation of suitable blocks of the boundary element collocation matrix through the Adaptive Cross Approximation algorithm, which allows to approximate the selected block by computing only few of the original entries. This leads to reduced assembly time, which for the anisotropic BEM is generally relevant, coupled with reduced memory storage requirements. Additionaly, the hierarchical structure of the resolving matrix together with the use of iterative solvers, leads remarkable reduction of the computational solution time. The effectiveness of the proposed method for time-harmonic elastodynamic analysis of anisotropic solids is numerically demonstrated in the reported applications.

139

140


Numerical model The displacement boundary integral equation for a point x0 of a 3-D elastic body Ω, bounded by the surface Γ and subjected to a time-harmonic load with circular frequency ω, is given by [2, 9] ∗ Uij (x0 , x, ω) tj (x, ω) dΓ (1) cij (x0 ) uj (x0 , ω) + Tij∗ (x0 , x, ω) uj (x, ω) dΓ = Γ

Γ

∗ where Uij (x0 , x, ω) and Tij∗ (x0 , x, ω) are the time-harmonic displacement and traction fundamental solution kernels, whereas uj (x, ω) and tj (x, ω) denote the j-th components of displacements and boundary tractions. The time-harmonic fundamental solution for 3-D solids of general anisotropy was derived by Wang and Achenbach who, employing the Radon transform, obtained the expression of the kernels in terms of an integral over the surface of a unit sphere and proposed a numerical scheme for their computation [7]. The time-harmonic fundamental solution kernels Kij (x0 , x, ω), where K stands for U or T , can be decomposed as the sum of R S (x0 , x, ω), which goes to zero as ω does, ii) a term Kij (x0 , x), which is intwo terms: i) a regular part Kij dependent from ω and corresponds to the static anisotropic fundamental solution. The frequency independent S and TijS present singularities of order O(r−1 ) and O(r−2 ) respectively, with r = dist(x0 , x), so terms Uij that the integrals in Eq. (1) have to be carefully evaluated. The expressions of the anisotropic time-harmonic fundamental solutions can be found in reference [7]. The numerical solution of the boundary integral equation is obtained by the Boundary Element Method [9]. After discretization, numerical integration and application of the boundary conditions, the BEM leads to a system of equations of the form

A (ω) X (ω) = Y (ω)

(2)

where the vector X collects unknown components of the boundary displacements or tractions and the vector Y is obtained as a linear combination of the columns of the influence matrices corresponding to the prescribed displacements and tractions nodal values. In the present work constant boundary elements are employed for the discretization; regular and weakly-singular integrals are computed by Gauss quadrature and Kutt’s quadrature is used for the computation of strongly-singular integrals. Hierarchical BEM for elastodynamics Some recent developments have dealt with the representation and solution of BEM systems, of the form of Eq. (2), by means of hierarchical matrices coupled with iterative solvers, allowing to reduce both assembly and solution computational time as well as memory storage. The application of the hierarchical matrices to anisotropic static boundary element problems has been proposed by the authors [8] and is here extended to time-harmonic elastodynamics. The hierarchical representation of a boundary element matrix consists of a collection of blocks, which correspond to sets of collocation nodes and integration elements. The subdivision of the matrix into blocks is based on a hierarchical partition of the mesh nodes aimed at grouping subsets of nodes, corresponding to contiguous collocation nodes and integration elements, on the basis of some computationally efficient geometrical criterion. The blocks generated from the hierarchical partition of the mesh nodes can be classified in: i) non-admissible blocks, which need to be represented entirely; ii) admissible blocks, which can admit a special compressed representation. The admissibility of a block depends on the following geometrical relation [10] min(diam Ωx0 , diam Ωx ) ≤ η · dist(Ωx0 , Ωx )

(3)

where Ωx0 denote the set of collocation nodes, Ωx the set of elements over which the integration is carried out to compute the block coefficients, and η > 0 is a parameter. An admissible block can be represented in low rank format provided that the asymptotic smoothness of the kernel functions is satisfied [10, 11], as it holds for ∗ (x0 , x, ω) and Tij∗ (x0 , x, ω). The low rank representation of the m × n admissible block C is Uij C Ck = Q · WT =

k i=1

qi · wTi

(4)


141

(b) Mesh and examined block.

Figure 1: Bar geometry and mesh. where the matrices Q and W are of order m × k and n × k, respectively, being k the rank of the new representation. The approximating block Ck satisfies the relation C − Ck F ≤ εc CF , where · F represents the Frobenius norm and εc is the set accuracy. The low rank blocks are built by computing and storing only some of the entries of the original blocks, which allow to compute the columns qi and wi of the representation (4) through the Adaptive Cross Approximation algorithm (ACA) [10, 11]. As pointed out in reference [8], the application of ACA to the approximation of blocks stemming from numerical computation of the kernels can be affected by the employed computational scheme. As the complex exponential term appearing in the time-harmonic fundamental solutions exhibits oscillatory behavior, this topic deserves particular attention in the present approach. The anisotropic time-harmonic fundamental solutions are computed according to the scheme proposed by Wang and Achenbach [7]. This scheme transforms the required integration over the surface of a unit sphere into the integration over the domain [0, 1] × [0, 2π], to which standard Gauss quadrature of order M × N is applied. The order of quadrature employed influences the accuracy of the fundamental solution, which can be assessed introducing the following error criterion [12] err (Kref , K) =

Kref − KF Kref F

(5)

where K = [Kij ] is the considered kernel, Kref is its convergence value. Tables 1 and 2 report the kernel error for different dimensionless circular frequencies ω = ω · L/c, where c and L are reference phase velocity and length respectively, taken as the corresponding maximum in the problem. The configuration assumed is shown in Fig. 1(a) and the material properties are set for a trigonal material [8]. The presented results are representative of many performed tests, which show, as expected, that the requested quadrature order grows with the degree of anisotropy of the material, the excitation frequency and the maximum geometric dimension of the examined problem. The behavior of the fundamental solution influences the convergence of the approximation of the low-rank blocks through ACA. The ACA algorithm computes the approximating low-rank block by using some of the entries of the original block, which are obtained by integrating the fundamental solution kernels. Therefore, it should be expected that the accuracy in the computation of the fundamental solutions affects the performance of ACA. This effect is shown in Figure 2 where the study on the convergence of a low rank anisotropic block

Table 1: Error for the Uij∗ (x0 , x, ω) kernel. Integration ω 0.2π π 2π 4π 5.4π

6 × 12 10−4

9.22 × 2.65 × 10−3 3.18 × 10−2 4.22 18.00

12 × 24 10−6

1.34 × 5.45 × 10−6 8.90 × 10−6 1.06 × 10−3 5.72 × 10−2

18 × 30 10−9

1.44 × 8.21 × 10−9 2.48 × 10−8 1.81 × 10−7 1.57 × 10−5

30 × 60 10−15

6.42 × 2.75 × 10−14 7.85 × 10−14 4.23 × 10−13 6.08 × 10−13

60 × 120 6.38 × 10−15 2.20 × 10−14 1.45 × 10−14 1.78 × 10−14 6.70 × 10−14

142

Eds: E L Albuquerque & M H Aliabadi Table 2: Error for the Tij∗ (x0 , x, ω) kernel. Integration ω

6 × 12

12 × 24

18 × 30

30 × 60

60 × 120

0.2π π 2π 4π 5.4π

2.66 × 10−4 3.48 × 10−3 3.25 × 10−2 2.87 16.23

4.56 × 10−7 7.77 × 10−6 1.23 × 10−5 1.31 × 10−3 4.61 × 10−2

5.40 × 10−10 1.19 × 10−8 3.86 × 10−8 2.93 × 10−7 2.11 × 10−5

2.62 × 10−15 2.42 × 10−14 1.53 × 10−13 8.89 × 10−13 1.16 × 10−12

1.62 × 10−15 9.95 × 10−15 1.40 × 10−14 2.52 × 10−14 8.49 × 10−14

of size 1740 × 1740 is presented in terms of its rank and memory ratio (low rank memory storage over full rank memory storage). The study is carried out analyzing the convergence with respect to the accuracy of the fundamental solution for different excitation frequencies and ACA requested accuracies. It is worth noting that the convergence threshold depends on the accuracy required to ACA. The higher the accuracy the higher the convergence rank and the higher the convergence threshold. The presented study refers to the single block corresponding to the sets of collocation point and integration elements shown in Figure 1(b), but a similar behavior is found for each low rank block and then for the entire hierarchical matrix. For the sake of brevity, no results are reported for the influence of the admissibility parameter η and the characteristics of the mesh hierarchical partition. Their effect is similar to that pointed out in reference [8]. Summarizing, the accuracy in the numerical computation of the fundamental solutions represents a crucial issue as the overall performance of the approach, in terms of computational time and memory storage, depends on it. Once the hierarchical representation of the collocation matrix has been built, the solution of the system can be conveniently computed by means of the GMRES iterative solver, which allows to exploit the advantages of the matrix-vector multiplication in hierarchical matrices. Details on the solution procedure can be found in [8]. Integration Order

Memory Storage %

εc=10-5 ; η=2

18x36

30x60 800

εc=10-3 ; η=2

600

600

400

400

200

200

0 100 80

εc=10

-5

;

η=2

εc=10

-3

;

0 100

η=2

80

60

60

40

40

20

20

0

0 0.2π

1π

2π

4π

5.4π

Non-dimensional Frequency

0.2π

1π

2π

4π

5.4π

Non-dimensional Frequency

Figure 2: Block rank and memory storage.

Block Rank

Block Rank

800

Memory Storage %

12x24

6x12


8.0E-011

143

-5.0E-012

HBEM {εc=10-5}

4.0E-011 -1.0E-011

u1

0.0E+000

u2

u3

Displacement [m]

Displacement [m]

HBEM {εc=10-3} FEM

-1.5E-011

-4.0E-011 0

0.2

0.4

x3/L

0.6

0.8

1

0.4

0.425

x3/L

0.45

Figure 3: Hierarchical BEM solution accuracy. Numerical results In the present section, some representative results obtained by applying the proposed fast Hierarchical BEM (HBEM) are presented. First the accuracy of the solution obtained by the HBEM was checked. The clamped bar shown in Fig.1(a), having dimensions H = W = 0.2 m and L = 1 m and subjected to a time-harmonic axial traction with amplitude t¯ = 100 P a acting over the left end surface, was analyzed; elastic coefficients for trigonal material have been considered [8]. Fig. 3 shows the displacements along the x3 parallel line at x1 = 0.2 and x2 = 0.15. The present results have been obtained with a mesh of 2200 elements, at a dimensionless circular frequency ω = 4π and different settings for the ACA accuracy parameter εc . A comparison with converged FEM results is also presented, to assess the accuracy of the present technique. To investigate the efficiency of the approach in terms of computation time and memory storage, the clamped bar previously described was solved for different meshes at different excitation frequencies. The time-harmonic HBEM performances are analyzed in terms of: (i) assembly and solution speed-up ratios, defined as the ratio between the computation time of the HBEM and that of the standard collocation BEM; (ii) memory storage ratio, defined as the ratio between the storage memory for the resolving matrix required by the HBEM and that required by the standard collocation BEM, and (iii) percentage of the resolving matrix approximated through ACA. Figure 4 shows the results obtained for ω = π (solid line) and ω = 4π (dotted line). It is observed that the HBEM reduces both assembly and solution time as well as memory storage. Its efficiency improves with the problem size (number of DoFs) and slightly depends on the excitation frequency, as this parameter affects the ACA behavior. Conclusions In this paper a time-harmonic BEM based on hierarchical matrices has been presented for the analysis of anisotropic three-dimensional problems. The ACA algorithm is used to approximate the admissible blocks and an iterative GMRES solver is employed for the solution in conjunction with the hierarchical representation of the solution matrix. It is found that the performance of the technique depends critically on the accuracy of the scheme used to compute the fundamental solution kernels. It has been shown that the HBEM provides accurate solutions. The analysis has also shown the efficiency of the approach, with respect to the conventional BEM, in terms of assembly time, solution time and storage memory, for which the technique allows remarkable savings. The efficiency of the approach grows with the size of the problem so it is appealing to solve large scale problems in anisotropic elastodynamics.


1

80%

0.6

60%

0.4

40%

0.2 0 4000

20%

8000

12000

16000

20000

24000

28000

4000

8000

12000

20000

24000

0% 28000

DoF

DoF Solution Speed-UP

16000

1

100%

0.8

80%

0.6

60%

0.4

40%

0.2 0 4000

Memory Storage

100%

0.8

20%

8000

12000

16000

DoF

20000

24000

28000

4000

8000

12000

16000

20000

24000

ACA Matrix %

Assembly Speed-UP

144

0% 28000

DoF

Figure 4: Hierarchical BEM performances. References [1] G. Manolis and D. E. Beskos. Boundary Element Methods in Elastodynamics. Unwin-Hyman (1988). [2] J. Dominguez. Boundary elements in dynamics. CMP and Elsevier Applied Science (1993). [3] A. Sáez and J. Domínguez. International Journal for Numerical Methods in Engineering, 44, 1283–1300 (1999). [4] A. Sáez and J. Domínguez. Engineering Analysis with Boundary Elements, 25, 203 – 210 (2001). [5] M. P. Ariza and J. Dominguez. Computer Methods in Applied Mechanics and Engineering, 193, 765 – 779 (2004). [6] Y. Niu and M. Dravinski. International Journal for Numerical Methods in Engineering, 58, 979–998 (2003). [7] C. Y. Wang and J. D. Achenbach. Proceedings of the Royal Society A, 449, 441–458 (1995). [8] I. Benedetti, A. Milazzo and M. H. Aliabadi. International Journal for Numerical Methods in Engineering, 80, 1356–1378 (2009). [9] M. H. Aliabadi. The Boundary Element Method, volume 2. Wiley (2002). [10] M. Bebendorf. Numerische Mathematik, 86, 565–589 (2000). [11] M. Bebendorf and S. Rjasanow. Computing, 70 (2003). [12] M. Dravinski and Y. Niu. International Journal for Numerical Methods in Engineering, 53, 445–472 (2002).


145

Coupled Evolution of Damage and Fluid Flow in a Mandel-type Problem Eduardo T Lima Junior1, Wilson S Venturini2 (i.m.), Ahmed Benallal3 1

Scientific Computing and Visualization Laboratory, Federal University of Alagoas – Maceió, Brazil [email protected] 2 3

São Carlos School of Engineering, University of São Paulo – São Carlos, Brazil

Laboratoire de Mécanique et Technologie, ENS de Cachan/CNRS – Cachan, France [email protected]

Keywords: saturated porous media, isotropic damage, consolidation, BEM.

Abstract. Some considerations on the numerical analysis of brittle rocks are presented in this paper. The rock is taken as a poro-elastic domain, in full-saturated condition, based on the Biot’s Theory. The solid matrix of this porous medium is considered to be susceptible to isotropic damage occurrence. 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 elastostatics 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. A geomechanical problem is analyzed in order to illustrate the efficiency of the implemented formulation.

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.

146


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 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 Let us assume the free energy potential per unit volume of a saturated porous medium subject to damage, as,

U\( jk , D, I I0 )

2 1 1 (1 D) jk E djklm lm b 2 M ª¬Tr jk º¼ 2 2

1 2 M I I0 bM I I0 Tr jk 2

(1)

where the constants M and b represent the Biot modulus and Biot coefficient of effective stress, respectively. In full-saturated condition, the lagrangian porosity I measures the variation of fluid content per unit volume of porous material. The bulk density is described by U . The 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 . E djklm represents the isotropic elastic tensor in drained condition, defined

§ dr 2G · ¨K ¸ GkjGlm 2GI kjlm 3 ¹ ©

E dr kjlm

(2)

The bulk modulus K dr and the shear modulus G refer to the drained material and can be obtained experimentally. The fourth order identity tensor is represented by I kjlm . It can be observed that one of the possible sets of parameters for the characterization of porous material is formed by M , b , K dr and G . 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

(3)

(4)

(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

(5)

Using equations (3) and (4) the total stress tensor is written as

V jk

E jklm Hlm DE jklm Hlm b p p0 G jk

(6)

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 .


147

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)

(7)

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: Y A D

(8)

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 ¼º

Qk

(9)

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

(10)

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

(11)

(12)

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 fields that keep a linear and proportional relationship between them. 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

*

*

:

(13)

:

³ V jk (q) V jk (q) bG jk p(q) Hijk (s, q)d: ³ H jk (q)Vijk (s, q)d: d

:

*

*

(14)

:

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.

148


In elastostatics, one applies the well-known Kelvin fundamental solutions. By applying the divergence theorem to equation (14), and considering the transient nature of the problem, one obtains the following integral equation for displacements on the boundary points S:

³ T k (Q)uik (S, Q)d* ³ Tik (S, Q)u k (Q)d*

Cik u k (S)

*

*

*

*

(15)

H*ijk (S, q)d: ³ V djk (q)H*ijk (S, q)d: ³ bG jk p(q) :

:

The stresses at internal points are obtained by differentiating equation (15), now written for internal points, and applying Hooke's law, which leads to

V ij (s)

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: *

*

:

(16)

@ 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: *

:

*

(17)

:

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: *

*

:

K indicates the outward normal direction to the boundary. Assuming Qk,k

(18)

:

I (see (10)) 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: *

*

(19)

:

For convenience, it is possible to take the derivative I(q) from (4), 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 * * :

(20)

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 (15), (16) and (20) along the interval 't , leading to the following set of equations, in terms of the variable increments:


149

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)

*

*

*

*

*

:

(21)

³ 'Vdjk (q)H*ijk (S, q)d: :

³ Sijk (s, Q)'u k (Q)d* ³ Dijk (s, Q)'Tk (Q)d* ³ R ijkl (s, q)'Vdkl (q)d:

'Vij (s)

*

TLij ª¬ 'Vdkl (s) º¼ c(s)p(s)

*

:

(22)

³ bGkl R ijkl (s, q)'p(q)d: TLij >bGkl 'p(s)@

:

³ Q*K (s, Q)p(Q)d* ³ p* (s, Q)Q K (Q)d* *

*

(23)

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 (21)-(23), 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)

(24) (25)

1 b ªQP(i) º¼ ^'p(i) ` ª¬QP(i) º¼ > Tr @^'H` M 't ¬ 't

(26)

> @ 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) `

(27)

1 ª ª ºº «¬> I@ M 't ¬QP (i) ¼ »¼ ^'p (i) `

(28)

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 `

(29)

150


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`

(30)

1 ª º b2 1 ª ªQP (i) º º ªQP (i) º > Tr @» «> E @ ª¬QSº¼ > IK @ «> I@ » ¬ ¼ ¬ ¼ t t M ' ' ¬ ¼ ¬« ¼»

(31)

Due to the presence of correction terms associated with damage, equation (29) is non-linear at each time increment, and can be written:

^ `

^Y ^'H ` `

ª¬ E º¼ ^'H n ` > 'Ns @ Np ª¬QSº¼ ^'Vd n ` 0

n

(32)

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:

^Y ^'H ` ` i n

where the derivative

^

w Y ^'Hin ` w ^'H

^

w Y ^'Hin ` w ^'Hin `

i n

`

`

^G'H ` i n

0

(33)

` is the consistent tangent operator.

Numerical Example The validation of the implemented formulation was presented in Lima Junior [20] and Lima Junior et al. [21], based on benchmark cases concerning poroelasticity and damage evolution. It is proposed in this paper the analysis of a plane problem, as shown in Figure 1. It consists of a rectangular area, with 2 m wide and 1 m in height. A load of 20 MN is applied monotonically over 2 s, on impermeable plates placed on the top and bottom faces. The flow occurs only through the lateral faces. The boundary conditions of the problem are inspired by the problem of consolidation proposed by Mandel [22]. The constituent material is Berea sandstone whose properties are defined in Table 1. The discretization used contains 24 boundary elements and 32 domain cells. The four possible material behaviors are considered, being the uncoupled elasto-damage and poroelastic cases and the coupled poro-damage regime. 20 MN

draining surface

draining surface

1m

2m

20 MN

Figure 1 - Problem definition, adopted cells mesh


Table 1 - Parameters of the Berea sandstone

Parameter

G X Xu Ks

I0 k P

Value 6000 MPa 0.2 0.33 36000 MPa 0.19 1.9 x 10-13 m2 1 x 10-9 MPa.s

The central point of the domain is taken as reference for the analysis of the problem. Initially, we observe the behavior in the vertical direction along which the load is applied. Based on the graphs concerning to damage and porodamage regimes in Figure 2, it can be seen the influence of the fluid as mitigation in the evolution of the strains on the solid skeleton, in the presence of damage.

Figure 2 - Vertical strain evolution at the central point

The analysis of Figure 3 allows to visualize that the coupled behavior (porodamage) is governed initially by the poroelastic regime, going to suffer the effects of damage, which starts at around 0.6s analysis (Figure 5).

Figure 3 - Vertical effective stress evolution at the central point

151

152


From around 0.6 s the pore-pressure starts to evolve coupled to the damage level on the material, as shown in Figures 4 and 5 in which it can be seen that the damage initiation, as well as its intensity, are delayed along the time, in the porodamage regime.

Figure 4 – Pore-pressure evolution at the central point

Figure 5 – Damage parameter evolution at the central point

Consider now the problem response along the horizontal direction, also measured at the center of the domain. Figure 6 shows the evolution of horizontal strain over time, considering the different behaviors.

Figure 6 – Horizontal strain evolution at the central point


Considering that this is not the direction of load application, the effects of loading are manifested only partially in the horizontal direction, due to Poisson's effect. However, the fluid flows preferentially along horizontal direction, due to the imposed boundary conditions. The comparison between the strain curves regarding the damage and porodamage regimes in Figure 6, allows the verification of the predominance of the effects due to the presence of fluid. The horizontal strains induced in the poroelastic case are higher than those caused in the damage case over the major part of the analysis. The values of effective stress in horizontal direction are negligible, considering the boundary conditions of the problem. From Figure 7 we observe the increase in effective stress caused by the consideration of the damage in poroelastic problem.

Figure 7 – Horizontal effective stress evolution at the central point

In order to illustrate conclusively the difference between the measured responses in the central point along the two coordinate directions, it is presented in Figures 8 and 9 the evolution of the parts of stress tensor, admitting the porodamage coupled regime. The predominance of the poroelastic behavior along the horizontal direction becomes clear.

Figure 4.47 – Stress balance in the horizontal direction, at the central point

153

154


Figure 4.48 – Stress balance in the vertical direction, at the central point

Conclusions and Perspectives A BEM formulation to poro-elasto-damaged material was applied to a Mandel-type problem. The model has shown a reasonable level of coupling between the damage and the fluid seepage. The predominance of each process becames clear in the two different directions. 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, specially about the permeability. Some developments in this way are being made in the presented model, in order to improve the solid-fluid 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).


155

[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]E.T.Lima Junior Isotropic damage phenomena in saturated porous media: a bem formulation Ph.D. thesis, University of São Paulo/Laboratory of Mechanics and Technology at Cachan (2011). [21]E.T.Lima Junior, W.S.Venturini and A.Benallal On the numerical analysis of damage phenomena in saturated porous media International Conference on Boundary Element and Meshless Techniques, 2010, Berlim. Advances in Boundary Element Techniques v.XI. Eastleigh: EC ltd 498-507 (2010). [22]J. Mandel, Consolidation des sols (étude mathématique). Geotechnique 3 287–299 (1953).

156


Stability analysis of composite laminate plates under non-uniform stress fields by the boundary element method P. C. M. Doval1 , E. L. Albuquerque2 , and P. Sollero3 1 Department of Mechanical and Material, Federal Institute of Maranhão Av. Get´ ulio Vargas,04, São Lu´ıs, Maranh˜ ao, Brazil, [email protected] 2

Faculty of Technology, University of Bras´ılia, Campus Universitário Darcy Ribeiro Bras´ılia, DF, Brazil, CEP 70.910-900, [email protected] 3

Department of Computational Mechanics , State University of Campinas Rua Mendeleiev, 200, Campinas, S˜ ao Paulo, Brazil, CEP 13.083-970, [email protected]

Keywords: Stability of structures, buckling, composite laminate plates, radial integration method, boundary element method. Abstract. This paper presents a boundary element formulation to investigate the onset of instability of composite laminate plates with arbitrary geometry. Stresses caused by external loads are calculated by the formulation of plane elasticity boundary element method. Then, these stresses are introduced as body forces in the classical formulation of plates. The domain integrals due to body forces are transformed into boundary integrals using the radial integration method. In this method, body forces are approximated by a sum of radial basis functions, called approximation functions, multiplied by coefficients to be determined. Functions used for approximations are known as thin plate splines. Some numerical examples are analyzed in which critical loads, buckling modes, and coefficients of buckling are calculated. The accuracy of the proposed formulation is assessed by comparison with results from literature.

Introduction An understanding of buckling of structural components under compressive load has become particularly important with the introduction of steel and high-strength alloys in engineering structures, which resulted in more optimized components than those used in previous projects. Buckling analysis of compression panels also is particularly important in aerospace structures. Structures built with these materials and slender members may fail when subjected to compressive loads in your plan. In some cases these failures are not by direct compression, but for lateral buckling. The finite element method (FEM) is currently one of the most used tools by researchers to study the engineering problems of buckling of plates. Potentially powerful and relatively new, the numerical method of boundary elements (BEM) has also shown excellent results in the study of buckling of plates. Syngellakis and Elzein [1] present solutions for the buckling of plates by boundary element method based on Kirchhoff’s theory in different load conditions and support. Nerantzaki and Katsidelakis [2] developed a boundary element method for analysis of buckling of plates with variable thickness. Buckling analysis of isotropic plates under non-uniform stress fields by the boundary element method isotropic plates was presented in [3]. To the best of author’s knowledge, the only work that presents a boundary element formulation applied to non-isotropic plates is due to [4] who presented an orthotropic formulation with a domain discretization. The cited articles either discretize the domain into cells and compute the domain integrals by direct integration over the area of each cells or use dual reciprocity boundary element method to transform domain integrals into boundary integrals. Both approaches have some drawbacks. The


157

discretization of the domain into cells represents the lost of the main advantage of boundary element formulations that is the boundary only discretization. In the dual reciprocity boundary element method, particular solutions are necessary to be computed. Although particular solutions are already available in literature for many approximation functions, it requires lots of modification on the code when new approximation functions are used. In this paper, a boundary element formulation for the stability analysis of general isotropic plates with no domain discretization is presented. Classical plate bending and plane elasticity formulations are used and the domain integrals due to non-uniform body forces are transformed into boundary integrals using the radial integration method. For the first time the radial integration method is used in the analysis of stability of plates with non-uniform stress fields. As it will be seen, the formulation does not require neither domain discretization nor computation of particular solutions. Numerical results for a plate with a square hole are presented to assess the accuracy of the method. Buckling coefficients computed using the proposed formulation is compared with results available in literature.

Governing equations Basically, the classic problem of buckling is a geometrically nonlinear problem described by a set of three differential equations which can be uncoupled and linearized in the case of elastic critical loads. In the absence of body forces equations that describe the buckling of plates are given by:

Nij,j = 0,

(1)

D11 u3,1111 + 4D16 u3,1112 + 2(D12 + D66 )u3,1122 + 4D26 u3,1222 + D22 u3,2222 = Nij u3,ij ,

(2)

where i, j, k = 1, 2; uk is the displacement in directions x1 and x2 , u3 stands for the displacement in the normal direction of the plate surface; Nij are the in-plane stress components, D11 , D22 , D66 , D12 , D16 , and D26 are the anisotropic thin plate stiffness constants.

Boundary integral equations The determination of in-plane stress resultants in the domain is the first step in the solution of plate buckling. The in-plane boundary integral equation for displacements, obtained by applying the reciprocity and Green theorems in equation (1), is given by [5]:

cij uj (Q) +

Γ

t∗ik (Q, P )uk (P )dΓ(P ) =

Γ

u∗ik (Q, P )tk (P )dΓ(P ),

(3)

where ti = Nij nj is the traction in the boundary of the plate in the plane x1 − x2 , and nj is the normal at the boundary point; P is the field point; Q is the source point; and asterisks denote fundamental solutions. The anisotropic plane elasticity fundamental solutions can be found, for example, in [6]. The constant cij is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. The in-plane stress resultants at a point Q ∈ Ω are written as:

cik Nkj (Q) +

Γ

∗ Sikj (Q, P )uk (P )dΓ(P ) =

Γ

∗ Dijk (Q, P )tk (P )dΓ(P ),

(4)

where Dikj and Sikj are linear combinations of the plane-elasticity fundamental solutions. Due to stress concentrations in the geometry, stress resultants are non-uniform over the domain. The plate buckling equations are derived from the plate bending equations. Critical load factors are are introduced into the equations as multiplication factors of body forces or transverse loads. Critical buckling loads are loads at which plates suddenly undergo considerable deflections in the transverse

158


direction due to loads applied in the plane of the plate. The relation between the applied load and critical loads are given by the critical load factor λ by the following equation: Nijc = λNij

(5)

Nijc

where are critical stress resultants that are obtained when critical loads are applied. The integral equation for the plate buckling formulation, obtained by applying reciprocity and Green theorems at equation (2), is given by:

Ku3 (Q) +

=

Nc i=1

Γ

Rci (P )u∗3ci (Q, P ) +

+λ

Vn∗ (Q, P )w(P ) − m∗n (Q, P )

Ω

u3 Nij u∗3,ij dΩ +

Γ

Γ

Nc ∂w(P ) dΓ(P ) + Rc∗i (Q, P )u3ci (P ) ∂n i=1

Vn (P )u∗3 (Q, P ) − mn (P )

∂u∗3 (Q, P ) dΓ(P ) ∂n

ti u∗3 u3,i − ti u3 u∗3,i dΓ ,

(6)

where ∂() ∂n is the derivative in the direction of the outward vector n that is normal to the boundary Γ; mn e Vn are, respectively, the normal bending moment and the Kirchhoff equivalent shear force on the boundary Γ; Rc is the thin-plate reaction of corners; u∗3ci is the transverse displacement of corners; λ is the critical load factor; the constant K is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. As in the previous equation, an asterisk denotes a fundamental solution. Fundamental solutions for anisotropic thin plates can be found, for example, in [7]. As can be seen in equation (6), domain integrals arise in the formulation owing to the contribution of in-plane stresses to the out of plane direction. In order to transform these integrals into boundary integrals, consider that a body force b is approximated over the domain as a sum of M products between approximation functions fm and unknown coefficients γm , that is: b(P ) ∼ =

M

γ m fm .

(7)

m=1

The approximation function used in this work is: fm = r2 log r,

(8)

The approximation function (8) is known as thin plate splines, where r is the distance between the centre of the radial basis function and integration points. It has already been shown in literature [8] that this approximation functions presents faster convergence when compared to other approximation functions. Equation (7) can be written in a matrix form, considering all boundary and domain source points, as: b = Fγ

(9)

γ = F−1 b.

(10)

Thus, γ can be computed as:

Body forces of integral equation (6) depend on displacements. So, using equation (10) and following the procedure presented by [9], domain integrals that come from these body forces can be transformed into boundary integrals. As can be seen in equation (6), the body force that generates domain integrals is given by:


159

b = Nij u3 .

(11)

So, we need to compute Nij in each integration points. However, we have only the values of Nij at nodes and internal points. Values of Nij in integration points is computed by: Ni (x1 , x2 ) = f (r)F−1 Nij .

(12)

The implementation of the buckling formulation is quite similar to shells. Details on the implementation of the radial integration method for shells are given by [10].

Matrix Equations After the discretization of equations (4) and (6) into boundary elements and collocation of the source points in all boundary nodes, a linear system is generated. It is worth notice that the only loads considered in the linear buckling equations are that related to the in-plane stress Nij and tractions ti that are multiplied by the critical load factor λ. Furthermore, all the known values of u3 , ∂u3 /∂n, Mn , Vn , wci , Rci (boundary conditions) are set to zero. Dividing the boundary into Γ1 e Γ2 , this linear system can be written as:

H11 H12 H21 H22

w1 w2

−

G11 G12 G21 G22

V1 V2

=λ

M11 M12 M21 M22

w1 w2

,

(13)

where Γ1 stands for the part of the boundary where displacements and rotations are zero and Γ2 stands for the part of the boundary where bending moment and tractions are zero. Indices 1 and 2 stand for boundaries Γ1 and Γ2 , respectively. Matrices H, G, and M are influence matrices of the boundary element method due to integral terms of equations (4) and (6). As w1 = 0 and V2 = 0, equation (13) can be written as: H12 w2 − G11 V1 = λM12 w2 , H22 w2 − G21 V1 = λM22 w2

(14)

ˆ 2 = λMw ˆ 2, Hw

(15)

ˆ = H22 − G21 G−1 H12 , H 11 ˆ M = M22 − G21 G−1 11 M12 .

(16)

or,

ˆ e M, ˆ are given by: where, H

The matrix equation (15) can be rewritten as an eigen vector problem 1 w2 , λ

(17)

ˆ ˆ −1 M. A=H

(18)

Aw2 = where,

Provided that A is non-symmetric, eigenvalues and eigenvectors of equation (17) can be found using standard numerical procedures for non symmetric matrices.

160


Numerical results The numerical results are presented in terms of the dimensionless parameter Kcr which is given by: Ncr a2 (19) D22 where, Ncr is the critical load and a is the edge length of the square plate. In this work, at the first moment, it is considered a continuous square graphite/epoxi plate under different boundary conditions, only to compare the results presented in this work with the results presented by [11]. The ratio between length a and thickness h of the square plate is a/h = 100, and the material properties are: elastic moduli E1 = 181 GP a and E2 = 10.3 GP a and Poisson ratio ν12 = 0.28, and shear modulus G12 = 7.17 GP a. At the second moment it is considered a perforated plate (a/b = 5), where b is the edge length of the hole, with the same ratio between length and thickness (a/h = 100), but with three different material properties: quasi-isotropic, with elastic moduli E1 = 210 GP a and E2 = 209 GP a and Poisson ratio ν12 = 0.30, and shear modulus E1 G12 = 2.(1+ν = 80.76 GP a; orthotropic, with elastic moduli E1 = 210 GP a and E2 = 105 GP a and 12 ) Kcr =

.E2 Poisson ratio ν12 = 0.30, and shear modulus G12 = E1 +EE21+2E = 60 GP a; and anisotropic, with 2 ν12 elastic moduli E1 = 181 GP a and E2 = 10.3 GP a and Poisson ratio ν12 = 0.28, and shear modulus G12 = 7.17 GP a. The mesh used has 24 quadratic discontinuous boundary elements (12 elements of equal length at the external boundary and 12 elements of equal length at the hole) with 49 uniformly distributed internal points, for the first case, and 48 for the second case (perforated plate). The plate is under uniformly uniaxial compression and the critical load parameter Kcr is computed considering all edges simply-supported (SSSS) and all edges clamped (CCCC). The results of the critical load parameters Kcr obtained by the boundary element formulation under different boundary conditions, for a square composite laminate plate - first case - are shown in Table 1 together with results obtained by [11] using a boundary element formulation. Table 2 shows the critical load parameters Kcr , obtained by the boundary element formulation under different boundary conditions, for a perforated quasi-isotropic, orthotropic an composite laminate square plates.

Table 1: Critical load parameter Kcr for a continuous square composite laminate plate Case 1 2 3 4

Boundary Conditions SSSS SSSS CCCC CCCC

Loadings N1 N2 N1 N2

= 0 = 0 = 0 = 0

K [11] 130.82 71.53 493.70 168.37

K This Work 128.21 75.83 486.29 164.48

Error % 2.00 6.00 1.50 2.30

Table 2: Critical load parameter Kcr for a graphite/epoxy plate with a square hole Case 1 2 3 4

Boundary conditions SSSS SSSS CCCC CCCC

Loadings N1 N2 N1 N2

= 0 = 0 = 0 = 0

Quasi Isotropic 35.30 21.54 31.24 22.81

This work Orthotropic 34.35 21.86 41.78 23.22

This work Anisotropic 112.30 13.29 265.11 13.46

The figure 3 show the first buckling mode for a simply-supported composite laminate plate with compression load in the longitudinal direction; the figure 4 show the first buckling mode for the same plate with loading in the transverse direction. It is noticed that the shape of the first buckling mode,

Advances in Boundary Element and Meshless Techniques XII with a transverse load (N2 = 0) is the same as the first buckling mode to a longitudinal load (N1 = 0), and the values for the buckling coefficient and the values of buckling modes are well below those presented in [11], as was expected due to weakening of the plate caused by the hole.

Conclusions This paper presented a boundary element formulation for the instability analysis of composite laminate plates with non-uniform stress field. Domain integrals are transformed into boundary integrals by the radial integration method. As the radial integration method does not demand particular solutions, it is easier to implement than the dual reciprocity boundary element method. The formulation is applied for a square plate with a square hole. As it can be seen, there is a good agreement between the results obtained in this work and those presented in literature, for a square composite laminate plate without hole. For the square composite laminate plate with hole, the results obtained with the proposed formulation are in good agreement with expected results.

References [1] S. Syngellakis and E. Elzein. Plate buckling loads by the boundary element method. International Journal for Numerical Methods in Engineering, 37:1763–1778, 1994. [2] M. S. Nerantzaki and J. T. Katsikadelis. Buckling of plates with variable thickness, an analog equation solution. Engineering Analysis with Boundary Element, 18:149–154, 1996. [3] P. C. M. Doval, E. L. Albuquerque, and P. Sollero. Stability analysis of elastic plates under nonuniform sress filds by the boundary element method. In E. N. Dvorkin, M. B. Goldschimit, and M. A. Storti, editors, Proc. CILAMCE/MECOM 2010, pages 699–705, Buenos Aires, Argentina, nov 2010. [4] G. Shi. Flexural vibration and buckling analysis of orthotropic plates by the boundary element method. J. of Solids and Structures, 26:1351–1370, 1990. [5] M. H. Aliabadi. Boundary element method, the application in solids and structures. John Wiley and Sons Ltd, New York, 2002. [6] P. Sollero and M. H. Aliabadi. Fracture mechanics analysis of anisotropic plates by the boundary element method. Int. J. of Fracture, 64:269–284, 1993. [7] 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. [8] P. W. Partridge. Towards criteria for selection approximation functions in the dual reciprocity method. Engineering Analysis with Boundary Elements, 24:519–529, 2000. [9] 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. [10] E. L. Albuquerque and M. H. Aliabadi. A boundary element formulation for boundary only analysis of thin shallow shells. CMES - Computer Modeling in Engineering and Sciences, 29:63– 73, 2008. [11] E. L. Albuquerque, P. M. Baiz, and M. H. Aliabadi. Stability analysis of composite plates by the boundary element method. In R. Abascal and M. H. Aliabadi, editors, Int. Conf. on Boundary Element Techniques - 2008, page Eletronic media, Seville, Spain, July 2008.

161

162


Expedite Implementation of the Boundary Element Method Ney Augusto Dumont and Carlos Andrés Aguilar Marón Department of Civil Engineering Pontifical Catholic University of Rio de Janeiro, PUC-Rio 22451-900, Brazil. e-mail: [email protected] Keywords: Boundary elements, variational methods, meshless methods, large-scale problems

Abstract. The present developments combine the variationally-based, hybrid boundary element method with a consistent formulation of the conventional, collocation boundary element method with the aim to establish a computationally less intensive procedure, although not necessarily less accurate, for large-scale, twodimensional and three-dimensional problems of potential and elasticity. A previous paper had focused on the mathematical fundamentals of the formulation. In the present contribution, the code implementation features are addressed. Both the double-layer and the single-layer potential matrices, H and G, respectively, whose evaluation requires dealing with singular and improper integrals, are obtained in an expedite way that circumvents almost any numerical integration - except for a few regular integrals. The evaluation of results at internal points is straightforward, as the fundamental solutions of the boundary element method are used as the domain trial functions. Boundary-layer effects are adequately taken into account, although special domain functions should be required for the simulation of stress gradients related to notches and cracks, for instance. The main issues related to the solution of large scale problems are addressed, suggesting that the present implementation can be at least as effective as procedures such as the fast multi-pole method. As in this latter method, although both H and G are fully populated matrices, special solution schemes can be conceived to dramatically decrease the storage allocation required in the iterative solution of the matrix system as well as to speed up the evaluation of results at internal points. A convergence study is shown to assess the applicability of the method, its computational effort and some convergence issues. 1.

Introduction

The collocation boundary element method (CBEM), whenever applicable, is a simple, powerful numerical analysis tool [1]. The present contribution is an attempt to show that the CBEM can be still more efficient and powerful – and still easier to implement computationally. (A not lesser contribution is the demonstration that simplicity can be achieved without resorting to exotic concepts such as node displacements from corner points or regularizations.) Some precursory works have already been published on the subject [8] or are being prepared [10]. However, this is the first attempt to summarize the basic concepts that lead to the expedite boundary element method (EBEM) and to show its main features and possibilities of application in an outline that is meant to be itself expedite. 2.

Problem Formulation

An elastic body is submitted to body forces bi in the domain Ω and traction forces ti on part Γσ of the boundary. Displacements ui are known on the complementary part Γu of Γ. The task is to find an adequate approximation of the stress field that satisfies equilibrium in the domain, σ ji, j + bi = 0 in Ω

(1)

also satisfying the boundary equilibrium and compatibility equations, σ ji n j = ti

along Γσ ,

ui = ui

on

Γu

(2)

where n j is the outward unit normal to Γ. Indices i, j, (also k, l) may assume values 1, 2 or 3, as they refer to the coordinate directions x, y or z, respectively, for a general 3D analysis. Summation is indicated by repeated indices. Particularization to 2D analysis as well as to potential problems is straightforward.


163

Stress and displacement assumptions. Three independent fields are used in the following developments. The displacement field is explicitly approximated along the boundary by udi , where ( )d means displacement assumption, in terms of polynomial functions uim with compact support and nodal displacement parameters d d = [dm ] ∈ Rn , for nd displacement degrees of freedom of the discretized model. An independent stress field p σisj , where ( ) s stands for stress assumption, is given in the domain in terms of some particular solution σi j plus ∗ a series of fundamental solutions σ∗i j m with global support, multiplied by force parameters p∗ = [p∗m ] ∈ Rn applied at the same boundary nodal points m to which the nodal displacements dm are attached (n∗ = nd ). Displacements uis are obtained from σisj . Then, udi = uim dm

σisj

=

σ∗i jm

p∗m

+

⇒

on

p σi j

Γ

udi = u¯i ∗ σ jim, j = 0 and p ui + uris C sm p∗m

such that

such that

uis = u∗im p∗m +

on Γu p σ ji, j

and

= bi

(3)

in Ω

(4)

in Ω

(5)

where u∗im are displacement fundamental solutions corresponding to σ∗i jm . Rigid body motion is included in r ∗ terms of functions uris multiplied by in principle arbitrary constants C sm ∈ Rn ×n , where nr is the number of rigid body displacements (r.b.d.) of the discretized problem [6, 9]. The fundamental solutions σ∗i jm are used as weight functions in the CBEM. In the variational BEMs and in the EBEM, in particular, they represent domain interpolation functions. The third independent field is used to approximate traction forces along the boundary by tit , where ( )t means traction assumption, as required in the conventional boundary element method, given as tit =

|J|(at ) ui t ≡ ti t |J|

(6) t

where ui are polynomial interpolation functions with compact support and t = [t ] ∈ Rn are traction-force parameters. The index i refers to the coordinate directions whereas the index refers to any of the nt tractionforce degrees of freedom of the problem (thus denoting both location and orientation), for nodes adequately distributed along boundary segments of Γ. The interpolation functions ui have the same properties of uin , as presented in eq (3). Equation (6) holds as ti = ti t along Γσ , in particular, according to eq (2). In the above equation, |J|(at ) is the value of the Jacobian of the global (x, y, z) to natural (ξ, η) coordinate transformation at the nodal point and the term |J|(at ) /|J| features a term in the denominator that cancels the Jacobian term of the infinitesimal boundary segment dΓ = |J|dξdη in the numerator of two integral expressions introduced in eqs (9) and (15). This not only improves the capacity of tit to represent the traction forces along curved boundary segments but also simplifies the numerical integration of the related terms. The numbers of degrees of freedom for traction forces nt and displacements nd are not necessarily the same, since more than one traction-force parameter are needed to represent tractions that are not single valued at the boundary surface, generally at nodes where adjacent boundary segments present different outward normals [7]. Then, it results that nt ≥ nd , as t in eq (6) are traction-force attributes on boundary segments, whereas uin in eq (3) are displacement attributes at nodal points. The fact that nt ≥ nd leads to some rectangular matrices – the same eqs (15) and (9) of the CBEM, which have been just referred to, plus a third one, introduced in eq (11). Boundary approximation of the particular solution. Although neither conceptually nor formally necessary, the following approximation may render all subsequent equations simpler and more elegant [6]. Given a p p sufficiently refined boundary mesh, the displacements ui and the traction forces ti related to an arbitrary particular solution of the non-homogeneous governing eq (1), whenever available, can be approximated accurately d t p p enough by nodal displacement parameters d p = [dn ] ∈ Rn and traction force parameters t p = [t ] ∈ Rn , respectively, in terms of the interpolation functions of eqs (3) and (6): p

p

ui ≈ uin dn ,

p

p

ti ≈ ti t

on Γ

(7)

It is assumed with the above equations that a particular solution for the domain forces bi in eq (1) is known in p p terms of displacements ui and stresses σi j . The means to obtain such particular solutions other than in close form are not discussed herein (see, for instance, Partridge et al [14]).

164

3.


Conventional and modified boundary element methods

The matrix equation of the CBEM [1] may be expressed as [7] H d − dp = G t − tp

(8)

nd ×nd

nd ×nt

is a kinematic transformation matrix [4, 6, 9] and G = [Gm ] ∈ R is a where H = [Hmn ] ∈ R flexibility-like matrix (that is in general rectangular, as proposed). The formal definition of these matrices is Hmn = σ∗jim n j uin dΓ , Gm = ti u∗im dΓ (9) Γ

Γ

where n j are the projections of the unit outward normal to the boundary. The double-layer and single-layer potential matrices Hmn and Gm comprise in their definition singular and improper integrals, respectively, when source (m) and field (either n or ) indexes refer to the same nodal points. The singular integrals can be always evaluated mathematically in correspondence to simple mechanical meanings [1]. A conceptual assessment of eq (9) is given in Reference [7]. 4.

Approximation of displacements and traction forces on the boundary

In the present EBEM, stress and displacement results at internal points are given directly by eqs (4) and (5) in terms of force parameters p∗m that still remains to be evaluated after the solution of eq (8). This kind of evaluation of results, which circumvents the computational intensive use of the Somigliana’s identity of the CBEM, is borrowed from the hybrid boundary element method – HBEM [5, 6, 7, 8, 9]. According to that, eqs (5) and (4) may be applied to the boundary nodes [6, 9]: U∗ p∗ = d(p∗ ) ∗ ∗

u.p.f. and r.b.d excluded ∗

T p = t(p )

u.p.f. excluded

(10) (11)

where u.p.f. and r.b.d. mean unbalanced point forces and rigid-body displacements, which cannot take part in the outlined linear transformations. The definition of r.b.d. is intuitive and straightforward. The definition of u.p.f. is not intuitive, but also straightforward in terms of linear algebra [11, 10]. The terms in brackets in the above equations, (p∗ ), indicate the the nodal displacement and traction-force attributes are functions of the point-force paramenters of the fundamental solution. The above equation are very simple statements, except that there are embedded amounts of rigid-body displacements and of unbalanced forces that cannot be transformed. Moreover, the coefficients of the displacement ∗ ] ∈ Rnd ×nd as well as of the traction-force matrix T∗ = [T ∗ ] ∈ Rnt ×n∗ are undefined (and not matrix U∗ = [Umn m infinite!) when their indices refer to the same nodal point [10]. 5.

Some virtual-work statements

Several virtual-work statements are unconditionally needed in the justification of the following developments [11, 10]. Some of them have already been dealt with in relation with the hybrid boundary element method [5, 6, 7, 8, 9]. In this paper, one attempts to keep full consistency of the equations while being not too formal. A couple of virtual-work statements are given next for operational purposes only. Displacement virtual-work statement. Part of the Hellinger-Reissner potential [4, 9] leads to the equilibrium equation p Hmn p∗m = pn − pn or HT p∗ = p − p p (12) ∗

in which H = [Hnm ] ∈ Rn ×n is the same double layer potential matrix of the collocation boundary element d d p method [1], already introduced in eq (9). Moreover, p = [pn ] ∈ Rn and p p = [pn ] ∈ Rn , defined as p p σ ji n j uin dΓ , pn = σ ji n j uin dΓ (13) pn = d

Γ

Γ

are vectors of equivalent nodal forces corresponding respectively to applied boundary tractions, as given in eq (2) and to the particular solution of eq (4).

Virtual-work relation between traction forces and equivalent nodal forces. It may be convenient to express the boundary traction approximations of eq (6) in terms of equivalent nodal forces, obtained from the virtual work statement: δdm pm (t) = δdm Γ uim ti dΓt (14) ⇒ pm (t) = Lm t or p(t) = LT t where the interpolation functions of eqs (3) and (6) were used, thus defining t d ti uim dΓ L = [Lm ] ∈ Rn ×n = Γ

(15)

According to eq (6), the Jacobian of dΓ = |J|dξdη, as for a 3D problem, cancels with the denominator of ti , such that the coefficients of Lm become pre-defined numbers that are independent from the problem’s geometry. 6.

The expedite boundary element method

Equation (8) is repeated in indicial notation, for the purpose of clarity, p p σ∗jim n j uin dΓ dn − dn ti u∗im dΓ t − t Γ

Γ

(16)

where means congruence in terms of weighted residuals, and as there is an inherent approximation error in the equation [7, 10]. Using the boundary interpolation functions of eqs (3) and (6) for the fundamental solutions themselves, the above equation is approximated as p p ∗ ∗ T m (17) ti uin dΓ dn − dn ≈ Unm ti uin dΓ t − t Γ

Γ

This is at first view a bold initiative that requires a careful justification as well as the disclosure of one key ∗ stems from the HBEM via the application of a virtual-work exception. The approximation involving Unm statement. The only issue, in this case, is the adequate evaluation of the coefficients when m and n refer to ∗ is more difficult the same nodal point, which is carried out in Section 6.2.. The approximation involving T m to justify and in principle questionable, although a virtual-work statement can be resorted to [11, 10]. The key issue – and this is the main achievement of the present developments – is to concede that the proposed approximation does not apply to boundary elements that contain a singularity, that is, when m and belong to the same boundary element (not only to the same node!). A consistent means of handling with this issue is proposed in the next Section and some numerical examples are given to check the adequacy of the developments. ∗ , T ∗ and L Using the definitions of the matrices Unm m in eqs (10), (11) and (15), eq (17) can be written in m matrix format, and also substituting = for ≈, as the expedite boundary element method – EBEM, T∗T L d − d p = U∗T LT t − t p

(18)

which is a reasonable approximation of eq (8) for the CBEM, provided that the puzzle of obtaining the still undefined coefficients of U∗ and T∗ is solved. If an equation in terms of equivalent nodal forces is preferred, as in the finite element method, eq (18) can be alternatively written, according to eq. (14), as T∗T L d − d p = U∗T p − p p

(19)

which is an additional, operational advantage of the proposed EBEM. 6.1.

Evaluation of the undefined coefficients of T∗ L

The traction-force matrix T∗ is rectangular. However, the undefined coefficients of the square matrix given as the product T∗T L above are the actual subjects of interest. The matrix L, as defined in eq (15), has the same numbers of rows and columns as T∗ , but is banded, with non-zero coefficient Lm only if the nodal displacement δdm and the traction-force attribute t refer to the same boundary segment (element). Figure 1 shows on the left a triangle with six nodes, for a discretization in terms of three quadratic elements with one d.o.f per node. The

166


corresponding matrices T∗ and L are given schematically in the following, together with the already integrated coefficients of a diagonal submatrix Lbl of L, obtained in terms of the Jacobians |J|(at ) of eq (6): 1 2 3

T∗=

4 5 6 7 8 9

⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣

1

u ∼ ∼ × × × ∼ ∼ u

2

∼ u ∼ × × × × × ×

3

∼ ∼ u u ∼ ∼ × × ×

4

× × × ∼ u ∼ × × ×

5

× × × ∼ ∼ u u ∼ ∼

6

× × × × × × ∼ u ∼

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ , L= ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦

1 2 3 4 5 6 7 8 9

⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣

1

× × × 0 0 0 × × ×

2

× × × 0 0 0 0 0 0

3

× × × × × × 0 0 0

4

5

6

0 0 0 × × × 0 0 0

0 0 0 × × × × × ×

0 0 0 0 0 0 × × ×

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎡ 2|J| 1 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ 15 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢ ⎥⎥⎥ , Lbl = ⎢⎢⎢⎢ |J|2 ⎢⎢⎢ 15 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ −|J| ⎥⎥⎥ 3 ⎣ ⎥⎥⎥ ⎥⎥⎥ 30 ⎥⎥⎥ ⎥⎦

|J|1 15 8|J|2 15 |J|3 15

−|J|1 30 |J|2 15 2|J|3 15

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎦

(20)

In these schemes, ” × ” stands for values that are given directly, and ”u” means undefined. The coefficients assigned with ” ∼ ” are actually known, but they correspond to nodes that are adjacent to undefined coefficients and, in the product T∗T L, lead to coefficient results that involve an undefinedness. Moreover, they correspond to the case to which the approximation in terms of ti in eq (17) does not apply. The partial solution consists in assigning to these coefficients the corresponding values of the matrix H in eq (9), which leads to 1 2

T∗T L =

3 4 5 6

⎡ 1 ⎢⎢⎢ u ⎢⎢⎢ ⎢⎢⎢ H2,1 ⎢⎢⎢ H ⎢⎢⎢ 3,1 ⎢⎢⎢ × ⎢⎢⎢ ⎢⎢⎢ H5,1 ⎣ H6,1

2

3

H1,2 H1,3 u H2,3 H3,2 u × H4,3 × H5,3 × ×

4

5

× H1,5 × × H3,4 H3,5 u H4,5 H5,4 u × H6,5

6

H1,6 × × × H5,6 u

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ , ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎦

1 2

U∗=

3 4 5 6

⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎣

1

u × × × × ×

2

× u × × × ×

3

× × u × × ×

4

× × × u × ×

5

× × × × u ×

6

× × × × × u

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎦

(21)

U∗ .

as shown side by side with the scheme of The indicated coefficients of H can be evaluated using a GaussLegendre quadrature, since there is no singularities involved. The unknown coefficients are obtained by using rigid body displacements, as done in the CBEM. The following algorithm summarizes the procedure [11, 10]. Algorithm for the evaluation of the undefined coefficients of T∗T L 1. If the indices (m, n) of T∗T L refer to a boundary segment that is not adjacent to a singularity, then just evaluate the coefficient as the indicated product. 2. If the indices (m, n) refer to a node that is adjacent to a singularity, then replace the coefficient with the corresponding value of H, eq (9), which requires the evaluation of a regular integral (uin = 0 at the singularity point). 3. If the indices (m, n) refer to a node directly affected by a singularity, evaluate the coefficient by forcing the matrix to be orthogonal to rigid-body displacements. (For an unbounded domain, use the complementary, bounded domain. In case of symmetries, when the number of r.b.d. is not sufficient, additionally apply the problem to a simple analytical solution.) 6.2.

Evaluation of the undefined coefficients of U∗

Once the undefined coefficients of the product T∗T L are evaluated, the best (and only) way of obtaining the undefined coefficients of U∗ , as illustrated in eq (21), is to apply either eq (18) or (19) to a sufficient number of simple, analytical solutions (Da , Ta ) or (Da , Pa ) and then write the least-squares statement ∗T a T LD − U∗T LT Ta = min or T∗T LDa − U∗T Pa = min (22) For potential problems, the number of constant fluxes is either two or three, for 2D or 3D problems, and just one unknown per node. For general elasticity problems, there are either three or six constant stress states, for 2D or 3D problems, and either 2 × 2 or 3 × 3 unknowns (if symmetry is not enforced). This solution scheme is similar to the one adopted in the HBEM for the evaluation of the undefined coefficients of the flexibility matrix F [2, 3, 4, 9, 10, 12, 13].


6.3.

167

Solution of the problem’s equation and evaluation of results at internal points

Given a general mixed-boundary problem, either eq (18) or (19) can be solved for the problem’s unknowns and, afterwards, results obtained at internal points by using the Somigliana’s identity. However, in the present framework the results can be obtained as functions of the point-force parameters p∗ , according to eqs (4) and (5), which circumvents any further integrations. The evaluation of p∗ can be carried out by solving eq (12), for instance (its singularity poses no actual difficulties [2, 3, 4, 9, 10, 12, 13]). A more efficient way of handling the problem posed in the above paragraph is to try to evaluate all the nodal unknowns – nodal displacements and forces, besides the point-force parameters p∗ – by solving a single matrix system. There are several linear algebra concerns on the subject, as whether one is dealing with a very large system of equations (to be solved in terms of GMRES). The basic idea, which is not further explored for space restrictions, is to work, for instance, with eqs (10) and (12) together, here repeated, for clarity, and including a particular solution: U∗ p∗ = d − d p , HT p∗ = p − p p (23) If one takes from these equations only the known coefficient values of d and p, it is possible to construct a system such as Ap∗ = y (24) solve for p∗ and then, using eq (23), obtain the unknown coefficients of d and p as well as results at internal points by directly using eqs (4) and (5). Numerical example Figure 1 shows on the right an irregularly shaped domain discretized with a total of 124 nodes, for which some of the equations and concepts outlined in this paper are assessed numerically for the solution of the 2D Laplace equation. The present results are im part complementary to the ones shwon in Reference [7]. The figure has corner coordinates (0, 0), (10, 20), (20, 0), (15, 35), (0, 20), (17, 19), (16, 22), (21, 24) and (22, 20). The four curved boundary segments have radii of curvature 20, 15, 4 and (−4). The problem is modeled using linear, quadratic and cubic elements with increasingly refined meshes, in a Fortran code with double precision, for the total numbers of nodes shown in Table 1.

7

6 1

5

6

8

9

61

35 30

109

25

1

101 17

20

2

5

4

C

117

93 15

124

10 2

5

3

3

4

B

A

0 −5

92

1 0

5

10

15

33 20

25

30

Figure 1: Triangle with six nodes and three quadratic boundary elements (left) and irregularly-shaped figure for a convergence study.

On the left of Figure 2 are plotted the errors, in terms of Euclidean norms, T∗T L = T∗T L − H | H| , U∗ LT = U∗ LT − G | G|

(25)

obtained with the approximations of the matrices H and G, as proposed in Section 6.. Although the undefined coefficients of U∗ are obtained only after the evaluation of the undefined coefficients of T∗T L, according to eq

168


Element type Linear Quadratic Cubic

Total number of nodal points 31 62 124 248 496 992 1984 – – 62 124 248 496 992 1984 3968 – 93 186 372 774 1488 2976

Table 1: Total numbers of nodal points for the numerical model on the right of Figure 1.

(22), they result in better approximations in all cases. This can be explained as there is a sound variational basis for the proposed approximation in terms of U∗ [2, 9, 10, 12, 13]. The graphics on the right of Figure 2 show the error norm (Hd − Gt) = | Hd − Gt | /| Hd|

(26)

for a pair of boundary potential and normal gradients (d, t) corresponding to a logarithmic source ln r/2π centered at point C on the right of Figure 1. The results with label “Con” correspond to the CBEM, in which the traction-force interpolation function of eq (6) is a polynomial, and not as proposed – a modified, slightly improved BEM with results labeled “Mod”. In this example, the results with the traction forces as in eq (6) are almost indistinguishable from the ones of the CBEM [7, 11]. The results labeled “Exp” correspond to the EBEM. The Figure shows the expected convergence pattern of a consistently formulated numerical method up to an error norm ≈ 10−6 , when numerical integration errors tend to prevail and accuracy hardly improves with increasing mesh refinement. The results with the EBEM are initially comparable to the ones of the CBEM. (In the example shown, the results with the EBEM for a coarse mesh are actually better, but no general conclusions can be drawn.) However, the convergence rate is smaller for the EBEM than in the case of the CBEM. The best results with the EBEM, in this and in other examples for 2D potential problems, are obtained in the implementation with quadratic elements. The results of the EBEM are consistently more accurate than in the implementations of the CBEM using linear elements, a pattern that is also observed in other numerical examples. It is worth observing from the error values of the graphics on the left and right of Fig. 2 that the EBEM leads to matrices that are not as accurately an approximation of the ones of the CBEM as numerical results in general show. The pattern of results of this Figure has been observed for other numerical examples [10]. −1

10

0

10

Hl

Conl≡Modl

Hq Hc −2

10

| T*TL − G |

Gc −2

Log(errors)

Log(errors)

Gl Gq

10

Expl

−4

10

Conq≈Modq

Expc Exp

q

−6

10

*T T

|U L −G|

Con ≈Mod c

−3

10

c

−8

1

10

2

3

10 10 Number of Nodal Points

10

2

3

10 10 Number of nodal points

4

10

Figure 2: Left: Error norms of the matrices that approximate H and G, according to eq (25), for linear, quadratic and cubic elements. Right: Error norm according to eq (26) for a logarithmic field given by a potential source at point C shown on the right of Figure 1.


Conclusions In the proposed expedite formulation of the boundary element method, no integrations are required, except for a few regular ones for a narrow band of coefficients above and below the main diagonal of the matrix that approximates the double-layer potential matrix H. The banded, auxiliary kinematic/equilibrium transformation matrix L is formulated in a way that circumvents any integration. For mixed boundary conditions, the matrix system is structured in such a way that an efficient iterative solver (GMRES) of very large equation systems can be used for the evaluation of the problem’s unknowns, at the same time that the parameters for the domain interpolation functions (fundamental solutions) are obtained. This enables the straightforward evaluation of results at internal points with no need of further integrations. As proposed, the EBEM promises to be superior to the fast multi-pole methods in concept, implementation and computational efficiency – very much in the direction of a meshless method. Application of the formulation to time-dependent problems in the frequency domain is straightforward. An extended version of the present manuscript is being prepared, in which numerical examples of three-dimensional problems are also shown. Acknowledgments This project was supported by the Brazilian agencies CAPES, CNPq and FAPERJ. References [1] C. A. Brebbia, J. F. C. Telles, and L. C. Wrobel. Boundary Element Techniques. Springer-Verlag, Berlin, 1984. [2] R. A. P. Chaves. The Simplified Hybrid Boundary Element Method Applied to Time-Dependend Problems (in Portuguese). PhD thesis, Pontifical Catholic University of Rio de Janeiro, 2003. [3] M. F. F. de Oliveira and N. A. Dumont. Conceptual completion of the simplified hybrid boundary element method. In E. J. Sapountzakis and M. H. Aliabadi, editors, BETeq 2009 - International Conference on Boundary Element Techniques, pages 49–54, Athens, Greece, July 2009. [4] N. A. Dumont. The hybrid boundary element method: an alliance between mechanical consistency and simplicity. Applied Mechanics Reviews, 42(11):S54–S63, 1989. [5] N. A. Dumont. An assessment of the spectral properties of the matrix G used in the boundary element methods. Computational Mechanics, 22:32–41, 1998. [6] N. A. Dumont. Variationally-based, hybrid boundary element methods. Computer Assisted Mechanics and Engineering Sciences (CAMES), 10:407–430, 2003. [7] N. A. Dumont. The boundary element method revisited. In C. A. Brebbia, editor, Boundary Elements and Other Mesh Reduction Methods XXXII, pages 227–238, Southampton, U.K., 2010. WITPress. [8] N. A. Dumont. From the collocation boundary element method to a meshless formulation. In M. A. Storti E. N. Dvorkin, M. B. Goldschmit, editor, Mecánica Computacional, MECOM 2010 – IX Argentinean congress on Computational Mechanics and II South American congress on Computational Mechanics, XXXI CILAMCE – XXXI Iberian Latin-American Congress on Computational Methods in Engineering, pages 4635–4659 (on CD), Buenos Aires, Argentina, 2010. [9] N. A. Dumont. The hybrid boundary element method – fundamentals (to be submitted). Engineering Analysis with Boundary Elements, 2011. [10] N. A. Dumont. Toward a meshless formulation of the simplified hybrid boundary element method (to be submitted). 2011. [11] N. A. Dumont and C. A. Aguilar. The expedite boundary element method (accepted). In C. A. Brebbia, editor, BEM/MRM 2011 33rd International Conference on Boundary Elements and other Mesh Reduction Methods, page 12 pp, New Forest, UK, June 2011. WIT. [12] N. A. Dumont and R. A. P. Chaves. General time-dependent analysis with the frequency-domain hybrid boundary element method. Computer Assisted Mechanics and Engineering Sciences, (10):431–452, 2003. [13] M. F. F. Oliveira. Conventional, hybrid and simplified boundary element methods (in Portuguese). Master’s thesis, Pontifical Catholic University of Rio de Janeiro, 2004. [14] P. W. Partridge, C. A. Brebbia, and L. C. Wrobel. The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton, 1992.

169

170


Use of Generalized Westergaard Stress Functions as Fundamental Solutions Ney Augusto Dumont and Elvis Yuri Mamani Vargas Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, 22453-900, Brazil. e-mail: [email protected] Keywords: Westergaard stress functions, Hellinger-Reissner potential, hybrid boundary element.

Abstract. In the traditional boundary element methods, the numerical modeling of cracks is usually carried out by means of hypersingular fundamental solutions. A more natural procedure should make use of fundamental solutions that represent the square-root singularity of the gradient field around the crack tip, which at most leads to improper integrals. Such a representation is best accomplished in a variationally-based framework that also addresses a convenient means of evaluating results at internal points. This is the subject of the present paper, with the use of generalized Westergaard stress functions for the numerical simulation of two-dimensional problems that may be completely unrelated to fracture mechanics. Problems of general topology can be modeled, such as in the case of unbounded and multiply-connected domains. The formulation is naturally applicable to notches and generally curved cracks. It also provides an easy means of evaluating stress intensity factors. For a general-purpose code, Kelvin’s and Westergaard-type fundamental solutions can be combined. A simple, validating numerical example is presented. Introduction Tada et al [14, 15] proposed a simple and efficient method of developing Westergaard stress functions for the analysis of displacement-prescribed and stress-prescribed crack problems that was restricted to the mathematical means of arriving at the stress functions and the illustration of several forms of crack openings – always in terms of analytical developments. The present paper generalizes Tada et al’s method and proposes Westergaard stress functions as fundamental solutions of a two-dimensional boundary element method for potential [4, 11] as well as elasticity problems [10, 12]. As illustrated in Fig. 1, such fundamental solutions correspond to cleavage (cracking) actions applied successively along straight segments of the boundary of an in principle finite domain. On the other hand, the formulation can be directly and advantageously applied to fracture mechanics [9, 12], when the cracking takes place along line segments inside the domain, as represented on the right of Fig. 2. 3

a3

a3

a4 3

3

a4

represents

5

3 4

4 5

Figure 1: Representation of two semicracks that compose the crack element # 3 related to node # 4.

Brief outline of the hybrid boundary element method The hybrid boundary element method (HBEM) was introduced in 1987 on the basis of the Hellinger-Reissner potential and as a generalization of Pian’s hybrid finite element method [13, 2]. The formulation requires evaluation of integrals only along the boundary and makes use of fundamental solutions (Green’s functions) to interpolate fields in the domain. Accordingly, an elastic body of arbitrary shape may be treated as a single finite macro-element with as many boundary degrees of freedom as desired. In the meantime, the formulation has evolved to several application possibilities, including time-dependent problems, fracture mechanics, nonhomogeneous materials and strain gradient elasticity [7, 9, 6, 8].


171

Problem Formulation. An elastic body is submitted to tractions t¯i on part Γσ of the boundary Γ and to displacements u¯i on the complementary part Γu . For the sake of brevity, body forces are not included [5]. The task is to find the best approximation for stresses and displacements, σi j and ui , such that σ ji, j = 0 ui = u¯i

along Γu

in the domain Ω, ti = σi j n j = t¯i along Γσ

and

(1) (2)

in which n j is the outward unit normal to the boundary. Indicial notation is used. Stress and Displacement Assumptions. Two independent trial fields are assumed [13, 2]. The displacement field is explicitly approximated along the boundary by udi , where ( )d means displacement assumption, in terms d of polynomial functions uim with compact support and nodal displacement parameters d = [dm ] ∈ Rn , for nd s displacement degrees of freedom of the discretized model. An independent stress field σi j , where ( ) s stands for stress assumption, is given in the domain in terms of a series of fundamental solutions σ∗i j m with global ∗ support, multiplied by force parameters p∗ = [p∗m ] ∈ Rn applied at the same boundary nodal points m to which ∗ d the nodal displacements dm are attached (n = n ). Displacements uis are obtained from σisj . Then, udi = uim dm

on

Γ

σisj = σ∗i jm p∗m ⇒

such that such that

udi = u¯i

on Γu

σ∗jim, j = 0 in

uis = u∗im p∗m + uris C sm p∗m

in Ω

u∗im

where terms of

and

Ω

are displacement fundamental solutions corresponding to σ∗i jm . Rigid functions uris multiplied by in principle arbitrary constants C sm [3, 5].

(3) (4) (5)

body motion is included in

Governing Matrix Equations. The Hellinger-Reissner potential, based on the two-field assumptions of the latter Section, as implemented by Pian [13] and generalized by Dumont [2], leads to two matrix equations that express nodal equilibrium and compatibility requirements. The simplest, and still mathematically consistent, means of laying out these equations is in terms of two independent virtual work principles [5]. Two virtual-work statements, one in terms of displacements and the other in terms of stresses, lead respectively to the matrix equilibrium and compatibility equations HT p∗ = p ,

F ∗ p∗

=

Hd

(6)

d ∗ Rn ×n

is the same double layer potential matrix of the collocation boundary element in which H = [Hnm ] ∈ d method [1], and p = [pn ] ∈ Rn are equivalent nodal forces obtained as in the finite element method. Moreover, ∗ ] ∈ Rn∗ ×n∗ is a symmetric, flexibility matrix. The matrices H and F∗ may be compactly defined as F∗ = [Fnm ∗ (7) σ∗i jm n j uin u∗in dΓ Hmn Fmn = Γ

Solving for p∗ in eq (6), one arrives at the matrix system HT F∗(−1) Hd = p, where HT F∗(−1) H ≡ K is a stiffness matrix. The inverse F∗(−1) must be evaluated in terms of generalized inverses, as F∗ is singular for a finite domain Ω [5]. Results at internal points are expressed in terms of eqs (4) and (5) after evaluation of p∗ . Basics on a rotated semicrack Tada et al [14] show that, given a crack opening of shape f (x) in the interval [x1 , x2 ] along the x axis and symmetric with respect to this axis in the Cartesian coordinate system (x, y), it is possible to define a potential function Φ(z) of the complex argument z = x + iy, x2 1 f (x) Φ(z) = − dx (8) 2π x1 z − x and then obtain the corresponding stress and displacement expressions, as a generalization of Westergaard’s initial proposition. Several crack and stress configurations are investigated by Tada et al [14, 15], as translation and superposition of effects can always be applied to compose intrincated crack patterns.

172


A very simple, although apparently original generalization is obtained for a semicrack of length a1 along a straight line that is rotated in the counter clock direction by an angle θ1 , as shown on the left of Fig. 2, with which it is possible to compose curved and kinked cracks of any length [4]. Although the crack shape may be rather general, as given by Tada et al [14] and as already investigated in the present framework for several configurations [4], the ensuing developments are given for the elliptic semicrack. The corresponding expression of eq (8) for the semicrack 1 is ⎛ ⎞⎞ ⎛ ⎜⎜ ⎜⎜⎜ 1 + 1 − Z 2 ⎟⎟⎟⎟⎟⎟ ⎜⎜⎜ 1⎟ 1 ⎜⎜⎜⎜ Z1 ⎟⎟⎟⎟⎟⎟⎟ 2 ⎜⎜1 − 1 − Z1 ln ⎜⎜⎜− (9) Φ1 ≡ Φ(Z1 ) = − − ⎟⎟⎟⎟⎟⎟ ⎜⎝ 4 2π ⎜⎜⎝ Z1 ⎠⎠ already given as argument of Z1 = zT 1 ≡

z −iθ1 x + iy −iθ1 r e ≡ e ≡ ei(θ−θ1 ) a1 a1 a1

(10)

from which the definition of the rotation and normalization term T 1 is inferred.

x1

ϯ

y

Ŷнϭ ŶнϮ

Ϯ

y1

ϭ

a1 ș1

0.5

͘͘͘

Q ;ĂĚĚŝƚŝŽŶĂůŶŽĚĞͿ

x 0.5

Figure 2: Semicrack of length a1 rotated by an angle θ1 (left) and fictitious node n + 3 for the simulation of a crack with n segments.

General displacement and stress expressions A semicrack expressed by eq (9), as illustrated in Fig. 2, presents insurmountable singularities at the origin. However, the adequate combination of two singularities, as firstly proposed for potential problems [4, 11], leads to potential and gradient expressions that are not only finite at z = 0, but also single valued. These developments have been recently extended to elasticity [10, 12] by the combination of the mode I and mode II effects of two juxtaposed semicracks, with results the are shown in the following. The general expressions of displacement and stresses at a point (x, y) are, for the composed crack illustrated in Fig. 1, ⎧ ⎫

∗

∗ ⎪ σx ⎪ ⎪ ⎪ ⎪ p p u ⎨ ⎪ ⎬ σy ⎪ = [U1 − U2 ] ∗x , ⎪ = [S1 − S2 ] ∗x (11) ⎪ ⎪ ⎪ ⎪ py py v ⎩τ ⎭ xy in terms of two force parameters p∗x and p∗y whose meanings are unveiled after eq (21). For the semicrack 1, (12) U1 = UIm Φ1 Im Φ1 + URe Φ1 Re Φ1 + UIm Φ1 y1 ImΦ1 + URe Φ1 y1 ReΦ1 (13) S1 = SIm Φ1 Im Φ1 + SRe Φ1 Re Φ1 + SIm Φ1 y1 ImΦ 1 + SRe Φ1 y1 ReΦ1 with the introduced displacement and stress matrices defined as 2(1 − ν2 ) 1 0 (1 + ν)(1 − 2ν) 0 1 UIm Φ1 = , URe Φ1 = 0 1 −1 0 E E

(14)


173

1 + ν sin 2θ1 − cos 2θ1 1 + ν cos 2θ1 sin 2θ1 , URe Φ1 = Ea1 − cos 2θ1 − sin 2θ1 Ea1 sin 2θ1 − cos 2θ1 ⎡ ⎤ ⎡ ⎤ 3 2 sin θ1 cos θ1 ⎥⎥ ⎢ cos θ1 ⎢−3 sin θ1 − sin 3θ1 2 cos θ1 cos 2θ1 ⎥⎥⎥ ⎥⎥⎥ 2 ⎢⎢⎢⎢ 2 1 ⎢⎢⎢⎢ ⎥ 3 ⎢ ⎥ ⎢ sin θ1 ⎥⎥ , SRe Φ1 = 3 cos θ1 − cos 3θ1 ⎥⎥⎥⎥ = ⎢sin θ1 cos θ1 ⎢ 2 sin θ1 cos 2θ1 ⎦ ⎦ a1 ⎢⎣ 2a1 ⎢⎣ 2 2 2 cos θ1 cos 2θ1 2 sin θ1 cos 2θ1 sin θ1 cos θ1 sin θ1 cos θ1 ⎡ ⎤ ⎡ ⎤ sin 3θ1 ⎥⎥⎥ ⎢ sin 3θ1 − cos 3θ1 ⎥⎥⎥ ⎢ cos 3θ1 1 ⎢⎢⎢ 1 ⎢⎢⎢ ⎥ ⎥ SIm Φ1 = 2 ⎢⎢⎢⎢ − sin 3θ1 cos 3θ1 ⎥⎥⎥⎥ , SRe Φ1 = 2 ⎢⎢⎢⎢− cos 3θ1 − sin 3θ1 ⎥⎥⎥⎥ ⎦ a1 ⎣− cos 3θ − sin 3θ ⎦ a1 ⎣ sin 3θ − cos 3θ1 1 1 1 UIm Φ1 =

SIm Φ1

(15)

(16)

(17)

The expressions in Eq. (11) for the semicrack 2 are obtained from the Eqs. (12)–(17) by substituting the subscripts 2 for the subscripts 1. Check that displacements and stresses are single valued when r tends to zero. According to the expressions given above, both lim u and lim v are finite and independent from the direction θ: r→0

r→0

⎡ sin 2θ − sin 2θ 1 2 ⎢⎢

− (1 − ν)∆12 1 + ν ⎢⎢⎢⎢⎢ u lim = ⎢⎢⎢ 1 − 2ν 4 a cos 2θ4 − cos 2θ1 1 r→0 v πE ⎢⎣− + ln 2 a2 2

1 − 2ν a1 cos 2θ4 − cos 2θ1 ⎤⎥⎥ ⎥⎥⎥ p∗ ln + ⎥⎥⎥ x 2 a2 2 ⎥⎥ p∗ sin 2θ2 − sin 2θ1 − (1 − ν)∆12 ⎥⎦ y 4

(18)

The term ∆12 = θ2 − θ1 ± 2π depends on whether r = 0 is approached from inside or outside the domain, which indicates a displacement jump between two crack faces. It may also be checked after some tedious manipulations of eq (11) that the stresses are finite and single valued: ⎡ cos θ + cos 3θ cos θ2 + cos 3θ2 1 1 ⎢ − ⎧ ⎫ ⎢⎢⎢⎢⎢ 8a1 8a2 ⎪ ⎪ σ ⎢ x ⎪ ⎪ ⎢ ⎪ ⎨ ⎪ ⎬ ⎢⎢⎢ 3 cos θ1 − cos 3θ1 3 cos θ2 − cos 3θ2 ⎢ σ lim ⎪ = − y⎪ ⎢ ⎪ ⎪ r→0 ⎪ 8a1 8a2 ⎩τ ⎪ ⎭ ⎢⎢⎢⎢⎢ xy ⎢⎢⎣ sin 3θ1 − sin θ1 8a2

3 sin θ2 + sin 3θ2 3 sin θ1 + sin 3θ1 − 8a2 8a1 sin θ2 − sin 3θ2 sin θ1 + sin 3θ1 − 8a2 8a1 cos θ1 + cos 3θ1 cos θ2 + cos 3θ2 − 8a1 8a2

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ∗ ⎥⎥⎥ p x ⎥⎥⎥ p∗ ⎥⎥⎥ y ⎥⎥⎦

(19)

Behavior of the traction forces along a crack surface. One obtains from eq (11)

⎫ ⎧

∗ ⎪ σx ⎪ ⎪ ⎪ px Tx n x 0 ny ⎪ ⎨ ⎪ ⎬ σ [T ] = − T = ⎪ ⎪ y 1 2 ⎪ ⎪ p∗y Ty 0 ny n x ⎪ ⎭ ⎩τ ⎪ xy

(20)

the traction-force expressions along the boundary. If the traction-force matrix T1 , related to the semicrack 1, is applied to this crack surface, the result Re Φ1 1 0 T1 |θ=θ1 = − (21) along the crack surface θ = θ1 a1 0 1 shows that, on θ = θ1 , T x and T y are uncoupled functions of p∗x and p∗y , respectively, which leads to the mechanical interpretation of these force parameters, as introduced in eq (11) [10, 12]. A validating numerical example A horizontal point force of unit intensity is applied at a node of coordinates (−10, 25) of an unbounded twodimensional continuum. An irregular figure is cut out, as illustrated on the left of Figure 3, and displacements and traction forces due to the point force, as measured along the drawn boundaries, are applied to create a problem of known analytical solution that, due to the lack of convexity of the cut-out domain, is difficult to solve numerically. The figure is composed of a total of 170 nodes and linear segments that are equally spaced between the indicated corner nodes and whose coordinates are given in Table 1. A series of 51 points along the line segment AB are also generated for the representation of some numerical results in the domain.

174


Node x y

1 0 0

25 10 15

40 20 10

74 15 35

103 0 20

130 10 20

142 11 21

154 12 20

A 5 20

B 15 18

Table 1: Cartesian Coordinates of the nodes that constitute the cut-out model on the left of Fig. 3.

The simplest problem that can be solved in this example is for Neumann boundary conditions, when only the matrix H of eq (6) needs be evaluated. Although H is a singular matrix for a bounded domain, the equivalent nodal gradients p of eq (6) are in balance, and the posed linear algebra problem admits of just one solution p∗ , to be obtained in the frame of generalized inverse matrices [2, 9, 12]. Once p∗ is evaluated, gradients and potentials can be obtained according to eqs (4) and (5). Figure 3 shows on the right both analytical and numerical stress values obtained along the line segment AB (the values of σ xx are multiplied by -1). Accuracy of results obtained in a formulation that uses Kelvin fundamental solution is almost matched, in this particular numerical example [10]. However, the boundary layer effect is larger in the case of fundamental solutions defined in terms of generalized Westergaard functions: It is possible to observe in the graphics that the simulated hole in the figure disturbs the numerical values of σ xy and σyy , which slightly undulate about the analytical results. This is expected,√as the gradient singularity 1/r for Kelvin’s solution is more pronounced and localized than the singularity 1/r of the Westergaard function. In spite of that, good accuracy can be still achieved after some post-processing of the results for points close to or on the boundary, as done for the case of Kelvin fundamental solutions [9, 5]. The numerical example of Fig. 3 is shown with the purpose of validating the theoretical developments of this paper, and not with the intention of proposing an alternative to the Kelvin fundamental solutions, as the adequacy of the conventional boundary element method to solve general problems is unquestionable. In fact, the best numerical simulation of problems of fracture mechanics seems to be achieved by conveniently combining Kelvin fundamental solutions and the proposed generalized stress functions [9, 10]. 74

0.016 0.014 0.012

130

Stress values

142

103

154

σxx analytic (−1) σyy analytic

0.01

σxy analytic σxx (−1)

0.008

σyy 0.006

25

40

σxy

0.004 0.002 0

1

0

5

10

15 20 25 30 35 Points along the segment AB

40

45

50

Figure 3: Irregularly shaped domain submitted to a horizontal point force at (−10, 25) and, on the right, comparison of analytical and numerical stress results along the line segment AB.

Conclusions The main attempt of this paper is to obtain a formulation for the modeling of cracks, as illustrated on the right of Fig. 2, which can also be applied to holes and notches and leads to a straightforward evaluation of stress


intensity and concentration factors, as a generalization of the developments of Reference [9]. The scheme of the curved crack on the right of Fig. 2, for instance, is topologically a hole with n nodes and segments, similar to the one with nodes 130 . . . 170 of Fig. 3, except that, after evaluation of the problem’s matrices and vectors, rows and columns referring to nodes assigned as 1, n + 2 and n + 3 are removed from the equation systems. The paper presents a novel development of fundamental solutions that are generalizations of the stress functions proposed by Westergaard for mode I and mode II deformation problems. Completeness of the formulation in terms of the Kolosov-Muskhelishvili potentials is guaranteed for general domain topology and boundary conditions, provided that some spectral properties of the resultant matrices are adequately taken into account [10]. Although this outline has been developed in a variational framework, it is possible to use the proposed fundamental solutions in the conventional, collocation boundary element method and resort to some concepts of the hybrid boundary element method for the evaluation of results at internal points. The evaluation of stress intensity factors in fracture mechanics problems can be carried out easily and more accurately in the present context than using existing finite element or boundary element codes. The specific issues of fracture mechanics – which are the motivation of the present developments – are dealt with in a forthcoming paper. Acknowledgments This project was supported by the Brazilian agencies CAPES, CNPq and FAPERJ. References [1] C. A. Brebbia, J. F. C. Telles, and L. C. Wrobel. Boundary Element Techniques. Springer-Verlag, Berlin, 1984. [2] N. A. Dumont. The hybrid boundary element method: an alliance between mechanical consistency and simplicity. Applied Mechanics Reviews, 42(11):S54–S63, 1989. [3] N. A. Dumont. Variationally-based, hybrid boundary element methods. Computer Assisted Mechanics and Engineering Sciences (CAMES), 10:407–430, 2003. [4] N. A. Dumont. Dislocation-based hybrid boundary element method. Draft paper, 2008. [5] N. A. Dumont. The hybrid boundary element method – fundamentals (to be submitted). Engineering Analysis with Boundary Elements, 2011. [6] N. A. Dumont, R. A. P. Chaves, and G. H. Paulino. The hybrid boundary element method applied to problems of potential of functionally graded materials. International Journal of Computational Engineering Science (IJCES), 5:863–891, 2004. [7] N. A. Dumont and R. de Oliveira. From frequency-dependent mass and stiffness matrices to the dynamic response of elastic systems. International Journal of Solids and Structures, 38(10-13):1813–1830, 2001. [8] N. A. Dumont and D. Huamán. Hybrid finite/boundary element formulation for strain gradient elasticity problems. In E. J. Sapountzakis and M. H. Aliabadi, editors, BETeq 2009 - International Conference on Boundary Element Techniques, pages 295–300, Athens, Greece, July 2009. [9] N. A. Dumont and A. A. O. Lopes. On the explicit evaluation of stress intensity factors in the hybrid boundary element method. Fatigue & Fracture of Engineering Materials & Structures, 26:151–165, 2003. [10] N. A. Dumont and E. Y. Mamani. Use of generalized Westergaard stress functions in the hybrid boundary element method (to be submitted). 2011. [11] N. A. Dumont and E. Y. Mamani. A variational boundary element method based on generalized Westergaard stress functions (accepted). In MecSol 2011 - International Symposium on Solid Mechanics, 12 pp, Florianópolis, Brazil, May 2011. [12] E. Y. Mamani. The hybrid boundary element method based on generalized Westergaard stress functions (in Portuguese). Master’s thesis, Pontifical Catholic University of Rio de Janeiro, 2011. [13] T. H. H. Pian. Derivation of element stiffness matrices by assumed stress distribution. AIAA Journal, 2:1333–1336, 1964. [14] H. Tada, H. Ernst, and P. Paris. Westergaard stress functions for displacement-prescribed crack problems - I. International Journal of Fracture, 61:39–53, 1993. [15] H. Tada, H. Ernst, and P. Paris. Westergaard stress functions for displacement-prescribed crack problems - II. International Journal of Fracture, 67:151–167, 1994.

175

176


Direct Solution of Differential Equations Using a Wavelet-Based Multiresolution Method Rodrigo Bird Burgos1, Raul Rosas e Silva1 and Marco Antonio Cetale Santos2 1

2

Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente, 225, Gávea, Rio de Janeiro, Brasil, [email protected], [email protected] LAGEMAR, UFF, Av. Gen. Milton Tavares de Souza, s/n - Niterói, RJ, Brasil, [email protected]

Keywords: Wavelets, Multiresolution

Daubechies,

Interpolets,

Wavelet-Galerkin

Method,

Differential

Equations,

Abstract. The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs). Therefore, the use of wavelets as basis functions in computational analysis holds some promise due to their compact support, orthogonality, localization and multiresolution properties, especially for problems with local high gradient, which would require a dense mesh in tradicional methods, like the FEM. Another possible advantage is the fact that the calculation of the integrals of the inner products of wavelet basis functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of approximating the integral by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness, mass and geometry matrices. In this work, the Galerkin Method has been adapted for the direct solution of differential equations in a meshless formulation using interpolating wavelets (Interpolets). This approach enables the use of a multiresolution analysis. One and two-dimensional examples are proposed. Introduction The use of wavelet-based numerical schemes has become popular in the last two decades. Wavelets have several properties that are especially useful for representing solutions of partial differential equations (PDEs), such as orthogonality, compact support and exact representation of polynomials of a certain degree. Their capability of representing data at different levels of resolution allows the efficient and stable calculation of functions with high gradients or singularities [1]. Compactly supported wavelets have a finite number of derivatives which can be highly oscillatory. This makes the numerical evaluation of integrals of their inner products difficult and unstable. Those integrals are called connection coefficients and they appear naturally when applying a numerical method for the solution of a PDE. Due to some properties of wavelet functions, these coefficients can be obtained by solving an eigenvalue problem. Working with dyadically refined grids, Deslauriers and Dubuc (1989) obtained a new family of wavelets with interpolating properties, later called Interpolets [2]. Unlike Daubechies’ wavelets [3], Interpolets are symmetric, which is especially interesting in numerical analysis. The use of wavelets as interpolating functions in numerical schemes holds some promise due to their compact support, localization and multiresolution properties. The approximation of the solution can be improved by increasing either the level resolution or the order of the wavelet used. Two examples were used for validating the proposed method. In a one-dimensional scheme, a beam with a concentrated load was used to test the method’s ability to capture singularities. In a second example, the method was then applied for a thin plate with excellent results. Wavelet Theory and Method Forumulation Multiresolution Analysis. Multiresolution analysis using orthogonal, compactly supported wavelets has become increasingly popular in numerical simulation. Wavelets are localized in space, which allows the analysis of local variations of the problem at various levels of resolution. In the following expression, known as the two-scale relation, ak are the filter coefficients of the wavelet scale function. In general, there are no analytical expressions for wavelet functions, which can be obtained using iterative procedures.


M ( x)

N 1

177

N 1

¦ a M (2 x k ) ¦ a M k

k

k 0

k

(2 x)

(1)

k 0

Interpolets. The basic characteristics of interpolating wavelets require that the mother scaling function satisfies the following condition [4]:

M (k ) G 0,k

1, k 0 , k ] ® ¯0, k z 0

(2)

The filter coefficients for Delauriers-Dubuc Interpolets can be obtained by an autocorrelation of the Daubechies filter coefficients. Interpolets satisfy the same requirements as other wavelets, specially the twoscale relation, which is fundamental for their use as interpolating functions in numerical methods. Fig. 1 shows the Interpolet IN8. Its symmetry and interpolating properties are evident. There is only one integer abscissa which evaluates to a non-zero value.

Figure 1: Interpolet IN8 scaling function with its full support Connection Coefficients. Assuming that a function f is approximated by a series of interpolating scale functions, the following may be written:

f [

¦D M k

k

[

(3)

k

The process of solving a differential equation requires the calculation of the inner products of the basis functions and their derivatives. These inner products are defined as connection coefficients:

*id,1j, d2

³M

( d1 )

([ i )M ( d2 ) ([ j )d [

(4)

The values for the limits of the integral in eq (4) depend on which method is used to impose boundary conditions. In this work, the limits are given by [0,2m], where m is the wavelet level of resolution. This method allows the use of Lagrange multipliers to deal with boundary conditions, similarly to what is usually done in a meshless scheme [5]. Connection coefficients at level m can be obtained through the calculation at level 0 thus avoiding its recalculation while increasing the level of resolution. Wavelet dilation and translation properties allow the calculation of connection coefficients within the interval [0 1] to be summarized by the solution of an eigenvalue problem based only on filter coefficients [6].

1 § · d1 ,d2 = 0, Pi , j:k ,l ¨ P - d1 d2 1 I ¸ 2 © ¹

ak 2i al 2 j ak 2i 1al 2 j 1

(5)

Since eq (5) leads to an infinite number of solutions, there is the need for a normalization rule that provides a unique eigenvector. This unique solution comes with the inclusion of the so-called moment equation, derived from the wavelet property of exact polynomial representation [7].

178


¦¦ M i

k i

(k !) 2 (k d1 )!(k d 2 )!(2k d1 d 2 1)

M kj * id,1j, d2

j

M ij

j § j · j k k 1 § k · l § N 1 k l · 1 ¦ ¨ ¸ i ¦ ¨ ¸ M 0 ¨ ¦ ai i ¸ j 1 2 2 k 0©k ¹ l 0©l ¹ ©i 0 ¹

(6)

Application to the Bending of a Thin Plate The bending of a thin plate with thickness t is modeled by the following DE:

§ w4 w w4w w4 w · Et 3 D ¨ 4 2 2 2 4 ¸ q( x, y ), D x x y y w w w w 12 1 Q 2 © ¹

(7)

Displacement w(x,y) is modeled using bi-dimensional wavelets, which are products between onedimensional wavelets:

w( x, y )

¦¦ d i

ij

I ( x i)I ( y j )

(8)

j

As done in a traditional Galerkin approach, eq (8) is substituted in the DE and integrated, leading to a system of equations which contains the wavelets’ connection coefficients [8]. kd = f ,

k = D ª 00 22 Q 20 02 02 20 [ 0 ,2m ] [ 0 ,2m ] [ 0 ,2m ] [ 0 ,2m ] [ 0 ,2m ] «¬ [ 0 ,2m ]

º, 22 00 2 1 Q 2 11 11 [ 0 ,2m ] [ 0 ,2m ] [ 0 ,2m ] [ 0 ,2m ] ¼

(9)

1

f

³ q( x)

T

dx

0

The symbol indicates Kronecker product. The system is solved using the stiffness and load matrices provided by eq (9) and imposing essential boundary conditions with Lagrange multipliers:

ª k G T º ½ f ½ « »® ¾ ® ¾ 0 ¼ ¯ ¿ ¯0 ¿ ¬ G

(10)

In eq (10), G is a matrix associated with boundary conditions and O is a vector of Lagrange multipliers. The unknowns in vector D are the interpolating coefficients of the basis functions instead of nodal displacements. Examples Fig. 2 shows a simple example of a beam subjected to a concentrated load at its midpoint. This example was formulated in order to verify the ability of the wavelet method to deal with singularities, since the load generates a discontinuity in the shear force diagram. This example is easily solved by dividing the beam in two elements and applying the load as a nodal force. In this work, since degrees of freedom don’t have a fixed position, the load is transformed into the wavelet space:

q([ )

1· § PG ¨ [ ¸ o 2¹ ©

1

³ q([ )M ([ i)d[ 0

§1 · PM ¨ i ¸ ©2 ¹

(11)


179

Figure 2: Beam with concentrated load The example was solved using the IN8 Interpolet at different levels of resolution and the results for bending moment and shear force diagrams are shown in Fig. 3. It is clear that higher levels of resolution are necessary in order to capture the singularity that occurs where the load is applied. Nevertheless, results are considerably good, since the solution is obtained in wavelet space and no discretization was performed. The discontinuity in the slope of the bending moment is captured even for a low level of resolution. 0.8

-0.2

level 0 level 4 level 8 exact

0.6

-0.15

0.4 -0.1

0.2 M/ P

V/ P

-0.05

0

0

-0.2 0.05

0.15 0

-0.4

level 0 level 4 level 8 exact

0.1

0.2

0.4

0.6

0.8

-0.6

1

-0.8 0

0.2

0.4

x/L

0.6

0.8

1

x/L

Figure 3: Bending moment and shear force using IN8 Finally, to test the possibility of extending the method to two-dimensional problems, a thin plate was modeled using the equations developed in previous sections. Fig. 4 shows a square plate with all edges clamped subjected to a concentrated load applied at its center. The plate was modeled using the IN6 Interpolet at level 3, leading to a total number of 289 degrees of freedom. The result for the central displacement was w = 0.00557 PL2/D which represents an error of 0.5% when compared to the exact solution w = 0.00560 PL2/D. Results were extremely good, considering that a FE mesh using 32x32 plate elements with 12 degrees of freedom each gives an error of 0.7% in the central displacement. Fig. 5 shows the results for the bending moments Mx, My and the twisting moment Mxy. Displacements and moments distribution were obtained using the wavelet’s second derivatives. The errors in the bending moments Mx and My at the center point were 4%.

Figure 4: Clamped plate subjected to a concentrated load at the center

180


Different types of boundary conditions and loadings were tested for a square plate and the values obtained for central displacement are summarized in Table 1. Results were compared with exact solutions given by [9]. Boundary Conditions and Loading Type

Exact

Clamped / Uniform

0.00126 qL4 D 2

WGM

Error

0.00126 qL4 D

0.4 %

Clamped / Concentrated

0.00560 PL D

0.00557 PL2 D

0.5 %

Simply Supported / Uniform

0.00406 qL4 D

0.00406 qL4 D

0.1 %

Simply Supported / Concentrated

0.01160 PL2 D

0.01156 PL2 D

0.3 %

Table 1: Results for different types of boundary conditions and loadings

Figure 5: Results for moments Mx, My, Mxy and displacement w

Conclusions This work presented the formulation and validation of the Wavelet-Galerkin Method using DeslauriersDubuc Interpolets. It was also shown that wavelets have the ability of capturing discontinuities without the need to place nodes where they occur. As in the traditional FEM and other numerical methods, the accuracy of the solution can be improved either by increasing the level of resolution or the wavelet order. Sometimes, lower order wavelets at higher resolutions can give better results than higher order wavelets at lower resolutions.


For two-dimensional problems, results for displacements and bending moments were extremely good, although only regular geometry problems were studied. The extension of the method to irregular geometries is still a challenge. Since the unknowns of the method are interpolation coefficients instead of nodal displacements, it is possible to obtain a smooth representation of bending moments even with a reduced number of degrees of freedom.

References [1] Qian, S and Weiss, J., Wavelets and the numerical solution of partial differential equations, Journal of Computational Physics, 106: 155-175, 1992. [2] Deslauriers, G. and Dubuc, S., Symmetric iterative interpolation processes, Constructive Approximation, 5: 49-68, 1989. [3] Daubechies, I., Orthonormal bases of compactly supported wavelets, Communications in Pure and Applied Mathematics, 41: 909-996, 1988. [4] Shi, Z., Kouri, D. J., Wei, G. W. and Hoffman, D. K., Generalized symmetric interpolating wavelets, Computer Physics Communications, 119: 194-218, 1999. [5] Nguyen, V. P., Rabczuk, T., Bordas, S. and Duflot, M., Meshless methods: a review and computer implementation aspects, Mathematics and Computers in Simulation, 79: 763-813, 2008. [6] Zhou, X. and Zhang, W., The evaluation of connection coefficients on an interval, Communications in Nonlinear Science & Numerical Simulation, 3: 252-255, 1998. [7] Latto, A., Resnikoff, L. and Tenenbaum, E., The evaluation of connection coefficients of compactly supported wavelets, Proceedings of the French-USA Workshop on Wavelets and Turbulence, 76-89, 1992. [8] Chen, X., Yang, S., Ma, J. and He, Z., The construction of wavelet finite element and its application, Finite Elements in Analysis and Design, 40: 541–554, 2004. [9] Timoshenko, S. and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill, 1959.

181

182


A Study of Dual Reciprocity for Three Dimensional Models Applied to the Solution of Pennes Bioheat Equation Fabrício Ribeiro Bueno(1) and Paul William Partridge(2). (1)

(2)

Programa de Pós Graduação em Estruturas e Construção Civil, Universidade de Brasília, Brasília-DF, Brasil .email fabrí[email protected]

DepartamentodeEngenhariaCivileAmbiental,UniversidadedeBrasília,BrasíliaDF,Brasil. emailpaulp@unb,br.

Keywords: Boundary Element Method, Dual Reciprocity, Augmentation functions, Tumours, Pennes bioheat equation.

Abstract. A system for the computational simulation of three dimensional models for heat transfer using the Pennes Bioheat equation was developed. The model is intended to be used to forecast temperatures and to be used in the diagnosis of tumours. The bioheat equitation is solved using the Boundary Element Method with Dual Reciprocity. In the Dual Reciprocity Method some approximating functions were implemented using augmentation functions. Given that the bioheat equation is an association of a Poissons equation (term which is a function of the space variables) with a Helmholtz equation (term which is dependent on the problem variable) the results for different sizes and locations of tumours determine which approximation functions will obtain best results.

Introduction In several previous studies, for example , [1, 2, 3, 4], it has been shown that the temperature on the skins surface is controlled by the local metabolism, by heat exchange with the environment and principally by the blood flow. There is no doubt that skin or breast tumours cause an increase in the local blood flow with a consequent increase in local temperature. A distribution of temperature and heat flux in a region of skin can be simulated using Pennes bioheat equation. Examples using the Boundary Element method, with Dual Reciprocity, considering a model of the skin in two dimensions can be found in [2, 5, 6] . Here results will be obtained for the solution of Pennes bioheat equation in three dimensions and different Dual Reciprocity approximation functions will be investigated to determine which are most appropriate. Later the study of which these results are a part will be continued including the diagnosis of tumours by thermal means using the genetic algorithm as was done for the two dimensional case [5,6].

The Pennes Bioheat Equation The Pennes bioheat equation can be written in the following form (Partridge and Wrobel, 2007):

Uc

wT wt

2T

Zb U b cb k

Ta T Q k

(1)

Where U , c and k represent respectively density, specific heat and thermal conductivity of tissue; U b , cb are the density and specific heat of blood respectively, Zb is blood perfusion, Ta is the blood temperature and Q is a spatial heating term. Equation (1) is problems: i) temperature prescribed T

q

subject to the usual boundary conditions for thermal

T ; ii) heat flux prescribed q q ; iii) convection h0 T T0 , where h0 is the coefficient of heat transfer de and T0 is the temperature of a surrounding


183

VH Tr4 T 4 , where V is the Stefan-Boltzman constant and H is the radioactive interchange factor between the surace and the exterior ambient. Tr [5]. Equation (1) can be

fluid and iv) radiation q

written in a steady state way as

2T

Zb U b cb k

Ta T Q

(2)

k

or

c1 Ta T c2

2T

(3)

Where c1 and c2 are defined as:

Z b U b cb

c1

k Q k

c2

Carrying out the multiplication equation (3) can also be written as

2T

c1Ta c1T c2

b

(4)

Given that Ta is constant equation (4) can be written:

2T

c3 c1T

(5)

where

c3

c1Ta c 2

In equation (5), considering only the constant c3 , the problem becomes one of the solution of a Poisson equation (the term being a function of space) and when one considers c1T

the problem becomes a

Helmholtz equation (the term being dependent on the problem variable T, temperature). Here these equations are solved using Boundary Elements with Dual Reciprocity.

Application of the Dual Reciprocity Method to the solution of the Bioheat Equation These equations Poisson and Helmhotz, are solved separately in [7], where a fundamental solution for a three dimensional isotropic medium is used u * 1 /( 4Sr ) , which is employed to deal with the term on the left hand side of equation (5) and the non homogeneous conditions are taken to the boundary using the Dual Reciprocity Method as in [8] leading to a system of matrix equations

Hu Gq

Huˆ Gqˆ F 1c3

(6)

Huˆ Gqˆ F 1uc1

(7)

For the Poisson equation this is

Hu Gq And for the Helmholtz equation

In this way the matrix equation for the solution of the bioheat equation is

Hu Gq

Huˆ Gqˆ F 1c3 Huˆ Gqˆ F 1uc1

Where the matrix F can be calculated once the approximating function is defined. Putting

S

Huˆ Gqˆ F 1

(8)

184

Eds: E L Albuquerque & M H Aliabadi Hu Gq

Sc3 Sc1u

(9)

Gq Sc3

(10)

or

H Sc1 u

For a well posed problem, a boundary condition is known for each boundary point and the tree dimensional model is done as considered below. Boundary conditions can be applied to produce the usual system of equations

Ax

y

(11)

The values of the parameters necessary for calculating c1 and c3 in equation (5) were obtained from [4]:

k 0.5W / mº C , U b 1000kg / m³ , cb 4000 J / kg º C , Zb 0.0005mlb / mlt s and Q 420 J / m³s for healthy tissue, while the values for the tissue with the tumour are Zb 0.002mlb / mlt s and Q 4200J / m³s , in which the subscripts b and t represent blood and tissue, respectively. The temperature of arterial blood is Ta 37 º C .

Approximation Functions In the original paper on DRM [9] the linear radial basis function f r was employed. After much research in order to find better approximation functions for use with the method, alternative functions such as such as the cubic radial basis function f r 3 and the polyharmonic splines which are generalizations of the form r 2 n logr became available. In spite of this, many recent papers which involve applications of the method continue using the linear function, which has the advantage of being simple to use. Here the following approximation functions were tested:

f

r 2 log r , and f

f

r, f

r3 ,

f

1 r , f

1 r3,

r 4 log r .

Augmentation functions were first suggested in a paper by Golberg and Chen [10] and first implemented in a paper on elasticity by Bridges and Wrobel [11], where it was concluded that they can produce a considerable improvement in accuracy.. Here tests were carried out without augmentation functions, and with the linear augmentation functions 1 , x , y and z.

ThreeDimensionalModel In fig. 1 a) a representation of a three dimensional model of skin tissue is shown. The section is delimited by the sides formed by the points A,B,C,D,E,F,G and H which is called the boundary *2 . In fig 1b, delimited by the points I,J,K,L,M,N,O and P, is the three dimensional model of the tumour called boundary *1 . Boundary *1 .is in a position within the boundary *2 . Considering the boundary *2 the face EFGH is an internal surface on which the temperature is considered to be constant at T 37º C (internal body temperature ). The boundary ABCD is considered to be the external surface on which initially the boundary condition q 0 is assumed, considering for example that the skin is bandaged. Another more realistic boundary condition for the boundary ABCD would be to consider convection, that is that heat is transmitted to the external medium. All of the remaining parts of *2 are artificial internal boundaries because the model represents only a section of tissue, where the skin is in contact with other


parts of the body and on these boundaries one can consider realistically the boundary condition q is no exchange of heat on these surfaces).

185

0 (there

Fig 1: a) Three dimensional model of healthy tissue b) Three dimensional model of tumour. When the boundary *1 is treated as a tumour internal to the boundary *2 , there exists two compatibility boundary conditions which will be that the temperature at a node on *1 is equal to the temperature on the corresponding node on *2 ( T1

q1

T2 ) and that there exists an exchange of heat between the two regions

q2 .

Here it is considered that the section of skin being considered has a rectangular surface of 0,08 x 0,08 m and a depth of 0,03m. The boundary element discretization is done considering 288 constant triangular elements on the boundary *2 and 48 constant triangular elements on the boundary *1 . Initially a tumour with dimensions 0,02 x 0,02 x 0,01 m is considered with its centre localized at coordinates (0,00; 0,00; 0,01) as shown in fig. 2 .

Fig 2: Tumour within healthy tissue, discretized using constant triangular boundary elements. Results In order to determine the dual reciprocity approximation function with the best performance various simulations were done using each of the function cited, combined or not with the augmentation functions.

186


As a criterion for judging the results found those of Partridge and Wrobel [5] were used. In this paper reference is made to Chan [2], where the two dimensional model is studied in detail. Partridge and Wrobel [5] concluded that the function f r 3 with linear augmentation functions 1 , x , y is the best choice for the two dimensional model. Table 1: Results for the approximation functions considered without considering augmentation functions. Position 2D inX reference 0,035 0,030 0,025 0,020 0,015 0,010 0,005 0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035

37,1782 37,1855 37,1989 37,2205 37,2523 37,2926 37,3284 37,3424 37,3284 37,2926 37,2523 37,2205 37,1989 37,1855 37,1782

f=r Value 33,3305 33,2193 33,1081 33,2636 33,4192 33,9662 34,5133 34,5170 34,5206 33,9821 33,4436 33,2983 33,1530 33,2775 33,4020

Error 10,3% 10,7% 11,0% 10,6% 10,3% 8,9% 7,5% 7,6% 7,5% 8,9% 10,2% 10,5% 10,9% 10,5% 10,2%

Withoutaugmentationfunctions f=1+r f=r³ Value Error Value Error 36,8382 0,9% 81,7657 119,9% 36,8414 0,9% 69,2333 86,2% 36,8447 1,0% 56,7009 52,4% 36,8057 1,1% 47,2696 27,0% 36,7667 1,3% 37,8383 1,6% 36,7256 1,5% 34,2271 8,2% 36,6846 1,7% 30,6159 18,0% 36,6844 1,8% 30,1468 19,3% 36,6842 1,7% 29,6777 20,5% 36,7255 1,5% 32,2195 13,6% 36,7669 1,3% 34,7613 6,7% 36,8070 1,1% 42,8674 15,2% 36,8472 0,9% 50,9735 37,0% 36,8452 0,9% 62,0809 66,9% 36,8431 0,9% 73,1882 96,9%

f=1+r³ Value Error 37,0934 0,2% 37,0626 0,3% 37,0318 0,4% 37,0359 0,5% 37,0399 0,6% 37,1361 0,4% 37,2323 0,3% 37,2303 0,3% 37,2283 0,3% 37,1289 0,4% 37,0295 0,6% 37,0230 0,5% 37,0166 0,5% 37,0468 0,4% 37,0770 0,3%

Table 2: Results for the approximation functions considered with the use of augmentation functions Position 2D inX reference 0,035 0,030 0,025 0,020 0,015 0,010 0,005 0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035

37,1782 37,1855 37,1989 37,2205 37,2523 37,2926 37,3284 37,3424 37,3284 37,2926 37,2523 37,2205 37,1989 37,1855 37,1782

f=r Value 33,4134 33,3207 33,2279 33,4049 33,5819 34,1180 34,6541 34,6577 34,6614 34,1348 33,6083 33,4462 33,2841 33,4014 33,5187

Error 10,1% 10,4% 10,7% 10,3% 9,9% 8,5% 7,2% 7,2% 7,1% 8,5% 9,8% 10,1% 10,5% 10,2% 9,8%

Withaugmentationfunctions f=1+r f=r³ Value Error Value Error 36,9161 0,7% 117,4509 215,9% 36,9179 0,7% 95,1818 156,0% 36,9197 0,8% 72,9127 96,0% 36,9468 0,7% 55,5844 49,3% 36,9739 0,7% 38,2562 2,7% 37,0417 0,7% 32,2441 13,5% 37,1095 0,6% 26,2320 29,7% 37,1098 0,6% 25,7724 31,0% 37,1100 0,6% 25,3128 32,2% 37,0428 0,7% 30,2586 18,9% 36,9756 0,7% 35,2045 5,5% 36,9495 0,7% 51,1732 37,5% 36,9234 0,7% 67,1420 80,5% 36,9233 0,7% 87,9008 136,4% 36,9232 0,7% 108,6596 192,3%

f=1+r³ Value Error 37,3486 0,5% 37,3106 0,3% 37,2726 0,2% 37,2524 0,1% 37,2321 0,1% 37,2393 0,1% 37,2464 0,2% 37,2453 0,3% 37,2443 0,2% 37,2347 0,2% 37,2250 0,1% 37,2420 0,1% 37,2590 0,2% 37,2932 0,3% 37,3274 0,4%

In order to make the comparison, a section through the three dimensional results was considered which is thus identical to the two dimensional results cited. Tables 1 and 2 are obtained considering the results at the sections and comparing with the two dimensional results Table 1 is done for the cases without augmentation functions and table 2 is done considering the functions.. On the basis of these results it is possible to eliminate the use of the functions f

r and f produce inaccurate results and indicates the necessity to study in more detail the functions f f 1 r 3 , with and without the augmentation functions.

r 3 which 1 r and

In order to judge the behaviour of these approximation functions for the different possible cases of position of tumour to be analysed, a non symmetric model with a tumour of size (0,02 x 0,02 x 0,01 m) and with its


187

centre located at the coordinates (0,015; 0,00; 0,01) was considered. The tumour has be displaced in the x , coordinate but continues to have a section equivalent to the two dimensional case for the comparison of the results given below. Table 3: Comparison of the approximating functions which produced best results. Position 2D inX reference 0,035 0,030 0,025 0,020 0,015 0,010 0,005 0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035

37,158756 37,162018 37,165279 37,17386 37,18244 37,201995 37,221551 37,259488 37,297426 37,324825 37,352224 37,330444 37,308664 37,279711 37,250758

Withoutaugmentationfunctions f=1+r Value Error 36,90132 0,7% 36,9145 0,7% 36,92769 0,6% 36,91588 0,7% 36,90408 0,7% 36,86183 0,9% 36,81959 1,1% 36,7445 1,4% 36,66942 1,7% 36,64842 1,8% 36,62743 1,9% 36,63352 1,9% 36,63962 1,8% 36,6624 1,7% 36,68518 1,5%

f=1+r³ Value Error 37,06672 0,2% 37,03418 0,3% 37,00165 0,4% 36,97559 0,5% 36,94953 0,6% 36,94474 0,7% 36,93994 0,8% 37,0195 0,6% 37,09906 0,5% 37,18993 0,4% 37,2808 0,2% 37,23147 0,3% 37,18215 0,3% 37,13639 0,4% 37,09063 0,4%

Withaugmentationfunctions f=1+r Value Error 36,98842 0,5% 36,97689 0,5% 36,96537 0,5% 36,95514 0,6% 36,94492 0,6% 36,95242 0,7% 36,95992 0,7% 37,01189 0,7% 37,06386 0,6% 37,09333 0,6% 37,12281 0,6% 37,05741 0,7% 36,99201 0,8% 36,92168 1,0% 36,85136 1,1%

f=1+r³ Value Error 37,38048 0,6% 37,34503 0,5% 37,30959 0,4% 37,27379 0,3% 37,238 0,1% 37,2187 0,0% 37,19941 0,1% 37,21284 0,1% 37,22626 0,2% 37,24607 0,2% 37,26589 0,2% 37,26936 0,2% 37,27283 0,1% 37,28879 0,0% 37,30474 0,1%

Fig. 3: Graphical comparison of the results in table 3. In fig. 3 it can be seen that “ f 1 r without augmentation functions” and “ f 1 r with augmentation functions” show sections where the behaviour is different from that expected considering the 2D reference model in such a way that they cannot be accepted as representing the physical model. 3

The functions which show a behaviour similar to that obtained in 2D are “ f functions” and “ f

1 r without augmentation functions”. 3

1 r with augmentation

188


Next the three dimensional behaviour of these two functions is compared

Fig. 4: Temperature distribution for the model of fig. 3 using f

Fig. 5: Temperature distribution for the model of fig. 3 using f

In fig. 5 it is observed that the function “ f the behaviour at the edges of the model.

1 r with augmentation functions.

1 r 3 without augmentation functions

1 r 3 without augmentation functions” was not able to capture

Conclusions The model using the Boundary Element method with Dual Reciprocity can be used to represent the behaviour of skin temperature distributions in the presence of tumours. The model using the Dual Reciprocity Method with the best results is that which uses the approximation function f 1 r with linear augmentation functions 1 , x , y and z. The approximation functions f

r and f

r ³ without augmentation give inaccurate results.

Advances in Boundary Element and Meshless Techniques XII Here the “ Polyharmonic Spline” functions generalized in the form r 2 n logr ,were also tested considering the cases with n 1 e n 2 , however the results obtained with these functions were not accurate as in the case of the functions f r and f r ³

References [1]Deng, Z. S. and Liu, J. Eng Anal Boundary Elem 28 97–108 (2004). [2]Chan, C.L. Trans ASME, J Biomech Eng 114 358–365 (1992). [3]Deng, Z.-S. and Liu, J. Med Eng Phys 22 693–702 (2000). [4] Lui, J and Xu, L. S. Int. J. Heat Mass Transfer 43 2827-2839 (2000) [5]Partridge, P. W. and Wrobel, L. Eng. Analysis with Boundary Elements 31 803-811 (2007). [6]L. C. Wrobel, P. W. Partridge, L. C. L. B. de Castro, F. R. Bueno, 7th UK conference on Boundary Integral Methods, 7th UKBIM, Proceedings, 99-106, (2009) [7] Brebbia, C.A. and Domingues, J. Boundary Elements: An Introductory Course. Computational Mechanics Publications and McGraw-Hill Book Company, (1989). [8]Partridge, P.W. Wrobel L.C. and Brebbia, C.A. The dual reciprocity boundary element method, Elsevier, (1992). [9]Nardini, D. and Brebbia, C.A. Appl Math Modell 7 157–162 (1983). [10]Golberg, M.A. and Chen, C.S. Boundary Elem Commun 5 57–61 (1994). [11] Bridges, T.R. and Wrobel, L.C. Commun. in Numerical Methods Eng 12, 209–220 (1996)

189

190


STRESS ANALYSIS OF THIN PLATE COMPOSITE MATERIALS UNDER DYNAMIC LOADS USING THE BOUNDARY ELEMENT METHOD L. S. Campos1, K. R. P. Sousa2, A. P. Santana2, A. dos Reis3, E. L. Albuquerque5 and P.Sollero6 University of Brazilia – UNB Faculty of Technology 70910-900, Brasilia, Bsb, Brazil [email protected] 1

2 Federal Institute of Maranhão Department of Mechanical and Materials 65025-000, São Luis, MA, Brazil {kerlles,andre}@ifma.edu.br 3

University of Brazilia – UNB Faculty of Technology 70910-900, Brasilia, Bsb, Brazil [email protected] 4

5

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

Keywords: Boundary element method, radial integration method, plates, composite materials,dynamic of plates.

Abstract. . This work presents a dynamic formulation of the boundary element method for the computation of stresses on the boundary of anisotropic thin plates. The formulations uses elastostatic fundamental solutions 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 the RIM, the augmented thin plate spline is used as the approximation function. The time integration is carried out using the Houbolt method. Stresses on the boundary are computed by a procedure that uses integral equations for the first transversal displacement derivatives, derivatives of shape functions, and constitutive relations. Only the boundary is discretized in the formulation. Numerical results show good agreement with results available in literature. Introduction. The boundary element formulation for plate bending has included the analysis of anisotropic problems. Shi and Bezine [1] presented a boundary element analysis of plate bending problems using fundamental solutions proposed by [6] based on Kirchhoff plate bending assumptions. Rajamohan and Raamachandran [12] proposed a formulation where singularities were avoided by placing source points outside the domain. Albuquerque et al [9] 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 [10]. In [1], this formulation was extended for dynamic problems. Shear deformable shells have been analyzed using the boundary element method by [3] with the analytical fundamental solution proposed by [4]. Wang and Huang [2] presented a boundary element formulation for orthotropic shear deformable plates. Later, in Wang and Schweizerhof [5], the previous formulation was extended to laminate composite plates.


191

Stress and moment computation by the BEM has been addressed by some works in literature. For example, Zao [8] and Zao and Lan [7] have discussed the computation of stresses in plane elastic problems, Knopke [11] presented and discussed the integral formulation for computation of stresses in isotropic thin plate, Rashed et al [13] presented an stress integral formulation in the BEM for Reissner plate bending problems. This paper proposes a numerical procedure to compute stresses at internal points and at the boundary of composite laminated plates using a dynamic boundary element plate formulation that follows the Kirchhoff hypotheses. Computation of stresses on the boundary. The equations for anisotropic thin plates problems are given by two integral equations, for displacement and rotation. These equations present domain integrals that are

originated by the body forces. For the purpose of transforming these integrals into boundary integrals, the radial integration method (RIM) is used and time integration is carried out using the Houbolt method, as shown in Sousa et al [15]. Due to the lack of space, integral formulation will not be shown here. The stress components in each layer can be calculated using the strain, as shown in: ߪ௫ ߳௫ ൥ ߪ௬ ൩ ൌ ܳത௞ ൥ ߳௬ ൩ǡ ߬௫௬ ߛ௫௬

(1)

where ܳത௞ is: (2)

ܳത௞ ൌ ܶ ିଵ ܳ௞ ܶ ି௧ ǡ

the matrix ܳ௞ is: ‫ܧ‬௅ ‫ۍ‬ ͳ െ ‫ݒ‬௅் ‫்ݒ‬௅ ܳ௞ ൌ ‫ݒ ێ‬௅் ‫ܧ‬௅ ‫ ͳێ‬െ ‫ݒ‬௅் ‫்ݒ‬௅ Ͳ ‫ۏ‬

‫ݒ‬௅் ‫ܧ‬௅ ͳ െ ‫ݒ‬௅் ‫்ݒ‬௅ ‫ܧ‬௅ ͳ െ ‫ݒ‬௅் ‫்ݒ‬௅ Ͳ

Ͳ ‫ې‬ ‫ۑ‬ Ͳ ‫ۑ‬ ‫ۑ‬ ‫ܩ‬௅் ‫ے‬

(3)

and the transformation matrix ܶ is: ݊ଶ ʹ݊݉ ݉ଶ ܶ ൌ ൥ ݊ଶ ݉ଶ െʹ݊݉ ൩ǡ െ݊݉ ݊݉ ݉ଶ െ ݊ଶ

(4)

with ݉ ൌ ሺߠሻ and ݊ ൌ ሺߠሻ, where ߠ is the angle between the fibers and the chosen system of reference. The strain components are written in terms of transversal displacement as:

192


߲ଶ‫ݓ‬ ‫ ۍ‬െ‫ݖ‬ ‫ې‬ ߲‫ ݔ‬ଶ ‫ۑ‬ ‫ێ‬ ߳௫ ଶ ߲ ‫ۑ ݓ‬ ‫ێ‬ ൥ ߳௬ ൩ ൌ ‫ ێ‬െ‫ ݖ‬ଶ ‫ۑ‬Ǥ ߲‫ݕ‬ ߛ௫௬ ‫ێ‬ ଶ ‫ۑ‬ ‫ێ‬െʹ‫ۑ ݓ ߲ ݖ‬ ‫ۏ‬ ߲‫ےݕ߲ݔ‬

(5)

Therefore the second derivatives of transverse displacement are necessary to compute de strain components, as shown in equation (5), and consequently the stress. For a point inside the domain, these derivatives are computed by integral formulation, as can be seen at [14]. On the boundary, a different approach is proposed, since the integral formulation for second derivatives of displacement presents some integrals that are more than hypersingular (1/r3). The second derivatives, in tangential and normal coordinates, can be computed using quadratic discontinuous shape functions, as: ሺଵሻ

ሺଶሻ

ሺଷሻ

݀ܰ ߲‫ݓ‬ଵ ݀ܰௗ ߲‫ݓ‬ଶ ݀ܰௗ ߲‫ݓ‬ଷ ݀‫ݐ‬ ߲ଶ‫ݓ‬ ൌ൭ ௗ ൅ ൅ ൱ ൬ͳൗ ൰ ݀ߦ ߲‫ݐ‬ ݀ߦ ߲‫ݐ‬ ݀ߦ ߲‫ݐ‬ ݀ߦ ߲‫ ݐ‬ଶ

ሺଵሻ

ሺଶሻ

(6)

ሺଷሻ

݀ܰ ߲‫ݓ‬ଵ ݀ܰௗ ߲‫ݓ‬ଶ ݀ܰௗ ߲‫ݓ‬ଷ ݀‫ݐ‬ ߲ଶ‫ݓ‬ ൌ൭ ௗ ൅ ൅ ൱ ൬ͳൗ ൰ ݀ߦ ݀ߦ ߲݊ ݀ߦ ߲݊ ݀ߦ ߲݊ ߲߲݊‫ݐ‬ డమ ௪

(7)

డమ ௪

డమ ௪

డ௡

The last derivative, డ௡మ , cannot be calculated in the same manner as డ௧ మ and డ௡డ௧ since డ௧ is not known. The alternative is to use the moment equation in the ݊‫ ݐ‬system, given, in matrix form, by: ߲ଶ‫ݓ‬ ‫ۍ‬ ‫ې‬ ଶ ‫ۑ ߲݊ ێ‬ ‫ܯ‬௡௡ ଶ ‫ۑݓ ߲ێ‬ ൥ ‫ܯ‬௧௧ ൩ ൌ ‫ܦ‬Ԣ ‫ێ‬ ଶ ‫ۑ‬ ‫ܯ‬௡௧ ‫ݐ߲ ێ‬ ଶ ‫ۑ‬ ‫ۑݓ ߲ێ‬ ‫ےݐ߲߲݊ۏ‬

(8)

where ‫ܦ‬Ԣ is the stiffness matrix given in ݊‫ ݐ‬coordinates, as: ‫ ܦ‬ᇱ ൌ ܶ ିଵ ‫ି ܶܦ‬௧ Ǥ

Equation (8) can be rewritten conveniently as:

(9)

߲ଶ‫ݓ‬ ‫ۍ‬ ‫ې‬ ‫߲݊ ێ‬ଶ ‫ۑ‬ ‫߲ ێ‬ଶ‫ۑ ݓ‬ ‫ ێ‬ଶ ‫ۑ‬ ‫ݐ߲ ێ‬ ଶ ‫ۑ‬ ‫ۑݓ ߲ێ‬ ‫ےݐ߲߲݊ۏ‬

‫ܯ‬௡௡ ܵԢ ൥ ‫ܯ‬௧௧ ൩ ൌ ‫ܯ‬௡௧

(10)

where ᇱ ܵଵଵ ᇱ ܵ ᇱ ൌ ቎ܵଵଶ ᇱ ܵଵ଺

ᇱ ܵଵଶ ᇱ ܵଶଶ ᇱ ܵଶ଺

ᇱ ܵଵ଺ ᇱ ܵଶ଺ ቏ ൌ ‫ ܦ‬ᇱିଵ Ǥ ᇱ ܵଷ଺

The unknown variables, ‫ܯ‬௧௧ , ‫ܯ‬௡௧ and

ᇱ െͳ ܵଵଶ ᇱ ቎ Ͳ ܵଶଶ ᇱ Ͳ ܵଶ଺

(11)

பమ ௪ ப௡మ

, can be isolated in matrix form as in:

െܵ ᇱ ‫ܯ‬ ‫ ۍ‬ଶ ଵଵ ௡௡ ‫ې‬ ߲ଶ‫ݓ‬ ᇱ ܵଵ଺ ߲ ‫ݓ‬ ‫ێ‬ ‫ۑ‬ ᇱ ଶ ᇱ ܵଶ଺ ቏ ൦ ߲݊ ൪ ൌ ‫ ݐ߲ ێ‬ଶ െ ܵଵଶ ‫ܯ‬௡௡ ‫ۑ‬Ǥ ‫ܯ‬௧௧ ‫ ێ‬ଶ ‫ۑ‬ ᇱ ܵଷ଺ ‫ ݓ ߲ ێ‬െ ܵᇱ ‫ۑ ܯ‬ ‫ܯ‬௡௧ ௡௡ ଵ଺ ‫ݐ߲߲݊ۏ‬ ‫ے‬

(12)

Once the system is solved, derivatives in the chosen reference system can be obtained by the coordinate transformation: ߲ଶ‫ݓ‬ ‫ۍ‬ ‫ې‬ ଶ ‫ۑ ݔ߲ ێ‬ ଶ ‫ۑݓ ߲ێ‬ ‫ ݕ߲ ێ‬ଶ ‫ ۑ‬ൌ ‫ ێ‬ଶ ‫ۑ‬ ‫ۑݓ ߲ێ‬ ‫ےݕ߲ݔ߲ۏ‬

߲ଶ‫ݓ‬ ‫ۍ‬ ‫ې‬ ‫߲݊ ێ‬ଶ ‫ۑ‬ ଶ ‫ۑݓ ߲ێ‬ ܶ‫ێ‬ ଶ ‫ۑ‬ ‫ݐ߲ ێ‬ ‫ۑ‬ ଶ ‫ۑݓ ߲ێ‬ ‫ےݐ߲߲݊ۏ‬

(13)

The stress is then given by equations (1) and (5). Numerical results. Consider a square clamped-plate under a uniformly distributed step load applied at time ɒ଴ ൌ Ͳ with amplitude ൌ ʹǡͲ͹ǤͳͲ଺ Ȁଶ. The plate is orthotropic with the following material properties: ଶ ൌ ͸ͺͻͷ, ଵ ൌ ʹଶ , ଵଶ ൌ ʹ͸ͷͳǤͻ,ଵଶ ൌ ͲǤ͵, ɏ ൌ ͹ͳ͸͸Ȁଷ . The edges of the plate have length ൌ ʹͷͶ and thickness ൌ ͳʹǤ͹. This problem is equivalent to a problem proposed by Sladek et al. [16] which was analyzed using the MPLG. Twelve quadratic discontinuous boundary elements (three per edge) with equal length and time steps οɒ ൌ ͹͵ߤ‫ ݏ‬are used in the discretization of space and time, respectively. Nine internal points are used in the simulation. The results obtained for ߪ௬௬ at the middle of the edge are shown in Figure 1.

194


Figure 1: Stress at the middle of the edge of the plate. Due to the lack of similar results in the computation of stress at the boundary of anisotropic plates in literature, this outcome could not be compared directly. However when the moments are calculated by the procedure shown in [15], results are found to be in good agreement with [16] and [17]. Provided that moments and stresses are both functions of the second derivatives of transverse displacement, it is coherent to assume that the results found are valid. Conclusion. This paper presented successfully a procedure to compute stresses on the boundary of anisotropic plates under dynamic loads. A different approach to the computations of the second derivatives of transverse displacement on the boundary is proposed, avoiding the treatment of more than hypersingular integrals and making possible to compute both stresses and moments. References [1] 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). [2] J. Wang and M. Huang. Boundary element method for orthotropic thick plates, Acta Mechanica Sinica, Vol. 7 (3), pp. 258–266,(1991). [3] 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). [4] 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). [5] 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).

Advances in Boundary Element and Meshless Techniques XII [6] 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). [7] Z. Zhao and S. Lan. Boundary stress calculation - a comparison study, Computers & Structures, Vol. 71, pp. 77-85, (1999). [8] Z. Zhao. On the calculation of boundary stress in boundary elements, Engineering Analysis with Boundary Elements, Vol. 16, pp. 317-322, (1995). [9] 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. [10] 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). [11] 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). [12] C. Rajamohan and J. Raamachandran. Bending of anisotropic plates charge simulation method, Advances in Engineering Software, Vol. 30, pp. 369–373, (1999). [13] 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). [14] 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. [15] K. R. Sousa, A. P. Santana, E. L. Albuquerque, and P. Sollero. Computation of Moments in thin Plates of Composite Materials under Dynamic Load using the Boundary Element Method In: Beteq, 2010, Berlim. International Conference on Boundary Element and Meshless Techniques, (2010). [16] 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. [17] J. Useche. Shellcomp v3.4: Finite Element Analysis Program for Linear Static and Dynamic Analysis of Composite Shell Structures. Universidade Tecnológica de Bolivar, Cartagena, Colmbia, 2008.

195

196


Deformation Analysis of Thin Plate with Distributed Load by Triple-Reciprocity Boundary Element Method

Yoshihiro OCHIAI㸨1 and Tomohiro SHIMIZU㸨1 *1

Kinki University, Department of Mechanical Engineering, 3-4-1 Kowakae, Higashi-Osaka, 577-8502 Japan, E-mail: [email protected]

.H\:RUGV%RXQGDU\(OHPHQW0HWKRG6KHOO6WUXFWXUH&RPSXWDWLRQDO0HFKDQLFV7KLQ3ODWH

In general, internal cells are required to solve the deformation of a thin plate with an arbitrary distributed load using a conventional boundary element method (BEM). However, in this case, the merit of the BEM, which is the easy preparation of data, is lost. In this paper, it is shown that the deformation analysis of a thin plate with an arbitrary distributed load can be performed without the use of internal cells by using the triple-reciprocity BEM. The distribution of an arbitrary load is interpolated using boundary integral equations. The problem of the thin plate, in accordance with Kirchhoff’s theory, is formulated by means of two coupled Poisson equations, which are expressed in integral form using the second theorem of Green in the classical way. A new computer program was developed and applied to several problems. 1. Introduction The finite element method (FEM) requires a finite element mesh for the deformation analysis of a thin plate. Thin-plate problems with an arbitrary distributed load can be solved by a conventional boundary element method (BEM) using internal cells for domain integrals. In this case, however, the merit of the BEM, which is the ease of data preparation, is lost. On the other hand, several countermeasures have been considered. For example, Nowak and Neves have proposed the conventional multiple-reciprocity boundary element method (MRBEM). In the conventional MRBEM, the distribution of load must be given analytically, and fundamental solutions of higher order are used to make solutions converge. Accordingly, this method is not suitable for thin-plate analysis with an arbitrary distributed load. A dual-reciprocity BEM has been proposed to reduce the dimensionality, which is an advantage of the BEM. However, it is difficult to select a suitable function for thin-plate problems with an arbitrary distributed load. Sladek V. et al. applied the local boundary element method to thin plate problems without internal cells [1]. Ochiai proposed the triple-reciprocity BEM (improved multiple-reciprocity BEM) without using internal cells for elastoplastic problems [2, 3]. Using this method, a highly accurate solution can be obtained using only fundamental solutions of low orders while reducing the need for data preparation. In this paper, it is shown that the deformation analysis of a thin plate with an arbitrary distributed load can be solved without the use of internal cells by using the triple-reciprocity BEM. The distribution of an arbitrary load is interpolated using boundary integral equations. The problem of a thin plate, in accordance with Kirchhoff’s theory, is formulated by means of two coupled Poisson equations, which are expressed in integral form using the second theorem of Green in the classical way [4, 5]. 2. Theory P L 2.1 Thin plate The deformation of a thin plate with a point load f1 or line load f1 can be easily solved using the conventional BEM. We use w to denote the transversal displacement of the plate and f1 ( x) to denote the distributed transversal load. According to Kirchhoff’s theory, the following equations must be solved:

2 M1

f1 ( x)

(1)


197

M1 , (2) D where E , Q , h and D are Young’s modulus, Poisson’s ratio, the thickness of the plate and the bending rigidity of the plate, respectively, where Eh3 . (3) D 12(1 Q 2 ) From Eqs. (1) and (2)the following boundary integral equations are obtained: 2w

wT ( x, [ ) M * ww( x) d*³ 1 w1 ( x)d * ³ 1 T1 ([ , x)d: , (5) wn wn D *

Cw1 ([ )

*

³ T1

( x, [ )

where C 0.5 on the smooth boundary and C 1 in the domain. * and : represent the boundary and domain, respectively. n is the unit outward vector to the boundary in the outward direction. [ and x are the observation point and loading point, respectively. The function T1* ([ , x) in eqs (1) and (2) is given as follows: 1 1 * (6) T1 ([ , x) [ln( ) B] , 2S r where B is an arbitrary constant that can be set to B 0 . Using the function T1* , the polyharmonic function T f* ([ , x) is considered, which is defined as

2T f* ([ , x) T f*1 ([ , x) .

(7)

The polyharmonic functions T f* ([ , x) can be obtained using the following equation. T f* ([ , x)

³

1 [ ³ rT f*1 ([ , x)dr ]dr r

(8)

Using Eq. (7), eq (5) becomes Cw1 ([ )

³ T1 ( x, [ ) *

1 1 wT ( x, [ ) ww( x) wT ( x, [ ) wM 1 ( x) * d*³ 1 w1 ( x) d * ³ T2 ( x, [ ) d* ³ 2 M 1 ( x)d * D D wn wn wn wn (9) *

*

1 * ³ f1 ( x)T1 ([ , x)d: D where T f* and its normal derivative wT f* / wn are given by

T f* ([ , x)

wT f* ([ , x) wn

f 1 1 1 r 2 f 2 {[ln( ) B ] sgn( f 1) ¦ } 2 r 2S [(2 f 2)!!] e 1e

(10)

f 1 r 2 f 3 r ,i ni 1 1 {2( f 1)[ln( ) B ] 1 2( f 1) ¦ } 2 r 2S [(2 f 2)!!] e 1e

.

(11)

Moreover, we set r ,i wr / wxi . The ith component of the unit normal vector is denoted by ni . Equations (4) and (9) contain domain integrals for the distributed load. 2.2 Interpolation for arbitrary distributed load To avoid the domain integrals in eqs (4) and (9), an P L interpolation method is introduced. The point load f1 and line load f1 are easily obtained by the conventional BEM. The distributed load is defined as f1S (q ) , and the total load is f1 ( x ) f1S (q)

S P L f1 ( x) f1 f1 .

can be interpolated using the following Poisson equations [6-8]㸬 2 f1 ( q ) S

f 2 (q) S

(12) (13)

198

Eds: E L Albuquerque & M H Aliabadi M

¦ f 3P ( xm )

2 f2S

(14)

m 1

Using eqs (13) and (14), the following boundary integral equations are obtained. Cf1 ([ )

³ [T1 ( x, [ ) *

wf1 ( x) wT ( x, [ ) wf ( x) wT2 ( x, [ ) * d* 1 f1 ( x)]d* ³ T2 ( x, [ ) 2 d * ³ f 2 ( x)d * wn wn wn wn *

M

*

(15)

¦ f 3P ( xm )T3 ([ , xm ) *

m 1

Cf 2 ([ )

³ T1 ( x, [ ) *

M wT ( x, [ ) wf 2 ( x) * f1 ( x)d * ¦ f 3P ( xm )T3 ([ , xm ) d*³ 2 wn wn m 1 *

(16) By solving the boundary integral equations (15) and (16), unknown values can be obtained for the interpolation. 2.3 Triple-reciprocity boundary element method Using eqs (13) and (14) and Green's second identity twice, eqs (4) and (9) become CM 1 ([ )

wT ( x, [ ) s wf ( x) wT ( x, [ ) S wM 1S ( x) * f1 ( x)d * d*³ 2 M 1 ( x)d * ³ T2 ( x, [ ) 1 d*³ 1 wn wn wn wn * s M wT ( x, [ ) S wf ( x) * (17) ³ 2 T3 ( x, [ )d * ³ 3 f 2 ( x)d * ¦ f 3P ( xm )T3* ([ , xm )

*

s

*

³ T1 ( x, [ )

wn

wn

*

m 1

* ww( x) wT ( x, [ ) 1 wM S ( x) 1 wT ( x, [ ) S [) Cw1 ([ ) ³ d*³ 1 w1 ( x)d * ³ T2* ( x, [ ) 1 d * ³ 2 M 1 ( x)d * D wn D wn wn wn * s S 1 1 wT ( x, [ ) S wf ( x) 1 wf ( x) * * d* ³ 3 f1 ( x)d * ³ T4 ( x, [ ) 2 ³ T3 ( x, [ ) 1 d* D D D wn wn wn (18) * 1 wT ( x, [ ) S 1 M * ³ 4 f 2 ( x)d * ¦ f 3P ( xm )T4 ([ , xm ) . D wn Dm 1

*

* T1 ( x,

Using eqs (17) and (18), unknown values are obtained. Differentiating eq (18), the gradient of the plate ww1 ([ ) / wxi at internal point is given as

ww1 ([ ) wxi

* S 2 * wT1 ( x, [ ) ww( x) w 2T1 ( x, [ ) d*³ w1 ( x)d * 1 ³ T2 ( x, [ ) wM 1 ( x) d * 1 ³ w T2 ( x, [ ) M 1S ( x)d * wxi wn wxi wn D wxi wn D wxi wn

*

³

*

1 wT3 ( x, [ ) wf1 ( x) 1 w 2T3 ( x, [ ) S 1 wT ( x, [ ) wf 2 ( x) d* ³ f1 ( x)d * ³ 4 d* ³ wx w w w wxi wn D n D x n D i i

wT ([ , xm ) 1 w 2T4 ( x, [ ) S 1 M f 2 ( x)d * ¦ f 3P ( xm ) 4 , ³ wxi wn wxi D Dm 1

s

*

S

*

*

*

*

(19)

where wT f* ([ , x)

r 2 f 3 r , i

wxi

2S [(2 f 2)!!]2

f 1 1 1 {2( f 1)[ln( ) B] 1 2( f 1) ¦ } r e 1e

w 2T f* ([ , x)

w 2T f*n j

r 2 f 4n j

wxi wn

wxi wx j

2S [(2 f 2)!!]2

1 r

f 1

(20)

1

G ij {2( f 1)[ln( ) B] 1 2( f 1) ¦ }] e 1e

f 1 1 1 2r ,i r , j {2( f 2)( f 1)[ln( ) B] (2 f 3) 2( f 2)( f 1) ¦ } r e 1e

(21) .

The moment of the plate can be obtained using the following relationships between the moment M ij and the transversal displacement of the plate w . M xx

D(

w2w w2w Q 2 ) (22) wx 2 wy


M yy M xy

D(

w2w w2w Q 2 ) wy 2 wx

w2w wxwy

D(1 Q )

199

(23)

(24)

The bending and twisting moments M ij at internal point are given by the following boundary integral equation㸬 M ij ([ )

wM ij*[1] ( x, [ ) 1 ww( x) wM 1S ( x) d*³ w1 ( x)d * ³ M ij*[ 2] ( x, [ ) d* D wn wn wn *[ 2 ] 1 wM ij ( x, [ ) ³ M 1S ( x ) d * D wn

³ M ij

*[1]

( x, [ )

*[3] 1 wf s ( x) 1 wM ij ( x, [ ) S 1 wf S ( x) d* ³ f1 ( x)d * ³ M ij*[ 4] ( x, [ ) 2 d* ³ M ij*[3] ( x, [ ) 1 D D D wn wn wn

*[ 4 ] 1 wM ij ( x, [ ) S 1 M f 2 ( x)d * ¦ f 3P ( xm ) M ij*[ 4] ([ , xm ) ³ D wn Dm 1

(25)

The function M ij*[ f ] ([ , x) is obtained using the following equations. w 2T f*

*[ f ] ([ , x) M xx

D(

M *yy[ f ] ([ , x)

D(

*[ f ] M xy ([ , x)

D(1 Q )

wx 2 w 2T f* wy 2

Q Q

w 2T f* wy 2 w 2T f*

w 2T f* wxwy

wx 2

) (26) ) (27)

(28)

Differentiating eq (20), w 3T f* / wxi wx j wxk , which is necessary to obtain the normal derivative wM ij*[ f ] / wn , is obtained. f 1 w 3T f* ([ , x) 1 1 r 2 f 5 ( f 2)[G ij rk 2( f 3)r ,i r , j r , k r ,i G jk r , j G ki ]{2( f 1)[ln( ) B] 1 2( f 1) ¦ } wxi wx j wxk S [(2 f 2)!!]2 r e 1e ( f 1)[G ij r , k 2(2 f 5)r ,i r , j r , k r ,i G kj r , j G ki ]

(29) Using eq (2), the shear force Qi ([ ) is obtained as Qi ([ )

D

w w2w w2w ( ) wxi wx 2 wy 2

w M. wxi

(30)

Concretely, Qi ([ ) at internal point is given by the following boundary integral equation.

³

Qx ([ ) ³

wT1* ( x, [ ) wM 1S ( x) w 2T1* ( x, [ ) S wT * ( x, [ ) wf1s ( x) w 2T2* ( x, [ ) s d*³ M 1 ( x )d * ³ 2 d*³ f1 ( x)d * wxi wn wxi wn wxi wn wxi wn

* w 2T3* ( x, [ ) S wf 2 s ( x) wT3 ( x, [ ) d*³ f 2 ( x)d * wn wxi wxi wn M

¦ f 3P ( xm )

m 1

(31)

wT3* ([ , xm ) wxi

Using two Poisson equations, the problem of a thin plate under any type of boundary condition including a curved boundary can be solved using the following relationship [4, 5]: w2w wt 2

w2w ww N (s) wn ws2

where N (s) is the curvature of the boundary and t denotes the tangential to the boundary direction.

(32)

200


3. Numerical Examples The deformation of a simply supported square plate with side length A 100 mm under the following sinusoidal load is obtained as a numerical example㸬 Sx Sy (33) ) sin( ) f1 ( x, y ) q0 sin( A

A

A plate thickness of h 1 mm , and a load of q0 1N / mm 2 , a Young’s modulus of E 210GPa and a Poisson’s ratio of Q 0.3 are assumed㸬As shown in Fig.1, the numbers of boundary elements and internal points are 200 and 81, respectively. Figure 1 shows a comparison of the deflection obtained by this method and the exact solution. Figure 2 shows the gradient of the plate and the bending moment distribution of the plate. The deflection of a simply supported circular plate with radius R 50 mm , as shown in Fig. 3, under the following quadratic distributed load is obtained as another numerical example. f1 (r )

q0 (

A2 r 2 A2

) (34)

A plate thickness of h 1 mm , load of q0 1N / mm 2 , a Young’s modulus of E 210GPa and a Poisson’s ratio of Q 0.3 are assumed. As shown in Fig.3, the numbers of boundary elements and internal points are 72 and 131, respectively. Figure 3 shows a comparison of the deflection obtained by this method and the exact solution. Fig. 4 shows the moment distribution and the moment distribution in the case of a clamped circular plate with the exact solutions. 4. Conclusion It was shown that thin-plate analysis can be carried out, without the use of internal cells, using the triple-reciprocity BEM. The theory is formulated by means of two coupled Poisson equations, therefore the formulation process is very simple. Fundamental solutions for thin-plate analysis were shown. In this method, the strong singularity that appears in the calculation of shear force by the conventional BEM becomes weak. Using numerical examples, the effectiveness and accuracy of this method were demonstrated. In this method, the merit of the BEM, which is the ease of data preparation, is not lost because internal cells are not necessary. References (1) Sladek, J., Sladek, V. and Mang, H. A., Meshless Formulations for Simply Supported and Clamped Plate Problem, International Journal for Numerical Methods in Engineering, No. 55, pp.359-375 (2002). (2) Ochiai, Y., Meshless Thermo-Elastoplastic Analysis by Triple-Reciprocity BEM, Transactions of the Japan Society of Mechanical Engineers, Vol.74, No.743, pp.939-945 (2008). (3) Ochiai, Y., Three-Dimensional Thermal Stress Analysis by Triple-Reciprocity Boundary Element Method, International Journal of Numerical Methods in Engineering, Vol.63, No.12, pp.1741-1756(2005). (4) Paris, F. and Leon, S., Thin Plates by the Boundary Element Method by Means of Two Poisson Equations, Engineering Analysis with Boundary Elements No.17, pp.111-122 (1996). (5) Katsikadelis, J. T. , Boundary Elements – Theory and Applications, Elsevier, London (2002).

Figure.1 Displacement of simply supported square plate with sinusoidal load

Advances in Boundary Element and Meshless Techniques XII (6) 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.1879 -1891 (2001). (7) Ochiai, Y. and Kobayashi, T., Initial Stress Formulation for Elastoplastic Analysis by Improved Multiple-Reciprocity Boundary Element Method, Engineering Analysis with Boundary Elements, Vol. 23, pp. 167-173, (1999). (8) Ochiai, Y. and Yasutomi, Z., Improved Method Generating a Free-Form Surface Using Integral Equation, Computer Aided Geometric Design, Vol. 17 (2000), pp.233-245.

Figure.2 Gradient and moment M xx of square plate with sinusoidal load

Figure.3 Displacement of simply supported circular plate under nonuniform load

Figure.4 Moment of simply supported and clamped circular plate under quadratic load

201

202


$Q$FWLYH1RLVH&RQWURO6HQVLWLYLW\)RUPXODWLRQIRU7KUHH'LPHQVLRQDO %(0 $%UDQFDWL0+$OLDEDGLDQG90DOODUGR

'HSDUWPHQWRI$HURQDXWLFV,PSHULDO&ROOHJH/RQGRQ6RXWK.HQVLQJWRQ6:$=8.

DEUDQFDWL#LPSHULDODFXN

PKDOLDEDGL#LPSHULDODFXN

2QOHDYHIURP'LSGL$UFKLWHWWXUD8QLYHUVLWjGHJOL6WXGLGL)HUUDUD,WDO\POY#XQLIHLW

.H\ZRUGV $FWLYH 1RLVH &RQWURO &RQWURO 9ROXPH 6HQVLWLYLW\ DQDO\VLV 2SWLPXP VHFRQGDU\ VRXUFH ORFDWLRQRULHQWDWLRQ$GDSWLYH&URVV$SSUR[LPDWLRQ+LHUDUFKLFDOPDWUL[IRUPDW

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


203

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

)LJ 2SWLPL]DWLRQSURFHVVDSSOLHGWRWKHFRQWUROYROXPHD RSWLPXP&9ORFDWLRQEF IL[HGFRQWUROVRXUFHV

)LJ 2SWLPL]DWLRQSURFHVVDSSOLHGWRWKHFRQWUROVRXUFHD RSWLPXP ORFDWLRQE RSWLPXPRULHQWDWLRQF IL[HGFRQWUROYROXPH

7KHWRWDOILHOG8LVJHQHUDWHGE\WKHXQZDQWHGSULPDU\QRLVH8SDQGWKHFRQWUROVRXUFHILHOGV8V'XHWR WKH OLQHDULW\ RI WKH +HOPKROW] HTXDWLRQ WKH SULPDU\ DQG VHFRQGDU\ ILHOGV DUH HYDOXDWHG VHSDUDWHO\ DQG VXPPHGWRJHWKHUWRREWDLQWKHUHVXOWLQJILHOGDVLQWKHIROORZLQJH[SUHVVLRQ ZKHUH Į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İ>@ 7KHGHVLJQYDULDEOHYHFWRUDRUE DWWKHNLWHUDWLRQFDQEHHYDOXDWHGE\WKHYDOXHRIWKHYHFWRUDWWKH NWKLWHUDWLRQSOXVDYDULDWLRQDN DNǻDNWKDWFDQEHYLHZDVǻDN ĮNGNZKHUHGNLVWKHGHVLUDEOHVHDUFK GLUHFWLRQWRVHHNWKHRSWLPXPVROXWLRQDQGĮ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

204


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±%(0FRXSOLQJ DQG VWUXFWXUDO±DFRXVWLF VHQVLWLYLW\ DQDO\VLV IRUVKHOOJHRPHWULHV&RPSXWHUVDQGVWUXFWXUHV

208 >@ >@ >@ >@ >@ >@

Eds: E L Albuquerque & M H Aliabadi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t$IDVW'GXDOERXQGDU\HOHPHQWPHWKRGEDVHGRQKLHUDUFKLFDO PDWULFHV,QWHUQDWLRQDO-RXUQDORI6ROLGVDQG6WUXFWXUHV


209

Transient Heat Conduction by the Boundary Element Method: An Alternative D-BEM Approach J. A. M. Carrer1, M. F. Oliveira2, A. L. Ferreira3 1

PPGMNE: Programa de Pós-Graduação em Métodos Numéricos em Engenharia, Universidade Federal do Paraná, Caixa Postal 19011, CEP 81531-990, Curitiba, PR, Brasil 1 email: [email protected] 2 email: [email protected] 3 email: [email protected]

Keywords: transient heat conduction, time weighting D-BEM

Abstract. This work is concerned with the development of an alternative D-BEM approach for the solution of 2D diffusion problems. The proposed approach starts by weighting, with respect to time, the basic D-BEM equation, under the assumption of linear and constant time variation for the temperature and for the heat flux, respectively. A constant time weighting function is adopted. The time integration reduces the order of the time-derivative that appears in the domain integral; as a consequence, the initial condition is directly taken into account. In order to verify the applicability of the proposed approach, four examples are presented and the D-BEM results are compared with the corresponding analytical solutions. Introduction This work is concerned with the development of an alternative D-BEM approach for the solution of twodimensional diffusion problems. The solution of time-dependent problems can be accomplished with or without the use of time-dependent fundamental solutions. In first case, the BEM formulations are designated TD-BEM (TD meaning time-domain), e.g. Wrobel [1]. In the second case, when the fundamental solution is not time dependent, two formulations arise, according to the treatment given the domain integral that appears in the BEM equations: the transformation of the domain integral into boundary integrals, by means of suitable interpolation functions, generates the so-called DR-BEM formulations (DR means dual reciprocity), e.g. Tanaka et al. [2]; the maintenance of the domain integral, on the other hand, generates the D-BEM formulations (D means domain), e.g. Vanzuit [3]. In the approach presented in this work, the basic D-BEM equation for the diffusion problem 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 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 [4], Finlayson [5]. Linear and constant time approximations are assumed, respectively, to the temperature and to the heat flux in the interval tn t tn+1. This approach will be called DS-BEM (D for the D-BEM formulation and S for the subdomain method) and is based on a similar approach developed for the solution of the scalar wave equation, see Carrer and Mansur [6]. The reliable and accurate results presented in Reference [6] encouraged the development of the proposed DS-BEM. It is important to mention that the time integration reduces the order of the time derivative of the potential in the domain integral; as a consequence, after time integration only the temperature appears in the domain integral and, in this way, the initial conditions can be imposed directly. Four examples are presented and discussed at the end of the article, with the aim of verifying the potentialities of the proposed formulation. Constant Time Weighting of the D-BEM Equation The basic integral equation of the D-BEM formulation is written as follows:

´ µ ¶*

´ µ ¶*

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

. 1´ µ T*([,X) T(X,t) d:(X) Nµ ¶:

(1)

where T*([,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 and Mansur [6].

210


In order to solve eq (1), linear time variation for the potential and constant time variation for the flux are assumed 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) T([,t)dt = µ T*([,X)µ W(t)p(X,t)dt d*(X) µ ¶*

µ ¶t0

µ ¶t0

. ´ ´ ´tF ´tF µ p*([,X)µ W(t) T(X,t)dt d*(X) 1 µ T*([,X) µ W(t)T(X,t)dt d:(X) Nµ 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) = 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 the basic DS-BEM equation (as mentioned before, S means subdomain):

´ 't c([) 2 ª¬Tn+1([) + Tn([)º¼ = µ T*([,X) 't pn+1(X) d*(X) µ ¶* ´ ´ µ p*([,X) 't ªTn+1(X) + Tn(X)º d*(X) 1 µ T*([,X) ª Tn+1(X) Tn(X)º d:(X) ¬ ¼ ¬ ¼ 2 N 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 ª Hbb 0 º ° Tn+1 + Tn « db » ® d ¬ H I ¼ °¯ Tn+1 + Tnd

½° ª 2 Gbb º b ¾=« » ® pn+1 °¿ ¬ 2 Gdb ¼ ¯

b

½ ¾ ¿

b

bb ° Tn+1 Tn Mbd º 2 ªM ® d « » db dd d N't ¬ M M ¼° ¯ Tn+1 Tn

½° ¾ °¿

(7)

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. After rearranging eq (7), one has the final matrix form of the DS-BEM written as eq (8):


ª §©N't Hbb + 2Mbb·¹ «§ ¬ ©N't Hdb + 2Mdb·¹

º ° Tbn+1 »® d §N't I + 2Mdd· ¯° Tn+1 © ¹¼

ª §©N't Hbb 2Mbb·¹ «§ ¬ ©N't Hdb 2Mdb·¹

º ° Tnb »® §N't I 2Mdd· ¯° Tdn © ¹¼

2Mbd

2Mbd

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

½ ¾ ¿

211

°½ ¾ ¿°

(8)

As usual in D-BEM formulations, the unknowns in eq (8) are the temperature and the flux at the boundary * and the temperature at the domain :. Note that the time-step length plays an important role in the analysis. In order to provide a measure of the time-step, the empirical formula to determine the critical value of 't, proposed by Onishi and Kuroki [7] and adopted by Wrobel [1], was taken here. According to this formula, one has: 'tcrit =

"2 2N

(9)

where " is the length of the smallest boundary element. Examples In what follows, the DS-BEM results are compared with the corresponding analytical solutions, see Stephenson [8], Kreyszig [9]. One-dimensional bar with heat flux prescribed at one end. This example consists of a one-dimensional bar defined over 0 d x d L subjected to the boundary conditions below: _ _ T(0,t) = 0 and p(L,t) = 1

(10)

The analytical solution to this problem, in the absence of initial conditions, is given by: f 8L T(x,t) = x + 2 S

¦

(2n 1)S 2 n N § 2L · t ( 1) © ¹ sin §(2n 1)Sx· 2e © 2L ¹ (2n 1)

(11)

n=1

Treated as a 2D problem, the original one-dimensional bar is substituted by a rectangular domain of length equal to L and height equal to L/2, under the assumption that the top and the bottom of the domain are thermally insulated. The discretization, depicted in Figure 1, employed 48 boundary elements and 256 internal cells.

Figure 1. One-dimensional bar: boundary and domain discretization. The results for the temperature at point A(L;L/4) are presented in Figure 2 and the results for the flux at node B(0,L/4) are presented in Figure 3. Note a good agreement between numerical and analytical solutions.

212


T

12.0

8.0 analytical DS-BEM: k = 0.25 DS-BEM: k = 1.00 DS-BEM: k = 4.00 4.0

0.0 0

250

500

750

1000

1250

t

Figure 2. One-dimensional bar with heat flux prescribed: temperature at A(L,L/4).

p

0.0

analytical DS-BEM: k = 0.25 DS-BEM: k = 1.00 DS-BEM: k = 4.00

-0.4

-0.8

-1.2 0

250

500

750

1000

1250

t

Figure 3. One-dimensional bar with heat flux prescribed: flux at B(0,L/4).

One-dimensional bar with time variable boundary condition. The one-dimensional bar of the previous example is now subjected to the to the boundary conditions given by eq (12): _ T(0,t) = 0 and

_ T(L,t) = TH (1 cos\t)

with

TH = constant

(12)

The analytical solution to this problem, in the absence of initial conditions, is given by: f x T(x,t) = TH ªL (1 cos \t) + w(x,t)º

¬

¼

where

w(x,t) =

¦ w (t) sin §©rSxL·¹ r

r=1

with:

(13)


° ª §rS·2 º \ «N sin \t \ cos\t» + \ ¼ 2 cos(rS) ® ¬ © L ¹ wr(t) = rS rS·4 § °¯ N L + \ © ¹ 2

2

213

rS 2 N§L· t © ¹ e

½° ¾ °¿

2

(15)

In the analysis presented here, \ = S/100. A discussion concerning this example can be found in the article by Ochiai et al. [10], which employs time-dependent fundamental solution together with the triplereciprocity formulation. The accurate results presented by Ochiai et al. [10] were the main motivation for including this example in this work; besides, this example is very interesting, as the steady-state condition, due to time dependent boundary condition given by eq (12), is not reached. The same mesh of the previous example was adopted under the assumption that the top and bottom of the domain are thermally insulated. The results for the temperature at point C(L/2;L/4) are presented in Figure 4; the results for the flux at node B(0,L/4) are presented in Figure 5. A good agreement between numerical and analytical solutions is observed again. T/TH


1.2

1.0

0.8

0.6

0.4

0.2

0.0 0

100

200

300

400

500

600

t

Figure 4. One-dimensional bar with time variable boundary condition: temperature at C(L/2,L/4). analytical DS-BEM: k = 0.25 DS-BEM: k = 1.00 DS-BEM: k = 4.00

p/TH

0.00

-0.05

-0.10

-0.15

-0.20 0

100

200

300

400

500

600

t

Figure 5. One-dimensional bar with time variable boundary condition: flux at B(0,L/4).

214


One-dimensional bar with initial condition prescribed over the entire domain. This example also deals with a one-dimensional bar; now, zero temperature is prescribed at both ends and the domain is subjected to the initial condition below: T0(x) = x(L x)

(16)

The analytical solution is given by: 8L2 T(x,t) = 3 S

f

¦

(2n 1)S·2 N§ t 1 L © ¹ sin §(2n 1)Sx· 3e L © ¹ (2n 1)

(17)

n=1 The results for the temperature at point C(L/2,L/4) are depicted in Figure 6, while the results for the flux at boundary node A(L,L/2) are depicted in Figure 7. Again, the comparison between numerical and analytical results shows a good agreement between them. T 36.0

30.0

24.0 analytical DS-BEM: k = 0.25 DS-BEM: k = 1.00 DS-BEM: k = 4.00

18.0

12.0

6.0

0.0

-6.0 0

100

200

300

400

500

600

t

Figure 6. One-dimensional bar with prescribed initial condition: temperature at C(L/2,L/4). p 0.0

-3.0


-6.0

-9.0

-12.0

-15.0 0

100

200

300

400

500

600

t

Figure 7. One-dimensional bar with prescribed initial condition: flux at B(0,L/4).


215

Square domain with constant temperature prescribed along the boundary. This example deals with a square membrane, defined in the domain 0 d x d L, 0 d y d L, subjected to a constant temperature TH = 100 prescribed along the boundary. The general analytical solution to this problem is given by: f 16TH T(x,y,t) = TH 2 S

f

2

¦¦

T t e mn §mSx· §nSy· mn sin © L ¹ sin © L ¹

(18)

m=1 n=1

where: 2

Tmn =

NS2 § 2 m + n2·¹ with odd subscripts m and n L2 ©

(34)

In this analysis, the boundary discretization employed 80 elements and the square domain, 800 cells, see Figure 8. The results corresponding to the temperature at point A(L/2,L/2) are shown in Figure 9 and are in good agreement with the analytical solution.

Figure 8. Square membrane: boundary and domain discretization. T

100.0

75.0


50.0

25.0

0.0 0

40

80

120

160

200

t

Figure 9. Square domain with constant temperature prescribed at the boundary: temperature at A(L/2,L/2).

216


Conclusions The alternative 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. Besides, due to the time integration, the initial condition is 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] L. C. Wrobel. Potential and viscous flow problems using the boundary element method. Ph.D. Thesis, University of Southampton (1981). [2] M. Tanaka, T. Matsumoto, S. Takakuwa. Dual reciprocity BEM based on time-stepping scheme for solution of transient heat conduction problems. In: Boundary elements XXV, Southampton, 299-308 (2003). [3] R. J. Vanzuit. Two-dimensional heat flux analysis by the boundary element method with fundamental solutions not time-dependent (in portuguese), M.Sc. Thesis, Universidade Federal do Paraná, Brasil (2007). [4] O. C. Zienkiewicz, K. Morgan. Finite Elements & Approximation. John Wiley & Sons, Inc., New York (1983). [5] B. A. Finlayson. The Method of Weighted Residuals and Variation Principles. Academic Press, New York (1972). [6] J. A. M. Carrer, 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). [7] K. Onishi, T. Kuroki. Boundary element method in transient heat transfer problems, Report no. 3, Civil Engineering and Applied Mathematics, The Institute for Advanced Research, Fukuoka University, Japan (1980). [8] G. Stephenson. An Introduction to Partial Differential Equations for Science Students, Longman, 1970. [9] E. Kreyszig. Advanced Engineering Mathematics, John Wiley & Sons, Inc. (1999). [10] Y. Ochiai, V. Sladek, J. Sladek. Transient heat conduction analysis by triple-reciprocity boundary element method. Engineering Analysis with Boundary Elements, 30, 194-204 (2006).


217

Building three-dimensional analysis considering bending plates by a BEM/FEM coupling Luttgardes de Oliveira Neto Departament of Civil Engineering, University of Engineering–UNESP Av. Eng. Luiz E. C. Coube, 14-01- ZIP CODE 17033-360, Bauru, SP, Brazil e-mail: [email protected] y Keywords: BEM/FEM coupling, Multi-storey buildings, Transverse stiffness of slabs, Boundary Element Method, Plate bending.

Abstract. The aim of this work is to present a BEM/FEM coupling formulation to calculate efforts and displacements of three-dimensional structures of multi-storey buildings, subjected to vertical and lateral loads. A bending plate BEM alternative formulation, with three displacement nodal parameters, is adopted in floor discretization, responsible for considering the bending stiffness contribution of slabs for building analysis. The plate is coupled to the three-dimensional structure of beams and columns, formulated by FEM. The serial and parallel techniques are applied for analysis of global stiffness matrix. Simple examples are presented to evaluate the formulation bring effective contributions to this analysis. Integral equations for plate bending using three nodal parameters. Many complete formulations of the direct BEM formulations for plate bending have been published, involving two integral equations, with the transverse displacement w and the normal derivative ww/wn. An alternative formulation of the boundary element method, proposed by [1], considered three displacement parameters (w , ww/wn and ww/ws) for the nodes on the boundary. In that formulation the three nodal values were approximated by linear functions. To refine the approximation, the transverse displacement w can be represented, along the boundary element, by a cubic polynomial function ([2]). Consider a thin plate with domain :, the integral equation obtained by BETTI's theorem is:

K (S) w (S) ww ww ª º (Q) m ns (S, Q) (Q)»d*(Q) «q n (S, Q) w (Q) m n (S, Q) *¬ ws wn ¼ º ª ww

(S, Q)»d*(Q) «Vn (Q) w (S, Q) m n (Q) *« wn ¼» N¬

³ ³ ¦ R ci (Q) w ci (S, Q) ³ : c

i 1

in which, K(s) = 1 K(S) = E/(2S) K(S) = ½

(1)

g (q ) w (S, q)d: g (q)

g

for internal points s; for a point S at a boundary corner, with internal angle E; for a point S on a smooth boundary;

The integral representation of the directional derivative of the transverse displacement has similar shape with respect to direction m. In this alternative formulation of the BEM with 3 nodal parameters, transverse displacement w and directional derivates ww/wn and ww/ws, two kinds of approximation have been used. In the new approach, w is approximated by cubic functions M([) and, as a consequence, ww/ws is approximated by a quadratic function (whose coefficients are functions of the nodal values of w and ww/ws at the extremities of the element); the geometry, the traction mn and the normal derivative ww/wn by linear functions; and the equivalent shear force Vn is assumed to be concentrated in the nodal points, as proposed by [3]. The transverse displacement w and tangential derivative ww/ws are expressed:

218


w([) = M1([) w1 + M2([) (ww/ws)1 +M3([) w2 + M4([) (ww/ws)2

(2)

ww/w[ ([) = M’1([) w1 + M’2([) (ww/ws)1 + M’3([) w2 + M’4([) (ww/ws)2

(3)

Rewriting the integral eqs. (1) to take into account the discretization of the plate boundary into elements and the approximation functions of the variables across these elements, results in matrix form: >H@{w * } >G @{V* } {p} (4) in which, § ww · § ww · § ww · § ww · ½ § ww · § ww · wT ¸ ¨ ¸ ¨ ¸ ... w n ¨ ¸ ¨ ¸ w2¨ ¸ ¾ * ®w 1 ¨ w w w w n s n s ¹1 © © wn ¹ n © ws ¹ n ¿ ¹2 ¹2 © ¹1 © ¯ ©

V*T

^Vn1

m n1 Vn 2 m n 2 ... Vnn m nn `

[H] and [G] contain the resultants of the integrations performed on the boundary elements and {p} is the vector of known loads. Equivalent Stiffness Matrix. The procedure adopted in this work combines the mentioned numeric methods treating the bending plate analyzed by BEM as a “finite element”, considering an equivalent stiffness matrix at the global system of the three-dimensional structure of the building. The usual system of BEM, eq (4), can be rewritten: (5) [G]-1[H]{w} = {V} + [G]-1{P} The system takes the similar form to that obtained by FEM, as it proceeds: [K]{w} = {S} + {F}

(6)

The equivalent stiffness matrix K is complete, not symmetrical and has columns that do not add zero, which implies that the balance is not satisfied. The lines add zero, due to the exact integrations, which makes possible the imposition of rigid body displacements ([4]). Eventually, a symmetrization process is proposed as presented in the following way, [K2] = (1/2){[K] + [K]T} ([5], [6]) Three-dimensional analysis of structures of buildings with combination MEC/MEF ([4]). For an effective resolution of a computational system with very many unknowns, the sub-structuring techniques are used, that analyze each substructure independently to solve the structure completely. The sub-structuring techniques use parallel and serial processes. Parallel sub-structuring. In the first step, the equivalent stiffness matrix and the vector of nodal forces of each pavement are condensed to the local coordinates of the respective substructure (Figure 1). The substructure system is represented by the horizontal elements (beams and floor slabs), contained in the superior pavement, and the vertical elements (columns), that link to the inferior floor. The stiffness matrix of the substructure is, therefore, obtained with the contribution of the rigidities of all structural components. Parallel sub-structuring

Figure 1: Parallel sub-structuring


219

Choleski’s method. The traditional Choleski’s decomposition method is used, in which the global stiffness matrix of the building has the partial liberation of the internal coordinates (Figure 1). The substructure matrix of the pavement can be written as ª >R II @ >R IE @º ^D I `½ ^FI `½ (7) «>R @ >R @» ®^D `¾ ®^F `¾ EE ¼ ¯ E ¿ ¬ EI ¯ E ¿ where I is index that indicates the internal parameters of the substructure and E is index that indicates the external parameters of the substructure. This method can be formulated starting from the decomposition of the substructure matrix in a triple matricial product. ª >R II @ >R IE @º ª >L@ >0@º ª>D@ >0@ º ª>L@T >RT@T º (8) «>R @ >R @» «>RT@ >I@» « >0@ R * » « >I@ »¼ ¬ ¼¬ ¼ ¬ >0@ EE ¼ ¬ EI being [L] a triangular inferior matrix with unitary terms in the main diagonal; [RT] a rectangular matrix; [D] a diagonal matrix; [0] a null matrix; [I] an identity matrix; [R*] a condensed symmetrical matrix; and

> @

ª>D@ « >0@ ¬

>0@ º^D I * `½

> @ ^ `

¾ ® R * »¼ ¯ D E * ¿

^ ` ^ `

FI * ½ ® *¾ ¯ FE ¿

(9)

[R*] and {FE*} represent the stiffness matrix and the vector of forces , respectively, refering to external nodes of the substructure. Serial sub-structuring. The global system is expressed as: >R @N, N 1 >0@ º >D@N ½ ª >R @N , N »° «>R @ ° R R > @ > @ N 1, N 1 N 1, N 2 » °>D@N 1 ° « N 1, N »° « ° »° « ° > @ > @ > @ R R R D > @ K , K 1 K ,K K , K 1 K ¾ »® « ° »° « ° »° « >R @1,2 >R @1,1 >R @1,0 » ° >D@1 ° « « >0@ >R @0,1 >R @0,0 »¼ °¯ >D@0 °¿ ¬

>F@N ½ °>F@ ° ° N 1 ° ° ° ° ° > @ F ® K ¾ ° ° ° ° ° >F@1 ° °> @ ° ¯ F0 ¿

(10)

Isolating the vector [D]N from the first line (equation) of the system (10), the expression that is obtained is substituted at the next line. The substitution process forward can be done for the complete system and the last line results in: (11) >R @1*,1 >D@1 >R @1,0 >D@0 >F@1* With the displacements known in the connection structure-foundation ([D]0), it returns to the previous equations where, through the retro-substitution process, the displacements of all the elements are calculated in each substructure, from the base to the top of the building. Considering the stiffness of the component elements of each pavement, the corresponding efforts in each one is obtained. Numerical examples. Example 1. Single square pavement linked to 4 beams on border and 4 columns at the corners, uniform vertical distributed load and wind load. Numeric data are: square plate dimensions 1200 cm and thickness 15 cm; stiffness beams Iv = 106666 cm4 and columns cross section 20x20 cmxcm and 300 cm height; the uniform distributed load is 4,0 kN/m2; the material properties are Q=0,30 and E=2000 kN/cm2. The FEM discretizations are 144 quadrangular elements and 288 triangular elements; the BEM discretization is 48 linear boundary elements.

220


Figures 2 to 4 show some values obtained by both formulations along boder beam considering vertical load distributed over the plate. displacement w - border -600

-400

-200

0

200

400

600

transversal displacement w (cm)

0,0E+00 -2,0E+00 -4,0E+00 -6,0E+00 -8,0E+00

FEM BEM

-1,0E+01 -1,2E+01 -1,4E+01 -1,6E+01 border axis

Figure 2- Vertical displacement w along beam (plate border). shear forces - beam -600

-500

-400

-300

-200

-100

bending moment - beam 0

-600

-500

-400

-300

-200

-100

0,0E+00

1,8E+04 1,5E+04

M(kN.cm)

-1,0E+01

FEM

-2,0E+01

V (kN)

0

BEM

-3,0E+01

1,2E+04

FEM

9,0E+03

BEM 6,0E+03

-4,0E+01

3,0E+03 -5,0E+01

0,0E+00

-6,0E+01

-3,0E+03

border axis

border axis

Figure 3- Shear forces along beam

Figure 4- Bending moment along beam

Figures 5 to 7 show values obtained by both formulations considering horizontal wind load of 200 kN acting at the simetric axis and at plate level displacement w - border -400

-200

0

200

400

600 1,0E-01

transversal displacement w (cm)

-600

8,0E-02 6,0E-02 4,0E-02

FEM

2,0E-02 0,0E+00

BEM

-2,0E-02 -4,0E-02 -6,0E-02 -8,0E-02 -1,0E-01 border axis

Figure 5- Vertical displacement w along beam. wind loading.


shear forces - beam -600

-500

-400

-300

-200

-100

bending moment - beam 0

-600

-500

-400

-300

-200

0,0E+00

0

-2,5E+02

-2,0E+00

BEM

M (kN.cm)

FEM

-3,0E+00

-100

0,0E+00

-1,0E+00 V (kN)

221

-5,0E+02

FEM

-7,5E+02

BEM

-1,0E+03 -4,0E+00

-1,3E+03

-5,0E+00 border axis

-1,5E+03 border axis

Figure 6- Shear forces along beam, wind load Figure 7- Bending moment along beam, wind load There are very good agreement between BE and FE results, but we can note the low number of boundary elements used to obtain good results in a simple example. Example 2. Tubular building structure with 9 pavements, subjected to wind horizontal load. Figure 8 shows the pavement plant of a building with 24 columns, subjected to a wind horizontal load as a resultant load of 43,2 kN acting along symmetric axis and at all floor levels. The numeric data are the same used in example 1, changing only the columns cross section (30x30 cmxcm) and 290 cm height; the uniform distributed load is 4,0 kN/m2. Four stiffness conditions are considered, FE plate formulation with bending stiffness (QL) and with no stiffness (Q0) and BE plate formulation with bending stiffness (CL) and with no stiffness (C0). The models which disregard the bending stiffness of the plates show higher horizontal translations when compared with models with actual bending stiffness, as expected.

beam

Fwind

Figure 8- Tubular building structure. Pavement plant Figures 9 to 16 show several results obtained by four conditions with both numerical codes. The variation of the efforts (shear forces and bending moments) on columns P1 and P2 and on beam V1 (branch between P22 and P23, paralell to the wind direction) presents similar shape when the four models are compared. BE results are quite higher then FE values on columns, but presenting lower values on beams, in order to 20%.

222


9

9

8

QL

8

7

Q0

7

QL

6

CL

5 4

C0

3

CL

4 3

CDL

2

Q0

5

floors

floors

6

C0

2

CD0

1

1

0

0

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

-20

0

20

40

60

translation - dir. y (cm)

80

100

120

140

160

N (kN)

Figure 10- Axial efforts – column P1

Figure 9- Horizontal translation, Wind loading 9

9

8

8 7

7 QL

6

CL

3

C0

4

CL

3

2

C0

2

1

1

0

0 0

5

10

15

20

25

30

35

-3000

V (kN)

-2500

-2000

-1500

-1000

-500

0

M (kN.cm)

Figure 11- Shear efforts – column P1

Figure 12- Bending moments – column P1 9

9

8

8 7

7

QL

6

4

CL

3

floors

Q0

5

floors

Q0

5

floors

floors

4

QL

6

Q0

5

QL

5

Q0

4

CL

3

C0

2

6

C0

2

1

1

0 0

5

10

15

0

20

-5000

-4000

-3000

V (kN)

-2000

-1000

0

M (kN.m)

Figure 14- Bending moments – column P2

Figure 13- Shear efforts – column P2

9 9

8

QL

8

7

7

Q0

4 3

floors

5

floors

6

TL

6

CL

2

C0

1

TL

4

Q0

3

CL

2

C0

1

0

0 0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000

M (kN.cm)

Figure 15- Bending moments – V1 (branch 5)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

QL

5

0

V (kN)

Figure 16- Shear efforts – V1 (branch 5)


223

We can be show the main advantage presented by BE formulation in which referes to the time of processement. The same tubular structure modeled by both numerical formulations were carried out in a PC with AMD Athlon numerical processor, with 1.2GHZ and 256 MB RAM. Table 1 shows the time processing spent by both computational codes from data reading to the results writing. We can be note that even considering the inverse matrix process necessary to include the BE bending plate contributions at the building FE stiffness matrix, which is not necessary in FE formulation, the BE formulation spend a third part of time processing spent by FE code.

Table 1- Time numeric processing – Example 2 Time numeric processing

Plate

with

no

stiffness

bending Plate

with

bending

stiffness

FEM/ quadrangular element

9.050 seconds

9.040 seconds

FEM/ triangular element

6.030 seconds

6.090 seconds

BEM/ linear element

2.070 seconds

2.070 seconds

Considerations. This work is an application of the alternative bending plate BEM formulation that uses the three nodal parameters for displacements, presents an association of plate bending formulated by BEM with another involving structural elements, formulated by FEM, seeking the practical use for analysis of buildings floor slabs. This formulation also uses the equivalent shear forces considering as concentrated forces in BEM formulation for plates with discrete supports or still in the association with beams and columns. References [1] PAIVA, J.B; OLIVEIRA NETO, L. An alternative boundary element formulation goes plate-bending analysis. In: Aliabadi, MH; Brebbia CA, Dular, P; Nicolet, A, editors. Boundary Element Technology X (BETECH 95). Liège, Belgium. Springer-Verlag, 1995. 1-8. [2] OLIVEIRA NETO, L.; PAIVA, J.B. Elastic plate BEM analysis using a cubic approximation for the transverse displacement. In: BeTeQ/2002 - Third International Conference on Boundary Element Techniques. Tsinghua University Press & Springer-Verlag. (2002). [3] PAIVA, J.B. Boundary Element Formulation for plate analysis with special distribution of reactions along the boundary. Adv. in Eng. Soft. and Workstations, 13, 162-168. (1991). [4] OLIVEIRA NETO, L. An alternative BEM/FEM coupling for building three-dimensional analysis. In: BeTeQ/2002 - Third International Conference on Boundary Element Techniques. Tsinghua University Press & Springer-Verlag.(2002). [5] ZIENKIEWICZ, O.C.; KELLY, D.W.; BETTESS, P. The coupling of the finite element method and boundary solution procedures. Int. J. for Num. Methods in Eng., 11, 355-375. (1977). [6] BREBBIA, C. A. ; GEORGIOU, P. Combination of boundary and finite elements for elastostatics, J. Appl. Math. Modeling, 3, 212-220. (1979).

224


'DPDJH'HWHFWLRQLQ%HDPV8VLQJ%RXQGDU\(OHPHQWV DQG:DYHOHW7UDQVIRUP 5DPRQ6@RUQRQGHWHUPLQLVWDOJRULWKPV>@:KHQVXFKUHVLGXDOIXQFWLRQLVDWDPLQLPXPRQH FRQFOXGHVWKDWWKHPDWKHPDWLFDOPRGHOLVZHOOUHSUHVHQWLQJWKHGDPDJH6XFKPHWKRGVJDLQHGSRSXODULWLHVGXHWR VLJQLILFDQWDGYDQFHVLQWKHH[SHULPHQWDOPHWKRGVDQGWHFKQRORJLHVIRUPRQLWRULQJVWUXFWXUHV>@,QWKLVDUWLFOH


225

KRZHYHU WKH PHWKRGV EDVHG RQ ZDYHOHW WUDQVIRUP DVVRFLDWHG ZLWK WKH ERXQGDU\ HOHPHQW WHFKQLTXH ZLOO EH FRPELQHGZLWKWKHVFRSHWRGHWHFWWKHSRVVLEOHSRVLWLRQRIDGDPDJHMXVWXVLQJWKHGDPDJHGUHVSRQVHRIWKHDFWXDO VWUXFWXUH$IWHU SUHVHQWLQJ WKH IRUPXODWLRQ QXPHULFDO DSSOLFDWLRQV IRU GHWHFWLQJ GDPDJHV LQ EHDPV XQGHU VWDWLF ORDGV DUH SUHVHQWHG DQG GLVFXVVHG +RZHYHUVXFK DSSURDFKHV QHHG D FRPSDULVRQ EHWZHHQ WZR VWDWHV ZKLFK PHDQVPRUHZRUNDQGWKHVWRUDJHRIWKHGDWDEHWZHHQWKHWZRVWDWHV0HWKRGVZKLFKFDQGHWHFWGDPDJHZLWK RQO\ LQIRUPDWLRQ REWDLQHG IURP WKH GDPDJH FRQGLWLRQ RI WKH VWUXFWXUH DUH PRUH DGYDQWDJHRXV VLQFH WKHLU FRQGLWLRQ EHIRUH GDPDJH LV UDUHO\ NQRZQ ,Q WKLV FRQWH[W WKH DSSOLFDWLRQ RIWKH ZDYHOHWEDVHG PHWKRGV FDQ EH XVHIXO7KHVHPHWKRGVGHWHFWWKHVLQJXODULWLHVSUHVHQWLQWKHVWDWLFDQGPRGDOUHVSRQVHFDXVHGE\GDPDJH DQGWKHUHIRUHWKH\GRQRWUHTXLUHWKHFRQGLWLRQRIWKHVWUXFWXUHEHIRUHGDPDJH ,WLVDOVRQHFHVVDU\WR GHWHUPLQHDQXPHULFDOSURFHGXUHIRUWKHVLPXODWLRQRIFUDFNHGVWUXFWXUHV,QWKLVZD\HYDOXDWLRQRIWKHGDPDJH GHWHFWLRQPHWKRGVFDQEHSUDFWLFDO>@ 7KLV SDSHU SUHVHQWV RQH QXPHULFDO H[DPSOH XVLQJ ERXQGDU\ HOHPHQWV DQG :DYHOHW 7UDQVIRUP IRU GDPDJH GHWHFWLRQ LQ D FDQWLOHYHU EHDP 7KH PRGHOLQJ RI GDPDJH LV GRQH LQ ERXQGDU\ HOHPHQWV E\ PHDQV RI SURJUDP (ODVWBTXD>@GHYHORSHGLQ0$7/$%7KHVWDWLFUHVSRQVHRIWKHVWUXFWXUHZLWKVLPXODWHGGDPDJHLVXVHGLQWKH DQDO\VLVWRGHWHFWWKHORFDWLRQRIGDPDJHXVLQJWKH:DYHOHW7UDQVIRUP 7+(%281'$5@FDQEHHDVLO\REWDLQHGDV

න ܾ௞‫ݑ כ‬௞ ݀π ൅ ර ‫ݐ‬௞‫ݑ כ‬௞ ݀Ȟ ൌ න ܾ௞ ‫ݑ‬௞‫݀ כ‬π ൅ ර ‫ݐ‬௞ ‫ݑ‬௞‫݀ כ‬Ȟ π

୻

π

୻

,Q(T W N X N DQG * DUHUHVSHFWLYHO\ORDGVGLVSODFHPHQWVDQGERXQGDU\RIWKHERG\8VLQJWKH'LUDFGHOWD IXQFWLRQPDWKHPDWLFDOSURSHUWLHV(T FDQEHUHZULWWHQDV ‫כ‬ ‫כ‬ ‫ݑ‬௜ ൌ ර ‫ݑ‬௜௝ ‫ݐ‬௝ ݀Ȟ െ ර ‫ݐ‬௜௝ ‫ݑ‬௝ ݀Ȟ ୻

୻

226

Eds: E L Albuquerque & M H Aliabadi @ &21&/86,216 7KLV SDSHU SUHVHQWHG D QHZ PHWKRGRORJ\ DSSOLHG WR WKH LQYHUVH SUREOHP RI GDPDJH GHWHFWLRQ XVLQJ ERXQGDU\ HOHPHQWVDQGZDYHOHWWUDQVIRUP

7KH ZDYHOHW WUDQVIRUP DSSOLHG LQ WKH VWDWLF VLJQDOV FRXOG GHWHFW WKH H[DFW SRVLWLRQ RI GDPDJHV DQG DOVR VLQJXODULWLHVGXHWRJHRPHWULFGLVFRQWLQXLWLHV

7KH XVH RI ERXQGDU\ HOHPHQW PHWKRG FRXOG EH DQ DOWHUQDWLYH WR REWDLQ WKH QXPHULFDO UHVSRQVH RI GDPDJHG VWUXFWXUHWREHXVHGLQWKHGDPDJHGHWHFWLRQSURFHVV 5HIHUHQFHV >@3(*LOO:0XUUD\0+:ULWH³3UDWLFDO2SWLPL]DWLRQ´/RQGRQ$FDGHPLF3UHVV >@5/)R[³2SWLPL]DWLRQ0HWKRGIRU(QJLQHHULQJ'HVLJQ´0DVVDFKXVHWWV$GGLVRQ:HVOH\ >@0'RULJR76WXW]OH³$QW&RORQ\2SWLPL]DWLRQ´:LOH\,67( >@( 6 (VWUDGD ³'DPDJH GHWHFWLRQ PHWKRGV LQ EULGJHV WURXJK YLEUDWLRQ PRQLWRULQJHYDOXDWLRQ DQG DSSOLFDWLRQ´3K'7KHVLV8QLYHUVLW\RI0LQKR >@56@%'RUUL³6ROXWLRQRI,QYHUVH+HDW&RQGXFWLRQ3UREOHPV8VLQJ%RXQGDU\,QWHJUDO0HWKRG´*HQHUDO(OHWULF &5'5HSRUW >@& $ %UHEELD - 'RPLQJXH] ³%RXQGDU\ (OHPHQWV $Q ,QWURGXFWRU\ &RXUVH´:,7 3UHVV&RPSXWDWLRQDO 0HFKDQLFV >@)$*/RXUHQoR³$QiOLVH GH 5HSDURV GH 7ULQFDV $WUDYpV GR 0pWRGR GRV (OHPHQWRV GH &RQWRUQR´ 3K' 7KHVLV8QLYHUVLW\RI&DPSLQDV >@$92YDQHVRYD³$SSOLFDWLRQVRIZDYHOHWVWRFUDFNGHWHFWLRQLQIUDPHVWUXFWXUHV´3K'7KHVLV8QLYHUVLW\RI 3RUWR5LFRS >@ 5 6 0 kn δn (x) δn (x) ≤ 0

(1)

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. kn and kt denote the normal and tangential stiffnesses of the spring distribution. The traction modulus t is defined as t(x) = σ 2 (x) + τ 2 (x). The ERR of the linear elastic interface at a point x is defined as: G(x) = GI (x) + GII (x) if σ > 0 and G(x) = GII (x) if σ ≤ 0. The fracture mode mixity angle ψσ at an interface point x is defined by tan ψσ (x) = τ (x)/σ(x), see also Figure 1. It is assumed that the crack tip at x advances (or an interface point x breaks) when the corresponding ERR G(x) reaches the critical ERR value Gc (ψσ (x)), that is G(x) = Gc (ψσ (x)). The functional dependence of Gc on the fracture mode mixity angle ψσ is defined similarly as in [6] with a slight modification [11]. This interface fracture criterion can be rewritten in terms of the critical interface stresses as follows. The critical stress modulus tc is defined as tc (ψσ (x)) = σc2 (ψσ ) + τc2 (ψσ ), and the general expressions of the critical normal and tangential stresses as functions of ψσ are: ˜ cos ψ, ˜ ¯c 1 + tan2 [(1 − λ)ψ] (2) σc (ψσ ) = σ kt ˜ sin ψ, ˜ τc (ψσ ) = σ ¯c 1 + tan2 [(1 − λ)ψ] (3) kn ˜ σ ) is a smooth continuous function defined as where ψ˜ = ψ(ψ ⎧ kn ⎪ |ψσ | < π/2 ⎪ ⎨ arctan{ kt tan ψσ } ˜ ±π/2 ψ ψ(ψσ ) = σ = ±π/2 ⎪ ⎪ ⎩ π sgn ψσ + arctan{ kn tan ψσ } π/2 < |ψσ | ≤ π kt

(4)

where sgn is the signum function, λ is a fracture-mode-sensitivity parameter (usually obtained from the best fit of experimental results) and σ ¯c is the critical tension in mode I, i.e. σ ¯c = σc (ψσ = 0). Typical range 0.2 ≤ λ ≤ 0.35 characterizes interfaces with moderately strong fracture mode dependence [6]. Further details of the above criterion deduction are presented in [9, 10, 16]. Cylindrical inclusion under biaxial loads The problem of an elastic cylindrical inclusion embedded in an elastic matrix with and without a partial debond along its interface and subjected to a remote uniform tension perpendicular to the debond has been studied in depth by many researchers, see references in [7, 8, 10, 17]. In the present study an infinitely long cylindrical inclusion, with a circular section of radius a, is considered embedded in an infinite matrix, Figure 1. 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. The uniform remote loads σx∞ and σy∞ are parallel to the x-axis and y-axis, respectively. Remote loads σx∞ > 0 and σy∞ = ησx∞ , where η is the load biaxility ratio, −1 ≤ η ≤ 1, are considered in the present work. The semidebond angle is denoted as θd . A plane strain state is assumed in the bimaterial system.


(a)

(b)

Figure 1: Inclusion problem configuration under biaxial remote transverse loads (a) without and (b) with a partial debond. Numerical Results Let us consider an inclusion bonded to the matrix along its lateral surface through a continuous distribution of springs that behave according to the LEBI model introduced. BEM model. A typical bimaterial system among fibre reinforced composite materials is chosen for this study: epoxy matrix (m) and glass fibre (f ). The elastic properties of matrix and fibre are Em = 2.79 GPa, νm = 0.33, Ef = 70.8 GPa and νf = 0.22. The corresponding Dundurs’ bimaterial parameters are α = 0.919 and β = 0.229, and the harmonic mean of the effective elasticity moduli in plane strain E ∗ = 6.01GPa, see [7, 8, 17]. The fracture and strength properties of the interface assumed in the present work are: the fibrematrix interface fracture toughness in mode I, GIc = 2Jm−2 , and the critical tension σ ¯c = 90MPa. These values are chosen because they are in the range of values found in the literature [17–19], and trying to simulate a brittle behavior as stated in [10], making the hypothesis of the LEBI model to represent appropriately a possible real composite material behavior. A fixed relation between kn and kt (kt = kn /4) is chosen. This value is obtained, see Equation (7) in [10], assuming that a very thin elastic layer of a fictitious material, whose Poisson’s ratio is ν = 0.33 (identical to the matrix νm ), placed between the fibre and matrix is represented by the present interface model. The 2D BEM model represents a circular inclusion with a radius a =7.5 µm inside a relatively large square matrix with a 1 mm side. 1472 continuous linear boundary elements [5] 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 load biaxility ratio η. In the present study, a value of the fracture-mode-sensitivity parameter λ = 0.25 is taken. Numerical results for different values of the load biaxility ratio η = σy∞ /σx∞ (η = −1, −0.5, 0, 0.5, 1) are presented in Figure 2. η characterizes different combinations of biaxial transversal loads, η = 0 corresponding to the case of uniaxial loading (σy∞ = 0). In Figure 2(a) the applied remote stress in the x-axis, σx∞ , is plotted as a function of the normal relative displacement (opening), δn , evaluated at point A(a, 0◦ ) as defined in Figure 1(a). The remote ∞ , and corresponds to stress value that is needed to initiate crack growth is called critical stress, σcx the local maximum of the functions shown in Figures 2(a) and (b). It can be also observed in these ∞ , the crack growth becomes unstable, requiring smaller figures that after reaching the critical stress, σcx values of the remote tension that causes the further crack growth, thus an instability phenomenon called snap-through takes place under constant load. In Figure 2(b) the (minimum) remote stress, σx∞ , needed to cause crack growth is plotted versus the semidebond angle θd (defined in Figure 1(b)). Note that from these figures it is possible to obtain an estimation of the critical semidebond angle, θc , as the value reached by the semidebond angle θd ∞ , which produces the growth. after the initial unstable crack growth, and also of the critical load, σcx In the cases where tension is applied in both axes, the critical angles are larger than 90◦ , implying

251

252


that the initial debond is produced in a large part of the interface. With reference to the influence of the load biaxility ratio η, it should be noticed that when the compression load σy∞ is increasing while keeping constant the tension load σx∞ (η tends to larger ∞ ) is smaller. In other words, negative values), the load necessary to initiate crack debonding (σcx compressions in the y-direction lead to a crack onset with lower loads. This conclusion agrees with the previous experimental and numerical studies in [1–4]. Additionally, the critical semidebond angle (θc ) becomes lower when the compression load σy∞ increases, see Table 1. When the tension load σy∞ > 0 is increasing for a fixed value of σx∞ > 0 (η tends to larger positive ∞ is also increasing. Thus, the presence of tensions in values), the load necessary for crack onset σcx σy∞ > 0 makes necessary larger loads to cause crack onset. Additionally, θc becomes higher when η increases, eventually producing a very large debond of the fibre (unstable growth with θc > 90◦ ) for the highest values of η, see Table 1.

(a)

(b)

Figure 2: BEM results (a) applied stress σx∞ with respect to the normal relative displacements δn at the point A, see Figure 1, and (b) applied stress σx∞ with respect to the semidebond angle θd , for different biaxial loads combinations, with kn /kt =4 and λ=0.25. ∞ , and the critical semidebond Table 1: Applied stress in the x-direction necessary for crack onset, σcx angle, θc , for different η values, with kn /kt =4 and λ=0.25.

∞ (MPa) σcx ∞ /¯ σcx σc θc (◦ )

η =-1 51.9 0.573 58.25

η = -0.5 56.6 0.629 63.25

η=0 62.3 0.692 72.75

η = 0.5 69.2 0.769 95.25

η=1 77.8 0.864 146.0

Failure curve. From all the cases solved numerically by BEM in the previous subsection, it is possible to obtain points of a failure curve for a circular inclusion under biaxial transversal loads. It is also noticeable, that another way to obtain these failure curves is by the use of the algorithm developed in [11] which combines an analytical solution introduced in [10] and the LEBI failure criterion. In the results presented in Figure 3, the BEM results are denoted by dots while the continuos line is obtained using the algorithm proposed in [11]. The excellent agreement between the BEM results and those obtained by the algorithm proposed in [11] is also noticeable. Nevertheless, it should be stressed that BEM has the capability to characterize the subsequent unstable crack growth and determine the critical semidebond angle θc , which is not possible using the available analytic solution. Moreover, the BEM code developed will be able to analyze more realistic problems involving a higher number of fibres.


Figure 3: Failure curves of a circular inclusion under biaxial transversal loads for different values of the load biaxiality ratio η, with λ=0.25. Conclusions A new linear elastic-brittle interface (LEBI) model (a kind of continuous spring distribution model) is used and analyzed in the present work to study fibre-matrix debond problems in composite materials Specifically, this model has been used to characterize the onset and growth of the debond in a single fibre-infinite matrix system subjected to far field biaxial transverse loads. The problem of a circular inclusion under biaxial transverse loading 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. From the numerical results it can be seen that after reaching a critical load, the crack growth becomes unstable. Then, an instability phenomenon called snap-through takes place. A parametric study of the influence of the load biaxility ratio η has also been carried out. BEM results show that compressions in the y-direction allow a crack onset with lower tension loads in the x-direction. The critical semidebond angle θc also becomes lower when the compression load in the y-direction increases for a constant tension in the x-direction. The presence of tensions in the ydirection requires larger loads in the x-direction to cause crack onset. Also, θc becomes larger when the tension load in the y-direction increases, eventually producing a very large debond of the fibre (unstable growth even for θd > 90◦ ) for η 0.5. From all the cases solved numerically by BEM it is possible to obtain points of a failure curve for a circular inclusion under biaxial transversal loads. The present numerical results confirm the conclusions of the previous experimental and numerical studies of fibre-matrix debonding under biaxial transverse loading in [1–4]. The main novelty of the present model based on the LEBI formulation consists in its capability of quantitative predictions about the fibre-matrix debond onset and mixed-mode growth. Acknowledgments The work was supported by the Junta de Andaluc´ıa and Fondo Social Europeo (Proyectos de Excelencia TEP 1207 and 4051) and the Spanish Ministry of Education and Science (Project MAT2009-14022). References [1] F. Par´ıs, E. Correa, and J. Ca˜ nas. Micromechanical view of failure of the matrix in fibrous composite materials. Composites Science and Technology, 63:1041–1052, 2003. [2] E. Correa. Micromechanical study of the “matrix failure” in fiber reinforced composites (in Spanish). PhD. Thesis, Universidad de Sevilla: Sevilla, 2008.

253

254


[3] E. Correa, F. Par´ıs, and V. Mantiˇc. Tension dominated inter-fibre failure under bi-directional loads. Micromechanical approach. In: COMATCOMP 2009, V International Conference on Science and Technology of Composite Materials and 8◦ Congreso Nacional de Materiales Compuestos, San Sebasti´ an, pages 915–918, 2009. [4] E. Correa, F. Par´ıs, and V. Mantiˇc. Interfacial fracture mechanics approach to the tension dominated inter-fibre failure under bi-directional loads. In: ICCM-17, 17th International Conference on Composite Materials (CD), The British Composites Society, Edinburgh, 10 pages, 2009. [5] F. Par´ıs and J. Ca˜ nas. Boundary Element Method, Fundamentals and Applications. Oxford University Press: Oxford, 1997. [6] JW. Hutchinson and Z. Suo. Mixed mode cracking in layered materials, volume 29 of Advances in Applied Mechanics. Academic Press: New York, 1992. [7] V. Mantiˇc, A. Blázquez, E. Correa, and F. Par´ıs. Analysis of interface cracks with contact in composites by 2D BEM. In: Fracture and Damage of Composites, M. Guagliano and M. H. Aliabadi (Eds.) WIT Press: Southampton, 2006. [8] F. Par´ıs, E. Correa, and V. Mantiˇc. Kinking of transverse interface cracks between fiber and matrix. Journal of Applied Mechanics, 74:703–716, 2007. [9] L. T´ avara. Damage initiation and propagation in composite materials. Boundary element analysis using weak interface and cohesive zone models. PhD Thesis. Universidad de Sevilla: Sevilla, 2010. [10] L. T´ avara, V. Mantiˇc, E. Graciani, and F. Par´ıs. BEM analysis of crack onset and propagation along fiber-matrix interface under transversetension using a linear elastic-brittle interface model. Engineering Analysis with Boundary Elements, 35:207–222, 2011. [11] V. Mantiˇc, L. Távara, A. Blázquez, E. Graciani, and F. Par´ıs. Crack onset and growth at fibre-matrix interface under biaxial transverse loads using a linear elastic-brittle interface model. Composites Science and Technology (submitted). [12] E. Graciani, V. Mantiˇc, F. Par´ıs, and A. Bl´ azquez. Weak formulation of axi-symmetric frictionless contact problems with boundary elements: Application to interface cracks. Computer and Structures, 83:836–855, 2005. [13] L. Távara, V. Mantiˇc, E. Graciani, and F. Par´ıs. A BEM analysis of the fibre size effect on the debond growth along the fibre-matrix interface. In: Advances in Boundary Element Techniques XI, EC Ltd, Eastleight, pages 474–481, 2010. [14] S. Lenci. Analysis of a crack at a weak interface. International Journal of Fracture, 108:275–290, 2001. [15] A. Carpinteri, P. Cornetti, and N. Pugno. Edge debonding in FRP strengthened beams: Stress versus energy failure criteria. Engineering Structures, 31:2436–2447, 2009. [16] L. Távara, V. Mantiˇc, E. Graciani, J. Ca˜ nas, and F. Par´ıs. Analysis of a crack in a thin adhesive layer between orthotropic materials. An application to composite interlaminar fracture toughness test. Computer Modeling in Engineering and Sciences, 58(3):247–270, 2010. [17] V. Mantiˇc. Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion. International Journal of Solids and Structures, 46:1287–1304, 2009. [18] H. Zhang, ML. Ericson, J. Varna, and LA. Berglund. Transverse single-fiber test for interfacial debonding in composites: 1. Experimental observations. Composites Part A: Applied Science and Manufacturing, 28A:309–315, 1997. [19] J. Varna, LA. Berglund, and ML. Ericson. Transverse single fiber test for interfacial debonding in composites 2: Modelling. Composites Part A: Applied Science and Manufacturing, 28:317–326, 1997.


Three-Dimensional Fundamental Solution for Unsaturated Poroelastic Media under Dynamic Loadings Pooneh Maghoul 1, Behrouz Gatmiri 2, 3, Denis Duhamel 3 1

Department of Civil and Water Engineering, Faculty of Sciences and Engineering, Laval University, Quebec City, Quebec, Canada E-mail : [email protected] 2 University of Tehran, Departments of Civil Engineering, Tehran, Iran Scientific Division, National Radioactive Waste Management Agency 3 E-mail : [email protected]; E-mail : [email protected]

Keywords: Boundary element method; Boundary integral equations; Fundamental solution; Frequency domain; Unsaturated soil; Porous media; Dynamic behaviour

Abstract. This paper aims at obtaining a 3D fundamental solution for unsaturated soils under dynamic loadings in Laplace transform domain using the method of Hörmander. These solutions can be used, afterwards, in a convolution quadrature method (CQM)-based boundary element formulations in order to model the wave propagation phenomena in such media in time domain. 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. Wave propagation in unsaturated soils and the dynamic response of such media are of great interest in geophysics, soil and rock mechanics, and many earthquake engineering problems. However, in geomechanics, the behavior of such media including more than two phases is not consistent with the principles and concepts of classic saturated soil mechanics. 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 this paper first of all, a set of fully coupled governing differential equations of hydro-mechanical behaviour of unsaturated porous media including the equilibrium, air and water transfer equations subjected to dynamic loadings is presented based on 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 ‫ݑ‬௜ , water pressure ‫݌‬௪ and air pressure ‫݌‬௔ are presumed to be independent variables. Secondly, the associated fundamental solution in Laplace transform domain is presented using the method of Hörmander (1963) [5] for 3D ‫ݑ‬௜ െ ‫݌‬௪ െ ‫݌‬௔ formulation of unsaturated porous media. In this case that the fundamental solution is known only in the frequency domain and it seems too difficult to obtain the time-dependent fundamental solution in an explicit analytical form by an inverse transformation of the frequency domain results; the convolution integral in the BIE can be numerically approximated by a new approach called “Operational Quadrature Methods” developed by [1,2]. 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, 7]. Governing Equations. Governing differential equations consist of mass conservation equations of liquid and gaseous phases, the equilibrium equation of the skeleton associated with water and air flow equations and constitutive relation. The assumption of infinitesimal transformation and incompressibility of solid matrix is considered.

255

256


Solid Skeleton. The equilibrium equation and the constitutive law for the soil’s solid skeleton including the effect of suction are written [3]: (V ij G ij pa ), j pa ,i f i U ui (1)

(V ij G ij pa ) (OG ijH kk 2 PH ij ) Fijs ( pa pw )

(2)

where P , O are Lame coefficients, pD w,a is the water or air pressure, G ij is the Kronecker delta and Fijs is the suction modulus matrix: F s D.D suc 1 (3) in which D suc is a vector obtained from the state surface of void ratio ( e ) which is a function of the independent variables of ߪ െ ‫݌‬௔ and ‫݌‬௔ െ ‫݌‬௪ . D suc 1 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(K0 , Et ) D(V pa , pa pw )

(5)

where Et is tangent elastic modulus which can be evaluated as Et

El Es

(6)

in which El is the elastic modulus in absence of suction and Es ms ( pa pw )

(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 K 01 (1 e)we / w(V pa ) (8)

Mass Conservation of Water. The mass of the water in a representative elementary volume can be written as (9) w iw,i S wHii Cww p w Cwa p a where Cww (ng1 Cwn S w ); Cwa Caw ng1 . In this equation, wiD w,a is the displacement of water or air relative to solid, SD w, a is the degree of saturation relative to water or air, CD is the compressibility of water or air CD w,a d UD / ( UD d pD ) and ݃ଵ ൌ ݀ܵ௪ Ȁ݀ሺ‫݌‬௔ െ ‫݌‬௪ ሻ. Mass Conservation of Air. With the same approach presented before, the mass conservation of the air can be written as (10) w ia,i SaHii Cwa p w Caa p a where Caa (ng1 Ca n Sa ); Cwa Caw ng1 . Flow Equation for the Water. Based on generalized Darcy’s law for describing the balance of the forces acting on the liquid phase of the representative elementary volume, the water velocity in the unsaturated soil takes the following form: w w / kw U w g pw , i U w u (11) in which k w denotes the water permeability in an unsaturated soil. Flow Equation for the Air. With the same approach presented for the water based on generalized Darcy’s law, the air velocity in the unsaturated soil takes the following form: w a / ka U a g pa , i U a u (12) in which ka denotes the air permeability in an unsaturated soil. Summary of the Governing Differential Equations in Laplace Domain. By introducing (2) into (1), (11) into ࢇ (9) and (12) into (10) and by applying the Laplace transform assuming ࢛࢏ሺ࢚ୀ૙ሻ ൌ ࢝࢝ ࢏ሺ࢚ୀ૙ሻ ൌ ࢝࢏ሺ࢚ୀ૙ሻ ൌ ૙ and ࢖࢝ሺ࢚ୀ૙ሻ ൌ ࢖ࢇሺ࢚ୀ૙ሻ ൌ ૙, we obtain the final set of governing equations in Laplace transform domain: (O P )u E ,DE P uD , EE F s p w,D (1 F s ) p a ,D U .s 2 .uD fD 0 (13) s.T1 .uD ,D kw p w,DD Cww .s. p w Cwa .s. p a 0 (14) s.T 2 .uD ,D Cwa .s. p w ka p a ,DD Caa .s. p a

0

where T1 ( S w U w .k w .s ) and T 2 ( Sa U a .ka .s ) .

(15)


257

We would like to rewrite compactly the transformed coupled differential equation system Eqs. (13), (14) and (15) into the following matrix form: T B >uD , p w , p a @ ª¬ fD , 0, 0 º¼

T

(16)

0

with the not self-adjoint operator B : ª(P ' U s2 )Gi j (O P )wi w j F s wi (1 F s )wi º « kw ' Cww s Cwa s » sT1w j B « Cwa s ka ' Caa s ¼» T s w 2 j ¬

(17)

In equation (16), the partial derivative ( ), i is denoted by w i and ' w i i is the Laplacian operator. Note the operators B in (17) are not self adjoint. Therefore, for the deduction of fundamental solutions, the adjoint operator to B has to be used: ª(P ' U s2 )Gi j (O P)wi w j sT1wi sT2w j º B « kw ' Cww s Cwa s » F s w j (18) « » (1 F s )w j Cwa s ka ' Caa s ¼» ¬« Fundamental Solutions. Here, the fundamental solution associated with the operator (18) 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 transform fundamental solutions. In this study, because the operator type of the governing equations is an elliptical operator, the explicit 3D Laplace transform domain fundamental solution can be derived by using the method Hörmander [5]. The idea of this method is to reduce the highly complicated operator given in (18) to simple well known operators. In this method, in the Laplace transform domain, the first stage is to find the matrix of cofactors 1 B co / det( B ) ). For the second stage, we assume that M is a B co to calculate the inverse matrix of B ( B scalar solution to the equation )IM IG ( x [ ) 0 l BB coM G ( x [ ) 0 det(B (19) Consequently, we get B coM G (20) Following Hörmander’s idea, first, the determinants of the operator B are calculated: ) P 2 (O 2 P )k k (' O 2 ) 2 (' O 2 )(' O 2 )(' O 2 ) det( B (21) w a 1 2 3 4 in which the coefficients O12 , O22 , O32 and O42 are the roots, where one of its roots is the O12

U s 2 / P which is

related to the shear wave velocity propagating through the medium. The remained three roots O22 , O32 , O42 must be determined as these which satisfy U s 2 F s U w s 2 U a (1 F s )s 2 Caa s Cww s S w Fs s S (1 F s )s O22 O32 O42 a (O 2 P ) ka kw ( O 2 P )k w (O 2 P )k a 2 U Caa s 3 U Cww s 3 U (F sCaa (1 F s )Cwa )s 3 (CwwCaa Cwa )s 2 w (O 2 P )k a (O 2 P )k w (O 2P )ka k wk a U ( F sCwa (1 F s )Cww )s 3 S w (F sCaa Cwa (1 F s ))s 2 Sa ( F sCwa Cww (1 F s ))s 2 a (O 2 P )k w (O 2P )k wka (O 2 P )k wk a

O22O32 O22O42 O32O42

O22O32O42

(22)

2 U (CwwCaa Cwa )s 4 (O 2P )k wka

These three roots correspond to the three compressional waves which are affected by the degree of saturation and the spatial distribution of fluids within the medium. Secondly, by introducing the determinant, the scalar equation corresponding to (19) is given by (23) ( ' O12 )( ' O22 )( ' O32 )( ' O42 )) G ( x [ ) 0 in which ) is an interim operator, i.e. (24) ) P 2 ( O 2 P ) k w k a ( ' O12 )M

258


Equation (23) can be expressed as either of four equations (25), (26), (27) and (28): (25) ( ' O12 )M1 G ( x [ ) 0; M1 ( ' O22 )( ' O32 )( ' O42 )) (26) ( ' O22 )M 2 G ( x [ ) 0; M 2 ( ' O12 )( ' O32 )( ' O42 )) (27) ( ' O32 )M 3 G ( x [ ) 0; M 3 ( ' O12 )( ' O22 )( ' O42 )) (28) ( ' O42 )M 4 G ( x [ ) 0; M 4 ( ' O12 )( ' O22 )( ' O32 )) The above differential equations are of the familiar Helmholtz type. The fundamental solution of Helmholtz differential equations for an only r-dependent fully symmetric two-dimensional domain is Mi exp( Oi r ) / 4S r, i 1,2,3,4 (29) By definition of M1 , M 2 , M3 and M 4 , it is deduced: ª M 3 M 2 M 3 M1 M 4 M1 M 4 M 2 º 1 « » (O32 O42 )(O22 O12 ) ¬ O32 O22 O32 O12 O42 O12 O42 O22 ¼

)

(30)

Replacing equation (29) into (30), one obtains exp( O1r ) exp( O2 r ) ½ 1 °°(O12 O32 )(O12 O42 )(O12 O22 ) (O22 O42 )(O22 O32 )(O22 O12 ) °° M ® ¾ exp( O3r ) exp( O r ) 4S r ° 2 2 2 42 2 2 ° 2 2 2 2 2 2 ¯°(O3 O2 )(O3 O1 )(O3 O4 ) (O4 O1 )(O4 O2 )(O4 O3 ) ¿° in which the argument r

(31)

x [ denotes the distance between a load point and an observation point.

Finally, we can determine the components of fundamental solution tensor by applying the matrix of co to the scalar function M which are: cofactors B Displacement caused by a Dirac force in the solid: 1 § (O P ) / 2 (O12 K ss2 1 )(O12 K ss2 2 ) G ij ( R1 R2O1 R3O12 )exp( O1r ) ¨ 4SP © U s2 (O12 O32 )(O12 O42 )(O12 O22 )

(O P ) / 2 (O22 K ss2 1 )(O22 K ss2 2 ) ( R1 R2O2 R3O22 )exp( O2 r ) 2 2 Us (O2 O32 )(O22 O42 )(O22 O12 )

(32a)

(O P ) / 2 (O32 K s2s1 )(O32 K ss2 2 ) ( R1 R2O3 R3O32 ) exp( O3r ) 2 2 (O3 O22 )(O32 O12 )(O32 O42 ) Us

G · (O P ) / 2 (O42 K ss2 1 )(O42 K ss2 2 ) ( R1 R2O4 R3O42 )exp( O4 r ) ¸ ij exp( O1r ) (O42 O12 )(O42 O22 )(O42 O32 ) 4 r U s2 SP ¹ with R1

3 xD xE / r 5 G DE / r 3 , R2 3 xD xE / r 4 G DE / r 2 , R3 xD xE / r 3 , / 2

K ss2 1 K ss2 2

S w Fs s Sa (1 F s )s (k wCaa Cwwka )s U wk w F s s 2 U a ka (1 F s )s 2 k w ka (O P )k w (O P )k a (O P )k w (O P )k a

2 ss1

K K

2 ss 2

s s § C 2 C C S C (1 F s ) Sa F Cwa Cww (1 F ) U w k w F sCaa s U a ka F sCwa s · S FC ¨ wa ww aa w s aa w wa ¸ (O P )k w k a (O P ) k w k a (O P ) k w k a (O P ) k w k a (O P ) k w k a ¸ k w ka s2 ¨ s s ¨ U w k wCwa (1 F )s U a ka Cww (1 F )s ¸ ¨ ¸ (O P )k w k a © (O P )k w k a ¹

Water pressure caused by a Dirac force in the solid: G 4 j

U s 2 / (O 2 P ) and

(32b)

§ (1 rO2 )exp( O2 r ) § 2 Caa F s s Cwa (1 F s ) s · (1 rO3 )exp( O3r ) § 2 Caa F s s Cwa (1 F s ) s · · O O ¨ 2 ¸ 2 ¸ ¸ 2 2 2 ¨ 2 2 2 2 ¨ 3 F s ka F s ka F s r,i ¹ (O3 O2 )(O3 O4 ) © ¹ ¸ ¨ (O2 O3 )(O2 O4 ) © ¸ 4S (O 2 P )k w r 2 ¨ (1 rO4 )exp( O4 r ) § 2 Caa F s s Cwa (1 F s ) s · ¨ ¸ O ¸ s ¨ (O 2 O 2 )(O 2 O 2 ) ¨ 4 ¸ F k a 2 4 3 © ¹ © 4 ¹

Air pressure caused by a Dirac force in the solid:

(32c)


G 5 j

259

§ (1 rO2 ) exp( O2 r ) § 2 Cww (1 F s ) s Cwa F s s · (1 rO3 ) exp( O3 r ) § 2 Cww (1 F s ) s Cwa F s s · · O O ¨ 2 ¸ 2 ¸¸ 2 2 2 ¨ 2 2 2 2 ¨ 3 (1 F s )k w (1 F s )k w ¨ (O2 O3 )(O2 O4 ) © ¹ (O3 O2 )(O3 O4 ) © ¹¸ 2 ¨ ¸ s s 4S (O 2 P )ka r ¨ (1 rO4 ) exp( O4 r ) ª O 2 Cww (1 F ) s Cwa F s º ¸ « » 4 s 2 2 2 2 ¨ ¸ (1 F )k w ¼ © (O4 O2 )(O4 O3 ) ¬ ¹ (1 F s ) r,i

Displacement caused by a Dirac source in the water fluid: (1 rO3 )exp( O3r ) 2 § (1 rO2 )exp( O2 r ) 2 · ¨ (O 2 O 2 )(O 2 O 2 ) O2 KD w (O 2 O 2 )(O 2 O 2 ) O3 KD w ¸ ( U ) . S k s s r 2 3 2 4 3 2 3 4 w w w i , ¨ ¸ G i 4 ¸ 4S (O 2 P )k w r 2 ¨ (1 rO4 )exp( O4 r ) 2 K O Dw ¨ (O 2 O 2 )(O 2 O 2 ) 4 ¸ 2 4 3 © 4 ¹

(32d) where KD w

Cwa (Sa U a ka s)s Caa (Sw U wkw s)s / ka (Sw U wkw s) .

Displacement caused by a Dirac source in the air fluid: O3r ) 2 § (1 rO2 ) exp( O2 r ) 2 · O KD a ¸ O2 KDa (1(O2 rOO3 )exp( 2 2 2 3 ( Sa ka U a s ) s.r,i ¨ (O22 O32 )(O22 O42 ) 3 2 )( O3 O4 ) ¨ ¸ Gi 5 ¸ 4S (O 2 P )ka r 2 ¨ (1 rO4 )exp( O4 r ) 2 ¨ (O 2 O 2 )(O 2 O 2 ) O4 KD a ¸ 4 2 4 3 © ¹

where KDa

(32e)

Cwa (Sw U wkw s)s Cww (Sa U a ka s)s / kw (Sa U a ka s) .

Water pressure caused by a Dirac source in the water fluid: 1 § exp( O2 r ) O3r ) G 44 O22 K w2 O22 / 2w (O 2 exp( O 2 K w2 O32 / 2w ¨ O22 )(O32 O42 ) 3 4S k w r © (O22 O32 )(O22 O42 ) 3

(32f)

· exp( O4 r ) O42 K w2 O42 / 2w ¸ (O42 O22 )(O42 O32 ) ¹

S a 1 Fs s Caa s U a ka 1 Fs s 2 U s2 . O 2P ka ka O 2P ka O 2P Air pressure caused by a Dirac source in the air fluid: O3r ) 1 § exp( O2 r ) G 55 O22 K a2 O22 / a2 (O 2 exp( O 2 K a2 O32 / 2a ¨ 2 O O32 O42 ) 3 4S ka r © (O22 O32 )(O22 O42 ) )( 3 2

with K w2 / 2w

U Caa s 3 / O 2P ka and K w2 / 2w

(32g)

· exp( O4 r ) O42 K a2 O42 / 2a ¸ (O42 O22 )(O42 O32 ) ¹

with

2 a

/ 2a

U Cww s 3 / O 2P kw and K a2 / 2a

S w Fs s C s U k F s2 U s2 . ww w w s O 2 P kw kw O 2P kw O 2P

(32h) Air pressure caused by a Dirac source in the water fluid: § · exp( O2 r ) s 2 2 (O 2 P )Cwa (S w U w k w s )(1 F ) O2 U Cwa s ¸ ¨ 2 2 2 2 ¨ (O2 O3 )(O2 O4 ) ¸ ¨ ¸ exp( O3r ) s 2 s 2 ( O 2 P ) ( U )(1 ) O U s G 54 C S k F C s ¨ 2 ¸ 3 wa w w w wa 2 2 2 4S (O 2 P )k w ka r ¨ (O3 O2 )(O3 O4 ) ¸ ¨ ¸ exp( O4 r ) s 2 2 (O 2 P )Cwa (S w U w k w s )(1 F ) O4 U Cwa s ¸ ¨ 2 2 2 2 © (O4 O2 )(O4 O3 ) ¹ Water pressure caused by a Dirac source in the air fluid: (32i)

260

Eds: E L Albuquerque & M H Aliabadi exp( O2 r ) § 2 s 2 ¨ (O 2 O 2 )(O 2 O 2 ) (O 2 P )Cwa (Sa U a ka s )F O2 U Cwa s s 2 3 2 4 ¨ exp( O3r ) 4S O 2 P k w ka r ¨ 2 s 2 ¨ (O 2 O 2 )(O 2 O 2 ) (O 2 P )Cwa (Sa U a ka s )F O3 U Cwa s 2 3 4 © 3 · exp( O4 r ) (O 2 P )C wa (S a U a ka s )F s O42 U C wa s 2 ¸ (O42 O22 )(O42 O32 ) ¹

G 45

Analytical verification of the fundamental solutions. Limiting Case: Elastodynamic: Having derived the fundamental solution, at this stage, it is of interest to verify the validity of these solutions in the limiting case of elastodynamic. Letting k w and k a approach infinity and U w , U a and Fs equal zero, the unsaturated fundamental solutions presented in this study take the form of the elastodynamic fundamental solutions [8]:

G i j

G i j

G D 4

G D 5

where a

3C2 / s.r2

C22

xi x j · 1 § ¨ aG ij b 2 ¸ 4S U C22 © r ¹ G 4 j

G 5 j

(33)

0

(34)

1/ r C / s.r C / s .r exp s.r / C C C / s.r C / s .r exp s.r / C / C , b 1/ r 3C / s .r exp s.r / C C 1/ r 3C / s.r 3C / s .r exp s.r / C / C , C O 2 P / U and 2

2

2 2

2

2 2

2

3

2 2

2

3

2

2 2

2

1

2

2 1

1

2 1

2

2

3

2 1

1

3

1

2 1

2 1

P/U.

Conclusion. In this paper, firstly coupled governing differential equations of a porous medium saturated by two compressible fluids (water and air) subjected to dynamic loadings are presented based on the poromechanics theory in the frame of the suction-based mathematical model presented by Gatmiri [3] and Gatmiri et al. [4]. After that, the associated fundamental solution in Laplace transformed domain is presented by the use of the method of Hörmander for 3D ui pw pa formulation of unsaturated porous media. The derived Laplace transform domain fundamental solutions can be directly implemented in time domain BEM in which the convolution integral is numerically approximated by a new approach so-called “Operational Quadrature Methods” developed by Lubich [1, 2] to model the dynamic behaviour of unsaturated porous media. This enables one to develop more effective numerical hybrid BE/FE methods to solve 3D non-linear wave propagation problems in the near future. References

[1] [2] [3] [4] [5] [6] [7] [8]

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). L. Hörmander Linear Partial Differential Operators (1963). P.Maghoul, PhD dissertation. Université Paris-Est (2010). P.Maghoul, B.Gatmiri, D.Duhamel Soil Dynam. Earthquake Eng. Accepted (2010). J. Chen. Int. J. Sol. Struct. 31, 169–202 (1994).


BEM implementation of energetic solutions for quasistatic delamination problems C. G. Panagiotopoulos1, V. Mantiˇc1, T. Roub´ıˇcek2,3 1

Group of Elasticity and Strength of Materials, Department of Continuum Mechanics, School of Engineering, University of Seville, Seville, ES-41092, Spain 2 Mathematical Institute, Charles University, Czech Republic, Prague, CZ-18675 3 Institute of Thermodynamics of the ASCR, Czech Republic, Prague, CZ-18200 [email protected], [email protected], tomas.roubicek@mff.cuni.cz

Keywords: Interface fracture, rate-independent quasistatic model, adhesive contact, energetic solutions, delamination, debonding, interface damage, Boundary Element Method.

Abstract. The problem of the onset and propagation of an elastic-brittle delamination is considered. We study delamination processes for elastic bodies glued by an adhesive to each other or to a rigid outer boundary. The interfacial adhesive is assumed to store and also dissipate a specific amount of energy during the delamination process. Damage along the interfaces is taken into account by introducing an interface damage variable. The present approach is based on the so-called energeticsolution concept. After introducing an implicit time discretization and a spatial discretization along the boundaries, the boundary element method (BEM) is utilized to solve the pertinent recursive boundary-value problems arising at each time step and compute the stored elastic energy. The whole solution process is based on the global minimization of the sum of the elastic potential energy in the solids and adhesive layer, defined in terms of the displacements along boundaries and the damage parameter along the interfaces, and of the dissipated energy at each time step. Numerical solution of a delamination problem is presented to demonstrate the capabilities of this new approach. Introduction A basic model for quasistatic delamination, proposed by Frémond [1, 2], involves a damage-type variable, called a delamination parameter and denoted by z in what follows, reflecting the destruction of the bonds in the a-priori given delamination surfaces or part of outer boundary. This basic model incorporates merely the Griffith concept [3], namely the idea that the crack grows as soon as the so-called energy release rate during the delamination propagation is greater or equal than the phenomenologically prescribed activation energy (per unit area of a new crack surface), called also fracture energy. This activation energy is, in fact, equal to the dissipated energy. As such, the energy needed (and disipated) for the delamination is assumed to be rate-independent (after neglecting all inertial/viscous/thermal effects), which corresponds to the quasistatic model. The weak surface undergoing delamination can be considered either purely brittle or elastic-brittle, reflecting that the adhesive gluing the two sides of this weak surface has a certain elastic response. The conctact is assumed unilateral and, for simplicity, frictionless and the delamination is assumed fracture-mode insensitive. This approach was first developed in [4–8], cf. also [9, Chap.14]. The goal of this work is to present briefly, from the theoretical and numerical viewpoint, some basic features of the present approach. As the damage processes occur only on boundaries, the spatial discretization can advantageously be made by the BEM. The implementation of the collocation BEM [10, 11] is described first. Then, the computational procedure is validated by a two-dimensional simulation of a delamination process, showing that it can really produce the expected responses. For the sake of brevity, the implementation of the above methodology will only briefly be presented, full details as well as further numerical simulations and discussion are incorporated into a forthcoming paper [12]. A more thorough theoretical presentation of such models may be found in [13], where also a finite-element method (FEM) implementation and numerical simulations are presented.

261

262


Γ2N Γ2D

Γ1N

Ω2 Γ23 Ω3

Γ12

Γ01

Ω1

Γ2N Γ1N

Γ1D

Γ1N

Figure 1: Schematic illustration of the geometry and of the notation for a two-dimensional case of three mutually contacted subdomains, i.e. d=2 and N =3. Theoretical background, quasistatic rate-independent evolution, delamination model We will consider the evolution on a fixed finite time interval [0, T ] governed by a stored energy functional E : [0, T ] × U × Z → R ∪ {∞} and a dissipated energy functional R : X → [0, ∞], where U stands for the linear space of displacement field u, Z for that of damage parameter field z while X for the linear space of time derivatives of z. The rate-independent evolution we have in mind is governed by the following system of doubly nonlinear degenerate parabolic/elliptic variational inclusions: . (1) ∂u E (t, u, z) 0 and ∂R z + ∂z E (t, u, z) 0,

.

where z := dz dt and the symbol “∂” refers to a (partial) subdifferential, relying on that R(·), E (t, ·, z), and E (t, u, ·) are convex functionals. Let us recall that the subdifferential ∂f (x) of a convex function f : X → R ∪ {∞} at a point x is defined as the convex closed subset ∂f (x) := {x∗ ∈X ∗ ; ∀v∈X : f (x) + x∗ , v−x ≤ f (v)} of the dual space X ∗ . Let Ω ⊂ Rd (d = 2, 3) be a bounded Lipschitz domain, and let us consider its decomposition into a finite number of mutually disjoint Lipschitz subdomains ¯ Further denote Γij = ∂Ωi ∩ ∂Ωj Ωi , i = 1, ..., N , see Figure 1. We denote formally Ω0 := Rd \ Ω. the (possibly empty) boundary between Ωi and Ωj , i, j = 1, ..., N . For i, j = 1, ..., N , Γij represents a prescribed (d−1)-dimensional surface, which may undergo delamination also called debonding. We ¯ i ). We assume also consider a possible debonding on some parts of the outer boundary Γ0i ⊂ (∂Ω ∩ Ω that the rest of the outer boundary ∂Ω, i.e. ∂Ω \ ∪N Γ , is the union of two disjoint subsets: ΓD of a i=1 0i Γ we impose timenon zero measure and ΓN . On the Dirichlet part of the boundary ΓD and on ∪N 0i i=1 dependent boundary displacements wD (t), while the Neumann part of the boundary ΓN is considered d free. Therefore, any admissible displacement u : ∪N i=1 Ωi → R has to be equal to a prescribed “hardΓ device” loading wD (t) on ΓD and (u−wD (t))·ν ≥ 0 on ∪N i=1 0i . Delamination (or debonding), can be developed on the surface Γij , (2) ΓC := 0≤i 0. A fruitful concept of a certain weak solution to the doubly nonlinear inclusion with degree-1 homogenous dissipation potential R, called energetic solutions, was developed by Mielke et al. [15, 16], cf. also [17] for a survey. In the convex case, this concept is essentially equivalent to the conventional weak-solution concept, while in our case where E (t, ·, ·) is non-convex this concept represents a generalization which is well amenable to mathematical analysis and numerical implementation and applicable to engineering problems, too. Definition.1 (Energetic solutions [15, 16]) The process (u, z) : [0, T ] → U ×Z is called an energetic solution to the initial-value problem (1) given by (U ×Z , E , R) and the initial condition (u0 , z0 ) if u ∈ B([0, T ]; U ), z ∈ B([0, T ]; Z ) ∩ BV([0, T ]; X ), and (i) the energy equality holds: T Et (t, u, z) dt + E (0, u0 , z0 ) . (5) E (T, u(T ), z(T )) + DissR (z, [0, T ]) = 0 stored energy at time t = T

where

energy dissipated during [0, T ]

DissR (z, [0, T ]) := sup

N

work done by mechanical load

stored energy at time t = 0

R(z(tj ) − z(tj−1 )),

(6)

j=1

with the supremum taken over all partitions 0 ≤ t0 < t1 < ... < tN −1 ≤ tN ≤ T . (ii) the following stability inequality holds for any t ∈ [0, T ]: ∀(˜ u, z˜) ∈ U ×Z :

E (t, u, z) ≤ E (t, u˜, z˜) + R(˜ z −z),

(7)

(iii) the initial conditions u(0) = u0 and z(0) = z0 hold. Here U = W 1,2 (Ω\ΓC ; Rd ), X := L1 (ΓC ), and Z := L∞ (ΓC ). If wD has an extension uD belonging to C 1 ([0, T ]; W 1,2 (Ω; Rd ) and the initial condition (u0 , z0 ) is stable in the sense that E (0, u0 , z0 ) ≤ E (0, u˜, z˜) + R(˜ z −z0 ) for all (˜ u, z˜) ∈ U ×Z, one can show that the energetic solutions do exist.

264


Discretization and numerical implementation We make an implicit time discretization by using, for simplicity, an equidistant partition of [0, T ] with a time-step τ > 0, assuming T /τ ∈ N. This leads to a recursive minimization problem: minimize (u, z) → E (kτ, u, z) + R(z−zτk−1 ) (8) subject to (u, z) ∈ U ×Z , to be solved successively for k = 1, ..., T /τ , starting from u0τ = u0 and zτ0 = z0 . By the standard direct method, existence of solutions to (8) is due to weak lower semicontinuity of E (t, ·, ·) and coerciveness of E (t, ·, ·) + R(·−zτk−1 ). A solution in a time step k is then denoted as (ukτ , zτk ). Comparing the energy value of a solution at a time step k with that of a solution (uτk−1 , zτk−1 ) of the incremental problem (8) at the time step k − 1, we may derive an upper estimate of the energy balance in the time step k. Moreover, writing the stability condition (7) at the time step k − 1 and testing it by (ukτ , zτk ) gives a lower estimate of the energy balance in the time step k. Defining also E τ (t, u, z) := E (kτ, u, z), uτ (t) = ukτ , z τ (t) = zτk , uτ (t) = uτk−1 , and z τ (t) = zτk−1 for t ∈ ((k − 1)τ, kτ ], the following two-sided energy estimate may be established: s s Et t, uτ (t), z τ (t) dt ≤ E τ s, uτ (s), z τ (s) +R(zτk −z0 )−E τ (0, u0 , z0 ) ≤ Et t, uτ (t), z τ (t) dt (9) 0

0

for any s = kτ , k = 1, ..., T /τ ; here we used also that simply DissR (z τ , [0, s]) = R(zτk −z0 ). This implicit time discretization serves as a theoretical tool to prove existence of the energetic solutions themselves. Moreover, the two-sided energy estimate (9) may facilitate the numerical solution of the global-optimization problem (8), as it can be used for a solution improvement by backtracking if the numerical minimization procedure did not work successfully, as described, for example, in [13]. To implement the scheme computationally, one must make still some spatial discretization. Here we use the boundary element method (BEM). The BEM is closely related to the map between the prescribed boundary conditions in displacements or tractions and the unknown boundary displacements or tractions. In pure Dirichlet and Neumann BVPs these maps are called Steklov-Poincaré and Poincaré-Steklov maps, respectively, and BEM can be considered as an approach to discretize these maps. Hereinafter in this section, we denote the shifted displacement as u ˜, i.e. u ˜ = u − uD (t), for the sake of clarity. In the present computational procedure, the role of the BEM analysis applied to each subdomain Ωi separately (which, in fact, makes this problem very suitable for parallel computers) is to solve the corresponding BVPs on Ωi . Then, by using the computed values of boundary displacements and tractions the elastic strain energy stored in all Ωi is evaluated in each time step and in each iteration of the minimization algorithm. For this goal, we numerically solve the Somigliana displacement identity [10]: tk (x)Ukl (x, ξ) dS, (10) ckl (ξ)uk (ξ) + − uk (x)Tkl (x, ξ) dS = ∂Ωi

∂Ωi

where ξ ∈ ∂Ωi , uk (x) and tk (x) denote the k-component of displacement and traction vector, respectively. The weakly singular integral kernel Ukl (x, ξ), two-point tensor field, given by the Kelvin fundamental solution (free-space Green’s function) represents the displacement at x in the k-direction originated by a unit point force at ξ in the l-direction in the unbounded elastic medium. The strongly singular integral kernel Tkl (x, ξ), two-point tensor field, represents the corresponding tractions at x in the k-direction. The coefficient-tensor ckl (ξ) of the free-term is a function of the local geometry of the boundary ∂Ωi at ξ, and may be evaluated by a closed analytical formula for isotropic elastic solids [18]. The symbol − stands for the Cauchy principal value of an integral. Consider a discretization of the boundary ∂Ωi by a boundary element mesh, which is also used to define suitable discretizations of the boundary displacements uh (x) ≈ u and the corresponding boundary tractions th (x) by interpolations of their nodal values. By imposing (collocating) the Somigliana


265

identity (10) at all boundary nodes (called collocation points) we set the BEM linear system of equations. The solution of this system defines the unknown nodal values of uh and th along ∂Ωi denoted as ui and ti , respectively. Then, the BEM system is usually written as Hi ui = Gi ti [10, 11]. Note that the nodal values of boundary tractions can be written in terms of the nodal values of boundary e map. displacements as ti = G−1 i Hi ui , which is in fact a discretized form of the Steklov-Poincar´ Then, to compute an approximation of the elastic energy stored in the bulk Ωi by using the obtained approximations of the shifted boundary displacements u˜h ≈ u ˜, the boundary displacements uh and the boundary tractions th along ∂Ωi , at each time step, we utilize the following general relation, ˜h defined on ΓC : involving the boundary data uD (t), with uc being the vector of nodal values of u EΩh (t, uc ) =

N i=1

EΩi h (t, uc )

with

EΩi h (t, uc ) =

1 2

∂Ωi

th uh dS.

The energy stored in the adhesive along the dissipative interfaces can be evaluated by

κ

κt

2 2 n zh u ˜h n + u ˜h t EΓh (uc , z) = dS, 2 2 ΓC

(11)

(12)

where an approximation of the damage variable zh ≈ z by using the same boundary element mesh as for displacements and tractions is considered, and z is the vector of nodal values of zh defined on ΓC . Similarly, the dissipated energy can be evaluated by putting Rh (z) := R(zh ) with zh determined by z. According to equations (11)-(12), the discretized minimization problem has now a boundary only form and may be given in terms of the boundary nodal values as follows, minimize (uc , z) → Eh (kτ, uc , z) + Rh (z−zk−1 ) (13) k−1 subject to BI uc ≥0, 0≤z≤z , where Eh (t, uc , z) = EΩh (t, uc ) + EΓh (uc , z) while BI represents the non-penetration Signorini conditions. Problem (13) is in general a difficult non-convex optimization problem that has to be numerically solved repeatedly at each time step. Several techniques have been used to solve efficiently this problem, such as the alternating minimization algorithm and a heuristic back-tracking algorithm which is based on the discrete form of the two-sided inequality (9). These techniques [13] have been modified appropriately for this work to fit in the current BEM implementation. Numerical Results The above introduced formulation has been implemented in a two-dimensional BEM code [19] using continuous piecewise linear boundary elements [10,11]. With reference to Figure 1, only one subdomain (i.e. N = 1) is used to model in a simple way an experimental test motivated by the pull-push shear test used in engineering practice [20]. The geometry of the problem is shown in Figure 2. The length and height of the rectangular domain Ω, respectively, are L = 250 mm and h = 12.5 mm. The length of the initially glued part ΓC placed at the bottom side of Ω is Lc = 0.9L. The elastic material of the bulk is an aluminium with the elasticity modulus E = 70 GPa and the Poisson’s ratio ν = 0.35. Elastic plain strain state is considered. The adhesive layer is represented by the normal stiffness κn =150 GPa/m, the tangential stiffness κt = κn /2 and the mode I fracture toughness aI = 187.5 J/m2 . The critical stress for the normal direction is defined by σc = 2κn aI =0.237 GPa. A hard device loading is assumed by prescribing horizontal and vertical displacements, respectively, wx and wy =0, at the right-hand side of the rectangle ∂Ω, defining the Dirichlet boundary ΓD . All the other boundary parts are considered to be traction free, defining the Neumann boundary ΓN , except for the contact surface ΓC . The uniform boundary element mesh used to discretize the boundary ∂Ω has 210 elements, 100 and 5 elements along each horizontal and vertical side, respectively. Thus, ΓC is discretized by 90 elements. The problem evolution is described in terms of a fictitious time t which is assumed to have unit value for a horizontal displacement wx =0.1368mm.

266


y

L

ΓN

Ω ΓD

h Lc

ΓC

x

ΓN

Figure 2: Geometry and boundary conditions of the problem considered. 4.5e+01

7.0e-06

(4,5,6)

Energies (J)

3.5e+01 3.0e+01 2.5e+01 2.0e+01

(1)

1.5e+01

Coordinate y (m)

4.0e+01

(a)

5.0e-06 4.0e-06 3.0e-06 2.0e-06

1.0e+01 5.0e+00

t=0.5 t=0.7

6.0e-06

(2)

1.0e-06

(3)

0.0e+00 0.0e+001.0e-01 2.0e-01 3.0e-01 4.0e-01 5.0e-01 6.0e-01 7.0e-01 8.0e-01 9.0e-01 Fictitious time t

0.0e+00 0.0e+00

(b)

5.0e-02

1.0e-01 1.5e-01 2.0e-01 2.5e-01 Coordinate x (m)

3.0e-01

Figure 3: a) Energy evolution for (1) bulk, (2) surface, (3) dissipated, (4) lower-estimation, (5) total, (6) upper-estimation. b) Deformed shape of the upper side of the adhesive layer for two time steps, in an undamaged and a damaged state. Figure 3a) shows the evolution of different energies with the fictitious time computed. In particular, the energy stored in the elastic bulk, in the adhesive layer and the dissipated energy. Also the total energy, which is actually minimized in the time stepping procedure, together with the lower and upper estimates of energy (9) are shown. As it can be seen in Figure 3a), the global minimization procedure defines as the end point, where the whole adhesive zone is debonded, the point where the sum of the energies in the bulk, the adhesive layer and the dissipated energy, is equal to the energy needed for the total delamination of the whole adhesive zone. In Figure 3b), the deformed shape of ΓC , or equivalently, the upper side of the adhesive layer, is plotted. Two contact zones can be observed in these plots, where Signorini contact conditions are valid. The resulting deformed shapes are shown for two time steps, for an undamaged and a partially damaged state, t=0.5 and 0.7, respectively. Figure 4 presents the components of the traction vector along the adhesive zone ΓC . As it can be observed there, very good agreement exists between the computed tractions in the elastic bulk by BEM and that computed in the adhesive layer, although the equilibrium has not been imposed directly but results as a consequence of the energy minimization. Progressive extension of the traction free portion of the original ΓC because of appearance of the total damage (z=0) can be observed in Figure 4a. It should be mentioned here that the portion of the adhesive zone which is totally damaged is still kept as a part of the minimization procedure, where nodal displacement values participate as part of the set of unknowns in the minimization procedure and their values are used in the BEM solution of the pertinent boundary value problem. For this reason an approximation of the developed traction-free zone is computed. Obviously other algorithms might be used where after total damage a change in the type of the boundary condition might be taken into account in the BEM computation. Nevertheless, we have been interested in the results obtained by the present simple procedure. In the plots of Figure 4b, one may observe the compression zones, where zero values of tractions are obtained in the adhesive. This is due to the fact that the rigid obstacle undertakes these forces.

1.0e+06 0.0e+00

t=0.5 t=0.5 t=0.7 t=0.7

1.5e+06

Bulk Adhesive Bulk Adhesive

Normal tractions ty (Pa)

Tangential tractions tx (Pa)


-1.0e+06

(t=0.5)

-2.0e+06 -3.0e+06

(t=0.7)

-4.0e+06 -5.0e+06 0.0e+00

(a)

1.0e+06

t=0.5 t=0.5 t=0.7 t=0.7

Bulk Adhesive Bulk Adhesive

267

(t=0.5)

5.0e+05

(t=0.7)

0.0e+00 -5.0e+05

-1.0e+06 5.0e-02

1.0e-01 1.5e-01 2.0e-01 Coordinate x (m)

-1.5e+06 0.0e+00

2.5e-01

(b)

5.0e-02

1.0e-01 1.5e-01 2.0e-01 Coordinate x (m)

2.5e-01

6.0e+05 5.0e+05

1.4e+04

(A) (B)

(C)

4.0e+05 3.0e+05 2.0e+05 1.0e+05

(D)

Resultant vertical force(N/m)

Resultant horizontal force(N/m)

Figure 4: Tractions along the adhesive zone ΓC in a) tangent and b) normal direction computed by BEM for the bulk and those in the adhesive layer.

(a)

1.0e+04 8.0e+03

(B) (C)

6.0e+03 4.0e+03 2.0e+03

(D) 0.0e+00 0.0e+001.0e-01 2.0e-01 3.0e-01 4.0e-01 5.0e-01 6.0e-01 7.0e-01 8.0e-01 9.0e-01

0.0e+00 0.0e+001.0e-01 2.0e-01 3.0e-01 4.0e-01 5.0e-01 6.0e-01 7.0e-01 8.0e-01 9.0e-01

Fictitious time t

(A)

1.2e+04

(b)

Fictitious time t

Figure 5: Evolution of the horizontal and vertical resultant force on the Dirichlet part of the boundary ΓD . Letters (A)–(D) indicate characteristic points of the overall behaviour. Finally, in Figure 5 the resultant forces acting at ΓD with respect to the displacements applied therein are shown. These two plots have some similarities in the behaviour that may come up through the characteristic points (A)–(D). Up to point (A) the linear elastic behaviour manifests in both the solid and adhesive, at this point the first damage appears in the first 6 boundary elements that are situated on the right-hand side of the adhesive layer. This new crack length results in a “jump down” of the resultant forces up to point (B). Then, up to point (C) the damaged zone is progressively extended and finally after point (C) the remaining adhesive zone is damaged instantaneously. The problem evolution ends up at point (D), where rigid body motion of the elastic body takes place. We have to mention here that for the damage variable approximation zh , as well as for the displacements and tractions, linear distribution over the elements has been considered, which results in some difficulty in the definition of the crack length. Therefore, an implementation of the present approach with piecewise constant distribution of damage variable approximation zh along ΓC is under development and will be presented in a forthcoming work. Conclusions A new approach to the problems of the onset and propagation of an elastic-brittle or brittle delamination or debonding has been proposed in this work. A numerical procedure based on the energeticsolution concept has been developed and implemented in a two-dimensional collocational BEM code.

268


Numerical results for an elastic-brittle delamination problem have briefly been presented and discussed to show some basic features of this new approach. Acknowledgments VM and CGP acknowledge the support by the Junta de Andaluc´ıa and Fondo Social Europeo (Proyecto de Excelencia TEP-4051). VM also acknowledges the support by the Ministerio de Ciencia e Innovación (Proyecto MAT2009-14022). CGP also acknowledges the hospitality of Charles University, where this work has partly ˇ ˇ been accomplished, supported by the “Neˇcas center for mathematical modeling” LC 06052 (MSMT CR). TR acknowledges the hospitality of Universidad de Sevilla, where this work has partly been accomplished, covered by Junta de Andaluc´ıa through the project IAC 09-III-6321, as well as partial support from the grants A 100750802 ˇ ˇ ˇ ˇ (GA AV CR), 201/09/0917 and 106/09/1573 (GA CR), and MSM 21620839 (MSMT CR), and from the research ˇ plan AV0Z20760514 (CR).

References [1] Frémond, M., Dissipation dans l’adhérence des solides, Comptes rendus de l’Acadmie des sciences, Paris, Série 2, 300(1985) 709–714. [2] Frémond, M., Contact with adhesion, In: Topics in Nonsmooth Mechanics, J. Moreau and P. Panagiotopoulos (Eds.), Birkh¨ auser, 1988. [3] Griffith, A.A., The phenomena of rupture and flow in solids, Philos. Trans. Royal Soc. London Ser. A. Math. Phys. Eng. Sci., 221(1921) 163–198. [4] Koˇcvara, M., Mielke, A., Roub´ıˇcek, T., A rate-independent approach to the delamination problem, Mathematics and Mechanics of Solids, 11(2006) 423–447. [5] Point, N., Unilateral contact with adherence, Mathematical Methods in the Applied Sciences, 10(1988) 367–381. [6] Point, N., Sacco, E., A delamination model for laminated composites, Mathematical Methods in the Applied Sciences, 33(1996) 483–509. [7] Point, N., Sacco, E., Mathematical properties of a delamination model, Mathematical and Computer Modelling, 28(1998) 359–371. [8] Roub´ıˇcek, T., Scardia, L., Zanini, C., Quasistatic delamination problem, Continuum Mechanics and Thermodynamics, 21(2009) 223–235. [9] Frémond, M., Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. [10] Par´ıs, F., Ca˜ nas, J., Boundary Element Method, Fundamentals and Applications, Oxford University Press, Oxford, 1997. [11] Aliabadi, M.H., The Boundary Element Method: Applications in Solids and Structures, Wiley, West Sussex, 2002. [12] Mantiˇc, V., Panagiotopoulos, C.G., Roub´ıˇcek, T., Rate-independent approach of delamination with energetic solutions and BEM implementation (in preparation). [13] Roub´ıˇcek, T., Kruˇz´ık M., Zeman J., Delamination and adhesive contact models and their mathematical analysis and numerical treatment. In: Mathematical Methods and Models in Composites, Mantiˇc V. (Ed.), Imperial College Press, London, 2011. [14] Távara, L., Mantiˇc, V., Graciani, E., Par´ıs, F., BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model, Engineering Analysis with Boundary Elements, 35(2011) 207–222. [15] Mielke, A., Theil, F., On rate-independent hysteresis models, Nonlinear Differential Equations and applications (NoDEA), 11(2004) 151–189. [16] Mielke, A., Theil, F., Levitas, V.I., A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162(2002) 137–177. [17] Mielke, A., Evolution in rate-independent systems, Ch. 6, In: Handbook of Differential Equations, Evol. Equations, Vol. 2, C.M.Dafermos, E.Feireisl (Eds.), Elsevier, Amsterdam, 2005, pp.461–559. [18] Mantiˇc, V., A new formula for the C-matrix in the Somigliana identity, Journal of Elasticity, 33(1993) 191–201. [19] Panagiotopoulos, C.G., Open BEM Project, http://www.openbemproject.org/, 2010. [20] Cornetti, P., Carpinteri, A., Modelling the FRP-concrete delamination by means of an exponential softening law, Engineering Structures (in press).


269

The Dual-Boundary-Element Formulation Using the Tangential Differential Operator and Incorporating a Cohesive Zone Model for Elastostatic Cracks P. C. Gonçalves1, L. G. Figueiredo1, L. Palermo Jr.1, S. P. B. Proença2 1

Universidade Estadual de Campinas, Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Brazil, [email protected]. 2 Universidade de São Paulo, Escola de Engenharia de São Carlos, Brazil

Keywords: Boundary element method, Dual boundary element method, Linear elastic fracture mechanic, Cohesive models.

Abstract. A crack analysis for a cohesive material was performed with the dual boundary element method (DBEM) using the tangential differential operator (TDO) in the boundary integral equation (BIE) for tractions. A single edge crack employing a linear cohesive model in terms of two parameters was used in the iterative procedure. The results obtained with DBEM employing the TDO are compared with those obtained without using the TDO in traction BIE. Isoparametric linear boundary elements were used with nodal parameters fixed at the ends. The collocation points were positioned inside elements and conformal interpolations were employed along crack surfaces. Introduction The dual boundary element formulation for two dimensional problems of linear fracture mechanics using TDO in traction BIE was studied in [1]. The use of the tangential differential operator (TDO) in conjunction with the integration by parts is an interesting procedure to reduce the order of singularity in fundamental solution kernels of traction BIE when Kelvin type fundamental solutions are employed. Kupradze [2] first presented an application using the tangential differential operator (TDO) and Sladek [3] employed the TDO in a curved crack solution. Regularized boundary element formulations employing TDO in BIE for gradients in potential problems and in stress BIE for elasticity problems, including fracture mechanics formulations, were presented by Bonnet in [4]. The strategy presented in [1], which allowed the application of TDO in problems using non-conformal interpolations, was extended in [5] for the traction BIE in three dimensional elasticity. It is necessary to note the boundary element meshes and positions for collocation points used in [1] were first tested for traction BIEs containing strong singularities in [6] and the results were not changed with reference to those available in the literature. The present model of the fracture process zone was motivated by the fact that in some materials such as concrete, brittle polymers, fiber-reinforced composites, tough ceramics and some alloys, the crack surfaces are usually not separated completely behind the (fictitious) crack tip. There is a relatively long extension of the crack –commonly called the wake zone, the bridging zone, or the cohesive zone – where tractions can be transferred across the crack line. The main assumption in this model is that the material softening beyond the peak load is located in a narrow layer behind the fictitious crack tip, whose volume is negligible and whose action is replaceable by cohesive forces. Typically, two types of constitutive laws are used in literature for cohesive materials: one is characterized by a traction-displacement relationship and the other one by a material constitutive law defined in terms of stress and strain accompanied by a layer thickening law. Traction boundary integral equation using the tangential differential operator A short explanation on how to obtain the TDO in traction BIE will be presented next. The BIE for the gradient at an internal point x can be written using the differentiation in terms of field variables: u i , m x

³T

ij, m

*

x , y u j y d* y ³ U ij,m x , y t j y d* y

(1)

*

Uij (x, y) and Tij (x, y) are the displacement and the traction, respectively, in the direction j at the boundary point y due to a singular load in the direction i at the collocation point x, according to the Kelvin solution for two-dimensional problems; uj(y) and pj(y) are the displacement and the traction at the field, respectively.

270


The first and the second integrals of eq (1) are regular for internal points and exhibit singularities of order 1/r2 and 1/r, respectively, when the field point approaches the collocation point. The TDO is introduced in the integral containing 1/r2 singularity of eq (1):

³T

ij, m

x, y u j y d*y

*

³n

b

³n

b

y V ibj,m x, y u j y d*y

*

y V ibj,m x, y u j y d*y

*

³ ^D >V bm

ibj

*

x, y @ n m y V ibj,b x, y ù j y d*y

(2) (3)

Dbm( ) is the tangential differential operator and has the following definition:

D bm >f y @ n b y f ,m y n m y f ,b y

(4)

The second term in the integral of the right member of eq. (3) is null at points y not coincident with x when the Kelvin solution is used. The application of the integration by parts on the resultant term of the integral of the right member of eq. (3) yields:

³ D >V bm

x, y @u j y d*y

ibj

*

³V

ibj

x, y D mb >u j y @d*y

(5)

*

The final BIE for the gradient using the TDO has the following expression: u i , m x

³V

ibj

*

x , y D mb >u j y @d* y ³ U ij,m x , y t j y d* y

(6)

*

The integrals of eq (6) are regular for internal points and exhibit singularities of order 1/r when the field point approaches the collocation point. The BIE for stresses is obtained from eq. (6) using the Hooke tensor for isotropic material and the symmetry property of Uij,m (x, y): V ak x

>

@

C aki m V ibj x , y D mb u j y d* y V jak x , y t j y d* y

³

³

*

(7)

*

§ 2Q · P¨ G ak G im G ai G km G am G ki ¸ © 1 2Q ¹ The stress BIE at a boundary point is defined as the limiting form of the corresponding BIE at an internal point when it is led to a point on the boundary. The BIE for stresses at the point x’ on a smooth boundary is given by: C akim

1 V ak x c 2

>

@

C aki m ³ V ibj x c, y D mb u j y d * y ³ V akj x c, y t j y d* y *

(8)

*

It is important to note the continuity requirement for the derivative of the displacement function at the collocation point x´. The traction BIE can be obtained from eq (8) when the stress tensor obtained at the boundary point x’ is multiplied by direction cosines of the outward normal at this point (n’a) and the corresponding integral equation is given by:

1 t k x c 2

>

@

n ca x c C aki m ³ V ibj x c, y D mb u j y d* y n ca x c ³ V akj x c, y t j y d * y *

(9)

*

The Dual Boundary Integral Equations

The dual equations of the method are the displacement and the traction boundary integral equations. The traction BIE using the TDO, equation (9), is used with the displacement BIE that is written as follows:

c ij x ' .u j x ' ³ Tij x ' , y .u j y .d*y *

³ U x' , y .t y .d*y ij

j

(10)

*

cij is given by Gij/2 (Gij is the Kronecker delta) when the collocation point is placed on a smooth boundary and is equal to Gij for internal points. The integrals of eq (10) are regular for internal points and exhibit singularities of order 1/r and ln(1/r) for points on the boundary. The displacement BIE is applied to one of the crack surfaces and the traction equation to the other to solve general mixed-mode crack problems with a single domain formulation. Although the integration path is still the same for coincident points on the crack surfaces, the respective boundary integral equations are now distinct. The collocation point needed to perform the traction boundary integral equation and the strategy used to treat improper integrals are the essential features of the formulation. The collocation points must be positioned to satisfy the continuity requirements for each BIE. The continuity of the displacement function at x’ is the necessary condition for displacement BIE and it is


271

satisfied when the collocation point is placed at the ends of the boundary element or inside the element. The continuity of the displacement derivative at x’ is required for stress BIE and it is satisfied when the collocation point is placed inside the element [7]. Boundary Elements and Internal Collocation Points

Linear shape functions were used to represent displacements and efforts in the boundary elements. The same shape function was used for conformal and non-conformal interpolations with nodal parameters positioned at the ends of the elements. The collocation points were shifted to the interior of the element at a distance of a sixth part of its length starting from the end. The collocation point position (’), in range (-1, 1), was: i) ’=0.67 for continuous elements; ii) ’=-0.67 and ’=+0.67 for discontinuous elements. The number of collocation points in the element was defined by the computer code according to the condition of the last node, which means that elements with discontinuity at the first node had one collocation point. The improper integrals were handled by the classical singularity subtraction method and natural definitions of ordinary finite-part integrals were reached [7]. Analytical expressions were used to evaluate singular integrals and Gauss-Legendre scheme used for regular integrals. The present numerical implementation was studied in [6] for fracture problems and it was shown that conformal interpolations on the crack surfaces could be applied without losing the accuracy of the dual formulation. The diagonal terms were directly obtained using the collocation point position on the element and the shape function. The use of non-conformal interpolations required a revision of the result obtained from eq. (5) and the effect of the ends from the integration by parts had to be considered. The integration by parts presented in eq (5) is repeated next, including the effect of the ends:

³ D >V x, y @ u y d*y >e bm

i bj

j

*e

@

>

@

V ibj x, y u j y 0 ³ V ibj x , y D mb u j y d*y *e

3 bm

*e

(16)

*e is an open line, eijk is the permutation symbol. Cohesive Zone Model

The cohesive zone is an extension of the crack where the material softening beyond the peak load is located in a narrow layer behind a fictitious crack tip, whose volume is negligible and whose action is replaceable by cohesive forces. The cohesive zone was represented in this study by a crack region containing springs connecting coincidental points. Thus, the points, originally coincidental on opposite sides of the crack line, separate into distinct points, were connected by the cohesive zone material (springs). Continued straining increases the separation between these two points and eventually leads to cracking. It is necessary to point out that the nodes of the elements on the crack surfaces were connected by springs in the numerical implementation. The present study used a traction-displacement relationship that is suitable for quasistatic loading and considered a simplified model, taking into account the normal separation distance (or the cohesive zone displacement opening) component.

ȁ‫ ݊݌‬ȁ

py k ȁ‫ ݊ݓ‬ȁ wf

Figure 1. - The two parameter constitutive law for the cohesive zone material used in the simulations in terms of the traction and displacement jump. The py value for normal traction in Figure 1 corresponds to the limit without opening displacement in the cohesive zone and wf is the opening displacement limit where the traction disappears. In the present study, a simplified maximum principal stress criterion was used to determine crack increment. If the maximum principal stress at a fictitious crack tip reaches the critical py value, this tip is ready to run under

272


further loading, however, using the constitutive law according to Figure 1. The direction in which the tip advances is perpendicular to the direction of the maximum principal stress at that point, and the extension is such that the maximum principal stress, at the new tip position, is kept at the critical py, value during continued loading. However, it is important to note that, structural instability may occur while a crack is advancing; i.e., extension of a crack may enhance the stress state at a crack tip rather than release it. In this case, the crack may run fast and inertia effects may have to be included. Numerical Example

A single-edge crack in a rectangular specimen was studied under a uniform far field tensile loading, taking into account a displacement controlled elongation, as shown in Figure 2. The specimen was a length l and height h rectangular region, considered equal to l in these simulations. All length quantities are normalized by l; while this is not a natural length scale for the fracture problem, it is convenient and easily interpreted. The initial length of the single edge crack (a0) was 0.2, the cohesive zone (ZPC) was 0.5, the Poisson ratio was 0.3 and the plane strain condition was adopted.

h/2 = 0.5

h/2 = 0.5

a0

ZPC

w = 1.0

Figure 2 - Rectangular specimen with a single edge crack of initial length a0, under controlled displacement elongation. ZPC is the cohesive zone. According to the cohesive model used, the physical crack tip was positioned at the end of the initial length a0 where the opening could be wf and the fictitious crack tip was positioned at the end of the cohesive zone ZPC, where the normal traction value could be py (or lower than py). The constitutive law of the cohesive crack shown in Figure 1 is represented by a straight line. The two parameter py and wf were equal to 0.01 and 0.001, respectively. It is necessary to note that all stress and traction quantities were normalized by shear modulus (). The unloaded specimen was initially in a stress free state. It was loaded incrementally in a direction which was perpendicular to the top and bottom boundaries under displacement controlled elongation. The increment of the loading displacement was taken to be 1E-5 in tension and a limit of 2000 load steps was used. The element length of the rectangular boundary was 0.05. Double nodes were introduced at corners and at the fictitious crack tip. The element length of the initial crack length (a0) was 0.05, i.e. 4 elements used on each crack surface. The element length of the cohesive zone (ZPC) was 0.025, i.e. 20 elements used on each crack surface. According to the number of elements on each surface at the cohesive zone, 21 springs connecting opposite nodes at the cohesive zone were used. The analyses were carried out until the first element on the cohesive zone was cracked, i.e. the initial crack length was increased by 0.025 or one-element length. Two iterative procedures were used in the analysis: i) the obtained forces in the cohesive region at the end of each loading step were introduced in the next loading step, as the existing forces in the cohesive region; ii) the obtained forces in the cohesive region were computed according to the actual separation value of the crack surfaces at the loading step and considered to be the existing forces in the cohesive region. The


273

second iterative procedure worked with two nested iterations: a global iteration was required for the loading step and a local iteration was required to obtain the cohesive forces according to the actual separation value of the crack surfaces. 0,0040

0,00003 ; 0,0036446 0,00003 ; 0,0036166 0,00012; 0,0035022

; Cohesive TDO ; Cohesive ; Iterative-Cohesive Iterative-Cohesive TDO

0,00012;0,0034749

0,0030

Normalized Load

A0 = 0,2 ; ZPC = 0,5 ; E = 10 A0 = 0,2 ; ZPC = 0,5 ; E = 10 A0 = 0,2 ; ZPC = 0,5 ; E = 10 A0 = 0,2 ; ZPC = 0,5; E = 10;

0,00106 ; 0,0015425 0,00106 ; 0,001521

0,0020

0,00105 ; 0,0014587 0,00104 ; 0,0014808

0,0010

0,0000 0,0000

0,0001

0,0002

0,0004

0,0005

0,0006

0,0007

0,0008

0,0010

Elongation

Cohesive force (Pn)

Figure 3. - The elongation of the specimen according to the normalized load at the top (bottom) boundary.

Spring Figure 4. – Normal traction in each spring of the cohesive zone (ZPC) at the end of the iterative procedure.

Normalopening (wn)

0,0010 coesivo coesivo iterativo não coesivo

0,0005

0,0000 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

Spring Figure 5. – Normal opening in each spring of the cohesive zone (ZPC) at the end of the iterative procedure.

274


Table 1. – Comparison of obtained results according to Young modulus

Non cohesive

Steps Elongation Cohesive Steps Elongation Iterative -Cohesive Steps Elongation

E = 21,000 E = 10 98 98 0.00098 0.00098 98 106 0.00098 0.00106 98 105 0.00098 0.00105

Conclusion

Figure 3 presented the elongation of the specimen according to the normalized load either at the top or at the bottom boundary. The results labeled as cohesive and iterative-cohesive correspond to the analysis including the cohesive material according to the first and the second iterative procedures, respectively. Two values for Young modulus were used in simulations: 10 and 21,000. The obtained results with both values of Young modulus were similar including the changes according to the iteration procedure, whereas only the numbers of iteration steps were changed. The precision was not lost with the use of the TDO. The increase in the value of Young modulus reduced the number of steps for cracking to occur in the first element in the cohesive zone. The obtained values for the normal opening were lower using the first iterative procedure because the cohesive forces in the previous step, and applied in the next step, were higher than the actual cohesive forces in the step (see Figure 4). The behavior of the specimen shown in Figure 3, using both iterative procedures, shows the same feature, i.e. a higher load was obtained in the second procedure due to the action of actual cohesive forces, whereas more elongation was obtained in the first procedure with reference to the second at the same external load, due to the “delay” in introducing actual cohesive forces in the first procedure. Low differences were noted using both iterative procedures, because iteration steps with low incremental loadings were adopted. No significant differences appeared in the normal opening and in the cohesive forces when results using the TDO were compared with those without using the operator (the wellknown BIE for tractions). Furthermore, Figure 5 presented values for the normal opening from the analysis without considering the cohesive material and results were labeled as non-cohesive. The use of the TDO in traction BIE when Kelvin type fundamental solutions are employed simplifies the formulation for traction BIE due to the use of similar fundamental solution kernels with reference to the displacement BIE, reduces the strong singularity of the general formulation for traction BIE without losing the accuracy of the dual boundary element method. References

[1] Palermo, Jr., L., Almeida, L.P.C.P.F. and Gonçalves, P.C., The Use of the Tangential Differential Operator in the Dual Boundary Element Equation, Structural Durability & Health Monitoring, vol.2, no.2, pp.123-130, Tech Science Press, 2006. [2] Kupradze, V.D., Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North Holand, 1979. [3] Sladek, J.; Sladek, V., Three-dimensional curved crack in an elastic body, Int. J. Solids Struct., 19, 425436, 1983. [4] Bonnet, M. Boundary Integral Equation Methods for Solids and Fluids, John Wiley & Sons Ltd, 1999. [5] Palermo, Jr., L., Almeida, L.P.C.P.F., On the Use of the Tangential Differential Operator in the Traction Boundary Integral Equation of the Dual Boundary Element Method for Three Dimensional Problems, ICCES, vol.7, no.2, pp.83-87, 2008 [6] Almeida, L.P.C.P.F.; Palermo Jr., L. On the Implementation of the Two Dimensional Dual Boundary Element Method for Crack Problems, 5th Int. Conference on Boundary Elements Techniques, Lisboa, Portugal, 2004. [7] Portela, A, Aliabadi, MH, Rooke, DP, The dual boundary element method: Effective implementation for crack problems, International Journal of Numerical Methods in Engineering, 33, 1269-1287, 1992.


Slow viscous migration of a solid particle near a plane wall with a slip condition 1

3

N. Ghalia1 , A. Sellier2 , L. Elasmi3 and F. Feuillebois4 Laboratory of Engineering Mathematics, Polytechnic School of Tunisia, University of Carthage. BP 743, La Marsa, Tunisia e-mail: [email protected] 2 LadHyx. Ecole polytechnique, 91128 Palaiseau C´ edex, France e-mail: [email protected] Laboratory of Engineering Mathematics, Polytechnic School of Tunisia, University of Carthage. BP 743, La Marsa, Tunisia e-mail: [email protected] 4 LIMSI - CNRS, BP 133, 91403 Orsay C´ edex, France e-mail: [email protected]

Keywords: Stokes flow, Navier slip condition, Boundary-integral equation, Green tensor.

Abstract. A new method is proposed to examine, within the widely-employed creeping flow assumption, the migration of a solid particle immersed in a Newtonian liquid in the vicinity of a plane, motionless and impermeable wall where a Navier slip condition holds. The advocated technique resorts to a few relevant boundary-integral equations solely involving the particle surface and requires the determination of the so-called Green tensor associated with the boundary conditions of the relevant quasi-steady creeping flow problem. It then permits one to deal with the migration of a solid and arbitrarily-shaped particle either experiencing a prescribed rigid-body motion in a quiescent liquid or being freely-suspended in an ambient flow field. The encountered boundary-integral equations are numerically inverted using a boundary element collocation method and comparisons for a spherical particle against results previously obtained elsewhere by the bipolar coordinates method are given. Introduction The macroscopic behavior (such as effective viscosity) of a suspension made of a collection of solid particles is dictated by the imposed ambient flow and the occurring particle-particle interactions. For small enough particles and negligible inertial effects the widely-adopted framework is the creeping flow approximation for which a large literature is available. In pratice, suspensions are however bounded and thus particle-boundary interactions must also be taken into account. Such additional interactions deeply depend upon the boundary shape and nature. For applications, one mainly encounters three different types of motionless boundaries: (i) The usual solid boundary at which one requires the fluid velocity to vanish. (ii) The free surface at which one this time requires zero normal velocity and tangential stress components. (iii) The case of a slipping and impermeable boundary for which one requests zero normal velocity and another additional and so-called Navier [1] condition. This is the counterpart of the case of a particle with slipping surface moving near a solid plane wall where a no-slip condition holds (such circumstances have been recently handled in [2] for a spherical particle). Henceforth, we confine our attention to a plane boundary although other shapes have been also considered in the literature, at least for type (i). In contrast to types (i)-(ii), type (iii) actually received a little attention in the available papers (for instance, one can cite [3-5]). However, in the last decade the use of new materials suggested to revisit and further investigate type (iii), as achieved in [6]. Recently, [7-8] also investigated in such type (iii) the migration of a spherical and solid particle in the vicinity of a plane, impermeable and slipping wall appealing to the semi-analytical bipolar coordinates approach previously-employed in [4-5]. Actually, [7-8] either deal with a sphere experiencing a prescribed translation or rotation parallel with the wall or with the case of a sphere freely-suspended

275

276


in a prescribed ambient linear shear flow tangent to the wall. When carefully implemented the bipolar coordinates technique permits one to obtain very accurate results for a large range of sphere-wall gap h. For instance, for a sphere with radius a it is even possible to deal with small values of h/a of order (10−3 ) or less depending upon the plane boundary slip length λ (see (2) and (4)). Unfortunately, the bipolar coordinates technique is by essence solely able to cope with a spherical particle! Since nonspherical particles are of course encountered in practice, is this hightly desirable to propose another efficient approach valid for arbitrarily-shaped particles. Such a challenging issue is addressed in the present work. Governing equations and available method As sketched in Fig. 1, we consider a solid and arbitrarily-shaped particle P immersed above the x3 = 0 plane Σ boundary in a Newtonian liquid with uniform density ρ and viscosity µ > 0. We adopt Cartesian coordinates (O, x1 , x2 , x3 ) and assume that the particle has length scale a, smooth surface S with unit outward normal n and attached point O . In addition, the plane boundary Σ with unit normal n = e3 directed into the liquid is a motionless, impermeable and slipping surface characterized by its prescribed slip length λ > 0 introduced in [1].

x3 S

D ρ, µ

n O

P

Σ(x3 = 0)

O

x1 Figure 1. A solid and arbitrarily-shaped particle P near the x3 = 0 motionless, impermeable and slipping plane boundary Σ. In absence of particle the liquid has velocity ua and pressure pa in the x3 > 0 half space. The solid particle experiences a rigid-body migration described by its translational velocity U (the velocity of its O ) and angular velocity Ω. The disturbed flow about the particle has velocity ua + u and pressure pa + p in the liquid domain D. The previous velocity fields have scale V and all inertial effects are negligible, i. e. Re = ρV a/µ 1. Under these assumptions, the governing problems read µ∇2 ua = ∇pa and ∇.ua = 0 for x3 > 0, ∂ua .ei for i = 1, 2 on Σ, ua .e3 = 0 and ua .ei = λ ∂x3 µ∇2 u = ∇p and ∇.u = 0 in D, (u, p) → (0, 0) as |OM| → ∞, ∂u.ei for i = 1, 2 on Σ, u.e3 = 0 and u.ei = λ ∂x3 u = −ua + U + Ω ∧ O M on S

(1) (2) (3) (4) (5)

where (2) and (4) are the famous mixed-type Navier boundary conditions [1] and λ > 0 designates the given boundary slip length. The flow (u, p) has stress tensor σ and three Cases occur in practice: Case (1): Prescribed rigid-body migration (U, Ω) of the particle in a quiescent liquid (ua = 0 and pa = 0). The resulting hydrodynamic net force Fh and torque Γh (about O ) applied on the particle are Fh = σ.ndS = Fh (U, Ω), Γh = O M ∧ σ.ndS = Γh (U, Ω). (6) S

S


277

Case (2): Particle held fixed (i. e. U = Ω = 0) in the ambient flow (ua , pa ). The disturbed flow (u + ua , p + pa ) then exerts on the particle the net force Fa and torque Γa about O . Case (3): Particle freely suspended in the the ambient flow (ua , pa ). Neglecting the particle inertia, the requirement of a force-free and torque-free migration of the particle with unknown rigid-body motion (U, Ω) yields the equations Fh (U, Ω) + Fa = Γh (U, Ω) + Γa = 0

(7)

which permit one to determine (U, Ω) once previous Cases (1)-(2) are treated. It is possible to solve (1)-(5) for a spherical particle by employing the bipolar coordinates method (for a review, see [8]). This has been done in [4] for Case (1) when both U and Ω are normal to the plane boundary Σ and in Cases (1)-(3) in [5,7] for U and Ω parallel to Σ and for the ambient “linear shear” flow ua = (x3 + λ)e1 , pa = 0. For a non-spherical particle another treatment is needed and proposed below. Advocated boundary approach We present in this section a new procedure to solve the previous Cases (1)-(3) by solely inverting a few boundary-integral equations on the particle’s surface. Key surface quantities and linear system (i) (i) (i) For i = 1, 2, 3 and l = t, r let us introduce the Stokes flows (ul , pl ) with stress tensor σl obeying (3)-(4) and the boundary conditions (i)

ut = ei and ur(i) = ei ∧ O M on S. (i)

(8)

(i)

On the particle surface S we introduce the vector fl = σl .n which is thus the traction arising on the particle when it translates or rotates at the velocity ei if l = t or l = r, respectively. Adopting henceforth the usual tensor summation convention and setting U = Uj ej and Ω = Ωj ej , it immediately follows that relations (7) become t r Uj + Bij Ωj }ei Fh = {Atij Uj + Arij Ωj }ei , Γh = {Bij

(9)

with for l = t, r the definitions Alij =

(i)

S

l ei fl dS, Bij =

(i)

S

ei .[O M ∧ fl ]dS.

(10)

For two Stokes flows (u, p) with stress tensor σ and (u , p ) with stress tensor σ satisfying (3) the famous reciprocal identity [10-11] reads S

u.σ .ndS =

S

u .σ.ndS +

Σ

[u .σ.n − u.σ .n]dS.

(11)

If u and u moreover obey (4) it is straightforward to show that the last integration over the boundary Σ on the right-hand side of (11) vanishes. Selecting solution (u, p) to (3)-(5) for U = Ω = 0 and (i) u = ul then immediately shows that for Case (2)

Fa = −[

S

(i)

ua .ft dS]ei , Γa = −[

S

ua .fr(i) dS]ei .

(12)

Combining (7) and (12) yields in Case (3) the key linear 6-equation system for the Cartesian components of the translational velocity U and angular velocity Ω of the freely-suspended particle Atij Uj + Arij Ωj =

S

(i)

t r ua .ft dS, Bij Uj + Bij Ωj =

S

ua .fr(i) dS.

(13)

278

Eds: E L Albuquerque & M H Aliabadi (i)

(i)

It is possible to prove that (13) has a real-valued, symmetric (invoke (11) for ut and ur ) and negative definite 6 × 6 square matrix and therefore a unique solution (U, Ω). Clearly, it is sufficient to gain the (i) (i) surface tractions ft and fr in solving Cases (1)-(3). Relevant boundary-integral equations and Green tensor Any velocity field u solution to (3) admits in the entire liquid domain D the following integral representation [11] u(x).ej = −

S∪Σ

[ei .σ.n](y)Gij (y, x) − [u(y).ei ]Tijk (y, x)nk (y) dS(y), x in D.

(14)

In (14), G = Gij (y, x)ei ⊗ej denotes any Green tensor for the problem (3) but in the entire x3 > 0 half space with associated stress tensor T = Tijk (y, x)ei ⊗ ej ⊗ ek . More precisely, v(j) (y) = Gij (y, x)ei is the velocity produced at the point y by a concentrated point force with strength ej located at x. Moreover, remind that at that stage one does not specify any boundary condition on the surface Σ for the possible Green tensor G arising in (14). Here the trick consists in getting ride of the integrals over the unbounded surface Σ by adequately selecting a specific Green tensor in applying (14). One here takes the Green tensor Gc which obeys the boundary conditions (4) and designates by Tc the associated stress tensor. For u subject to (4) it is then possible to show that (14) then becomes u(x).ej = −

S∪Σ

[ei .σ.n](y)Gcij (y, x)dSy) +

S

c [u(y).ei ]Tijk (y, x)nk (y)dS(y), x in D.

(15)

(i)

Note that ul is a rigid-body velocity on the surface S. Accordingly [10], the double-layer integral on (i) the right-hand side of (15) vanishes and one therefore gets for the velocity field ul the simple integral representation (i)

ul (x).ej = −

(i)

S

fl (y).ek Gckj (y, x)dS(y), x in D.

(16)

It is possible to determine versus the slip length λ the key Green tensor Gc fulfilling the boundary conditions (4). For a sake of conciseness, the results are not displayed here and will be given at the oral presentation. At this stage, it is sufficient to remark that Gc = Gfree−space + R with Gfree−space the usual and weakly singular Oseen-Burgers free-space tensor and R a tensor regular in the entire x3 > 0 half-space. Exploiting this decomposition it follows that (16) is still valid as the point x tends onto the particle surface S. One thus ends up with the following Fredholm boundary-integral equation of the first kind (i) (i) (17) ul (x).ej = − fl (y).ek Gckj (y, x)dS(y), for x on S. S

In summary, one has to invert for i = 1, 2, 3 and l = t, r six boundary-integral equations (17) to gain (i) (i) the required surface tractions ft and fr on the particle boundary S. Advocated boundary element method and numerical comparisons This section briefly describes the adopted numerical strategy and also presents benchmark tests against the results available for a spherical particle in the literature [3,5]. Numerical implementation Each boundary-integral equations (17) is numerically inverted by using a collocation point method [12]. More precisely, we mesh the particle surface S by using 6 − node curved triangular boundary elements. For a N −node mesh on S the discretized counter-part of (17) is a linear system AX = Y with 3N × 3N dense and non-symmetric influence square matrix A. This linear system is then solved by Gaussian elimination. Numerical comparisons for a spherical particle Henceforth, the solid particle is a sphere with radius a and center O with OO = he3 and h > a. In


279

Case (1) there is no ambient flow field and the sphere rigid-body motion (U, Ω) is prescribed. Symmetries than show that it is sufficient to introduce the so-called friction coefficients f33 , c33 , f11 , c12 , f21 and c22 such that Fh = −6πµaf33 U and Γh = 0

Γh = −8πµa3 c33 Ω and Fh = 0

U ∧ e3 = Ω = 0,

for

Fh = −6πµaf11 U and Γh = −8πµa2 c12 U ∧ e3 3

(18)

Ω ∧ e3 = U = 0,

for 2

Γh = −8πµa c22 Ω and Fh = −6πµa f21 Ω ∧ e3

for

U ∧ e1 = Ω = 0,

for

Ω ∧ e2 = U = 0.

(19) (20) (21)

The coefficients f33 and c33 have been calculated in [4] using the bipolar coordinates method. Comparisons are made in Table 1 against these results for h/a = 1.543 versus the number N of collocation points put on the sphere’s surface S. Clearly, a nice agreement is found as N increases. Note that not Table 1: Computed friction coefficients f33 and c33 for h/a = 1.543 and λ = 2.5. f33 c33

N = 74 2.11545 0.99321

N = 242 2.12474 0.98947

N = 1058 2.12572 0.98934

[4] 2.11843 0.98835

[4]Corrected 2.12568 0.98835

accurate enough calculations reported in [4] have been corrected by the present authors by running a Code using the bipolar coordinates approach (the corrected results are also given in Table 1). Finally, we give comparisons against [5,7] for the friction coefficients f11 , c12 , f21 and c22 obtained when the solid sphere either translates or rotates parallel with the slipping wall Σ. Table 2: Computed friction coefficients f11 , c12 , f21 and c22 for h/a = 1.5 and λ = 1. N 74 242 1058 [5, 7]

f11 1.28628 1.28981 1.29020 1.29023

c12 0.026335 0.026200 0.026285 0.026293

f21 -0.034849 -0.035023 -0.035054 -0.035057

c22 1.03624 1.03560 1.03548 1.03548

Again, the computations perfectly match the results obtained by the bipolar coordinates procedure in [5,7]. Observe that using the reciprocal identity (17) yields the theoretical relation f21 = 4c12 /3. Conclusions A new approach has been proposed to investigate the migration of a solid and arbitrarily-shaped particle suspended in a liquid above a plane, motionless, impermeable and slipping surface. Owing to the determination of the Green tensor complying with the so-called Navier boundary conditions at the surface, it has been possible to reduce the task to the treatment of six boundary-integral equations solely involving the particle’s boundary. The numerical implementation appeals to a collocation technique and permits one to retrieve for a spherical particle the results proposed earlier by other authors. New results for a non-spherical particle will be reported and discussed at the oral presentation. References [1] C. L. M. H. Navier Mémoire sur les lois du mouvement des fluides. Mémoire de l’Académie Royale des Sciences de l’Institut de France, VI, 389-440 (1823).

280


[2] H. Luo and C. Pozrikidis Effect of slip on the motion of a spherical particle in infinite flow and near a plane wall. J. Eng. Math.,62(1), 1-21. (2008). [3] N.V Churaev, V.D Sobolev and A. N. Somov Slippage of liquids over lyophobic solid aurfaces. J. Colloid Int. Sci.,97, 574-581 (1984). [4] M. E. O’Neill and B. S. Bhatt Slow motion of a solid sphere in the presence of a naturally permeable surface. Q. J. Mech. Appl. Math.,44, 91-104 (1991). [5] A.M.J Davis, M.T. Kezirian and H. Brenner On the Stokes-Einstein model of surface diffusion along solid surfaces: Slip boundary conditions J.Colloid Interface Sci.,1065, 129-140 (1994). [6] J. Baudry, E. Charlaix, A. Tonck and D. Mazuyer Experimental evidence for a large slip effect at a nonwetting fluid-solid interface. Langmuir,17, 5232-5236 (2001). [7] H. Loussaief Ecoulement de suspensions avec condition de glissement sur la paroi. PHD Thesis. Laboratory of Engineering Mathematics, Polytechnic School of Tunisia, BP 743, La Marsa, Tunisia. (2008). [8] F. Feuillebois, H. Loussaief and L. Pasol Particles in Creeping Flow Near a Slip Wall. CP1186, Applications of Mathematics in Technical and Natural Sciences Editors M. D. Todorov and C.I. Christov. American Institute of Physics, 3-14 (2009). [9] L. Pasol, A. Sellier and F. Feuillebois Creeping flow around a solid sphere in the vicinity of a solid wall. In Theoretical Methods for Micro Scale Viscous Flows. Editors Fran¸cois Feuillebois and Antoine Sellier. Transworld Research Network, 105-126 (2009). [10] J. Happel, H. Brenner Low Reynolds number hydrodynamics, Martinus Nijhoff, (1973). [11] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992). [12] M. Bonnet Boundary Integral Equation Methods for Solids and Fluids, John Wiley & Sons Ltd, (1999).


281

Green tensor for a general non-isotropic slip condition A. Sellier1 and N. Ghalia2 LadHyx. Ecole polytechnique, 91128 Palaiseau Cédex, France e-mail: [email protected] Laboratory of Engineering Mathematics, Polytechnic School of Tunisia, University of Carthage. BP 743, La Marsa, Tunisia e-mail: [email protected] 1

2

Keywords: Stokes flow, Non-isotropic slip condition, Green tensor.

Abstract. The Green tensor for Stokes flows above a plane, motionless, impermeable and slipping wall where a non-isotropic extended slip condition holds is introduced and determined by using a decomposition involving the usual free-space Green tensor and analytically solving in the Fourier space a system of linear equations for one unknown regular contribution. Each Cartesian component of the Green tensor is then obtained by performing a two-dimensional Fourier inverse transform; a key step which requires an efficient and accurate numerical treatment. Introduction In predicting the behavior (rheology, effective viscosity) of a bounded suspension both particle-particle and particle-boundary interactions play a key role. However, for sufficiently-dilute suspensions these interactions reduce, at first order in the weak volume fraction, to the ones prevailing for one particle located near a boundary. Those particle-boundary interactions not only strongly depend upon the particle and boundary shapes and locations but also upon the boundary nature which dictates the conditions to impose at the boundary for the liquid flow about the particle. For negligible inertial effects, this flow, with pressure p and velocity u, is a creeping flow and we moreover confine our attention to a plane, motionless and impermeable boundary Σ with attached Cartesian coordinates (O, x1 , x2 , x3 ) such that Σ has equation x3 = 0. Setting ui = u.ei , we then require the following and possibly non-isotropic boundary conditions u3 = 0, u1 = b1

∂u1 ∂u2 , u2 = b2 on Σ(x3 = 0) ∂x3 ∂x3

(1)

with given and constant quantities b1 ≥ 0 and b2 ≥ 0. According to (1), no liquid flows across the plane boundary Σ which furthermore presents possibly different slipping behaviors in its tangential e1 and e2 directions. As positive and homogenous to a length b1 and b2 actually designate the boundary slip length in the e1 or e2 directions, respectively. The general boundary condition (1) might be useful in future to modelize the flow above non-isotropic surfaces with specific properties (for isotropic ones see, for instance, [1-2]). It actually encompasses the following basic cases: (i) Case 1: the usual no-slip condition u = 0 here retrieved for b1 = b2 = 0. (ii) Case 2: the case of a free surface with conditions of zero normal velocity and zero tangential stree here obtained for b1 = b2 = ∞. (iii) Case 3: the usual Navier [3] slip condition for which b1 = b2 = λ with λ the slip length. Note that [3-7] handled this case for a sphere located in the vicinity of the plane boundary Σ. Here, we are interested in the general setting (b1 , b2 ). As it is well known [8], it is possible to determine the net force and torque exerted on the particle and the liquid flow about the particle by resorting to the Green tensor approach and the associated efficient boundary-integral representation and equations. Such a powerful method, detailed in [9] for the b1 = b2 case, applies for arbitraryshaped particles but however rests on the determination of a Green tensor Gc which complies with the boundary conditions (1). This task has been solely achieved in [10] for the solid plane wall, i. e. for b1 = b2 = 0. The present work therefore addresses the challenging case of arbitrary quantities b1 ≥ 0 and b2 ≥ 0.

282


Governing equations and relevant boundary formulation This section introduces the problem for the Stokes flow about a solid particule and a suitable boundary approach. Creeping flow problem, Green tensors and resulting boundary formulation We consider, as illustrated in Fig. 1, a solid and arbitrarily-shaped particle P, with smooth surface S, embedded in a Newtonian liquid with viscosity µ > 0 above the motionless and impermeable x3 = 0 plane boundary Σ.

x3 D µ

n

S P

O

n = e3

Σ(x3 = 0) x1

Figure 1. A solid and arbitrarily-shaped particle near the x3 = 0 motionless, impermeable and slipping plane boundary Σ. Cartesian coordinates (O, x1 , x2 , x3 ) are employed and we introduce on the liquid boundary S ∪ Σ the unit outward normal n pointing into the fluid domain D. For the present work we confine the analysis to a prescribed rigid-body migration of the particle in a quiescent liquid (as seen in [9], this problem also permits one to gain the net force and torque exerted on the particle when held fixed in a prescribed ambient flow). The pressure p and velocity u of the creeping flow about the particle then obey µ∇2 u = ∇p and ∇.u = 0 in D, (u, p) → (0, 0) as |OM| → ∞, ∂u.e1 ∂u.e2 and u.e2 = b2 on Σ, u.e3 = 0, u.e1 = b1 ∂x3 ∂x3 u = urb on S

(2) (3) (4)

where urb is a prescribed rigid-body velocity and b1 ≥ 0 or b2 ≥ 0 denotes the slip length of the wall Σ in its e1 or e2 direction, respectively. Note that for b1 = b2 one retrieves the famous Navier [3] isotropic boundary conditions. Green tensors and specific Green tensor Let us consider for a so-called pole y in the y3 > 0 half-space and for k = 1, 2, 3 the Stokes flows with pressure p(k) , velocity v(k) and stress tensor σ(k) , such that µ∇2 v(k) = ∇p(k) − δ3d (x − y)ek , ∇.v(k) = 0 in the x3 > 0 half-space

(u(k) , p(k) ) → (0, 0) as |OM| → ∞,

(5) (6)

with δ3d (x − y) = δd (x1 − y1 )δd (x2 − y2 )δd (x3 − y3 ) and δd the Dirac pseudo-function. Note that neither the previous flows due to a concentrated point force of strength ek placed at the point y, nor the associated Green tensor G, with Cartesian components Gjk (x, y) = v(k) (x, y).ej , are unique since there is no prescribed boundary conditions on the x3 = 0 plane! For instance, the usual [11] free-space Oseen-Burgers tensor Gfree−space and associated pressure fields pf ree−space,(k) such that free−space (x, y) = Gjk

δjk 1 (x − y).ek [(x − y).ej ][(x − y).ek ] , pf ree−space,(k)(x, y) = + (7) 8πµ |x − y| |x − y|3 4π


283

with δ the Kronecker delta symbol, satisfy the problem (5)-(6). Clearly, Gfree−space is weakly singular as the observation point x approaches the source point y. In addition, any Green tensor G solution to (5)-(6) then reads G = Gfree−space + R with R a tensor regular in the entire x3 > 0 half-space. Henceforth, we select a specific Green tensor Gc such that each flow (v(k) , p(k) ) satisfies (5)-(6) and the specific boundary conditions (3) on the plane boundary, i. e. the relations v(k) .e3 = 0, v(k) .e1 = b1

∂v(k) .e1 ∂v(k) .e2 and v(k) .e2 = b2 ∂x3 ∂x3

on Σ.

(8)

Associated boundary formulation and boundary-integral equation The flow (u, p) governed by (2)-(4) has stress tensor σ and exerts on the particle surface S the traction f = σ.n. Adopting the usual tensor summation convention, the velocity u then admits in the entire liquid domain D the integral representation (see [9]) u(x).ej = −

S

f (y).ek Gckj (y, x)dS(y), x in D

(9)

which also yields the following Fredholm boundary-integral equation of the first kind for the unknown traction f (10) urb .ej = − f (y).ek Gckj (y, x)dS(y), x on S. S

The results (9)-(10) deserve the following remarks: (i) The integrals in (9)-(10) are single-layer terms because the potential double-layer contributions vanish for u being a rigid-body motion on the surface S. (ii) In (9)-(10) this is the specific Green tensor Gc (y, x) produced at point y by a source point located at y which arises. For two flows (u, p) and (u , p ) satisfying the boundary conditions (3) and having stress tensors σ and σ it is straightforward to check the nice relation Σ

u.σ .ndS =

Σ

u .σ.ndS.

(11)

As a basic consequence (mimick the treatment employed in [8] for the different boundary conditions u = u = 0 on Σ), the following key symmetric property holds Gckj (y, x) = Gcjk (x, y).

(12)

One can exploit (12) to rewrite (9)-(10). Therefore it is sufficient to determine the quantities Gcjk (x, y), i. e. the specific Green tensor Gc (x, y) with source at the point y and associated velocity v(k) = Gcjk (x, y)ej and pressure pc,(k) fields solution to (5)-(6) and (8) for k = 1, 2, 3. Determination of the specific Green tensor Gc This section shows how to obtain the required Green tensor Gc . Adopted decomposition and resulting system in the Fourier space We introduce, as shown in Fig. 2, the symmetric point y with respect to the plane, impermeable and slipping boundary Σ of the selected pole y in the y3 > 0 half-space. Accordingly, y3 = −y3 and yi = −yi f or i = 1, 2.

284


x •

x3 y

•

O

h = y3 > 0

Σ(x3 = 0) x1

• y

Figure 1. A pole y and its symmetric y with respect to the x3 = 0 plane Σ. Moreover, we adopt the following decompositions for k = 1, 2, 3 (k)

f ree−space f ree−space (x, y) − Gjk (x, y ) + wj (x, y) for j = 1, 2, 3 Gcjk (x, y) = Gjk

(13)

pc,(k) (x, y) = pf ree−space,(k)(x, y) − pf ree−space,(k)(x, y ) + s(k) (x, y).

(14)

In other words, we obtain the flow (Gcjk (x, y)ej , pc,(k) (x, y)) by putting a Stokeslet with strength ek at the pole y, a Stokeslet with strength −ek at the symmetric point y and a regular Stokes flow with ve(k) locity wj (x, y)ej and pressure s(k) (x, y) to be determined using (2) and the boundary conditions (3). Results in the Fourier space At that stage it is convenient to introduce the vectors R = x − y, R = x − y and the variable h = y3 such that R3 = x3 +h. For a function g(R1 , R2 , R3 ) we also define the following two-dimensional Fourier transform gˆ(λ1 , λ2 ; R3 ) =

∞ ∞

1 2π

−∞ −∞

g(R1 , R2 , R3 )ei(λ1 R1 +λ2 R2 ) dR1 dR2

(15)

with complex i such that i2 = −1. Setting ξ = {λ21 + λ22 }1/2 and enforcing the equations and boundary conditions (2), one then gets h [2µB (k) ]e−ξ(R3 −h) , 4πµ h λ1 δ1j + λ2 δ2j (k) [B + i( + δj3 )(R3 − h)B (k) ]e−ξ(R3 −h) , = 4πµ j ξ

sˆ(k) = (k)

w ˆj

B (k) =

(k) iλ1 B1

+

(k) iλ2 B2

+

(k) ξB3

(16) (17) (18)

(k)

with unknown functions B (k) and Bj depending upon (λ1 , λ2 ; R3 ). Those fucntions are determined using the relation (18) and expressing the boundary conditions (3) which result in the additional equalities (k)

B3

= i(λ1 δ1k + λ2 δ1k )g1 (ξ),

Bj

− iδk3 (λ1 δ1j + λ2 δ2j )g1 (ξ) =

(k)

i(λ1 δ1j + λ2 δ2j ) (k) (k) B − ξBj + aj3 g3 (ξ) + aj5 g5 (ξ) − λj λk aj1 g1 (ξ)] (j = 1, 2), ξ e−hξ e−hξ 1 + hξ −hξ ]e , , g3 (ξ) = , g5 (ξ) = [ g1 (ξ) = ξ h 3h3 bj [

aj1 = (1 − δj3 )(1 − δk3 ), aj3 = δjk − 2δj3 δk3 + δjk (1 − δj3 )(1 − δk3 ), aj5 = 3h2 δj3 δk3 .

(19)

(20) (21) (22)

Advances in Boundary Element and Meshless Techniques XII (k)

(k)

285

(k)

The relation (19) gives the fucntion B3 whereas the functions B1 , B2 and B (k) are determined by inverting the 3-equation linear system made of (18) and (21). Using the Maple Software, one ends up with (3)

iλ1 g1 (ξ)[b2 ξ 2 + ξ + (b2 − b1 )λ22 ] , 2ξ 3 b1 b2 + (b2 + 2b1 )ξ 2 + ξ + (b2 − b1 )λ22 2 2 iλ2 g1 (ξ)[(2b1 − b2 )ξ + ξ + (b2 − b1 )λ2 ] = 3 , 2ξ b1 b2 + (b2 + 2b1 )ξ 2 + ξ + (b2 − b1 )λ22 g1 (ξ)ξ 2 [b2 ξ 2 + ξ + (b1 − b2 )λ22 ] =− 3 2ξ b1 b2 + (b2 + 2b1 )ξ 2 + ξ + (b2 − b1 )λ22

B1 = (3)

B2

B (3)

(23) (24) (25)

for k = 3 and with (1)

2b1 [b2 g1 (ξ)ξ 4 + g1 (ξ)ξ 3 − b2 ξ 2 (g3 (ξ) + g1 (ξ)λ22 ) − g1 (ξ)ξλ22 − b2 g3 (ξ)λ22 ] , 2b1 b2 ξ 3 + (b2 + 2b1 )ξ 2 + ξ + (b2 − b1 )λ22 2 2b2 λ1 λ2 [b1 g1 (ξ)ξ + g1 (ξ)ξ + b1 g3 (ξ)] =− , 2b1 b2 ξ 3 + (b2 + 2b1 )ξ 2 + ξ + (b2 − b1 )λ22 iλ1 ξ[b2 g1 (ξ)ξ 2 + (g1 (ξ) + 2b1 b2 g3 (ξ))ξ + 2b1 g3 (ξ) + (b1 − b2 )g1 (ξ)λ22 ] = 2b1 b2 ξ 3 + (b2 + 2b1 )ξ 2 + ξ + (b2 − b1 )λ22

B1 = − (1)

B2

B (1)

(2)

(26) (27) (28)

(2)

for k = 1. Of course, the functions B1 , B2 and B (2) are obtained by replacing upperscripts (1) with (2) and switching subscripts 1 and 2 in (26)-(28). Solution by inverse Fourier transform (k) The required functions wˆj are finally obtained by combining (17)-(18) with (23)-(28). Once this (k)

is done applying the inverse transform finally gives wj (x, y). For b1 = b2 simple results occur with inverse transforms reducing to one-dimensional integrations. By contrast, as soon as b1 = b2 the calculation of the inverse tranform is more tricky and results in numerical evaluation of two-dimensional integrals. Such a task, not displayed here for a sake of conciseness, will be detailed at the oral presentation. Conclusions A new and non-isotropic boundary condition on a plane, impermeable and slipping surface is proposed and can be seen as an extension of the famous Navier condition [3]. The Green tensor which complies with this boundary condition is determined by using a decomposition involving two stokeslets and an extra regular term whose Fourier transform is analytically obtained using the Maple Software. The final computation of the Green tensor is then based on the accurate and efficient numerical evaluation of two-dimensional integrations in the Fourier space. References [1] N.V Churaev, V.D Sobolev and A. N. Somov Slippage of liquids over lyophobic solid aurfaces. J. Colloid Int. Sci.,97, 574-581 (1984). [2] J. Baudry, E. Charlaix, A. Tonck and D. Mazuyer Experimental evidence for a large slip effect at a nonwetting fluid-solid interface. Langmuir,17, 5232-5236 (2001). [3] C. L. M. H. Navier Mémoire sur les lois du mouvement des fluides. Mémoire de l’Académie Royale des Sciences de l’Institut de France, VI, 389-440 (1823).

286


[4] M. E. O’Neill and B. S. Bhatt Slow motion of a solid sphere in the presence of a naturally permeable surface. Q. J. Mech. Appl. Math.,44, 91-104 (1991). [5] A.M.J Davis, M.T. Kezirian and H. Brenner On the Stokes-Einstein model of surface diffusion along solid surfaces: Slip boundary conditions J.Colloid Interface Sci.,1065, 129-140 (1994). [6] H. Loussaief Ecoulement de suspensions avec condition de glissement sur la paroi. PHD Thesis. Laboratory of Engineering Mathematics, Polytechnic School of Tunisia, BP 743, La Marsa, Tunisia. (2008). [7] F. Feuillebois, H. Loussaief and L. Pasol Particles in Creeping Flow Near a Slip Wall. CP1186, Applications of Mathematics in Technical and Natural Sciences Editors M. D. Todorov and C.I. Christov. American Institute of Physics, 3-14 (2009). [8] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992). [9] N. Ghalia, A. Sellier, L. Elsami and F. Feuillebois Slow viscous migration of a solid particle near a plane wall with a slip condition International Conference on Boundary Element and Meshless Techniques, Brasilia, 13-15 July 2011. [10] J. R. Blake A note on the image system for a Stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc., 70, 303-310 (1971). [11] J. Happel, H. Brenner Low Reynolds number hydrodynamics, Martinus Nijhoff, (1973).

Application of the OMLS interpolation to evaluate volume integrals arising in static and time–dependent elastoplastic analysis via D–BEM K. I. da Silva1, J. C. F. Telles2 and F. C. de Araújo3 1

Federal University of São João Del–Rei, Campus Alto Paraopeba, C.P. 131, 36420–000, Ouro Branco–MG, Brazil, [email protected]

2

Federal University of Rio de Janeiro, COPPE/UFRJ, Programa de Engenharia Civil, C.P. 68506, 21945–970, Rio de Janeiro–RJ, Brazil, [email protected] 3

Federal University of Ouro Preto, Department of Civil Engineering, 35400–000, Ouro Preto–MG, Brazil, [email protected]

Keywords: Boundary Element Method, elastoplasticity, orthogonal moving least squares interpolation, static and transient analysis.

Abstract. In this work the Boundary Element Method is applied to solve 2D static and time–dependent elastoplastic problems. Since the boundary–element formulation adopted for solving transient problems is based on static fundamental solutions (D–BEM), domain integrals due to inertial and damping forces have to be explicitly evaluated. Hence, in elastoplastic BE analysis, domain contributions have to be calculated not only for considering elastoplastic stresses in yielded zones but also for taking into account dynamic effects. In order to avoid using domain integration cells for evaluating these integrals, a cell free strategy based on the OMLS (Orthogonal Moving Least Squares) interpolation, typically adopted in meshless methods, is here implemented. In this approach the definition of points to compute the interpolated value of a function at a given location only depends on their relative distance, without need to define any element/cell connectivity. In addition, for time–domain analyses, Houbolt and Newmark integration schemes have been used, and a problem is solved to validate the strategy. BE responses obtained by using integration cells are confronted with those calculated with the present technique. Introduction While applying domain–based BEM formulations (D–BEM) to analyze inelastic transient problems, cell meshes are in general required to evaluate both inelastic and inertial terms (domain integrals) [1, 2, 3]. In elastoplastic analysis, domain discretization is naturally needed to take into account the residual stresses in yielded zones of the solid, and especially concerning time–dependent problems, this apparent disadvantage (domain discretization) is in fact compensated by the use of simpler elastostatic fundamental solutions. In this work, one aims to apply the OMLS (Orthogonal Moving Least Squares) strategy, typically used to develop meshless methods, to approximate the domain integrals arising from plastic and inertial terms in D–BEM formulations. In other words, a way to avoid dealing with domain integrals (supposed to be disadvantageous or more awkward) is proposed. The OMLS strategy is derived from the MLS (Moving Least Squares) method, according to which a function value to be reconstructed at a certain point of its definition domain only depends on the distance from this point to the points contributing to that particular function value. However, unlike the MLS method, in the OMLS approach orthogonal weighting functions are used as basis functions, what causes the resulting approximation function to have a delta–distribution characteristic, and allows for working with real values of the physical quantities to be interpolated. Furthermore, the system of equations obtained for determining the OMLS interpolation parameters is well–conditioned, and compared to the MLS method, less coefficients are employed in the process [4, 5, 6]. By applying the OMLS strategy, plastic and inertial effects are included into the regular algebraic BE system of equations with no need for integration cells. In the particular case of time–dependent problems, the Newmark and Houbolt time integration methods [7] are employed to obtain the time–domain responses. To illustrate the performance of the technique proposed, transient elastoplastic responses over a circular cavity under Heaviside loading is carried out using both OMLS–based and cell–based techniques.

288


The OMLS technique for elastoplastic problems The boundary integral equations for expressing the time–dependent displacement and stress components in elastoplastic problems via the D–BEM are given (by neglecting body forces other than inertial ones) by cij ξ ui ξ, t

³ uij ξ, x p j x, t d * x ³ pij ξ, x u j x, t d * x

*

*

U ³ uij ξ, x u j x, t d : x ³ H jki ξ, x V jkp x, t d : x :

,

(1)

,

(2)

:

V ij ξ, t ³ uijk ξ, x pk x, t d * x ³ pijk ξ, x uk x, t d * x *

*

U ³ uijk ξ, x uk x, t d : x ³ H ijkl ξ, x V klp x d : x gij V klp :

:

which obviously simplifies for static elastoplastic problems by excluding the integrals associated to the inertial–force terms [1, 2]. In general, the numerical solution of boundary integral equations (1) and (2) is carried out by interpolating the field variables at hand by means of boundary elements and domain cells so as to convert these equations into an algebraic system of equations. Unlike other approaches, in this work, the OMLS technique [4–6] is employed to evaluate the domain integrals. In the OMLS approach, the approximation of a function u x , x : (its definition domain), is based on its OMLS definition domain, : x , which is defined by the set of points xi , i 1, 2,

, n , wherein n is the ,n

number of points in : that contribute to the interpolation of u x . The OMLS approximation is expressed by uh x Φ x u

,

(3)

where u is a vector containing the real quantities to be interpolated, and Φ x pT x A1 x B x is a vector containing the OMLS interpolation functions, with

A x

PT W x P , B x

PT W x , P

pT x1 ½ ° T ° °p x 2 ° ¾ and W ® ° ° °pT x ° n ¯ ¿

ª w x x1 « « 0 « « « « 0 ¬

0

w x x2

0

º » » 0 » . » » w x xn » ¼ nu n 0

(4)

In the expressions above, wI x w x x I is the weighting function associated with the x I point, which has the following characteristics: its support, S ª¬ wI x º¼ , is radial with radius rI , and 0 wI x d 1 for all x S ª¬ wI x º¼ . In this work, the Gaussian function §r · §¨ di ·¸ ¨ i ¸ ° e ¨© ci ¸¹ e ¨© ci ¸¹ ° , 0 d di d ri 2 §r · ® ¨ i ¸ ¨c ¸ ° 1 e © i ¹ ° ¯0, di t ri 2

w x xi

2

(5)


289

is employed as weighting function (Fig. 1). The other function in (4), p x , is the interpolation basis, which is composed by a set of polynomials mutually orthogonal with regard to the weighting functions, derived

a) ci

ri

b) ci

0.5 ri

c) ci

0.3 ri

Fig. 1 Gaussian weighting function from a monomial basis p x by applying the Gram–Schmidt orthogonalization process [8]. For 2D problems, the following bases are usually considered:

1, x , x 1, x , x , x , x x , x

pT x p

T

1

x

1

2

2 1

2

1

2 2

2

(linear basis),

(6a)

(quadratic basis).

(6b)

Using the OMLS interpolation, equations (1) and (2) are expressed in discretized form as ne § nnoel ne § nnoel · · cij ξ ui ξ, t ¦ ¨ ³ pij ª¬ξ, x K º¼ ¦ hq K d * ª¬ x K º¼ ¸ u (jql ) ¦ ¨ ³ uij ª¬ξ, x K º¼ ¦ hq K d * ª¬ x K º¼ ¸ p (jql ) l 1 * q 1 l 1 * q 1 © ¹ © ¹ NT § n n · NT § ·

p U ¦ ¨ ³ uij ξ, x ¦ ) I x u j x I , t d :d x ¸ ¦ ¨ ³ H jki ξ, x ¦ ) I x V jk x I , t d :d x ¸ d 1 : I 1 I 1 © ¹ d 1© : ¹ l

l

d

§

nnoel

l 1 * l

q 1

d

ne § nnoel · (l )

¦ ¨ ³ uijk ª¬ξ, x K º¼ ¦ hq K d * ª¬ x K º¼ ¸ pkq q 1 © ¹ © ¹ NT § n n · NT § ·

p U ¦ ¨ ³ uijk ξ, x ¦ ) I x uk x I , t d :d x ¸ ¦ ¨ ³ H ijkl ξ, x ¦ ) I x V kl x I , t d :d x ¸ gij V klp d 1 : I 1 I 1 © ¹ d 1© : ¹ ne

·

(l ) ª¬ξ, x K º¼ ¦ hq K d * ª¬ x K º¼ ¸ ukq V ij ξ, t ¦ ¨ ³ pijk

d

(7)

l 1 * l

d

(8)

where ne is the total number of boundary elements of the model, nnoel is the number of element nodes, and NT is the number of triangular subdomains used for effecting the domain integrations. By writing then Eqs. (7) and (8) for all boundary nodes and inner points, the systems of algebraic equations ª Hbb « db ¬H

ub ( m 1) ° ½ 0º ° » ® d ( m 1) ¾ I¼ ¯ °u ° ¿

ª σb ( m1) º « d ( m1) » ¬σ ¼

ªGbb º b ( m 1) ªMbb « db « db » p ¬G ¼ ¬M

^

`

ub ( m 1) ° ½ ªQbb Mbd º ° « db dd » ® d ( m 1) ¾ M ¼¯ °u ° ¬Q ¿

ªGcbb º b ( m 1) ª Hcbb º ª 0 « db » ub ( m 1) « db « db » p c c G H ¬Mc ¬ ¼ ¬ ¼

^

`

^

`

pb ( m 1) ½ Qbd º °σ ° dd » ® pd ( m 1) ¾ Q ¼¯ °σ ° ¿

b ( m 1)) bb ½ 0 º °u ° ªQc ® d ( m 1)) ¾ « db Mcdd ¼» ¯ c Q °u ° ¿ ¬

,

pb ( m 1) ½ º 0 °σ ° » ® pd ( m 1) ¾ Qcdd Ecdd ¼ ¯ °σ ° ¿

(9)

(10)

are obtained [9, 10], from which the elastoplastic dynamic response for the model at the time point tm 1 is determined. In these equations, superscripts b e d stand for boundary and domain respectively. Time–marching scheme To integrate the Eqs. (9) and (10), the Newmark and Houbolt methods [8], based on the acceleration interpolations given by

290


4 't 2 1 't 2

u ( m1)

u ( m1)

4 ª u ( m1)) u ( m) º u ( m) u ( m) ¬ ¼ 't

(Newmark), and

(11)

ª 2 u ( m1) 5 u ( m) 4 u ( m1) u ( m2) º ¬ ¼

(Houbolt),

(12)

are applied. Substituting Eqs. (11) or (12) in Eq. (9) and (10), and rearranging the terms, the effective systems of Eqs. H Hu( m1) Gp( m1) σ

( m1)

( m1)

Gc p

h( m ) Q Qσ p ( m1) ( m1)

Hc u

hc

( m)

Q σ

p ( m1)

,

(13)

,

(14)

are obtained, where the effective matrices in Eq. (13) are given by H

't 2 H D M ,

with D

h( m ) h

( m)

G

't 2 G ,

Q

4 for the Newmark method, and D

M ª¬4 u( m) 4 't u( m) 't 2 u( m) º¼ , M ª¬5 u ( m) 4 u ( m1) u ( m2) º¼ ,

't 2 Q ,

(15)

2 for the Houbolt one, and the vector h( m ) is

(Newmark),

(16)

(Houbolt).

(17)

The terms Hc Hc , G Gcc and hc( m ) in Eq. (14) are analogous to those in relations (15)–(17), except that they are obtained for the stress–based fundamental kernels. To solve the dynamic elastoplastic problem, the explicit incremental technique presented in [1] is adopted. It is based on the following expressions: 'σe ( m1) 'n( m1)

Sσ p ( m1) 'n( m1) n( m1) n( m)

,

(18)

,

(19)

The time steps adopted in the analyses are chosen by making the relationship Et cL 't l not so quite different from the unit, wherein l is the minimal element length of the model, and cL is the longitudinal wave propagation velocity in the solid.

Application To show how the technique proposed works, the circular cavity under a Heaviside load in Fig. 2 (plane strain state) is analyzed. The following problem data are considered: E 94.6769 ksi , Q 0.2308 , U 3.5 slug ft3 , R 3.048 ft and p 1 ksi .

Fig. 2. Circular cavity


291

Because of the symmetry only one quarter of the cavity is discretized, and the analysis was carried out by considering a finite domain. Thus, the fictitious boundary has been placed far enough (Fig. 3) so as to avoid spurious waves associated with wave reflections on it. The adopted models are presented in Fig. 3, wherein ne is the number of boundary elements, nc is the number of cells, and np is the number of inner points.

a) With cells ( ne

52 e nc

b) Without cells ( ne

244 )

52 e np

209 )

Fig. 3 – Discretization adopted 0.5

1.5

-0.5 -1.0

Point A - Cells Point B - Cells Point C - Cells Point A - OMLS Point B - OMLS Point C - OMLS

1.0

Point A - Cells Point B - Cells Point C - Cells Point A - OMLS Point B - OMLS Point C - OMLS

VT

VR

0.0

0.5 0.0

-1.5

-0.5

0.0 2.0 4.0 6.0 8.0 10.0 12.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0

cL t / R

cL t / R

Fig. 4 – Radial ( V R ) and circumferential ( V T ) stresses (elastic transient analysis) In Fig. 4, the radial ( V R ) and circumferential ( V T ) stresses at points A ( R,0) , B (2.02R ,0) and C (3.45R,0) are plotted as a function of the dimensionless time for a transient elastic analysis while in Fig. 5 the corresponding results for an elastoplastic analysis are shown. The Mohr–Coulomb fracture criterion (for brittle materials) with cohesion cc 0.9 and angle of friction I c 30º is considered for the elastoplastic analysis, and the motion equations are integrated with the Houbolt method with a time step of 't 0.1s . In the graphs in Figs. 4–5, the stresses obtained using cell integrations and the cell–free OMLS–based strategy are confronted, and show excellent agreement with one another. Notice that responses for the elastic case are kept in the graphs in Fig. 5 to show the possible differences between elastic and elastoplastic analyses. Point B

Point C

0.50

0.50

1.0

0.25

0.25

0.5

0.00

0.0

0.00

V

V

V

Point A 1.5

-0.25

-0.25

-0.5

-0.50

-0.50

-1.0

-0.75

-0.75

-1.5

-1.00

-1.00

0.0 2.0 4.0 6.0 8.0 10.0 12.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0

cL t / R

cL t / R

cL t / R

VR elastic VT elastoplastic - Cells

VT elastic

VR elastoplastic - OMLS

VR elastoplastic - Cells

VT elastoplastic - OMLS

Fig. 5 – Radial ( V R ) and circumferential ( V T ) stresses (elastoplastic transient analysis)

292


Conclusions In this work, the OMLS interpolation technique has been employed to approximate residual plastic stresses and inertial forces present in domain–based boundary–element (D–BEM) formulations for transient inelastic problems. In this technique the function values at any point are reconstructed from isolated points spread inside its definition domain. The excellent agreement with regard to the comparison of the numerical responses obtained with both integration cells and the cell–free OMLS technique shows that the latter might be regarded as an interesting alternative to evaluate BEM domain integrals. Of course, an advantage of OMLS technique is to make the model generation considerably easier, with no need of furnishing cell connectivity, but just the coordinates of the domain points. However, depending on the number of points considered as well as on their position inside the domain, unsatisfactory results may be obtained, so that further studies to establish a criterion for determining the OMLS point data are still needed. 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]

J.C.F. Telles, The Boundary Element Method Applied to Inelastic Problems: Lecture Notes in Engineering, Springer–Verlag (1983).

[2]

J.C.F. Telles and J.A.M. Carrer, Engineering Analysis with Boundary Elements, 14 (1), 65–74 (1994).

[3]

D. Soares Jr., J.A.M. Carrer and W.J. Mansur, Engineering Analysis with Boundary Elements, 29 (8), 761–774 (2005).

[4]

S.N. Atluri and T. Zhu, 2000; Computational Mechanics, 25 (2–3), 169–179 (2000).

[5]

S.N. Atluri and S. Shen, 2002, CMES – Computer Modelling in Engineering and Sciences, 3 (1), 11– 51 (2002).

[6]

K.M. Liew, Y. Cheng and S. Kitipornchai, International Journal of Numerical Methods in Engineering, 65 (8), 1310–1332 (2006).

[7]

K.J. Bathe, Finite Element Procedures, Prentice Hall, Englewood Cliffs, New Jersey (1996).

[8]

D.L. Kreider, R.G. Kuller, D.R. Ostberg and F.W. Perkins, An Introduction to Linear Analysis, v. 2, Addison–Wesley Publishing Company Inc. (1966).

[9]

J.C.F. Telles and J.A.M. Carrer, Computers & Structures, 45 (4), 707–713 (1992).

[10]

J.A.M. Carrer and W.J. Mansur, Computational Mechanics, 34 (5), 387–399 (2004).


A Dual Reciprocity Boundary Element Formulation for Transient Dynamic Analysis of Shallow Shells Jairo F. Useche Department of Mechanical Engineering Universidad Tecnológica de Bolivar, Cartagena, Colombia [email protected] Abstract In this work, the transient dynamic analysis of shear deformable shallow shells, using a dual reciprocity boundary element method formulation, based on elastostatic fundamental solutions, is presented. The formulation is obtained by coupling boundary element formulation of shear deformable plate and two-dimensional plane stress elasticity, considering flexural and rotary inertia effects. Domain integrals related with inertial terms were treated using the Dual Reciprocity Boundary Element Method. In order to obtain the time evolution of the solution, the Houbolt time-integration method was used. Numerical examples are presented and results were compared with those obtained using finite element models. Key words: Boundary element method, shallow shells, dual reciprocity boundary element method, dynamic analysis.

1

Introduction

The dynamic analysis of shallow shells problems appears on civil, mechanical, aerospatial, naval and electronics applications. The complexity involved in the dynamic response of shells, turns this problem in a challenging ones from a mathematical point of view. For the analysis of problems involving simply geometry, boundary and initial conditions, analytical approaches based on energy and superposition methods, has been proposed [3], [2]. However, in the general case, dynamic analysis of shells involving complex geometries, loading and boundary conditions, numerical methods represents the only way to obtain approximate solutions. Numerical methods such as the the finite element method (FEM) and the finite difference methods (FDM) are well developed. However, the use of these methods, requires a refined discretization of the shells since the lengths of the elements should be proportional to the size of the wavelength. This means an increase in the number of degrees of freedom, which requires a significant computational effort. Alternatively, the BEM requires only boundary discretization, leading to considerable lower computational effort than FEM or FDM models in dynamic plates bending analysis. Thus, nowdays BEM emerging as an accurate and efficient numerical method for plate dynamic analysis [4], [5], [6], [7]. BEM solutions to dynamic plate analysis problems are usually obtained by using three basic approaches: BEM formulations based on elastodynamic fundamental solution of the problem, BEM formulations based on Laplace or Fourier transformations of the elastodynamic fundamental solution of the problem into frequency domain and the direct BEM formulations based on static fundamental solutions. The direct BEM formulations based on elastostatic fundamental solutions creates domain integrals due to the presence of the inertia terms in addition to the boundary ones. Thus, domain discretization is required in addition to the boundary one [7]. In spite of this interior discretization, the simplicity of the real valuated static fundamental solution as compared to the complicated complex valued elastodynamic one results in a more efficient scheme. In this approach, in order to threat domain integrals, the Cell Integration Method (CIM), the Dual Reciprocity Method (DRM) and the Radial Integration Method (RIM), have been used. In this work, the transient analysis of shear deformable elastic shalow shells using a direct BEM based on a direct time-domain formulation using the elastostatic fundamental solution of the problem, is presented.

1

293

294


Shells were modeled by coupling the boundary element formulation for shear deformable plates based on the Reissner plate theory and two-dimensional plane stress elasticity, as presented in [1]. Effects of shear deformation and rotatory inertia are included in the formulation. The elastostatic fundamental solution and plane elasticity, were used. The Dual Reciprocity Boundary Element Method for the treatment of domain integrals involving distributed domain applied loads and those related with inertial mass forces, was used. Houbolt time-integration method for the time-integration of the solution, was used. A Numerical example is presented and results were compared with those obtained using finite element models.

2

Shallow shell dynamic equations

Consider a shallow shell of uniform thickness h andmass density ρ, occupying the area Ω, in the x1 x2 plane, bounded by the contour Γ = Γw Γq with Γ = Γw Γq ≡ 0. The dynamic bending response for the shallow shell was modeling using the classical Reissner plate theory. In this way, the equations of motion for an infinitesimal plate element under a distributed transverse loading, q3 and distributed in-plane loads, qα , are given by [10]: D ∂w2 ∂w3 ∂ ∂w1 + − − Cw1 − C (1 + ν) 2 ∂x2 ∂x2 ∂x1 ∂x1 ∂w2 ∂w3 D ∂ ∂w1 + D∇2 w2 − Cw2 − C − (1 + ν) 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂w1 ∂w2 ∂ 2 u1 2 C∇ w3 + C +C + q3 − B(κ11 + νκ22 ) ∂x1 ∂x2 ∂x1 2 ∂ u2 2 2 −B(νκ11 + κ22 ) − B(κ11 + κ22 + 2νκ11 κ22 )w3 ∂x2 B ∂u2 ∂w3 ∂u1 ∂ B∇2 u1 + (1 + ν) + − + q1 + B(κ11 + νκ22 ) 2 ∂x2 ∂x2 ∂x1 ∂x1 ∂u2 ∂w3 B ∂ ∂u1 2 + B∇ u2 + q2 + B(νκ11 + κ22 ) − (1 + ν) 2 ∂x1 ∂x2 ∂x1 ∂x2 D∇2 w1 +

= =

= = =

∂ 2 φ1 ∂ 2 u1 + I1 2 2 ∂t ∂t ∂ 2 φ2 ∂ 2 u2 I2 2 + I1 2 ∂t ∂t I2

∂ 2 w3 ∂t2 ∂ 2 w1 ∂ 2 φ1 I0 + I1 2 ∂t2 ∂t ∂ 2 w2 ∂ 2 φ2 I0 + I1 2 ∂t2 ∂t I1

(1)

where, B = Eh/(1 − ν 2 ), D = Eh3 /12(1 − ν 2 ) and C = D(1 − ν)λ2 are the tension, bending and shear stiffness of the plate, respectively; λ2 = 10/h is called the shear factor, ν is the Poisson’s ratio and καβ represents curvature tensor of the shell surface; I1 = I2 = 1/12ρh3 δαβ and I0 = ρh. Indicial notation is used throughout this work. Greek indices vary from 1 to 2 and Latin indices takes values from 1 to 3. Einstein’s summation convention is used unless otherwise indicated. In these equations, wα represents rotations about x1 and x2 axis, respectively and w3 transverse deflection; w ¨α denote angular accelerations about x1 and x2 axis, respectively and w ¨3 represent transverse acceleration. Simillary, uα represents displacements and u ¨α denote linear accelerations along x1 and x2 axis, respectively.

3

Boundary integral formulation for shallow shells

The derivation of the intergral formulation for equations (1) is based on application of the boundary element method to the Reissner plate theory as presented in [11], were the integral representations related to the governing equations for bending and transverse shear stress resultants are derived by using the weighted residual method, and making use of the Green’s identity. In this work, only uniform distributed constant pressure is considered as external load acting on the shell. Thus, the integral formulation for equations (1) are defined by: cij wj (x ) + Pik (x , x)wj (x)dΓ = Wij (x , x)pj (x)dΓ Γ Γ 1−ν 2ν − καβ B uγ (x)nγ δαβ Wi3 (x , x)dΓ uα (x)nβ + uβ (x)nα + 2 1−ν Γ 1−ν 2ν καβ B + uγ (X)Wi3,γ (x , X)δαβ dΩ uα (X)Wi3,β (x , X) + uβ (X)Wi3,α (x , X) + 2 1−ν Ω καβ B[(1 − ν)καβ + νδαβ κγγ ]w3 (X)Wi3 (x , X)dΩ − Ω

2

Advances in Boundary Element and Meshless Techniques XII + Ω

Wi3 (x , X)q3 (X)dΩ +

Ω

Wij (x , X)Iij w ¨j (X)dΩ

295

(2)

and, cθα (x )uα (x ) +

Γ

Tθα (x , x)uα (x)dΓ +

Uθα,β (X , X)B[καβ (1 − ν) + νδαβ κγγ ]w3 (X)dΩ Uθα (x , x)tα (x)dΓ + Uθα (X , X)I0 u ¨α dΩ =

Ω

Γ

(3)

Ω

In these equations, Wik and Pik are the fundamental solutions for shear deformable plates; Tθα and Uθα are the fundamental solutions for plane stress; Iij tensor is defined as: Iαβ = 1/12ρh3 δαβ and I33 = I0 = ρh; nα is the normal vector to the boundary at field point.

4

Boundary element formulation

4.1

Transformation of domain integrals

Domain integrals involving curvature terms in equations (2) and (3), are numerically evaluated using the Cell Integration Method as presented in [1]: 1−ν 2ν uα (X)Wi3,β (x , X) + uβ (X)Wi3,α (x , X) + καβ B uγ (X)Wi3,γ (x , X)δαβ dΩ = 2 1 − ν Ω Nc Nc 1 − ν k +1 +1 1 − ν k +1 +1 καβ B Wi3,β (x , X)Jk dξdη + καβ B Wi3,α (x , X)Jk dξdη uα uβ 2 2 −1 −1 −1 −1 k=1 k=1 +1 +1 Nc + καβ Bνukγ Wi3,α (x , X)Jk dξdη −1

k=1

Ω Nc

καβ B[(1 − ν)καβ + νδαβ κγγ ]w3 (X)Wi3 (x , X)dΩ =

καβ B[(1 − ν)καβ + νδαβ κγγ w3k

Ω Nc

+1

−1

k=1

=

−1

+1 −1

Wi3,β (x , X)Jk dξdη

Uθα,β (X , X)B[καβ (1 − ν) + νδαβ κγγ ]w3 (X)dΩ

B[καβ (1 − ν) + νδαβ κγγ ]w3k

k=1

+1 −1

+1 −1

Wi3,α (x , X)Jk dξdη

(4)

where Ne and Nc are number of boundary elements and internal cells respectively and Jn is the jacobian of transformation for boundary elements. Otherwise, by applying the divergence theorem, the domain integral in equation (2) related with the q3 term can be transferred to boundary integral to give [12]: Ω

Wi3,α (x , X)q3 (X)dΩ = q3

Ne k=1

+1 −1

Vi,α (x , X)nα Jn dξ

(5)

In this work, the Dual Reciprocity Method was used to transform domain integrals related to inertial terms into boundary integrals [8]. In this way, last integrals in equations (2) and (3) can be written as: Ω

Wij (x , X)Iij w ¨j (X)dΩ =

N DR

ˆ m − Wik (x , x)Pˆ m dΓ + Pik (x , x)W ˆ m dΓ β¨lm (t) cik W lk lk lk Γ

m=1

3

Γ

(6)

296


and, Ω

Uθα (X , X)I0 u ¨α dΩ =

N DR

m m m ˆ lk ˆlk α ¨ lm (t) cik W − Uik (x , x)Tˆlk dΓ + Tik (x , x)U dΓ Γ

m=1

(7)

Γ

In this equation, N DR represents the number of total dual reciprocity collocations points used in the plate; ˆ m and Pˆ m are the particular solutions to the equivalent homogeneous equations (1), considering the W lk lk 3 function fm = 1 − λ2 rm /9 for the approximation of angular velocities and the function fm = 1 + rm for the approximation of the transversal and in-plane accelerations. Coefficients α ¨ lm and β¨lm are related to I0m u ¨k and m Iik w ¨k , respectively, through expressions: I0 u ¨k (t) = Fkl α ¨ lm (t),

l = 1, 2 . . . , N DRM

Iik w ¨k (t) = Skl β¨lm (t),

l = 1, 2 . . . , N DRM

(8)

where Skl and Fkl are matrices of coefficients. Quadratic discontinuous boundary elements were used to ˆ m , Tˆm , Pˆ m and W ˆ m at the elements. In order to modeling the geometry of approximate uα , tα , wk , pk , U lk lk lk lk the elements, continuous quadratic elements, were used.

4.2

Boundary elements equations

Replacing expressions (4) to (5) into equations (2) and (3) and (8) into expressions (6) and (7) and these into equations (2) and (3) and applying these equations at each collocation point succesively, we obtain a set of equations that can be expressed in compact matrix form as: p

p ¨ Mp M u w H Hu 0 w p G q + = + (9) ¨ u 0 Gs M w Ms Hw Hs u t 0 where u = {u1 , u2 }T , w = {w1 , w2 , w3 }T , p = {p1 , p2 , p3 }T and t = {t1 , t2 }T are displacement and traction ¨ = {¨ ¨ = {w vector for plane stress and plate bending formulations respectively; u u1 , u ¨2 }T , w ¨1 , w ¨2 , w ¨3 }T are the in-plane and bending acceleration vectors; q = {0, 0, q3 }T is the domain load vectors; Ms , Mp are the mass matrices for plane stress and plate bending formulations respectively; Hs , Hp , Gs and Gp are boundary element influence matrices for plane stress and plate bending formulations respectively, and Mu , Mw , Hu , Hw are matrices which contain coupled terms between shell bending and plane stress formulations. The boundary integral equations discussed above contain integrands with several different orders of singularities. These singular integrals are treated separately based on their order of singularity. In this work, all of the regular integrals are evaluated numerically using the standard Gauss quadrature formulae. The influence matrix G contain weakly singular integrals, which are treated using Telles’s nonlinear coordinate tranformation method. However, for better numerical accuracy, a suitable number of element sub divisions must be used with the Telles tranformation. In this work, four element subdivisions are used. The influence matrix H contains strongly singular integrals, and in this work these integrals are computed indirectly by considering the generalised rigid body movements.

5 5.1

Numerical examples Transient analysis response

In order to obtain the time response using equation (9) the Houbolt integration scheme is used in this work. The most important aspect of this method when compared with others time integration methods based on central difference approximations or Newmark scheme is its introduction of artificial damping, which truncates the influence of higher modes in the response [8]. The Houbolt integration scheme is an explicit unconditionally stable algorithm based on backward-type finite difference formula with error of order O(∆τ 2 ), in which the acceleration is approximate in the form, w ¨ τ +∆τ =

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

(10)

where ∆τ represents time-step. Writing equation (9) at time level τ + ∆τ : Mw ¨ τ +∆τ + Hwτ +∆τ = Gpτ +∆τ + fτ +∆τ 4

(11)


297

Figure 1: Boundary element model for circular shallow shell 0.03

0.025

0.02

w

0.015

0.01

0.005 FEM model BEM model

0

−0.005

0

1

2

3

4 5 6 Time − seconds

7

8

9

10

Figure 2: Transversal displacement at the centre of the plate comparing BEM and FEM solutions and substituting the equation (10), we have: (2M + ∆τ 2 H)wτ +∆τ −

1 (Gpτ +∆τ + f τ +∆τ ) = M(5wτ − 4wτ −∆τ − wτ −2∆τ ) ∆τ 2

(12)

The above equation allows the calculation of the distribution of w at time level τ + ∆τ by using the boundary conditions at that time and information form three previous time steps. Special starting procedure in which initial conditions (2) are employed to calculate w1 and w2 .

5.1.1

Circular shallow shell under impulsive pressure load

Consider a clampled circular shallow shell under a variable ⎧ 2t if ⎨ 8t(t − 0.5) + 1 if p(t) = ⎩ 4(t − 1) if

pressure load p(t) given by: 0 ≤ t ≤ 0.5s 0 ≤ t ≤ 0.5s 0 ≤ t ≤ 0.5s

(13)

The geometric properties of plate are as follows: a = 5.0, h/a = 0.02, a/R = 0.05, κ11 = κ22 = 1/R = 0.01, and elastic constants E = 210M P a, ν = 0.33 and mass density ρ=500 Kgm/m−3 . Boundary conditions employed are, w3 = 0 and Mn = 0 along the boundary. The mesh has 32 quadratic discontinuous boundary elements with equal length, 100 cells and 197 DRM points as showed in figure 1. The time steps is ∆τ = 0.001s. In order to validate BEM solution, a FEM analysis using the commercial software package R was carried out. Shell93 shear deformable element type, with three degrees of freedom per ANSYS 10.0, 5

298


node in conjuction with a lumped mass model, were used. Figure 2 shows the history of central deflection for time interval 0s to 10s compared with a FEM solution using 192 elements. It can be seen that, for this case, the thick plate results are in good agreement results from FEM model.

6

Conclusions

This paper presented a boundary element formulation for the transient analysis of shear deformable shallow shells. Domain integrals which come from inertial terms were transformed into boundary integral using the Dual Reciprocity Boundary Element Method. The formulation was applied to a circular shallow shell under impulsive load. From results, we can conclude that the formulation proposed present a good performance, allowing good agreement with results of FEM models.

7

Acknowledgement

The authors are grateful to Research Office of Universidad Tecnológica de Bolivar for supporting their research work on dynamic analysis of shear deformable shallow shells.

References [1] Dirgantara, T. and Aliabadi, M. H. A new boundary element formulation for shear deformable shells analysis. International Journal for Numerical Methods In Engineering, 45: 1257-1275, 1999. [2] D. J. Gorman. Vibration Analysis of Plates by the Superposition Method: Series on Stability, Vibration and Control of Systems, Vol. 3 Word Scientific, New Jersey, 1999. [3] R. R. Szilard. Theories and Applications of Plate Analysis: Classical Numerical and Engineering Methods Wiley, New York, 2004. [4] J. Dominguez. Boundary Elements in Dynamics Computational Mechanics, New York, 1993. [5] L. C. Wrobel, M. H. Aliabadi. The Boundary Element Method Volume 2: Applications in Solid and Structures Wiley, New York, 2002. [6] F. M. Duddeck. Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform Springer, New York, 2010. [7] C. P. Providakis, D. E. Beskos. Dynamic analysis of plates by boundary elements Appl Mech Rev, 52(7):213-236, 1999. [8] P. W. Partridge, C. A. Brebbia, L. C. Wrobel. The Dual Reciprocity Boundary Element Method Computational Mechanics Publications, Southampton, 1992. [9] T. W. Davies, F. A., Moslehy. Modal analysis of plates using the dual reciprocity boundary element method Eng Anal Boundary Element, 14:357-362, 1994. [10] L. Palermo. On the harmonic response of plates with the shear deformation effect using the elastodynamic solution in the boundary element method Eng Anal Boundary Element, 31:176-183, 2007. [11] F. Vander Wee¨ en. Application of the direct boundary integral equation method to Reissner’s plate model Int J Num Meth Eng, 67:1-10, 1982. [12] Y. Rashed. Boundary Element formulation for thick plates WIT Press, Southampton, Boston, 2000.

6


A coupled boundary and finite element methodology for solving fluid-structure interaction problems Manuel Barcelos1, Carla T. M. Anflor 2 and Éder L. Albuquerque3 1e2 Faculdade do Gama, Campus da Universidade de Brasília no Gama PO Box 8114, 72405-610, Gama – DF – Brasil, [email protected] / [email protected] 3 Faculdade de Tecnologia, Universidade de Brasília, Campus Universitário Darcy Ribeiro 70910-900, Brasília – DF – Brasil, [email protected]

Keywords: Boundary Element Method, Fluid-Structure Interaction Problem, Finite Element Method, Aeroelasticity.

Abstract. This work aims the development of an efficient and robust numerical methodology to study the mechanical behavior of structures under the influence of external flow fields. In order to achieve this goal, it is necessary to simplify and make reliable the exchange of information between two numerical domains. Therefore, two efficient and robust numerical methodologies are coupled over matching meshes to guarantee the quality of the exchanging process. Thus, the external flow field is solved by using a viscous laminar equation model and a finite element method (FEM), while the airfoil structure is solved by using an elastic equation model and a boundary element method (BEM). The coupling between both numerical techniques allow for the simulation of the fluid flow over the airfoil as well as its structural behavior such as a realistic fluid-structure interaction problem. A NACA0012 airfoil with a specific set of mechanical properties and free stream flow configurations is analyzed to illustrate the FSI framework. Introduction. The progress we have experienced in experimental and numerical analysis of fluid-structural interaction (FSI) has not been enough to become it a less complex problem. This is mainly due to the physical phenomenon of interaction of fluid-structure that generates dynamic structural behavior of high complexity, giving highly non-linear responses. Therefore, FSI is an open issue and a research field very active today. On the other hand, the increase in processing power of computer systems in last decade and the use of computer models are presented today as an attractive alternative to analytical and experimental approaches traditionally used for analysis of problems of fluid-structure interaction, FSI in the naval, automotive and aerospace industries. However, a major disadvantage of the approach is its high computational cost and storage requirements, mainly. This is essentially happens because most currently used formulations are based on discretization methods of the whole domain, both for the structural problem and the flow problem. The computational analysis of FSI represents a fundamental tool in the design of ship and aircraft structures. The increase in recent years of computational processing power has allowed the use of high fidelity computer models. However, computational cost is still important on aeroelastic and hydroelastic problems because fluid domains are, in general, too large. The numerical development of new formulations that reduce the computational cost for aeroelastic and hydroelatic analyses represents one of the most active research areas in engineering, today. These new formulations can reduce the computational resources necessary for its implementation or to increase the accuracy of analysis. The FSI formulation applied in this work deals with fluid and structure as separate domains. This formulation is called staggered or partitioned scheme, allowing for the fluid and the structures problems to be solved by different numerical methodologies. The shortcoming of this approach is the flux of information over the fluid-structure boundary. For simplicity and

299

300


robustness, in this work matching meshes domains are chosen. Thus, data is transferred directly through the meshes without the necessity of projection or interpolation procedures. The main advantage of the staggered approach is simplicity and flexibility. Different numerical methods can be used to solve specific problems and few modifications have to be done to previous numerical schemes to adapt them to a FSI framework. Fluid-Structure Analysis. In a general fluid-structure framework, the governing equations are based on the three field formulation [3], as described: F (u, p, Ȥ ) = 0 (1) M ( Ȥ , d) = 0 (2) S (d, u, p) = 0 (3) where F is the state equation of the fluid, M is the equation which governs the motion of the fluid mesh and S is the state equation of the structure. The following state variables: u , p , Ȥ and d represent, respectively, the fluid velocity field, the fluid pressure field, the fluid mesh displacement and the structural displacements. As most of the design requirements are effectively computed by using stead state responses, a quasi-static solution methodology is proposed to solve the coupled problem. Therefore, in order to employ a quasi-static solution methodology some terms related to time derivatives are neglected. Equations (1) to (3) are coupled through the transmission conditions on the fluid-structure interface Γ fs , which represent the equilibrium of forces, equation (4), the compatibility of the displacement, equation (5), and the velocity, equation (6), between the fluid and the structure domains, σ S ⋅ n = τ ⋅ n − p ⋅ n on Γ fs (4)

Ȥ = d on Γ fs

(5)

∂Ȥ ∂d = on Γ fs (6) ∂t ∂t where p and τ are, respectively, the flow pressure and shear stress tensor, σ s is the structure stress tensor and n is the normal at a point on Γ fs . Flow Problem. By considering a time dependent viscous laminar and incompressible flow problem, the flow governing equations F are described as: ∇. u = 0 (7)

∂ 2u 1 + (∇u )uˆ = − ∇p + ∇. (ν f ∇u ) + f f ∂t ρf

(8)

where u(x, t ) e p (x, t ) , respectively, the velocity and the pressure fields of the flow are function of the mesh position and the time, ρ f and ν f are in this order the density and the viscosity of the fluid problem, and f f is a given force function. Due to the moving boundary problem the continuity and momentum equations are modified to satisfy the arbitrary Lagrangian-Eulerian (ALE) formulation. Therefore, the relative velocity term uˆ is introduced and represents the diơerence between flow and mesh velocities. The boundary conditions are defined on Γ f = Γd Γ fs Γo . The subset Γd represents the boundary values that are constant where the Dirichlet boundary condition for the velocity field u d is prescribed. Γ fs is the moving boundary, where the fluid velocity u Γ fs is equivalent to the domain velocity. Γo is the outlet boundary condition where a reference pressure pref is prescribed. The flow solution


301

methodology is based on the semi-explicit iterative solution of the systems of equations (6) and (7) after a time and a spatial discretization, and considering a projection method framework [4, 5, 6]. Once a flow state u n +1 and p n +1 is computed, the fluid forces acting over the structure is determined by: f fsn+1 = τ n+1 ⋅ n − p n+1 ⋅ n on Γ fs (9)

Mesh Motion Problem. In this work for the fluid mesh motion problem M , a quasi-static _

model is used as described in equation (10), and the fictitious stiffness matrix K constructed by an improved spring analogy method [1, 2, 7]. _

is

_

KȤ = R (10) The kinematic compatibility dictates how the position Ȥ Γ fs of fluid mesh on the boundary Γ is related to the fluid mesh boundary velocity u Γn +fs1 by:

Ȥ Γ fs = ∆t u Γn +fs1

u Γn +fs1 =

(11)

n +1

d ∆t

(12)

where ∆t is fluid problem time step and d n +1 is the displacement field of the structure. The direct method is preferred to solve equation (24) because of its simplicity and robustness.

The mesh position x n+1 is updated by summing the mesh displacement vector to the previous configuration of the mesh position vector x n , such as follows: ªȤ Ω º (13) x n+1 = x n + « fs » ¬« Ȥ Γfs ¼»

Structure Problem. Now consider an elastic plate of thickness h occupying the area Ω S , bounded by the contour ΓS , in the x1 x2 plane. The governing equations for the quasi-static response, considering plane stress conditions, are given by [8]: G Gd i , jj + d i , ji = − f i (14) 1 − 2ν where G is the shear modulus, ν is the Poisson ratio and d i represents in-plane displacements and f i in-plane body forces. The boundary integral formulation for equation (14) is given by: 1 (15) cij d + ³ Tij*d j dΓS = ³ Dij*l j dΓS + ³ Dij* f j dΩ S ΓS ΓS h ΩS where Tij* and Dij* are the fundamental solutions for plane stress elasticity, respectively [8],

l j is the load at the boundary and cij is a constant that depends on the geometry at the collocation point. Applying the boundary element method and considering discontinuous quadratic elements to discretize the boundary of the airfoil, the following equation is obtained: Hd n +1 = Gl n +1 + Bf n +1 (16) In this work, a matching mesh problem approach is chosen to illustrate the fluid-structure coupling concept. Therefore, the exchange of force and displacement over the fluid and structure boundaries is executed through every pair of matched points between the two mesh

302


domains, without the need of projection and interpolation procedures. The exchange of force and displacement over matching meshes is represented by: l n +1 = I f fsn +1 on Γ fs (17)

Ȥ nfs+1 = I d n +1 on Γ fs

(18)

where I is the identity matrix. Results. The test case studied is a NACA0012 airfoil of 1 m of chord. The fluid problem is solved by a finite element code for laminar and incompressible flows. The finite element mesh has 5,712 nodes and is composed by 10,952 triangular elements. The mesh region close to the boundary layer has a high ratio of refinement. As a first step, a preliminary flow solution is computed in order to use it for starting point to solve the coupled problem. Thus, a flow air with Reynolds number of 1,000, angle of attack of 5 degrees and free stream kinematic viscosity of 1.33 10-5 m2/s and density of 1.293 kg/m3 was set to run with a time step of 0.01s and a maximum number of flow solution iterations of 10,000. A preliminary steady state flow solution was obtained with zero velocity boundary condition on the airfoil and a velocity error target of 10-4.

(a)

(b) Figure 1: Fluid mesh: original configuration (a) and deformed configuration (b). In the second step, the coupled problem was set to start from the preliminary steady state solution and the flow properties remained the same. The structure problem is solved by a boundary element method with a mesh of 204 nodes over the airfoil contour line. As the resulting boundary force computed from the flow solution has a small magnitude, the structure properties were chosen to make it flexible when under the influence of this type of load. So, pseudo material was defined with an elasticity modulus of 700Pa and a Poisson


ration of 0.3 to satisfy these conditions. In order to have a smooth solution, the structure code employs a numerical integration scheme with 10 Gauss points over each element. For structural boundary condition, the last 20 points on the airfoil trailing, 10 on the upper surface and 10 on the lower surface, were clamped. The staggered solver was set to run for 10 coupled problem iterations to allow it to converge to the most reasonable fluid-structure response and for 1000 fluid iterations in order to have a converged steady state flow solution (with a velocity error target of 10-6) at each coupled problem iteration. Despite the flow problem is running a time dependent solver to determine a steady state solution, the fluid mesh motion and the structure problems are running in a quasistatic mode to simplify the whole FSI modeling. The exchange of force and displacement data over the fluid-structure boundary is done directly through the meshes, because the fluid and the structure domains have matching meshes.

(a)

(b) Figure 2: Structure mesh: original configuration with boundary conditions (a) and deformed configuration with internal point displacements (b). The steady state flow solution reached a lift, C L , and drag, C D , coefficients of 0.031 and 0.011 , respectively. The maximum displacement happened at the leading edge, about 0.044 m. Figures (1) and (2) show the fluid and structure meshes before and after the FSI solution strategy. One can remark that the resulting lift force deformed the airfoil as if was a clamped beam. This behavior happened as expected, because an airfoil under the flow influence is a device that generates lift and drag forces. If there are forces over a structure, then its most natural mechanical response is to deform. This deformation changes the flow path, and after a set of cyclic interactions, a steady state is reached and a final deformed state is defined. The main goal of this work was achieved, because a numerical FSI framework was built and tested with success. This is the proof that is possible to improve the simulation fidelity with a few set of steps and reorganizing preexistent numerical schemes. Now, by the application of isolated numerical schemes is possible to tackle more complex coupling problems. Conclusion. The work here presented has as goal to show the great potential of application of staggered numerical schemes to solve coupled fluid-structure problems. The main advantage of the staggered scheme is its simplicity and flexibility, once the fluid and structure problems are solved in a separate fashion. One can use different methodologies and numerical algorithms to tackle each individual problem. This allow for applying numerical methods with the best characteristics to solve specific fluid-structure coupling problems.

303

304


The bottleneck of the staggered approach lies on the fact that is necessary the development of preprocessing tools to project, interpolate and transfer force and displacement data over the fluid-structure boundary. The methodology employed in this work avoided this difficulty by using matching meshes that allowed for transferring information between domains directly, without the need of a preprocessing step. The solution of the coupling problem by employing a finite element method for the fluid domain and a boundary element method for the structure domain was remarkably successful. Few modifications had to be executed to prepare the exchange of information between de codes. Also, as the fluid code had the ALE and mesh motion featured already installed and tested, there is no difficulty of adapting it to fit in a fluid-structure coupling framework.

References [1] C. Degand and C. Farhat. A three-dimensional torsional spring analogy method for unstructured dynamic meshes. Computers and Structures, vol. 80, pp. 305-316, (2002). [2] C. Farhat, C. Degand, B. Koobus and M. Lesoinne. Torsinal springs for two dimensional dynamic unstructured fluid meshes. Comput. Methods Appl. Mech. Engrg., vol 163, pp. 231245, (1998). [3] C. Farhat, P. Geuzaine and G. Brown. Application of a three-field nonlinear fluid-structure formulation to the prediction of the aeroelastic parameters of an F-16 fighter. Computers and Fluids, vol 32, pp. 3–29, (2003). [4] D. Goldberg and V. Ruas. A numerical study of projection algorithms on finite element simulation of three-dimensional viscous incompressible flows. Int. J. Numer. Meth. Fluids, vol 30, pp. 233-256, (1999). [5] J. L. Guermond, P. Minev and J. Shen. An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg., vol. 195, pp. 6011-6045, (2006). [6] R. Lohner, C. Yang, J. Cebral, F. Camelli, O. Soto and J. Walts. Improving the speed and accuracy of the projection-type incompressible flow solvers. Comput. Methods Appl. Mech. Engrg., vol. 195, 3087-3109, (2006). [7] B. Koobus, C. Farhat, C. Degand and M. Lesoinne. An improved method of spring analogy for dynamic unstructured fluid meshes. In Proceedings of the 39th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference and exhibit, April 20-23, Long Beach, California, (1998). [8] C. A. Brebbia and J. Dominguez. Boundary Elements: An Introductory Course. WIT Press, Southampton, (1993).


Analysis of Acoustic Wave Propagation in Shallow Water using the Method of Fundamental Solutions J. A. F. Santiago1, E. G. A. Costa1,L. Godinho2 and A. Pereira2 1

Program of Civil Engineering, COPPE - Federal University of Rio de Janeiro, CP 68506, CEP 21945-970, Rio de Janeiro, RJ, Brazil, [email protected], [email protected].

2

CICC, Department of Civil Engineering, University of Coimbra, Pinhal de Marrocos, 3030-788 Coimbra, Portugal, [email protected], [email protected].

Keywords: Method of Fundamental Solutions, Green’s Functions, Shallow Water

Abstract. In this paper, an efficient numerical frequency domain formulation is proposed to investigate the 2D acoustic wave propagation in shallow water. The analyzed configuration combines sets of regions with a sloping bottom with those regions where the bottom is flat and assumes also a wedged shoreline. The numerical approach used here is based on the Method of Fundamental Solutions (MFS). In this model only the vertical interface between regions of different configuration is discretized, as Green’s functions that take into account the presence of flat rigid and free surfaces and that provided by a wedge shape, are incorporated. It was found that the model is able to efficiently capture the acoustic wave propagation pattern generated in wedge coastal domains. 1.

Introduction

The sound field generated by an acoustic source in an ocean environment can be obtained by solving either the wave equation or the Helmholtz equation. The solution of these equations may be difficult to obtain due to the complexity of the environment. In addition to that, sound is widely used in undersea applications such as detection of ships and submarines, seismic profiling, echo-sounding, high resolution imaging, communications and acoustic tomography. The classical book by Jensen et al. [1] discusses in detail the different acoustic environments, which are divided in deep, shallow water and range dependent regions. Among these, the shallow water regions are the part of the ocean lying over the Continental Shelf where the water depth is less than 200 m. Here, the sound field may be seen in terms of a succession of reflections that travel back and forth between the boundaries as sound propagates along the channel. For modelling homogeneous unbounded domains, the Boundary Element Method is usually preferred since the far field conditions are automatically satisfied and only the boundaries of the interfaces need to be discretized, which considerably reduces the time consuming effort for the mesh generation. However, the application of the BEM is often limited to the prior knowledge of the fundamental solutions or of the singular integrals. In spite of these difficulties, the method has been applied with success in shallow water acoustics, firstly by Dawson and Fawcett [2], and later by in 1998, by Grilli et al [3], who firstly proposed a hybrid numerical model which combines the standard method in an inner region with varying bathimetry and an eigenfunction expansion in the outer region of constant depth. Santiago and Wrobel [4] proposed the sub-region technique in boundary element formulation for the analysis of two-dimensional acoustic wave propagation in a shallow water region with irregular seabed topography. In a subsequent work [5], they discuss the use of several Green’s functions that satisfy either the free surface boundary condition or both the boundary conditions on the free flat surface and rigid flat bottom. Godinho et al. [6] also analyzed wave propagation using a Boundary Element formulation, generated by point sources in a 2D fluid channel with a rigid deformation on its floor, where the bottom is modelled using Neumann-Dirichlet boundary conditions. Subsequently, António et al. [7] and Tadeu et al. [8] used BEM formulations integrating Green’s functions for the case of a waveguide with an elastic bottom, to study the scattering of waves by a buried or by a submerged object. Recently, Pereira et al. [9] presented a BEM formulation that allows the prediction of wave propagation in waveguides containing sharp discontinuities, such as a step or a slope, connecting two flat regions. Their formulation allowed a simple description of the media, making use of Green’s functions that account for

305

306


the flat parts of the system, and requiring only the discretization of the discontinuity and of a virtual interface between the flat regions. In recent years, meshless techniques have been interesting the scientific community leading with wave propagation problems, as they do not require explicit domain discretization, they are computational efficient and they provide simpler mathematical formulation [10, 11]. Among these techniques, the Method of Fundamental Solutions seems to be straightforward when solving wave propagation problems with regular geometries, even outperforming the BEM [12]. The present paper analyses shallow water acoustic propagation using a frequency domain formulation based on the Method of Fundamental Solution. The configuration analyzed combines sets of wedge shaped regions with flat ones, where the bottom is always assumed flat rigid and the surface free. Here, we make use of the sub-region technique and Green’s functions for confined domains [13, 14, 15] are employed, which allow to deal with only a limited number of virtual interfaces. The advantages of the proposed model such as the stability and accuracy are here discussed by performing comparisons with other well established methods such as the Boundary Element Method. 2.

Governing Equation Problem

Consider the problem of acoustic wave propagation in a bi-dimensional region : of infinite extent along the z direction, with irregular rigid seabed topography and flat free surface. The propagation domain is described using a number of sub-regions and excited by a linear pressure load which may be placed in any sub-regions with inclined or flat bottom, as shown in Fig. 1. y *F

Free surface Source

: Bottom *B

x

Figure 1: Geometry of the problem.

If the source of acoustic disturbance is time-harmonic, the sound velocity is constant and the medium in the absence of perturbations is quiescent, the governing Helmholtz equation for this problem can be written as: 2I k 2I

NS

¦ Q G ( l

f l

, ),

in region :

(1)

l 1

where NS is the number of sources in the domain; I is the velocity potential; Ql is the magnitude of the existing sources lf located at ( xf , y f ) ; is a domain point located at ( x[ , y[ ) ; G ( lf , ) is the Dirac delta generalized function; and k Z c is the wave number, with Z being the angular frequency and c the speed of sound in the medium. In this problem the following boundary conditions were assumed: Dirichlet condition in the boundary of the free surface * F ; Neumann condition in the boundary of the rigid bottom * B and Sommerfeld radiation condition at infinity. 3.

Method of Fundamental Solutions

In the MFS formulation here developed, the acoustic domain is assumed to be divided into a set of n sub-regions : j (with j 1,..., n respectively), as illustrated in Fig. 2. Within each of those sub-regions, it is possible to define Green’s functions taking into account the following considerations about boundary conditions: (a) in sub-regions with inclined bottom, a fundamental solution that directly satisfies the sloping rigid bottom and free surface conditions is assumed, and (b) in sub-regions with flat bottom, a solution that directly satisfies the rigid and free flat surface conditions is employed. The use of those functions only requires a set of fictitious vertical boundaries ( *1 , * 2 ,…, * n ), connecting adjacent sub-regions along which collocation points ( CP1 , CP2 , …, CPn ) will be located (see Fig. 2). Within each sub-region, the MFS allows


307

the response to be obtained as a linear combination of fundamental solutions, simulating the field within that sub-region by means of sets of virtual sources l (with l 1,! ,VSn ) placed outside the sub-region and at a fixed distance to the vertical interface ( VS1 , VS 2 , …, VS n ) as illustrated in Fig. 2. y

2VS n

2VS1 Source Points

2VS 2

Free surface

:n

:1

:2 Source

*2 *n

Bottom

CP2

*1

CP1 Collocation Points

x

CPn

Figure 2: Geometry of the MFS model.

For each sub-region, the velocity potential at an internal point xk can then be written as:

I ( xk )

VS1

¦a

:1 l

Gkl:1 ( xk , [l ) (- 1)Gkf:1 ( xk , [ f ) , within :1

(2a)

l 1

I ( xk )

VS j 1 VS j

¦

:

:

:

al j Gkl j ( xk , [l ) (- j )Gkf j ( xk , [ f ) , within : j ( j 2,..., n 1 )

(2b)

Gkl:n ( xk , [l ) (- n)Gkf:n ( xk , [ f ), within : n

(2c)

l 1

I ( xk )

VS n

¦a

:n l

l 1

where f is the real source with coordinates ( xs , ys ) ; l refers to the lth virtual source point placed along a :j

fictitious boundary; al

is the amplitude to be determined for each of the lth source points of : j ; - is the

sub-region in which the real source is positioned and is the logical “AND” operator, so that - j

-

:j kf

1 if

j and 0 otherwise; G ( xk , [ ) is the incident field regarding the velocity potential generated by the f

:

real source when placed in the sub-region : j ; Gkl j ( xk , [l ) refers to the Green’s function for sub-region

: j whose functions are given in the next section. By adequately deriving expressions (2), the normal particle velocity with respect to the interface between sub-regions may be obtained. It is important to note that, for this case, the normal direction always corresponds to the horizontal (x) direction. If we impose, at each collocation point, continuity of velocity potentials and its normal velocities, the following 2 CP1 CP2 ... CPn u 2 VS1 VS2 ... VSn system of equations is obtained. Once the system of equations is solved, then the response at any internal point of the domain can be obtained by using expressions (2). 4.

Green’s Functions 4.1.

Analytical Solution for a Region with Flat Rigid and Free Surface

The Green’s function, that satisfies the boundary conditions at the flat free surface and at the flat rigid bottom, as displayed in Fig. 3a, is constructed by an eigenfunction expansion scheme given by [14]: ik

x x

i f e xm [ (3) sin ª¬ k ym YF y[ º¼ sin ª¬ k ym YF y º¼ , ¦ Hm1 k xm where H refers to the depth of the flat waveguide, YF refers to the y coordinate of the free surface, and the G

parameters k ym respectively.

1· S § and k xm ¨m ¸ 2¹ H ©

2 k 2 k ym are the horizontal and vertical wavenumbers,

308


y

xr c

y

xr Free surface

Free surface

yr

R

yr c

( x, y )

H

S

T

R

r' S

( x , y )

T'

r

Bottom

T

x

Bottom

x

(b)

(a)

Figure 3: Geometry of the individual sub regions, for which fundamental solutions are employed: a) waveguide with rigid bottom and free surface; b) wedge coastal region.

It is important noting that the exponential term in eq. (4), makes the series decrease rapidly when k xm an imaginary number becomes. Note also that when the exponential term of the function is equal to one ( x x[ ), the series converges slowly. 4.2.

Analytical Solution for an Ideal Wedge

The analytical solution, for an ideal wedge (as displayed in Fig. 3b), used here, exactly satisfies the boundary conditions on the flat free surface and on the sloping bottom of the wedge. This solution was obtained from the inhomogeneous Helmholtz equation and is given by the following expressions [13, 15]: i f G JQ (kr ) HQ(1) (kr! ) u sin(QT )sin(QT c) . (4)

T0 ¦ m 1

In this equation, T 0 is the wedge angle; JQ is the Bessel function of the first kind of order Q ; YQ is the Bessel function of the second kind of order Q ; HQ(1) is the Hankel function of the first kind of order Q ;

r

min( r , r c) , r!

max(r , r c) are the distances from the receiver and source to the apex of the wedge; T

and T c are the angular depths of the receiver and source measured about the apex (as defined in Figure 3b); the orders of the Bessel and Hankel functions for rigid bottom are given by Q m 12 T0 . It is important to note that this fundamental solution poses some difficulties in its implementation. In fact, when the real order Q becomes large in relation to the fixed argument N , (i.e. when Q o f ) the function YQ (N ) o f while JQ (N ) o 0 . This occurs when r and r ' are very close to each other, for which case a large number of terms (and very high orders Q ) becomes necessary to attain convergence. However, in the present work as the MFS is applied to model the proposed system, the source point is always positioned away from the receiver point (i.e. r and r ' are markedly distinct). 5.

Behaviour of the MFS Model

The MFS algorithm developed in this work was implemented and verified by comparing the results with a standard BEM model, where all interfaces need to be discretized. Firstly, a simple geometry (Geometry 1) was chosen to verify the proposed MFS formulation, consisting of a flat waveguide region 20.00m deep connected to a wedge region 40.00m long, with an apex angle of 26.56o , as illustrated in Fig. 4a. The acoustic medium allows the propagation of pressure waves with a velocity of 1500.00 m/s. The system was subjected to a linear pressure load applied near the rigid bottom at point (0.00m;0.05m). The responses were calculated at receiver R1 placed in region : 2 at (21.00m;11.00m) and at receiver R2 placed in region :1 at (29.00m;11.00m) . Computations were performed for frequencies up to 256.0 Hz, with a frequency step of 4.0 Hz. Complex frequencies with an imaginary part of (] 0.7 'Z ) were assumed.

20.00 m


c=1500.00 m/s

Source (0.00m, 0.05m)

309

(21.00m, 11.00 m) (29.00m, 11.00 m) R2 R1 :1 :2 *1 25.00m

x

40.00m

a) 0.50

0.60 MFS Model - real part BEM Model - real part BEM Model - imag. part MFS Model - imag. part

MFS Model - real part BEM Model - real part BEM Model - imag. part MFS Model - imag. part

0.40 0.30

Velocity Potential (m²/s)

Velocity Potential (m²/s)

0.40 0.20 0.00 -0.20

0.20 0.10 0.00 -0.10 -0.20 -0.30

-0.40

-0.40

-0.60

-0.50

0

20

40

60

80

100 120 140 160 180 200 220 240 260

Frequency (Hz)

0

b)

20

40

60

80

100

120

140

160

Frequency (Hz)

180

200

220

240

260

c)

Figure 4: Verification of the numerical model: (a) using Geometry 1; (b) responses at receiver R1; (c) responses at receiver R2.

When using the BEM model the number of boundary elements is defined as a function of the frequency, by using a relation between the incident wavelength and the length of the boundary element, equal to a minimum of R=8. As in the BEM model, the number of collocation points used in the interfaces of the MFS model is defined by fixing the relation between the incident wavelength and the distance between collocation points. For this case, this relation was fixed to R=5. Additionally, for the MFS, the distance (D) between the fictitious sources and the interface was fixed at D=5 times the distance between collocation points. In order to illustrate the responses obtained in this verification, Fig. 4b and 4c display the velocity potential computed at receivers R1 and R2 respectively, using both models. Observation of the results confirms an excellent agreement between both models. Although for the previous case the computed responses revealed a good performance of the method, it is known that, for the MFS, the quality of the results can depend on the definition of a good position for the virtual sources. This difficulty has been extensively discussed in the literature, and researchers have proposed strategies to control the error associated with the position of those sources. For this reason, it becomes important to assess the influence of the position of the virtual sources in the quality of the results provided by the model proposed here. Finally, a second geometry (Geometry 2), composed of four sub-regions connected by vertical interfaces, was also analyzed to assess the accuracy of the proposed method in modelling more complex configurations. This geometry is illustrated in Fig. 5a, combining two waveguides (with depths of 20.00m and 30.00m respectively) and two wedge regions (with lengths of 40.00m and 20.00m , respectively). The stability of the model was also assessed for this case, for varying distances between the fictitious sources and the vertical interfaces. Here, we illustrate the response computed at receivers R1 and R2, assuming distances between the fictitious sources and the vertical interfaces of D=5 and D=7.5, for a fixed relation between the incident wavelength and the distance between collocation points of R=5. To further verify the model, responses computed with a standard BEM model, as described for the previous case, were also included in the presented results. The curves of Fig. 5b and 5c display an excellent agreement, indicating that even for more complex geometries the proposed model provides stable results for different positions of the virtual sources.

310


30.00 m

c=1500.00 m/s :3

:4

20.00 m

y (21.00 m, 11.00 m) R1

Source (0.00m,0.05m)

(29.00m, 11.00 m) R2 :1

:2 *1

*2

x *3

15.00 m

20.00 m

ba)

0.4

0.4 Imag MFS D7.5 Real MFS D7.5 Imag MFS D5 Real MFS D5 Imag BEM Real BEM

0.2 2

2

0.2

Imag MFS D7.5 Real MFS D7.5 Imag MFS D5 Real MFS D5 Imag BEM Real BEM

0.3

Velocity potential (m /s)

0.3 Velocity potential (m /s)

40.00m

0.1 0 -0.1

0.1 0 -0.1

-0.2

-0.2

-0.3

-0.3

-0.4

0

20

40

60

-0.4

80 100 120 140 160 180 200 220 240 Frequency (Hz)

0

20

40

60

80

100

120

140

160

180

200

220

240

Frequency (Hz)

b)

c)

Figure 5: Verification of the numerical model: (a) using Geometry 2; (b) responses at receiver R1; (c) responses at receiver R2.

6.

Conclusions

In this paper, a numerical formulation in the frequency domain based on the Method of Fundamental Solutions has been applied to study the wave propagation in a coastal region confined by a free surface and by a seabed which combines sub-regions of inclined rigid bottom with sub-regions with flat rigid bottom. The formulation makes use of appropriate Green’s functions, therefore only the vertical interfaces between the wedged regions and the flat regions are virtually discretized. With the performed numerical examples, it was found that the present model provides accuracy and stability when modelling coastal regions. Acknowledgement The second author would like to thank CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior) for providing financial support to this research. References [1] F.B. Jensen, W.A. Kuperman, M.B. Porter and H. Schmidt, Computational Ocean Acoustics, American Institute of Physics, Woodbury, New York (2000). [2] T.W. Dawson and J.A. Fawcett, A Boundary Integral Equation Method for Acoustic Scattering in a Waveguide with Nonplanar Surfaces, Journal of the Acoustic Society of America, 87, 1110-1125 (1990). [3] S. Grilli, T. Pedersen and P. Stepanishen, A Hybrid Boundary Element Method for Shallow Water Acoustic Propagation over an Irregular Bottom, Engineering Analysis with Boundary Elements, 21, 131-145 (1998). [4] J.A.F. Santiago and L.C. Wrobel, A Boundary Element Model for Underwater Acoustics in Shallow Water, Computer Modelling in Engineering & Sciences, 1, 73-80 (2000).


[5] J.A.F. Santiago and L.C. Wrobel, Modified Green’s Functions for Shallow Water Acoustic Wave Propagation, Engineering Analysis with Boundary Elements, 28, 1375-1385 (2004). [6] L. Godinho, A. Tadeu and F. Branco, 3D Acoustic Scattering from an Irregular Fluid Waveguide via the BEM, Engineering Analysis with Boundary Elements, 25, 443-453 (2001). [7] J. António, A. Tadeu and L. Godinho, 2.5D Scattering of Waves by Rigid Inclusions Buried Under a Fluid Channel via BEM, European Journal of Mechanics /A Solids, 24, 957-973 (2005). [8] A. Tadeu, L. Godinho and J. António, Dynamic response of a three dimensional fluid channel bounded by an elastic floor in the presence of a submerged inclusion via BEM, Journal of Computational Acoustics, 13 203-227 (2005). [9] A. Pereira, A. Tadeu, L. Godinho and J. Santiago, 2.5D BEM Modelling of Underwater Sound Scattering in the Presence of a Slippage Interface Separating two Flat Layered Regions, Wave Motion, 47, 676-692 (2010). [10] G. Fairweather, A. Karageorghis and P. A. Martin, The Method of Fundamental Solutions for Scattering and Radiation Problems, Engineering Analysis with Boundary Elements, 27, 789-769 (2003). [11] H. Cho, M. Golberg, A. Muleshkov and X. Li, Trefftz Methods for Time Dependent Partial Differential Equations, CMC: Computers, Materials and Continua, 1, 1-38 (2004). [12] L. Godinho, A. Tadeu and N. Simões, Accuracy of the MFS and BEM on the Analysis of Acoustic Wave Propagation and Heat Conduction Problems, In: J. Sladek, V. Sladek (Eds.), Advances in Meshless Methods, Tech Science Press (2006). [13] S. Stotts, Coupled-Mode Solutions in Generalized Ocean Environments, Journal of the Acoustical Society of America, 111, 1623-1643 (2002). [14] T. Pederson, Modeling Shallow Water Acoustic Wave Propagation, MSc Thesis, University of Rhode Island, USA, (1996). [15] M. J. Buckingham and A. Tolstoy, An Analytical Solution for Benchmark Problem 1: The “ideal wedge”, Journal of the Acoustic Society of America, 88, 1511-1513 (1990).

311

312


Modal Analysis of Thick Plates Using Boundary Elements W. P. Paiva1, P. Sollero2, E. L. Albuquerque3 1

Faculty of Mechanical Engineering, State University of Campinas – SP – Brazil [email protected]

2

Faculty of Mechanical Engineering, State University of Campinas – SP – Brazil [email protected] 3

Faculty of Technology, University of Brasilia – DF – Brazil [email protected]

Keywords: Boundary element method; Dual reciprocity method; Dynamic problems; Modal analysis; Plate bending.

Abstract. This paper presents a numerical formulation for analysis of dynamic problems of thick plate bending. The bending behavior follows Mindlin hypotheses. The formulation is based on the direct boundary element method. The problem is simplified by using the elastostatic fundamental solution of an infinite plate. Domain integrals arising from inertial terms are transformed into boundary integrals using the radial integration technique. Boundary integrals are discretized and evaluated numerically. Natural frequencies for free vibration are obtained and the respective mode shapes are shown. The accuracy of numerical results obtained is assured by comparison with analytical or finite element results. Introduction Vibration of plates is a very important area to be studied. This importance comes from application of plate in most diverse engineering fields. Many authors have presented solutions for Kirchhoff’s thin plate bending analysis using the Boundary Element Method (BE). First works on indirect BEM were presented by Altiero and Sikarskie [1] and Tottenham [2]. Direct BEM formulation for arbitrary boundary conditions was presented by Bézine [3] and later by Hartmann and Zotemantel [4], among others. Since Weeën [5] presented his original work on BEM applied to analysis of Reissner’s thick plate bending an amount of works has appeared in the literature [6, 7, 8, 9]. Following Weeën [10], de Barcellos [11] proposed a formulation for treat Mindlin’s model. More references on Reissner/Mindlin’s plate bending problems can be seen on [12]. It is well known that body forces of plate formulations lead to domain integrals in BEM formulation. Domain integrals can be computed by exact transformation, by discretizing domain into cells, or using only boundary formulations as Dual Reciprocity Method (DRM) or Radial Integration Method (RIM) [13, 14, 15]. This paper presents an application of RIM for numerical analysis of Mindlin’s thick plate free vibration problems. Analysis is carried out using BEM with elastostatic fundamental solution and domain integrals that came from inertial terms are transformed into boundary integrals using the RIM. Natural frequencies and mode shapes are investigated for a fully clamped square plate and for a more general boundary condition SCSF. Numerical results show a good agreement with results available in literature. Mindlin’s Plate Theory Mindlin’s plate theory assumes that the three-dimensional displacement components U D are linearly proportional to the thickness coordinate x3 (Fig. 1).


313

Figure 1: Plate geometry. Equilibrium equations for the plate are written in indicial notation as

M DE , E QD QD ,D

U h3 12 U h Ü3

ÜD

(1) (2)

where M DE are the moments, QD are the shear forces that relate displacements and slopes U is the material density and h is the plate thickness, U D are rotations and U 3 are deflections of the plate in x3 direction. Greek indices range from 1 to 2 and Latin indices range from 1 to 3. Eqs. (1) and (2) comprises three equations that contain five unknowns. Two required additional equations are obtained from stressdisplacement relationship and application of boundary conditions. Boundary Element Formulation Boundary Integral Equation. The boundary integral equation for Reissner/Mindlin’s thick plate bending is obtained using Rayleigh-Green identity. It is derived from governing Eqs. (1) and (2) and is given by

ci j (Q) U j (Q) ³ * Pi*j (Q, P) U j ( P) d*( P) ³ * U i* j (Q, P) Pj ( p) d*( P)

U h 3Z 2 12

³:U iD (Q, P)ÜD ( P) d: U h Z ³:U i3 (Q, P)Ü 3 ( P) d: *

2

*

(3)

where ³ denotes a Cauchy Principal Value integral, Q and P are the source and field points, respectively, the symbol * stands for the fundamental solution and c is introduced in order to consider that the point Q can be placed in the domain, in the boundary, or outside the domain. When point Q is located on a smooth boundary, ci j (Q)

G i j / 2 . U i*j (Q, P) and Pi*j (Q, P) are the two-point fundamental solutions for

displacements and tractions, respectively. Fundamental Solutions. The expressions for displacement fundamental solutions U i*j (Q, P ) and traction fundamental solutions Pi *j (Q, P ) are given by [5]

U D* E

1 8 S D (1Q )

U D* 3

U 3*D

U 3*3

^> 8 B( z ) (1 Q ) (2 ln z 1)@GDE >8 A( z ) 2(1 Q )@r, a r,b `

1 8S D

1 8 S D (1Q ) O2

(2 ln z 1) rr,D

> (1 Q ) z (ln z 1) 8 ln z @ 2

(4)

(5)

(6)

314


PJ*D

1 4S r

> (4 A( z ) 2 zK1 ( z ) 1 Q ) (GDJ r,K r, a nJ )

(4 A( z ) 1 Q ) rJ nD 2(8 A( z ) 2 zK1 ( z ) 1 Q ) r,D r,J r,K O2

>B( z) nJ A( z) r,J r,K @

PJ*3

2S

P3*D

1Q 8S

P3*3

1 r 2S r ,K

(8)

> 2 11QQ ln z 1 nD 2 r,D r,K @

(9)

(10)

O r , r is the distance from source to

where A( z ) and B ( z ) are function of modified Bessel functions, z field point, r,D

rD / r , rD

(7)

xD ( P) xD (Q) and r,K

r,D nD [16].

Radial Integration Method. Lets consider that each domain term of eq. (3) is approximated over the domain as a sum of the M products between radial basis function f m and unknown coefficients J m M

b( P )

¦J

m

fm (11)

m 1

The domain integrals of eq. (3) can be written as a boundary integral, given by

T '(Q) T "(Q)

h2 * U iD (Q, P ) ÜD ( P) d : : 12

U hZ 2 ³

U hZ 2 ³ U i*3 (Q, P ) Ü 3 ( P ) d : :

M

F 'm (Q) n r d* * r

¦J ³ m

m 1

M

F "m (Q) n r d* * r

¦J ³ m

m 1

(12)

(13)

Eq. (3) can be written in matrix form as

Pu=Up+S where S

>t'

(14)

t " @ , t ' and t " are the boundary integrals of eqs. (12) and (13), respectively, and is the

vector that contains coefficients J m . Body forces terms can be written in a matrix form, considering all source points, as

b = F

(15)

= F -1 b

(16)

T = S F -1 b

(17)

Thus can be computed as Then For free vibration dynamic problems, the body force vector is given by

b U hZ2 U

(18)

where

U Substituting eq. (18) in eq. (17), results

ÜD ½ ® ¾ ¯Ü 3 ¿

(19)


315

T = Z 2 U h S F -1 U

(20)

T = Z2 M U

(21)

M = U h S F -1

(22)

or where M is the mass matrix given by Finally, eq. (14) for dynamic problems can be written as

P u = U p +Z2 M U

(23)

Modal Analysis Eq. (23) is transformed in an eigenproblem. The boundary is divided into *1 (stands for constrained degrees of freedom) and * 2 (stands for free degrees of freedom). Thus, eq. (23) can be written as

ª P11 «P ¬ 21

P12 º u1 ½ ª U11 ® ¾P22 »¼ ¯u 2 ¿ ¬« U 21

U12 º p1 ½ M11 M12 º U1 ½ 2 ª ® ¾ Z « »® ¾ U 22 ¼» ¯p 2 ¿ M ¬ 21 M 22 ¼ ¯U 2 ¿

(24)

where indices 1 and 2 stand for boundaries *1 and * 2 , respectively. As u1

0 and p 2

0 , Eq. (24) can be written as ˆ U Pˆ u 2 = Z 2 M 2

(25)

A u2 = O U2

(26)

ˆ -1 M ˆ A= H

(27)

that results in the following eigenproblem where

O is the eigenvalue that can be written as

O

1

Z2

(28)

and U 2 is the eigenvector. Eigenvalues and eigenvectors of eq. (26) can be obtained using standard numerical procedures for nonsymmetric matrices.

Numerical Results Two test problems were solved to verify the accuracy of the proposed method. Fully clamped plate was the first analysis carried out. A plate with the more general boundary condition SCSF was the second analysis carried out. In both cases it was considered a steel square plate of side length a 1 m and thickness

h 0, 01 m. The problems were discretized with a mesh of 20 quadratic boundary elements of equal length and 25 internal points (fig. 2b). Numerical integrations of RIM used 12 Gauss points.

Clamped Plate. First analysis considers a fully clamped plate (fig. 2a). Table 1 shows the first five natural frequencies computed by the RIM compared to results obtained in the literature [17].

316


a b Figure 2: (a) Steel square plate with clamped edges. (b) Boundary elements mesh and internal points.

present ref. [17] difference %

O1

O2

O3

O4

O5

36.058 35.988 0.19

72.325 73.398 -1.46

72.606 73.398 -1.08

109.89 108.26 1.51

123.41 131.89 -6.43

vibration modes Table 1: First five frequencies and respective vibration modes for a square plate with all sides as clamped.

As shown in Table 1, results are in good agreement. For first four mode shapes, differences are bellow 2 percent. SCSF Boundary Condition. Second analysis considers a plate with supported-clamped-supported-free boundary condition (fig. 3a).

a b Figure 3: (a) Steel square plate with SCSF boundary condition. Table 2 shows the first five natural frequencies computed by the RIM compared to results obtained in the literature [17, 18].

present ref. [17] difference %

O1

O2

O3

O4

O5

12.601 12.687 -0.68

32.763 33.067 -0.92

41.285 41.714 -1.03

63.263 63.260 0.01

72.744 73.870 -1.52

vibration modes Table 2: First five frequencies and respective vibration modes for a square plate with SCSF boundary condition.


As shown in Table 2, also in this case, results are in good agreement. All first five mode shapes present differences bellow 2 percent. Conclusions This work presented a boundary element formulation for the analysis of dynamic problems of Reissner’s thick plates. It used elastostatic fundamental solutions and domain terms were computed using RIM. Two numerical examples were shown. Results obtained using this approach were compared to results available in literature. Comparisons show a good agreement, with differences about 2 percent. Acknowledgements. The author are thankful to References [1] N. J. Altiero and D. L. Sikarskie. A boundary integral method applied to plates of arbitrary plan form. Computer and Structures, 9:163-168, 1978. [2] H. Tottenham. The boundary element method for plates and shells. In: P. K. Banerjee and R. Butterfield (Eds.), Development in Boundary Element Methods-I. Applied Science Publications, London, 1979. [3] G. Bézine. Boundary integral formulation for plate flexure with arbitrary boundary conditions. Mechanics Research Communications, 5(4): 197-206, 1978. [4] F. Hartmann and R. Zotemantel. The direct boundary element method in plate bending. International Journal for Numerical Methods in Engineering, 23:2049-2069, 1986. [5] V. F. Weeën. Application of the boundary integral equation method to Reissner’s plate model. International Journal for Numerical Methods in Engineering, 18:1-10, 1982. [6] V. J. Karam and J. C. F. Telles. On boundary elements for Reissner’s plate theory. Engineering Analysis with Boundary Elements, 5:21-27, 1988. [7] J. T. Katsikadelis and A. J. Yotis. A new boundary element solution of thick plates modelled by Reissner’s theory. Engineering Analysis with Boundary Elements, 12:65-74, 1993. [8] X. Yan. A new BEM approach for Reissner’s plate bending. Computer and Structures, 54:1085-1090, 1995. [9] Y. F. Rashed, M. H. Aliabadi and C. A. Brebbia. On the evaluation of the stresses in the BEM for Reissner plate bending problems. Applied Mathematical Modelling, 21:155-163, 1997. [10] V. F. Weeën. Application of the boundary integral equation method to Reissner’s plate model. International Journal for Numerical Methods in Engineering, 18:1-10, 1982. [11] C. S. de Barcellos and L. H. M. Silva. A boundary element formulation for the Mindlin’s plate model. In: C. A. Brebbia and W. S. Venturini (Eds.), Boundary Element Techniques: applications in stress analysis and heat transfer. Computational Mechanics Publication, Southampton, 1987. [12] T. Westphal Jr., E. Schnack and C. S. de Barcellos. The general fundamental solution of the sixth-order Reissner and Mindlin plate bending models revisited. Computational Methods Applied in Mechanical Engineering, 166:363-378, 1998. [13] P. W. Partridge, C. A. Brebbia and L. C. Wrobel. The Dual Reciprocity Boundary Element Method. Computational Mechanics Publication, Southampton, 1992. [14] A. J. Nowak and A. C. Neves. The Multiple Reciprocity Boundary Element Method. Computational Mechanics Publication, Southampton, 1994. [15] P. H. Wen, M. H. Aliabadi and D. Rooke. A new method for transformation of domain integrals to the boundary integrals in boundary element method. Communications in Numerical Methods in Engineering, 14:1055-1065, 1998. [16] Y. F. Rashed. Boundary Element Formulations for Thick Plates. WIT Press, CMP, Southampton, Boston, 2000. [17] S. Chakraverty. Vibration of Plates. CRC Press, Boca Raton, 2009. [18] S.A. Sadrnejad, A. Saedi Daryan and M. Ziaei. Vibration equations of thick rectangular plates using Mindlin plate theory. Journal of Computer Science, 5 (11): 838-842, 2009.

317

318


Analytical Expressions for Radial Integration BEM Xiao-Wei Gao1, Kai Yang2 and Jian Liu1 1

2

State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China. Email: [email protected]

Department of Engineering Mechanics, Southeast University, Nanjing 210096, China. Email: [email protected]

Keywords: Boundary element method; radial integration method; heat conduction; fourth-order spline RBF

Abstract. In this paper, a new approach using analytical expressions in the radial integration boundary element method (RIBEM) is presented for solving variable coefficient heat conduction problems. This approach can improve the computational efficiency considerably and can overcome the time-consuming deficiency of RIBEM in computing involved radial integrals. The fourth-order spline RBF is employed to approximate unknowns appearing in domain integrals arising from the varying heat conductivity. The radial integration method is utilized to convert domain integrals to the boundary resulting in a pure boundary discretization algorithm. A numerical example is given to demonstrate the efficiency of the presented approach.

Introduction Boundary element method (BEM) has been successfully applied to linear homogeneous heat conduction problems [1,2]. However, its extension to nonhomogeneous and nonlinear [3] problems is not straight forward, since usually there are domain integrals involved in the resulting integral equations, which make BEM lose its distinct advantage of only boundary discrimination. To circumvent this deficiency, the commonly used method is to transform the domain integrals into equivalent boundary integrals. In this method, the dual reciprocity method (DRM) developed by Nardini and Brebbia [4] is extensively used. However, DRM requires particular solutions to the basis functions, which restricts its application to complicated problems. Recently, a new transformation method, the radial integration method (RIM), has been developed by Gao [5,6], which not only can transform any complicated domain integrals to the boundary without using particular solutions, but also can remove various singularities appearing in domain integrals [7]. Although RIBEM is very flexible in treatment of the general nonlinear and non-homogeneous problems [8], evaluation of radial integrals numerically is very time-consuming, especially for large three-dimensional problems. In this paper, a new type of boundary-only integral equation analysis technique is developed for nonhomogeneous heat conduction problems based on the use of the fundamental solution of homogeneous problems. Unlike in the existing method (e.g. [6]) where the radial integrals of RIM are numerically evaluated, the radial integrals are analytically integrated based on the use of the fourth-order spline radial basis function (RBF). Through use of the analytical expressions in RIBEM, the computational efficiency can be improved considerably. Two numerical examples are given to demonstrate the efficiency of the presented method. Though the formulations are derived on the heat conduction background, they can also be applied to other physical problems [8].

Review of no homogeneous BEM in heat conduction problems [6] The governing equation for heat conduction problems in isotropic media with a spatially varying thermal conductivity can be expressed as

w wT ( x ) (k (x) ) Q (x) w xi w xi

0

where xi is the i -th component of the spatial coordinates at point x, k(x) the thermal conductivity, T(x) the temperature, and Q(x) the heat-generation rate. The repeated subscript i represents the summation through its range which is 2 for 2D and 3 for 3D problems.

(1)


319

Applying the weighted residual technique to eq (1) and using Gauss’s divergence theorem, the following boundary-domain integral equation can be obtained [6]

wG ( x, y ) ~ T ( x ) d *( x ) wn ~ ³ G ( x, y )Q ( x )d :( x ) ³ V ( x, y )T ( x )d :( x ) : : ~ ~ where q (x) is the heat flux, T ( x) and k ( x) are the normalized temperature and thermal conductivity, ~ T ( y)

³ G ( x , y ) q ( x ) d *( x ) ³ *

*

respectively, i.e.

q( x )

~ T ( x)

k ( x )

wT ( x ) , wn

V ( x, y )

~ k ( x)

k ( x)T ( x) ,

~ wG ( x, y ) wk ( x) wxi wxi

ln k ( x)

(2)

(3) (4)

Green’s function G ( x, y ) and its derivatives for equation can be expressed as [1]

1 1 °° 2S ln( r ) G ( x, y ) ® 1 ° °¯ 4S r r, i wG ( x, y ) 2SDr D wxi

for

2D (5)

for

3D (6)

where D 1 for 2D and D 2 for 3D problems, r is the distance between the source point y and the field point x, and r, i wr / wxi ( xi yi ) / r . Integral equation (2) is valid only for internal points. For boundary points, a similar integral equation can be obtained by letting y o * as is done in conventional BEM books [1-3].

Transformation of domain integrals to the boundary using RIM The two domain integrals involved in eq (2) are transformed into equivalent boundary integrals using the radial integration method (RIM) [5-7]. Usually, the heat generation rate Q(x) is a known function of the coordinates x and RIM can be directly used to transform the first domain integral in eq (2) to the boundary as follows

³

:

G ( x, y )Q( x)d :( x)

³

*

1 wr F ( z , y ) d *( z ) r D ( z , y ) wn

(7)

where

F ( z, y )

³

r( z,y)

0

G ( x, y )Q( x) r D dr

(8)

The radial integral shown in eq (8) can be evaluated analytically or numerically by using the following variable transformation relationship:

xi

yi r, i r

(9)

It is noted that r, i and yi are constants for the radial integral (8). This is an important characteristic of RIM. It makes the evaluation of the radial integral in equation (8) possible and easy for complicated functions Q(x).

~ wk ( x) are involved, which are wxi ~ wk ( x ) ~ T is unknown’s priories, the RIM formulations cannot be directly applied. To solve this problem, the term wxi ~

For the last domain integral in eq (2), since the normalized temperature T and

approximated by a series of prescribed radial basis functions (RBFs). Thus,

320


~ wk ( x ) ~ T ( x) wxi

NA

¦D A 1

NA

¦D

E

E

E

j 1

j 1 k j

I A ( R ) ¦ aij x j ¦¦ aijk x j xk ai0

A i

NA

¦D

A i

A 1

¦D

s i

)s

(10)

s 1

NA

A i

¦D

x Aj

A 1

A i

x Aj x kA

0

(11)

A 1

where NA is the total number of application points consisting of all boundary nodes and some selected internal points, D A , a i and a ij are coefficients to be determined, and Dis is a vector consisting of the coefficients D iA and

aij , aijk . The coefficients, D A , a i and a ij in eqs. (10) and (11) can be determined by collocating the application point A at all nodes. A set of algebraic equations can be written in the matrix form as

~ wk ~ ½ T¾ ® ¯ wxi ¿

[) s ]^D is `

(12)

If no two nodes share the same coordinates, the matrix [) s ] is invertible and thereby

~ ª wk º ~ « »T ¬ wxi ¼ ~ ~ wk ~ ½ wk ~ ~ ~ T ¾ and T are vectors consisting of values of T and T at all nodes, respectively, and where ® wxi ¯ wxi ¿

^D ` >) @

s 1

s i

~ wk ~ ½ T¾ ® ¯ wxi ¿

^`

>) @

s 1

^`

(13)

~ ª wk º « » ¬ wxi ¼

is a diagonal matrix. Substituting eq (10) into the last domain integral in eqs. (2)᧨(3) and (6), and then using RIM, the following results can be obtained.

~

³ V ( x, y )T ( x )d :( x ) ¦ D ³ s i

:

s

*

1 wr s Fi ( z, y )d *( z ) r D ( z, y ) wn

where

Fi s ( z, y )

³

r( z, y )

0

wG ( x, y ) s D ) r dr wxi

r( z, y )

r, i

0

2SD

³

) s dr

(14)

(15)

For the radial integral, r,i is a constant, thus

Fi s ( z, y ) R.

(30)

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), α2 R cos(np) sin(αq) , αR

(31)

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

In (αR) Kn (αR) − . I (αR K (αR)

(32)

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

(33) (34)

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 . (35) 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) = 2α2 A sin(η) cos(nη) cos(αζ) − αR 2 2wp (η, ζ) = −α AVn cos(nη) sin(αζ) ,

2uP (η, ζ) = 2α2 A cos(η) cos(nη) cos(αζ) +

(36) (37) (38)


Figure 2: Maximum absolute error for different grid sizes where Vn =

In (αR) Kn (αR) + I (αR K (αR)

(39)

The choice for γ1 (33) and γ2 (34) will produce the velocity components given above. This, then, provides a test case for the numerical integration of the boundary integrals (14).

4

Numerical Results

A specific test case is chosen with A = R = n = 1. Since the test case has a periodicity of 2π/α along the cylinder, the infinite integration in q may be replace by a finite range through the use of the method of images. Specifically, ∞ 2π/α ∞ K(p, q) dq = K(p, q + 2kπ/α) dq −∞

0

k=−∞

Ewald summation is used to accelerate the convergence of this sum. The surface parameters p and q are divided into N and M evenly spaced intervals with spacings h1 = 2π/N and h2 = 2π/(αM ). With the choice α = 1, the maximum absolute error on the surface in the numerical calculation of the velocity (14) is shown in Fig. 2 for various choices of the grid spacings. The evidence is strong that ln(error) is decreasing linearly in M when large enough, and hence the method is exponentially accurate. The case M = 2N has slightly worse errors and is due to the lower number of points in the azimuthal angle p. Keeping the spacing comparable in size seems to provide the best results. The exponential rate of convergence does not depend on the range in the transition function as demonstrated in Fig. 3. The results from the selection of ranges suggest that its width r2 − r1 is the most important factor. More experimentation would be helpful in further clarifying the choice of the transition function and its influence on the errors.


Figure 3: Maximum absolute error for different transition zones.

5

Conclusion

Converting the surface integration into polar coordinates allows a spectrally accurate calculation of the surface integral typically found in applications of boundary integral methods for free surface flow. The approach does require interpolation but by performing the integration first in the radial coordinate it is possible to restrict all interpolations to locations on grid lines, thus limiting the costs. Employment of a transition function also reduces the amount of interpolation to a disk around the point of singularity in the integrand. Acknowledgements: The research was supported by NSF (grant OCE-0620885). References: [1] J.T. Beale, A convergent boundary integral method for three-dimensional water waves, Math. Comput., 70, 2001, p. 977. [2] G.R. Baker, Boundary Element Methods in Engineering and Sciences, Chapter 8, Imperial College Press, 2010. [3] P.G. Saffman, Vortex Dynamics, Cambridge University Press, 1992. [4] 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. [5] 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. [6] G.R. Baker and J.T. Beale, Vortex blob methods applied to interfacial motion, J. Comput. Phys., 196, 2004, pp. 233–258. [7] G.R. Baker and H. Zhang, Blob regularization of boundary integral integrals, in Advances in Boundary Element Techniques XI, Berlin, Germany, 2010. [8] O. Bruno and L. Kunyansky, A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications, J. Comput. Physics, 169, 2001, p. 80.

Inverse analysis of solidification problems using the mesh-free radial point interpolation method Khosravifard1, M. R. Hematiyan2 1,2

Department of Mechanical Engineering, Shiraz University, Shiraz 71345, Iran 1 [email protected] 2 [email protected]

Keywords: Inverse analysis, Solidification, Meshless methods, pseudo heat source Abstract. An inverse method for optimal control of the freezing front motion in the solidification of pure materials is presented. The inverse technique utilizes the idea of a pseudo heat source to account for the latent heat effects. The numerical formulation of this inverse method is based on a formerly introduced meshless technique. The proposed method is based on the sequential regularization technique and Beck’s future time steps. In this method, the solid and liquid regions are modeled simultaneously. Introduction In a direct solidification problem, boundary conditions on the boundaries of the problem domain are specified, and the objective is to find the time-dependent location of the moving liquid-solid interface. However, in an inverse solidification problem, the boundary conditions, at least on some parts of the boundary are usually unknown. The inverse solidification problems are often considered as design problems, in that, the objective is to find the unknown boundary conditions in order to obtain a specific pattern of solidification front. It is well known that controlling the morphology of the solid-liquid interface during the solidification processing is of utmost importance to the formation of microstructure in the solidified materials [1]. In case of solidification of a molten metal, controlling the speed and pattern of the solidification front can have a great impact on the mechanical, as well as metallurgical properties of the solidified metal. By controlling the velocities at the freezing front, one can optimize the solidification time for different parts in a mold for a desired characteristic. Numerical methods are usually used for the inverse analysis of applied inverse solidification problems. In these methods, two conventional strategies can be utilized, i.e. the fixed grid techniques, and the moving or deforming grid techniques. The former incorporates the enthalpy formulation to account for the latent heat effects, while the latter continuously tracks the moving boundary by considering deforming grids or elements. There is also a third technique for the inverse analysis of solidification problems, which has found less attention in the literature. In this technique, the latent heat effect is accounted for, by considering a pseudo heat source near the moving boundary. The idea of using a pseudo heat source for consideration of the latent heat effects can reduce or eliminate the nonlinearity of the inverse solidification problem. By this approach, the inverse analysis of a solidification problem reduces to a simple inverse heat conduction problem, involving a heat generation source [2]. Although this method is not as accurate as the other two techniques, it can yield acceptable results with a little computational labor. The finite and boundary element methods are the most important numerical tools for the analysis of the inverse solidification problems. Katz and Rubinsky were among the first ones who used the FEM for the inverse analysis of solidification processes in one-dimensional problems [3]. Zabaras et al. utilized the BEM for the design of casting processes. Their method was aimed at finding a specific boundary flux history that would result in a desired solidification front motion that is required to control the liquid feeding to the front [4]. Zabras also provided an FEM methodology for the solution of several one-dimensional solidification problems [5]. Voller presented an enthalpy-based inverse algorithm for the inverse analysis of two-phase Stefan problems [6]. Later, Zabaras et al. presented a methodology for finding the boundary flux or temperature for obtaining a desired motion of the freezing interface in two-dimensional problems [7]. Okamoto and Li presented a design algorithm for optimal computation of boundary heat flux in solidification processing systems [1]. In their method, the Tikhonov regularization method, along with the L-curve method is used to select an optimal regularization parameter. Recently, Nowak et al. proposed a three-dimensional


numerical solution of the inverse boundary problem for a continuous casting process of an aluminum alloy [8]. In this paper, a pseudo heat source inverse method for design of 2D solidification of pure metals is presented. A previously developed improved mesh-free radial point interpolation method (IRPIM) [9, 10], is employed for sensitivity analysis. The latent heat effects are implemented by introducing a pseudo heat source near the interface. By using this method, the inverse solidification problem changes into a new inverse transient heat conduction problem that can be solved much more efficiently. The flux and the velocity of the liquid-solid interface are treated as secondary variables and the liquid and solid domains are modeled simultaneously. Some examples are also provided to demonstrate the effectiveness of the presented method. The inverse solidification problem based on the concept of a pseudo heat source In this paper, the latent heat effects of a solidifying medium are taken into account by considering a pseudo heat source near the moving solid-liquid interface. By considering the concept of the pseudo heat source and based on an improved meshless RPIM, a methodology is presented for the computation of an optimal distribution of heat flux history on the boundary, which results in a desired motion of the solid-liquid interface. In the present method, the fluxes on the moving boundary and its velocity are regarded as secondary variables, while the primary unknown in the inverse problem is the flux on the fixed boundaries of the problem. The modeling of the problem can be regarded as that of fixed domain, since both the solid and liquid phases are modeled simultaneously. As a result, the equilibrium and compatibility constraints are automatically satisfied at the moving boundary. To account for the latent heat effects a moving pseudo heat source is considered near the solid-liquid interface. The position of the heat source is updated at each time step of the problem analysis, but the interface position is fixed during each time step. In order to obtain stable and smooth results, primary and secondary regularizations are performed. Problem statement. A liquid, which is initially at a temperature equal or above its melting temperature and occupies a domain is considered. Figure 1 depicts the problem geometry and the terminology used in this paper. A part of the fixed boundary on which prescribed boundary conditions are applied is denoted by * p , and the rest of the boundary with the unknown boundary heat flux is referred to as *q . The moving

boundary, i.e. the solid-liquid interface, at any instant of time is denoted by *I t . The moving boundary initially coincides with the fixed boundary *

*p *q . Starting at the time t

0 the boundary portion *q

is cooled and the interface proceeds inwards, generating a two-phase medium. A part of the domain that is solidified up to time t is denoted by : s and the part of the domain which is still liquid is denoted by :l . In the numerical formulation of the inverse problem, n 1 discrete time steps are considered. The objective of the problem is to find the distribution of the heat flux on *q at the mentioned time steps. The part of the domain which solidifies during the time step from t M 1 to t M is referred to as the transition zone and denoted by :T .

Figure 1: Geometry and terminology of the inverse solidification problem

For obtaining the distribution of the heat flux on *q at each time step, we adopt the function estimation approach, i.e. N q discrete points on the boundary *q are selected and a value of heat flux at each point of the boundary is obtained by the inverse analysis. In the formulation presented in this paper, *I at the n 1

discrete time steps (t0, t1, …, tn) is prescribed, and the boundary heat flux qB ti , i 1, 2, , n is to be so determined as to achieve the desired solid-liquid interface motion. Suppose that the inverse problem is analyzed up to the time t M 1 and the heat flux qB t and the

t1 , t2 , , t M 1 . Now t M . The vector q B contains

temperature distribution within the domain is obtained at the preceding time steps t it is required to obtain the heat flux vector q B on *q for the new time step t

the values of the heat flux at the N q points on the boundary *q . Considering r future time steps [11], the vector q B is written as follows:

qB

>q

qi

>q

with

1

i,M

q 2 q Nq

@

T

(1)

qi , M 1 qi , M r @

(2)

where qi,k in Eq. (2) is the heat flux on the ith point on the boundary *q at the time tk.

At each time step of the inverse analysis, the position of the moving boundary *I is assumed fixed at *I tM for all the r time steps. The computations are done for the r time steps, however only the fluxes corresponding to the first time step are preserved. These r future time steps are considered to obtain smooth results. In the analysis of the problem from t t M 1 to t t M a pseudo heat source is applied in the transition region to model the latent heat effects. At each time step of the inverse analysis, the value of the heat source which is considered to be uniformly distributed over :T is selected such that it can provide as much energy

as required for the solidification of the amount of material which is enclosed in :T . In this way, the intensity of the pseudo heat source can be expressed as

g

UL

(3)

't

where U is the density of the material, L is the latent heat, and 't

t M t M 1 .

To obtain the vector of heat fluxes q B , a least squares function corresponding to the inverse heat conduction problem is formed as follows [11]:

S

Y T T Y T Dq TB q B

Y

Y1 ½ °Y ° ° 2° ® ¾; T ° ° °¯YJ °¿

T1 ½ °T ° ° 2° ® ¾ °° °¯TJ °¿

(5)

Yi

Tm ½ °T ° ° m° ® ¾; Ti °° °¯Tm °¿

Ti , M ½ ° °T ° i , M 1 ° ¾ ® ° ° °¯Ti , M r °¿

(6)

(4) where the vectors Y and T contain, respectively, the values of the desired and estimated temperature at the J points on the moving interface *I tM . These vectors are written as

in which

In Eq. (4), is a regularization parameter. The temperature field can be represented by a Taylor’s series expansion in terms of an arbitrary vector of ~ as follows: heat fluxes q B


~ ~ T Xq B q (7) B ~ The vector T is a temperature vector, obtained by an analysis of a direct problem with the vector of heat ~ applied at the boundary * . The matrix X in Eq. (7) is the sensitivity matrix of the problem with fluxes q B q T

respect to the heat flux components, and is expressed as

X

ª X11 «X « 21 « « ¬« X J 1

X12 X1N q º » X 22 » » » X JN q ¼»

(8)

Each element of the matrix X is a lower triangular matrix of the following from:

Xij

ª wTi , M « wq j,M « « « « « « wTi , M r 1 « wq j,M ¬

º » » » » » 0 » wTi , M r 1 » wq j , M r 1 »¼

0 wTi , M 1 wq j , M 1

0

(9)

The simplest method for the computation of the elements of the sensitivity matrix is to use the finite difference approximation for evaluation of the derivatives. After substituting the vector T from Eq. (7) into Eq. (4) and minimizing the least squares function S with respect to heat flux components, the vector q B is obtained as follows [2]:

qB

>X

T

X DI

@ >X Y T~ X 1

T

T

~ Xq

@

(10)

Secondary regularization. As the value of in Eq. (4) increases, the oscillations of the heat flux on the boundary decrease. However, increasing the value of increases the difference of the desired and estimated values of the heat flux. A remedy for this problem is proposed by Hematiyan and Karami [2]. In their method, after an appropriate value for is selected, a so-called secondary regularization is performed on the results, which further smooth the heat flux. In this method, to obtain a vector Vc with smooth elements from a vector V with oscillatory elements, one can use the following formula [2]:

Vc

S S T

1

S TK

(11)

where

S

ª I º «J H »; K ¼ ¬

V ½ ® ¾ ¯0¿

(12)

in which

H

0 0 ª0 0 «1 2 1 0 « «0 1 2 1 « 1 2 «0 0 «¬0 0 0 0

0º 0»» 0» » 1» 0»¼

(13)

In general H is an N×N matrix, where N is equal to the number of elements of V. However, Eq. (13) gives H for the special case with N=5. J in Eq. (12) is a regularization parameter, with the usual values between 0.5 and 5. Hematiyan and Karami stated that the summation of the elements of the vectors V and Vc are equal. For instance, if the vector V describes the distribution of heat flux intensity with respect to time, then the resulted vector Vc imposes the same amount of total energy to the domain as the vector V does.

The meshless IRPIM formulation for the general transient heat conduction problem including heat sources. The general transient heat conduction equation for an isotropic medium is given as follows:

k ( x, T )T ( x, t ) g ( x, T , t )

U ( x , T ) c ( x, T )

wT ( x, t ) wt

(14)

where is the density, c is the specific heat, k is the thermal conductivity, and g is the heat generation per unit volume. The initial and boundary conditions for Eq. (14) are T (x,0) T0 (x) in the domain (:) ,

T T on the boundaies with the essential condition (1 ) , k T n q on the boundaries with the natutal condition ( 2 ) , where n is the unit outward normal vector to the boundary, T0 is the initial condition, T is the applied temperature on the boundary, and q is the applied normal heat flux over the boundary. On using the interpolation functions of the RPIM in the Galerkin weak form of Eq. (14), one obtains the following system of equations: K (T )T F(T , t ) M (T )T (15) where:

M ij

³

K ij

³

Fi

:

³

:

U (x, T )c ( x, T )IiI j d:

ª wI wI j wIi wI j wIi wI j º k (x, T ) « i » d: wz wz ¼ ¬ wx wx wy wy g ( x, T , t )Ii d: ³ q Ii d*

:

*2

(16) (17) (18)

In Eqs (16) to (18), i and j are the RPIM shape functions corresponding to the nodes i and j, respectively. In the meshless IRPIM of this paper, the domain integrals in Eqs (16) to (18) are evaluated with a meshless integration technique, namely the Cartesian transformation method (CTM). For instance, the evaluation of the mass matrix in Eq. (16) by the CTM is as follows [9, 10]:

¦ W R u C N

M ij

2D

l

l

l

u S i ,l u S j ,l

(19)

l 1

where Wl

2D

is the integration weight of the CTM corresponding to the lth integration point, and there are a total of N

integration points in the domain. The expression for other elements of Eq. (19) and similar expression for Eqs (17) and (18) can be found in [9, 10].

Numerical examples To study the effectiveness of the proposed method, two example problems are analyzed. In the first example, a physically feasible motion of the solid-liquid interface is considered, and the corresponding history of heat flux on the fixed boundary is obtained. In the second example, a freezing front motion which violates the governing equations, and therefore is not physically possible, is selected. A heat flux history that results in an interface motion that best matches the desired freezing front motion is obtained in this example. Example 1: A unidirectional solidification problem, with varying interface velocity. In this example, solidification of a casting in a long and narrow mold with a varying front velocity is simulated and designed. The problem geometry and boundary conditions are shown in Fig. 2 (a). The design objective of this example is to achieve a unidirectional motion of the freezing front, parallel to the line AB, by computation of an optimal value for the heat flux on the boundary line AB. It is required that the velocity of the front vary uniformly from 10-4 m/s at the beginning of the process to 5×10-4 m/s when the whole casting is solidified. It is also required that the whole solidification process takes place in 300 seconds. The material properties of the casting are k=240 W/m°C, c=950 J/kg°C, =2700 kg/m3, and L=397 kJ/kg. Both the melting and initial temperatures of the casting are 660 °C.


Figure 2: Configuration of example 1: (a) the geometry and boundary conditions; (b) the nodal arrangement of the meshless IRPIM The inverse problem is analyzed once by considering 5 time steps ( 't 60 s ) and once by 10 time steps ( 't 30 s ), and the results are compared. Three and five future time steps are considered for the cases with 't 60 s and 't 30 s , respectively. 15 sampling points are selected on the moving boundary, and 6 points are selected on the boundary line AB for the sake of computation of the required heat flux. It should be stated that as the moving front gets further away from the fixed boundary line AB, the control of the front motion by application of heat flux on the boundary becomes more difficult. The numerical solutions are performed with the meshless IRPIM [9, 10]. The obtained values for the average heat flux output from the boundary line AB is given in Table 1. In this table, by tend we mean the time at the end of each time step. Table 1: The average value of heat flux (KW/m2) on the boundary line AB, example 1. tend 30 60 90 120 150 180 210 240 270 300 5 steps 159 159 214 214 354 354 503 503 677 677 10 steps 124 173 176 255 357 368 523 504 706 726 Figure 3, depicts the location of the freezing front at various instances of time, as obtained by applying the computed values of the heat flux at the boundary line AB. To obtain the designed front position, a direct enthalpy-based solidification analysis is performed by ANSYS.

Figure 3: The solid-liquid interface position obtained by the present method, example 1

Example 2: Design of a continuous casting process. In this example, a continuous casting process is designed. The problem geometry and boundary conditions are shown in Fig. 4. In this figure, the desired motion of the solid-liquid interface is shown by the dashed lines. The solid-liquid interface moves inwards parallel to the fixed boundaries at a constant speed of 5×10-4 m/s. It should be noted that the desired solidification front is physically impossible because of the steep corners. The aim of this example is to find a heat flux history that results in an interface motion that best matches the desired freezing front motion.

Figure 4: The problem specification of example 2 The material properties of the casting are k=34 W/m°C, c=475 J/kg°C, =7800 kg/m3, and L=271 kJ/kg. Both the melting and initial temperatures of the casting are 1450 °C. In this example, one can make use of the existing symmetry of the problem, and only one eighth of the geometry can be modeled. The geometrical model of the numerical method, along with the nodal distribution of the meshless IRPIM is shown in Fig. 5.

Figure 5: The geometrical model and nodal distribution of the numerical method For the inverse analysis of this example, 5 time steps are used, and the values of the heat fluxes are computed at the beginning of each time step. Also, six points on the boundary line AB (Fig. 5) are chosen, at which the values of heat flux are computed. Figure 6, depicts the computed values of the heat flux on the boundary line AB at various instances of time. Figure 7, depicts the location of the freezing front at various instances of time, as obtained by applying the computed values of the heat flux at the boundary line AB. To obtain the estimated front position, a direct enthalpy-based solidification analysis is performed by ANSYS.


Figure 6: Heat flux on the boundary line AB at various instances of time, example 2

Figure 7: The solid-liquid interface position obtained by the present method, example 2 Conclusions Based on the concept of pseudo heat source method and the use of an improved meshless RPIM, a procedure for optimal calculation of heat fluxes in solidification processes was presented. The presented method, converts an inverse solidification problem into a much simpler inverse heat conduction problem. As a result, the computational cost of the presented procedure is much lower than other inverse solidification methods. Sequential and secondary regularizations are performed to damp the oscillations. Two examples were presented that demonstrate the usefulness of the presented technique. References [1] K. Okamoto and B.Q. Li International Journal of Heat and Mass Transfer, 50, 4409–4423 (2007). [2] M.R. Hematiyan and G. Karami Computational Mechanics, 31, 262–271 (2003). [3] M.A. Katz and B. Rubinsky Numerical Heat Transfer, 7, 269–283, (1984). [4] N. Zabaras, S. Mukherjee and O. Richmond Journal of Heat Transfer, 110, 554–561, (1988). [5] N. Zabaras International Journal for Numerical Methods in Engineering, 29, 1569–1587, (1990). [6] V.R. Voller Numerical Heat Transfer, Part B, 21, 41–55, (1992). [7] N. Zabaras, Y. Ruan and O. Richmond Numerical Heat Transfer, Part B, 21, 307–325, (1992). [8] I. Nowak, J. Smolka and A.J. Nowak Applied Thermal Engineering, 30, 1140–1151, (2010). [9] A. Khosravifard and M.R. Hematiyan, Advances in Boundary Element techniques XI, BETEQ 2010, 12-14 July, Berlin, (2010). [10] A. Khosravifard, M.R. Hematiyan and L. Marin Applied Mathematical Modelling (2011), doi: 10.1016/j.apm.2011.02.039. [11] J.V. Beck, B. Blackwell and C.R. St. Clair Inverse Heat Conduction: Ill posed Problems, Wiley Interscience (1985).