Advances in Boundary Element & Meshless

ISBN 978-0-9576731-0-6

Advances in Boundary Element & Meshless Techniques XIV

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. Previous conferences devoted to Boundary Element and Meshless 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), Berlin, Germany (2010), Brasilia, Brazil (2011) and Prague, Czech Republic (2012)

EC ltd

Advances in Boundary Element & Meshless Techniques XIV

Edited by A Sellier M H Aliabadi

Advances In Boundary Element and Meshless Techniques XIV

Advances In Boundary Element and Meshless Techniques XIV

Edited by A Sellier M H Aliabadi

EC

ltd

Published by EC, Ltd, UK Copyright © 2013, Published by EC Ltd, P.O.Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

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ISBN: 978-0-9576731-0-6

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 otherwise, 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 XIV 16-18 July 2013, Paris, France

Organising Committee: Prof. Antoine Sellier LadHyX. Ecole Polytechnique. 91128. Palaiseau cedex. France. Tel: +33 (0)1 69 33 52 77 [email protected] Prof. Ferri M H Aliabadi Department of Aeronautics Imperial College, South Kensington Campus London SW7 2AZ [email protected] International Scientific Advisory Committee Abascal,R (Spain) Abe,K (Japan) Baker,G (USA) Benedetti,I (Italy) Beskos,D (Greece) Blasquez,A (Spain) Chen, Weiqiu (China) Chen, Wen (China) Cisilino,A (Argentina) Darrigrand, E (France) De Araujo, F C (Brazil) Denda,M (USA) Dong,C (China) Dumont,N (Brazil) Estorff, O.v (Germany) Gao,X.W. (China) Garcia-Sanchez,F (Spain) Hartmann,F (Germany) Hematiyan,M.R. (Iran)

Hirose, S (Japan) Kinnas,S (USA) Liu,G-R (Singapore) Mallardo,V (Italy) Mansur, W. J (Brazil) Mantic,V (Spain) Marin, L (Romania) Matsumoto, T (Japan) Mesquita,E (Brazil) Millazo, A (Italy) Minutolo,V (Italy) Ochiai,Y (Japan) Panzeca,T (Italy) Perez Gavilan, J J (Mexico) Pineda,E (Mexico) Prochazka,P (Czech Republic) Qin,Q (Australia) Saez,A (Spain) Sapountzakis E.J. (Greece) Sellier, A (France) Semblat, J-F (France) Seok Soon Lee (Korea) Shiah,Y (Taiwan) Sladek,J (Slovakia) Saldek, V (Slovakia) Sollero.P. (Brazil) Stephan, E.P (Germany) Taigbenu,A (South Africa) Tan,C.L (Canada) Telles,J.C.F. (Brazil) Wen,P.H. (UK) Wrobel,L.C. (UK) Yao,Z (China) 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), Berlin, Germany (2010), Brasilia, Brazil (2011) and Prague, Czech Republic (2012). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the LadHyX. Ecole Polytechnique, Paris, th France during 16-18 July 2013. 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. Editors

CONTENT Boundary element method for hyperbolic problems: recent advances A. Temponi, F. Valvona, A. Salvadori and A. Carini Inverse heat source recovery by the Green element method A. E. Taigbenu A meshless approach for the mandible bone tissue remodelling analysisdue to the insertion of dental implants A.S.S.L. Ferreira, J. Belinha, L.M.J.S. Dinis and R.M.Natal Jorge Energetic BEM-FEM coupling for wave propagation in layered media A. Aimi, L. Desiderio, M. Diligenti, A. Frangi, C. Guardasoni Performance of CMRH for the simulation of a vibroacoustic problem based on coupled finite and boundary element method A. Alia, H. Sadok Multidomain formulation of BEM analysis applied to large-scale polycrystalline materials A. F. Galvis, R. Q. Rodriguez, P. Sollero, E.L Albuquerque Gravity-driven migration of bubbles and/or solid particles near a free surface M. Guemas, A. Sellier and F. Pigeonneau Vortex Patches G. Baker A cohesive boundary element approach to material degradation in three-dimensional polycrystalline aggregates I. Benedetti, M.H. Aliabadi Macro-zones SGBEM approach for static shakedown analysis of convex optimization T. Panzeca, M. Salerno and L. Zito Strain energy evaluation in structures having zone-wise physicalmechanical T. Panzeca, F. Cucco and S. Terravecchia Dual boundary element method for fatigue crack growth: implementation of the Richard’s criterion F. Bonanno, I. Benedetti, A. Milazzo and M.H. Aliabadi Analysis of stiffened panels using coupled DBEM and FEM Z. Sharif Khodaei and M. H. Aliabadi Application of BEM for internal propulsor flows S.A. Kinnas and A. Valsaraj A rapid approach to BEM-BEM acoustic-structural coupling A. Brancati, M.H. Aliabadi and A. Alaimo

1 7 13 21 27 32 38 44 50 56 62 70 78 82 89

Performance of the recursive procedure in elastic problems Modeled by the Boundary Element Method C. F. Loeffler, A. B. Freitas, L.Valoto A special boundary element for elastodynamic analysis of anisotropic plates with cracks C. Hwu and Y.C. Chen A rapid approach to BEM quadratic formulation for acoustic problems A. Brancati, M.H. Aliabadi and A. Milazzo A quadrature rule for Hadamard finite-part integrals L. S. Campos and E.L. Albuquerque Series expansion of anisotropic plane elasticity fundamental solutions A. dos Reis, E. L. Albuquerque, and C.Tatiana Mota Anflor A boundary element method formulation applied to the dynamic analysis of Timoshenko beams J. A. M. Carrer, S. A. Fleischfresser, L. F. T. Garcia, W. J. Mansur One dimensional modelling of scalars transport C. L. N. Cunha, J. A. M. Carrer, M. F. Oliveira, W. J. Mansur Calculation of boundary and interfacial stresses in 3D solids by employing a SBS-based BEM F. C. de Araújo and C. R. da Silva Jr. Development of compressible-incompressible link to efficiently model bubble dynamics near floating body C-T Hsiao, G.L. Chahine Exact computation of drilling rotations with the boundary element method D. I. G. Costa, E. L. Albuquerque, P. M. Baiz A boundary element formulation with boundary only discretization for the stability analysis of perforated thin plates P. C. M. Doval, E. L. Albuquerque and P. Sollero Combining analytic preconditioner and fast multipole method for the 3-D Helmholtz equation M. Darbas, E. Darrigrand, Y. Lafranche Nonlinear BEM Formulation based on tangent operator applied to cohesive crack growth modelling H. L. Oliveira and E. D. Leonel Treatment of singularities in fundamental solutions of orthotropic thick plates A.P. Santana, E.L. Albuqerque, L.S. Campos, D.I.G. Costa Analysis of folded plates by the boundary element method K. R. P. Sousa, E. L. Albuquerque, D. I. G. Costa, S. Hoefel

95 101 107 112 117 124 130 135 141 147 153 167 171 178 184

The boundary element method applied to the analysis of Fluid extraction from a reservoir A. B. Dias Jr, E. L. Albuquerque, E. Fortaleza Application of the boundary element method to creep with plasticity E. Pineda León, M.H. Aliabadi, A. Rodríguez-Castellanos and J. Zapata Derivation of 3D anisotropic heat conduction Green’s functions using a 2D complex-variable method F. C. Buroni, A. Sáez, M. Denda A non-linear BEM formulation for plate bending with quadratic convergence G. R. Fernandes, E. A. de Souza Neto Application of dual boundary element method in active sensing F. Zou, I Benedetti, M. H. Aliabadi Combining regularization and desingularization techniques in the method of fundamental solutions C Gáspár Analysis of multiple inclusion potential problems by the adaptive cross approximation method R. Q. Rodriguez, A.F. Galvis, P. Sollero and E. L. Albuquerque Analysis of solidification involving finite heat wave speed by a meshless method A. Khosravifard, M. R. Hematiyan Material tailoring in functionally graded rods under torsion F. Liaghat, M. R. Hematiyan, A. Khosravifard The radial basis function methods for wave propagation analysis H. Zheng, Ch Zhang, W. Chen, J. Sladek, V. Sladek Air cushioning in drop impact on solid surface C. Josser and L Duchemin Bending of the circular porous piezoelectric plate J. Sladek, V. Sladek, Ch Zhang and l Wünsche Crack opening path prediction using a meshless method J. Belinha, J.M.C. Azevedo, L.M.J.S. Dinis and R.M. Natal Jorge A quadratic implementation for 2D elasticity topology optimization using BEM K.L. Teotônio, C. T.M. Anflor, É.L Albuquerque A new method to reduce the stiffness of interfacial flows with surface tension L Duchemin Application of the time-domain boundary element method to analysis of flow-acoustic interaction in expansion chamber silencer models

190 195 201 207 213 219 225 231 237 243 251 257 263 270 275 280

Mikael A. Langthjem, Masami Nakano On explicit expressions of 3D elastostatic Green's functions and theirOn Explicit Expressions of 3D Elastostatic Green's Functions and Their Derivatives for Anisotropic Solids L Xie, Ch Zhang, Y Wan and Z Zhong Quadratic programing for minimization of the total potential energy to solve contact problems using the collocation BEM C. G. Panagiotopoulos, V. Mantic, I.G. Garcıa, E. Graciani An efficient BEM approach to quasi-static viscoelasticity of KelvinVoigt materials C. G. Panagiotopoulos, V. Mantic The application of BEM to analysis of elastic phononic solids with local resonance H.F. Gao, T. Matsumoto, T. Takahashi, H. Isakari BEM modelling of interface cracks in a group of fibres under biaxial transverse loads L. Tavara, V. Mantic, E. Graciani, F. Parıs Explicit evaluation of integrals arising in Galerkin BEM M. Lenoiry, N. Sallesy Density results and the method of fundamental solutions for Cauchy data reconstruction. Carlos J.S. Alves and Nuno F.M. Martins Elastoplastic dynamic analysis of beam-foundation systems employing BEM A. E. Kampitsis and E.J. Sapountzakis Diffraction of in-plane (P, SV) and anti-plane (SH) waves in a half – space with cylindrical tunnels S. Parvanova, P. Dineva, G. Manolis, F. Wuttke DRBEM solution for the incompressible MHD equations in terms of magnetic potential B. Pekmen and M. Tezer-Sezgin Application of the ACA compression technique for the scattering of Periodic Surfaces R. Perrussel, J.-R. Poirier Numerical solution of 2D steady-‐state thermoelastic problems through a new and simple meshless Local Boundary Integral Equation (LBIE) method in combination with the Boundary Element Method(BEM) T. Gortsas, S.V. Tsinopoulos, E.J. Sellountos, D.Polyzos A detailed boundary element analysis of the flow field outside a growing immiscible viscous fingering within a Hele-Shaw cell H. Power, D. Stevens and A. Cliffe

286

292 298 304 311 317 323 329 335 341 347 352

358

A solution procedure to 3D integral nonlocal elasticity: Coupling isotropic-BEM with strong form local radial point interpolation. R Kouitat Njiwa, N Taha Niane, M Schwartz Solutions for free vibration analysis of thick square plates by the boundary element method W.L.A. Pereira, V.J. Karam, J.A.M. Carrer, W.J. Mansur Numerical simulation of heat and mass transport S Gümgüm The method of fundamental solutions of MHD pipe flow in an exterior region S. Han Aydın Quadratic B-splines in the analog equation method for the nonuniform torsional problem of bars E.J.Sapountzakis and I.N.Tsiptsis Arbitrary Stokes flow about a fixed or freely-suspended slip particle A. Sellier Gravity-driven migration of bubbles and/or solid particles near a free surface M. Guemas, A. Sellier and F. Pigeonneau Elastoplastic analysis of structures using the NNRPIM S.F. Moreira, J. Belinha, L.M.J.S. Dinis and R.M. Natal Jorge A simple BEM based solution procedure for some extended continuum mechanical problems: application to a microdilatation medium N Thurieau, R Kouitat Njiwa, M Taghite Efficient BEM stress analysis of 3D generally anisotropic elastic solids with stress concentrations and cracks Y.C. Shiah, C.L. Tan and Y.H. Chen Optimization of cathodic protection systems combining genetic algorithms and the method of fundamental solutions W. J. Santos, J. A. F. Santiago and J. C. F. Telles A level set-based topology optimization method using 3D BEM T. Yamada, S. Shichi, T. Matsumoto, T. Takahashi, H. Isakari Mesh-free solutions for bending of thin plates with variable bending stiffness V. Sladek, J. Sladek and L. Sator An SGBEM implementation with quadratic programming for solving contact problems with Coulomb friction R.Vodicka, V. Mantic Nonlocal elasticity analysis by local integral equation method P.H. Wen, X.J. Huang, M.H. Aliabadi

364 370 376 382 388 394 400 406 412

418 426 432 438 444 450

Boundary element analysis of polymer composites under frictional contact conditions L. Rodrguez-Tembleque, F.C. Buroni, R. Abascal and A. Saez Efficient FFT–MFS algorithms for boundary value problems in two-dimensional linear thermoelasticity A. Karageorghis and M. Liviu Coupling boundary integral and shell finite element methods to study the fluid structure interactions of a microcapsule in a simple shear flow C. Dupont, A-V Salsac, D Barthès-Biesel, M Vidrascu, P Le Tallec Far field Green’s functions for time-harmonic loading in magnetoelectroelastic materials G Hattori, A Saez and A Bostrom A meshless approach for the mandible bone tissue remodelling analysis due to the insertion of dental implants A.S.S.L. Ferreira, J. Belinha, L.M.J.S. Dinis and R.M.Natal Jorge Porosity effects on elastic properties of polycrystalline materials: a three-dimensional grain boundary formulation F Trentacoste, I Benedetti, M H Aliabadi (Abstract) Effects of voids and flaws on mechanical properties, and on intergranular damage and fracture for polycrystalline materials G Geraci, I Benedetti, M H Aliabadi (Abstract) DRBEM solution of Liquid Metal MHD Flow in a Staggered Double Lid-Driven Cavity M. Tezer-Sezgin and B. Pekmen

457 463 469

475 481 489 490 491

Advances in Boundary Element Techniques XIV

Boundary element method for hyperbolic problems: recent advances A. Temponi1, F. Valvona2, A. Salvadori1 and A. Carini1 1

Ce.Si.A. - DICATAM, University of Brescia, Italy, [email protected], [email protected], [email protected] 2

Department of Engineering and Geology, University of Chieti-Pescara, Italy, [email protected]

Keywords: hyperbolic problems, time domain BEM, energetic weak formulation, analytical integrations

Abstract. The present note aims at giving a further contribution in the numerical approximation of hyperbolic problems via Boundary Element Methods (BEM). A time-dependent BEM, based on a weak formulation (known as energetic), is formulated and implemented. The formulation covers a wide range of classical dynamic problems governed by the equations of scalar waves as well as linear elastodynamics. Complex 3D bodies are considered, made of different materials (multimaterials) and containing stationary cracks. Some numerical benchmarks validate the proposed approach. Introduction Boundary Integral Equations (BIEs), and their approximated solution through Boundary Element Methods (BEMs), have been successfully used for decades in the propagation and scattering of acoustics, electromagnetic and elastic waves. Several modern research endeavours and applications have dealt with them: to cite but a few, recent publications have considered the analysis of active noise control [1] and the study of wave propagations within soils, modelled as fully or partially saturated poroelastic media [2, 3]. Let Ω ∈ R3 , be an open domain with compact boundary Γ = Γu ∪ Γp . One considers the problem of a multimaterial body: Ω is constituted by N subdomains made of isotropic linear elastic material, denoted by Ωn , with n ∈ IN = {1, 2, ..., N }; the Dirichlet and Neumann boundaries of each subdomain are defined as: ¯ n ∩ Γu ; ¯ n ∩ Γp Γnp = Ω Γnu = Ω n n ¯ where Ω denotes the closure of the open domain Ω . The subdomains are connected by Mi perfect interfaces (interfaces with continuous primal field along them), the m-th interface is denoted by Γm i , with m ∈ IMi = {1, 2, ..., Mi }. Besides Mc traction free stationary cracks (cracks with no motion of the crack front) are present inside the domain; Γlc , with l ∈ IMc = {1, 2, ..., Mc }, denotes the l-th crack. The response of such a system to time-dependent external actions is the reference problem of the paper. Under the assumptions of small displacements and strains, vanishing initial conditions, absence of body forces, and unpressurized cracks, the initial boundary value problem reads: ⎧ ün (x, t) − Lx [un (x, t)] = bn (x, t) ; ⎪ ∀x ∈ Ωn ; ∀t ∈ (0, ∞) ; ∀n ∈ IN ⎪ ⎪ ⎪ n (x, 0) = un (x) ; ⎪ ⎪ u ∀x ∈ Ωn ; ∀n ∈ IN 0 ⎪ ⎪ ⎪ n (x, 0) = u n (x) ; ⎪ ˙ ˙ ⎪ ∀x ∈ Ωn ; ∀n ∈ IN u 0 ⎨ (1) ¯n (x, t) ; un (x, t) = u ∀x ∈ Γnu ; ∀t ∈ [0, ∞) ; ∀n ∈ IN ⎪ ⎪ n; ⎪pn (x, t) = p¯n (x, t) ; ∀x ∈ Γ ∀t ∈ [0, ∞) ; ∀n ∈ I ⎪ N p ⎪ ⎪ ⎪ ⎪ wm (x, t) = 0 ; ∀x ∈ Γm ; ∀t ∈ [0, ∞) ; ∀m ∈ IMi ⎪ i ⎪ ⎪ ⎩l t (x, t) = 0 ; ∀x ∈ Γlc ; ∀t ∈ [0, ∞) ; ∀l ∈ IMc where wm (x, t) represents the jump of the primal field along the m-th interface, while Lx is the differential operator of the second order defined as: c2 Δu(x, t) for scalar wave problems Lx [u(x, t)] = c21 ∇ ∇ · u(x, t) − c22 ∇ × ∇ × u(x, t) for elastodynamics

1

2

which allows a unified representation of scalar wave problems and elastodynamics. Energetic integral formulation and integration issues Formulation. The integral formulation of problem (1) (see e.g. [4, 5]) can be obtained by using the time dependent fundamental solutions (or Kernels) and Graffi reciprocity theorem [6]. The compact integral form of problem (1) reads: L [ z(x, t) ] = f (x, t) (2) where: a) z(x, t) is the unknown vector, it is made of: dual actions pn on the n-th Dirichlet boundary Γnu , primal field unknowns un on the n-th Neumann boundary Γnp , dual tm and primal um actions l l on the m-th perfect interface Γm i and primal field jumps w on the l-the stationary crack Γc b) the vector f (x, t) gathers all the boundary data; it contains integrals over the Dirichlet Γnu and ¯(x, t) and ¯ p(x, t) respectively Neumann Γpu boundaries of the data u c) L is an integral operator with a block-wise structure, each block represents the effects on a certain sub-domain or interface due to the unknowns acting on each surface, either an external (Dirichlet or Neumann) boundary or internal (interface or crack) boundary By defining a suitable extension of the bilinear form AE , defined in [7] for a single domain, to the problem of multimaterial cracked body (see [5] for details), one obtains the energetic weak form of the problem (2): ⎡ ⎛ ⎞⎤ ⎛ ˙ u,n ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ˙ u,n ⎞ ς ς ς f f p,n p,n ⎢ ⎜ ν˙ ⎟⎥ ⎜ f ⎟ ⎜ν˙ ⎟ ⎜ν˙ ⎟ ⎜f ⎟ ⎟ ⎢ ⎜ ⎟⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ i,m ⎟ ⎢ ⎜ ⎟⎥ ⎜ i,m ⎟ ⎜ ⎟ ⎜ ⎟ Given ⎜ ⎜g˙ ⎟ find z(x, t) ∈ W : AE ⎢z(x, t), ⎜ ι ⎟⎥ = ⎜g˙ ⎟ , ⎜ ι ⎟ ; ∀ ⎜ ι ⎟ ∈ W ⎣ ⎝υ˙ ⎠⎦ ⎝gi,m ⎠ ⎝υ˙ ⎠ ⎝υ˙ ⎠ ⎝gi,m ⎠ gc,l ϑ˙ ϑ˙ ϑ˙ gc,l Discretization. The discretization of the unknown fields is achieved by space-time polynomial shape functions on a trapezoidal tassellation of Γ and a uniform decomposition of segment (0, T ). Shape functions are taken of tensor product form; they read: ξ u (y, τ ) = Ψu (τ ) ⊗ Φu (y) ;

ξ p (y, τ ) = Ψp (τ ) ⊗ Φp (y)

(3)

where Ψ(τ ) = {Ψm (τ ) , (m = 1, 2, ..., NT )} is a basis of a space of finite elements on the time interval, and Φ(y) = {φn (y) , (n = 1, 2, ..., NΓ )} is a basis of a space of finite elements on boundary Γ. Using (3) the discrete form of the energetic formulation comes out. It requires the computation of the following fundamental integrals: t T ∂ Gus Ψsm (τ ) dτ dt dΓy (4) ϕω(j,n) (y) Ψph (t) ∂t 0 0 Qj j T t ˙ u (t) Gps Ψsm Ψ ϕω(j,n) (y) (τ ) dτ dt dΓy (5) h j

Issues.

Qj

0

0

The evaluation of integrals (4) and (5) requires some care because of:

a) The time dependent fundamental solutions have a distributional definition; for this reason time integrals must be seen as the action of these distributions against the time shape functions, which do not belong to C0∞ ; to evaluate the action of the distributions against constant and linear piece-wise shape functions Ψ(τ ) a smooth approximation of the shape functions and a limit process are required.


3

b) The fundamental solutions are singular; after the double analytical integration in time the order of singularity of the time dependent Kernels is the same as the corresponding steady state problem (elasticity or Laplace problem). Singular integrals have been computed analytically, following the path of reasoning shown in [8] for statics. c) The presence of the wave front which travels within the body is reflected on discontinuities of the integrand functions of the space integrals: the space integrals have to be computed over the intersection between the elements and suitable spherical shells. This integration can be performed working on a polar reference system centered on the projection of the source point over the element plane. For paucity of space the reader is referred to [5] for a complete description on how these issues have been dealt with. Numerical benchmarks Longitudinal vibration of a rod The problem of a rod fixed at one end, and excited by a pressure jump at the free end, is almost a classical test concerning the validation of time dependent BEM (see [9]). Consider a prismatic rod of square cross section (1m × 1m) and length 3m, fixed at one end, and excited by an unit pressure jump at the free end. The numerical solution of the problem is computed adopting three different spatial discretizations, obtained by triangular linear elements and considering three different time step sizes for each mesh; the space and time discretizations adopted in this test are summarized in Table 1, where β = cΔT lc is the Courant Friedrichs Levy number (CFL number). This parameter connects the time step size ΔT with the characteristic length of the spatial discretization le via the wave velocity c and it governs stability properties of any numerical method. Analysis

Elements

le

Time Steps

β

type

number

[m]

size [s]

number

A.1 A.2 A.3

Linear Triangular

56

0.80

8.20 · 10−5 4.10 · 10−5 1.37 · 10−5

122 244 730

0.5 0.3 0.1

B.1 B.2 B.3

Linear Triangular

224

0.40

4.10 · 10−5 2.05 · 10−5 7.00 · 10−6

245 490 1430

0.5 0.3 0.1

C.1 C.2 C.3

Linear Triangular

504

0.27

2.73 · 10−5 1.36 · 10−5 4.55 · 10−6

370 740 2200

0.5 0.3 0.1

Table 1: Characteristics of the rod discretizations adopted in the numerical solution via BEM.

Fig. 1 and Fig. 2 show the result of the numerical analysis of the longitudinal vibrations of the rod compared with the analytical solution of the 1D problem. The numerical results allow stating that the energetic time domain BEM is highly stable: numerical instability never appears also for small time steps as in the case β = 0.1. As expected, the analysis C.3, which makes use of the finest mesh with the smallest time steps, shows more accurate results. A second general consideration is that displacements are computed more accurately with respect to tractions, because of the jumps in the traction solution. Another common feature of the results is the accuracy reduction when the analysis advances over time: the errors propagate together with the wave reflections along the bar. The plots of Fig. 1 and Fig. 2 lead to some comments on results dependency by spatial and time discretization: the phase shift of the numerical results with respect to the analytical solution decreases by adopting smaller space discretizations (see Fig. 1), while smaller time steps lead to reductions in the numerical damping, (see Fig. 2).

4

Tim e Tim e

s 2.5

s 0

2

1.5 10 11

Pa

Displacem ents

1. 10 11

1.5

Tractions

m

5. 10 12

1

0.5

2. 10 11

0

2.5 10 11 0

0.002

0.004

0.006

0.008

0.01

0

(a) Displacement solution at centroid of the free end.

0.002

0.004

0.006

0.008

0.5 0.01

(b) Traction solution at centroid of the fixed end.

Figure 1: Comparison between the analytical solutions (dotted lines) and the numerical results for β = 0.1 and different meshes (Mesh A dot-dashed lines, Mesh B continuous lines, Mesh C dashed lines).

Tim e Tim e

s 2.5

s 0

2

1.5 10 11

Pa

Displacem ents

1. 10 11

1.5

Tractions

m

5. 10 12

1

0.5

2. 10 11

0

2.5 10 11 0

0.002

0.004

0.006

0.008

0.01

0

(a) Displacement solution at centroid of the free end.

0.002

0.004

0.006

0.008

0.5 0.01

(b) Traction solution at centroid of the fixed end.

Figure 2: Comparison between the analytical solutions (dotted lines) and the numerical results for mesh C and different time step size (β = 0.5 dot-dashed lines, β = 0.3 continuous lines, β = 0.1 dashed lines).

s

Tim e

s

2.

2.

1.5

1.5

1.

Prim al field jum p

Prim al field jum p

Tim e

1.

0.5

0

2

4

6

(a) Mesh A: 56 elements.

8

0 10

0.5

0

2

4

6

8

0 10

(b) Mesh B: 396 elements.

Figure 3: Time history of the primal field jump at point (0, 0, 0) for the two different meshes. For each mesh the results obtained with different time step sizes (dot-dashed lines Δt = 0.100, dashed lines Δt = 0.050, continuous lines Δt = 0.025) are compared with the static analytical solution (dotted lines).


5

Analysis

Penny shaped crack within an infinite medium The aim of this numerical example, which considers the scalar counterpart of the well known problem of a penny shaped crack embedded in an infinite medium (whose analytical static solution will be used for comparison) is to validate the formulation when considering a stationary crack. Consider the whole 3D elastic space Ω∞ = R3 , made of a fictitious material with unit Young mod ulus and density. Within Ω∞ is defined the circular surface Γw = x ∈ R3 : x3 = 0 ; x21 + x22 1 , which is a locus of possible primal field discontinuities w(x, t). One seeks the jump of the primal field over Γw due to the dual action p¯(x, t) = H(t), applied on Γw (where H(t) is the Heaviside step function). Due the BEM features, the surface Γw is the only one which has to be discretized in this problem. For the numerical solution one adopts two non uniform meshes, made of linear triangular elements. Table 2 summarizes the characteristics of the discretizations. Since the adopted space decompositions are non uniform, the CFL numbers vary significantly on the same mesh once the time step is fixed. The magnitude of this difference reduces by adopting smaller time step (it is evident observing table for mesh B). All the discretizations used in these analysis are set such that the average values of the CFL numbers are smaller than one. Space discretization Elements

Time discretization Lengths

Time steps

CFL numbers

type

num.

lemin

lemax

size

num.

β min

β max

βˆ

A.1 A.2 A.3

Linear Triangular

64

0.30

0.44

0.100 0.050 0.025

100 200 400

0.23 0.11 0.06

0.33 0.17 0.08

0.28 0.14 0.07

B.1 B.2 B.3

Linear Triangular

396

0.08

0.23

0.100 0.050 0.025

100 200 400

0.43 0.22 0.11

1.25 0.62 0.31

0.84 0.42 0.21

Table 2: Space and time discretizations of the crack surface adopted in the different BEM analysis.

In Fig. 3 the time history solution in the centroid of the discontinuity surface is presented, considering different space and time discretizations. The numerical solutions are essentially independent on the time step size, since the curves concerning different time steps, depicted in Fig. 3, overlap perfectly; this probably happens because no wave reflections occur in the considered problem. The dependency of the results on the spatial discretization is more evident, since finer meshes better approach the static analytical solution in their asymptotic behaviour. Although convergence improvements clearly appears by refining the spatial discretization, the difference between the asymptotic numerical solution, obtained with the finest mesh, and the static solution is still larger than expected. This is probably due to two issues. The first one is the approximation of the circumference which bounds the surface: although this curve is better approximate using fine mesh, the triangular linear elements can never represent exactly this circumference. The second issue is the singularity of the dual field along the circular edge of Γw : the asymptotic behaviour cannot be captured with the adopted linear shape functions. To solve this problem special elements are needed. One remarks again the numerical stability of the proposed BEM approach, since no instability appears even in analysis A.3, performed with a very small CFL number (βˆ = 0.07). One notes also that non-uniformity of the mesh does not invalidate the performance of the numerical procedure: this is particularly evident in analysis C.1, characterized by a strong difference between the minimum and maximum value of the CFL number (β min = 0.43 and β max = 1.25), whose results are not affected by any particular degradation with respect to the other analysis.

6

Conclusions In this work an energetic time domain Boundary Element Method has been validated with some numerical tests. The results look promising, in particular the method shows a good stability, also for small sizes of the time step. Ongoing works deal with the implementation of the elastodynamics within the code and the development of special elements for half space problems and enhanced quarter point elements for crack front discretization in dynamic fracture mechanics (see [5, 10] for some preliminaries). Future developments concern the optimization and parallelization of the code as well as the use of fast techniques (ACA, FMM) in order to be able to solve problems with a large number of degrees of freedom. Another interesting development is the coupling between BEM and Finite Element Method (FEM), with the aim to study soil-structure interaction problems. Acknowledgements The numerical analysis have been performed using a cluster of the Interuniversity Consortium CILEA (nowadays part of CINECA), thanks to initiative LISA 2010 - 2012. Authors wish to thank the CILEA HPC group for their technical support, and Regione Lombardia which sponsored the initiative. A.T. developed part of the present work during his stay at Graz University of Technology, he wishes to thank Prof. M. Schanz and his research group for their kind hospitality and fruitful discussions and Cariplo foundation for their economic support. References [1] A. Brancati, M.H. Aliabadi, and V. Mallardo. A BEM sensitivity formulation for threedimensional active noise control. International Journal for Numerical Methods in Engineering, 90:1183–1206, 2012. [2] M. Messner and M. Schanz. A regularized collocation boundary element method for linear poroelasticity. Computational Mechanics, 47:669–680, 2011. [3] P. Li and M. Schanz. Wave propagation in a 1-D partially saturated poroelastic column. Geophysical journal international, 184:1341–1353, 2011. [4] M.H. Aliabadi. The Boundary Element Method, volume 2 Applications in Solids and Structures. John Wiley & Sons, 2002. [5] A. Temponi. A Time-Domain BEM for Dynamic Fractures in Brittle Multimaterials. PhD thesis, Università degli Studi di Brescia, 2013. [6] D. Graffi. Sul teorema di reciprocit` a nella dinamica dei corpi elastici. Memorie della Accademia delle Scienze dell’Istituto di Bologna, 10:103–109, 1947. [7] T. Ha Duong. On retarded potential boundary integral equations and their discretizations. In P. Davis, D. Duncan, P. Martin, and B. Rynne, editors, Topics in computational wave propagation: direct and inverse problems. Springer, 2003. [8] A. Salvadori. Analytical integrations in 3D BEM for elliptic problems: Evaluation and implementation. International Journal for Numerical Methods in Engineering, 84:505–542, 2010. [9] M. Schanz. Wave propagation in Viscoelastic and Poroelastic Continua: A Boundary Element approach. Springer, 2001. [10] F. Valvona. Infinite Boundary Elements for 3D Problems. PhD thesis, Universit` a degli Studi di Chieti Pescara, 2013.


7

Inverse heat source recovery by the Green element method Akpofure E. Taigbenu School of Civil and Environmental Engineering, University of the Witwatersrand. P. Bag 3, Johannesburg, WITS 2050. South Africa. [email protected] Keywords: Green element method, inverse heat source, Tikhonov regularization, Singular value decomposition.

Abstract The transient inverse heat source problem is simulated in 2-D spatial dimensions by the Green element method. The over-determined, ill-conditioned discrete equations, arising from the element-by-element implementation of the singular integral equations, are solved in the least square sense using the singular value decomposition (SVD) technique in conjunction with the Tikhonov regularization. The accuracy of the current formulation is demonstrated with two numerical examples in which the recovery of the strength of the heat source is predicted. It is significant to note that good estimates of the heat source are obtained with coarse discretization of the computational domain. Introduction One class of inverse heat conduction problems (IHCP), that has applications in many areas of the sciences and engineering, is the recovery of heat (mass, energy, contaminant) sources. It typically involves using available field data to predict the spatial and temporal distribution of heat sources/sinks in the medium. The modeling of such problems presents challenges of unstable and non-unique solutions as a result of the illconditioned nature of the system of discrete equations which arises from the numerical discretization. The instability of the numerical solutions is further exacerbated by measurement errors that may be inherent in the observed data. The inverse modeling of the recovery of heat sources had been addressed by many numerical approaches [1-4]. In this paper, the Green element method (GEM) is used to predict the heat source distribution that is assumed to be only time-dependent. The GEM formulation is based on that of Taigbenu [5] in which the internal normal fluxes are approximated by a second-order polynomial relationship in terms of the field variable. This formulation had been shown to exhibit comparable accuracy as the flux-based formulation, and also readily lends itself for the solution of inverse problems. The system of discrete equations, which is over-determined and ill-conditioned, is solved in the least square sense using the singular value decomposition technique and Tikhonov regularization. Two numerical examples of transient inverse heat source problems are solved, and excellent results are obtained with the current formulation. Governing Equation and Numerical Formulation The governing differential equation for the heat source problem addressed in this paper is wT K 2T c Q(t ) (1) wt where is the 2-D gradient operator with spatial variables x and y, t is time, T is the temperature field, K and c are respectively the thermal conductivity and heat capacity of the medium, and Q represents heat sources and sinks that are assumed to have only temporal variation. The heat sources can be uniformly distributed and/or point sources, represented as Ns

Q(t )

¦ H n (t )G (r rn )

(2)

n 1

where Hn is the strength of the nth heat source located at rn=(xn,yn), and Ns is the number of these point heat sources. The specified boundary conditions are:

8

T ( x, y, t ) T1 on Γ1

(3a)

KT n

(3b)

q2 on Γ2

and the initial condition is: T ( x, y, t 0) T0 on Ω

(3c)

where n is the unit outward pointing normal on the boundary. The differential equation applies to a domain Ω with boundary Γ= Γ1 Γ2. The recovery of the temporal variation of the heat source strength is sought using the prescribed conditions and available temperature data Tm=T(xm,ym) at Ni internal points. The integral equation of eq (1) is obtained by applying Green’s theorem º ª wT OTi ³ TG n GMq dS ³³ G «) MQ» dA 0 t w ¼ ¬ * :

(4)

where Φ = c/K, φ = 1/K, G = ln(r–ri) is the fundamental solution of the Laplacian operator, q

KT n ,

the subscript i denotes the source or collocation point ri = (xi,yi) and O is the nodal angle at ri obtained from the Cauchy part of the integration of the Dirac delta function at the source node. The boundary and domain integrals in eq (4) are implemented in the Green element sense over elements that are used to discretize the computational domain [6]. On these elements, Lagrange-type interpolations are prescribed, that is T ≈ NjTj (Nj are the interpolation functions that have been chosen to be linear). Introducing the interpolation into eq (4) results in the discrete element equations applicable to each element denoted as Ω e. That is dT j Rij T j Lij Mq j )Wij Fi 0 (5) dt where

Rij

³ N G n ds G O , j

i

ij

Lij

*e

³ N G ds, j

*e

i

Wij

³³ G N i

j

dA.

(6)

:e

The expression for the vector Fi resulting from the contribution of heat sources depends on whether they are distributed or point sources. For distributed sources, it has the expression Fi

Q(t )M ³³ ln(r ri ) dA

(7a)

:e

and, for point sources, the expression is Ns

Fi

M ¦ H n ln(rn ri )

(7b)

n 1

Aggregating eq (5) over all the elements results in the matrix equation dTj (8) EijT j Bij q j Cij Fi 0 dt The temporal derivative term is approximated as dT/dt≈[T(2) – T(1)]/∆t at t=t1+β∆t, where 0≤β≤1, is the difference weighting factor, and ∆t=t2–t1 is the time step. Using the approximation for the temporal derivative transforms eq (8) to C · C · § § ¨¨ EEij ij ¸¸T j( 2) EBij q (j2) EF j( 2) ¨¨ ZEij ij ¸¸T j(1) ZBij q (j1) ZF j(1) (9) 't ¹ 't ¹ © © where ω=β–1 and the bracketed superscripts represent the times at which the quantities are evaluated. In matrix form eq (9) is (10) Ap b Where p={T,q,Q}tr is an N×1 vector of unknowns (T and/or q at external nodes, T at internal nodes and the heat sources, Q) and tr denotes the transpose. The matrix A is an M×N matrix, where M is the number of nodes in the computational domain and M ≥ N, and b is an M×1 vector of known quantities. Eq (10) is over-


9

determined and generally ill-conditioned. It is solved by the least square technique. The M×N matrix A is decomposed by the singular value decomposition (SVD) method that is expressed as N

¦\ u v

UDV t

A

tr i i i

(11)

i 1

where U is an M×M matrix and V is an N×N matrix; both are orthogonal square matrices, and D is an M×N diagonal matrix with N non-negative diagonal elements (D = diag(ψ1, ψ2, ..., ψN)) which satisfy the condition: ψ1 > ψ2 > ···> ψN > 0. N is the rank of the matrix A, and ui and vi are the ith column of the matrices U and V, respectively. By the least square solution of eq (10), the Euclidian norm ║Ap–b║2 is minimized to give the solution for the unknowns p p

A A A b 1

tr

tr

(12)

Introducing the expression for A from eq (11) into eq (12) gives N

p

u tr b

¦ \i i 1

(13)

vi

i

The small singular values cause the solution for p to be unstable and hence the need for regularization. Adopting the widely known Tikhonov regularization approach, which is a smoothening technique, the Euclidian norm of ║Ap–b║2+α2║p║2 is minimized. This yields the solution for p that is given by p(D )

N

\

¦ D 2 i\ 2 uitr bvi i 1

(14)

i

where α is the regulation parameter. The factor ψi/(α2+ψi2) in eq (14) serves to dampen the contribution of the small singular values. The choice of α has to be carefully done so that it is not too small to retain the instability of the numerical solution or too large to have smooth solutions that do not reflect the physics of the problem being addressed. As suggested by Hansen [7], the singular values are normalized by their largest one ψ1 such that ψN 1 V u RN (m)@ (15) where σ denotes the magnitude of the error and RN is a random number ϵ [-1,1] and generated with the IMSL program routine RNNOR. Example 1 In this example, the temperature distribution, which satisfies eq (1) in 1-D spatial domain xϵ[0,1] with K=1 and c=1, is prescribed T ( x, t )

x 2 2t sin(2St )

That implies that the analytical expression for the strength of the heat source is Q(t ) 2S cos(2St )

(16) (17)

The temperature, obtained from eq (16), is specified on the boundary along x=0 and x=1, and within the domain at x=0.5 for all times. The GEM simulations use 10 linear rectangular elements with no-heat flux boundaries imposed along y=0 and y=103, a uniform time step ∆t=0.025 and β=1. The temperature data at x=0, 0.5 and 1 are perturbed randomly with noise levels of σ=1, 3 and 5%. The GEM and exact solutions for the heat source are presented in Fig. 1 for various noise levels. The values of the regularization parameter

10

used in the GEM simulations are 2.2×10-4 for σ=0% and 3.2×10-4 for the other noise levels of 1, 3 and 5%. The GEM results reproduce the general trend of the exact solution, but they exhibit oscillations which increase with noise level. The GEM and exact solutions for the temperature at x=0.3 and the flux at x=0 are plotted in Fig. 2. The GEM solution for the temperature is virtually oscillation free, but not for the flux when the noise levels are 3% and 5%.

Figure 1: GEM and exact solutions of the heat source for various noise levels of Example 1.

(a)

(b)

Figure 2: GEM and exact solutions of Example 1 for various noise levels – (a) T(x=0.3,t), (b) q(x=0,t)

Advances in Boundary Element Techniques XIV Example 2 In this example, also previously simulated by Yan et al. [3] using the MFS, the IHCP governed by eq (1) in 1-D spatial domain xϵ[0,1] is solved with zero temperature specified at both ends of domain. Initially the temperature is zero everywhere in the domain, and the IHCP is to recover the step-wise heat source distribution expressed as: t [0,0.25) 1, ° 1, t [0.25,0.5) ° (18) Q(t ) ® t [0.5,0.75) ° 1, t [0.75,1] ¯° 1, Because no analytical solution exists, the temperature distribution is generated by solving the direct problem with GEM using fine spatial and temporal discretizations of 40 linear rectangular elements and time step ∆t=2.5×10-3. The direct GEM solution is presented in Fig. 3. The inverse modeling with GEM used the specified boundary conditions: T(x=0,t) = T(x=1,t) = 0, the initial condition: T(x,0) = 0, and the generated temperature data at x=0.5. Ten linear rectangular elements are used in the inverse GEM simulations with noheat flux boundaries imposed along y=0 and y=103. A uniform time step ∆t=2.5×10-2 and time differencing scheme β=1 are used in the GEM simulations. Because of the homogeneous boundary conditions, only the temperature data at x=0.5 are affected when they are randomly perturbed with noise levels of 1, 3 and 5%. The GEM solutions for the heat source are presented in Fig. 4 for various noise levels. The range of regularization parameter values used in the GEM simulations are 2.2×10-4 for σ=0% and 3.2×10-4 for the other noise levels. The numerical results are excellent for all noise levels, considering the coarse discretization that is used to recover this discontinuous heat source. These results are superior to those obtained by Yan et al. [3] who used the method of fundamental solutions. Fig. 5 shows the numerical solutions for T(x=0.3,t) and q(x=0,t) obtained by the direct and inverse GEM simulations. There is excellent agreement between both numerical solutions.

Figure 3: Solution of Example 2 from direct GEM simulations

Conclusion Inverse modeling by the GEM has been carried out to predict the transient heat source distribution arising from heat conduction in a 2-D spatial domain. The predictive capability of the current formulation is excellent, considering that for the two examples to which it has been applied, coarse grid of elements has been used. The formulation is also able to accommodate situations where noise is inherent in the observed measurements.

11

12

Figure 4: Recovery of the heat source for various noise levels of Example 2

(a)

(b)

Figure 5: GEM and exact solutions of Example 1 for various noise levels – (a) T(x=0.3,t), (b) q(x=0,t) References [1] T. Wei and J.C.Wang, Engrg. Anal. Bound. Elem. 36, 1848–1855 (2012). [2] C-H. Huang, J-X. Li and S. Kim, App. Math. Mod. 32, 417-431 (2008). [3] L. Yan, C-L. Fu and F-L. Yang, Engrg. Anal. Bound. Elem. 32, 216-222(2008). [4] A. Farcas and D. Lesnic, J. Engrg. Math., 22, 1289-1305, 2006. [5] A.E. Taigbenu, Engineering Analysis with Boundary Elements, 35, 125-136 (2012). [6] A.E. Taigbenu, The Green Element Method, Kluwer, Boston, USA (1999). [7] P.C. Hansen, Regularization Tools: Numer. Algorithms, 6, 1-35 (1994).


A meshless approach for the mandible bone tissue remodelling analysis due to the insertion of dental implants A.S.S.L. Ferreira1, J. Belinha2, L.M.J.S. Dinis3 and R.M.Natal Jorge4 1

Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s/n 4200-465 Porto – Portugal, [email protected] 2 Instituto de Engenharia Mecânica, Rua Dr. Roberto Frias, s/n 4200-465 Porto – Portugal, [email protected] 3 Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s/n 4200-465 Porto – Portugal, [email protected] 4 Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s/n 4200-465 Porto – Portugal, [email protected] Keywords: Bone remodelling, Dental implants, Radial Point Interpolation Method, Natural Neighbours, Meshless Methods

Abstract: Based on deformation energy methods a topology optimization model was developed for predicting bone density distribution on the mandible bone due to the insertion of dental implants. The bone structure is assumed to be a self-optimizing anisotropic biological material which maximizes its own structural stiffness. A recently developed mathematical law for the biologic behaviour of the bone anisotropic material is assumed, based on experimental data available in the literature. This mathematical law permits to correlate the bone density with the obtained level of stress. The numerical method used in the analysis is the Natural Neighbour Radial Point Interpolator Method (NNRPIM), an efficient and flexible meshless method, presenting various advantages in the topologic analysis comparing with other numerical methods such as the Finite Element Method. The NNRPIM uses the Voronoï concept to force the nodal connectivity and to construct an integration mesh, both completely node-dependent. The interpolation functions possess the delta Kronecker property, which simplifies the boundary condition imposition. The viability and efficiency of this method were tested on a cross section of the mandible bone with a dental implant subject to single and multiple loading conditions. The main goal of this work was to obtain similar trabecular architecture when compared with real clinical cases. State of Art Bone, as a living tissue, is continuously responding to internal and external signals [1]. These external stimulus, which may be biological, cellular and chemical, or mechanical in nature, are in the origin of the bone remodelling process; the process by which bone tissue is capable of self renovation, modulating both its mass and microstructure [2]. This process allows the maintenance of the shape, quality and appropriate size of the skeleton through the repairing of microdamages, generated during normal and daily activities, and adaptation of microstructure according to the mechanical demands [3,4]. Therefore, when the mechanical environment changes, the bone internal structure is rearranged appropriately [5]. The bone density variation prediction, and its corresponding mechanical responses using computational techniques, gradually has been regarded as a necessity for the study of the long term success of bone prostheses. In opposition to clinical trials, numerical methods are non-invasive and time-efficient [6]. Therefore several remodelling algorithms have been proposed. Since the bone remodelling is a complex multifaceted process guided by several factors, different approaches were proposed by several researchers. Some algorithms take into consideration biological aspects in their formulations [7-9], cellular or chemical factors such as the bone cells activity, matrix mineralization and damage accumulation. The approach proposed by Martínez [10] is based on the activity of basic multicellular units (BMUs), temporal associations of cells that take part into the bone resorption and formation. Within the model these units govern changes in porosity, more specifically, changes in the pores’ shape and orientation. Thus porosity determines the directional dependence of the bone tissue mechanical properties.

13

14

It is possible to find in the literature other algorithms assuming that external loads, and resulting bone stress and strain fields, are the main factor governing the remodelling process. This numerical approach has led to results that do not seem very far from the reality and is also considered in the present work [1,6]. Wolff (1892) was the first to suggest that the bone modifies its internal structure according to the external loads applied, more precisely, that the trabecular architecture tends to assume the orientation of the principal stresses installed [11]. Following Wolff’s qualitative observations, several investigators have been developing increasingly sophisticated theoretical and numerical models that aim to predict and quantify the bone remodelling process [12-14]. Huskies[15] suggested a model considering the strain energy density (SED) as the mechanical stimulus for both superficial and internal remodelling of bone. Additionally, a homeostatic range of strains, named as ‘lazy zone’, is defined, for which no remodelling takes place. The idea, initially proposed by Carter in 1984 [16], is to define a range corresponding to the normal physiological loading conditions, which do not trigger a structural change, representing a balance between osteoclastic resorption and osteoblastic apposition. Although based on Huskies algorithm, Mullender’s [17] model considers the bone remodelling to be a selforganizational control process. The value of the stimulus in a particular bone region does not depend anymore on the deviation from the homeostatic state in that region only, but also on the deviation found in other nearby regions. In this work the implemented bone tissue remodelling algorithm uses a new material law that, following the work of Zioupos [18], correlates the bone apparent density with the Young modulus, considering its anisotropic behaviour and representing both the cortical and trabecular bone. The remodelling algorithm is an adaptation of Carter’s algorithm [14], initially proposed in 1987 and later applied and modified by various authors [19-21].In the model the bone gradually transits from an initial uniform density distribution to a final trabecular arrangement. This is accomplished with an iterative process (forward Euler scheme) that obtains the density distribution and trabeculae orientation resulting from the stress-strain state caused by the external loads, which in turn, originate new material properties, and, consequently, a new stress-strain field. The process ends when the medium bone density reaches a predefined value. In order to determine the stresses and strains involved, a numerical tool is required, commonly the FEM [2,22]. Nevertheless, in this work a meshless numerical method is used, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). This method presents several advantages such as the low computational cost, compared to other meshless methods, and the generation of smooth and accurate stress and strain fields. Intending to overcome some drawbacks and limitations of the finite element method, meshless methods arose as a numerical alternative and have been under development and improvement [23-30]. In this numerical approach the field function is approximated within an influence-domain rather than an element [1]. These numerical methods present several advantages. The most relevant are the ability to provide more accurate approximations than FEM for structures with complex geometries, such is the case of biological ones, and the capacity to deal with large deformation problems [26] due the natural remeshing flexibility. The NNRPIM [31] is the meshless method used in this work. It combines the radial point interpolators (RPI)[32], used in the construction of the shape functions, with the natural neighbours geometric concept[33], used to define the nodal connectivity and obtain the background integration mesh. Therefore, despite being considered a meshless method, the NNRPIM creates a background integration mesh [34]. The NNRPIM was already used in several studies, for example, in static [35] and dynamic [36,37] analysis and applied in plates, laminates [38,39] and shell structures [40]. The performance of the method was also tested in more demanding situations such as the nonlinearity of the material [41] and the large deformation analysis [42].In addition, the NNRPIM was one of the few meshless methods applied in bone remodelling, which is the subject of this work [43-46]. The major advantages of this method are the low computational cost, compared to other meshless methods, and the generation of smooth and accurate stress and strain fields [43].The combination of the implemented numerical method, the NNRPIM, with the proposed remodelling algorithm will be analyzed in terms of the potential to predict the bone remodelling process following a dental implantation. The proposed meshless method The numerical method implemented in this work is the Natural Neighbour Radial Point Interpolation Method (NNRPIM) [31]. First the domain is discretized in a nodal mesh, not necessarily structured, afterwards a nodal dependent background integration mesh is constructed in order to numerically integrate the weak form equations.


15

While in the FEM the nodal connectivity is directly determined by the creation of the element-mesh, in meshless methods the nodal connectivity is established after the domain nodal discretization, based on the nodal spatial distribution. Therefore, the NNRPIM relies on the natural neighbour concept to both construct the integration mesh and determine the referred nodal connectivity. In this method the influence-domain is substituted by the concept of influence-cell, which is obtained from the Voronoï cell, and the nodal connectivity is imposed by the overlapping of the various influence-cells. In order to create a nodedepending background integration mesh for the interpolation functions, the duality between the Voronoï cell and the Delaunay triangle is used. This total dependency of the integration mesh on the nodal mesh allows the NNRPIM to be considered a truly meshless method. Contrary to the FEM, where geometrical restrictions on the elements are imposed for the convergence of the method, in the NNRPIM there is no need for such restrictions, allowing a total random node distribution. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed similarly to the ones of RPIM, existing some differences that modify the method performance. Consider the function u x

defined in the domain : , discretized by N nodes and that passes through all of them using a radial basis function (RBF). The value of the function associated to the point of interest xI is determined by the n nodes within its influence-cell and can be obtained by: n

¦ R (x ) a (x )

u( xI )

i

I

i

I

RT ( xI ) a

(1)

i 1

where Ri xI is the RBF and ai xI are the non-constant coefficients respectively associated. Considering the nodal values Eq.(1) becomes: u( xI )

RT ( xI ) RQ 1uS M( xI ) uS

(2)

where M xI refers to the value of the interpolation function on the point of interest that allows the determination of the displacement value, u( xI ) , from the nodal ones, us . The partial derivative of the interpolation function can be easily obtained. In addition, early works on the radial point interpolators confirm that these interpolation functions possess the delta Kronecker property, which simplifies the imposition of the boundary conditions, and that the partition of unity is satisfied [32]. Bone tissue material law In order to develop a bone remodelling model, a relationship between the bone properties and its mechanical behaviour must be firstly established. The material laws implemented are based on the experimental works of Zioupos [18] and Lotz [47].In 2008, the work of Zioupos allowed to obtain the bone tissue elasticity modulus based uniquely on the bone tissue apparent density. Following his work, the implemented material law permits to correlate the bone tissue elasticity modulus with the bone tissue apparent density. The Young modulus in the axial direction, in GPa, is related to the apparent density, in g/cm3, by: 3

E axial

¦ a U j

app

j

(3)

j 0

For the bone tissue presenting Uapp d 1,3g / cm3 , and by: 3

E axial

¦ b U j

app

j

(4)

j 0

For bone tissue with Uapp ! 1,3g / cm3 . This theoretical curve possesses a 95% correlation with the experimental data obtained by Zioupos. Earlier, in 1991, Lotz had suggested distinct mathematical laws for cortical and trabecular bones relating in two directions (axial and transversal) the Young modulus and the ultimate compressive stress with the bone

16

apparent density. In the present study, the laws relating the Young modulus in the transversal direction and the ultimate compressive stress in the axial and transversal direction, all expressed in GPa, with the bone apparent density, in g/cm3, were developed based on his work and also represent the behaviour of both cortical and trabecular bone in a single curve: 3

E trans

¦ c U j

app

j

c , V axial

j 0

3

¦ d U j

app

j

c , V trans

j 0

3

¦ e U j

app

j

(5)

j 0

All the coefficients associated to Eqs (3) to (5) can be found in J. Belinha (2012) [1]. In order to determine t c c the ultimate tensile stress it is considered: V i D V i 0,5 V i . The remodelling algorithm The proposed remodelling algorithm initially assumes a uniform apparent density Uapp 2.1g / cm3 for the entire discretized problem domain. Therefore the initial bone tissue mechanical properties can be determined using Eqs (3) and (4). Once the loads are applied on this uniform and isotropic material the Von Mises effective stress field can be determined, using the obtained principal stresses, as well as the strain energy density (SED) field for the loading case considered, resorting to the obtained stress and strain fields. Afterwards, based on Carter’s idea, the global variables combining the various loading cases are calculated through a weighted average as follows:

[

l

¦ i 1

mi [i l

(6)

¦m

j

j 1

where m represents the number of cycles corresponding to the loading case considered. After obtaining the global values for the variables effective stress field, principal stress field, principal direction field, strain field and SED field, and in order to assure that the model is not experiencing stresses much higher than the real ultimate stress, the resulted effective stress field is compared with the admissible effective stress field defined as:

Vadj

Vaxial Uapp j Vtrans Uapp j

(7)

where V axial and V trans result from equation Eq (5) for the compressive case. This equation is obtained considering the bone as an orthotropic material with two preferential directions whose trabecular structure is orientated according to the principal stresses. It is considered that V1 V axial and V 2 V 3 V trans . To determine the maximum admissible effective stress, the maximum apparent density possible in this model, 2,1 g/cm3, is considered in the anterior expression. Thereby, the stress in the model should not overcome this value. In order to guarantee that this does not occur, the global field variables are scaled using the maximum ratio between the effective stress and the admissible effective stress for each interesting point j . With the principal stresses scaled the next step is to substitute back into Eqs (3) and (4) to determine the corresponding apparent densities. Since in this work the density is considered isotropic, equal in the axial and transversal direction, the following expressions apply:

V axial j V1 j

and V trans j

max V 2 , V 3 j

(8)

The new value of apparent density to be used for the considered interest point j in the next iteration step is the maximum between the one obtained for the axial and the transversal directions. The material matrix is also re-orientated accordingly to the principal directions that resulted from the present iteration. Although for all the interest points in each iteration the material matrix is re-orientated according to the principal


17

directions, the density is not always altered. Only the points with lower SED will have their density modified. This process ends when the medium bone density achieves a stable value of 0,1 g/cm3, being determined as follows: med Uapp

1 n ¦ Uapp j n j1

(9)

where n refers to the total number of interest points. Bone square patch example First it is presented a benchmark example to validate the bone tissue remodelling algorithm. Consider a square bone patch submitted to a triangular load and with the essential boundary conditions indicated in Figure 1(a). It is assumed in the beginning of the analysis a uniform and constant apparent density for the entire square domain, Uapp 2.1g / cm3 . The initial Young Modulus is obtained with Eq(4)and it is considered a Poisson ration of X 0.3 . The solid domain is discretized in two distinct meshes, a regular and an irregular mesh, Figure 1(b) and (c).

(a)

(b)

(c)

(d)

Figure 1 – a) Plate patch with essential and natural boundary conditions, b) regular and c) irregular meshes used and d) FEM solution [48].

(a)

(b)

Figure 2 – a) Trabecular architecture obtained for the 2D patch considering a regular mesh and b) an irregular mesh. In Figure 2 the trabecular architecture obtained with the proposed method can be observed for two different types of meshes. The results seem to be in good agreement with the ones from the literature [48] and don’t differ significantly whether the mesh used is regular or irregular.

18

In addition, a mandibular patch nearby a dental implant, Figure 3(a), was considered in order to study the interface region between the two entities, being loaded with several load cases according to Figure 3(a). The studied load cases intend to represent a successful osteointegration of the dental implant in the mandible bone. The results obtained with the meshless method are presented in Figure 3 (b), and, as confirmed by histological plates, Figure 3(c), the trabecular architecture seems to acquire an oblique direction in relation to the implant surface.

(a)

(b)

(c)

Figure 3 – (a) 2D implant patch and boundary conditions applied. (b) NNRPIM remodelling results. (c) Histological results obtained by Watzak G. et all in 2005 [49]. Conclusions The experience acquired with the developed work permits to assert the following conclusions: i. The proposed bone anisotropic material law permits a smooth transition between the cortical bone stage and the trabecular bone condition. ii. The developed remodelling algorithm combined with the NNRPIM accuracy permits to predict correctly the secondary trabecular structures, which are very important in the stability of the principal structures. iii. The preliminary results on a small patch surrounding the implant suggest that this numerical approach can be improved to predict with great detail the trabecular remodelling of the bone tissue near the implant.

References [1] [2]

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Belinha J, Jorge RMN, Dinis LMJS. Bone tissue remodeling analysis considering a radial point interpolator meshless method. Eng Anal Boundary Elem 36 (2012) 1660-1670. Wang H, Ji B, Liu XS, Guo XE, Huang Y, Hwang KC. Analysis of microstructural and mechanical alterations of trabecular bone in a simulated three-dimensional remodeling process. J Biomech 45 (2012) 2417-2425. Fernández JR, García-Aznar JM, Martínez R. Numerical analysis of a piezoelectric bone remodeling problem. Eur J Appl Math (2012). Hadjidakis DJ, Androulakis II. Bone remodeling. Ann. New York Acad. Sci. 1092, 385-396. Lian Z, Guan H, Ivanovski S, Loo YC, Johnson NW, Zhang H. Effect of bone to implant contact percentage on bone remodeling surrounding a dental implant. Int. J. Oral Maxillofac. Surg. 2010; 39: 690-698. Lin D, Li Q, Li W, Duckmanton N, Swain M. Mandibular bone remodeling induced by dental implant. J Biomech 43 (2010) 287-293. Mullender MG, Huiskes R, Weinans H. A physiological approach to the simulation of bone remodeling as a selforganizational control process. J Biomech 27(11) (1994) 1389–1394. Hart RT, Davy DT. Theories of Bone Modeling and Remodeling, in Bone Mechanics, Cowin. SC (1989), CRC Press Boca Raton. 253-277.

Advances in Boundary Element Techniques XIV [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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Garcia JM, Rueberg T, Doblare M. A bone remodelling model coupling microdamage growth and repair by 3D BMU-activity. Biomech Model Mechanobiology 4(2-3) (2005) 147-167. Martínez RJ, G.J., Domínguez J, Doblaré M, A bone remodelling model including the directional activity of BMUs. Biomechanics and Modeling in Mechanobiology, 2009. 8: p. 111–127. Wolff J. The law of bone remodeling (Das Gesetzder Transformationder Knochen, Hirschwald, 1892). Berlin Heidelberg New York: Springer; 1986. Cowin SC, Hegedus DH. Bone remodeling I: a theory of adaptive elasticity. J Elasticity 1976; 6:313-326. Hart RT, Davy DT, Heiple KG. Mathematical modeling and numerical solutions for functionally dependent bone remodeling. Calcif Tissue Int 1984; 36:S104-S109. Carter DR, Fyhrie DP, Whalen RT. Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy. J Biomech 1987; 20 (8): 785-794. Huiskes R, Weinans H, Dalstra M. Adaptive bone remodeling and biomechanical design considerations for noncemented total hip arthroplasty. Orthop (1989) 12, 1255-1267. Carter DR. Mechanical loading histories and cortical bone remodeling.Calcif Tissue Int. 1984;36 Suppl 1:S19-24. Mullender, M.G., Huiskes, R., Weinans, H.: A physiological approach to simulation of bone remodelling as a selforganizational control process. J. Biomech. 27, 1389–1394 (1994). Zioupos P, Cook RB, Hutchinsonc JR. Some basic relationships between density values in cancellous and cortical bone. J Biomech 41 (2008) 1961-1968. Starke GR, M.C.e.a., Some aspects of adaptative bone growth as a result of total hip joint replacement. in ABACUS User's Conference, Providence, RI, 1992. Pettermann H, R.T., Rammerstorfer FG, Computational simulation of internal bone remodeling. Archives in Computational Methods in Engineering, 1997. 4(4): p. 295–323. Beaupré GS, O.T., Carter DR, An approach for timedependent bone modelling and remodelling: Theoretical development. Journal of Orthopaedic Research, 1990. 8(5): p. 651–661. Quental C, Folgado J, Fernandes P, Monteiro J. Bone remodeling analysis of the humerus after a shoulder arthroplasty. Med Eng Phys 34 (2012) 1132-1138. Gingold RA, M.J., Smoothed particle hydrodynamics: theory and allocation to non-spherical stars. Mon Not Roy Astron Soc, 1977. 181: p. 375-89. Nayroles B, T.G., Villon P, Generalizing the finite element method: Diffuse approximation and diffuse elements. Comp Mech, 1992. 10: p. 307-318. Liu WK, J.S., Zhang YF, Reproducing kernel particle methods. Int J Numer Meth Fluids, 1995. 20(6): p. 1081-1106. Atluri SN, Z.T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech, 1998. 22: p. 117-127. Liew KM, Zhao X, Ferreira AJM. A review of meshless methods for laminated and functionally graded plates and shells. Composite Struc 93 (2011) 2031-2041. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless Methods: An overview and recent developments. Comput Methods Appl Mech Eng 139 (1) (1996) 3-47. Gu YT. Meshfree methods and their comparisons. Int J Comput Methods 2 (4) (2005) 477-515. Nguyen VP, Rabczuk T, Bordas S, Duflot M. Meshless methods: A review and computer implementation aspects. Math Comput Simulation 79 (2008) 763-813. Dinis LMJS, Jorge RMN, Belinha J. Analysis of 3D solids using the natural neighbour radial point interpolation method. Comput Meth Appl Mech Eng 196 (2007) (13-16) 2009-2028. Wang JG, L.G., A point interpolation meshless method based on radial basis functions. Int J Num Meth Eng, 2002. 54: p. 1623-1648. R, S., A vector identity for the Dirichlet tessellation. Math Proc Cambridge Philos Soc, 1980. 87(1): p. 151-155. Liu GR, GU YT, Wu YL: A MeshFree Weak-Strong-form (MWS) method. Int Workshop MeshFree Meth (2003). Dinis LMJS, Jorge RMN, Belinha J. Static and Dynamic Analysis of Laminated Plates Based on an Unconstrained Third Order Theory and Using a Radial Point Interpolator Meshless Method. Comput Struc 89 (19-20) (2011) 1771-1784. Dinis LMJS, Jorge RMN, Belinha J. The Natural Neighbour Radial Point Interpolation Method: Dynamic Applications. Eng Comput 26(8) (2009) 911-949.

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Dinis LMJS, Jorge RMN, Belinha J. The dynamic analysis of thin structures using a radial interpolator meshless method, in Vibration and Structural Acoustics Analysis, Vasques CMA and Rodrigues JD, Springer (2011) 1-20. Dinis LMJS, Jorge RMN, Belinha J. Analysis of plates and laminates using the natural neighbour radial point interpolation method. Eng Anal Bound Elemen 32(3) (2008) 267-279. Dinis LMJS, Jorge RMN, Belinha J. Composite Laminated Plates: A 3D natural neighbour radial point interpolation method approach. J Sandwich Struct Mat 12(2) (2010) 119-138. Dinis LMJS, Jorge RMN, Belinha J. A 3D Shell-Like approach using a Natural Neighbour Meshless Method: isotropic and orthotropic thin structures. Composite Struct 92(5) (2010) 1132-1142. Dinis LMJS, Jorge RMN, Belinha J. Radial Natural Neighbours Interpolators: 2D and 3D Elastic and Elastoplastic Applications, in Progress on Meshless Methods, Ferreira AJM et al., Springer (2008). Dinis LMJS, Jorge RMN, Belinha J. Large Deformation Applications with the Radial Natural Neighbours Interpolators. Comput Model Eng Sciences 44(1) (2009) 1-34. Belinha J, Jorge RMN, Dinis LMJS. Bone tissue remodeling analysis considering a radial point interpolator meshless method. Eng Anal Bound Elements 36 (11) (2012) 1660-1670. Doblaré M, Cueto E, Calvo B, Martínez MA, Garcia JM, Cegoñino J. On the employ of meshless methods in biomechanics. Comput Meth Appl Mech Eng 194 (2005) 801-821. Fernandez JW, Das R, Thomas CDL, Cleary PW, Sinnott MD, Clement J. Strain reduction between Cortical Pore Structures Leads to Bone Weakening and Fracture Susceptibility: An investigation Using Smooth Particle Hydrodynamics. IFMBE Proceedings 31 (2010) 784-787. Fernandez JW, Das R, Cleary PW, Hunter PJ, Thomas CDL, Clement JG. Using smooth particle hydrodynamics to investigate femoral cortical bone remodeling at the Haversian level. Int J Num Meth Biomed Eng (2012) In Press. Lotz JC, G.T., Hayes WC, Mechanical properties of metaphyseal bone in the proximal femur. J Biomech, 1991. 24(5): p. 317-329. Xinghua Z., He G., Dong Z., and Bingzhao G., A study of the effect of non-linearities in the equation of bone remodeling. Journal of Biomechanics, 2002. 35: p. 951–960. Watzak G, Zechner W, Ulm C, Tangl S, Tepper G, Watzek G. Histologic and histomorphometric analysis of three types of dental implants following 18 months of occlusal loading: a preliminary study in baboons. Clin. Oral Impl. Res. 2005. 16: 408–416.


Energetic BEM-FEM coupling for wave propagation in layered media A. Aimi1, L. Desiderio2, M. Diligenti3 , A. Frangi4, C. Guardasoni5 1

Dept. of Mathematics and Computer Science, Univ. of Parma, Italy, [email protected]

2


3


4

Dept. of Civil and Environmental Engineering, Polytechnic of Milan, Italy, [email protected]

5


Keywords: wave propagation, multi-domain, non-overlapping domain decomposition, energetic BEM-FEM coupling

Abstract. Starting from a recently developed energetic space-time weak formulation of boundary integral equations related to wave propagation problems defined on single and multi domains, a coupling algorithm is presented, which allows a flexible use of finite and boundary element methods as local discretization techniques, in order to efficiently treat bounded or unbounded three-dimensional multidomains. Partial differential equations associated to boundary integral equations will be weakly reformulated by the energetic approach and a particular emphasis will be given to theoretical and experimental analysis of the stability of the proposed method.

1

Introduction

Time-dependent problems that are frequently modeled by hyperbolic partial differential equations can be dealt with the boundary integral equations (BIEs) method. The transformation of the differential problem to a BIE follows the same well-known method for elliptic boundary value problems. For the discretization phase Boundary Element Methods (BEMs) are successfully applied in seismology, in particular for the study of the soilstructure interaction, in acoustic and in the analysis of the electromagnetic scattering (see e.g. [5, 8, 6]), taking advantage of dimensionality reduction and of the implicit enforcement of radiation conditions at infinity. When one deals with regions having different material properties (e.g. layered soils, [9, 4]) or even different physics (e.g. in solid-fluid coupling [7] or wave-soil-structure interaction [10]) domain decomposition is needed. In this framework, BEM is nowadays understood to be complementary rather than concurrent to finite element method (FEM). The BEM, also when formulated directly in the space-time domain, has attracted particular interest for: its high accuracy; the simplicity of imposing the interface conditions in problems defined on multi-domains: the continuity and compatibility conditions that have to be satisfied on the interface respectively by the primal unknown function and its derivative can be simply incorporated because they both appear directly in the boundary integral formulation; the implicit fulfillment of the infinity radiation conditions; the low cost of discretization when problems are defined over unbounded domains and the classical numerical methods (finite difference, finite element) cannot efficiently determine the solution having to insert artificial boundaries and consequently non-reflective conditions in order to try to significantly reduce spurious reflections of the wave propagating from the fictitious boundary towards the interior of the domain. The use of BIEs and BEMs, however, is complex and not particularly efficient in presence of non-linearities localized in small parts of the domain. In this case, the classical differential models and numerical techniques, such as the finite difference method (FDM) and FEM help to efficiently deal with the nonlinear part of the problem, but require,

21

22

in general, a fine discretization of the entire domain with a significant increase in computational cost. In this context, BEM and FEM methods for the approximation of boundary integral equations systems and systems of partial differential equations are complementary and a suitable coupling of these two techniques can take advantage of what both offer. In this paper, starting from a recently developed energetic space-time weak formulation of BIEs related to wave propagation problems defined on single and multi domains (see in particular [2, 3] and references therein), a coupling algorithm is presented, which allows a flexible use of FEM and BEM as local discretization techniques, in order to efficiently treat bounded or unbounded multidomains. Partial differential equations associated to BIEs will be weakly reformulated by the energetic approach and a particular emphasis will be given to the experimental analysis of the stability of the proposed method.

2

Model problem

Let Ω ⊂ R3 be an open bounded domain, with a sufficiently smooth boundary ∂Ω. Let Ω1 ∪ Ω2 = R3 \ Ω be a decomposition of R3 \ Ω, with Ω1 unbounded and Ω2 bounded non-overlapping subdomains such that ¯ 2 = Γ, as depicted in transversal section in Figure 1. Note that the boundary of the unbounded subdomain ¯ 1 ∩Ω Ω Ω1 is just the interface Γ. Having denoted with ui (x,t) the unknown function in the i-th subdomain, which

Figure 1: Transversal section of the spatial domain for the model problem. ∂ui (x,t), which depends on a unitary (outward) normal depends on space and time, and with pi (x,t) := μi ∂n x vector and on μi , a typical constant related to the material constituting Ωi , we want to solve the following wave propagation model problem:

Δui (x,t) − c−2 i uï (x,t) = f i (x,t),

x ∈ Ωi , t ∈ [0, T ], i = 1, 2

(2.1)

ui (x, 0) = 0,

x ∈ Ωi , i = 1, 2

(2.2)

u˙i (x, 0) = 0,

x ∈ Ωi , i = 1, 2

(2.3)

x ∈ ΓN := ∂Ω, t ∈ [0, T ],

(2.4)

x ∈ Γ, t ∈ [0, T ],

(2.5)

¯ p2 (x,t) = p(x,t), u1 (x,t) = u2 (x,t), p1 (x,t) = −p2 (x,t),

where overhead dots indicate derivatives with respect to time, ci is the propagation velocity of a perturbation in the i-th subdomain, p(t) ¯ is a given function, the assigned PDE right-hand sides f1 (x,t) ≡ 0 and f2 (x,t) are suitably connected and continuity and equilibrium conditions for the solutions are imposed at the interface between the two subdomains. The unknown functions ui are understood in a weak sense, i.e. ui ∈ H 1 ([0, T ]; H 1 (Ωi )). Since the goal of this work is to approximate u1 using a BEM approach and u2 using a FEM technique, we have to obtain a boundary integral reformulation of the problem (2.1)-(2.5) in Ω1 . Let us consider the boundary integral representation of u1 (x,t), for x ∈ Ω1 , t ∈ [0, T ]: u1 (x,t) =

1 μ1

t Γ 0

G(r,t − τ)p1 (ξ, τ)dτ dγξ −

t ∂G Γ 0

∂nξ

(r,t − τ)u1 (ξ, τ) dτ dγξ ,

(2.6)


23

c1 where r = r2 = x − ξ2 and G(r,t − τ) = 4π r δ[c1 (t − τ) − r] is the forward fundamental solution of the three dimensional wave operator, with δ[·] the Dirac distribution. With a limiting process for x tending to Γ we obtain the space-time BIE

t

1 1 u1 (x,t) = 2 μ1

Γ 0

G(r,t − τ)p1 (ξ, τ) dτ dγξ −

t ∂G Γ 0

∂nξ

(r,t − τ)u1 (ξ, τ) dτ dγξ .

(2.7)

Further, we consider a second space-time BIE, obtained from (2.6) by taking the normal derivative with respect to nx and operating a limiting process for x tending to Γ: 1 p1 (x,t) = 2

t ∂G Γ 0

∂nx

(r,t − τ)p1 (ξ, τ)dτdγξ − μ1

t Γ 0

∂2 G (r,t − τ)u1 (ξ, τ)dτdγξ . ∂nx ∂nξ

(2.8)

The problem (2.1)-(2.5) in the subdomain Ω1 can be rewritten in a strong form as a system, with obvious meaning of notation, of two BIEs in the boundary unknowns the functions p1 (x,t) and u1 (x,t) on Γ: ⎧ ⎨ 1 u (x,t) = 1 (V p )(x,t) − (Ku )(x,t) 1 1 2 1 . (2.9) μ ⎩ − 1 p (x,t) =1−(K ∗ p )(x,t) + μ (Du )(x,t) 2

1

1

1

1

Of course, this problem has to be coupled with the differential one specified for Ω2 , under the coupling conditions (2.5) at the interface. In particular, here we are interested in a direct space-time weak formulation for the coupling of the integro-differential problem on Ω1 ∪ Ω2 , and this will be done in the next Section.

3

Energetic weak formulation for the coupling

We start remarking that the solution of (2.1)-(2.5) in Ω1 satisfies the following energy identity:

EΩ1 (u1 , T ) :=

1 2

1 Ω1

c21

T 1 u˙21 (x, T ) + |∇u1 (x, T )|2 dx dt = u˙1 (x,t)p1 (x,t) dt dγx μ1 Γ 0

(3.10)

which can be obtained multiplying by u˙1 equation (2.1) specified for i = 1 and integrating by parts over Ω1 × [0, T ]. Then, the energetic weak formulation of the system (2.9) is finally defined as follows: 1

find u1 ∈ H 1 ([0, T ]; H02 (Γ)) and p1 ∈ L2 ([0, T ]; H − 2 (Γ)) such that 1 1 ˙ ˙ 2 < u˙1 , q1 >= μ1 < (V p1 ), q1 > − < (Ku1 ), q1 > , − 12 < p1 , v˙1 >= − < K ∗ p1 , v˙1 > +μ1 < Du1 , v˙1 > 1

(3.11)

where < ·, · >=< ·, · >L2 ([0,T ]×Γ) and q1 (x,t), v1 (x,t) are suitable test functions, belonging to the same functional space of p1 (x,t), u1 (x,t), respectively. In particular, the first equation in (2.9) has been differentiated with respect to time and projected with the L2 ([0, T ] × Γ) scalar product by means of functions belonging to 1 L2 ([0, T ]; H − 2 (Γ)), while the second equation in (2.9) has been projected with the L2 ([0, T ] × Γ) scalar product 1

by means of functions belonging to H 1 ([0, T ]; H02 (Γ)), derived with respect to time. For the energetic weak formulation in Ω2 , let us multiply the differential equation (2.1) for the time derivative of test functions v2 (x,t) ∈ H 1 ([0, T ]; H 1 (Ω2 )) and integrate by parts in space obtaining, after a multiplication by the coefficient μ2 : −μ2 A (v2 , u2 )+ < v˙2|Γ , p2|Γ >= μ2 F (v2 )− > ,

(3.12)

24

where

A (v2 , u2 ) :=

T 0

Ω2

∇v˙2 (x,t) · ∇u2 (x,t) +

F (v2 ) :=

T 0

Ω2

1 v˙2 (x,t) u¨2 (x,t) dx dt , 2 c2

v˙2 (x,t) f2 (x,t) dx dt

(3.13) (3.14)

and >=>L2 ([0,T ]×ΓN ) . Now, remembering (2.5) and using the further coupling condition at interface for test functions v1 (x,t) = v2|Γ (x,t) ,

(3.15)

combining (3.12) with the second weak BIE in (3.11), we finally obtain the following energetic weak formulation of the coupled problem 1 1 ˙ ˙ μ1 < (V p1 ), q1 > − < (Ku2|Γ ), q1 > − 2 < u˙2|Γ , q1 >= 0 − 12 < p1 , v˙2|Γ > − < K ∗ p1 , v˙2|Γ > +μ1 < Du2|Γ , v˙2|Γ > −μ2 A (v2 , u2 ) = μ2 F (v2 )− > . (3.16) At every time instant, the unknowns are p1 over the interface Γ and u2 in Ω2 . Let us conclude this Section with some energy considerations. At first, let us consider system (3.11) with q1 = p1 and v1 = u1 and change the sign in the second equation; then, summing up the two equation and remembering (3.10) one obtains: < u˙1 , p1 >=

1 < (V ˙p1 ), p1 > −μ1 < Du1 , u1 >= μ1 EΩ1 (u1 , T ) . μ1

(3.17)

On the other side, considering v2 = u2 in (3.13), one gets:

A (u2 , u2 ) = EΩ2 (u2 , T ) .

(3.18)

Following similar arguments, starting from (3.16) the following relation appears μ1 EΩ1 (u1 , T ) + μ2 EΩ2 (u2 , T ) = −μ2 F (u2 )+ > ,

(3.19)

from which one can easily deduce a-priori stability estimates for regular solutions.

4

Space-time Galerkin discretization

To simplify the notation, from now on, we will drop the subscripts from the unknowns p1 and u2 , being clear the spatial subdomain to which they are referred. For time discretization we consider a uniform decomposition of the time interval [0, T ] with time step Δt = T /NΔt , NΔt ∈ N+ , generated by the NΔt + 1 time-knots: tk = k Δt, k = 0, · · · , NΔt , and we choose temporally piecewise constant shape functions for the approximation of p and piecewise linear shape functions for the approximation of u, although higher degree shape functions can be used. Note that, with this particular choice, temporal shape functions for the approximation of p and for the approximation of u will be defined, respectively, for k = 0, · · · , NΔt − 1, as ψkp (t) = H[t − tk ] − H[t − tk+1 ] ,

ψuk (t) = R(t − tk ) − R(t − tk+1 ),

where H[τ] is the Heaviside function and R(τ) = Δtτ H[τ] is the ramp function. For the space discretization, we consider the bounded subdomain Ω2 (suitably approximated by a domain) of polyhedral type and a mesh Th = {e1 , · · · , eMh } constituted by Mh tetrahedra, with diam(ei ) ≤ h, ei ∩ e j = 0/ if


25

h i = j and such that M i=1 ei = Ω2 . The mesh TΓ,h on the interface will be the restriction of Th to Γ, therefore constituted by, let us say, M1 non-overlapping triangles. The functional background compels one to choose spatially shape functions belonging to L2 (Γ) for the approximation of p and to C0 (Ω2 ) for the approximation of u. Hence, we will choose piece-wise constant basis functions ϕ pj (x), j = 1, · · · , M1 related to TΓ,h for the approximation of p over the interface and piece-wise linear continuous functions ϕuj (x), j = 1, · · · , M2 related to Th for the approximation of u in Ω2 . The approximate solutions of the problem at hand will be expressed as

NΔt −1 M1

p(x,t) ˜ :=

NΔt −1 M2

∑ ∑ α p j ϕ pj (x) ψkp (t) , (k)

u(x,t) ˜ :=

k=0 j=1

∑ ∑ αu j ϕuj (x) ψuk (t). (k)

k=0 j=1

The Galerkin BEM-FEM discretization coming from energetic weak formulation (3.11) produces the linear system Eα = b, (4.1) where matrix E has a block lower triangular Toeplitz structure, since its elements depend on the difference th − tk and in particular they vanish if th ≤ tk . Each block has dimension M := M1 + M2 . If we denote by E() the block obtained when th − tk = ( + 1) Δt, = 0, . . . , NΔt − 1, it has a symmetric 2 × 2 block sub-structure of the type ⎡ ⎤ () () EΓ,FEM EΓ () ⎦ (4.2) E =⎣ () () EFEM,Γ EFEM where we can recognize the contribution of the coupling on the interface Γ and of the pure energetic FEM inside Ω2 , and it presents a highly sparse structure. The solution of (4.1) is obtained with a block forward substitution, i.e. at every time instant t = ( + 1) Δt, = 0, · · · , NΔt − 1, one computes

z() = b() − ∑ E( j) α(− j) j=1

and then solves the reduced linear system E(0) α() = z() .

(4.3)

Procedure (4.3) is a time-marching technique, where the only matrix to be inverted is the non-singular block E(0) , therefore the LU factorization needs to be performed only once and saved. All the other blocks are used to update at every time step the right-hand side. Owing to this procedure we can construct and store only the blocks E(0) , · · · , E(NΔt −1) with a considerable reduction of computational cost and memory requirement.

5

Numerical results

Let’s consider a spherical cavity of radius RC = 3 embedded in an infinite linear and homogeneous medium. The Ω2 region is defined by RC < r < RI , with RI = 5, the Ω1 region is defined by r > RI and the interface Γ is the spherical surface r = RI . Here, μ1 = μ2 = 1 and c1 = c2 = 1. Meshes in Ω2 and over Γ are conformal, i.e. triangles for Γ coincide with faces of tetrahedra for Ω2 . The cavity surface is subjected to a uniform traction p¯ = H[t]. The analytical solution can be obtained following the same scheme as in [3]: # 2 $ " RC RC R2 ! R2 p(r) = e(r−RC −t)/RC − t > r − RC . (5.4) u(r) = C 1 − e(r−RC −t)/RC , − 2C , 2 r r r r The time interval of the analysis has been set to [0, 40]. The mesh adopted has an average side length h = 0.5 and h = 1 on the cavity surface and on the interface, respectively. In Figure 2 time histories of tractions at

26

interface are shown for a broad range of time steps: no reflections are observed, and even if the approximate solution presents some oscillations for small Δt, these never explode into instabilities.

Figure 2: History of interface tractions for different time steps

References [1] T. Abboud, P. Joly, J. Rodriguez, I. Terrasse, J. Comput. Physiscs, 230 (15), 5877–5907, (2011). [2] A. Aimi, S. Gazzola, C. Guardasoni, Mathematical Methods in the Applied Sciences, 35, 1140–1160, (2012). [3] A. Aimi, M. Diligenti, A. Frangi, C. Guardasoni, Engineering Analysis with Boundary Elements, 36, 1756–1765, (2012). [4] V.S. Almedida, J.B. Paiva, Adv. Eng. Software, 38, 835-845, (2007). [5] H. Antes, G. Beer, W. Moser, Comput. Mech., 36 (6), 431-443, (2005). [6] A. Buffa, M. Costabel, C. Schwab, Numer. Math., 92 (4), 679–710, (2002). [7] O. Czygan, O. von Estorff, Engineering Analysis with Boundary Elements, 26, 773–779, (2002). [8] T. Ha Duong, B. Ludwig, I. Terrasse, Int. J. Numer. Methods Engrg., 57, 1845–1882, (2003). [9] M. Sari, I. Demir, J. Appl. Sciences, 6 (8), 1703–1711, (2006). [10] J.L. Wegner, M.M. Yao, X. Zhang, Computer and Structures, 83, 2206–2214, (2005). [11] G.Y. Yu, Journal of Applied Mechanics, 70, 451–454, (2003).


27

Performance of CMRH for the simulation of a vibroacoustic problem based on coupled finite and boundary element method Ahlem ALIA1, Hassane Sadok2 1

LML, université Lille1, Cité Scientifique, Villeneuve d’Ascq, France, [email protected] 2

LMPA, Université du Littoral, 50 Rue F. Buisson B.P. 699, 62228 Calais Cedex, France, [email protected]

Keywords: Vibroacoustic coupling, BEM, iterative method, CMRH.

Abstract. In order to simulate a simple strong vibroacoutic coupling problem, the boundary element method (BEM) applied to acoustics is combined to the finite element method (FEM) usually used for structures. The obtained system involves a sparse real submatrix due to the structure, a dense complex submatrix for the acoustic part of the problem as well as two coupling submatrices. In this paper, the corresponding linear system is solved more efficiently than common direct solvers by using the iterative technique CMRH (Changing Minimal Residual method based on Hessenberg process). In this method, the generation of the basis vectors of the Krylov subspace is based on the Hessenberg process instead of Arnoldi's one that the most known GMRES (Generalized Minimal RESidual) solver uses. The efficiency of this iterative method is compared to GMRES. The results are presented in term of storage requirements, errors and residuals. Introduction In the last years, the noise generated by vibrating structures has attracted the interest of the industry expressing a stong demand on the limitation of noise machines such as vehicles, airplanes... Consequently, research focuses on methods that predict these vibroacoustic phenomena. In many cases, it is not necessary to consider a strong coupling between fluid and structure if the structure is never affected by the acoustic waves propagating in the fluid. Therefore, the problem simulation consists in two decoupled small systems. Abiously, when the fluid is sufficiently heavy it affects the vibratory behaviour of the structure and consequetly, the strong coupling must be taken into account [1]. Generally, there is a well-suited method for each application. Among the different numerical methods used for vibroacoustic simulation, FEM is the most used technique for both acoustic and the structure [1,2]. In this case, the eigen-modes of the structure and the fluid cavity in vaccum are usually prefered in order to accelerate the calculations because of the reducing of the system size. This method stills the most suited for internal vibroacoustic problems. Another way to simulate external vibroacoustic problems is to apply BEM for acoustic and FEM for the structure [3]. In this case, there is no need to use absorbing boundary conditions on the numerical domain boundaries since the Somerfeld radiation is already included in BEM formulation via Green’s function. However, it appears less efficient than FEM in term of CPU time since it needs the computation of surface integrals and involves dense complex linear system. The resolution phase, in which the computation time is related to the nodes number, and the integration phase that depends on the total number of elements are time consumming especially for very large-scale problems. Recently, many efforts have been devoted to optimize the resolution phase of the linear system arising from BEM formulation by using Krylov subspace iterative solvers [4]. The most common used one is GMRES that is based on Arnoldi’s process. In this paper, the performance of CMRH [5,6], which is based on Hassenberg process, is tested with respect to GMRES. The most advantage of this technique is that it needs less arithmetic operations because it constructs a lower trapezoïdal basis. Moreover, when implemented efficiently, it requires less memory storage since

28

the original matrix of the linear system serves to store both Krylov vector basis and Hassenberg matrix. In this paper, CMRH is used to solve the linear system resulting from a simple vibroacoustic coupling problem. It is organized as follows. In the second section, the governing equations of the problem are etablished. The third section represents a brief introduction to CMRH. In the last section, some numerical results and comparisons with GMRES are given to show the efficiency of this method. Governing Equations in Vibroacoustic Problem In this paper, the acoustic radiation inside a closed cavity of volume (f) is simulated. The fluid occupying the cavity is assumed homogeneous and inviscid. The cavity is partially elastic on the boundary ( s) and it is considered as rigid on ( f). The surface ( sf) denotes the interface between fluid and the elastic structure. BEM for the fluid. Hemholtz equation governs the propagation of acoustic waves in inviscid, homogeneous and perfectly compressible fluid: (1) Δp + k 2 p = 0 in Ω f where p is the pressure, k = ω c = 2πf c is the wave number, c is the sound velocity in the fluid and f is the excitation frequency. The application of divergence theorem to eq (1) leads to the following integral equation [7]: (2) ∂g C (r ) p (r ) = − iρ f ωVn g + p dΓ ∂n where g (r , ry ) = exp − ik r − ry 4π r − ry is the Green's function, p(r) is the pressure at any field

(

)

point r, ry is the position vector of a source point located at acoustic domain boundary, C is the jump term resulting from the treatment of singular integral involving Green's function and ∂p ∂p = −iρ f ωVn . In particular, = 0 on the rigid boundary (Γf). ∂n ∂n By applying BEM to eq (2), the discretized equation can be written as : (3) Cp + H * p = iρ f ωGVn Hp = iρ f ωGVn where the global matrices H* and G are obtained from the elementary ones: (4) ∂g h *i = N i dΓ, g i = − gN i dΓ, ∂n FEM for the structure. The vibratory behavior of an elastic, linear and isotropic structure occupying the bounded domain (Ωs) without any body forces, is governed by the following equation [8]: (5) σ ij , j (u ) + ρ sω 2ui = 0 in Ω s where ui is the displacement in the ith direction, ρs is the density and is the stress. On (Γs), a surface force is prescribed: (6) σ ij (u )n j = f i on Γ s When modeled by finite element method, the structure response is given by the following equation: (7) K − ω2M x = F

(

)

where K and M represent, respectively, stiffness and mass matrices, ω is the pulsation, x denotes the vector of unknowns (displacement and rotations) and F is the mechanical load vector. FEM-BEM for vibroacoustic coupling. In a coupled problem, the BE model is divided into two parts: (Γsf) for which the BE are connected to the FE and (Γf) representing the remainder

Advances in Boundary Element Techniques XIV surface of the fluid domain. The continuity, at the interface, of the normal displacement and stress leads to the following conditions: (8) ∂p = ρ f ω 2un on Γ sf ∂n σ ij n j = − pni on Γ sf Finally, the resulting coupling system can be written as: (9) K − ω2M L 0 x F 2 ρ f ω G11T H 11 H 12 psf = iρ f ωG12TV f 2 ρ f ω G21T H 21 H 22 p iρ ωG TV 22 f f f

(

)

CMRH Method Let consider a linear ( N × N ) system Ax = b and an initial approximation x0 of the solution x corresponding to an initial residual r0 = b − Ax0 . CMRH is an iterative algorithm based on Hassenberg process for solving general linear systems [5,6]. In this process, a set of Krylov vectors l1 + l2 + forming the columns of a unit lower triangular matrix Lk are computed. These columns which constitute

the basis vectors of Krylov subspace K k ( A, r0 ) = span{r0 , Ar0 , , Ak −1r0 } are orthogonal to unit basis

vectors ek( n ) . The columns of Lk are calculated by the following recurence equation starting with a vector l1 : (10) where l1 = r0 β β = r0 (1) k where k = 1, ,m hk + 1,k lk + 1 = Alk − h j ,k l j j =1 Eq (10) leads to a matrix equation given by ALk = Lk +1 H k where H k is ( k + 1 ) × k upper Hassenberg matrix and L is a matrix such as the kth component of lk equals one and the first (k-1) components of lk are zero [6]. The computed matrix Lk by Hassenberg process is used by CMRH to calculate the correction zk = Lk yk . The kth iterate of CMRH is defined by xk = x0 + Lk yk where yk minimizes r0 e1( k + 1 ) − H k yk . In CMRH, the stopping criterion is based on upper bound as an estimation of the residual beacause the exact formula of the residual in CMRH are too expensive in term of computational time. [5] Numerical Results Let consider a rigid cavity with one simply supported flexible face considered as a plate made from brass (density s=8500 kg/m3, Young modulus E=107 GPa, Poisson ratio =0.34, thickness t=0.9144mm). The cavity is filled with air (density f=1.21 kg/m3, sound velocity c=343m/s). A harmonic pressure load of 1 psi is uniformly distbuted on the elastic plate. In this application, 10-5 is adobted as stopping criterion for both GMRES and CMRH techniques. Fig. 1 and Fig. 2 represent, respectively, the acoustic pressure at the cavity centre and the velocity of the centre of the elastic plate. Both GMRES and CMRH give similar result in agreement with direct solver for the pressure and the velocity.

29

30

Fig.1 Pressure at the centre of the cavity

Fig.2 Velocity at the centre of the elastic plate

In order to study the performance of CMRH with respect to GMRES, the error was analyzed by choosing the right hand side b is such way that the exact solution x* is known, where x* is the solution obtained by the direct solver and b is calculated by A x*=b. Hence, the error norm is considered as ||x- x*|| where x is the solution given by either GMRES or CMRH.

Fig.3 Variation of residual norm with iterations

Fig.4 Variation of error norm with iterations

In Fig. 3, the residual norm is monitored in logarithmic scale for a frequecy of 800 Hz. It can be seen that both methods present the same behavior. However, GMRES seems more efficient since it is a minimising residual method. Moreover, CMRH requires less iterations than GMRES. This can be explained by the fact that CMRH overestimates the residual via its stopping cretirion. In fact, the true residual is 9.621 10 −5 whereas the estimated one is 9.419 10−6 . Fig. 4 represents the variation of the error norm with respect to iterations. It shows the same behavior of error curve of both methods. These same behaviors have been observed when CMRH was applied to solve the linear system due to variational BEM in case of acoustic problem [10]. In term of memory storage, to store Hassenberg matrix as well as the Krylov basis vectors, CMRH uses the existing A matrix. In GMRES, however, extra memory is needed to store them. Hence, in CMRH the same behavior is obtained as GMRES but with much less memory.

Advances in Boundary Element Techniques XIV Conclusion In this paper, the CMRH performance is analysed in case of a simple vibroacoustic problem simulated by coupled boundary element and finite element method. The analysis was conducted in comparison with GMRES in term of pressure, velocity, residual and error. CMRH showed the same behavior as GMRES by producing convergence curves close to those provided by GMRES but by needing fewer operations and memory. This represents the main advantage of CMRH. References [1] (1995).

G. Sandberg International Journal for Numerical Methods in Engineering, 38, 357-370

[2]

A. Alia and M. Souli. M Journal AIAA, 48, 2196-2205 (2010).

[3] C.M. Lee, L.H. Royster and R.D. Ciskowski Engineering Analysis with Boundary Elements, 16, 305-315 (1995). [4] (2003).

S. Schneider and S. Marburg Engineering Analysis with Boundary Elements, 27, 751-757

[5]

H. Sadok Numerical Algorithms, 20, 303-321 (1999).

[6] (2008).

M. Heyouni and H. Sadok Journal of Computational and Applied Mathematics, 213, 387-399

[7] T.W. Wu Boundary element acoustics: Fundamentals and computer codes, Witpress, Southampton: Boston (2000). [8]

R. Ohayon and C. Soize Structural acoustics and vibration. Academic press, London (1998).

[9]

A. Alia and M. Souli Engineering Analysis with Boundary Elements, 36, 346-350 (2012)

31

32

Multidomain Formulation of BEM Analysis Applied to Large-Scale Polycrystalline Materials A. F. Galvis1, R. Q. Rodriguez1, P. Sollero1, E.L Albuquerque2 1

Faculdade de Engenharia Mecanica, Universidade Estadual de Campinas, Brazil {andres.galvis, reneqr87, sollero}@fem.unicamp.br 2

Faculdade de Tecnologia, Universidade de Brasilia, Brazil [email protected]

Keywords: Multidomain Boundary Element Method, Polycrystalline Materials.

Abstract. Polycrystalline structures are present on metal alloys. Therefore, it is necessary to understand and model the mechanical behavior of this media. Usually, this is accomplished by the use of different numerical methods. However, the analysis of polycrystalline materials lead to other type of problems, such as high computational requirements generated in order to get an efficient solution. In this work, the 2D polycrystalline structure is generated using an average grain size through the Voronoi tessellation method and discretized through simulations with random material, crystalline orientation and orthotropic behavior [1]. BEM discretization requires multidomain analysis and large-scale degrees of freedom [3, 4]. This technique demands a different strategy in order to get a faster response. Numerical examples were carried out to demonstrate the feasibility of the application of the method to large-scale polycrystalline problems. Results were compared with the conventional BEM solution for several set of loads. The analysis of the structure is performed using the proposed anisotropic multidomain BEM formulation [3, 4]. Introduction. The Boundary Element Method (BEM) is a powerful numerical method for the solution of different problems in engineering [5]. However, the BEM has some disadvantages when compared with other methods, meanly due to the fully populated matrices that are generated and the high computational load required during its process. Sfantos and Aliabadi [1] used the BEM for polycrystalline structure analysis, they investigated two-dimensional crack propagation along grain boundaries through a linear cohesive law, and mixed mode failure conditions. Benedetti and Aliabadi [6] apply techniques for homogenization of a three-dimensional model of cubic polycrystalline material. Sfantos and Aliabadi [11] proposed a Multi-scale BEM modellig for material and degradation fracture. Kokaly et. al [7] used the implementation of another numerical method to generate a two-dimensional finite element (FE) model of assembly with idealized microstructure and uniform grain size of polycrystalline alumina. Large-scale polycrystalline material modeling require the use of Muldomain BEM that is widely explained by Kane [3] and Katsikadelis [4], both suggested a different strategy to generate the array of hypermatrices. Recently, several numerical methods for solving these large systems of equations have been developed in order to reduce the time of execution. The Adaptive Cross Approximation (ACA) was used by Grytsenko and Peratta [8] as a solver for three-dimension singular domain BEM application. An OUT-CORE solver for


33

large, multi-zone boundary element matrices is presented by Rigby and Aliabadi [13] and Block Equation Solver are developed by Kane [3] and Crotty [9] that uses the Gaussian Elimination by blocks for linear elasticity BEM problems. In this paper the implementation of Multidomain formulation of BEM by Katsikadelis [4] is performed over polycrystalline structure generated with anisotropic fundamental solution and the application of Blocked Equation Solver proposed. Finally, conclusions are pointed out. Polycrystalline Structure Modeling. The material modeling used in this work is the generation of a random artificial structure with the Voronoi Tessellation method as in [1]; this approach defines the behavior of the structure with random orthotropic material and crystalline orientation. The simulation was performed as shown in Fig1.

Figure 1. Artificial structure generated for 20 and 40 grains with randomly distributed material orientation for each grain [1]. Due to the formulation used in this paper the material orientation coordinated axes 123 coincides with the geometrical coordinate system xyz, that means ߠ ൌ Ͳι. Different cases are taken into consideration when each axis coincides with the axis ‫ ݖ‬of the geometry coordinated system; thus case ͳ ‫ݖ ؠ‬, case 2‫ݖ ؠ‬ and case ͵ ‫ݖ ؠ‬. These cases are presented in three different colors in Fig 1. Grain material properties are modeled for plain strain and plain stress analysis with the following constitutive relations:

σ ij = cijkl ε ij

ε ij = sijklσ ij

(1)

where ܿ௜௝௞௟ is the stiffness tensor and ‫ݏ‬௜௝௞௟ is the compliance tensor using the Voigt notation that is defined by [1, 2, 10]

s = ª¬ sij º¼ , ( i, j = 1, 2,..., 6 )

(2)

34

For the case of orthotropic material with three mutually perpendicular symmetry planes, the compliance tensor is reduce to 9 components, since s14 = s15 = s16 = 0 , s24 = s25 = s26 = 0 , s34 = s35 = s36 = 0 and

s45 = s46 = s56 = 0 , see [1, 2]. Multidomain Boundary Element Method. In order to illustrate the multidomain formulation, Fig 2 shows the three-multidomain model.

Figure 2. Three-Multidomain model [3] Equation (4) shows the general hypermatrix applying the multidomain formulation in [3] to form a system of equations ሾ‫ܣ‬ሿሼ‫ݔ‬ሽ ൌ ሼܾሽ.

ª[ A ]1 « 1 = A [ ] « [0] « «¬ [ 0]

[ 0] [ 0] [ H ]12 [ H ]13 [ 0] − [G ]12 − [G ]13 [0] º» 2 2 2 2 2 [ A ]2 [ 0] [ H ]12 [ 0] [ H ]23 [G ]12 [ 0] − [G ]23 » » 3 3 3 3 3 [ 0] [ A ]3 [0] [ H ]13 [ H ]23 [ 0] [G ]13 [G ]23 »¼ 1

1

1

1

(4)

ሼ࢞ሽ is the vector that contains the unknown values of the external boundary and interfaces and ሼ࢈ሽ is the known vector.

{x} = {{x}1 {x}2 {x}3 {u}12 {u}13 {u}23 {t}12 {t}13 {t}23 } 1

2

3

1

1

2

1

{b} = {{b}1 {b}2 {b}3} 1

2

3 T

1

2

T

(5)

(6)

Blocked Equation Solver. The Multidomain BEM formulation generates a general system of equations where matrix ሾ࡭ሿ is a hypermatrix like eq (4), and its entries are smaller matrices [4]. These kind of matrices are sparse. For the polycrystalline structures that is large-scale problem, a big percentage of the matrix will be zero due to the majority of subdomains are internal grains with unknown boundary conditions. In order to solve the equation system, the use of conventional methods can be inefficient. Thus the Blocked Equation Solver [4] is proposed as another way to obtain the solution. The strategy treats the hypermatrix as a division per blocks and has three different phases the Block Factorization, Black-forward Reduction and Block-Back Substitution.


35

The division of the matrix in blocks depends on the problem; and appropriated hypermatrix is required in order to apply this method. Therefore is strongly necessary to have nonzero blocks in the main diagonal. This is possible using the presented array matrix (4) proposed by Kane [4] and Crotty [9]. However, its implementation can be more complex than the suggested by Katsikadelis [3], where part of the diagonal blocks are zero leading the method fail. the proposed method in this work is determined with the size of the overall matrix (4) and the size of the square matrix ሾࡴሿ for each subdomain located it in the main diagonal of the of the overall matrix. Completed system of equations divided per blocks is shown in (7) [4].

ª[ A11 ] « «[ A 21 ] «¬[ A 31 ]

[ A12 ] [ A13 ] º {x1} ½ {b1} ½ [ A 22 ] [ A 23 ]»» °®{x 2 }°¾ = °®{b 2 }°¾ [ A32 ] [ A33 ]»¼ °¯{x3 }°¿ °¯{b3}°¿

(7)

The method is based on the ሾ‫ܮ‬ሿሾܷሿ factorization on the main diagonal and used matrix multiply and subtract operation between different matrices and vectors from the overall system. The iterative process is shown in appendix A, and is used to develop a general algorithm for random polycrystalline structure in which the matrix has a random block division in the diagonal. An advantage of this method is that the only extra memory needed for the alterations of the blocks from the overall matrix is when the matrix ሾ‫ܦ‬ሿ calculated [4]. Numerical Results. Simulation performed in this work is for a Carbon/Epoxy unidirectional fiber composite material (AS4/3501-6) [10]. The components of the stiffness tensor are presented in Table 3 and the material

is generated with a virtual polycrystalline structure as abovementioned with random

orientation and orthotropic behavior shown in Figure 1. Table 3. Elastic constants for the Carbon/Epoxy unidirectional fiber composite material (AS4/3501-6) [10] ࡱ૚ ሺ ሻ ࡱ૛ ሺ ሻ ࡱ૜ ሺ ሻ ࡳ૚૛ ሺ ሻ ࡳ૛૜ ሺ ሻ ࡳ૚૜ ሺ ሻ 147

10.3

10.3

7

3.7

7

࢜૛૚

࢜૛૜

࢜૚૜

0.27 0.54 0.27

Figure 4 shows the boundary conditions of the analyzed numerical example.

Figure 4. Boundary Conditions for the polycrystalline structure.

36

The main idea is to obtain some data time for comparing the process. In Table 4 are presented the numerical results for each method. Table 4. Time require for different methods to solve the system Number of Grains Blocked Equation Solver (s) Solver Matlab (s) 20

%

2,21

0,1384

-

40

11,817

11,257

-

60

33,471

37,99

11,89

80

76,4844

91,187

16,12

100

125,809

173,825

27,62

120

208,904

283,740

26,37

Figure 5 shows the comparison between the Block Equation Solver and Matlab Solver. ϯϬϬ ϮϱϬ

Time (s)

ϮϬϬ ϭϱϬ

^

ϭϬϬ ϱϬ

^ŽůǀĞƌDĂƚůĂď

Ϭ Ϭ

ϱϬ

ϭϬϬ

Number of Grains

ϭϱϬ

Figure 5. Time comparison between the Block Equation Solver and Matlab Solver Conclusions. In this work, a Block Equation Solver (BES), based on Kane [4], was adapted to Multidomain Polycrystalline large-scale problems. This methodology showed to be efficient when compared with the conventional Matlab solver. This is mainly because a multidomain problem can be easily adapted to a block solver. However, the collocation matrix structure could be assembled by several approaches, such as [3, 4], and this fact affects directly the proposed strategy. The BES works better where more than 40 grains are analyzed Acknowledgment. The authors would like to thank “Coordenação de Aperfeiçoamento de Pessoal de Nível Superior” (CAPES) for the financial support of this work.

37 References [1] G. K. Sfantos and M. H. Aliabadi. A boundary cohesive grain element formulation for modeling intergranular microfracture in polycrystalline brittle materials. International Journal for Numerical Methods in Engineering. 69:1590-1626, 2007. [2] P. Sollero and M. H. Aliabadi. Fracture mechanics analysis of anisotropic plates by the boundary element method. International Journal in Fracture. 64: 269-284, 1993 [3] J. T. Katsikadelis. Boundary Elements Theory and Applications. Elsevier Science, Ltd Kidlington, 2002. [4] J. H. Kane. Boundary Element Analysis in Engineering Continuum Mechanics. Prentice-Hall, Inc New Yersey, 1994. [5] R. Q. Rodriguez, P. Sollero and E. L. Albuquerque. Analysis of Anisotropic Symmetric Plates by the Adaptive

Cross

Approximation.

International

Conference

on

Boundary

Element

and Meshless Techniques. 2012. [6] I. Benedetti and M. H. Aliabadi. A three-dimensional grain boundary formulation for microstructural modeling of polycrystalline materials. Computational Materials Science. 67: 249-260, 2013. [7] M.T. Kokaly, D. K. Tran, A. S. Kobayashi. X. Dai, K. Patel and K. W. White. Modeling of grain pullout forces in polycrystalline alumina. Materials Science and Engineering. A285: 151-157, 2000. [8] T. Grytsenko and A. Peratta. Adaptive Cross Approximation based solver for boundary element method with single domain in 3D. Boundary Elements and Other Mesh Reduction Methods. 2008. [9] J. M. Crotty. Ablock Equation Solver for Large Unsymmetric Matrices Arising in the Boundary Integral Equation Method. International Journal for Numerical Methods in Engineering. 18: 997-1017. 1982. [10] I. M. Daniel and O. Ishai. Engineering Mechanics of Composite Materials. Oxford University Press, Inc New York, 2006. [11] G. K. Sfantos and M. H. Aliabadi. Multi-scale boundary element modeling of material degradation and fracture. Computer Methods in Applied Mechanics and Engineering. 196: 1310-1329, 2007. [12] C. A. Brebbia, J. C. F. Telles and L. C. Wrobel. Boundary Element Techniques: Theory and Applications in Engineering. Springer-Verlag, Berlin, 1984. [13] R. H. Rigby and M. H. Aliabadi. OUT-OF-CORE Solver for Large, Multi-Zone Boundary Element Matrices. International Journal for Numerical Methods in Engineering. 38: 1507-1533, 1995.

Gravity-driven migration of bubbles and/or solid particles near a free surface M. Guémas1 ,2 , A. Sellier1 and F. Pigeonneau2 Ecole polytechnique, 91128 Palaiseau Cédex, France 2 Surface du Verre et Interfaces, UMR125 CNRS St Gobain, 39 quai Lucien Lefranc, BP 135, 93303 Aubervilliers, Cedex, France e-mail: [email protected] e-mail: [email protected] 1 LadHyx.

Keywords: Bubble, free surface, surface tension, Stokes flow, Boundary-integral equation, film drainage.

Abstract We investigate the challenging problem of bubble(s) and rigid particle(s) interacting near a free surface. The time-dependent bubble(s) and free surface shapes are determined for a large range of Bond number by solving the creeping flow induced by the bubble(s) and the particle(s) motion. This works extends the boundary-integral formulation handled in a recent work solely dealing with bubble(s) ascending toward a free surface. 1. Introduction The gravity-driven motion of bubble(s) interacting with solid particle(s) in a viscous liquid in presence of a free surface is of high interest in applications such as geophysics, chemistry, glass process, . . . This task is quite involved due to the interactions occurring between the different solid and evolving surfaces. The axisymmetric gravity-driven migration of bubble(s) ascending toward a free surface has been numerically investigated either for bubble with equal surface tension in [4] or unequal surface tension in [1]. In contrast, this work considers, still for axisymmetric geometry, the more-involved case of cluster made of both bubble(s) and solid particle(s). This problem is solved adopting regularized and carefully-selected boundary-integral equations enforced on the entire liquid domain. 2. Challenging time-dependent problem 2.1 Assumptions and relevant axisymmetric quasi-steady Stokes flow We consider a cluster made of M ≥ 0 bubbles Bm and/or N ≥ 0 solid particles Pn with M +N ≥ 1 immersed in a Newtonian fluid with uniform density ρ and viscosity μ. This liquid is bounded by a free surface and both the cluster and the liquid are subject to the uniform gravity g = −ge3 (with g > 0). The bubble Bm , the solid particle Pn and the free surface have smooth and time-dependent surfaces Sm (t) with uniform surface tension γn , Σn (t) and S0 (t) with uniform surface tension γ0 , respectively. As illustrated in Figure 1 for M = N = 1, all surfaces S0 (t), Sm (t) and /or Σn (t) admit unit normal vector n directed into the liquid domain D(t) and the same axis of revolution (O, e3 ) (axisymmetric problem). As the cluster migrates under the gravity, the shapes of the bubble(s) and free surface evolve in time. At initial time, each bubble is spherical with typical radius a and the free surface is the z = 0 plane. At any time t, the pressure p0 above the disturbed free surface S0 (t) and pm inside the disturbed bubble Bm are assumed to be constant. Each solid particle Pn with uniform density ρn has, for symmetry reasons, time-dependent velocity U (n) (t)e3 . In addition, the liquid flow has pressure p + ρg.x (here x = OM with O denoting the origin of our Cartesian coordinates) velocity u and stress tensor σ. We denote by a the bubble(s) and solid particles typical length scale and by V the typical magnitude of velocities u and U (n) (t). Assuming that Re = ρV a/μ 1, inertial effects are negligible

39

z γ0

S0 (t)

n γ1

D(t)

x

B1 S1 (t)

n

g = −ge3

P1 n

Σ1 (t)

Figure 1: One bubble B1 and a solid sphere P1 moving near a free surface S0 (t). and the flow (u, p) obeys the following quasi-steady creeping flow equations and boundary conditions ∇ · u = 0 and μ∇2 u = gradp in D(t), (u, p) → (0, 0) as |x| → ∞,

(1)

σ · n = (ρg · x − pm + γm ∇S · n) n on Sm (t) for m = 0, ..., M,

(2)

u = U (n) (t)e3 on Σn (t) for n = 1, ..., N

(3)

where H = [∇S · n]/2 is the local average curvature. Assuming bubbles with constant volume, one supplements (1)-(3) with the relations 1 u · n dS = 0 on Sm for m=0,...,M. (4) Sm (t)

For N ≥ 1 the velocities U (n) (t) are unknown. By symmetry, each solid particle Pn is torquefree. In addition, each Pn with negligeable inertia is force-free. This latter property results in the additionnal conditions e3 · σ · ndS = (ρn − ρ)Vn g for n = 1, ..., N (5) Σn (t)

where ρn and Vn designate the uniform density and volume of the particle Pn . The material surface(s) Sm (t) have velocity V. Since there is no mass transfer across the surfaces Sm (t), one has (6) V · n = u · n on Sm for m = 0, ..., M. 2.2 Proposed tracking algorithm for the time-dependent entire liquid boundary We compute the time-dependent shape of the free surface, the bubble(s) and particle(s) surface(s) by running at each time t the following steps : 1

Note that (4) indeed also holds for m = 0 because u is divergence-free and u → 0.

Step 1: From the knowledge at time t of the liquid domain D(t), one first computes the quantity ∇S · n on each surface Sm (t). Step 2: One then solves at time t the relations (1)-(5) to get the unknown velocities U (n) (t) and the fluid velocity u on each surface Sm (t). Step 3: The liquid boundary D(t + dt) at time t + dt is obtained by moving between times t and t + dt each surfaces Sm by exploiting the relation (6) and each solid surface Σn at the velocity U (n) (t)e3 . One should note that for such a procedure the following issues are of the utmost importance: (i) To accurately compute the local average curvature (σ · n)/2 on each surface Sm in Step 1. (ii) To efficiently and accurately solve the Stokes problem (1)-(5) in Step 2. (iii) To adequately select a time step at in Step 3. This work introduces a suitable treatment to cope with the previous issue (ii). 3. Advocated method This section presents a new procedure to appropriately solve the problem (1)-(5). 3.1 Auxiliary Stokes flows for cluster involving at least one solid particle. As soon as N ≥ 1, each velocity U (n) occuring in (1)-(5) is unknown. Fortunately, it is possible to determine U (1) (t), · · · , U (N ) (t) prior to obtain the liquid flow (u, p) ! The trick consists in introducing, for n = 1, · · · N , auxiliary Stokes flows (u(n) , p(n) ) obtained without stress on each Sm and when each solid surface Σq is motionless for q = m with the surface Σn of Pq which translates at the velocity e3 . In other words, the flow (u(n) , p(n) ) with stress tensor σ (n) satisfies (1) and the following boundary conditions u(n) = δnq e3 on Σq for q = 1, · · · , N σ

(n)

(7)

· n = 0 on Sm for m = 0, · · · , M.

In addition, one supplements (1), (7)-(8) with the additionnal conditions u(n) · n dS = 0 on Sm for m = 0, ..., M.

(8)

(9)

Sm (t)

Denoting by ∂D the liquid boundary, the reciprocal identity [2] for the flows (u, p) and (u(n) , p(n) ) reads u(n) · σ · n dS = u · σ (n) · n dS. (10) ∂D

∂D

Enforcing the relations (5) by exploiting the aformentionned identity (10) and the boundary conditions (2)-(3) and (7)-(8), one then arrives at the N -equation linear system " ! e3 · σ (n) · n dS U (q) (t) = (ρ − ρn )Vn g q≥1

Σq

+

m≥0 Sm

u(n) · (ρg · x − pm + γm ∇S · n) n dS

for n = 1, · · · , N.

(11)

Furthemore, the pressure pm is uniform in the bubble Bm which leads, in conjection with (5), to " ! e3 · σ(n) · n dS U (q) (t) = (ρ − ρn )Vn g q≥1

Σq

+

m≥0 Sm

u(n) · (ρg · x + γm ∇S · n) n dS for n = 1, · · · , N.

(12)

It is possible (and here admitted) to prove, invoking the energy dissipation in Stokes flow, that (12) is well-posed (i. e. presents a non-singular matrix). Note that, one solely needs to evaluate the surface quantities u(n) on each Sm and σ (n) ·n on each Σq to obtain the translational velocity U (q) (t)e3 of the particle Pq . As shown in the next subsection, those required key surface quantities u(n) and σ(n) · n are calculated by inverting relevant boundary-integral equations on the entire liquid boundary ∂D. 3.2 Relevant boundary-integral equations 3.2.1 Three-dimensionnal formulation For a Stokes flow (u, p) with stress tensor σ obeying (1) with prescribed values of the stress σ · n on each Sm and of the velocity u on each Σn , one has the key coupled regularized boundary-integral equations (see for instance [5]), μ[u(x) − u(x0 )] · T(x, x0 ) · n(x)dS − G(x, x0 ) · σ · n(x)dS −8μπu(x0 ) + m≥0 Sm

=

m≥0 Sm

n≥1 Σn

G(x, x0 ) · σ · n(x)dS

for x0 on Sm

(13)

and m≥0

Sm

μ[u(x)−u(x0 )] · T(x, x0 ) · n(x)dS − = +8μπu(x0 ) +

m≥0 Sm

n≥1 Σn

G(x, x0 ) · σ · n(x)dS

G(x, x0 ) · σ · n(x)dS

for x0 on Σn

(14)

where the second-rank tensor G and third-rank stress tensor T are defined as [3] I (x − x0 ) ⊗ (x − x0 ) + ; |x − x0 | |x − x0 |3 (x − x0 ) ⊗ (x − x0 ) ⊗ (x − x0 ) T (x, x0 ) = −6 |x − x0 |5 .

G(x, x0 ) =

(15) (16)

with I the identity tensor. Clearly, solving (13)-(14) permits one to get the unknown vectors u on Sm and σ · n on Σn from the knowledge of u on Σn and σ · n on Sm . 3.2.2 Axisymmetric formulation Since we restrict the analysis to the% axisymmetric configuration depicted in Fig.1, we adopt cylindrical coordinates (r, φ, z) with r = x2 + y 2 , z = x3 and φ the azimuthal angle in the range [0, 2π]. We set u = ur er + uz ez = uα eα (with α = r, z), f = σ · n = fr er + fz ez = fα eα and n = nr e + nz ez = nα eα and introduce the traces Ln of Σn and Lm of Sm in the φ = 0 half plane.

Integrating over φ the equations (13)-(14), then yields the equivalent coupled boundary equations μ[uβ (x) − uβ (x0 )] Cαβ (x, x0 )dl − Bαβ (x, x0 )fβ nβ (x)dl −8πuα (x0 ) + m≥0 Lm

=

m≥0 Lm

and

m≥0 L

n≥1 Ln

Bαβ (x, x0 )fβ nβ (x)dl

μ[uβ (x) − uβ (x0 )]Cαβ (x, x0 )dl − = 8πuα (x0 ) +

m≥0 Lm

for x0 on Lm

n≥1 Ln

(17)



for x0 on Ln

(18)

for α = r, z, the differential arc length dl in the φ = 0 plane and the so-called single-layer and doublelayer 2 × 2 square matrices Bαβ (x, x0 ) and Cαβ (x, x0 ) given in Pozrikidis [5]. Note that a summation over β = r, z holds in (17)-(18). 3.2.3 Resulting boundary-integral equations for the axisymmetric Stokes flow problem Dealing with our axisymmetric problem (1)-(4), we first evaluate for each axisymmetric flow (n) (n) (u(n) , p(n) ) the needed vectors u(n) = uα eα on each Sm and σ (n) = fβ eβ on each Σq . We per(n)

(n)

form this calculation by inverting (17)-(18) for uz = δnq and ur = 0 on each Σq and σ(n) · n = 0 on each Sm . Once both the velocity and stress vectors are known on each surfaces Σq and Sm , one then obtains each velocity U (q) (t) by solving the linear system (12). Finally, we gain the required velocity u = uα eα on each Sm by inverting one more time (17)-(18) using the boundary conditions (2)-(3), i. e. μ[uβ (x) − uβ (x0 )] Cαβ (x, x0 )dl − Bαβ (x, x0 )fβ nβ (x)dl −8πuα (x0 ) + m≥0 Lm

=

m≥0 Lm

and

m≥0 Lm

n≥1 Ln

Bαβ (x, x0 )[ρg · x + γm ∇S · n] nβ (x)dl

μ[uβ (x) − uβ (x0 )]Cαβ (x, x0 )dl − +

m≥0

Lm

n≥1 Ln

for x0 on Lm

(19)

Bαβ (x, x0 )fβ nβ (x)dl = +8πU (n) (t)e3 (x0 )


for x0 on Ln

(20)

for α = r, z and γm uniform on each surface Sm . In summary, our approach consists, for N ≥ 1 solid particle(s), in inverting N + 1 boundary-integral equations (17)-(18). 4. Numerical method The coupled boundary-integral equation (17)-(18) are numerically inverted by appealing to the following key steps (see for further details [4, 1]): (i) First, the entire contour L = Ln ∪ Ln is divided into Ne curved boundary elements with the L0 truncated free surface. Each boundary element has Nc collocation points spread with a uniform distribution. An isoparametric approximation is used for the components u and f = σ · n on each boundary element.

For the nodes located on Lm , the vectors U and Fd collect the unknown and prescribed components of u and f . In a similar fashion, F and Ud are the vectors associated with the unknown and given values of f and u at the nodes of the solid contours Ln . Finally, once the coupled boundary-integral equations (17)-(18) are discretized, these vectors satisfy indeed the 2Ne Nc -equation linear system U + C · U − B1 · F = −B2 · Fd

for x0 on ∪m≥0 Lm ,

(21)

C · U − B1 · F = −Ud + B2 · Fd

for x0 on ∪n≥1 Ln .

(22)

The matrices B1 , B2 and C involve integrations of the quantities Bαβ and Cαβ introduced in §3.2.2 over the entire contours ∪m≥0 Lm and ∪n≥1 Ln . . (ii) One finds the solution (U,F) of (21)-(22) by Gaussian elimination. (iii) The shape of each surface Sm and the position of each Σn if N ≥ 1 is tracked in time using the boundary condition (6) and solving the equation dx/dt = u(x, t) for each nodal point. A RungeKutta-Fehlberg method performs this task using a time-step selected by controlling the errors for the second and third-order schemes. Furthermore, as the distance between two surfaces tends to zero, the adjusted time step is then very small and the computations is stopped. 5. Conclusions Preliminary numerical results will be exposed at the oral presentation for a cluster made of one bubble and one spherical solid particle. Furthermore, this particular case will be dicussed and compared with the two-bubbles configurations studied in [4, 1]

References [1] M. Guémas, F. Pigeonneau, and A. Sellier. Gravity-driven migration of one bubble near a free surface: surface tension effects. In M. H. Aliabadi P. Prochazca, editor, Advances in Boundary Element & Meshless Techniques XIII, 2012. [2] J. Happel and H. Brenner. Low Reynolds number hydrodynamics. Martinus Nijhoff Publishers, The Hague, 1983. [3] S. Kim and S. J. Karrila. Microhydrodynamics. Principles and selected applications. ButterworthHeinemann, Boston, 1991. [4] F. Pigeonneau and A. Sellier. Low-reynolds-number gravity-driven migration and deformation of bubbles near a free surface. Phys. Fluids, 23:092302, 2011. [5] C. Pozrikidis. Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, Cambridge, 1992.

44

Vortex Patches Gregory Baker Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA [email protected] Abstract: A vortex patch is a bounded region of uniform vorticity in two-dimensional, incompressible, inviscid fluid flow. The streamfunction satisfies Poissons equation with the vorticity acting as a forcing term. The standard formulation is to write the streamfunction as a convolution of the vorticity with the two-dimensional free-space Greens function. A simple application of Greens theorem converts the area integral to a boundary integral. Numerical methods must then account for the singular nature of the boundary integral, and high accuracy is difficult when filamentation takes place, that is, when long, very thin filaments of vorticity erupt from the main boundary. A new boundary integral is derived based on a different viewpoint. A particular solution is readily known which represents solid body rotation. To the particular solution must be added a homogeneous solution, and the combination must satisfy the boundary conditions. A standard boundary integral can be used to solve Laplaces equation with Dirichlet boundary conditions. This approach leads to a boundary integral without singularity and easily approximated by the trapezoidal rule that ensures spectral accuracy. Results indicate that high accuracy is possible with even modest resolution. Key–Words: Vortex Flows, Contour Dynamics, Filamentation.

1

Introduction

Given the challenges in numerically simulating fluid flow at high Reynolds numbers, researchers often turn to idealized models that contain simple distributions of vorticity under the assumption that viscosity may be neglected. In two-dimensional flow, the simplest distribution of vorticity is a vortex patch. It is a uniform distribution of vorticity bounded by a closed curve. The motion of the patch is determined completely by the motion of its boundary and an evolution equation for the boundary can be derived as a boundary integral with reference only to the location of the boundary. This approach is referred to as contour dynamics, see [1] for a review of the early work. More recently, there has been much effort placed in extended the method to more complex flows, for example, in the presence of solid boundaries [2]. Perhaps the biggest success of the method of contour dynamics is the revelation of the phenomenon of filamentation [3], [4] whereby thin streams of vorticity are shed from the boundary of the vortex patch. The apparent consequence is that the vortex patch transforms into an essentially circular patch with uniform rotation as any perturbations are shed off as filaments. There is a companion phenomenon whereby a thin vortex layer (the infinite open vortex patch) rolls up into a central almost circular core; the thin layer appears to be thin filaments that are attached to the core [5]. What is clear in the numerical simulations of vortex patches and vortex layers is that the curvature at the points of contact of the filaments and the main core is extremely high. While the boundary cannot become singular [6], the high curvature introduces very small length scales and these small scales can dominate the energy spectrum. Thus it is worthwhile to examine the formation of very high curvature and to understand when it occurs. There is a natural mathematical perspective developed during studies of the formation of curvature singularities in vortex &sheets, the limit'of an infinitely thin vortex layer [7]. By representing the boundary of the layer in parametric form x(p, t), y(p, t) , it is possible to find branch-point singularities in the complex p-plane that move towards the real axis and reach it in finite time, at which moment the singularities becomes physically relevant. The question arises naturally whether there are such singularities in the complex spatial plane for the location of the boundary of a vortex patch. From experience in studying such singularities in vortex sheets [7] and other free surface flow problems [8], we know that highly accurate calculations of the motion of the boundary are needed to detect the presence of

45

singularities in the complex spatial plane. Indeed, spectrally accurate methods are necessary and that in turn requires a suitable formulation of the evolution equations for the boundary. The standard formulation used in contour dynamics is not completely suitable and an alternate approach, capable of further generalizations is presented here.

2

Evolution equations for contour dynamics

& ' Consider a finite region of vorticity ω = (0, 0, ω) with the boundary given in parametric form x(p, t), y(p, t) . Vorticity is assumed constant in this region. The motion of the vorticity is governed by the vorticity/streamfunction formulation: ∂ω ∂ω ∂ω +u +v = 0, (1) ∂t ∂x ∂y where the velocity is (u, v, 0) and the vorticity is ∂v ∂u − . ∂x ∂y

(2)

∂u ∂v + = 0, ∂x ∂y

(3)

ω= The flow is assumed incompressible,

which allows the introduction of the stream function ψ u=

∂ψ , ∂y

v=−

∂ψ ∂x

(4)

Upon substitution of (4) into (2), one obtains the Poisson equation for ψ given ω, ∇2 ψ = −ω .

(5)

The solution to (5) must satisfy the kinematic constraints that the velocity is continuous at the boundary. The appropriate solution to (1) is found by requiring the boundary to move with the velocity at the boundary, the standard definition of Lagrangian motion. & ' ∂x (p, t) = u x(p, t), y)p, t) , ∂t

& ' ∂y (p, t) = v x(p, t), y(p, t) . ∂t

The standard derivation of contour dynamics [9] starts with the solution to (5) ψ(x, y) = −ω G(x − ξ, y − η) dξ dη , A

(6)

(7)

where G(x, y) is the free space Green’s function for the Laplacian and the integration is over the area A of the vortex patch. Consequently the velocity is given by ∂G ∂G (x − ξ, y − η) dξ dη , v(x, y) = ω (x − ξ, y − η) dξ dη . (8) u(x, y) = −ω ∂y A A ∂x Since

∂G ∂G ∂G ∂G (x − ξ, y − η) = − (x − ξ, y − η) , (x − ξ, y − η) = − (x − ξ, y − η) , ∂y ∂η ∂x ∂ξ the integrals in (8) may be written as & ' (0, 0, 1) × ∇G(x − ξ, y − η) dξ dη . u(x, y), v(x, y) = −ω A

(9)

(10)

The integration can be reduced to integration around the boundary, and evaluating the result on the boundary produces ' & ' & '& (11) u(p, t), v(p, t) = −ω G x(p, t) − x(q, t), y(p, t) − y(q, t) xq (q), yq (q) dq .

46

Eds: A Sellier & M H Aliabadi

The subscripts q refer to differentiation. By substituting (11) into (6), the evolution equations for the boundary are complete. The particular pleasing aspect of the result is that the evolution of the boundary depends only on the location of the boundary and thus represents a reduction in spatial dimension. Since G contains the natural logarithm, the integral is weakly singular and is not conducive to highly accurate numerical integration. An alternate approach is to pick a particular solution for (5) valid inside the patch and add an homogeneous solution. The obvious choice is ω ψi = − r2 + ψ˜i (12) 4 inside the patch, while ωR2 ln r + ψõ (13) ψo = − 2 2 outside the patch. The area of the patch is written as A = πR . Both ψ˜i and ψõ must satisfy Laplace’s equation and the solutions are connected by the requirement that ψ and its normal derivative are continuous at the boundary. The continuity of ψ requires 2 & ' 1 R ψõ (p) − ψ˜i (p) = ω ln r(p) − r2 (p) (14) 2 4 where r2 (p) = x2 (p) + y 2 (p). The continuity of the normal derivative of ψ requires $ # ˜ ' ∂ ψo ∂ ψ˜i ω x(p) yp (p) − y(p) xp (p) & 2 sp (p) (p) − (p) = R − r2 (p) ∂n ∂n 2 r2 (p) where the subscripts p refer to differentiation and s2p (p) = x2p (p) + yp2 (p). Represent ψ˜ by a dipole distribution μ(p) and a source distribution σ(p) along the boundary. ( ) 1 1 μ(q) ψ˜ = zq (q) dq + σ(q) ln |z − z(q)| sq (q) dq 2πi z − z(q) 2π

(15)

(16)

where z = x + iy is a location in complex form and z(p) = x(p) + iy(p) marks the location of the boundary: zp (p) = xp (p) + iyp (p). The complex form of the boundary integrals proves useful for two-dimensional free surface flow in general [10]. This representation has the important following properties: the dipole strength is given by μ(p) = ψõ (p) − ψ˜i (p) (17) and accounts for the jump in the stream function in (14), and the source strength is given by ∂ ψ˜i ∂ ψõ (p) − (p) . (18) ∂n ∂n The combination is successful because the contribution form the dipole strength has a continuous normal derivative while the contribution from the source distribution is continuous across the boundary. ˜ = ψ˜ − iφ˜ is analytic and is There is a further advantage to (16). The harmonic conjugate to ψ˜ is −φ˜ so that Ψ given by μ(q) 1 ˜ = 1 zq (q) dq + ν(q) ln(z − z(q)) dq , (19) Ψ 2πi z − z(q) 2π σ(p) =

where ν(p) = σ(p)sp (p). We are now in a position to calculate the velocity at the boundary. Let w = u + iv be the complex form of the velocity with components (u, v). then dΨ w∗ = i (20) dz where the star superscript indicates complex conjugation. While the stream function has been decomposed into two parts in each region, the velocity is unique at the boundary and can be calculated by taking the average value of the two representations there. Thus w∗ (p) = −i

˜ i dΨ ω ∗ ωR2 1 z (p) − i + (p) 4 4 z(p) zp (p) dp

(21)

47

˜ where the derivative of Ψ(p) contains the principle value of the integrals that arise from (19). This derivative may be rewritten as ˜ μ(q) zq (q) 1 dΨ 1 ν(q) 1 (p) = − dq & '2 dq + zp (p) dp 2πi 2π z(p) − z(q) z(p) − z(q) μq (q) + iν(q) 1 = dq , (22) 2πi z(p) − z(q) where integration by parts has been used on the first integral. From (14) and eqrefeq15, # $ ωzp (p) R2 μp (p) + iν(p) = − z ∗ (p) 2 z(p) By the calculus of residues, the principle value of the integral zq (q) 1 1 1 dq = . 2πi z(p) − z(q) z(q) 2z(p) So the velocity becomes w∗ (p) = −i

ω ∗ ω z (p) − 4 4π

z ∗ (q) zq (q) dq , z(p) − z(q)

which may be rewritten in a more convenient form ∗ z (p) − z ∗ (q) ω zq (q) dq . w∗ (p) = 4π z(p) − z(q)

(23)

(24)

(25)

(26)

This form is the same as derived by integration by parts of (11) [11]. The main advantage in the derivation presented here is that this approach can be easily combined with methods to track free surface between fluids of different densities and moving solid boundaries [10].

3

Numerical method and results

Because the boundary of the vortex patch is closed, the integrand is periodic. The trapezoidal rule provides a spectrally accurate approximation but to apply it to (26), the limiting form of the integrand must be calculated as q → p. A simple calculation provides the limit zp∗ (p). On the other hand, it is easy to rewrite (26) in a form where the limit proves to be zero: $ # ' & ∗ zq∗ (q) zq (q) ∂z ∗ ω (27) (p) = − ∗ z (p) − z ∗ (q) dq ∗ ∂t 4π z(p) − z(q) z (p) − z (q) The boundary is partitioned into N points equally spaced in p and the derivatives in p are obtained by analytic differentiation of the Fourier series representation of z(p). The Fourier coefficients are obtained through the use of the Fast Fourier Transform. The trapezoidal rule is applied to (27). Thus the following system of ordinary differential equations in time is obtained. $ N −1# ∗ zq,k dzj∗ & ∗ ' zq,k ω = zj − zk∗ − ∗ dt 2N zj − z k zj − zk∗

(28)

z(p) = eip + δ e2ip

(29)

k=0

The initial condition

is chosen because it leads to the formation of a single filament and makes the study of the behavior of the curvature simpler. It also provides a challenging test cad elf the method. The choice ω = 1 and δ = 0.35 is made and the evolution of the boundary of the patch is shown in Fig. 1 as a series of snapshots at times t = π, 2π, 3π, 4π. A time step of t = π/160 is used with N = 2048 surface markers.

48


1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5 -1

-0.5

0

0.5

1

1.5

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1

-0.5

0

0.5

1

1.5

2

Figure 1: Time sequences of the vortex patch boundary from left to right, top to bottom: the times are t = π, 2π, 3π, 4π. Clearly visible is the formation of a filament. Two regions of high curvature are evident, one at the tip of the filament and the other where the filament joins the main part of the patch. The profile of the curvature xp (p)ypp − yp (p)xpp κ(p) = & '3/2 , x2p (p) + yp2 (p)

(30)

is shown in Fig. 2 at t = 4π. The very sharp spike is the curvature at the tip of the filament, while the shallower dip in curvature occurs near the attachment point. The profile of the curvature is well resolved by the number of markers used. An interesting perspective on the behavior of the curvature comes from previous studies of singularity or near singularity formation in free surface flows, [12] for example. It considers the analytic continuation of the curvature into the complex p-plane. # ∗ $ zpp zpp −i κ(p) = % ∗ (31) − ∗ . zp zp 2 z p zp Both theory and numerical simulations support the presence of 3/2-power branch point singularities that move about the complex p-plane while retaining their form for free surfaces between fluids of different densities [12]. For the curvature behavior of the vortex patch, there are two generic ways that the curvature exhibits singularities in the complex p-plane. One is the case where z(p) has a 3/2-power branch point and the other is a zero in zp (p). They are both present in the results shown in Fig. 2 with the zero in zp (p) being associated with the tip of the filament. The location of the zero is very close to the real p-axis which causes the sharp spike in the curvature. The other occurs near the attachment point but is much further away from the real p-axis and the dip in the curvature is less pronounced. It should be emphasized that the detection of the presence of the singularities in the complex p-plane depends on a form-fit to a highly accurate Fourier spectrum for the curvature which would be very difficult to achieve

49

700 600 500 400 300 200 100 0 -100 0

1

2

3

4

5

6

Figure 2: The curvature of the vortex patch boundary at t =. without the use of the form (29). References: [1] D.I. Pullin, Contour dynamics methods, Ann. Rev. Fluid Mech., 24, 1992, pp.. 89–115. [2] D. Crowdy and A. Surana, Contour dynamics in complex domains, J. Fluid Mech., 593, 2007, pp. 235–254. [3] G.S. Deem and N.J. Zabusky, Vortex waves:stationary ‘V’ states, interactions, recurrence and breaking, Phys. Rev. Lett., 40, 1978, pp. 859–62. [4] D.G. Dritschel, The repeated filamentation of two-dimensional vorticity interfaces, J. Fluid Mech., 194, 1988, pp.511–547. [5] G.R. Baker and M.J. Shelley, On the connection between thin vortex layers and vortex sheets, J. Fluid Mech., 15, 1990, pp. 161–194. [6] J.-Y. Chemin, Existence globalle pour le problème des poches de tourbillon, CR Acad. Sci. Paris Ser. I, 1991, pp.803–806. [7] S.J. Cowley, G.R. Baker and S.A. Tanveer, On the formation of Moore curvature singularities in vortex sheets, J. Fluid Mech., 378, 1999, pp. 233–267. [8] G.R. Baker and C. Xie, Singularities in the complex physical plane for deep water waves, J. Fluid Mech., [9] N.J. Zabusky, M.H. Hughes and K.V. Roberts, Contour dynamics for the Euler equation in two dimensions, J. Comput. Phys., 30, 1979, pp.96–106. [10] G.R. Baker, Boundary Element Methods in Engineering and Sciences, Chapter 8, Imperial College Press, 2010. [11] D.I. Pullin, The nonlinear behavior of a constant vorticity layer at a wall, J. Fluid Mech., 108, 1981, pp.401– 421. [12] G.R. Baker, R.E. Caflisch and M. Siegel, Singularity formation during Rayleigh-Taylor instability, J. Fluid Mech., 252, 1993, pp. 51–78.

50


A cohesive boundary element approach to material degradation in three-dimensional polycrystalline aggregates I. Benedetti1,2,a, M.H. Aliabadi1 1

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

2

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale e dei materiali, Università degli Studi di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy a

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

Keywords: Polycrystalline materials, Microstructure Modelling, Intergranular damage, Cohesive laws, Anisotropic Boundary Element Method.

Abstract. A new three-dimensional grain-level formulation for intergranular degradation and failure in polycrystalline materials is presented. The polycrystalline microstructure is represented as a Voronoi tessellation and the boundary element method is used to express the elastic problem for each crystal of the aggregate. The continuity of the aggregate is enforced through suitable conditions at the intergranular interfaces. The grain-boundary model takes into account the onset and evolution of damage by means of an irreversible linear cohesive law, able to address mixed-mode failure conditions. Upon interface failure, a non-linear frictional contact analysis is introduced for addressing the contact between micro-crack surfaces. An incremental-iterative algorithm is used for tracking the micro-degradation and cracking evolution. The behavior of a polycrystalline specimen under tensile load has been performed, to show the capability of the formulation. Introduction For modern structural applications (aerospace, automotive, off-shore, etc.), a deep understanding of materials degradation and failure is of crucial relevance. Fracture modelling can be considered at different length scales: it is nowadays widely recognized that the macroscopic material properties depend on the features of the microstructure [1]. Polycrystalline materials (metals, alloys or ceramics) are commonly employed in engineering structures. Their microstructure is characterized by features of the grains (morphology, size distribution, anisotropy and crystallographic orientation, stiffness and toughness mismatch) and by physical and chemical properties of the intergranular interfaces. These aspects have a direct influence on the initiation and evolution of microstructural damage, which is also sensitive to the presence of imperfections, flaws or porosity. The microstructure of polycrystalline materials, and its failure mechanisms, can be investigated using different experimental techniques (see references in [2] for a brief overview); these provide fundamental information and understanding but require sophisticated equipment, careful material manufacturing and preparation and complicated postprocessing, especially whenever a truly three-dimensional (3D) characterization is pursued. A viable alternative, or complement, to the experimental effort is offered by the Computational Micromechanics [3]. Several investigations have been devoted to modelling of polycrystalline microstructures and their failure processes [4] and there is currently an interest for the development of truly 3D models. Until recently, this has been hindered by excessive computational requirements. However, the present-day availability of cheaper and more powerful computational resources and facilities, namely high performance parallel computing, is favoring the advancement of the subject, especially in the framework of the Finite Element Method (FEM), see e.g. [5]. A popular approach for modelling both 2D and 3D fracture problems in polycrystalline materials consists in the use of cohesive surfaces embedded in a finite element (FE) representation of the microstructure. In this way, initiation, propagation, branching and coalescence of microcracks stem as an outcome of the simulation, without any assumptions. Several cohesive laws have been proposed in the literature [6].

51 An alternative to the FEM is the Boundary Element Method (BEM) that has proved effective for a variety of physical and engineering problems [7,8]. A cohesive boundary element formulation for brittle intergranular failure in polycrystalline materials was proposed by Sfantos and Aliabadi [9]. A 3D grain boundary formulation has been recently developed by Benedetti and Aliabadi for the material homogenization of polycrystalline materials [2]. In this work, a novel 3D grain-level model for the analysis of intergranular degradation and failure in polycrystalline materials is presented. The polycrystalline microstructure is represented as a Voronoi tessellation and the formulation is based on a grain-boundary integral representation of the elastic problem for the anisotropic crystals, that have random orientation in the 3D space. The integrity of the aggregate is restored by enforcing suitable intergranular conditions. The onset and evolution of damage at the grain boundaries is modeled using an irreversible cohesive linear law. Upon interface failure, a non-linear frictional contact analysis is used, to address separation, sliding or sticking between micro-crack surfaces. An incremental-iterative algorithm is used for tracking the degradation and micro-cracking evolution. A numerical test is presented to demonstrate the capability of the formulation. Grain boundary formulation for polycrystalline aggregates Artificial microstructure. For polycrystalline materials, Voronoi tesselations are widely used for the generation of the microstructural models [10]. The assignation of a specific orientation to each crystal of the aggregate completes the microstructure representation. Grain constitutive modelling. Each grain is modeled as a three-dimensional linear elastic orthotropic domain with arbitrary spatial orientation. This is not restrictive, as the majority of single metallic and ceramic crystals present general orthotropic behavior. Grain boundary element formulation. Each crystal is modeled using the BEM for 3D anisotropic elasticity [11]. The polycrystalline aggregate is seen as a multi-region problem, so that different elastic properties and spatial orientation can be assigned to each grain [2]. Given a volume bounded by an external surface and containing N g grains, two kinds of grains can be distinguished: the boundary grains, intersecting the external boundary, and the internal grains, completely surrounded by other grains. Boundary conditions are prescribed on the surface of the boundary grains lying on the external boundary, while interface equations and equilibrium conditions are forced on interfaces between adjacent grains, to restore the integrity of the aggregate. The boundary integral equation for a generic grain G k is written

cijk x u kj x

³

BC BNC

Ti kj x, y u kj y dB k y

³

BC BNC

U ikj x, y tjk y dB k y

(1)

where u kj and tjk represent components of displacements and tractions of points belonging to the surface of the grain G , the tilde refers to quantities expressed in a local reference system set on the grain surface, U k ij

k

and Ti kj are the 3D displacement and traction fundamental solutions for the anisotropic elastic problem. The integrals in Eq.(1) are defined over the surface of the grain, that is generally given by the union of contact interfaces BC and external non-contact surfaces BNC . The model for the polycrystalline aggregate is obtained by writing Eq.(1) for each grain and then complementing the system so obtained with the boundary conditions

ui

ui

or

ti

ti

on BNC

(2)

and with a set of suitable interface equations, expressing the different possible states of an interface. Interface model. The interface between two grains can be in three different possible states: pristine, when no damage is present and perfect bonding between the grains holds; damaged, when damage is present and intergranular tractions and displacement jumps are linked through a cohesive traction-separation law; failed, when the grains are completely separated and the laws of the frictional contract mechanics hold. Let us consider two adjacent grains G a and G b . When the interface between them is in pristine state, the following interface continuity equations hold

52


G uiab

uia uib

(continuity)

0

ti a

and

tiib

(equilibrium)

(3)

The previous equations express the absence of interface displacement opening and the equilibrium of the interface tractions. The equilibrium equations always hold during the analysis, regardless the interface state, so they are always assumed in the following, while the continuity equations are replaced by other laws expressing the interface state during the interface evolution. Damage is introduced at the interface when the value of a suitable effective traction overcomes the interface cohesive strength Tmax

teff

ª « tn ¬«

1

2

2 § E · º2 ¨ tt ¸ » t Tmax © D ¹ ¼»

(4)

In the previous equation, the local tractions are expressed in terms of local normal and tangential contribution, tn and tt . The parameters D and E give different weight to mode I and mode II loading. When the previous condition is fulfilled, the following traction-separation laws are introduced at the interface

ªD G utc 0 0 º ª G u1 º ª t1 º 1 d* « » « » c t T 0 D G u 0 » ««G u2 »» t « 2 » max d * « c « 0 «¬t3 »¼ 0 1 G un »¼ «¬G u3 »¼ ¬

(5)

where d * [0,1] is a damage parameter given by

d*

max {d }

Loading history

with

ª Gu n d=« c «¬ G un

1

2

2 § Gu · º2 E 2 ¨ ct ¸ » © G ut ¹ »¼

(6)

where G un and G ut are the normal and tangential opening displacements at the interface and G unc and G utc represent their critical values in pure mode I and II respectively. The parameter d is the effective opening displacement and the damage parameter is given by the maximum value reached by the effective displacement during the loading history. For d * 0 the interface is pristine, while d * 1 expresses the failure of the interface. Upon interface failure, the traction-separation laws are replaced by the laws of the frictional contact mechanics. In general, the micro-crack surfaces can be either separated or in contact; moreover, two surfaces in contact can either stick or slip over each other. The equations of frictional contact mechanics are not recalled here, but the interested reader is referred to the literature on the subject (see [8] and references therein). Discretization and numerical solution. The present formulation has the advantage that only meshing of the grain surfaces is required. The artificial microstructure is, in this context, a collection of flat convex polygonal faces. Plane triangular linear elements are used to discretize such faces. Linear discontinuous triangular elements are implemented for representing the unknown boundary fields. Since the Voronoi tessellations used for microstructure modelling have stochastic nature, care must be taken to ensure mesh consistency and homogeneity to the greatest extent [2]. After discretization and classical BEM treatment of the Eqs.(1), the following system can be written

ª A1 «0 « «0 « «[ ¬

0 % 0 d*

0 º ª º 0 »» « x1 » # » A Ng » « » «x N » g ¼ ¬ ] »¼

ª y1 º « # » « » «y Ng » « » ¬« ¼»

(7)

where the matrix blocks A k contain columns of the boundary element matrices H k and G k corresponding to the unknown displacements and tractions of the k-th grain, contained in the vector x k , and y k derive

53 from the applications of the known boundary conditions. The block d * implements the varying coefficients of the interface equations, i.e. the continuity, cohesive and frictional contact equations, for all the grain boundary interfaces. System (7) is sparse and special solvers can be used for its solution. To track the evolution o f a polycrystalline microstructure, an incremental/iterative algorithm is employed. A load increment is applied and the system solution is iterated until no violation of the interface equations is detected. At each iteration, all the interfaces are checked, to assess whether any violation of the assumed interface state is detected. For example, if the effective traction of a pristine interface overcomes the cohesive strength, then damage is initiated and the continuity equations are replaced by cohesive laws. Analogous checks are also done for interfaces in the cohesive or failed state. For the damaged interfaces, the cohesive law has to be updated if a loading condition exists, while no update is required in the cases of unloading or reloading. When convergence is reached a new load increment can be applied and the iterative search is restarted. In this work PARDISO [12] is used as iterative solver and a hybrid direct/iterative solution strategy is employed to speed up the numerical solution of the polycrystalline evolution problem.

Numerical simulation of a SiC micro-specimen under tensile load A prismatic polycrystalline specimen subjected to tensile load is considered. The specimen is comprised of N g 200 fully three-dimensional SiC grains; an uniform displacement is applied over the bases and it is directed along the longer side, Fig.1. The material properties for crystalline SiC are given in Table 1, the grain size is ASTM G=12 (calculated number of grains per mm3 : n / v 4.527 106 [13]). The specimen's size is 2W u 2W u 2 H with H W 2 , its volume is V 8 HW 2 N g Vgrain , where Vgrain is the estimated

average grain volume. The mesh density is specified by d m 0.5 (see [2] for further details about the meshing strategy). The properties of the interfaces are uniform and they are given in Table 2.

Figure 1: Polycrystalline SiC specimen with 200 grains subjected to tensile load. C11 502

C12 95

C13 96

C33 565

C44 169

C66 203.5

Table 1: Material constant for single SiC crystals.

54


Fig. 1 shows the microstructural crack pattern immediately before the complete failure of the specimen. The corresponding macroscopic stress-strain curve is shown in Fig.2. The curve reports the value of the relevant component of the averaged stress tensor versus the nominal strain, obtained from the value of the applied displacement over the relevant specimen size. It is apparent how the specimen softens before the complete failure. Table 3 reports some statistics about the considered microstructure.

Figure 2: Volume average stress component versus nominal strain for the considered specimen.

Tmax 500

D 1

E

GII GI

2

1

P 0.05

Table 2: Selected values for the interface properties.

N elements

N interfaces

DoFs

T'O

17,031

7,709

222,660

~5000s

Table 3: Some statistics about the considered polycrystalline specimen. The time per load increment was measured on 12-core nodes.

Summary A new three-dimensional formulation for the analysis of intergranular degradation and failure in polycrystalline materials has been developed. The polycrystalline microstructure is represented as a threedimensional Voronoi tessellation, able to retain the main morphological and crystallographic features of polycrystalline aggregates. The micromechanical model is expressed in terms of intergranular fields, namely displacement jumps and tractions. The nucleation and evolution of intergranular damage has been followed using an irreversible cohesive law at the intergranular interfaces: this resulted particularly straightforward, being the formulation itself expressed in terms of grain boundary variables. Upon complete intergranular failure the frictional contact analysis is introduced to follow the intergranular micro-cracking process, taking into account separation, contact and sliding between the micro-crack surfaces. A numerical test demonstrated the capability of the formulation to model the nucleation, evolution and coalescence of multiple damage and cracks. For its nature, the developed formulation appears particularly promising in the framework of grain boundary engineering.

55 Acknowledgements This research was supported by a Marie Curie Intra-European Fellowship within the 7th European Community Framework Programme (Project No 274161).

References [1] S. Nemat-Nasser, M. Hori, Micromechanics: overall properties of heterogeneous materials, NorthHolland, Elsevier, The Netherlands, second revised edition edition, (1999). [2] I. Benedetti, M. H. Aliabadi, A three-dimensional grain boundary formulation for microstructural modelling of polycrystalline materials, Computational Materials Science, 67, 249–260, (2013). [3] T. I. Zohdi, P. Wriggers, An introduction to computational micromechanics, Lecture Notes in Applied and Computational Mechanics, vol. 20, Springer, Berlin, (2005). [4] A. G. Crocker, P. E. J. Flewitt, G. E. Smith, Computational modelling of fracture in polycrystalline materials, International Materials Reviews, 50, 99–124, (2005). [5] I. Simonovski, L. Cizelj, Computational multiscale modeling of intergranular cracking, Journal of Nuclear Materials, 414, 243 – 250, (2011). [6] N. Chandra, H. Li, C. Shet, H. Ghonem, Some issues in the application of cohesive zone models for metal-ceramic interfaces, International Journal of Solids & Structures, 39, 2827–2855, (2002). [7] L.C. Wrobel, M. H. Aliabadi, The boundary element method: applications in thermo-fluids and acoustics., Vol. 1, John Wiley & Sons Ltd, England, (2002). [8] M.H. Aliabadi, The boundary element method: applications in solids and structures., Vol. 2, John Wiley & Sons Ltd, England, (2002). [9] G. K. Sfantos, M. H. Aliabadi, A boundary cohesive grain element formulation for modelling intergranular microfracture in polycrystalline brittle materials, International Journal for Numerical Methods in Engineering, 69, 1590–1626, (2007). [10] S. Kumar, S. K. Kurtz, J. R. Banavar, M. G. Sharma, Properties of a three-dimensional PoissonVoronoi tessellation: a Monte Carlo study, Journal of Statistical Physics, 67, 523–551, (1992). [11] R. B. Wilson, T. A. Cruse, Efficient implementation of anisotropic three-dimensional boundaryintegral equation stress analysis, International Journal for Numerical Methods in Engineering, 12, 1383– 1397, (1978). [12] O. Schenk, K. Gartner, Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO, Journal of Future Generation Computer Systems, 20, 475–487, (2004). [13] ASTM E112-10, Standard Test Methods for Determining Average Grain Size. ASTM International. DOI: 10.1520/E0112-10, (2010).

56


MACRO-ZONES SGBEM APPROACH FOR STATIC SHAKEDOWN ANALYSIS AS CONVEX OPTIMIZATION Panzeca T.1,a,Salerno M.2 and Zito L.1,b 1

Dip.to di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali, University of Palermo, Italy. 2 Dip.to di Costruzioni e Metodi Matematici in Architettura, University of Naples, Italy. 1a

[email protected], [email protected] 1b [email protected]

Keywords: symmetric BEM; substructuring; shakedown; convex optimization. Abstract: A new strategy utilizing the Multidomain SGBEM for rapidly performing shakedown analysis as a convex optimization problem has been shown in this paper. The present multidomain approach, called displacement method, makes 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 includes influence coefficients, which characterize the input of the quadratic constraints, it provides a nonlinear optimization problem solved as a convex optimization problem. Furthermore, the strategy makes it possible to introduce a domain discretization exclusively of zones involved by plastic strain storage, leaving the rest of the structure as elastic macroelements, consequently governed by few boundary variables. It limits considerably the number of variables in the problem and makes the proposed strategy extremely advantageous. The implementation of the procedure by the Karnak.sGbem code, coupled with optimization toolbox Matlab 7.6.0, made it possible to perform some numerical tests showing the high performance of the algorithm due to solution accuracy and low computational cost. Introduction A reformulation of the static approach to evaluate directly the shakedown multiplier by using the Symmetric Galerkin Boundary Element Method (SGBEM) for multidomain type problems [1,2] is shown in this paper. The present formulation utilizes the self-equilibrium stress equation [3-5] connecting the stresses at the Gauss points of each substructure (bem-e) to plastic strains through a stiffness matrix (self stress matrix) involving all the bem-elements in the discretized system. The numerical method proposed is a direct approach because it permits to evaluate the multiplier directly as lower bound through the static approach. The analysis has been performed as a constrained optimization problem is rephrased in the canonical form as a convex optimization (CO) problem with quadratic constraints, in terms of discrete variables within the SGBEM, and here implemented using the Karnak.sGbem code [7] coupled with an optimization toolbox by MatLab. In this procedure, through the use of the Multidomain SGBEM, it was possible to reduce the size of the problem, since this method allows one to introduce a domain discretization exclusively in the zones of potential storage of the plastic strains, the remaining part of the structure being considered as made up of macroelements having purely elastic behaviour, consequently governed by few boundary variables. This aspect makes the strategy proposed extremely advantageous by the point of view of the computational burdens. In order to prove the efficiency of the proposed approach, the response is compared with the elastoplastic iterative analysis via Multidomain SGBEM, an advantageous strategy developed by some of the present authors [5]. Finally, a numerical test where the limit multiplier value has been compared both with the value obtained by elastoplastic analysis and by literature available [4,5] is shown. The application shows a very important computational advantage to confine the domain discretization only in the potentially plastic zones and to leave the rest of the structure subdivided in elastic macroelements, these latter therefore governed by few boundary variables.

57 1. Shakedown analysis via SGBEM and convex optimization The multidomain Symmetric Boundary Element Method (SGBEM), developed by some authors [1,2], is utilized to riformulate the static shakedown theorem [3,4], which represents a powerful tool for providing directly, by means of mathematical programming techniques, the safety condition of a structure. The proposed strategy uses the self-equilibrium stress equation [3,5] to define the self-equilibrium stress field employed in the classical shakedown approach. This equation connects stresses, computed at each bem-e Gauss point, to plastic strains through an influence matrix (self-stress matrix). Then the shakedown multiplier is obtained through a constrained optimization problem within the canonical form of a CO in terms of discrete variables. 1.1 Self-equilibrium stress equation via multidomain SGBEM 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 Zp . Indeed, the following equation:

Z p 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, fullypopulated, non-symmetric and semi-definite negative. The evaluation of this matrix only involves the knowledge of the material elastic characteristics and of the structure geometry within a domain discretization process. The reader can refer to Zito et al. [5] for a more detailed discussion of the characteristics of this equation introduced for a multidomain SGBEM problem.

1.2 Shakedown analysis as CO problem In order to evaluate the shakedown multiplier directly, the classic shakedown approach was rephrased by means of SGBEM for multidomain type problems. In the hypothesis of a 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 nonlinear optimization techniques through the introduction of the self-equilibrium stress field, defined in eq.(1). According to the shakedown 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....v the basic load and i

ˆ ie p

(3)

ˆ ie , due to external actions, and the selfrepresenting the total stress as the sum of the elastic stress vector equilibrium stress vector p . The classical static approach makes it possible to obtain the shakedown factor E sh as the maximum of the shakedown factors E for which the structure does not fail:

max E E sh E , p ° ° ®s.t.: ° e p ° F ª¬ E ˆ i º¼ d 0 ¯

(4)

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

58


p

(5)

Zp

the optimization problem can be written as follows: E sh max E E ,p ° ° s.t.: ® ° e °¯ F ª¬ E ˆ i Zp º¼ d 0

(6)

or in explicit form: max E E sh E ,p1 ,"p m ° °s.t.: °° e ® F1 ¬ª E ˆ i1 Z11 p1 " Z1m p m ¼º d 0 ° °# ° F ª E ˆ e Z p " Z p º d 0 m1 1 mm m ¼ °¯ m ¬ im

(7)

where m is the bem-e number. 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: max E E sh E ,p1 ,.....p m ° ° s.t.: °° T e e 2 1 ® 2 E ˆ i1 Z11 p1 " Z1m p m M E ˆ i1 Z11 p1 " Z1m p m 1y d 0 ° °# T °1 e e 2 °¯ 2 E ˆ im Z m1 p1 " Z mm p m M E ˆ m1 Z m1 p1 " Z mm p m my d 0

(8)

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

(9)

where B is a symmetric positive matrix and y is the unknown quantity vector. The canonical form (9) is obtained by collecting in the B matrix the constant terms of eq.(8), i.e. for the j-th bem-e: Fij

T M T E p T1 " p Tm ˆ ije Z j1 " Z jm ˆ ije Z j1"Z jm E p T1 " p Tm 1 d 0

2V 2y T

y y

(10)

Bij

and in compact form:

Fij

y T B ij y 1 d 0

(11)

As this computational strategy has to be applied to practical engineering cases, it is appropriate to introduce suitable constraints on the plastic strains. Therefore the unknown vector y must also satisfy the following condition: yd y

with y

f q1T

" qTm

T

(12)

59 The values to be assigned as constraints on the strains have to be meaningfully analyzed from the point of view of practical engineering and obtained by experimental tests on the material ductility. The shakedown problem can be rewritten as follows:

min cT y ° y ° s.t.: ° T ° y B1 y 1 d 0 ® °# ° T °y B k y 1 d 0 °yd y ¯

(13) k = m* v

where the vector cT 1 0 " 0 has been introduced. Problem (6), in the form (13), was implemented by coupling the Karnak.sGbem code [7] with a Matlab 7.6.0 optimization toolbox. In this procedure, using multidomain SGBEM, it was also possible to reduce the size of the problem. Indeed, since this method introduces 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 computationally advantageous. 2. Numerical results To show the computational effectiveness of the proposal method the limit load multiplier ( v = 1 ) has been computed by static theorem (Section 1) and compared with incremental approaches [5,6]. At this purpose the frame of Figure 1a, subjected to a uniform load q 10 KN / m , was considered as a bidimensional body. Its domain was discretized using 149 bem-elements and 3 macro-elements. The material characteristics are Young’s modulus E 210000 MPa and Poisson’s ratio n 0.3 , whereas the uniaxial yield value is V y 250 MPa . 2000

200

q

2000

1517.2

A

282.8

200

a)

b)

Figure 1. Frame subjected to a uniform load: a) geometric description, b) adopted meshes. It is necessary to emphasize that only the potentially plastic zones were meshed and the rest was studied as made by elastic macro-elements, as shown in Figure 1b. The numerical solutions were obtained using the strategies proposed in the present paper. Moreover, the results were compared with those obtained by another incremental method employed by some of the present authors [5,6]. This strategy works in the field of a multidomain SGBEM and it is characterized by strongly recursive elastoplastic analysis.

60


In detail, the computational work contemplates two direct approaches: - the incremental elastoplastic analysis for active macrozones [5], - the incremental elastoplastic analysis [6]); and an indirect approach: - the lower bound limit analysis as CO (see Section 1), considering different constraints on the plastic strains, that is q 0 00 1.0 0 00 , 2.0 0 00 , 3.0 0 00 . In all these cases the tests were performed using as substructures, subjected to possible plastic strains, both bem-elements having four and eight nodes. 6

6

5

5 3.0 ‰ 2.0 ‰ 1.0 ‰

4

3 Lower bound limit analysis (CO)

3.0 ‰ 2.0 ‰ 1.0 ‰

4

3 Lower bound limit analysis (CO)

Elastoplastic analysis (active macro-zone)

2

2

Elastoplastic analysis (active macro-zone)

Elastoplastic analysis (single active bem-e) 1

Elastoplastic analysis (single active bem-e)

1

10

20

30

40

50

10

60 70 80 Displacement of point A [mm]

a)

20

30

40

50

60 70 80 Displacement of point A [mm]

b)

Figure 2. Load factor-displacement curves by elastoplastic analysis compared with lower bound via CO: a) four nodes bem-e, b) eight nodes bem-e.

Load factor 4 nodes bem-e

8 nodes bem-e

Limit analysis (CO, q 0 00

3.0 0 00 )*

4.531

4.342

Limit analysis (CO, q 0 00

2.0 0 00 )*

4.310

4.164

Limit analysis (CO, q 0 00 1.0 0 00 )*

4.060

3.951

Method

Elastoplastic analysis (active macro-zones)

4.4

3.9

Elastoplastic analysis (single active bem-e)

4.2

3.9

Table 1: Collapse load factor, obtained by present approaches (*) compared to other formulation. In Figure 2 the curves characterizing the load multipliers E E u , as functions of the displacements of the point A, are shown. In the step-by-step analyses the increment of the load is equal to 'E 0.1 . In Figure 2 the dropped line shows the lower bounds obtained through a CO analysis by imposing three different constraints on all the plastic strains, that is q 0 00 1.0 0 00 , 2.0 0 00 , 3.0 0 00 . The load multipliers are shown in Table 1, obtained both through direct analysis (using the active macrozone or the single active bem-e) and indirect analysis (CO). In the direct analyses the multipliers were taken by choosing those load values before attaining the displacement characterizing the structure as not usable. This parameter was fixed as the ratio between the maximum displacement u of the point A and the length of the horizontal beam, that is u L 2% , therefore choosing as maximum displacement u 40 mm . It is to be noted that the structure partially discretized by 4-node bem-elements appears more rigid and the load multiplier proves to be higher in comparison with the values obtained by the same discretization with 8node bem-elements in both the direct and indirect analyses, that is step-by-step elastoplastic analysis and CO analysis. Further, the incremental analyses performed on the system discretized by using 8-node bem-elements through either the active macro-zones or the single active bem-e proves to have the same value, whereas both

61 prove to be lower in comparison with that obtained within the CO analysis performed assuming a constraint on the plastic strains equal to q 0 00 1.0 0 00 . In conclusion, this application proves the high performance and effectiveness of the 8-node bem-e discretization, used in zones where the plasticity is potentially active. Its use allows one to obtain, using the direct or indirect approach, a useful solution regarding the safety of the structure. 3. Conclusions

The static shakedown approach of the classical plasticity theory is rephrased using the Multidomain Symmetric Galerkin Boundary Element Method, under conditions of plane and initial strains, ideal plasticity and associated flow rule. The new formulation couples a multidomain procedure with nonlinear programming techniques and defines the self-equilibrium stress field by an equation involving all the substructures (bem-elements) of the discretized system. The analysis is performed in a canonical form as a convex optimization problem with quadratic constraints, in terms of discrete variables, and implemented using the Karnak.sGbem code coupled with the optimization toolbox by MatLab. The numerical tests, compared with the iterative elastoplastic analysis via the Multidomain Symmetric Galerkin Boundary Element Method, developed by some of the present authors prove the computational advantages of the proposed algorithm. References [1] Panzeca T., Cucco F., Terravecchia S., (2002). “Symmetric boundary element method versus Finite

element method”. Comput. Meth. Appl. Mech. Engng., 191, 3347-3367. [2] Pérez-Gavilán J.J., Aliabadi M.H., (2001). “A Symmetric Galerkin Bem for multi-connected bodies: a

new approach”. Eng. Anal. Bound. Elem., 25, 633-638. [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”. Meth. Appl. Mech. Engng. 191, 1967-1982. [5] Zito, L., Cucco, F., Parlavecchio, E., Panzeca, T., (2012). “Incremental elastoplastic analysis for active

macro-zones”. Int. J. Num. Meth. Engng. 91, 1365-1385. [6] 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, 103-120. [7] Cucco, F., Panzeca, T., Terravecchia, S., “The program Karnak.sGbem Release 2.1”, Palermo

University. (2002).

62


STRAIN ENERGY EVALUATION IN STRUCTURES HAVING ZONE-WISE PHYSICAL- MECHANICAL QUANTITIES Panzeca T.1,a, Cucco F.2 and Terravecchia S.1,b 1

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali – Palermo University 2

Facoltà di Ingegneria e di Architettura, Enna University

Keywords: SGBEM; substructuring approach; displacement method; energy evaluation. Abstract: Among the possible aims of structural analysis inside some engineering spheres it can be useful to know the strain energy stored in all or in a part of the structure caused by assigned external actions, like the boundary and domain quantities. This serves to evaluate globally whether an assigned portion of structure undergoes an excessive store of energy able to compromise the stability of all the structure. This evaluation can be carried out through boundary work obtained using appropriate boundary generalized quantities connected to the results of the analysis on the whole structure. The advantage consists in using a very restricted number of quantities which, because of the characteristics of the method, are only evaluated on the boundary. Some strategies used to evaluate the error made are introduced through the computation of the external direct work and of the reciprocal works involving quantities only connected to the boundary of the complementary domain and quantities connected to either the real boundary of the structure or the boundary of its complementary domain. A reduction of this error is suggested. Introduction This paper has as its objective the computation of the elastic strain energy stored in a body or in a limited portion of it, and this can happen also when zone-wise variable physical and mechanical characteristics are present. In this case the body is subdivided into substructures and the analysis is performed by the displacement method within the Symmetric Boundary Element Method (SGBEM), according to the Galerkin hypotheses. Fig. 1 shows a system having zone-wise variable physical characteristics, subjected to displacements imposed on the constraint and known forces on the free boundary. One wants to know the elastic strain energy stored in the substructures marked by 1, 2, 3, 4.

Figure 1: System subjected to imposed displacements and known forces; substructures 1, 2, 3, 4 where the elastic strain energy has to be evaluated. Several researches have developed substructuring techniques [3-5]. Only in the last few years, in virtue of a particular approach for substructuring in [1], has the implementation of the SGBEM become possible, making this method competitive in comparison with other analysis methods. Through this method the elastic strain energy can be computed using domain integrals after evaluating the stresses and strains by Somigliana Identities (SI), but this strategy shows considerable computational difficulties. Moreover, the energy balance between elastic energy stored and boundary work in terms of punctual quantities, also difficult to evaluate for the same reasons, suggested introducing a type of work here defined as generalized through interpretation of the mechanical and kinematical boundary quantities as

63 layered quantities. This form of work was obtained in [2] as a direct product between boundary quantities, referred to as generalized and directly obtainable in the phase of post-analysis and nodal values, thus avoiding tedious evaluation of the SI and the use of the boundary integrals involving continuous boundary functions. The numerical equality between the boundary work in terms of continuous functions and the generalized work, here shown, justifies the procedure followed and makes it easy implementable within the Karnak.sGbem program, utilized for the numerical trials. This work is based on the employment of appropriate boundary quantities to be computed when the analysis is concluded. A first attempt at using these boundary quantities was made in [6,7] through the introduction of weighted residuals regarding the coefficients of Dirichlet and Neumann dual equations, relations not utilized to solve the analysis problem. But these quantities have not made it possible to apply the method based on computing the boundary work to the case of a system subdivided into substructures where the interface boundary also appears. The problem was overcome by giving an appropriate interpretation of the boundary quantities. Indeed, when the solid was embedded in the infinite domain, a boundary * of the : domain and a boundary * of the complementary domain :f / : are introduced and the boundary quantities, whether kinematical or mechanical, take on the meaning of layered quantities, that is to say of jumps between the actual boundaries * of the solid and the boundary * of the complementary domain. The use of these layered functions within the expressions of the work and the utilizations of the basic theory of the SGBEM led to the definition of a new type of generalized work in which, in addition to the direct work, the indirect works where the kinematical and mechanical quantities of the boundary * interact in energy meaning with the related quantities of * appear. This way of interpreting the work, together with a more cogent discussion of the boundary conditions leading to characterization of the algebraic operators of the single substructure within the displacement method, made it possible to evaluate in a fast and simple way the strain energy in all the infinite domain, the sum of the actual and complementary domains. Further, embedding of the single substructure in the infinite domain allows one to assert that, depending on the boundary discretization, the solution is reached when the complementary domain is unstrained and unstressed, or when the boundary work is all in the solid as strain energy. This can happen when the direct work on * and the reciprocal ones cancel each other out on all the boundary elements. On the basis of these remarks it is possible to create a strategy which reduces the error of the solution through appropriate boundary discretization where the scattered energy quantities are greater. 1. Boundary conditions In any mechanics problem, the boundary conditions of the continuum are the following:

t f (Dirichlet conditions) u u (Neumann conditions)

(1) (2)

In the Boundary Element Method, the solid of the : domain is embedded in the infinite domain :f having the same physical and mechanical characteristics. The infinite domain is characterized by a boundary * { * of : having external unit vector n n and by a boundary * of :f \ : with n n (Fig. 2).

Figure 2. Body embedded in the infinite domain and layered quantities

64


The quantities characterizing the boundary take on the meaning of layered quantities and in particular: x The forces acting on the boundary take on the meaning of single layer sources:

f(x) = -t + (x,n ) + t (x,n)

(3)

where t and t symbolize in :f the response in terms of traction evaluated in x on the boundaries * e * , respectively. x The displacements of the boundary take on the meaning of double layer sources:

+

v(x) = u+ (x) - u (x)

(4)

where u and u symbolize in :f the response in terms of displacements evaluated in x on the boundaries * and * , respectively. To obtain the solution of the elastic problem, it is necessary for the complementary domain :f / : to prove to be unstressed and unstrained, and this can be obtained by imposing the condition that all the points of the boundary * satisfy the following conditions: +

f = t v = -u

o o

t+ 0 u+ 0

(5a,b) (6a,b)

2. Boundary condition in weighted form In accordance with the SGBEM formulation, the boundary conditions of problem (5a) and (6a) have to be imposed in weighted form. Let us introduce the boundary discretization into boundary elements and the related modeling of the quantities through the introduction of the appropriate shape functions:

f

F,

t

u

u

(7a,b)

U

where F and U symbolize the nodal value of the forces and displacements, respectively. The weighting of the equilibrium (3) and compatibility (4) is achieved by using the shape functions as weighted functions, introduced in dual form according to the Galerkin hypotheses. One obtains:

³

T

u

*

f d* ³ *

T

T

t+ d* ³ *

P

P

³ t

u

(u) d *

*

³ t *

W

u

T

t d d* *

o

P P P

o

W

(8a,b)

P T

u+ d * ³ *

W

t

T

u d *

W W

(9a,b)

W

Eqs.(8,9) are the equilibrium and compatibility equations written in terms of weighted tractions P and P and weighted displacements W and W , respectively; on the analogy of the previous definitions, P and W symbolize layered generalized forces and distortions. The previous relations can be utilized for the corresponding boundary conditions (5,6) in weighted form.

P = 0 W

0

o

P= P

(10a,b)

o

W W

(11a,b)

which can be interpreted by saying that, when the solution is obtained, the weighed traction and displacement on * have to be equal to zero, whereas the solutions evaluated in terms of weighted quantities coincide with the weighted traction and displacement on * , respectively. If in eqs.(8,9) one introduces the modeling (7), and utilizes the positions

³ t

T u

d * Cut ,

*

one obtains

³ u

*

T t

d * Ctu ,

Ctu

CTut

(12a,b,c)

65

Ctu F

Ctu T+ Ctu T P

P

P

T + T

F

Cut U

Cutu U Cut U +

W

W

U

P Ctu (T+ T ) Ctu F

o

U U +

(13a,b,c)

W

W Cut (U + U ) Cut U

o

(14a,b,c)

where T and T are the nodal forces acting respectively on the nodes of the boundaries * and * whereas U + and U are the corresponding displacements. Eqs. (13b) and (14b) are the equilibrium and compatibility equations (3,4) written in terms of nodal variables. Eqs. (13c) and (14c) are respectively equations in which the matrix Ctu transforms the nodal forces into corresponding weighted tractions associated with the nodes, whereas the matrix Cut transforms the nodal displacements into the corresponding weighted displacements associated with the nodes. +

3. Generalized work In the infinite domain :f the classic energy balance is valid L+* L*

U ( :f \ :) U :

1 1 σ T ε d (:f \ :) ³ σ T ε d : 2 ( :f³\ :) 2:

(17)

where L+* and L* are respectively the work performed by the boundary quantities * of :f \ : and * of : , whereas U U (: :) U : is the total energy stored in :f . Since in the infinite domain the forces and the displacements take on the meaning of layered quantities (3,4), taking into account modeling (7) and the definition of the weighted quantities (8,9), the work can be expressed in the following alternative forms f

I

L*

1 T f u d* 2 ³*

1 (t t )T (u u ) d * 2 ³*

1 ª(T )T W (T )T W (T )T W (T )T W º¼ 2¬ II

L*

1 T u f d* 2 ³*

1 (u u )T (t t ) d * 2 ³*

1 ª(U )T P (U )T P (U )T P (U )T P º¼ 2¬

1 T U P 2

1 T F W 2

(18)

(19)

These are the same expressions as provided by Denda [8] in terms of nodal and weighted layered quantities. The previous expressions will be utilized depending on the goals to be reached. 4. Displacement method: algebraic operators The response on the boundary * of the solid, caused by known and unknown actions distributed on the same boundary, is given by the Somigliana Identities (SI) of the displacements and tractions. Introducing modeling (7) in the SI and executing the weighting second Galerkin hypotheses, one obtains an equation system in weighted form, which, through the introduction of the boundary condition and a reordering of the equations, provides the following block system

66


ª W2+ º ª A u2u2 « + » « « W1 { 0 » « A u1u2 + « W0 { 0 » « A u0u2 « + »=« « P2 { 0 » « Ef2u2 « P » « A 0 « » « f0u2 + «¬ P1 »¼ «¬ A f1u2

A u2u1 A u1u1 A u0u1 A f2u1 A f0u1 Ef1u1

A u2u0 A u1u0 A u0u0 A f2u0 Ef0u0 A f1u0

H u2f2 A u1f2 A u0f2 A f2f2 A f0f2 A f1f2

A u2f0 A u1f0 H u0f0 A f2f0 A f0f0 A f1f0

A u2f1 º ª F2 º « » H u1f1 »» « F1 » A u0f1 » « F0 » » »« A f2f1 » «-U 2 » « » A f0f1 -U 0 » » »« A f1f1 »¼ « -U1 » ¬ ¼

(20)

Making use of this block matrix equation, through appropriate block subdivisions [1] the weighed forcenodal displacement characteristic elasticity equation of the single substructure takes on the following form: P0

D0 U0 Pˆ 0

(21)

5. Galerkin residuals and boundary work In the previous section the structure of the algebraic relations leading to the writing of the characteristic equation of the elasticity of the single substructure utilized within the displacement method of the SGBEM formulation was analyzed. On the basis of the boundary conditions imposed one notes that: x The weighted displacements W2 and the weighted tractions P1 are not utilized in the relations of the solving system; as a consequence the solution of the analysis problem will give W2 z 0 and P1 z 0 . These boundary quantities are defined by Paulino [6,7] as Galerkin weighted residuals and their values have to be null depending on the convergence of the solution of the discrete toward the solution of the continuum. x Further, one notes that to write the characteristic elasticity equation the following conditions were imposed W1 0 , W0 0 , P2 0 ; these relations will always be verified when the solution is obtained. 5. 1 Boundary work evaluation The generalized work L* on the boundary will be computed taking into account, among the expressions defining the alternative forms (I,II), those best suited to examining the boundary type. 5.1.1 Free boundary Γ 2 Through the relation P2 Ct2u2T2 , the position P2 0 does not necessarily involve T2 0 , because it is possible to find on the boundary * 2 a system of auto-stressed nodal forces, whose weighted resultant is null. In [2] through an appropriate system the vector T2 was computed which by using the relation T2 F2 T2 allows one to obtain a system of nodal force such that P2 Ct2u2T2 0 is satisfied. x Utilizing the first form of work one finds that: W2 z 0 with the additional condition P2 0 and T2 z 0 provides ,) L*2

ª º T 1« T2 T2 W2 W2 » » 2« W2 F2 ¬« ¼»

(22) where (1/ 2)(T2 )T W2 is the direct work on * 2 with T2 obtained in [2] and W2 W2 Cu2t2 U2 . In order for the residual quantities on * 2 to be null, the terms on which it is necessary to work are (1/ 2)(T2 )T ( W2 ) , (1/ 2)(T2 )T ( W2 ) , (1/ 2)(T2 )T ( W2 ) .

5.1.2 Constrained boundary Γ1 Through the relation WXR Cu1t1U1 , the position WXR 0 does not necessarily involve U1 0 , T2 0 , because it is possible to find on the boundary *1 a system of nodal displacements, whose weighted resultant is null. In [2] through an appropriate system the vector U1 was computed which by using the relation

67 U1 U1 U1 allows one to obtain a system of nodal displacements such that WXR Cu1t1U1 satisfied. x Utilizing the second form of work one finds that:

0 is

PXR z 0 with the additional condition WXR 0 and U1 z 0 provides

,,) L*1

ª º T 1« (U1 U1 ) (P1 P1 ) » » 2« P1 U1 ¬ ¼

(23)

where (1/ 2)(U1 )T P1 is the direct work on *1 with U1 obtained in [2] and P1 P1 Ct1u1F1 . In order that the residual quantities on *1 are null, the terms on which it is necessary to work are (1/ 2)(U1 )T P1 , (1/ 2)(U1 )T P1 , (1/ 2)(U1 )T P1 with U1 U1 U1 5.1.3 Interface boundary Γ 0 The boundary conditions on * 0 are: W0

0 , P0

D0 U0 Pˆ 0 z 0

By the relation of compatibility and equilibrium in weighted form one writes: W0 0

P

W0 Cu0t0 U0

o U0

U0

and

U0

U0 U0

0

P Ct0u0 F0

x Utilizing the second form of work one finds that: P0 z 0 with the additional condition W0 0 and U0 ,,)

o U0

0

L*0

0 provides

º T 1 ª« ( U 0 U 0 ) (P0 P0 ) » 2« » U0 P0 ¬ ¼

(24)

where (1/ 2)(U0 )T P0 is the direct work on * 0 obtained as a product between U0 U0 and P0 D0 U0 Pˆ 0 . In order for the residual quantities on * 0 to be null, the terms on which it is necessary to perform the convergence is the reciprocal work (1/ 2)(U0 )T P0 with U0 U0 and P0 P0 Ct0u0 F0 . On the basis of the strategy employed, the summary table is shown. These expressions are alternative to the integrals containing the punctual functions and are obtainable by the coefficients of the block matrix (20), utilized in the analysis process. Boundary type

Boundary work type ª º T 1« T2 T2 W2 W2 » » 2« »¼ W2 F2 ¬«

Γ2

,) L*2

Γ1

,,) L*1

ª º T 1« (U1 U1 ) (P1 P1 ) » » 2« P1 U1 ¬ ¼

Γ0

,,)

º T 1 ª« ( U 0 U 0 ) (P0 P0 ) » 2« » U0 P0 ¬ ¼

L*0

Table 1: Type contour and boundary work type adopted for every boundary type. 6. Numerical example Let us consider the rectangular plate in Fig. 2 having dimension 2 u 4 of thickness s 1 cm, constrained at the lower side and subjected to the constant vertical load q 1 . The plate is divided into two substructures A,

68


B having dimension 2 u 2 . Each is discretized into 8 boundary elements having unit length with *A *2 *0 ed *B *2 *1 *0 . The boundary elements of each substructure are denoted by small letters a, b,....h and the nodes are numbered clockwise. The physical characteristics of both the substructures are E 1 , Q 0.3 . The analysis is performed by the displacement method using the Karnak.sGbem program [9]. In order to justify and numerically validate the expressions of the work shown in Table 2, for each substructure only the vertical forces and displacements are considered. Specifically, for substructure A the loaded side (a, b) and the interface one (e, f) are considered; similarly, for substructure B the interface side (a, b) and the constrained one (e, f). It is to note that the values of the work obtained by the integrals containing the function evaluated on the boundary through the SI prove to be equal to that obtained through the layered generalized quantities of the matrix (20). This particular aspect allows one to perform all the numerical assessments, as for example the check of the error in the structural solution, by using the generalized quantities.

Figure 3. Body subjected to a uniform load: a) geometric description and substucturing, b) meshes adopted.

Boundary work: by punctual quantities and by generalized quantities Substructure A, a and b on Γ 2

t 1 (t2 t2 ) (u2 u2 ) d * 2 ³*a ,b

t 1 (T2 T2 ) ( W2 W2 ) 3.5 3.58829 2

Substructure A, e and f on Γ 0

t 1 (u0 u0 ) (t0 t0 ) d * 2 ³*e , f

t 1 ( U0 U0 ) (P0 P0 ) -1.76838 -1.7 2

Substructure B, a and b on Γ 0 Substructure B, e and f on Γ1

f2

u2

u0

f0

t 1 (u0 u0 ) (t0 t0 ) d * 2 ³*e , f u0

f0

U0 { U0

f1

P0 Ct0u0 F0

t 1 1.79103 ( U0 U0 ) (P0 P0 ) 1.7 2

t 1 (u1 u1 ) (t1 t1 ) d * 2 ³*e , f u1

W2 Cu2t2 U 2

F2

U0 { U0

P0 Ct0u0 F0

t 1 (U1 U1 ) (P1 P1 ) 0 2 U1

P1 Ct1u1F1

Table 2: Energy evaluated on the boundary computed by the traction and displacement functions and by the generalized quantities. 7. Conclusions The example used shows the effectiveness and the remarkable simplicity of the method in order to compute the elastic strain energy through the boundary work in terms of generalized layered quantities. Moreover, the procedure introduces remarkable potentialities for developing an energy criterion to evaluate the error of the solution obtained in the analysis phase.

69 The presence of the direct and indirect terms, constituting the generalized work, offers within the sphere of substructuring the possibility of formulating a strategy having as aim to limit the error of the solution through an appropriate discretization checking in what boundary elements the energy scattered is greater. References [1] Panzeca T., Cucco F., Terravecchia S. (2002). “Symmetric boundary element method versus Finite

element method”. Comput. Meth. Appl. Mech. Engng., 191, 3347-3367. [2] Panzeca T., Cucco F., Terravecchia S. (2007). “Boundary discretization based on the energy residuals

using the SGBEM”. Int. J. Solids Struct., 44 (22-23), 7239-7260. [3] Pérez-Gavilán, J.J., Aliabadi, M.H. (2001). “A Symmetric Galerkin Bem for multi-connected bodies: a

new approach”. Eng. Anal. Bound. Elem., 25, 633-638. [4] Gray, L.J., Paulino, G.R. (1997). “Symmetric Galerkin boundary integral formulation for interface and

multi-zone problems”. Int. J. Numer. Meth. Eng. 40, 3085–3101. [5] Layton, J.B., Ganguly, S. Balakrishna, C., Kane, J.H., (1997). “A symmetric Galerkin multi-zone

boundary element formulation”. Int. J. Numer. Meth. Eng. 40, 2913–2931. [6] Paulino, G.H., Gray, L.J. (1999). “Galerkin Residuals for adaptive Symmetric-Galerkin Boundary

Element Methods”. J. Eng. Mech. 125, 575–585. [7] Paulino, G.H., Gray, L.J., Zarikian, V. (1996). “Hypersingular Residuals – a new approach for error

estimation in the boundary element method”. Int. J. Numer. Meth. Eng. 36, 2005–2009. [8] Denda, M., Dong, Y.F. (2001). “A unified formulation and error estimation measure for the direct and

indirect boundary element methods in elasticity”. Eng. Anal. Bound. Elem. 25, 557–564 [9] Cucco, F., Panzeca, T., Terravecchia, S. (2002). “The program Karnak.sGbem Release 2.1”, Palermo

University.

70


Dual Boundary Element Method for Fatigue Crack Growth: Implementation of the Richard’s Criterion F. Bonanno1,2, I. Benedetti1,2, A. Milazzo1 and M.H. Aliabadi2 1

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale e dei materiali, Università degli Studi di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy 2

Department of Aeronautics, Imperial College London, South Kensington Campus, SW7 2AZ, London UK [email protected], [email protected], [email protected], [email protected], [email protected] , [email protected],

Keywords: Dual Boundary Element Method (DBEM), Mixed-mode fracture, 3D crack growth, Richard’s criterion, crack propagation. Abstract. A new criterion for fatigue crack growth, whose accuracy was previously tested in the literature with the Finite Element Method, is here adopted with a Dual Boundary Element formulation. The fatigue crack growth of an elliptical inclined crack, embedded in a three dimensional cylindrical bar, is analyzed. In this way in addition to the propagation angle estimated by the Sih’s criterion, it is possible to take into account a twist propagation angle. The two propagation criteria are compared in terms of shape of the propagated crack and in terms of SIFs along the crack front. The efficiency of the Dual Boundary Element Method in this study is highlighted. Introduction It is well established that computational modeling may provide a formidable tool for design and maintenance of engineering structures. To date the boundary element method (BEM) has proved very effective for fracture mechanics problems, without the limitations of an extremely refined mesh in the crack tip, and the requirement of a continuous re-meshing for crack growth simulations typical of the finite element method (FEM) [1]. The Dual Boundary Element Method (DBEM), developed by Portela, Aliabadi and Rooke [2] for 2D problems and then by Mi and Aliabadi [3] for 3D problems, appears to be a more general and computationally efficient way to model crack problems with respect to the multi-region method developed by Blandford et al. [4]. The DBEM is a single region formulation, applying the displacement boundary equation on one crack surface and the traction boundary equation on the other. The advantage of the method is its robustness in modeling the crack growth, with need of little re-meshing of the original model. It is well-known that a sequence of increasing and decreasing loads, could lead to an increase in the crack front at each step, even though the maximum stress intensity factor may be much less than the critical value; in the framework of the linear fracture mechanics it is possible to use the Paris law [5], that nowadays has gained general acceptance, to predict the crack incremental size. On the other hand, there is no unique accepted criterion regarding crack growth direction, especially for 3D cases; the most common and used criterion, for its precision and versatility in numerical simulation but also for the possibility to use it both in two and three dimensions, is the minimum strain energy density criterion formulated by Sih [6]; even if it is the most used criterion, it has a some drawback: in three dimensional cases it is insensitive to Mode III stress intensity factors, and for this reason, the twist propagation angle is always equal to zero. In the present study the numerical simulation of the crack growth of an embedded inclined elliptical crack subjected to Mixed Mode load conditions is presented, adopting a new criterion by Richard [7], that being sensitive to Mode III, estimates the crack’s twisting. As results the propagated crack patterns for the two different criteria are shown. Furthermore, for the Richard’s criterion, the trend of the crack growth angles and stress intensity factors along the crack front are presented, together with comparison graphics between Sih’s and Richard’s results.

71

Dual Boundary Element Method Dual Boundary Integral Equation Formulation for Crack Problems. Let us consider a cracked body and let Ȟ ା and Ȟ ି be the upper and lower crack surfaces, and Ȟ ௘ the rest of the boundary. The DBEM formulation is obtained by collocating the displacement integral equation on the boundary and on one of the crack surfaces and the traction boundary integral equation on the other crack face. The equation collocated on the boundary is the classical one and it is not recalled here, while the equations written for the crack surfaces are: ͳ ͳ ‫ ݑ‬ሺ࢞ା ሻ ൅ ‫ݑ‬௜ ሺ࢞ି ሻ ൅ න ܶ௜௝‫ כ‬ሺ࢞ା ǡ ࢞ሻ‫ݑ‬௜ ሺ࢞ሻ݀Ȟሺ‫ܠ‬ሻ ൌ න ܷ௜௝‫ כ‬ሺ࢞ା ǡ ࢞ሻ‫ݐ‬௝ ሺ࢞ሻ݀Ȟሺ‫ܠ‬ሻሺͳሻ ʹ ௜ ʹ ୻ ୻ ͳ ͳ ‫כ‬ ‫כ‬ ሺ࢞ି ǡ ࢞ሻ‫ݑ‬௞ ሺ࢞ሻ݀Ȟሺ‫ܠ‬ሻ ൌ ݊௜ ሺ࢞ି ሻ න ܷ௜௝௞ ሺ࢞ି ǡ ࢞ሻ‫ݐ‬௞ ሺ࢞ሻ݀Ȟሺ‫ܠ‬ሻሺʹሻ ‫ ݐ‬ሺ࢞ି ሻ െ ‫ݐ‬௝ ሺ࢞ା ሻ ൅ ݊௜ ሺ࢞ି ሻ න ܶ௜௝௞ ʹ ʹ ௝ ୻ ୻ Where ݅ǡ ݆ ൌ ͳǡʹǡ͵Ǣܷ௜௝‫ כ‬ሺ‫ ܠ‬ା ǡ ‫ܠ‬ሻand ܶ௜௝‫ כ‬ሺ‫ ܠ‬ା ǡ ‫ܠ‬ሻ are respectively Kelvin displacement and traction ‫כ‬ ‫כ‬ ሺ‫ ܠ‬ା ǡ ‫ܠ‬ሻ and ܶ௜௝௞ ሺ‫ ܠ‬ା ǡ ‫ܠ‬ሻ are obtained from the derivatives of the fundamental fundamental solutions, ܷ௜௝௞ solutions; in eq.(1) the integral at the left-hand side is a Cauchy principal value integral as the integral at the right-hand side of eq.(2); while the first integral in eq.(2) stands for an Hadamard principal value integral. Crack Modelling Strategy. The strategy used for DBEM modelling is given in [3] and it can be summarized in the following points x x x x x x

The crack boundaries are modeled with discontinuous quadratic elements; The surfaces intersecting a crack surface are modeled with edge-discontinuous quadrilateral and triangular quadratic elements; Continuous quadratic elements are used along the remaining boundary; For collocation on the crack surface Ȟ ା the displacement equation in eq.(1) is applied; For collocation on the other crack surface Ȟ ି the traction equation in eq.(2) is applied; The usual boundary displacement equation is applied for collocation on all other surfaces.

For further details the interested reader is referred to [3]. Fatigue Crack Growth When a cracked body is subjected to a generic loading system, the movements of the upper and lower surfaces of the crack with respect to each other can be described using three basic modes: x x x

Mode I or opening mode, where the two crack surfaces are pulled apart; Mode II or sliding mode, where the two crack surfaces slide over each other along the crack line; Mode III or tearing mode, where the crack surfaces slide over each other perpendicular to the crack line.

With the superposition of these three basic modes any crack deformation can be described, and under the hypothesis of linear elastic fracture mechanics (LEFM), the determination of the rate of crack growth in a loaded structure subjected to fatigue loading is correlated only to the knowledge of the stress intensity factors, in this way the behavior of the cracked body can be fully described. The fatigue loading is a sequence of increasing and decreasing load (cyclic), that can lead to an increase in crack front even if the maximum stress intensity factor may be much less than the critical one. The goal for the designer is to establish the necessary number of cycles for a crack to extend from some initial length to a pre-imposed one. The typical Paris’ sigmoidal curve relates the rate of crack growth per load cycle: ݀ܽȀ݀݊, with the applied stress intensity factor range: ȟ‫ ܭ‬ൌ ‫ܭ‬௠௔௫ െ ‫ܭ‬௠௜௡ ; in this log-log plot can be recognized three different zones [5]:

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x x x

The first region, where the crack growth goes asymptotically to zero as ȟ‫ ܭ‬approaches a threshold value: ȟ‫ܭ‬௧௛ ; this value represents the fatigue limit, for stress intensity factors below ȟ‫ܭ‬௧௛ there is no crack growth; In the second region, the ሺ݀ܽΤ݀݊ሻ tends to vary linearly with respect to the ȟ‫;ܭ‬ In the third region it can be seen a drastic acceleration as ‫ܭ‬௠௔௫ approaches ‫ܭ‬௖ the fracture toughness of the material.

To describe the behaviour in the linear region, Paris et al. [8] developed an empirical formula: ݀ܽ ൌ ݂ሺȟ‫ܭ‬ሻሺ͵ሻ ݀݊ To take into account also other factors, among which: load frequency, environment and mean load; Paris and Erdogan [5] suggested another law depending on two empirical material constants: C and m. ݀ܽ ൌ ‫ܥ‬ȟ‫ ܭ‬௠ ሺͶሻ ݀݊ Equation (4) is generally called Paris law, and has gained worldwide acceptance in engineering practice. To obtain a generalized fatigue crack growth formula which takes into account the combined effect of Mode I and Mode II, Tanaka [9] proposed an expression for ȟ‫ܭ‬: ȟ‫ܭ‬௘௙௙ ൌ ሺ‫ܭ‬ூସ ൅ ͺ‫ܭ‬ூூସ ሻଵΤସ ሺͷሻ Criteria for Crack Growth Propagation Spatial Mixed Mode problems are characterised by the superposition of the three fracture modes. Exist only few fracture criteria to describe 3D Mixed Mode problems, two of them will be described in the following with a brief comparison. Minimum Strain Energy Density Criterion (S-Criterion). Formulated by Sih [6] to date is the most popular and used criterion for 3D problems, because it seems to be able to handle very well three dimensional crack growth, taking into consideration the three stress intensity factors. The explicit expression for the strain energy density around the crack front can be written as a function of the strain energy density factor: ܵ [7] ܹ݀ ܵሺ߮ǡ ߰ሻ ൌ ൅ ܱሺͳሻሺ͸ሻ ܸ݀ ‫ݎ‬ where ܵሺ߮ǡ ߰ሻ is given by ܵሺ߮ǡ ߰ሻ ൌ

ͳ

߰

ଶ ሺܽଵଵ ‫ܭ‬ூଶ ൅ ʹܽଵଶ ‫ܭ‬ூ ‫ܭ‬ூூ ൅ ܽଶଶ ‫ܭ‬ூூଶ ൅ ܽଷଷ ‫ܭ‬ூூூ ሻሺ͹ሻ

Where ܽଵଵ ǡ ܽଵଶ ǡ ܽଶଶ ǡ ܽଷଷ depend only by ߮ǡ ߭ and ߤ that is the shear modulus of elasticity and ߭ is the Poisson’s ratio. The crack angles ߮଴ and ߰଴ (elevation and twisting) shown in Fig.1 are derived by minimizing ܵ of eq.(7): ߲ܵ ߲ܵ ൌ Ͳ ฬ ൌ Ͳሺͺሻ ฬ ߲߮ ఝୀఝ ߲߰ టୀట బ

బ

It is apparent from eq.(7), that the minimum of ܵሺ߮ሻ always occurs in the normal plane of the crack front curve, namely when ߰ ൌ ͳ ֜ ߰ ൌ Ͳ, independently of the Mixed Mode combination; also the partial derivative of ܵ by ߮ is independent of ߰଴ as well as ‫ܭ‬ூூூ .

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Figure 1: The two different rotations:࣐ , ࣒.

Therefore this criterion is insensitive to Mode III [7]. Richard’s Criterion. In order to take into account the twist rotation (߰ angle), considering in this way also the Mode III, and to simplify the prediction of crack growth, Richard developing approximation functions, proposed a criterion whose efficiency is tested experimentally. The angles can be written as: ଶ

߮଴ ൌ ‫ ט‬൥‫ܣ‬

ȁ‫ܭ‬ூ ȁ ȁ‫ܭ‬ூூ ȁ ൅‫ܤ‬ቆ ቇ ൩ሺͻሻ ‫ܭ‬ூ ൅ ȁ‫ܭ‬ூூ ȁ ൅ ȁ‫ܭ‬ூூூ ȁ ‫ܭ‬ூ ൅ ȁ‫ܭ‬ூூ ȁ ൅ ȁ‫ܭ‬ூூூ ȁ

where ߮଴ ൏ Ͳι for ‫ܭ‬ூூ ൐ Ͳ and ߮଴ ൐ Ͳι for ‫ܭ‬ூூ ൑ Ͳ. ଶ

߰଴ ൌ ‫ ט‬൥‫ܥ‬

ȁ‫ܭ‬ூூூ ȁ ȁ‫ܭ‬ூூூ ȁ ൅‫ܦ‬ቆ ቇ ൩ሺͳͲሻ ‫ܭ‬ூ ൅ ȁ‫ܭ‬ூூ ȁ ൅ ȁ‫ܭ‬ூூூ ȁ ‫ܭ‬ூ ൅ ȁ‫ܭ‬ூூ ȁ ൅ ȁ‫ܭ‬ூூூ ȁ

where ߰଴ ൏ Ͳι for ‫ܭ‬ூூூ ൐ Ͳ and ߰଴ ൐ Ͳι for ‫ܭ‬ூூூ ൑ Ͳ. With ‫ ܣ‬ൌ ͳͶͲιǡ ‫ ܤ‬ൌ െ͹Ͳιǡ ‫ ܥ‬ൌ ͹ͺιǡ ‫ ܦ‬ൌ െ͵͵ι eq.(9,10) are in good agreement with the crack deflection angles predicted by another criterion: the Schöllmann criterion [7]. Richard proposes also an expression for an equivalent stress intensity factor: ‫ܭ‬௩ to evaluate if an unstable crack growth will occur:

‫ܭ‬௩ ൌ

ଶ ଶ ‫ܭ‬ூ ͳ ‫ܭ‬ூ஼ ‫ܭ‬ூ஼ ൅ ඨ‫ܭ‬ூଶ ൅ Ͷ ൬ ‫ܭ‬ூூ ൰ ൅ Ͷ ൬ ‫ܭ‬ூூூ ൰ ൒ ‫ܭ‬ூ஼ ሺͳͳሻ ʹ ʹ ‫ܭ‬ூூ஼ ‫ܭ‬ூூூ஼

where ‫ܭ‬ூ஼ ǡ ‫ܭ‬ூூ஼ and ‫ܭ‬ூூூ஼ are the fracture toughness; also for this expression, if ‫ܭ‬ூ஼ Τ‫ܭ‬ூூ஼ ൌ ͳǤͳͷͷ and ‫ܭ‬ூ஼ Τ‫ܭ‬ூூூ஼ ൌ ͳǤͲ ; eq.(11) is in a good agreement with the ‫ܭ‬௩ predicted by the Schöllmann criterion. In the same way the fatigue crack growth is possible, only if, defining the cyclic stress intensity factor as in eq.(12), it exceeds the threshold value ο‫ܭ‬ூǡ௧௛ and is smaller than ο‫ܭ‬ூǡ௖ .

ο‫ܭ‬௩ ൌ

ଶ ଶ ο‫ܭ‬ூ ͳ ο‫ܭ‬ூ஼ ο‫ܭ‬ூ஼ ൅ ඨο‫ܭ‬ூଶ ൅ Ͷ ൬ ο‫ܭ‬ூூ ൰ ൅ Ͷ ൬ ο‫ܭ‬ூூூ ൰ ሺͳʹሻ ʹ ο‫ܭ‬ூூ஼ ο‫ܭ‬ூூூ஼ ʹ

To test the efficiency of this criterion, simulations were performed using the Finite Element Method and the program system ADAPCRACK3D [10].

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Numerical Simulation of Crack Growth Experiments are surely a fundamental part to understand the crack behaviour, but they are in most cases very expensive; for this reason, becomes necessary the developing of numerical techniques which allow the engineer to predict the crack evolution. The Finite Element Method has been used in a lot of numerical simulations, by many researchers [11]; but the common drawback of this method is the need of a continuous crack re-meshing to follow the crack extension, especially when a 3D problem is described. The main advantage of the Dual Boundary Element Method, is that the need of a re-meshing procedure is practically negligible, by virtue of the boundary formulation, and of the intrinsic characteristic of the single region analysis; in this way both embedded and edge crack can be studied with only a localized re-meshing on the free surfaces on the breaking cracks as the growth takes place [3,12]. To illustrate this procedure a mixed mode 3D crack growth problem is considered here: the fatigue cracking process is generated by a constant amplitude cyclic tensile loading applied on the upper and lower surfaces of a cylindrical bar of radius ܴ and height ݄ with an embedded inclined elliptical crack [3]. Initially crack surfaces are defined and the DBEM, is used to analyze the stress system; the three stress intensity factors are evaluated in three nodes for each element that forms the crack front; for a total of 16 elements and 48 nodes. The incremental direction was calculated using both the Sih and Richard’s criterion, to show as the second is sensitive to the Mode III and thus to the twisting rotation. Four incremental steps were performed, fixing at each step the maximum incremental crack length at 0.2 times the crack semi-major axis. The incremental part of the crack is constructed using the incremental direction and size in the form of piecewise surfaces which vary linearly along the crack growth direction. After the necessary modification of the boundary mesh, the analysis carries on taking into account the new configuration. This method can be utilized until the predefined crack length is reached or the effective stress intensity factor has exceeded the fracture toughness of the material. Determination of Incremental Direction and Size. To perform the incremental analysis is necessary to know two different parameters: the direction and the size of the crack incremental extension. Using The Sih criterion, ߮଴ is evaluated in the local coordinate plane perpendicular to the crack front Fig.1, minimizing numerically ܵ with respect to ߮Ǥ The resultant propagation directions can then be referred to the global system of coordinates and expressed as propagation unit vectors. The size of the increment is evaluated using the Paris law eq.(4), and since linear elasticity is considered in this problem the maximum amount ȟܽ௠௔௫ of the increment corresponds to the crack front point where the maximum value of ȟ‫ ܭ‬is reached. Therefore, the incremental size at each step, and at each node can be evaluated by : ȟ‫ܭ‬ ȟܽ ൌ ȟܽ௠௔௫ ൬ ൰ሺͳ͵ሻ ȟ‫ܭ‬௠௔௫ where for ȟ‫ ܭ‬was used the expression proposed by Gerstle, that takes into account all the three stress intensity factors: ଶ ൌ ሺ‫ܭ‬ூ ൅ ‫ܤ‬ȁ‫ܭ‬ூூூ ȁሻଶ ൅ ʹ‫ܭ‬ூூଶ ሺͳͶሻ ȟ‫ܭ‬௘௙௙

The same procedure is adopted for the Richard’s criterion: for each node the three stress intensity factors are evaluated, and so, using eq.(9,10) ߮଴ and ߰଴ are known; thus the resultant propagation vector will take into account not only the propagation angle evaluated by Sih but also the twisting rotation as shown in Fig.1. In this way considering also the Mode III, the pattern of the deformed crack, will be surely more realistic and precise. A comparison between the Sih’s and the Richard’s deformed crack pattern, is shown in Fig.2.

75

Figure 2: Comparison between the crack growth path with the Sih’s criterion a) and the Richard’s Criterion b)

The deformed crack surfaces after the last analysis are shown in Fig.3; in Fig.4 are also presented the crack growth angles: ߮ and ߰, along the crack front computed with the Richard’s criterion, and in Fig.5 the three stress intensity factors: ‫ܭ‬ூ ǡ ‫ܭ‬ூூ and ‫ܭ‬ூூூ , corresponding to each analysis normalized by the value of the stress applied. It is worth to be notice that they are in good agreement with those found by Mi for the same case, using the Sih’s criterion [3].

Figure 3: Comparison between the deformed crack surfaces with the Sih’s criterion a) and the Richard’s Criterion b)

Figure 4: Crack growth angles along the front estimated with the Richard’s criterion

76


Figure 5: Trend of the normalized Stress Intensity Factors: ࡷࡵ ǡ ࡷࡵࡵ and ࡷࡵࡵࡵ

Figure 6: Comparison between the results obtained from Sih (solid lines) and Richard (markers)

Fig.(6-7) show respectively the comparison between the ߮ angles and the different stress intensity factors, found for each analysis with the Sih’s criterion, represented by the solid lines, and those estimated with the Richard’s criterion, represented by the markers.

77

Figure 7: Comparison between the results obtained from Sih (solid lines) and Richard (markers)

Summary A three-dimensional dual boundary element formulation for the analysis of the fatigue crack growth of an embedded inclined elliptical crack subjected to a fracture Mixed Mode has been adopted, to carry out a numerical simulation and thus, to estimate the new propagated crack pattern. Two different criteria have been used to calculate the crack growth direction, from them comparison a more realistic crack behavior is shown using the Richard’s one. In this article are also shown the trends and the comparisons of the crack angles and the three stress intensity factors along the crack front, for each propagation. References [1] M. H. Aliabadi and D. P. Rooke Numerical Fracture Mechanics. Computational Mechanics Publications and Kluwer Academic Publishers, (1991). [2] A. Portela, M. H. Aliabadi and D. P. Rooke Dual boundary element analysis of pin-loaded lugs. Boundary element Technology, 6, 381-392, (1991). [3] Y. Mi and M. H. Aliabadi Three Dimensional Crack Growth Simulation Using BEM. Computers & Structures, 52, 871-878, (1994). [4] G. E. Blandford, A. R. Ingraffea and J. A. Ligget Two-dimensional stress intensity factor computations using the boundary element method. International Journal for Numerical Methods in Engineering, 17, 387406, (1981). [5] P. C. Paris and F. Erdogan A critical analysis of crack propagation laws. Trans. ASME, J. Basic Engng., 85, 528-534, (1963). [6] G. C. Sih Mechanics of Fracture Initiation and Propagation. Kluwer Academic Publishers, (1991). [7] H.A. Richard, F. G. Buchholz, G. Kullmer and M. Scöllmann 2D- and 3D-Mixed Mode Fracture Criteria. Advances in Fracture and Damage Mechanics, Trans Tech Publications LTD, 251-260, (2003). [8] P. C. Paris, M. P. Gomez and W. P. Anderson A rational analytic theory of fatigue. The trend in Engineering, 13, 9-14, (1961). [9] K. Tanaka Fatigue crack propagation from a crack inclined to the cyclic tensile axis. Engng Fracture Mech., 6, 493-507, (1974). [10] M. Scöllmann, M. Fulland and H.A. Richard Development of a new software for adaptive crack growth simulations in 3D structures. Engineering Fracture Mechanics, 70, 249-268, (2002). [11] E. E. Gdoutos and N. Hatzitrifon Growth of three-dimensional cracks in finite-thickness plates. Engineering Fracture Mechanics, 6, 883-895, (1987). [12] A. Cisilino M. H. Aliabadi Dual boundary element assessment of three dimensional fatigue crack growth. Engineering Analysis with Boundary element, 28, 1157-1173, (2004).

78


Analysis of Stiffened Panels using Coupled DBEM and FEM Z. Sharif Khodaei1,a and M. H. Aliabadi1,b 1

Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW7 2AZ, London, UK., a

[email protected], [email protected],

Keywords:, DBEM, FEM, static condensation, stiffened panel. Abstract. This paper presents the application of Dual Boundary Element Method (DBEM) for determination of the static response of a stiffened plate under external load. The plate is formulated as 2D plate using BEM. The stiffeners attached to the panel are modeled as beams using FEM. The coupling between the plate and the stiffeners are modeled through defining an attachment coefficient based on the shear modulus of the adhesive. To couple only the required degrees of the freedom between the 2D plate and 2D beam, static condensation method is applied to the FEM beam formulation.

Introduction Fatigue damage in aircraft structure is of high importance, since the presence of crack and its propagation can significantly reduce the residual strength of the structure. Therefor accurate residual strength estimation of the components must be carried out to ensure safety. Moreover, the behavior of the damaged structure alters from that of the pristine case. Thus by monitoring the strain response of the structure by attached sensors, presence of damage can be detected. Aircraft components are mainly made of thin sheets of metal or composite reinforced by stiffeners which could be continuously bonded or riveted to the structure. The stiffeners provide and alternative path and part of the load is transferred through them which reduce the load in the skin and retards the crack propagation. The behavior of stiffened panels has been addressed by numerous authors. Considering that the skin, stiffener and the adhesive layer are thin, the problem can be solved using two dimensional elastostatic theory, i.e. the out of plane bending is ignored. A Boundary Element Method (BEM) for shear-deformable plate structures joined with rivets was formulated in [1]. The main advantage of BEM technique is that a complete solution is available in boundary terms only. However the technique can lose the attraction of its boundary terms only because non-homogenous terms are available in the formulation by means of domain integral. Dual Reciprocity Method (DRM) transforms the domain integral in boundary integrals in BEM analysis. A comprehensive coverage of the BEM formulation can be found in [2]. Modeling cracks using BEM is associated with difficulties, however Dual Boundary Element Method (DBEM) developed by Portela et. al. [3] overcomes the complications. Salgado and Aliabadi [4] extended the DBEM formulation to crack propagation in stiffened panels. The formulation has been validated by applying the method to a wide range of problems[5]. In this paper a DBEM/DRM formulation is presented for modeling a cracked 2D stiffened plate. The stiffeners model is based on an FEM formulation of a 2D Euler beam bonded with adhesive to the plate. The interaction between the plate and the stiffener is modeled through coefficient of a shear deformation of the adhesive layer as described in [4].

79 Numerical model of the plate using Dual Boundary Element Method DBEM is capable of analyzing any given geometry with any number or configuration of edge/embedded cracks. The displacement at any given source point x’ at the boundary of a finite sheet is given by:

ܿ௜௝ ሺ‫ ݔ‬ᇱ ሻ‫ݑ‬௝ ሺ‫ ݔ‬ᇱ ሻ ൅ ර ܶ௜௝ ሺ‫ ݔ‬ᇱ ǡ ‫ݔ‬ሻ‫ݑ‬௝ ሺ‫ݔ‬ሻ݀*ሺ‫ݔ‬ሻ ൌ න ܷ௜௝ ሺ‫ ݔ‬ᇱ ǡ ‫ݔ‬ሻ ‫ݐ‬௝ ሺ‫ݔ‬ሻ݀*ሺ‫ݔ‬ሻ *

*

(1)

൅ ඵ ܷ௜௝ ሺ‫ ݔ‬ᇱ ǡ ܺሻܾ௝ ሺܺሻ݀ߪሺܺሻ ఙ

where ܶ௜௝ ሺ‫ ݔ‬ᇱ ǡ ‫ݔ‬ሻ and ܷ௜௝ ሺ‫ ݔ‬ᇱ ǡ ‫ݔ‬ሻ are the Kelvin traction and displacement fundamental solution respectively, ‫ݑ‬௝ ሺ‫ݔ‬ሻ and ‫ݐ‬௝ ሺ‫ݔ‬ሻ are displacement and traction of a field point x on the boundary *, ܾ௝ ሺܺሻ is the body

force acting at the internal point X inside the domain :. ‫ ׯ‬Stands for Cauchy principal value integral and ܿ௜௝ is a coefficient determined from rigid body motion considerations. If we assume that the body force are distributed over a straight line inside it (stiffener axis), the body forces in equation (1) are reduced to line integrals over the forces loci:

ܿ௜௝ ሺ‫ ݔ‬ᇱ ሻ‫ݑ‬௝ ሺ‫ ݔ‬ᇱ ሻ ൅ ර ܶ௜௝ ሺ‫ ݔ‬ᇱ ǡ ‫ݔ‬ሻ‫ݑ‬௝ ሺ‫ݔ‬ሻ݀*ሺ‫ݔ‬ሻ ൌ න ܷ௜௝ ሺ‫ ݔ‬ᇱ ǡ ‫ݔ‬ሻ ‫ݐ‬௝ ሺ‫ݔ‬ሻ݀*ሺ‫ݔ‬ሻ *

*

൅ ෍ න ܷ௜௝ ே

*ೄ೅

ሺ‫ ݔ‬ᇱ

ǡ ܺሻ ܾ௝ ሺܺሻ݀*ௌ்

(2)

where *ௌ் stands for body forces loci and N is the number internal load paths (stiffeners). By

discretizing the boundary * and *ௌ் into sets of elements, by collocating at each boundary element node a system of linear equations is constructed with the unknowns being the boundary displacement or traction and the distributed internal load. The displacement of the nodes in the stiffener should be compatible with the displacement of the same nodes positioned in the plate through the shear coefficient of the adhesive. This can be expressed as: ௣

ο‫ݑ‬௜ െ ο‫ݑ‬௜௦௧ ൌ ) οܾ௜

(3)

௣

where ο‫ݑ‬௜ is the difference between the displacements of nodes in the plate along the stiffener with ௣ ௣ ௣ respect to a fixed point X0, i.e. ο‫ݑ‬௜ ൌ ‫ݑ‬௜ ሺܺ ௡ ሻ െ ‫ݑ‬௜ ሺܺ ଴ ሻ. The same holds for ο‫ݑ‬௜௦௧ which is the difference of the exact coinciding points but in the stiffener. οܾ௜ is the difference in nodal attachment forces and ) is the coefficient of the shear deformation of the adhesive layer. The relative displacements of the plate’s internal point X at the attachment can be determined from:

‫ݑ‬௜௦ ሺܺሻ ൌ ර ܷ௜௝ ሺܺǡ ‫ݔ‬ሻ‫ݐ‬௝ ሺ‫ݔ‬ሻ݀*ሺ‫ݔ‬ሻ െ න ܶ௜௝ ሺܺǡ ‫ݔ‬ሻ ‫ݑ‬௝ ሺ‫ݔ‬ሻ݀*ሺ‫ݔ‬ሻ *

*

൅ ෍ න ܷ௜௝ ሺܺǡ ‫ݔ‬ሻ ܾ௝ ሺܺሻ݀*ௌ் ே

(4)

*ೄ೅

The relative displacement of the stiffener is formulated through FEM of 2D beam and coupled with the boundary integral equation (4) through the compatibility condition (3).

80


Stiffeners – FE formulation

Plate – BEM formulation

y

Crack x

Figure 1. Stiffeners bonded to a cracked sheet Numerical model of the stiffener using Finite Element Method and static condensation The stiffener is modeled as a 2D beam element. That means the elementary beam theory is used following Euler-Bernoulli beam theory. The beam element has one node at each end with two Degrees of Freedom (DoF), namely lateral translation and rotation. However the stiffener can have lateral and transversal translation rotation in the plane of the plate. Therefore transversal translation DoF has been added to the 2D beam element. According to the compatibility condition, equation (3), the relative displacements of the stiffener nodes should be coupled to the relative displacement of the plate nodes through attachment force. As pointed out earlier, the out of plane bending of the plate and stiffeners are ignored due to their small thickness. Therefore, to solve the coupled BEM/FEM equations, the DoFs of the plate and stiffener must conform. To accomplish this, static condensation is applied to have the FEM formulation only in terms or lateral and transversal DoFs for each node. Static condensation is a technique used to reduce the number of DoFs to represent the full set of DoFs. There are various methods to accomplish the reduction ranging from intuitive to semi-rigorous. In this work Guyan reduction is applied. In Guyan reduction the DoFs of the FE model are designated as either masters or slaves. The slave DoFs are required to move as dictated by the displacement of the master DoFs and the stiffness matrix K. Only master DoFs appear in the reduced equation set. Starting with the set of equation ‫ܭ‬Ǥ ܷ ൌ ‫ ܨ‬and separate the master and slave DoFs: ‫ܭ‬ൌ൤

‫ܭ‬௠௠ ‫ܭ‬௦௦

‫ݑ‬௠ ‫ܭ‬௠௦ ‫ܨ‬௠ ் ൨ ǡܷ ൌ ቄ ‫ ݑ‬ቅ ǡ‫ ܨ‬ൌ ൜ ‫ ܨ‬ൠǤ ‫ܭ‬௠௦ ௦ ௦

(5)

where the subscript m refers to lateral and transversal displacement and subscript s relates to the condensed degree of freedom, i.e. rotation. For example the U vector in equation (5) for a single ‫ݑ‬௫ element comprises of ‫ݑ‬௠ ൌ ቄ‫ ݑ‬ቅ ǡ ‫ݑ‬௦ ൌ ሼ‫ݑ‬௠ ሽ. The lower partition for ‫ݑ‬௦ will be solved first: ௬ ் ሻିଵ ் ሻିଵ ‫ݑ‬௦ ൌ ሺ‫ܭ‬௠௦ Ǥ ‫ܨ‬௦ െ ሺ‫ܭ‬௠௦ Ǥ ‫ܭ‬௦௦ .‫ݑ‬௠ And substitute in the upper partition:

‫ܭ‬௠௠ Ǥ ‫ݑ‬௠ ൅ ‫ܭ‬௠௦ Ǥ ‫ݑ‬௦ ൌǤ ‫ܨ‬௠ Thus we obtain a smaller system in terms of ‫ݑ‬௠ only, same as the plate BEM formulation.

(6)

(7)

81 Finally the full matrix ‫ܣ‬Ǥ ‫ ݔ‬ൌ ‫ ܨ‬will be solved where A comprises of the BEM formulation of the plate, FEM formulation of the stiffeners and the compatibility condition between them. The unknown matrix ‫ ݔ‬consists of the unknown boundary displacement (BEM formulation), unknown internal point displacement (BEM formulation), unknown stiffener displacement (FEM formulation) and the compatibility between the FEM displacements and BEM displacements through shear coefficient of the adhesive layer. Discussion

In this paper a convenient DBEM/FEM formulation of static analysis of cracked stiffened panel has been presented and discussed. The plate was modeled using DBEM which allows for any geometry and number of cracks to be present in the structure. The proposed method for modeling the stiffener is using an FE formulation coupled with the DBEM formulation of the plate. The advantage of the FE modeling of the stiffener is that it can be extended easily. For example one application can be modeling sensors which can be used in structural health monitoring where the behavior of the structure in presence of damage. The next step will be to extend the model to a dynamic formulation for both plate and the stiffener in which case the model can be used as well for structural health monitoring. References

1.

2. 3.

4.

5.

Di Pisa, C., M. Aliabadi, and A. Young, Boundary element method analysis of assembled plate structures undergoing large deflection. The Journal of Strain Analysis for Engineering Design, 2010. 45(3): p. 179-195. Aliabadi, M., The boundary element method. Volume 2, Applications in solids and structures2002: Wiley London. Portela, A., M. Aliabadi, and D. Rooke, The dual boundary element method- Effective implementation for crack problems. International Journal for Numerical Methods in Engineering, 1992. 33(6): p. 1269-1287. Salgado, N. and M. Aliabadi, The application of the dual boundary element method to the analysis of cracked stiffened panels. Engineering Fracture Mechanics, 1996. 54(1): p. 91105. Salgado, N. and M. Aliabadi, Boundary element analysis of fatigue crack propagation in stiffened panels. Journal of aircraft, 1998. 35(1): p. 122-130.

82


Application of BEM for Internal Propulsor Flows Spyros A. Kinnas1 and Alokraj Valsaraj1 1

Ocean Engineering Group, Department of Civil, Architectural and Environmental Engineering The University of Texas at Austin, Austin, TX 78712, USA

Keywords Panel method, water-jet pump, Gauss quadrature

Abstract A numerical panel method based on the potential flow theory and utilizing hyperboloid panels has been developed and applied to the simulations of steady fully-wetted flows inside a water-jet pump. The source and dipole coefficients of the hyperboloid panels have been computed using Gauss quadrature. Finally, the hydrodynamic performance predictions for the water-jet pump from the present method are validated against existing experimental data or numerical results. Introduction Axial flow water-jet pumps are marine propulsion systems particularly suited for high speed vessels due to their balance between performance and robustness. Therefore, water-jet propulsion systems have become fairly popular for commercial and naval applications. The absence of appendages such as shafts, rudders and ducts under the waterline not only lowers the resistance but also makes water-jets a preferred solution for shallow water manoeuvring. Also, the occurrence of cavitation and debris damages to the propeller can be reduced inside water-jet pumps. However, the complicated geometry configurations, the inherent unsteadiness of the interaction between the rotor and stator, and the inevitable cavitation due to local pressure depressions, makes the simulation and analysis of flow fields inside water-jets significantly challenging. Several numerical and experimental approaches have been developed and applied to analyse the water-jet problem over past years. Kerwin [1] presented a comprehensive review of the issues in predicting the performance and in designing of water-jets. Presently, numerical methods for predicting the cavitating performance and assisting the design of water-jet propulsors are still limited. A more detailed description and literature review on water-jets can be found in the International Towing Tank Conference paper [2]. CFD tools, especially Reynolds Averaged Navier-Stokes (RANS) solvers, have gained increasing popularity for simulating flows inside water-jets. Chun et al. [3] used a RANS solver with a moving, non-orthogonal bodyfitted multi-blocked grid system for the interaction of the rotor and stator. The former component was considered in an unsteady sense and the latter component was accounted for in a circumferentially averaged sense. Brewton et al. [4] incorporated periodic boundaries and a mixing plane model into a RANS solver, and considered the interaction between the rotor and stator in the time-averaged sense. An intermediate approach which combines a potential flow method and a RANS method was applied to the prediction and design of water-jet components. Kerwin et al. [5] and Taylor et al. [6] applied a vortex lattice method (VLM) coupled with either an Euler equation solver or a RANS solver to include the effects of the hull and other appendages, and to analyse the global flow through the water-jet pump. Kinnas et al. [7, 8] applied a panel method to predict the hydrodynamic performance of a water-jet, and the interaction between the rotor and stator was considered in an iterative manner by taking into account the circumferentially averaged induced potentials from one to the other. Sun and Kinnas et al. [9] and Kinnas et al. [10] have used the viscous/inviscid interactive scheme successfully by coupling the panel method with a boundary layer solver XFOIL [11] to simulate the viscous flow around single, ducted and ONR-AxWJ1 water-jet propulsion systems, including boundary layer effects on the cavities over the blades. In this paper, a potential flow solver, based on the panel method by Kinnas & Fine [12] has been refined considerably by utilizing a non-planar numerical scheme to evaluate the dipole and source influence coefficients. This method is then used to analyse the steady fully-wetted performance of a water-jet propulsion system. Governing Equation and Boundary Conditions A panel method based on the potential flow theory is utilized to analyse 3-D steady flows around internal propulsors. The formulation of the rotor only problem of a water-jet pump is addressed here. Consider a water-jet, subject to a uniform inflow ୧୬ at the inlet of the pump and defined in a ship fixed coordinate system ሺ ୗ ǡ ୗ ǡ ୗ ሻ as shown in Figure 1. An impeller rotates with a constant angular velocity

83 vector ω. Thus, the inflow velocity can be defined as ୧୬ ൌ ୧୬ in a ship fixed coordinate system or as ୧୬ ൌ ୧୬ ൅ ɘu in a rotating coordinate system where X indicates a propeller fixed coordinate system. The flow is assumed to be inviscid, incompressible and irrotational, and thus the velocity field can be described as following: ࢗ௧ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ ൌ ࢂ௜௡ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ ൅ ‫߶׏‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ (1) whereࢗ௧ is the total velocity. For the rotor only problem, the perturbation potential at any point located either on the wetted surfaces of the rotor blades (SR) or on the hub and casing surfaces (SHC) must satisfy Green’s third identity as given: ʹߨ߶ ൌ න ቈ߶௤ ௌೃ

߲߶௤ ߲‫ܩ‬ሺ‫݌‬Ǣ ‫ݍ‬ሻ ߲‫ܩ‬ሺ‫݌‬Ǣ ‫ݍ‬ሻ െ ‫ܩ‬ሺ‫݌‬ǣ ‫ݍ‬ሻ ቉ ݀‫ ݏ‬൅ න ο߶ோೈ ݀‫ݏ‬ ߲݊௤ ߲݊௤ ߲݊௤ ௌೃ ೈ

൅න ௌಹ಴

ቈ߶௤

߲߶௤ ߲‫ܩ‬ሺ‫݌‬ǣ ‫ݍ‬ሻ െ ‫ܩ‬ሺ‫݌‬Ǣ ‫ݍ‬ሻ ቉ ݀‫ݏ‬ ߲݊௤ ߲݊௤

(2)

where p and q correspond to the variable points and the field points, respectively. ‫ܩ‬ሺ‫݌‬Ǣ ‫ݍ‬ሻ ൌ ͳȀܴሺ‫݌‬Ǣ ‫ݍ‬ሻ is the Green function and ܴሺ‫݌‬Ǣ ‫ݍ‬ሻ is the distance between the field point p and the variable point q. ݊௤ indicates the normal direction pointing into the flow field.ο߶ோೈ is the potential jump across the trailing wake sheets shedding from the rotor blade trailing edge.

Fig. 1: A water-jet rotor only problem In order to solve the rotor only problem, the following boundary conditions must be satisfied. The flow is tangent to the wetted rotor blades, hub and casing surfaces. (3) wI vin .n wn The Morino’s steady Kutta condition [13] is applied to ensure the fluid velocities are finite at the trailing edge of the blade. An iterative pressure Kutta (IPK) condition [14] is required to force a zero pressure jump between the pressure and suction sides at the blade trailing edge. The solution of the boundary value problem for a fully-wetted water-jet rotor is obtained by solving Equation (2) with the boundary conditions described above. It should be noted that the inlet panels of the water-jet are removed, and the perturbation potentials are set to be zero on those panels to obtain a unique solution of an internal boundary value problem. Forces and Parameters The definition of the non-dimensional coefficients used in this paper is explained. Γ is defined as:

*

'I / 2S R Uin2 (0.7nS D)2

(4)

where ο߶ indicates the potential jump at trailing edge of the propeller blade in the present method. The advance ratio ‫ܬ‬௦ and the Reynolds number Re are also given as: ܷ௜௡ ܷ௜௡ ܴ (5) ‫ܬ‬௦ ൌ ǡ ܴ݁ ൌ ݊‫ܦ‬ ߥ where ܷ௜௡ is the flow velocity at the inlet boundary, R represents the radius of the rotor and ν denotes the kinematic viscosity. The pressure coefficient Cp used in the present method is defined as follows: ܲ െ ܲ଴ ‫ܥ‬௣ ൌ (6) ͲǤͷߩ݊ଶ ‫ܦ‬ଶ

84


where ܲ௢ is the far upstream pressure on the shaft axis. Evaluation of dipole and source coefficients The numerical method (NRPAN) presented here discretizes the geometry of the body using hyperboloid panels. The dipole ‫ܣ‬௝௜ and source ‫ܤ‬௝௜ coefficients are given as

Aji

1 4S

1 )dsi

³ n .( r i

S Bi

B ji

ji

1 4S

³

S Bi

1 dsi rji

(7)

The above integrals have been evaluated on each panel using Gauss quadrature. The discretized form of the integrals is given as ௡ ௡ ௡ ௡ ͳ ͳ (8) ‫ ܣ‬ൌ ෍ ෍ ݊ሬԦ௜௝ Ǥ ߘ ቆെ ቇ ‫ݓ‬௜௝ ο‫ ݏ‬ǡ ‫ ܤ‬ൌ ෍ ෍ ቆെ ቇ ‫ݓ‬௜௝ ο‫ݏ‬ ‫ݎ‬௜௝ ‫ݎ‬௜௝ ௜ୀଵ ௝ୀଵ

௜ୀଵ ௝ୀଵ

Here J represents the Jacobian, w the weights associated with the Gaussian quadrature and n denotes the number of Gaussian points used to evaluate the integral. The present numerical method has n equal to 8 for calculating the integrals. Non-planar quadrilateral In order to validate the Gauss quadrature scheme, we use it to calculate the dipole and source influence coefficients of a non-planar quadrilateral as shown in Fig. 2. The control point is moved along the z axis.

Fig. 2 Non planar panel The error in calculating the source and dipole influence coefficients are shown in Fig 3. The error is defined ȁ௩ ି௩ಶ೉ಲ಴೅ ȁ as݁‫ ݎ݋ݎݎ‬ൌ ቀ ಿೆಾಶೃ಺಴ಲಽ ቁ. The exact value is obtained using MATLAB. ȁ௩ ȁ ಶ೉ಲ಴೅

Fig. 3 Error in calculating the influence coefficients It is observed that the error in evaluating the dipole influence coefficient by Gauss quadrature scheme increases rapidly as the control point is moved very close to the panel. This is because the numerical scheme in its present form cannot resolve the singularity in ‫ ܣ‬as‫ݎ‬௜௝ ՜ Ͳ. This error is mitigated by evaluating the dipole influence coefficients of a hyperboloid panel using the approach developed by Morino, Chen and Suciu [15]. However this scheme is valid only for control points very close to the panel. Henceforth for the calculation of the dipole influence coefficients we will use the Morino’s formula for control points very close to the panel (like self-influence coefficients) and Gauss quadrature for control points farther away.

85 Casing only problem After using the non-planar method to analyse a single panel, we move on to a simplified case, considering the casing only problem of a water-jet propulsion system. The casing selected was the casing of ONRAXWJ2 water-jet. The panelling on the casing of the water-jet follows the pitch of the rotor blade. For low pitch rotor blades we have highly skewed panels on the casing. The planar method used to evaluate the influence coefficients RPAN [16], which handles non-planar panels as planar panels resulting from the projection of the non-planar panel on the midpoint plane, requires a large number of panels to properly evaluate the dipole and source influence coefficients for low pitch cases. However using hyperboloid panels and the non-planar numerical method (NRPAN) allows us to significantly reduce the number of panels used to discretize the surface of the casing.

Fig. 4 Panelling on the casing for ܲ ൌ ͺͲ଴ case, only two quarters of the panels are shown In the study, the pitch angle considered isͺͲ଴. The total number of panels used to discretize the casing is 4000. The number of axial elements is 240 and number of circumferential elements is 40 on the casing. The results for ܲ ൌ ͺͲ଴ case are also compared with the results for ܲ ൌ Ͳ௢ case where both the planar and nonplanar methods work very well. Here we observe that the non-planar method works exceptionally well for ܲ ൌ ͺͲ଴ case while the planar panel method fails for the same case.

Fig. 5 Pressure distribution on the casing

Axial Flow Water-Jet Pump This section presents the analysis of an axial flow water-jet pump using the non-planar numerical scheme to evaluate the influence coefficients in the present BEM method for the rotor only case without the effect of the stator. The study considers the water-jet subject to a uniform inflow and only the steady simulations are considered here. The design advance ratio ୱ is 1.19 and the rotational frequency is 1400 rpm for the fully wetted condition. The panelled geometries used for the numerical calculation of the present method are shown in Fig. 6 for the rotor only problem.

86


Fig. 6 Geometry of the water-jet pump analysed with present BEM method The fully-wetted results including the pressure distributions on the rotor sections and casing are compared with those from the Reynolds Averaged Navier-Stokes (RANS) solver FLUENT. The present method takes 10 minutes for a fully-wetted analysis by using 120×40 panels on the blade, 90×20 panels on the hub and 200×20 panels on the casing. The first number denotes the number of panels in the chord-wise or axial direction and the second number denotes the number of elements in the span-wise on the blade or circumferential directions between two blades. A 3-D rotor only periodic mesh consisting of 5.94 million cells as shown in Fig. 7 is also used to analyse the rotor only simulation. An inviscid Reynolds Averaged Navier-Stokes (RANS) simulation is performed using FLUENT. The calculation takes about 8 hours by using 16 CPUs to finish 20,000 iterations on a cluster with 2.43 GHZ quad-core 64-bit Intel Xeon processors and 16 GB of RAM.

Fig. 7 3-D periodic computational domain used in FLUENT The circulation distribution on the blade from the present method is positive and is in good agreement with the values obtained from FLUENT (RANS) solver.

Fig. 8 Circulation distribution on the rotor blade The pressure distributions on the rotor blade are evaluated at radii of ‫ ݎ‬ൌ ͲǤ͹ͲͲǤͻͷ and compared with the results from FLUENT as shown in Fig. 9.

87

Fig 9 Pressure distributions at radii of 0.7 and 0.95 on the blade The pressure rise on the casing for the rotor only case is investigated. The prediction of the pressure distributions obtained from the present BEM method is compared with those from the 3-D FLUENT simulation. The comparison of pressure distributions on the casing surface is shown in Fig. 10. The two ‫ܥ‬௉ curves are the results of circumferentially averaged pressures of all stripes on the casing surface from 3D FLUENT simulation and those from the present BEM method.

Fig. 10 Pressure distribution on the casing Conclusions In this paper, a low-order panel method for internal flows has been improved and refined by utilizing Gauss quadrature to evaluate the dipole and source influence coefficients. The numerical method is then applied to analyse the hydrodynamic performance of a water-jet subject to uniform inflow. For the water-jet simulations, the predicted pressure distributions on the rotor blade and the casing are in very good agreement with the results of RANS calculations. Acknowledgement Support for this research was provided by the U.S. Office of Naval Research (Contract No. N00014-07-10616 and N0014-10-1-0931) and Phases V and VI of the “Consortium on Cavitation Performance of High Speed Propulsors” with the following current members: American Bureau of Shipping, Andritz Hydro GmbH, Daewoo Shipbuilding and Marine Engineering Co. Ltd., Kawasaki Heavy Industry Ltd., Rolls-Royce Marine AB, Rolls-Royce Marine AS, Samsung Heavy Industries Co. Ltd., SSPA AB, Sweden, VA Tech Escher Wyss GmbH, Wärtsilä Propulsion Netherlands B.V., Wärtsilä Propulsion Norway AS, Wärtsilä Lips Defense S.A.S., and Wärtsilä CME Zhenjiang Propeller Co. Ltd. References [1] Kerwin, J. E. Hydrodynamic Issues in Water-jet Design and Analysis, Proc. 26th Symposium on Naval Hydrodynamics, Rome, Italy (2006).

88


[2] ITTC Final Report and Recommendations to the 25th ITTC, Specialist Committee on Cavitation, Proc. 25th International Towing Tank Conference, Fukuoka, Japan, Volume II, 473-533 (2008). [3] Chun, H. H., Park, W. G., and Jun, J. G. Experimental and CFD Analysis for Rotor-Stator Interaction of a Water-jet Pump, Proc. 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan (2002). [4] Brewton, S., Gowing, S., and Gorski, J. Performance Predictions of a Waterjet Rotor and Rotor/Stator Combination Using RANS Calculations, Proc. 26th Symposium on Naval Hydrodynamics, Rome, Italy (2006). [5] Kerwin, J. E., Taylor, T. E., Black, S. D., and McHugh, G. P. A coupled lifting-surface analysis technique for marine propulsors in steady flow, Proc. Society of Naval Architects & Marine Engineers Propellers/Shafting '97 Symposium, Virginia Beach, VA, United States (1997). [6] Taylor, T. E., Kerwin, J. E., and Scherer, J. O. (1998). Water-jet Pump Design and Analysis Using a Coupled Lifting-Surface and RANS Procedure, Proc. Int. Conf. on Water-jet Propulsion, The Royal Institution of Naval Architects, London, UK [7] Kinnas, S. A., Lee, H. S., Michael, T. J., and Sun, H. Prediction of Cavitating Water-jet Propulsor Performance Using a Boundary Element Method, Proc. 9th Int. Conf. on Numerical Ship Hydrodynamics, Ann Arbor, Michigan (2007a). [8] Kinnas, S. A., Chang, S.-H., and Yu, Y.-H. Prediction of Wetted and Cavitating Performance of Waterjet,. Proc. 28th Symposium on Naval Hydrodynamics, Pasadena, CA (2010). [9] Sun, H. and Kinnas, S. A. Simulation of Sheet Cavitation on Propulsor Blades Using a viscous/inviscid interactive method, Proc. 6th Int. Symposium on Cavitation, CAV2006, Wageningen, The Netherlands (2006). [10] Kinnas, S. A., Lee, H. S., Sun, H., and He, L. Performance Prediction of Single or Multi-Component Propulsors Using Coupled Viscous/Inviscid Methods, Proc. 10th Int. Symposium on the Practical Design of Ships and other Floating Structures, PRADS2007, Houston, TX, United States (2007b). [11] Drela, M., XFOIL: An analysis and design system for low Reynolds number airfoils, In Lecture Notes in Engineering (Vol. 54, Low Reynolds Number Aerodynamics), New York, Springer-Verlag (1989). [12] Kinnas, S. A. and Fine, N. E., A Numerical Nonlinear-Analysis of the Flow around 2-Dimensional and 3-Dimensional Partially Cavitating Hydrofoils, J. Fluid Mech., 254, 151-181 (1993). [13] Morino, L. and Kuo, C.C., Subsonic Potential Aerodynamics for Complex Configuration: a general theory, AIAA Journal, 12, 191-197 (1974). [14] Kinnas, S. A. and Hsin, C.-Y., A Boundary Element Method for the Analysis of the Unsteady Flow around Extreme Propeller Geometries, AIAA Journal, 30(3), 688-696 (1992). [15] Morino, L., Chen Lee, Emil O.Suciu, Steady and Oscillatory subsonic and supersonic aerodynamics around complex configurations, AIAA Journal, Vol. 13(3), (1975). [16] Newman, J.N., Distributions of sources and normal dipoles over a quadrilateral panel, Journal of Engineering Mathematics, Volume 20, Issue 2, pp 113-126, (1986).

89

A Rapid Approach to BEM-BEM Acoustic-Structural Coupling A. Brancati1,2, M.H. Aliabadi3,4 and A. Alaimo1,5 1

Facoltà di Ingegneria e Architettura, Università di Enna “Kore”, Cittadella Universitaria, 94100 Enna, IT 2 3

[email protected]

Department of Aeronautics, Imperial College London, South Kensington, SW7 2AZ, UK, 4

[email protected] 5

[email protected]

Keywords: Acoustic-Structural Coupling, Adaptive Cross Approximation/Hierarchical matrix format.

Abstract. This paper presents a fast approach to the Boundary Element acoustic-structural coupling based on a Hierarchical GMRES solver for 3D elastodynamic and acoustic coupled problems. The coupling process of a structure with an internal pulsating cavity is described in details. The technique can be easily extended for problems with a different acoustic-structural interactions. The Adaptive Cross Approximation is utilised to generate both the coupled system matrix and the right hand side vector, whereas the Hierarchical matrix format for the hierarchical partition of the matrix into blocks, for memory storage reduction of approximated blocks and to accelerate the matrix-vector multiplication. Numerical result consists of a spherical structure with a spherical pulsating cavity and it demonstrates the accuracy and efficiency of the proposed strategy. 1.

Introduction

In last three decades researchers spent much attention on accurate evaluation of the structural-acoustic interaction. This phenomenon is caracterised by a continuous exchange of energy between the acoustic field and the elastic structure and thus the structure can be strongly affected when the acoustic and structural impedances are similar. Hence, the study of the acoustic-structural coupling phenomenon becomes mandatory to ensure the safety of the structure. Many researchers focused their efforts in analysing the acoustic-structural interaction and several techniques have been proposed. The most common way is to solve the structural part using the Finite Element Method (FEM) and the acoustic part by the Boundary Element Method (BEM) [1-3]. In fact the FEM is a well-established method to study the mechanical behaviour of a structure and the BEM is a wellrecognised method to study the acoustic propagation phenomenon, especially in open space. The main drawback of this approach consists on the necessity to perform two different discretisations, one for the FEM and the other for the BEM. To overcome this drawback, a few researchers solved the acoustic-structural coupling using a FEMFEM strategy [4-5], where both the structure and the fluid for acoustic propagation are discretised by finite elements. Nevertheless, the BEM is an efficient and accurate technique to solve the elastodynamics problems and thus a BEM-BEM coupling approach to coupling can be suitable for this purpose [6-8], further research on this regards is required. The main advantage of this approach consists on the fact that only a single discretisation is required. In fact, the structure is the same as the sound propagation medium boundary. However, the BEM is well known to generate fully populated and not symmetric system matrix and the solution time grows with quadratic law respect to the degrees of freedom. To overcome this drawback lots of technique has been proposed, such as block-based solvers [9], lumping techniques [10], iterative solvers [11], fast multiple method [12]. The approach proposed in this paper to accelerate the CPU time is the Adaptive Cross Approximation (ACA) [13] in conjunction with the Hierarchical matrix (H-matrix) format and the iterative solver GMRES [14-16]. Both the acoustic and structural system solutions are treated with this strategy resulting in a strong reduction of the solution time. The acoustic and elastodynamic BEM and the relationships for the fluid-solid interface are presented in the next section. Next the coupling process is described in details as well as the used speed-up solution time technique. Finally numerical results are described to demonstrate the accuracy and efficiency of the proposed approach.

90


2.

Boundary Element Method

Acoustic BEM Under the hypothesis of a uniform and irrotational fluid and time-harmonic wave motion, the wave propagation can be studied using the Helmholtz equation expressed in terms of the velocity potential p(X) at the point X

1)

where is the Laplacian operator, is related to the source presence inside the domain and is the wave number, with being the angular frequency and c0 the speed of sound in the medium. By considering a boundary (Γ) of a domain (Ωa), the problem is solved at the boundary in terms of the potential and the flux , i.e. the particle velocity with respect to the normal of the boundary surface at the considered node. The boundary integral equation for Helmholtz problem can be written as [23]

2)

where and are the pressure and flux fundamental solutions, respectively, and depends on the location of the point . The last term refers to the presence of sources within the domain Ω. The integral equation in (2) is discretised into Ne quadratic elements (six nodes triangular element) and Nd nodes and the resulting system of equations can be represented in matrix form as

3)

where and are coefficient matrices corresponding to integrals of the product of the Jacobian with boundary particle velocity and pressure fundamental solutions, respectively, and are the Nd×1 boundary pressure and flux vectors, respectively. Finally, the last integral in the equation (5) produces the vector , created by a number of Np sources within the domain Ω, such as monopoles and plane-waves. Structural BEM The wave equation govering an elastic domain, under the hypothesis of homogeneous, isotropic and linearly elastic behaviour with body force assumed to be negligible, and for time-harmonic problems, can be written as follows

4)

where and are the pressure wave (P-wave) and shear wave (S-wave) velocities, respectively; is the displacement at the point and it does not depends upon Ω. Let’s now consider the same boundary (Γ) of the acoustic domain as the boundary of the structural domain (Ωs). The structural problem is solved in terms of displacement and at the boundary. The boundary integral equation for structural problem can be written as [23]

5)

where and are the displacement and traction fundamental solutions, respectively, and the integral on the left hand side stands for a Cauchy principal value integral. The main advantage of the BEM-BEM acoustic-structural coupling is related to the fact that only a single mesh is required for both problems, hence the same discretisation discussed previously is utilised and the resulting system of equations can be represented in matrix form as

6)

91 where and are coefficient matrices corresponding to integrals of the product of the Jacobian with the boundary displacement and traction fundamental solutions, respectively, and are the boundary 3Nd×1 displacement and traction vectors, respectively. It is important to notice that the pressure and the displacement are unique for each node, whereas the flux and the traction depends upon the element under analysis. Fluid Solid Interface The compatibility and equilibrium conditions at the fluid-solid interface allow us to relate the traction with the acoustic pressure, and the displacement with the acoustic flux as follows

7a) 7b)

where n is the normal on Γ and nT is its transponse. 3.

Acoustic-Structure Coupling In this section, the problem of a structure with an internal pulsating cavity (see figure 1) is presented. The internal pressure is considered to be uniform and time-harmonic. By considering the first relation of the equation (7), the internal pressure generates an internal traction and no acoustic solution for the internal cavity is required. The coupled system of equation can thus be written as follows Fig. 1.Structure with an internal pulsating cavity immersed in a fluid..

8)

8) where is a 3Nd×Nd collecting the components of the normal along the three axes at each node, and the superscript e and i refers to the external and internal problem, respectively. Rearranging all relations in (8), the following matrix notation is obtained

9)

where and are two 3Nd×Nd matrices, and is a 3Nde×Nde matrix, with being a 33Ne×Ne matrix collecting all the components of the normal. 4.

H-format matrix and Adaptive Cross Approximation

One of the main advantages of the proposed method consists on the need of a single mesh on the surfaces of the structure under analysis. On the other side it is well know that the BEM generates a nonsymmetric and fully populated system matrix. Since the coupled system (9) is the results of two systems, for

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the acoustic and for structural problems, it is evident that an effective strategy to speed up the time solution is required for practical applications of the proposed method. Several techniques have been proposed to overcome these difficulties and to accelerate the BEM solution time, such as block-based solvers [9], lumping techniques [10], iterative solvers [11], fast multiple method [12]. The approach proposed in this paper to accelerate the CPU time is the Adaptive Cross Approximation (ACA) in conjunction with the Hierarchical matrix (H-matrix) format and the iterative solver GMRES. The ACA is a technique, firstly introduced by Bebendorf [13], that had received much attention in the last few decades due to the combination of its simplicity of implementation, its accuracy and its high CPU time reduction level. It has been applied in both structural [14] and acoustic [15] fields. The strategy consists on dividing each solving matrix of the systems (9) into two groups of blocks, full-rank and low-rank blocks. The former blocks are calculated entirely, while the latter blocks admit a reppresentation using only a few entries of the original block. The origin of the ACA representation stands on the asymptotic smoothness property of the block kernel that can be expanded by the value of the kernel itself at a few collocation points and field elements. Each low rank block fulfils an admissible criterion and the kernels κ(x’, x), referring to a group of contiguous elements whose separation distance is above a certain level with a group of collocation points, can be approximated by an expansion as follows [16]

10)

where the two functions and are the i-th row and column of the block under analysis and they depend only upon the collocation points and the field points, respectively; is the residuum and tends to zero when tends to the rank of the original block. The separation distance of two groups, the block size and the level of the selected accuracy drive the number of expands needed to represent a low rank block. The ACA takes full advantages when is in conjunction with the H-matrix format. The hierarchical partitioning of the matrix into blocks, the block-wise restriction of approximated blocks and the matrixvector multiplication with almost linear complexity are the three main factors responsible of a significant reduction of the solution time [24]. The first step to apply the ACA approach is the well know cluster tree generation which consists on dividing the whole geometry in order to collect contiguous nodes and elements. The procedure follows the same expansion of a tree which has an initial single root divided into two branches, each of which divided into other two branches and so on. Since a parametric formulation with quadratic geometry and unknowns is adopted in this paper, two cluster tree are required, one for the nodes and the other for the elements. The elements are firstly divided in order to have the same boundary conditions [16]. Hence, the geometry is further split till a minimum number, here called the cardinality, of nodes and elements is reached for each group. The cluster tree is afterward the base to populate the solving matrix through the constitution of a quaternary tree collecting all the matrix blocks and coefficients, namely the block tree. The classification between low rank and full rank blocks is accomplished with the assistance of a geometrical criterion based on the distance dist and diameter diam between a group of elements Ωr and a group of nodes Ωc as follows

11)

where is a fundamental parameter that influences the convergence and the acceleration ratio of the whole procedure. In the present study is equal to 50. Distances and diameters of groups are approximated using the box that contains each group whose corner points are easily evaluated by the maximum and minimum coordinates of the points that constitute the group. 5.

Numerical Results

In this section a simple example to prove the accuracy, efficiency and solution time reduction of the proposed method is presented. The structure consists on a sphere having a diameter equal to 1 m with a spherical cavity of 0,5 m, immersed in a fluid whose speed of sound is 1 m/s. The internal cavity is excited by a normal uniform radial time harmonic pressure which generates a traction equal to 1 N/m2 with 10 rad/sec as angular frequency. The real part of the shear modulus is equal to 106 N/m2, the density is equal to 100 kg/m3 and the Poisson’s

93 ratio equal to 0,25. The internal as well as the external bodies of the sphere are discretised into 382 nodes and 190 elements with quadratic variation of the geometry and the boundary variables as shown in figure 2. Figure 3 shows the deformation of the sphere and its internal cavity using a scaling factor equal to 6 106. Such deformation is uniform and regular along both sides of the boundaries, but in the internal edge is more evident as expected. The external acoustic pressure is also uniform and equal to (-0.3283E-05, 0.2603E-05).

Fig. 2. Discretisation of a sphere with a spherical internal cavity into 764 nodes and 380 quadratic elements.

6.

Fig 3. Deformation of the internal and external sphere.

Conclusion and future works

A fast BEM based approach on the structural acoustic coupling has been presented. The acoustic and the structural BEM has been described as well as the fluid solid interaction at the contact surface. The coupling strategy and the approach based on the ACA in conjunction with H-matrix format and the GMRES solver the has been presented. An example consisting on a sphere with a internal spherical pulsating cavity demonstrated the accuracy of the proposed approach. Comparison with a conventional approach on the level of the speed up rapid achieved by the proposed method will be further conducted and results presented in the near future.

References [1] D.Soares, W.J.Mansur. An Efficient Time-Domain BEM/FEM Coupling for Acoustic-Elastodynamic Interaction Problems. CMES, 8(2), 153-164 (2005). [2] H.Zheng, G.R.Liu, J.S.Tao, K.Y.Lam. FEM/BEM analysis of diesel piston-slap induced ship hull vibration and underwater noise. Applied Acoustics, 62, 341-358 (2001). [3] D.Fritze, S.Marburg, H.J.Hardtke. FEM–BEM-coupling and structural–acoustic sensitivity analysis for shell geometries. Computers and Structures, 83, 143-154 (2005). [4] G.C.Everstine. Finite element formulations of structural acoustics problems. Comput. Struct. 65, 307321 (1997). [5] I.Harari, K.Grosh, T.J.R.Hughes, M.Malhotra, P.M.Pinsky, J.R.Stewart, L.L.Thompson. Recent developments in finite element methods for structural acoustics. Archives of Computational Methods in Engineering, 3, 131-309 (1996). [6] S.H.Chen, Y.J.Liu. A unified boundary element method for the analysis of sound and shell-like structure interactions. I. Formulation and verification. J Acoust Soc Am, 103(3), 1247-1254 (1999). [7] D.Soares. Numerical modelling of acoustic–elastodynamic coupled problems by stabilized boundary element techniques. Comput Mech, 42, 787-802 (2008). [8] M.Tanaka, Y.Masuda. Boundary element method applied to certain structural-acoustic coupling problems. Computer Methods in Applied Mechanics and Engineering, 71, 225-234 (1988). [9] J.M.Crotty. A block equation solver for large unsymmetric matrices arising in the boundary integral equation method. International Journal for Numerical Methods in Engineering; 18(7), pp. 997-1017 (1982). [10] R.H.Rigby, M.H.Aliabadi. Out-of-core solver for large, multi-zone boundary element matrices. International Journal for Numerical Methods in Engineering. 38(9), 1507-1533 (1995). [11] W.J.Mansur, F.C.Araújo, J.E.B. Malaghini. Solution of BEM systems of equations via iterative techniques. International Journal for Numerical Methods in Engineering, 33(9), 1823-1841 (1992). [12] N.A.Gumerov, R.Duraiswami. Fast Multipole Methods for the Helmholtz Equation in Three Dimensions. Oxford, UK: Elsevier, 2005.

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[13] M.Bebendorf, S.Rjasanow. Adaptive low-rank approximation of collocation matrices. Computing 70(1), 1-24 (2003). [14] I.Benedetti, M.H.Aliabadi, G.Daví. A fast 3D dual boundary element method based on hierarchical matrices. International Journal of Solids and Structures 45(7-8), 2355-2376 (2007). [15] A.Brancati, M.H.Aliabadi, I.Benedetti. Hierarchical Adaptive Cross Approximation GMRES Technique for Solution of Acoustic Problems Using the Boundary Element Method. CMES, 43(2), 149-172 (2009). [16] A.Brancati, M.H.Aliabadi, A.Milazzo. An Improved Hierarchical ACA Technique for Sound Absorbent Materials. CMES, 78(1), 1-24 (2011). [17] J.H. Kane JH, Mao S, Everstine GC. A boundary element formulation for acoustic shape sensitivity analysis. Journal of the Acoustical Society, 90(1), 561-573 (1991). [18] R.D.Ciskowski, C.A.Brebbia. Boundary element methods in acoustics. Computational Mechanics Publications Elsevier Applied Science, 1991. [19] K.Guru Prasad, J.H.Kane. Shape reanalysis and sensitivities utilizing preconditioned iterative boundary solvers. Structural Optimization, 4,224-235 (1992). [20] N.Nemitz, M.Bonnet. Topological sensitivity and FMM-accelerated BEM applied to 3D acoustic inverse scattering. Engineering Analysis with Boundary Elements, 32(11), 957-970 (2008). [21] A.Brancati, M.H.Aliabadi. Boundary Element Simulations For Local Active Noise Control Using An Extended Volume. Engineering Analysis with Boundary Elements 36, 190–202 (2012). [22] W.H.Press. Numerical recipes: the art of scientific computing. Cambridge University Press (2007). [23] L.C.Wrobel, M.H.Aliabadi. The Boundary Element Method, UK John Wiley, New Jersey (2002). [24] W.Hackbush. A sparse matrix arithmetic based on H-matrices. Part I: introduction to H-matrices. Computing, 63, 89-108 (1999).

95

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V D!U U!V!"WV "!# F$%&WDW$"D' W!VWV F$"F!U""* W! &V! $+ W! U!F&UV-! VF!%! $+ W! E$&";DU&DW$"@ !V!FD''&DW$" U!D'FDW$" V !>&-D'!"W $+ D +&UW!U U!V;&D'V %"%_DW$" ! W!F">&! #DV W!VW!; !DU'!U V&FF!VV+&''j @!DU!W!F$"VWD"WV$+?D%`{D";j@ !OQUWUD"V+$U%VD%DW!%DWFD'%$;!'+$U%&'DW!;E&DW$"V""W!*UD'!>&DW$"V "-$'-"* $"'&DW$"V W!$U VW!!DU'!U%!"W$"!;F$"F!"WUDW!;'$D;D'!;W$W!F$$U;"DW!;U!FW$"V!"W!U%!;DU&DW$"DU!#!''^"$#"D";W$U$&*'&DW$"V!U!VV!;"D*!"!UD'+$U%#!U!"W!;E22, it is reasonable that higher natural frequencies are obtained for larger value of J 3

104


Based upon the natural frequencies and mode shapes obtained in the free vibration analysis, the forced vibration analysis can be made by employing the modal superposition method on the system of ordinary differential equations (2a). To include the complete spectrum of frequencies, in this example a Heaviside-type load is applied on the upper edge of the plate, and L=100 mm, a=25 mm, T =0o and G=50mm are set for Figure 1. The responses of the crack opening displacements, uA-uB (see Figure 1), obtained from BEM and ANSYS are shown in Figure 3. This figure shows that the periods of crack opening response are 0.55, 0.47 and 0.30 for the cases of J 0o, J 45o and J 0o, respectively, whose corresponding frequencies, 11.42, 13.37 and 20.94, are close to the second natural frequency shown in Table 4. This means that the crack opening displacement is dominated by the natural frequency of tensile mode. By comparing the results of BEM and ANSYS, we see that their amplitudes well agree each other, but their periods have a small difference. This is consistent with the difference shown in Table 4 for natural frequencies. From the amplitudes of crack opening displacements, it can be observed that the most dangerous state occurs on J 0o if only the fiber orientation can be varied. Table 1. Natural frequencies versus crack lengths (L=100 mm, a=0~25 mm, T =0o, G=50mm). Natural frequencies ratio of reduced fre. (%) BEM BEM BEM ANSYS BEM ANSYS BEM ANSYS [(1)-(2)]/(1) [(1)-(3)]/(1) (1) (2) (3) x100% x100% (no crack) (no crack) 0.3 0.3 0.5 0.5 0.3 0.5 2a/L mode 1 10.732ġ 10.725ġ 10.424ġ 10.449 9.896ġ 9.959ġ 2.811 7.768 mode 2 26.458ġ 26.295ġ 23.857ġ 24.027 19.899 20.408 9.893 24.851 mode 3 29.028ġ 28.771ġ 29.072ġ 28.626 28.394 28.111 0.288 2.615 46.219ġ 44.944ġ 44.981 41.626 41.205 4.137 11.217 mode 4 46.791ġ

Mode 2 80

60

60

40

40

20

20

x2 axis

x2 axis

Mode 1 80

0

0

-20

-20

-40

-40

-60

-60 -80

-60

-40

-20

0 20 x1 axis

40

60

80

-80

-60

-40

-20

40

60

40

60

80

Mode 4

80

80

60

60

40

40

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x2 axis

x2 axis

Mode 3

0 20 x1 axis

0

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-20

-20

-40

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-60

-60 -80

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0 20 x1 axis

40

60

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-80

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0 20 x1 axis

80

Figure 2. Vibration mode shapes of a plate with a crack (2a/L=0.5).

4

105

Table 2. Natural frequencies versus crack orientations (L=100 mm, a=25 mm, T =0o~90o, G=50mm). Natural frequencies ratio of reduced fre. (%) no crack 2a/L=0.5 [(1)-(2)]/(1) [(1)-(3)]/(1) [(1)-(4)]/(1) 0 45 90 T[degree] x100% x100% x100% (1) (2) (3) (4) mode 1 10.725ġ 9.896ġ 9.890ġ 9.990ġ 7.768 7.818 6.892 mode 2 26.477ġ 19.899ġ 22.934 26.151 24.851 13.379 1.236 2.615 2.980 2.651 mode 3 29.154ġ 28.394ġ 28.287 28.381 mode 4 46.885ġ 41.626ġ 35.274 30.423 11.217 24.767 35.114 Table 3. Natural frequencies versus crack locations (L=100 mm, a=25 mm, T =0o, G=25~75mm). Natural frequencies ratio of reduced fre. (%) no crack 2a/L=0.5 25 50 75 [(1)-(2)]/(1) [(1)-(3)]/(1) [(1)-(4)]/(1) G [mm] (1) (2) (3) (4) x100% x100% x100% mode 1 10.725ġ 9.695ġ 9.896ġ 10.392ġ 9.629 7.768 3.104 mode 2 26.477ġ 19.440ġ 19.899 22.180ġ 26.582 24.851 16.239 28.507ġ 11.706 2.615 2.214 mode 3 29.154ġ 25.742ġ 28.394 36.989ġ 3.422 11.217 21.112 mode 4 46.885ġ 45.283ġ 41.626 Table 4. Natural frequencies versus fiber orientations (L=100 mm, a=25 mm, T =0o, G=50mm, J: fiber orientation). Natural frequencies BEM ANSYS 0o 45o 90o 0o 45o 90o J mode1 5.849 7.211 7.886 5.848 7.305 7.907 mode2 11.300 13.521 21.049 11.526 13.288 21.713 mode3 16.139 20.514 23.958 15.990 20.669 24.430 mode4 28.824 25.598 28.306 28.154 26.499 28.428

Conclusions By using the anisotropic elastostatic fundamental solutions and employing the dual reciprocity method, a special boundary element was developed in this paper to perform elastodynamic analysis of isotropic/anisotropic elastic plates containing cracks. Since the fundamental solutions used in the present BEM satisfy the traction-free boundary conditions set on the crack surfaces, no meshes are needed along these boundaries, which is convenient for the parametric studies without involving re-mesh of the boundary elements. The effects of crack length, orientation, position, and fiber orientation shown in this paper, are therefore studied using the same element discretization. Moreover, to get an accurate result much fewer elements were used in the present BEM comparing with those in the traditional BEM or finite element method.

5

106


Crack opening displacement (uA-uB) [mm]

30 26

BEM (J =0o) BEM (J =45o) BEM (J =90o) ANSYS (J =90o)

T=0.55

22 18 T=0.47

14 10 T=0.30

6 2 -2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

time [sec]

Figure 3. Dynamic responses of crack opening displacements.

Reference [1] [2] [3] [4] [5] [6] [7]

D. Nardini and C.A. Brebbia A new approach to free vibration analysis using boundary elements, Proceedings 4th International Conference on Boundary Element Methods (1982). Y.C. Chen and C. Hwu Boundary Element Method for Vibration Analysis of Anisotropic Elastic Plates Containing Holes, Cracks or Interfaces, submitted for publication. C. Hwu Anisotropic Elastic Plates. Springer, New York (2010). T.C.T. Ting Anisotropic Elasticity. Oxford University Press, Oxford (1996). M. Kögl and L. Gaul A 3-D boundary element method for dynamic analysis of anisotropic elastic solids, Computer Modeling in Engineering & Sciences, 1, 27-43 (2000). C.A. Brebbia, J.C.F. Telles and L.C. Wrobel Boundary Element Techniques: Theory and Applications in Engineering. Springer-Verlag, Berlin (1984). L. Gaul, M. Kögl and M. Wagner Boundary element methods for engineers and scientists, Springer, New York (2003).

6

107

A Rapid Approach to BEM Quadratic Formulation for Acoustic Problems A. Brancati1,2, M.H. Aliabadi3,4 and A. Milazzo5,6 1

Facoltà di Ingegneria e Architettura, Università di Enna “Kore”, Cittadella Universitaria, 94100 Enna, IT 2 3

[email protected]

Department of Aeronautics, Imperial College London, South Kensington, SW7 2AZ, UK 4

5

[email protected]

Dipartimento di Ingegneria Civile, Ambientale e Aerospaziale, Facolta di Ingegneria, Università degli Studi di Palermo, 90100 Palermo, IT 6

[email protected]

Keywords: Adaptive Cross Approximation/Hierarchical matrix format, large-scale industrial acoustic simulations.

Abstract. This paper presents a novel and fast approach for acoustic simulations using a quadratic Boundary Element formulation. A 3D Boundary Element Method is used to solve the Helmholtz equation. The Adaptive Cross Approximation has been developed in conjunction with the Hierarchical Matrix format and a GMRES solver and they are utilised to generate and store both the solving matrix and the right hand side vector and to evaluate the system solution. The solution of the proposed method is compared with the analytical solution of a simple problem, i.e. a pulsating radiating sphere. The example proved the accuracy of the proposed procedure. The technique is efficient and can be suitable for large-scale industrial problems. 1.

Introduction

The Boundary Element Method (BEM) is an established technique to solve the Helmholtz equation and thus for simulate the acoustic propagation phenomenon [1]. In this contest the BEM is to prefer to the more popular Finite Element Method (FEM) for a number advantages: i) the mesh required is the discretised surface of the problem geometry, ii) the BEM involves the problem variables as well as their derivatives and results are more accurate then other methods, iii) the BEM integral equation automatically satisfies the Sommerfeld radiation condition. Hence, the FEM cannot be directly utilised for unbounded problems and it leads to larger system matrix than the BEM, but symmetric, sparse and banded, that grows almost linearly with the size of the problem. On the contrary, boundary element matrices are non-symmetric and fully populated, and their memory storage requirement is of O(N2), with N being the number of degrees of freedom (d.o.f.). Typically direct solvers require O(N3) operations while iterative solvers O(kN2), where k is the number of iterations. Several techniques have been investigated to overcome this drawback and speed up the BEM solution time including block-based solvers [2], lumping techniques [3], iterative solvers [4] and Fast Multiple Method (FMM) [5-7]. The Adaptive Cross Approximation (ACA) [8-9] is one of the most popular technique able to reduces effectively both the assembly time and the memory storage requirements. Although FMM techniques are efficient for fast solutions of boundary element problems, knowledge of the kernel expansion is required to carry out the integration process and this represents their main drawback; all the terms of the series needed to reach a given accuracy must be computed in advance and then integrated, which can lead to a significant modification of the integration procedures in standard BEM codes. The ACA in conjunction with Hierarchical matrix (H-matrix) format [10] can be easily implemented with iterative solvers, such as the Generalised Minimal Residual Method (GMRES), the most popular iterative solver for non symmetric linear systems, since it strongly speeds up the matrix-vector product. A BEM based code that utilises the ACA, the H-matrix format and the GMRES is very competitive and can strongly speed up the time required for simulations. In this paper a new technique based on the hierarchical ACA and the GMRES solver to solve the Helmholtz equation in a 3D space is presented. It is the progression of a couple of papers already published by the author [8-9] and herein a quadratic parametric

108


BEM formulation is utilised, instead. The procedure to generate the cluster tree based on the provided boundary conditions, and the procedure to generate the matrix solution are presented. The proposed technique has been also formulated to evaluate the potential and particle velocity at selected internal points. Results show the accuracy of the proposed formulation. 2.

Boundary Element Method

The wave equation, under the hypothesis of time-harmonic wave motion and a uniform and irrotational acoustic medium, is transformed into the Helmholtz equation expressed in terms of the velocity potential at the point as follows

1)

where is the Laplacian operator, is related to the source presence inside the domain and is the wave number, with being the angular frequency and the speed of sound in the medium. The acoustic boundary conditions (BCs) can be divided into three groups as follows

2a) 2b) 2c)

where is the flux at the boundary points, , and are three non-intersecting surfaces such that , is a boundary point, is the outward normal to the boundary. , and are three constants depending on the absorbing properties of the materials. In acoustic simulations representing real circumstances, the main quantity to be evaluated is the sound pressure level (SPL) at selected points for a discrete number of frequencies, generated by the action of acoustic sources, such as monopoles or planewaves, which act inside an enclosed space or around a certain object whose surfaces have a predefined value of absorbing coefficient (or impedance). The first two BCs, called Dirichlet (2a) and Neumann (2b), respectively, are quite rare and they represent, under certain conditions, hard, soft and vibrating surfaces. Most common absorbing materials are mathematically described with the third group of BCs, namely mixed Robin conditions. The sound propagation problem is solved by considering a boundary (Γ) of a domain (Ω) and the solution is calculated in terms of the potential and the flux , i.e. the particle velocity with respect to the normal of the boundary surface at the considered node. The boundary integral equation for Helmholtz problem can be written as [1]

3)

where and are the pressure and flux fundamental solutions, respectively, and depends on the location of the point . The last term refers to the presence of sources within the domain Ω. The equation (3) has an exact analytical solution only for a few problems having a simple geometry such as sphere. A more complex geometry requires the evaluation of the solution using a numerical method, thus, in general, the discretisation of the geometry. In this paper the parametric formulation using quadratic elements for both unknown and geometry is adopted. Using the BEM, the solution of the integral equation (3) is evaluated by discretising the geometry into Ne quadratic elements (six nodes triangular element) and Nd nodes and the resulting system of equations can be represented in matrix form as

4)

where and are coefficient matrices corresponding to integrals of the product of the Jacobian with boundary particle velocity and pressure fundamental solutions, respectively, and are the Nd×1 boundary pressure and flux vectors, respectively. Finally, the last term refers to a number of Np sources within the domain Ω, such as monopoles and plane-waves.

109 By applying the BCs, a system of equations is obtained and it can be written as follows

5)

where is a Nd ×1 vector that collect all the unknowns of the problem, is a Nd×Nd matrix composed by the columns of and that correspond to the unknowns, and is a Nd×1 vector evaluated by multiplying the columns of and by the corresponding BCs and it may include the effect of extra sources inside the domain. 3.

H-format matrix and Adaptive Cross Approximation

The BEM is well-established technique to solve acoustic problem, but this method is also well known to generate non-symmetric and fully populated system matrix. In the past several techniques have been proposed to overcome these difficulties and to accelerate the BEM solution time, such as block-based solvers [2], lumping techniques [3], iterative solvers [4], fast multiple method [5-7]. In this paper, the strategy adopted to accelerate the CPU time is the Adaptive Cross Approximation (ACA) in conjunction with the Hierarchical matrix (H-matrix) format and the iterative solver GMRES. The ACA has bee firstly introduced by Bebendorf [12], which had received much attention in the last few decades due to the combination of its simplicity of implementation, its accuracy and its high CPU time reduction level. The level of accuracy is preseted in advance and the strategy is able to reach that level adaptively. The strategy consists on dividing each solving matrix of the systems (5) into two groups of blocks, full-rank and lowrank blocks. The former blocks are calculated entirely and they do not contribute at CPU reduction time, while the latter blocks admit a representation using only a few entries of the original block and are responsible of the system solution reduction. The asymptotic smoothness property of the block kernel is at the origin of the ACA representation and it permits the expansion of the kernel at a few collocation points and field elements. In order to identify a block that has a low rank representation, an admissible criterion has to be fulfilled. The kernels , referring to a group of contiguous elements whose separation distance is above a certain level with a group of collocation points, can be approximated by an expansion as follows [12]

6)

where the two functions and are the i-th row and column of the block under analysis and they depend only upon the collocation points and the field points, respectively; is the residuum and tends to zero when tends to the rank of the original block. The block size, the level of the selected accuracy and the separation distance of two groups, drive the number of expands required to represent a low rank block. When in conjunction with the H-matrix format, the ACA takes full advantages from the hierarchical partitioning of the matrix into blocks, the block-wise restriction of approximated blocks and the matrixvector multiplication with almost linear complexity that are the three main factors responsible of a significant reduction of the solution time [10]. The cluster tree generation is the first step to apply the ACA. It consists on dividing the whole geometry in order to collect contiguous nodes and elements. The procedure follows the same expansion of a tree which has an initial single root divided into two branches, each of which divided into other two branches and so on. Since a parametric formulation with quadratic geometry and unknowns is adopted in this paper, two cluster trees are required, one for the nodes of the geometry and the other one for the elements. The elements are firstly divided in order to have the same boundary conditions [9]. Hence, the geometry is further split till a minimum number, here called the cardinality, of nodes and elements is reached for each group. Each cluster tree is afterward the base to populate the solving matrix through the constitution of a quaternary tree collecting all the matrix blocks and coefficients, namely the block tree. The classification between low rank and full rank blocks is accomplished with the assistance of a geometrical criterion based on the distance and diameter between a group of elements and a group of nodes as follows

7)

110

Eds: A Sellier & M H Aliabadi where is a fundamental parameter that influences the convergence and the acceleration ratio of the whole procedure. In the present study is equal to 50. Distances and diameters of groups are approximated using the box that contains each group whose corner points are easily evaluated by the maximum and minimum coordinates of the points that constitute the group. The main difference with the previous version of the acoustic ACA [8-9], that uses a superparametric formulation with constant unknown and linear geometry, is related to the treatment of the BCs and the creation of the right hand side vector. In the BEM parametric quadratic formulation, a single node can belong to different elements. Hence, in a conventional BEM formulation each coefficient of the solving matrix comes from the contribution of the elements to which the node belongs. The value of the right hand side vector is evaluated by multiplying each row of the matrix with the related BCs. When the ACA is utilised, a single node can belong to different cluster tree node and element groups, thus it is necessary to evaluate in which block the node belongs and to enumerate how many times it is recalled inside each group. The generation of the right hand side vector requires these enumerations to evaluate the contribution in an appropriate manner. 4.

Numerical Results

This section presents a simple example to demonstrate the accuracy and efficiency of the proposed method. The simulation consists on a pulsating sphere having a diameter equal to 1 m, immersed in a fluid whose speed of sound is 1 m/s. The analytical solution in terms of pressure p for the sound radiated by a uniform radiating pulsating sphere with velocity qr at a point with distance d from the centre of the sphere with radius a is given as follows

8)

where is the medium density.

Fig. 2. Discretisation of a pulsating sphere with 382 nodes and 190 quadratic elements.

Wave number k 1 2 3 4 5

5.

To determine the pressure on the surface of the sphere and to make the study as simple as possible, the radius , distance , normal uniform radial vibrating velocity and acoustic impedance are all considered equal to unity. The sphere body is discretised into 382 nodes and 190 elements with quadratic variation of the geometry and the boundary variables as shown in figure 2. Table 1 shows the comparison between the BEM and the analytical solutions for different wave numbers . As expected BEM solutions are in close agreement with the analytical values for all the wave numbers.

Analytical Solution Real Part Imaginary part 0.5000 0.5000 0.8000 0.4000 0.9000 0.3000 0.9412 0.2353 0.9615 0.1923

BEM Solution Real Part Imaginary part 0.5000 0.4999 0.8007 0.3998 0.9014 0.2996 0.9406 0.2364 0.9624 0.1977

Conclusion and future works

An improved and fast approach for acoustic simulations using a quadratic Boundary Element formulation to solve the Helmholtz equation in a 3D environment has been presented in this paper. The proposed procedure utilises the Adaptive Cross Approximation and it takes full advantages since it has been developed in conjunction with the Hierarchical Matrix format and a GMRES solver. The approach is used

111 to generate and store both the solving matrix and the right hand side vector and to evaluate the system solution. An example consisting of a pulsating radiating sphere, whose analytical solution is well known, has been utilised to evaluate the accuracy of the proposed strategy. Further simulations are required to evaluate the CPU time reduction compared with a conventional BEM formulation. References [1] L.C.Wrobel, M.H.Aliabadi. The Boundary Element Method, UK John Wiley, New Jersey (2002). [2] J.M.Crotty. A block equation solver for large unsymmetric matrices arising in the boundary integral equation method. International Journal for Numerical Methods in Engineering; 18(7), pp. 997-1017 (1982). [3] R.H.Rigby, M.H.Aliabadi. Out-of-core solver for large, multi-zone boundary element matrices. International Journal for Numerical Methods in Engineering. 38(9), 1507-1533 (1995). [4] W.J.Mansur, F.C.Araújo, J.E.B. Malaghini. Solution of BEM systems of equations via iterative techniques. International Journal for Numerical Methods in Engineering, 33(9), 1823-1841 (1992). [5] N.A.Gumerov, R.Duraiswami. Fast Multipole Methods for the Helmholtz Equation in Three Dimensions. Oxford, UK: Elsevier, 2005. [6] V.Rokhlin. Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60, 187-207 (1983). [7] W.C.Chew, J.M.Song, T.J.Cui, S.Velamparambil, M.L.Hastriter, B.Hu. Review of Large Scale Computing in Electromagnetics with Fast Integral Equation Solvers. CMES: Computer Modeling in Engineering & Sciences, 5(4), 361-372 (2004). [8] A.Brancati, M.H.Aliabadi, I.Benedetti. Hierarchical Adaptive Cross Approximation GMRES Technique for Solution of Acoustic Problems Using the Boundary Element Method. CMES, 43(2), 149-172 (2009). [9] A.Brancati, M.H.Aliabadi, A.Milazzo. An Improved Hierarchical ACA Technique for Sound Absorbent Materials. CMES, 78(1), 1-24 (2011). [10] W.Hackbush. A sparse matrix arithmetic based on H-matrices. Part I: introduction to H-matrices. Computing, 63, 89-108 (1999). [11] Y.Saad, M.H.Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3), 856-869 (1986). [12] M.Bebendorf, S.Rjasanow. Adaptive low-rank approximation of collocation matrices. Computing 70(1), 1-24 (2003).

112


A quadrature rule for Hadamard finite-part integrals L. S. Campos and E.L. Albuquerque University of Brasilia - Campus Darcy Ribeiro, North Wing - Brasilia, Brazil [email protected]

Abstract. In this paper a simple procedure is presented for computing a quadrature rule for integrals possessing high order singularities, to be interpreted in the Hadamard finite-part sense. On several Boundary Element Method problems it is common to find those types of singularities on the fundamental solution and its derivatives. The Hadamard finite-part integral will be addressed and the proposed quadrature rule will be used to compute its numerical value. A specific rule is created for each node on an element to overcome problems with the change of interval. Some numerical results for the computation of stresses on elastic problems are also provided. Keywords: quadrature rule, singular and hypersingular integrals.

1

Introduction

Numerical integration is one of the most recurrent procedures found in numerical methods. When the integrand is polynomial or at least smooth, the classical Gauss quadrature is found to be extremely efficient and therefore is widely used as a quadrature rule [7]. However, when singular functions are to be integrated, its performance becomes less than satisfactory [2]. On the Boundary Element Method it is usual to find a singular fundamental solution and the closer the integrand is to the source point the more ill-behaved it becomes. When the source point is part of the integration domain the classical Gauss quadrature is not able to correctly predict its value. And even when the source point is not part of the integration domain the Gauss quadrature can become inefficient: a high order rule must be used in order to reach a desired accuracy. This problem occurs because the Gauss quadrature is based on polynomial functions which are not able to interpolate the integrand adequately. The situation is further complicated when the integrand is hypersingular and the integral is not understood in the classical sense but rather in the finite-part sense [1, 3, 5, 6]. In this paper a quadrature rule for Hadamard finite-part integrals is proposed presenting not only a quadrature capable of evaluating singularities but a more efficient rule even when the singularity is not in the domain.

2

Definitions

Equations that involve integrals of the type b f (x) a

(x − c)

dx

(1)

are not integrable in the ordinary sense over any interval that includes the point x = c. It is regularized by the Cauchy principal value integral (CPV), as shown in [4]: $ # c−ε b b f (x) f (x) f (x) p.v. dx := lim dx + dx (2) ε→0 (x − c) a (x − c) a c+ε (x − c) where a < c < b. To cancel out the singularity in the CPV integral, the function f (x) needs to be at least C0 continuous on (a, b) so that around the singular point x = c the integrand is symmetric. Although the CPV integral is defined for a singularity interior to the interval (a, b), it can also be evaluated separately on both sides with the singularity as an end point:

113

p.v.

c f (x) a

p.v.

(x − c)

# ε→0

b f (x) c

c−ε

dx := lim

(x − c)

a

#

b

dx := lim

ε→0

c+ε

$ f (x) dx − f (c)ln(ε) (x − c) $ f (x) dx + f (c)ln(ε) . (x − c)

(3)

For integrals with higher order singularities the CPV integral does not exist[9]. Integrals of the type b a

f (x) dx (x − c)2

(4)

are not only divergent but also the principal value is non-existent, because if the limit presented in equation 2 is applied to this type of integral it would not be finite. The Hadamard finite part integral (HFP) is defined by disregarding the infinite part and keeping the finite part. The HFP can be considered a generalization of the CPV integral since both will yield the same results if the principal value exists. If a step of integration by-parts # $ b f (x) b f (x) f (x) dx = − + dx 2 (x − c) a a (x − c) a (x − c) and a division in the interval are applied b

# # $ c $ b f (x) c f (x) b f (x) f (x) f (x) dx = − + + dx − dx 2 (x − c) a (x − c) c a (x − c) a (x − c) c (x − c) the remaining integrals can be evaluated using Cauchy principal value and the remaining terms can be separated in a finite and an infinite part. Using the Hadamard definition and disregarding the infinite part we obtain: b b f (b) f (x) f (x) f (a) f .p. − + p.v. dx. (5) dx = 2 (a − c) (b − c) a (x − c) a (x − c) b

The function f (x) has to be C1 continuous as a direct consequence of the C0 requirement of the CPV integral.

3

Quadrature Rule

We are now going to replace the HFP integral by a n-order quadrature as follows f .p.

b a

n f (x) dx = ∑ wi f (xi ). 2 (x − c) i=1

(6)

The proposed quadrature is only able to exactly evaluate the integral when f (x) is a polynomial of a degree that is lower to the order of the quadrature. When f (x) is a general function, it will be approximated by a polynomial in the same way that the classical Gauss quadrature approximates the integrand. Good results are expected when f (x) is smooth. To obtain the nodes and weights of the quadrature a non-linear system of equations is proposed in the following form n

∑ wi P0 (xi ) = f .p.

i=1 n

∑ wi P1 (xi ) = f .p.

i=1 n

∑ wi P2 (xi ) = f .p.

b P0 (x) a

(x − c)2

a

(x − c)2

b P1 (x) b P2 (x)

(x − c)2 .. . b n P2n−1 (x) w P (x ) = f .p. . ∑ i 2n−1 i 2 a (x − c) i=1 i=1

a

(7)

114


where Pj (x) are polynomials of degree j. One can also use a quadrature rule with arbitrary nodes that will result on a linear system that is simpler to solve but will need more nodes to have the same order. The results presented in [8] shows surprisingly that these types of quadrature rules will converge, for most cases, including non singular ones, just as fast as Gauss quadrature. The non-linear system can be solved via Newton’s method or the simpler, lower order, linear system, where only the weights are unknowns, can be solved by a linear solver.

4

Results

*

*

1 sin(x) 1 sin(x) In order to prove the efficiency of the proposed quadrature, the regular integrals −1 x dx and .5 x2 dx will be evaluated using the classical Gauss quadrature and the proposed one. Notice that both integrals are not singular since for the first one the function sin(x) cancels out the singularity at x = 0 and the interval of the second integral does not contain the singularity at x = 0. With a small number of nodes the proposed quadrature achieves machine precision as can be seen in fig.1.

∫ sin(x)/(x) dx

0

10

error

proposed quadrature gaussian quadrature −10

10

−20

10

2

4

6

8

10 12 number of nodes

14

16

18

20

2

∫ sin(x)/(x) dx

0

10

error

proposed quadrature gaussian quadrature −10

10

−20

10

2

4

6

8


Figure 1: Error in the evaluation of

14

* 1 sin(x) −1

x

16

dx and

18

* 1 sin(x) .5

x2

20

dx

*

1 sin(x) dx is evaluated by only the proposed quadrature as the Gauss quadraThe singular integral −.5 x2 ture is not able to correctly evaluate this type of integral. This quadrature was also applied in the computation of stresses on elastic BEM code. The boundary integrals that are used to calculate the stresses at the boundary are hypersingular. Theses integrals were actually easier to evaluate considering that the f (x), in this problem, is polynomial and the quadrature can exactly evaluate this type of integrals. The problem analysed is a unitary squared body under unitary load with unitary Young’s modulus. Four discontinuous quadratic elements are used to discretise the problem. The results obtained are equivalent to the results using analytical integration, as was already expected. The maximum error in σx at the boundary was 0.09% as can be seen in table 1.

5

Conclusion

This paper presented an alternative quadrature to deal with integrals involving singularities. The resulting quadrature proved itself as a reliable and efficient integrating method, requiring a small number of nodes to correctly evaluate the singular integrals in the Hadamard finite part sense and having a better

115

∫ sin(x)/(x)2 dx

0

10

proposed quadrature −2

10

−4

10

−6

error

10

−8

10

−10

10

−12

10

−14

10

−16

10

2

4

6

8


14

Figure 2: Error in the evaluation of

16

* 1 sin(x) −.5

x2

18

20

dx

Table 1: Stresses on the boundary of a squared body under compression node 1 2 3 4 5 6 7 8 9 10 11 12 expected value

σx -1,0009E+00 -1,0000E+00 -1,0009E+00 -1,0006E+00 -1,0000E+00 -1,0006E+00 -1,0009E+00 -1,0000E+00 -1,0009E+00 -1,0006E+00 -1,0000E+00 -1,0006E+00 -1

σy 6,5064E-04 -1,7124E-05 6,5064E-04 7,4192E-04 4,7512E-06 7,4192E-04 6,5064E-04 -1,7124E-05 6,5064E-04 7,4192E-04 4,7512E-06 7,4192E-04 0

σxy 4,6409E-04 2,9946E-16 -4,6409E-04 -4,5318E-04 7,1553E-17 4,5318E-04 4,6409E-04 -3,6753E-17 -4,6409E-04 -4,5318E-04 -3,1433E-16 4,5318E-04 0

116


performance, when compared to the classical Gauss quadrature, even for integrals where the singularity is outside the domain.

References [1] K Atkinson and Ezio Venturino. Numerical evaluation of line integrals. SIAM journal on numerical analysis, 30(3):882–888, 1993. [2] Youn-Sha Chan, Albert C Fannjiang, Glaucio H Paulino, and Bao-Feng Feng. Finite part integrals and hypersingular kernels. Advances in Dynamical Systems, 14:264–269, 2007. [3] LJ Gray. Evaluation of hypersingular integrals in the boundary element method. Mathematical and Computer Modelling, 15(3):165–174, 1991. [4] Jacques Hadamard. Lectures on Cauchy’s problem: In linear partial differential equations. Courier Dover Publications, 2003. [5] P Kolm and V Rokhlin. Numerical quadratures for singular and hypersingular integrals. Computers & Mathematics with Applications, 41(3):327–352, 2001. [6] DF Paget. The numerical evaluation of hadamard finite-part integrals. Numerische Mathematik, 36(4):447–453, 1981. [7] A. H. Stroud and D. Secrest. Gaussian Quadrature Formulas. Prentice Hall, new jersey edition, 1966. [8] L. N. Trefethen. Is gauss quadrature better than clenshaw-curtis? Society for Industrial and Applied Mathematics, 50:67–87, 2008. [9] E. Venturino. On the numerical calculation of hadamard finite-part integrals. volume 53, pages 277–292, 1998.

117

Series Expansion of Anisotropic Plane Elasticity Fundamental Solutions Adriana dos Reis, Eder Lima Albuquerque, and Carla Tatiana Mota Anflor University of Bras´ılia - UnB Bras´ılia, DF, Brazil [email protected] [email protected] anfl[email protected]

Keywords: Fast Multipole Boundary Element Method, Boundary Element Method, Anisotropic Plates.

Abstract. In this work we presented the expansion of anisotropic plane elasticity fundamental solutions into Taylor series. These expansions will be used in a formulation of the Fast Multipole Boundary Element Method for solving large scale composite material problems. Fundamental solutions of plane elasticity are represented by complex functions from the classical 2D elasticity theory. The convergence of the series expansion to the fundamental solutions is analyzed considering different numbers of series terms and different distance from source point to field point. The series expansion will be used to evaluate the integrals in the elements that are far away from the source point, whereas the conventional approach will be applied to evaluate the integrals on the remaining elements that are closer to the source point.

1

Introduction

The efficiency of boundary element method (BEM) has been a serious problem with respect to the analysis of large-scale models. Proposed by [1] and [2] in the 1980s, the fast multipole method (FMM) can be used to accelerate the solution of BEM by several fold. Initially applied to physical or gravitational electrostatic potential governed by the Laplace equation in two or three dimensions, [3] extended the method to the two dimensional elastostatic problem. In this last work, formulations of the FMM are given for the Dirichlet, Neumann and mixed boundary value problems, including the evaluation of internal stresses. The method requires O(NlogN) work and memory. The main idea of the FMM is to translate the node-to-node (or element-to-element) interactions to cell-to-cell interactions, where cells can have a hierarchical (tree) structure with the smallest cells (leaves) containing a specified number of elements. The grouping of elements into cells and expansion of fundamental solutions in series cause reduction of the computational cost. Furthermore, if an iterative method is used for solving linear systems, generally GMRES - Generalized Minimal Residual Method, the storage of the coefficient matrix in the computer memory can be reduced. For 2D problems of elasticity, there are several applications FMM. For example, [4] and [5] developed a FMM formulation based on two-dimensional elasticity biharmonic equations. They applied the formulation of complex variable Sherman to solve the biharmonic equation and made applications in various large-scale problems. [6] developed a spectral multipole method (SMM) which has common characteristics with FMM. In this approach, a grid is generated and the Taylor series expansion of the integral is made on grid points. This approach is of complexity O(NLogN). [7] proposed a similar spectral method using the equations of displacement and traction in the regularized form. The representation in terms of complex variables and integral expansion in multipoles were originally proposed for two-dimensional problems by [8] and [9]. [10] showed a formulation in which the moments for two-dimensional elasticity are written in a compact form

118


with all symmetrical translations, which improve the efficiency of the method. [11] also studied fracture mechanics problems using the boundary element method with dual expansion in multipoles. In the work [12], the authors studied two-dimensional problems with multiple domains using only the displacement integral equation. To the best of authors knowledge, the FMM still hasn’t been applied to anisotropic elasticity problems. The purpose of this work is the expansion of anisotropic plane elasticity fundamental solutions into Taylor series. In a near future, these expansions will be used in a formulation of the Fast Multipole Boundary Element Method for solving large scale composite material problems. The convergence of the series expansion to the fundamental solutions is analyzed considering different numbers of series terms and different distance from source point to field point.

2

Fundamental solutions to anisotropic materials

Fundamental solutions for anisotropic materials in plane elastic problems, Ui j (x,y) and Ti j (x,y), can be represented in complex variables by (see [13]): Ui j (z , z) = 2Re qi1 A j1 log(z1 − z1 ) + qi2 A j2 log(z2 − z2 ) ,

(1)

and Ti j (z , z) = 2Re

1 1 gi1 (μ1 n1 − n2 )A j1 + gi2 (μ2 n1 − n2 )A j2 , (z1 − z1 ) (z2 − z2 )

(2)

where + zk

= +

zk =

z1 z2 z1 z2

,

+ =

,

+ =

x1 + μ1 x2 x1 + μ2 x2 x1 + μ1 x2 x1 + μ2 x2

, (3) , (4)

where μk are roots of the characteristic polynomial: a11 μ 4 − 2a16 μ 3 + (2a12 + a66 )μ 2 − 2a26 μ + a22 = 0,

x1 + μk x2

, (x1 , x2 ) are a field point coordinates, zk = x1 + μk x2 , zk = qi j , Ai j , gi j are material complex constants. Writing:

(x1 , x2 )

(5)

are source point coordinates,

G(zk , zk ) = log(zk − zk )

(6)

and

G (zk , zk ) =

∂ G(zk , zk ) 1 = ∂z (zk − zk )

(7)

we have: Ui j (z , z) = 2Re qi1 A j1 G(z1 , z1 ) + qi2 A j2 G(z2 , z2 ) , and

(8)

119

S0 n z zc r zc zL zL

z0

Figure 1: Complex notation and the related points for fast multipole expansions. Ti j (z , z) = 2Re G (z1 , z1 )gi1 (μ1 n1 − n2 )A j1 + G (z2 , z2 )gi2 (μ2 n1 − n2 )A j2 , .

3

(9)

Multipole Expansions

The key point in the FMM is the expansion of the fundamental solutions Ui j (x,y) and Ti j (x,y) around an expansion point zck (see Figure 1), where ,

+ zck =

zc1 zc2

+ =

xc1 + μ1 xc2 xc1 + μ2 xc2

, (10)

xck are coordinates of the expansion point that is near xk and far from x k . In order to derive the multipole expansion, let’s first rewrite G(zk , zk ) as:

G(zk , zk ) = log(zk − zck − zk + zck )

(11)

Multiplying and dividing by (zck − zk ), we have: + G(zk , zk ) = log

, + , (zck − zk ) (zk − zck − z + zck ) k − z − z + z ) = log (z − z ) (z . ck ck ck k k k (zck − zk ) (zck − zk )

(12)

Using the logarithm properties, we can write: +

G(zk , zk ) or

zc − zk (zk − zck ) = log(zck − zk ) + log + k (zck − zk ) (zck − zk )

, (13)

120


zc − zk G(zk , zk ) = log(zck − zk ) + log 1 − k (zck − zk )

(14)

It is worth noting here that the first term of the right hand side of equation (14) is not function of the field point zk . Calling # $ zck − zk ξ= , (15) zck − zk we can write:

G(zk , zk ) = log(zck − zk ) + log [1 − ξ ] .

(16)

Note that, when zk → zck , ξ → 0. Thus, the expansion of log(1 − ξ ) around ξ = 0, given by: log(1 − ξ ) = −

∞

ξm , m=1 m

∑

for |ξ | < 1,

(17)

zc −z is the same that the expansion of log 1 − k k around zck . zck −zk

So, from (17) and (15) we have: # log(1 − ξ ) = −

∞

zck −zck

$m

zck −zk

∑

m=1

for |zck − zk | < |zck − zk |

,

m

(18)

Thus, log(1 − ξ ) = −

∞

∑

(m − 1)! (zck − zk )m 'm m! zck − zk

&

m=1

(19)

From (16) and (19) we have:

G(zk , zk ) = log(zck − zk ) +

∞

∑

m=1

&

(m − 1)! (zck − zk )m 'm m! zck − zk

(20)

that can be written as: G(zk , zk ) =

∞

∑ Om (zk − zk )Im (zk − zk ) c

c

(21)

m=0

where Om (z) =

(m − 1)! , for m 1, zm

O0 (z) = − log(z),

(22) (23)

and Im (z) =

zm , for m 0. m!

(24)

121

Following similar procedure, we can show that (see [14]):

G (zk , zk ) =

∞ ∂ G(zk , zk ) 1 = = ∑ Om (zk − zkc )Im−1 (zk − zkc ). ∂z (zk − zk ) m=1

(25)

Substituting (21) and (25) into (8) and (9) we obtain the multipole expansion of the anisotropic fundamental solutions for plane elastic problems. As can be seen from equations (8) and (9), the multipole expansion of anisotropic material fundamental solutions is given by the expansion around two complex coordinate points zc1 and zc2 . This is the main difference if we compared with the expansion of isotropic material fundamental solutions where the expansion is around just one point.

4

Numerical Results

In order to assess the convergence of the expansion of fundamental solutions to the analytical results, consider a four layer composite material with the following material properties: E1 = 2, 2 × 106 Pa, E2 = 4.4 × 106 Pa, G12 = 0, 7692 × 106 Pa and ν12 = 0, 4286, with stacking sequence [45o / − 45o ]S . Figures 2, 3, and 4 show the convergence of the expansion of displacement fundamental solutions given by equation (8) and Figures 5, 6, and 7 show the convergence of the expansion of traction fundamental solutions given by equation (9). As it can be seen, there is a good convergence of all expansion to the analytical solution as the number of terms increases in the series. 0.35

0.0322 *

*

Expansão com 1 termo Expansão com 6 termos Expansão com 11 termos Expansão com 16 termos

0.25

*

Solução fundamental u21= u12

Solução fundamental u11

0.3


0.0322

0.0322

0.0322

*

*

u11

*

u21= u12

0.2 0.15 0.1 0.0322 0.05 0.0322 0 −0.05

0

0.2

0.4

0.6 r

Figure 2: U11

5

0.8

1

0.0322

0

0.2

0.4

0.6

0.8

1

r

Figure 3: U12 = U21

Conclusions

This paper presented the expansion of anisotropic plane elasticity fundamental solutions into Taylor series. Fundamental solutions of plane elasticity were represented by complex functions from the classical 2D elasticity theory. The convergence of the series expansion to the analytical fundamental solutions was analyzed considering different numbers of series terms and different distance from source point to field point. All expansions converge to the analytical solution. The main difference of the expansion of the anisotropic fundamental solutions in Taylor series when compared to isotropic fundamental solution is that in the anisotropic case the expansion demands two complex coordinate points of expansion while in the isotropic case the expansion is around one single complex coordinate point.

122


0.5

4 Solução fundamental t*

*

Solução fundamental u22

0.45

11

3.5


0.4 0.35

3 2.5

0.25

t*

*

11

0.3 u22


0.2

2 1.5

0.15 1 0.1 0.5

0.05 0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

0.6

r

0.8

1

0.8

1

r

Figure 4: U22

Figure 5: T11

1.6

1.4 Solução fundamental t*

Solução fundamental t*

22

12

1.4 1.2


1.2


1

1 22

t*

t*

12

0.8 0.8

0.6 0.6 0.4

0.4

0.2

0.2 0

0

0.2

0.4

0.6 r

Figure 6: T12

6

0.8

1

0

0

0.2

0.4

0.6 r

Figure 7: T22

Acknowledgments

The author are grateful to the National Council for Scientific and Technological Development (CNPq) for the financial support of this work.

123

7

References

[1]

Rokhlin, V., ”Rapid solution of integral equations of classical potential theory”. Journal of Computational Physics, vol.60, 187-207, 1985.

[2]

Greengard, L.F.,”A fast algorithm for particle simulations”. Journal of Computational Physics, vol.73, 325-348, 1987.

[3]

Yamada, Y., Hayami, K., ”A multipole boundary element method for two dimensional elastostatics”. Report METR 95-07, Department of Mathematical Engineering and Information Physics, University of Tokyo, 1995.

[4]

Greengard, L.F., Helsing, J., ”On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites”. Journal of the Mechanics and Physics of Solids v.46, 1441-1462, 1998.

[5]

Greengard, L.F., Kropinski, M.C. and Mayo, A., ”Integral equation methods for Stokes flow and isotropic elasticity in the plane”. Journal of Computational Physics vol.125, 403-414, 1996.

[6]

Peirce, A.P., Napier, J.A.L., ” A spectral multipole method for efficient solution of large-scale boundary element models in elastostatics”. International Journal for Numerical Methods in Engineering v.38, 4009-4034, 1995.

[7]

Richardson, J.D, Gray, L.J., Kaplan, T. and Napier, J.A., ”Regularized spectral multipole BEM for plane elasticity”. Engineering Analysis with Boundary Elements , vol.25, 297-311, 2001.

[8]

Liu, Y.J., Nishimura N., ”The fast multipole boundary element method for potential problems: a tutorial”. Engineering Analysis with Boundary Elements v.30, 371-381, 2006.

[9]

Greengard, L.F., ”The rapid evaluation of potential fields in particle systems”. MIT Press, Cambridge, 1988.

[10]

Liu, Y.J.., ”A new fast multipole boundary element method for solving large-scale twodimensional elastostatic problems”.International Journal for Numerical Methods in Engineering, vol.65, 863-881, 2005.

[11]

Wang, P., Yao, Z., ”Fast multipole DBEM analysis of fatigue crack growth”. Computational Mechanics, vol.38, 223-233, 2006.

[12]

Yao Z., Kong F., Wang H., Wang P., ”2D simulation of composite materials using BEM”. Engineering Analysis with Boundary Elements, vol. 28, 927-935, 2009.

[13]

Lekhnitskii, S. G. Anisotropic plates. New York: Gordon and Breach, 1968.

[14]

Liu, Y.J., ”Fast Multipole Boundary Element Method: Theory and Applications in Engineering”. Cambridge University Press, Cambridge , 2009.

124


A Boundary Element Method Formulation Applied to the Dynamic Analysis of Timoshenko Beams J. A. M. Carrer1, S. A. Fleischfresser2, L. F. T. Garcia3, W. J. Mansur4 1,2

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] 3,4

Programa de Engenharia Civil, COPPE/UFRJ, Universidade Federal do Rio de Janeiro, Caixa Postal 68506, CEP 21945-970, Rio de Janeiro, Brasil 3 email: [email protected] 4 email: [email protected] Keywords: Timoshenko beams, dynamics, D-BEM formulation

Abstract. A Boundary Element Method formulation of the type Domain Boundary Element Method is developed for the dynamic analysis of Timoshenko beams. Beside the typical domain integrals containing the second order time derivatives of the transverse displacement and of the rotation of the cross-section due to bending, additional domain integrals appear: one due to the load and the other two due to the coupled differential equations that govern the problem. The time-marching employs the Houbolt method. A pinnedpinned beam under uniformly distributed, concentrated, harmonic concentrated and impulsive load is analysed. The BEM results are compared with the corresponding analytical solutions. Introduction The Timoshenko theory of beams, e.g. Timoshenko [1, 2], Graff [3], Rao [4], takes into account the shear deformation and rotatory inertia, generating an improved theory that gives more reliable results than the classical one, especially for higher frequencies. For this reason, a great deal of attention was given to the development of Boundary Element Method (BEM) formulations based in this theory. For static analysis, the reader is referred to Antes [5]. Dynamic analysis of Timoshenko beams is carried out in the work by Antes et al. [6], in which the convolution quadrature method is employed to perform the convolution in the time dependent integral equation. The present work is concerned with the presentation of a D-BEM formulation for the vibration analysis of Timoshenko beams (those formulations that employ static fundamental solutions, instead of the timedependent ones, and that present, in the BEM integral equations, a domain integral whose integrand is the product of the fundamental solution by the second order time derivative of the basic variable, are called DBEM, D meaning domain). It is important to bear in mind that the problem is described by two coupled differential equations: the first equation has, as the main variable, the transverse displacement, u, and the second, the rotation of the cross-section due to bending, ψ. In order to obtain the D-BEM formulation, initially the differential equations are treated separately and, for each one, a specific fundamental solution is adopted. After following standard BEM procedures, the BEM equation associated to the first differential equation exhibits three domain integrals: one related to the load, as expected, another one that contains the second order time derivative of the transverse displacement, typical from the D-BEM formulations, and a third one, that contains the rotation ψ. This last domain integral is due to the coupling of the differential equations. The BEM equation associated to the second differential equation presents one domain integral that contains the second order time derivative of the rotation and another one that contains the transverse displacement u. The explanation is the same: the first domain integral is characteristic from the D-BEM formulation and the second one comes from the coupled differential equations that describe the problem. It is important to mention that domain integrals involving the coupled variables will appear even in the development of a time-domain BEM formulation, beside the one related to the load. The problem is essentially one-dimensional: the domain is represented by the segment [0,L], if L is the length of the beam, and the boundary is constituted by the nodes at x = 0 and x = L. The domain integrals require the domain

125 discretization, which is accomplished with quadratic cells. In the formulation developed here, the second order time derivative of the basic variables, i.e., of the variables u and ψ, was approximated by the Houbolt method [7]. A uniform pinned-pinned beam, that is, a beam with constant cross-section and constant material properties is taken into account. Four cases of the load are considered: i) uniformly distributed along the length of the beam; ii) concentrated; iii) harmonic concentrated and iv) impulsive. All the numerical results are compared with the corresponding analytical solutions. Good agreement is observed between them, demonstrating the applicability of the present D-BEM formulation. The D-BEM Formulation for the Timoshenko Beam Theory In the Timoshenko theory of beams, e.g. Timoshenko [1, 2], Graff [3], Rao [4], the effect of the shear deformation and rotatory inertia are taken into account, generating an improved theory. Plane cross-sections remain plane but not necessarily perpendicular to the neutral axis after bending. For the system of coordinates described in Fig. 1, the problem is governed by the coupled differential equations written below:

§∂ u ∂ψ· ∂2u κGA ¨ 2 − ¸ = − q(x,t) + ρA 2 ∂t ©∂x ∂x ¹ 2

and

∂2ψ ∂2ψ §∂u · κGA ¨ − ψ¸ + EI 2 = ρI 2 ∂x ∂t ©∂x ¹

(1)

Essential (eqs. (2)) and natural (eqs. (3)) boundary conditions are given by: − u(x,t) = u(x,t): transverse displacement

and

− ψ(x,t) = ψ(x,t): rotation due to bending

∂ψ(x,t) − = M(x,t): bending moment ∂x

and

∂u(x,t) − κGA §¨ − ψ(x,t)·¸ = Q(x,t): shear force (3) © ∂x ¹

− EI

(2)

In eqs. (1), A is the area and I is the moment of inertia of the cress-section; ρ is the mass density and E and G are, respectively, the Young’s and the shear moduli. The shear coefficient, κ, is an adjusting coefficient used to compensate the error introduced when the shear stresses are assumed to be functions only of the variable x, that is, when not taking into account their variation along the cross-section, see for instance, Borges [8], Cowper [9]. The load, q(x,t), is represented as a function of space and time.

Figure 1. Timoshenko beam: definition of the system of coordinates. The BEM equation that corresponds to the first eq. (1) is written as:

ª∂u*(ξ,x) u(x,t)º ª∂u*(ξ,x) u(x,t)º −« − » » ¬ ∂x ¼ x=L ¬ ∂x ¼ x=0

u(ξ,t) = «

Q(x,t)º ªu*(ξ,x) Q(x,t)º + ªu*(ξ,x) − κGA ¼ x=L ¬ κGA ¼ x=0 ¬ 1 ´ κGA µ

L

¶0

u*(ξ,x) q(x,t) dx +

ρ ´ κG µ

L

¶0

The fundamental solution is given by:

´ .. u*(ξ,x) u(x,t) dx − µ

L

¶0

∂u*(ξ,x) ψ(x,t) dx ∂x

(4)

126


u*(ξ,x) =

» x − ξ» 2

(5)

The BEM equation that corresponds to the second eq. (1) is written as:

ª∂ψ*(ξ,x) ψ(x,t)º ª∂ψ*(ξ,x) ψ(x,t)º −« + » » ¬ ∂x ¼ x=L ¬ ∂x ¼ x=0

ψ(ξ,t) = «

M(x,t)º ªψ*(ξ,x) M(x,t)º ª * EI ¼ x=L − ¬ψ (ξ,x) EI ¼ x=0 − ¬ β [ψ*(ξ,x) u(x,t)] x=L + β [ψ*(ξ,x) u(x,t)] x=0 + ρ´ Eµ

L

¶0

´ .. ψ*(ξ,x) ψ(x,t) dx + β µ

L

¶0

∂ψ*(ξ,x) u(x,t) dx ∂x

(6)

The fundamental solution, now, is given by: ψ*(ξ,x) =

sinh β» x − ξ» 2 β

κGA β = EI

with

(7)

In order to solve the problem, eqs.(4) and (6) are applied to the boundary nodes, located at x = 0 and at x = L, and to the internal points, generating an enlarged system of equations. The domain discretization is carried out with quadratic cells. For additional details, the reader is referred to Fleischfresser [10]. The Analytical Solution for the Pinned-Pinned Beam The analytical solutions for the pinned-pinned beam are assumed to be of the form: ∞

¦

u(x,t) =

∞

Um(t) sin§

mπx· © L ¹

m = 1;3;5

and

¦Q

ψ(x,t) =

m = 1;3;5

§mπx· L ¹

m(t) cos©

(8)

A general expression for the time dependent variable Um(t) is given by: ° .. °½ ª mπ 2 º Um(t) = ®ρIQm(t) + «EI § L · + κGA» Qm(t)¾

¬

¯°

©

¹

¼

¿°

1

(9)

κGA§

mπ· ©L¹

.. The expressions for Qm(t) and Qm(t) depend on the type of the load. For q(x,t) = q, one has: 3

Qm(t) =

4qL

ªθ2 cosδ t − δ2 cosθ t + (δ2 − θ2 )º m m m m ¼ m m

(10)

EI (mπ) (δm − θm) ¬ 4

2

2

The natural frequencies θm and δm are given by: θm =

αm − βm

and

δm =

αm + βm

(11)

where: ρA + ρI §1 + αm =

©

E · §mπ·2 κG¹ © L ¹

§ρ I· 2¨ ¸ ©κG¹ 2

(ρA) + 2 (ρA)(ρI) §1 + 2

and βm =

©

E · §mπ·2 E 2 mπ 4 2 + (ρI) §1 − · § · κG¹ © L ¹ © κG¹ © L ¹

§ρ2I· ¸ ©κG¹

2¨

(12)

127 The smaller frequency, θm, corresponds to the bending deformation mode and the larger one, δm, corresponds to the shear deformation mode, Rao [4]. For a concentrated load of intensity P applied at x = x0, q(x,t) = P δ(x − x0), one has: 2

Qm(t) =

2PL

ªθ2 cosδ t − δ2 cosθ t + (δ2 − θ2 )º sin§mπx0· m m m m ¼ m m © L ¹

EI (mπ) (δm − θm) ¬ 3

2

2

(13)

For a harmonic concentrated load of the type q(x,t) = P δ(x − x0) sinωpt, one has:

Qm(t) =

ª sinθ t ½° sinδmt º» sinωpt 2P mπ ° ωp mπx0· m « ® 2 ¾ sin§ 2 2 2 « 2 2 − 2 2 »+ 2 2 2 2 L ¹ © L § ρ I· ¯ (δ − θ ) θ (ω − θ ) δm (ωp − δm)¼ (ωp − θm)(ωp − δm) ¿ ° ¨ ¸° m m ¬ m p m ©κG¹

(14)

When ωp = θm or ωp = δm, resonance occurs. When ωp → δm, the following expression arises:

§mπx0· ª sinθmt δmt cosδmt − sinδmtº» sin© L ¹ 2P mπ « δm Qm(t) = 2 2 « 2 + 2 » (δ2 − θ2 ) L §ρ I· ¬(δm − θ2m) θm 2δm ¼ m m ¨ ¸ ©κG¹

(15)

When ωp → θm, the resulting expression is:

§mπx0· sinδmt sinθmt − θmt cosθmtº sin© L ¹ 2P mπ ª θm + Qm(t) = 2 2 « 2 » 2 2 2 L §ρ I· ¬(δm − θ2m) δm 2θm ¼ (δm − θm) ¨ ¸ ©κG¹

(16)

For the case of an impulsive load applied at x = x0 at t = 0, q(x,t) = P δ(x − x0) δ(t − 0), one has: mπx0· sin§ 2P mπ §sinθmt sinδmt· © L ¹ − Qm(t) = 2 2 ¨ ¸ 2 2 L §ρ I· © θm δm ¹ (δm − θm) ¨ ¸ ©κG¹

(17)

Numerical examples The following data were adopted: E = 50 GPa, ρ = 2500 kg/m3 and ν = 0.2 (concrete). The rectangular crosssection of height h = 0.6 m and width b = 0.2 m has I = 0.0036 m4 and A = 0.12 m2. The beam length is L = 4 M. The shear coefficient, for the rectangular cross-section, is given by, see Borges [8]: κ = 5/6. The loads are given by: q = 100.000 N/m (distributed), P = 1.000.000 N (concentrated) and P = 1000 Ns (impulsive). The domain discretization employed 8 cells of the same length and Δt = 0.000040 s, except for the case of the harmonic load with ωp = δ1 and for the case of the impulsive load: in these cases, the domain discretization required 64 cells and Δt = 0.000001 s. In what follows, BEM and analytical solutions related to the displacement at x = L/2 will be presented. See Fig. 2a for the case of the distributed load, Fig. 2b for the case of the concentrated load and Fig. 3a for the case of the concentrated harmonic load with frequency ωp = 1000 Hz. For the chosen data, the first natural frequencies are: θ1 = 461.73 Hz and δ1 = 15744.49 Hz. Results corresponding to ωp = θ1 are presented in Fig. 3b. For ωp = δ1, the BEM and the analytical results are plotted side-by-side, see Figs. 4a and 4b, due to the large amount of data. The same procedure was followed for the case of the impulsive load, see Figs. 5a and 5b. The overall conclusion is that the BEM results are very good, showing great accuracy when compared with the corresponding analytical solutions.

128


u

u

0.0040

0.0160

0.0032 0.0120 0.0024 0.0080 0.0016 0.0040 0.0008 0.0000

0.0000

analytical BEM

analytical BEM -0.0008 0.00

0.01

0.02

0.03

0.04

0.05 t

-0.0040 0.00

0.01

0.02

0.03

0.04

0.05 t

(a) (b) Figure 2. Displacements corresponding to distributed (a) and concentrated load (b). u

u

0.0800 0.0060

0.0400

0.0030

0.0000

0.0000

-0.0400

-0.0030

analytical BEM

analytical BEM -0.0060 0.00

0.01

0.02

0.03

0.04

0.05 t

-0.0800 0.00

0.01

0.02

0.03

0.04

0.05 t

(b) (a) Figure 3. Displacements corresponding to harmonic load with ωp = 1000 Hz (a) and with ωp = 461.73 Hz (b). u

u

0.00050

0.00050

0.00025

0.00025

0.00000

0.00000

-0.00025

-0.00025

-0.00050 0.00

0.01

0.02

0.03

0.04

0.05 t

-0.00050 0.00

0.01

0.02

0.03

0.04

0.05 t

(b) (a) Figure 4. Displacements corresponding to harmonic load with ωp = 15744.49 Hz: BEM (a) and analytical (b).

129

u

u

0.0050

0.0050

0.0025

0.0025

0.0000

0.0000

-0.0025

-0.0025

-0.0050 0.00

0.01

0.02

0.03

0.04

0.05 t

-0.0050 0.00

0.01

0.02

0.03

0.04

0.05 t

(a) (b) Figure 5. Displacements corresponding to impulsive load: BEM (a) and analytical (b). Conclusions This work is concerned with the dynamic analysis of Timoshenko beams. From the coupled differential equations that govern the problem, two BEM integral equations were obtained. For each one, a proper static fundamental solution was employed. As the fundamental solutions are not time-dependent ones, each integral equation presents a domain integral related to the second order time derivative of the basic variable, that is, the transverse displacement in the first equation and the rotation due to bending in the second equation. However, other domain integrals also appear in the BEM equations: one is due to the load term, and comes from the first differential equation; the other two are due to the coupling of the differential equations. The Houbolt method was chosen to approximate the second order time derivatives of the transverse displacement and of the rotation due to bending, that is, to perform the march in the time. For the chosen pinned-pinned beam, the BEM results were compared with the corresponding analytical solutions and a good agreement could be observed between them. This encourages the development of other formulations: the authors’ opinion is that the development of a time-dependent formulation is a task that must deserve attention. References [1] S.P.Timoshenko Philosophical Magazine, 41, 744-746 (1921). [2] S.P.Timoshenko Philosophical Magazine, 43, 125-131 (1922). [3] K.F.Graff Wave Motion in Elastic Solids, Dover Publications, Inc., New York (1991). [4] S.S.Rao Mechanical Vibrations, Addison-Wesley Publishing Company, Inc., 3rd edition (1995). [5] H.Antes Computers and Structures, 81, 383-396 (2001). [6] H.Antes, M.Schanz and S.Alvermann Journal of Sound and Vibration, 276, 807-836 (2004). [7] J.C.Houbolt Journal of the Aeronautical Sciences, 17, 540-550 (1950). [8] M.S.S.Borges Shear Effect Analysis in the Bending of Beams – General Formulation of the Problem and Determination of the Shear Coefficient (in portuguese), D. Sc. qualification seminar, COPPE/UFRJ, Rio de Janeiro, RJ, Brasil (1996). [9] G.R.Cowper Journal of Applied Mechanics, 33, 335-340 (1966). [10] S.A.Fleischfresser Shear Effect Analysis in the Bending of Beams: a Boundary Element Method Formulation for Timoshenko Beams (in portuguese), D. Sc. qualification seminar, PPGMNE/UFPR, Curitiba, PR, Brasil (2012).

130


One Dimensional Modelling of Scalars Transport C. L. N. Cunha1, J. A. M. Carrer2, M. F. Oliveira3, W. J. Mansur4 1,2,3

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] 4

Programa de Engenharia Civil, COPPE/UFRJ, Universidade Federal do Rio de Janeiro, Caixa Postal 68506, CEP 21945-970, Rio de Janeiro, Brasil 4 email: [email protected] Keywords: scalar transport, D-BEM

Abstract. In this paper, the transport of substances in a water body, that is, contaminants that are well mixed in the water column and in the transversal direction and that are applied as passive and non-conservative scalars, is simulated for one-dimensional problems. Two formulations are employed to solve the advective diffusive transport equation integrated in the vertical and in the transversal directions: the first one, presented in this work, consists of a typical D-BEM formulation, whereas the other one employs finite elements in the spatial discretization and finite differences in the time discretization. The validation of the models is carried out through the comparison of their results with the analytical solution for a typical problem. Introduction The main problem related to the hydrological systems is that due to the excess of pollution loading, domestic or industrial, released continuously or not. The control of the pollution of water resources is an important ally for the implementation of prevention measures concerning the population and the environment health, in view of the importance of these systems to human life. For this reason, the computational modeling constitutes a very important tool in the elaboration of the study and of the environmental management of production activities and transportation of hazardous, or not, materials and effluent discharge. In the literature, many models can be found. Among them, one can cite the QUAL2E, developed by the EPA (U. S. Environmental Protection Agency), which is one-dimensional transport model capable of simulating various water quality parameters, see Chapra et al. [1]. Another model is the MIKE11; for additional details, see reference [2]. This work is mainly concerned with the development of a Boundary Element Method formulation for the solution of one-dimensional advective diffusive transport problems. The formulation employs the static fundamental solution and, consequently, is of the type called D-BEM, D meaning domain, as it presents a domain integral that turns the discretization of the entire domain mandatory. Although time-dependent formulations (TD-BEM) are very attractive from the mathematical point of view, see Wrobel [3], providing accurate results, D-BEM formulations appear as alternative approaches, much simpler and equally capable of providing accurate results, see Carrer et al. [4] and Vanzuit [5]. The problem to be solved is one-dimensional; consequently, the boundary of the problem is constituted only by the extreme nodes of the finite domain, that is, the nodes at x = 0 and at x = L, if the interval [0, L] represents the domain. The presence of a domain integral, whose kernel is constituted by the product of the fundamental solution by the first order time derivative of the concentration of the scalar of interest, requires the discretization of the domain. By assuming a linear variation of the concentration inside each cell, the required integration can be carried out analytically. A backward finite difference, see Smith [6], is adopted to approximate the time derivative. An example is presented, in which the BEM results are compared with the analytical solution and with the Finite Element Method (FEM) results, see Cunha et al. [7]. A good agreement is observed between the numerical results and the analytical solution. Besides, an error analysis shows that the proposed formulation presents reliable results, even better than those provided by the FEM formulation, turning it attractive and even encouraging further developments.

131 The Mathematical Model The equation that describes the transport of a scalar for medium or great scale variables is given by, Bedford [8]: 2

∂C ∂C ∂C +U = Ex 2 + ΣR ∂t ∂x ∂x

(1)

where C(x,t) is the concentration of the scalar of interest, U is the velocity component in the x direction, ΣR represents the sources or loses of mass integrated in the vertical and transversal directions and Ex is the diffusivity coefficient (assuming the medium is homogeneous and isotropic). It is assumed that the kinetic reactions are of the first order and can be written as ΣR = − KC, where K is the decay or increasing constant. A well-posed problem is conditioned by adequate initial and boundary conditions. There are two kinds of horizontal boundaries: land boundaries and open boundaries. Land boundaries in general represent the margins of the water body and possible points with inflows or outflows. Open boundaries usually represent water domain limits, and not a physical boundary. Prescription of normal fluxes is associated with land boundaries, and prescription of concentration is associated with open boundaries. At the open boundaries, the concentration is imposed as an essential boundary condition such that: C(x,t) = C*(x,t)

(2)

where C* (x,t) is the prescribed concentration. For land boundaries points presenting significant inflows or outflows, such as a river or a small estuary that ends in a bay, one has: UN C − EN

∂C = f* ∂xN N

(3)

where subscript N represents the normal direction and f* is the flux prescribed at the boundary. Generally, along the land boundaries U and f* are taken as null.

The Numerical Model As mentioned at the Introduction to this work, two models were employed for the solution of the problem. The first employs a Finite Element Method (FEM) approach. For additional details concerning the model, the reader is referred to Cunha et al. [7]. The second model is based on a D-BEM formulation for 1-D problems and is briefly discussed in what follows. The basic BEM equation is written as: 1 1 C(ξ,t) = 2 C(x,t)⏐x=L + 2 C(x,t)⏐x=0 −

∂C 1 − e ∂x 2β

−β(x − ξ)

β(ξ − x) − 1

⏐x=L + ∂C e

⏐x=0 + ∂x 2β

β(ξ − x) −β(x − ξ) . 1 L 1 − e − 1 . 1 ξ e

C(x,t)dx + +

C(x,t)dx E Ex β 2 β 2 x

0

(4)

ξ

where U β=E . x

(5)

132


For the solution of the problem, eq.(4) is written for the boundary nodes, that is, for the nodes at the positions x = 0 and at x = L. Due to the presence of the domain integral, which turns the domain discretization mandatory, eq. (4) is also applied at the internal points. In the present formulation, linear cells were employed to perform the domain discretization and the integration over each cell is carried out analytically. The time derivative of C(x,t) is approximated by a standard backward finite difference scheme, see [5]. The time marching scheme is very simple: once the algebraic system of equations is formed, the boundary conditions are imposed and the unknowns determined. The unknowns are the values of C(x,t) and/or ∂C/∂x at the boundary nodes and the values of C(x,t) at the internal points. In the sequence, the values of C(x,t) are updated and so on.

Example In the example presented in what follows, a plane continuous source acts at x = 0, and the water flows in the channel at a constant velocity component in the x direction, U. For the simulation it was assumed the absence of kinetic reaction, that is, the substance is a conservative one. By taking into account these considerations, the transport equation is reduced to: 2

∂C ∂C ∂C +U = Ex 2 ∂t ∂x ∂x

(6)

with boundary conditions: C (0,t) = C0

0 G

i1

G i ,i 1

A ii

H i ,i 1 H in @

,i

1, n s

(3)

where the Q i matrices are straightforwardly formed from the subregion matrices of the model at hand. To store is preconditioner, 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, should be allocated.

137 Stress calculations Considering the O ( r 3 ) and O ( r 2 ) singularities of the fundamental kernels involved in the boundary integral expressions for evaluating the strain and stress tensors at a given point of a solid, the calculation of these quantities is a tough problem as special integration algorithms for dealing with the singularities at hand have to be devised. This is particularly difficult in case of the calculation of stresses at boundary nodes of any solid or in case of thin-walled domain problems, wherein all the points of the solid are either at the boundary or very close to it (Fig. 1). To avoid facing directly the singular integrals appearing in the stress integration kernels, the stress tensor can be directly determined from the boundary displacement field by means of the Hooke's law referred to a local system [12]. x2 m2

discontinuous elements

n

x3 outward normal unit vector

continuous elements

m1

tangential unit vectors

x3

x1 x2

x1

Fig. 1. Thin-walled domain discretized with boundary elements In the procedure presented in [12], one takes a local, mutually orthogonal - - coordinate system centered at the point where the boundary stresses should be calculated, and wherein the is defined by the outward normal vector, and and are two tangential vectors. After the boundary solution has been completely calculated, the stress components referred to the local system are then given by:

V 13 V 31 V 23 V 32 V 33 p3

p1 p2

.

(4)

Thus, only three more stress components have to be calculated, namely, those related to the tangential - plane, V 11 , V 22 , and V 12 , which are given by

1

>Qp3 2GH 11 QH 22 @

1

>Qp3 2GH 22 QH 11 @

V 11

1 Q

V 22

1 Q

V 12

,

(5)

2GH 12

wherein the local strain components H ij , i,j=1,2, in (5) are calculated by

H ij

1 §¨ wui x1 , x2 , x3 wu j x1 , x2 , x3 ·¸ , ¸ 2 ¨© wx j wxi ¹

wherein

(6)

138


wui (x) wx1

Oik wu k (x) Oik § nnoel whq (r , s ) J (r )

wr

¨¦ J (r ) ¨© q 1

wr

· u kq ¸¸ , ¹

(7)

and

wui wx2

1 wui wui ( m '12 ) , m '22 wx2c wx1

(8)

with

wui (x) wx2c

Oik wu k (x) Oik § nnoel whq (r , s) J (s)

ws

¨¦ J ( s) ¨© q 1

ws

· u kq ¸¸ . ¹

(9)

and hq ( r , s ) denotes the isoparametric shape function of the element, J (s ) is the Jacobian associated with its geometrical mapping into the natural coordinates, and nnoel is the number of nodes per element. The expression (8) is needed for xc2 is not necessarily perpendicular to x1 . The vector mc2 is the unit tangent vector m'2

dl 2 , where dl 2 dl 2

wx , expressed in relation to the local system. All the boundary stresses ws

are calculated at the geometrical contour of boundary elements, which are always continuous. In case of discontinuous elements, the displacement fields at the geometrical contour of the boundary elements are first determined via the interpolation functions for the discontinuous elements, before the stress calculation procedure described above be applied. The global stress tensor is obtained by rotating the local stress tensor, , to the referred to the global coordinate system via RR T . By taking then, say the global stress tensor, , at any boundary point, the corresponding principal stresses can be easily obtained by solving the eigenvalue problem x i Oi x i , wherein Oi and x i , i 1,2,3, are the principal stresses and corresponding principal directions. Results and discussions To show some results on the strategy for stress calculation on boundary nodes, a rod under axial load, a beam under shear load, and a carbon-nanotube-reinforced composite under axial deformation (see references [1-2] for description of this load) are analyzed. Rod under axial load. A simple rod submitted to a uniformly distributed unit load, p x 1.0 Ncm 2 , at one of its end is discretized with 3 subregions and number of elements shown in Fig. 2. The rod is in fact homogeneous with E 1.0 u 10 5 Ncm 2 , Q 0 , length l 2 m , and cross section 20 u 20 cm 2 . The BiCG solver with SBS preconditioning is used to solve the coupled problem, and displacement, boundary tractions, and stresses are calculated. All values calculated are in good agreement with the corresponding analytical values, and to show the calculation of boundary stresses, the V xx component is plotted in Fig. 2. Cantilever plate under distributed load at the upper surface. As 2nd application, the cantilever plate in Fig. 3 is analyzed. Here just 1 subregion is considered with the following material properties: E 2.9 u 103 MNm 2 , Q 0.0 . The geometrical dimensions of the plate are l 4 m (length, x axis), h 2 m (height, y axis), t 0.2 m (thickness, z axis), and the distributed load acting at the upper boundary of the plate is p y

50 kNm 2 . For this particular problem, in the linear-elastic case, the stress

component V xx goes to infinity at the cross-section upper and lower borders of the plate, totally diverting from the simple beam theory solution, given by V xx r M W . The FE (ANSYS) solution in this case has shown very slow convergence to the exact elasticity solution.

139

interfaces

y x z Fig. 2. Rod under axial load: V xx component

y x z

Fig. 3. Cantilever plate under distributed load: V xx component CNT-based composite. Finally, the CNT-fiber reinforced composite shown in Fig. 4 is analyzed. The long CNT fibers are geometrically defined by cylindrical tubes having outer radius r0 5.0 nm and inner radius ri

Q CNT

4.6 nm , and length l f

10 nm . Its material properties are ECNT

0.30 . For the matrix material, E m

1,000 nN nm 2 (GPa), and

100 nN nm 2 (GPa), and Q m

0.30 . This problem has been considered in [1] to characterize CNT composites with various fiber-packing patterns. In reference [2] all the details of the loadings considered for determining the equivalent material properties are given. Here, just a sample of response, namely the V zz component, is shown. Again, the block-diagonal SBS-based

preconditioned BiCG with tolerance number ] 10 8 is applied. The boundary stress values calculated are compatible with those obtained in previous papers [1, 2, 13]. In this model, discontinuous boundary elements with d 0.10 are used when needed. Conclusions and prospects In this paper, the BE SBS technique proposed in previous papers ([1], [2]) is incremented with routines for the calculation of stresses at the boundary nodes. The technique used, based on the direct application of Hooke's law, allows accurately and efficiently calculating strain and stress at boundary element nodes

140


without the need of evaluating the strongly singular and hypersingular boundary integrals involved in standard BEM formulations. This is very important for determining the stress-tensor fields in thin-walled domains, wherein in fact only the boundary stress fields are needed. Along with the whole boundary-

y z x

Fig. 4. CNT composite under shortening G z

1.0 : V zz component

element SBS technique, including its straightforward parallel-computing implementation, the strategy proposed in this paper may be fundamental for analyzing general composites, and for the microstructural analysis of materials.

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] F.C. Araújo, L.J. Gray Comp. Mod. Eng. Sci. 24(2), 103-121 (2008). [2] F.C. Araujo, E.F. d’Azevedo, L.J. Gray, Eng. Anal. Boundary Elements 35, 517–526 (2011) [3] F.C. Araújo, Time-domain solution of three-dimensional linear problems of elastodynamics by means of a BE/FE coupling process (in German), Ph.D. Thesis, T.U. Braunschweig, Germany, (1994). [4] F.C. Araújo, E.F. d’Azevedo, L.J. Gray, Computers & Structures 88, 773-784 (2010). [5] I. Benedetti, M.H. Aliabadi, Computational Materials Science, 67, 249-260 (2013). [6] R. Barrett, M. Berry, J. Dongarra, V. Eijkhout, C. Romine, J. Comp. Appl. Mathematics 74, 91-109 (1996). [7] G.L.G. Sleijpen, D.R. Fokkema Electronic Trans. Num. Methods Anal., 1, 11-32 (1993). [8] S.-L. Zhang Comp. and Appl. Math. 149, 297–305 (2002). [9] H. A. van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge University Press (2003). [10] K. Chen, Matrix Preconditioning Techniques and Applications, Cambridge, UK, Cambridge University Press, (2005). [11] T.J.R. Hughes, I. Levit, L. Winget, Comput. Methods Appl. Mech. Engrg. 36(2), 241-254 (1983). [12] C. A. Brebbia, J. C. F. Telles, L.C. Wrobel, Boundary element techniques: theory and applications in engineering. Springer-Verlag, Berlin (1984) [13]

X.L. Chen, Y.J. Liu Comput. Mat. Sci. 29, 1–11 (2004).

141

Development of Compressible-Incompressible Link to Efficiently Model Bubble Dynamics near Floating Body Chao-Tsung Hsiao1, Georges L. Chahine2 1 2

DYNAFLOW , INC. 10621-J Iron Bridge Road, Jessup, MD 20794, USA, [email protected]

DYNAFLOW , INC. 10621-J Iron Bridge Road, Jessup, MD 20794, USA, [email protected]

Keywords: Fluid Structure Interaction, boundary element method, bubble dynamics, free surface, piercing body

Abstract. The interaction between an explosion bubble and a floating body involves several physical phenomena that an accurate numerical simulation needs to capture. These include proper description of the initial shock wave and its propagation and interaction with the floating body and free surface and the subsequent growth, dynamics, collapse, and interaction of the resulting bubble with the structure. A numerical procedure which links a compressible finite difference flow solver with an incompressible boundary element method was applied to capture both shock and bubble phases efficiently and accurately. Both codes solved the fluid dynamics problem and coupled with a finite element structure code to simulate fluid-structure interaction. This numerical approach was validated by comparing the numerical results with available experimental measurements. The effects of including or neglecting some modeling aspects on the solution were considered.

Interface Interface

Flow Stage Fluid Code Shock Phase GEMINI Compressible-Incompressible Link (Handoff) Bubble Phase 3DYNAFS FS-B S BEM® Incompressible-Compressible Link(Handback) Rebound Phase GEMINI

Coupler

Time

Introduction Underwater explosion interaction with a structure has long been an area of interest in the fluid and structures community [1-2]. The complex dynamics results from a shock wave created by the explosion, which leaves behind it a very dynamic underwater bubble. The shock wave and the bubble strongly interact with the structure. The dynamics of the bubble result in long duration loads with time scales, which could match the structure natural frequency. Modeling of the initial times in the dynamics requires a compressible flow solver to capture shock wave emission and propagation. However, such a flow solver usually uses a finite difference method (FDM) which requires very fine spatial resolution and small time step sizes. This makes them not very efficient for resolving the relatively long duration bubble dynamics. To overcome this, an incompressible potential flow solver based on a boundary element method (BEM) can be used to model the bubble dynamics once the shock phase has been described properly. In this study we present the numerical procedures to transfer from the FDM compressible flow solver the flow field variables required to proceed with the BEM incompressible potential flow computations. In addition we describe coupling procedures between the BEM and a finite element method (FEM) based structure code, which are used to enable simulation of the full fluid-structure interaction between the bubble and the floating body. Unlike for a fully submerged body, a floating object pierces the free surface and only the wetted area of the body and the water free surface need to be discretized and solved for the BEM code. Since the body waterline changes dynamically, meshing, re-meshing and interpolation between BEM and FEM meshes needs to be done at each communication step for proper information exchange between the fluid and the structure codes. In this study we compare the numerical results with experimental measurements to validate the developed numerical procedure and the importance of various parameters which influence the accuracy of numerical simulations is addressed by isolating their effects. Structure Code

DYNA3D

Figure 1. Schematic diagram of the numerical approach used to simulate fluid structure interaction problem between an underwater explosion and a structure.

Numerical Approach Overview of the Approach. In the present study a hybrid scheme using three solvers was utilized to simulate the Fluid Structure Interaction (FSI) problem between an underwater explosion bubble and a floating body. This hybrid scheme combines the advantages of a compressible code, GEMINI [3], and an incompressible solver,

142


3DYNAFS-BEM© [4], to capture the shock and bubble phases. Both GEMINI and 3DYNAFS-BEM© are coupled with the structure dynamics code, DYNA3D [5], through an interface coupler. The schematic diagram shown in Figure 1 illustrates the hybrid numerical approach used in this paper. As illustrated in Figure 1, after the shock phase the flow field is “handed off” from GEMINI to 3DYNAFS-BEM© [6], which is then used to simulate most of the bubble period until the end of the bubble collapse where due to high speeds or to reentrant jet impact compressible flow effects prevail again. To continue the simulation into the rebound phase the solution is “handed back” [7] to GEMINI. In the current study, we will only focus on the shock and bubble phase during the bubble first period. FDM-Based Compressible Flow Solver. GEMINI is a multi-component compressible Euler equation solver developed by the Naval Surface Warfare Center, Indian Head division based on a finite difference scheme [3]. The code solves continuity and momentum equations for a compressible inviscid liquid in Cartesian coordinates. These can be written in the following format: wQ wE wF wG (1) S, wt wx wy wz ªU w º ª Uu º ªUv º ª0 º « » « 2 » « U vu » «0 » U u p U wu « » « » « » « » » « U uv » , F « U v 2 p » , G « U wv «0 » , (2) , Q S « » « » « » « » 2 « » « U uw » « U vw » Uw p » «U g » « « » « » «¬ U gw»¼ «¬ U et p w¼» «¬ U et p u »¼ «¬ U et p v »¼ where U is the liquid density, p is the pressure, u, v, and w are the velocity components in the x, y, z directions respectively (z is vertical), g is the acceleration of gravity , and et=e+0.5(u2+v2+w2) is the total energy with e being the internal energy. Eq (1) is applied to solve gas, liquid, or other media with an equation of state relating pressure, density, and energy for a given material or phase. GEMINI uses a high order Gudonov scheme. It employs the Riemann problem to construct a local flow solution between adjacent cells. The numerical method is based on single material higher order MUSCL scheme and tracks each material. To improve efficiency, an approximate Riemann problem solution replaces the full problem. The MUSCL scheme is augmented with a Lagrange re-map treatment of mixed cells. The code has been extensively validated against experiments [3]. ªU º « Uu » « » «Uv » , E « » «Uw » «¬ U et »¼

BEM-Based Incompressible Flow Solver. 3DYNAFS-BEM© is a potential flow solver based on a boundary element method. The code solves the Laplace equation, 2I 0 , for the velocity potential, I, with the velocity vector defined as u I . A boundary integral method is used to solve the Laplace equation based on Green’s theorem:

³: I G G I d : ³S n >IG GI @ dS. 2

2

(3)

In this expression : is the domain of integration having elementary volume d:. The boundary surface of : is S, which includes the surfaces of the bubble and the nearby boundaries with elementary surface element dS. n is the local normal unit vector. G 1/ x y is Green's function, where x is a fixed point in : and y is a point on the boundary surface S. Eq (3) reduces to Green’s formula with DS being the solid angle under which x sees the domain, :: wG wI ª º aSI (x) ³ «I (y) (x,y) G(x,y) (y) » dS , (4) S¬ wn wn ¼ where aS is the solid angle. To solve eq (4) numerically, the boundary element method, which discretizes the surface of all objects in the computational domain into panel elements, is applied. Eq (4) provides a relationship between I and wI/wn at the boundary surface S. Thus, if either of these two variables (e.g. I) is known everywhere on the surface, the other variable (e.g. wI/wn) can be obtained. For the current problem, at each time step I on the bubble and free surface will be obtained through the integration of Bernoulli equation: dI 1 1 2 (5) pf pl gz I , dt U 2

143 where pl is the liquid pressure at the bubble surface or free surface. For the floating body surface, wI/wn is provided when the code is coupled with a structure code. The velocity potential, I , and wI/wn derived from eq (5) on the bubble and free surfaces are used to compute the velocity vector, u. The locations of the bubble and free surfaces nodes are then updated with the coordinates of surface nodes, x, advanced according to dx / dt u . FEM-Based Structure Code. DYNA3D is a non-linear explicit structure dynamics code based on finite element method developed by the Laurence Livermore National Laboratory. DYNA3D is used here to handle the floating body motion and deformation with the loading provided by the fluid solver. DYNA3D uses a lumped mass formulation for efficiency. This produces a diagonal mass matrix M, to express the momentum equation as: d 2x (6) M 2 Fext Fint , dt where Fext represents the applied external forces, and Fint the internal forces. The acceleration, a dx2 / dt , for each element is obtained through an explicit temporal central difference method. Additional details on the general formulation can be found in [5]. Compressible-Incompressible Link. A key aspect of the hybrid numerical procedure used here is the need of BEM flow field initial conditions at the time of “handoff”. These are based on the FDM compressible flow solution which provides all flow properties such as the velocity, pressure and density at the volume grids and need to be interpolated to the BEM grid as illustrated in Fig. 2 and Fig. 3. The steps of the procedure can be described as follows: x The bubble and free surface liquid/gas interfaces are determined and the boundary element mesh generated based on the density distribution computed by the compressible solver (see Fig. 3). x The velocities on the boundary element surface nodes are interpolated from the compressible flow field solution (volume grid) and the normal components of the velocities, wI / wn , are deduced according to the normal vectors at the discretized surface. x The velocity potential values, I , on the boundary element mesh including bubble, free surface, and body surfaces are computed after solution of eq (4) using the known wI / wn at all surface nodes. x The values of I on the bubble and free surface nodes are used to initialize the BEM computation, which is then marched in time using eq (5).

Fig. 2: Pressure contours and velocity vectors from the compressible flow solution at the time of handoff.

Fig. 3: Boundary element mesh generated on the gas/liquid interfaces according to the density distribution of compressible flow solution.

Fluid Structure Interaction Computation. In order to include fluid/structure interaction (FSI) effects in the simulations both fluid codes are coupled with DYNA3D through a coupler interface. The coupling is achieved through the following steps: x Each of the fluid codes provides the structure code with the pressures at the structure surface nodes. x The structure code then computes deformations and velocities in response to the loading. x The coordinates and the velocities of the structure surface nodes are then sent back to the fluid code. x The fluid code then solves the flow field and deduces the pressures at the surface using the body boundary nodes positions and normal velocities. Additional details on the procedure concerning the exchange of information between the GEMINI volume grid and DYNA3D finite element mesh can be found in [3]. Concerning the coupling between 3DYNAFS-BEM© and DYNA3D, the information is exchanged using interpolations between values at the boundary element mesh and

144


values at the surface elements of the structure finite element model. Since the water contact line of the floating body changes dynamically, an automatic re-meshing scheme is implemented to modify the boundary element mesh on the floating body and adapt it at each time step to the position of the waterline. The following procedure is as follows: x A “Slicer” is used to cross-cut the structure at selected locations (see Fig. 4a). The intersection of the cutting planes and the body surface elements provides a set of nodes in each cutting plane. These nodes are then ordered and connected with straight lines to form a contour of the structure section (see Fig. 4b). x The position of the water-atmosphere contact points (blue nodes in Fig. 4c) are determined and all nodes above the water contact point except for the top corner point are removed. The nodes below the water contact point are coarsened, if necessary, and redistributed evenly (see Fig. 4c). x The contours of all sections are connected to form a boundary element mesh (see Fig. 4d).

(a)

(b)

(d)

(c)

Fig. 4: Illustration of the automatic meshing scheme for generation of BEM grid on the floating body from an FEM body grid.

Validation Problem Setup. In the present study we demonstrate the numerical scheme introduced above on the problem of the interaction of an explosion bubble and a Floating Shock Platform (FSP) which has a rectangular base of 4.9 m × 8.5 m and an initial draft of 1.3 m. The problem setup is shown in Fig. 5. An explosion bubble is simulated using sudden release of gas bubble of initial radius 0.34 m and gas pressure 120 MPa. The bubble center is initially located at 7.3 m below the free surface and 6.1 m horizontally away from the floating body.

Floating Body Free Surface

6.1 m

4.9 m

7.3 m

Explosion Bubble

Fig. 5: Geometrical setup of an explosion bubble located at a depth of 7.3 m and at 6.1 m away from a floating body

Compressible-Incompressible Link. One important factors which affects solution accuracy of the link procedure is the time to handoff. Ideally, the time to execute handoff should be when all compressible effects have become negligible. Away from boundaries, compressible effects on the flow field due to an explosion bubble die out after the initial shock propagates away from the bubble center. However, the presence of a free surface and a nearby body result in the refection of expansion and shock waves back towards the bubble. This expansion wave can also cause bulk cavitation near the free surface. To investigate the optimal handoff time we compare the solutions when the compressible-incompressible link is executed at four different times: 50, 60, 70 and 80 ms. The GEMINI pressure fields obtained by a grid of 6.1 million points with far field boundaries located at 400 m away from the bubble center at these four times are shown in Fig. 6. In the pressure contours, the white color regions indicate bulk cavitation (i.e. pressure below vapor pressure). It is seen that the water is still filled with large cavitation regions at 60 ms while only small cavitation regions near the bubble exist after 70 ms. This leads to selecting a handoff time selection larger than 70 ms and can be confirmed by comparing the time history of the floating body motion for these four handoff times. Fig. 7 compares the vertical velocity of the bottom center node on the FSP for the four handoff times with the velocity when the compressible code GEMINI is allowed to do the full computation and no link to incompressible is made (blue curve). It is seen that the results converge and become close to the no-handoff case for the handoff times 70 and 80 ms. Evaluation of Different Modeling Aspects. With the numerical approach described above, we are able to isolate some modeling aspects such as inclusion of shock phase and structure deformation easily to study their effects on

145 the accuracy of the solution. To isolate the shock wave effects, we compare the above FSI results with the computations using 3DYNAFS-BEM© and DYNA3D only after by neglecting the initial compressible shock phase. Fig. 8 compares the vertical velocity and displacement of the floating body obtained from the two numerical simulations to test measurements [8]. It is seen that, as expected, the inclusion of the shock phase is essential to accurately predict the initial response of the floating body to the initial explosive bubble growth but is less critical for the later motion of the FSP. 50 ms

60 ms

80 ms

70 ms

Fig. 6: Pressure contours obtained from a GEMINI simulation at four different times.

To study the effect of the structure deformation on the FSI simulation, we also conducted a simulation by treating the floating body as a rigid body. This is illustrated in Fig. 9 which compares the results of the two computations. One can see that ignoring the structure deformation shortens the bubble period and results in a smaller response (velocities) of the floating body during the bubble collapse than in the case of the deformable body. The numerical predictions of the vertical velocity obtained from the different computations are compared to the experimental observations in Fig. 10. It is seen that the hybrid scheme with handoff at 70 ms predicts the floating body response the best. The CPU time requirement for this case is 1/10 of that using only the compressible code for the whole bubble period.

All Gemini 60 ms 70 ms

50 ms

80 ms

Fig. 7: Effect of handoff time on the vertical velocity monitored at the bottom centre node of the floating body

Fig. 8: Effect of the inclusion or neglect of the shock phase on the vertical velocity (left) and the floating body bottom center node displacement.

146


Fig. 9: Effect of the inclusion or neglect of the structure deformation on the bubble dynamics (left) and the floating body bottom center node vertical velocity.

Conclusions A hybrid numerical procedure integrating three different solvers was developed to accurately and efficiently predict the response of a floating body to an underwater explosion bubble. It was found that shifting from the compressible to the incompressible simulation (link) should be performed after bulk cavitation has disappeared. Modeling the shock wave and the induced bulk cavitation is important for the simulation of the early history of the floating body response. Inclusion of the structure deformation was also shown to have a significant influence on the response of the floating body during the bubble expansion and collapse phases. Acknowledgments This study was conducted during the execution of an SBIR contract for the development of an alternative ship shock testing method. The support of The Naval Surface Warfare Center, Carderock Division (Dr. Fred Costanzo) and Indian Head Division (Mr. Greg Harris) is gratefully acknowledged.

Fig. 10: Boundary element mesh generated on the gas/liquid interfaces determined according to the density distribution of compressible flow solution at the handoff time.

References 1. G.L. Chahine, and K.M. Kalumuck, “BEM Software for Free Surface Flow Simulation Including Fluid Structure Interaction Effects,” Int. Journal of Computer Application in Technology, Vol. 3/4/5, 1998 2. K.M. Kalumuck, G.L. Chahine, G.L. and C.-T. Hsiao, “Simulation of Surface Piercing Body Coupled Response to Underwater Bubble Dynamics Utilizing 3DYNAFS, a Three-Dimensional BEM Code,” Computational Mechanics, 32, 319-326, 2003. 3. A.B. Wardlaw, J.A. Luton, J.R. Renzi, K.C. Kiddy, and R.M. McKeown, “The Gemini Euler Solver for the Coupled Simulation of Underwater Explosions,” Indian Head Technical Report 2500, November 2003. 4. G.L. Chahine, R. Duraiswami, and K.M. Kalumuck, “Boundary Element Method for Calculating 2-D and 3-D Underwater Explosion Bubble Loading on Nearby Structures,” Naval Surface Warfare Center, Weapons Research and Technology Department, Report NSWCDD/TR-93/46, September 1996. 5. DYNA3D, User manual “ A nonlinear explicit, three dimensional finite element code for solid and structural mechanics” , January 2005. 6. C.-T. Hsiao, J-K Choi, J-K and G.L. Chahine, “Analysis of the Phenomena influencing the Accurate Modelling of a Surface Vessel Response to an UNDEX,” 78th Shock and Vibration Symposium Philadelphia, PA, November 4-8, 2007. 7. C.-T. Hsiao, G.L. Chahine, “Incompressible-Compressible Link to Accurately Predict Wall Pressure,” 81th Shock and Vibration Symposium, Orlando, FL, October 24-28, 2010. 8. G.L Chahine, K.M. Kalumuck, M. Tanguay, J.P. Galambos, M. Rayleigh, R.D. Miller and C. Mairs, “Development of a Non-Explosive Ship Shock Testing System-SBIR Phase I Final Report,” Dynaflow, Inc. Report 2M3030-1 DTRC, March 15, 2004.

147

Exact Computation of Drilling Rotations with the Boundary Element Method D. I. G. Costa1 , E. L. Albuquerque1, P. M. Baiz2 1

University of Brasilia - Darcy Ribeiro Campus, North Wing - Brasilia - Brazil [email protected]

2

Imperial College London - South Kensington Campus, London SW7 2AZ - U.K

Keywords: Boundary Element Method, Plane Elasticity, Drilling Rotation, Singularities.

Abstract. In this paper, a boundary element formulation for plane elasticity that includes drilling rotation is implemented. Derivatives of plane elasticity boundary integral equations are used to compute this new degree of freedom that depends only on displacements and tractions already present on traditional formulations. Expressions obtained are directly applied to elasticity theory rotation formula. Derivatives of boundary integral equations present strong (O(r−1 )) and hyper singularities (O(r−2 )) when source point belongs to the element. In this work they are treated using analytical solutions. Finally, results obtained for benchmark problems are compared with those of literature. Introduction. Historically shell-like structures have always been important and due to its natural complexity in analysis, researchers were forced to develop numerical techniques in order to obtain solutions for many types of shell applications. Methods like finite elements are largely applied, but others are suitable as well, for instance the boundary element method and meshless techniques. During the assembly of shells or plates belonging to different planes, a difficulty will arise relative to satisfaction of equilibrium and compatibility conditions when using coupled plate-membrane formulation. That is so, because this approach produces only five degrees of freedom, three translational and two rotational, what makes difficult to couple rotations of different planes positioned in arbitrary directions. In the procedure used in finite elements, adjacent elements in the same plane can lead to rank deficiency. For this reason a fictitious stiffness about the normal to the plane of the element is often added [7]. Many researchers have dealt with this problem including in plane rotation by means of a particular plane element, which has the so-called drilling rotation as one of vertex connectors. Allman (1985) [3] has presented one of the first successful elements with this purpose that has become known as the Allman triangle. Bergan and Felippa (1985) [5] have also derived a triangular element with drilling rotations using a free formulation in which stiffness matrix is constructed as the sum of a basic and a higher order stiffness matrix. Others like [13] and [6] have improved Allman’s element concept and gave it a more physical significance, in order to make it equal to the true rotation of elasticity theory. Other numerical techniques have also made advances in this area and through the application of Partition of unity [10] to Allman’s triangle it was possible to extend the idea for meshless and boundary elements formulations. The first work to include drilling rotations for a boundary element formulation performed by Leung and Baiz [9] which used partition of unity to develop linear elements enriched with drilling rotations together with an additional functional based on the rotational residual. This paper presents a boundary element formulation for plane elasticity that includes an out-of-plane rotation, known as drilling rotation or out of plane rotational degree of freedom. This formulation consider this rotation as a secondary answer, in a similar way of computing stresses. This is just a matter of choice as the resulting equation system (including drilling rotation integrals) could be easily solved at once. Expressions for that purpose were obtained applying derivatives of the original boundary integral equations in such a manner to match the rotational field of elasticity theory. Because of that, hypersingular integrals appear in the formulation. Considering that linear discontinuous elements were used, analytical treatment was given to integrals in which higher order of singularities were found. The term exact has been used because the boundary integral equation obtained here, presents no approximation up to its discretization in boundary elements.

148


Boundary element equations. Boundary integral equation for plane elasticity can be found in many BEM basic textbooks [2, 8]. It is obtained using the equilibrium equation from elasticity theory and Betti’s reciprocal theorem [2]. Boundary integral equation for plane elasticity in absence of body forces is given by: ui (x ) = where Uisj

Γ

Uisj (x , x)t j (x)dΓ(x) −

Γ

Tisj (x , x)u j (x)dΓ(x)

(1)

Tisj

and are displacement and traction fundamental solutions for plane stress problems, respectively, given by [2]: # $ 1 1 (3 − 4ν ) ln δi j + r,i r, j 8πμ (1 − ν ) r ( ) ∂r 1 [(1 − 2ν )δi j + 2r,i r, j ] + (1 − 2ν )(ni r, j − n j r,i ) = − 4π (1 − ν )r ∂ n

Uisj =

(2)

Tisj

(3)

where μ = shear modulus, ν = Poisson ratio, r = distance from source to field point, and ni = unity normal vector. Taking x to the boundary in eq (1), we obtain: Cisj (x )ui (x ) =

Γ

Uisj (x , x)t j (x)dΓ(x) − − Tisj (x , x)u j (x)dΓ(x) Γ

(4)

where Cisj is added to allow different positions for the collocation point, inside domain or exactly at the boundary. Exact boundary integral equation for drilling rotations. Elastic solids will deform when subjected to loadings, and these deformations can be quantified by knowing the displacements field in the body. The continuum hypothesis establishes a displacement field at all points within the elastic solid. Applying linear theory of elasticity and some kinematic considerations it is possible to find that displacement gradient tensor is given by [11]: ⎡ ⎢ ui, j = ⎣

∂ u1 ∂ x1 ∂ u2 ∂ x1 ∂ u3 ∂ x1

∂ u1 ∂ x2 ∂ u2 ∂ x2 ∂ u3 ∂ x2

∂ u1 ∂ x3 ∂ u2 ∂ x3 ∂ u3 ∂ x3

⎤ ⎥ ⎦

(5)

This tensor can be divided into a symmetric and antisymmetric parts:

ei j =

ωi j =

1 2 (ui, j + u j,i ) 1 2 (ui, j − u j,i )

(6) (7)

which are strain and rotation tensors, respectively. A dual vector can be associated with the rotation tensor using ωi = −1/2εi jk ω jk . For out of plane rotation we have:

ω3 = ω21 =

1 2

#

∂ u2 ∂ u1 − ∂ x1 ∂ x2

$ (8)

which is the expression for drilling rotations in plane elasticity analysis. Equation (4) gives values for displacements at a specified collocation point, that can be placed in any point. Derivatives to the source point of this equation can be found in order to compute eq. (8), taking into account that only fundamental solutions are function of source point position, BIE derivative is given by: ∂Ts ∂ Uisj ∂ ui ij (x ) = 2 − (x , x)t j (x)dΓ(x) − = (x , x)u j (x)dΓ(x) ∂ xk Γ ∂ xk Γ ∂ xk

(9)

149 Derivatives of displacement and traction fundamental solutions, present in (9), are given by:

∂ Uisj ∂ xk ∂ Tisj ∂ xk

1 r,i δ jk + r, j δik − (3 − 4ν )r,k δi j + r,i r, j r,k 8πμ (1 − ν )r ( 1 ∂r = − [(1 − 2ν )δi j + 2r,i r, j ] + 4π (1 − ν )r ∂ xk ∂ n ' 2& ∂r (1 − 2ν )δi j + r,i δ jk + r, j δik − 2r,i r, j r,k ∂n r $ $) # # δ jk − r, j r,k δik − r,i r,k − nj +(1 − 2ν ) ni r r =

(10)

$ # δik − r,i r,k ∂r = ni (11) ∂ xk ∂ n r In this last expression summation convention of index notation holds. Substituting (9) into (8), we finally find the boundary integral equation for drilling rotation, eq. (12):

where

. . ∂ T2sj (x , x) ∂ T1sj (x , x) ∂ U2s j (x , x) ∂ U1s j (x , x) − − u j (x)dΓ(x) + − t j (x)dΓ(x) ∂ x1 ∂ x2 ∂ x1 ∂ x2 Γ Γ (12) When the source point belongs to the element being integrated there are two types of singularities ∂Ts in expressions (10). Those involving ∂ xikj terms are hypersingular of order O(r−2 ) and those involving

-

i ω12 (x ) = −=

∂ Uisj ∂ xk

are strong singular of order O(r−1 ). The approach used in this work for solving these integrals is the same used in [1] based on an analytical treatment. Numerical results. One of the most common linear tests used for evaluation of elements that include drilling degrees of freedom is the short cantilever depicted in Fig. 1(a). This problem has been used in several papers [9, 10, 13] and is designed for assessment of the performance of plane stress elements to in-plane bending behaviour within the finite element community [5]. It has also an analytical solution for both displacements and rotations making it ideal to verify accuracy of the presented formulation. y P x

E = 30000 ν = 0.25

A

12

48 (b)

(a)

Figure 1: Short cantilever and boundary element model with 20 linear discontinuous elements. The tip shear load has value P = 40 N and beam has thickness t = 1. Timoshenko analytical solution for tip vertical displacement and drilling rotation at point A (x = 48, y = 0) are: uy = 0.355333

(13)

ω = 0.010750

(14)

Analytical results and the ones obtained in [9] for fully discontinuous elements will be used for comparison and verification of convergence. Fig. 2 presents results with increasing number of collocation nodes, considering all nodes restricted in x and y directions at the edge x = 0 and vertical distributed

150


shear load at x = 48. Results obtained here for vertical displacements and the ones in [9] are almost identical, drilling rotations however show a different behaviour. Both of them converge with finer meshes but to different values. 0.36

0.0108 0.0107

0.355

0.0105 Drilling Rotation

Vertical displacement

0.0106 0.35

0.345

0.34

0.0104 0.0103 0.0102 0.0101

0.335

0.33

Present work Leung & Baiz (2013) [4] Analytical 0

50

100 150 200 Number of collocation nodes

250

300

Present work Leung & Baiz (2013) [4] Analytical

0.01 0.0099

0

50

100 150 200 Number of collocation nodes

(a)

250

300

(b)

Figure 2: Convergence results for short cantilever.

1

1

10

10

Present work Leung & Baiz (2013) [4]

Drilling rotation error (%)

Vertical displacement error (%)


0

10

−1

10

0

10 2

2

10 Number of collocation nodes


(a)

(b)

Figure 3: Short cantilever error plots for vertical displacements and drilling rotations in log-log scale. Table 1 shows a comparison of results for the most refined mesh used in the analysis considering the proposed formulation and the one given by [9]. Both results were found using totally discontinuous linear elements and meshes are exactly the same. It is possible to note a better agreement of the present results with the analytical solution for drilling rotations, with less than 1% of error achieved. Table 2 shows values of vertical displacements and drilling rotations at point A(x = 48, y = 0) for each used mesh. Table 1: Comparison of results nodes. Present work uy 0.355259 ω 0.010654

for short cantilever middle-axis at the free end, using 280 collocations Error (%) 0.0264 0.8941

Baiz & Leung (2013) [9] 0.355241 0.010520

Error (%) 0.0260 2.1396

Analytical [12] 0.355333 0.010750

Timoshenko analytical solution was obtained for a very specific set of boundary conditions [12]. Therefore, according to [4] reproducing the exact constraints and load distribution is essential to obtain

151

Table 2: Results for short cantilever. Collocation uy uy error ω Nodes (%) 40 0.333320 6.195 0.010033 80 0.349391 1.672 0.010489 120 0.352802 0.712 0.010585 160 0.354071 0.355 0.010621 200 0.354683 0.183 0.010638 240 0.355027 0.086 0.010648 280 0.355239 0.026 0.010654

ω error (%) 6.672 2.425 1.531 1.199 1.039 0.950 0.894

0.36

0.011

0.355

0.0108

0.35

0.0106

0.345

0.0104

Drilling Rotation

Vertical displacement

comparable results. Fig. 2 (E) in [4] shows the exact boundary conditions that should be used (note the reversed sign of the loading). An analysis of this problem under these restrictions has also been made. Results are presented and compared to analytical solutions.

0.34 0.335 0.33

0.01 0.0098


0.325 0.32

0.0102

0

50

100

150 200 250 300 Number of collocation nodes

(a)

350

400


0.0096 0.0094

0

50

100

150 200 250 300 Number of collocation nodes

350

400

(b)

Figure 4: Results for short cantilever with corrected boundary conditions. As it is possible to verify, results are closer to analytical solution in Fig. 4 than in Fig. 2, specially for drilling rotations. Showing that values obtained by previous works would be more accurate if boundary conditions by [12] were used. Conclusions. Formulation just developed in this paper presents an alternative for the computation of drilling rotations. The boundary integral equation (12) is an exact expression for the true rotation of elasticity theory and approximations are present only because of geometry and boundary conditions discretization. In the limit, when the number of elements goes to infinity, results are expected to match analytical ones. Numerical values obtained are in good agreement with theory and literature, showing the accuracy of the present formulation. Acknowledgements. The authors would like to thank CAPES for the financial support of this work.

152



1

10

Drilling rotation error (%)

Vertical displacement error (%)


1

10

0

10

2


(a)

0

10

2


(b)

Figure 5: Short cantilever error plots for vertical displacements and drilling rotations with corrected boundary conditions in log-log scale.

References [1] E. L. Albuquerque. Analysis of dynamic problems in anisotropic materials using the boundary element method. PhD thesis, State University of Campinas, Campinas, Brazil, 2001. [2] M.H. Aliabadi. The Boundary Element Method, Applications in Solids and Structures. The Boundary Element Method. Wiley, 2002. [3] D.J. Allman. A compatible triangular element including vertex rotations for plane elasticity analysis. Computers & Structures, 19:1–8, 1984. [4] C. E. Augarde and A. J. Deeks. The use of timoshenko’s exact solution for a cantilever beam in adaptive analysis. Finite Elements in Analysis and Design, 44:595–601, 2008. [5] P. G. Bergan and C. A. Felippa. A triangular membrane element with rotational degrees of freedom. Computer methods in applied mechanics and engineering, 50:25–69, 1985. [6] M. Huang, Z. Zhao, and C. Shen. An effective planar triangular element with drilling rotation. Finite Elements in Analysis and Design, 46:1031–1036, 2010. [7] T.J.R. Hughes. The finite element method: linear static and dynamic finite element analysis. Dover Civil and Mechanical Engineering Series. Dover Publications, 2000. [8] J.H. Kane. Boundary Elements Analysis in Engineering Continuum Mechanics. Prentice Hall, New Jersey, 1994. [9] H. Leung and P.M. Baiz. Partition of unity and drilling rotations in the boundary element method. International Journal of Solids and Structures, 50:379–395, 2013. [10] Tian R. and Yagawa G. Allman’s triangle, rotational dof and partition of unity. International Journal for Numerical Methods in Engineering, 69:837–858, 2007. [11] M.H. Sadd. Elasticity: Theory, Applications, and Numerics. Elsevier Science, 2009. [12] S. Timoshenko. Theory Of Elasticity 3E. Engineering societies monographs. McGraw-Hill, 1936. [13] K. Wisniewski and H. Turska. Enhanced allman quadrilateral for finite drilling rotations. Comput. Methods Appl. Mech. Engrg., 195:6086–6109, 2006.

153

A boundary element formulation with boundary only discretization for the stability analysis of perforated thin plates 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, linear buckling, elastic plates, radial integration method, boundary element method. Abstract. This paper presents a boundary element method to the analysis of buckling plates. Neither domain discretization, nor particular solutions are necessary in the proposed formulation. This becomes the proposed formulation different from the existent boundary element formulations applied to structural stability analysis. The method is applied to an important problem in computational engineering that is the stability of perforated and non perforated plates. The performance is assessed through comparison with finite element results. The proposed formulation agrees quite well with finite element. However, the stability analysis is a much smaller eigenvalue problem if boundary elements are used instead of finite elements, provided that only the domain and fewer internal points are necessary in the discretization.

1

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 Katsikadelis [2] developed a boundary element method for analysis of buckling of plates with variable thickness. Linear and local buckling analysis of thin plates using the boundary element method also can be found in Lin et al. [3] and [4]. Buckling and post buckling analysis of shear deformable shallow shells by the boundary element method can be found in P. M. Baiz and M. H. Aliabadi [5], [6] and [7]. Post buckling analysis of Raissner plates by the boundary element method and stability of Euler’s method for evaluating large deformation of shear deformable plates by dual reciprocity boundary element method com be found in P. H. Wen et al. [8] and J. Purbolaksono and M. H. Aliabadi [9], respectively. Buckling analysis of perforated thin plates subjected to compressive loads was presented by Purbolaksono and Aliabadi [10], S. Shimizu [11], Brown et al. [12], Khaled M. El-Sawy and Aly S. Nazmy [13] and T. M. Shakerley and C. J. Brown [14]. Most of the cited articles using the Boundary Element Method either discretize the domain into cells and compute the domain integrals by direct integration over the area of each cells or use dual

154


reciprocity boundary element method to transform domain integrals into boundary integrals. Both approaches have some drawbacks. The 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. This paper presents a boundary element formulation to investigate the onset of instability of elastic perforated and unperforated thin plates with rectangular geometry. Stresses caused by external loads are calculated by the formulation of plane elasticity boundary element method. These stresses are introduced as body forces in the classical formulation of plates [15]. 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. Numerical examples are analyzed in which critical loads, buckling modes, and coefficients of buckling are calculated to evaluating the effect of aspect ratio on the elastic buckling of uniaxially loaded perforated thin plates. The formulations presented does not require neither domain discretization nor computation of particular solutions. The accuracy of the proposed formulation is assessed by comparison with results from FEM analysis and literature.

2

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: ∂Nxx ∂x ∂Nxy ∂x ∂4w ∂x4

4

w + 2 ∂x∂2 ∂y 2 +

∂4w ∂y 4

=

1 D

+ + !

∂Nxy ∂y ∂Nyy ∂y

= 0, = 0.

2

(1) 2

2

"

∂ w Nxx ∂∂xw2 + 2Nxy ∂x∂y + Nyy ∂∂yw2 ,

(2)

where w is the displacement in the normal direction of the plate surface; D is the stiffness constant of the plate; Nij are the stress components; i, j = 1, 2 with 1 = x and 2 = y (repeated index means summation). The same notation apply to other indices throughout this work...

2.1

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 Aliabadi [16]:

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 x − y, 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 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)

155

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 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 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:

Kw(Q) + +

Γ

Nc i=1

Vn∗ (Q, P )w(P ) − m∗n (Q, P )

Rc∗i (Q, P )wci (P ) =

Nc i=1

∂w(P ) dΓ(P ) ∂n

∗ Rci (P )wci (Q, P )

∂w∗ (Q, P ) dΓ(P ) ∂n Γ + +., , ∂2w ∂2w ∂2w +λ w∗ Nxx 2 + Nyy 2 + 2Nxy dΩ , ∂x ∂y ∂x∂y Ω +

Vn (P )w∗ (Q, P ) − mn (P )

(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; wci 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. As it can be seen, equation (6) has a domain integral that contains unknown domain curvatures w,ij given by: +-

Id =

Ω

w∗

Nxx

∂2w ∂2w ∂2w + Nyy 2 + 2Nxy ∂x2 ∂y ∂x∂y

.,

dΩ.

(7)

As proposed by [17], this domain integral can be transformed into another domain integral that contains deflections instead of curvature in the following way:

#

$

#

∂ ∂w ∂w ∂ ∂w ∂w + Nxy + Nyy Nxx + w∗ Nxy ∂x ∂x ∂y ∂y ∂x ∂y Ω ∂Nxy ∂w ∂Nxy ∂w ∂Nyy ∂w ∗ ∂Nxx ∂w − w dΩ. + + + ∂x ∂x ∂x ∂y ∂y ∂x ∂y ∂y Ω

Id =

w∗

$

dΩ (8)

The second integral on the right hand side of equation (8) vanishes due to equilibrium equations (1). The first integral can be rewritten as: (

Id =

#

$

#

$)

∂ ∂w ∂w ∂w ∂w ∂ + Nxy + Nyy w∗ Nxx + w∗ Nxy ∂x ∂y ∂y ∂x ∂y Ω ∂x # $ # $ ∂w∗ ∂w ∂w ∂w ∂w ∂w∗ + Nxy + Nyy − Nxx + Nxy dΩ. ∂x ∂x ∂y ∂y ∂x ∂y Ω

dΩ (9)

156


As proposed by [18], the first integral of equation (9) can be transformed into a boundary integral by applying first Green identity and the second integral can be written as a sum of 3 other integrals as:

#

$

#

$

∂w ∂w ∂w ∂w dx − w∗ Nxy dy dΓ + Nxy + Nyy ∂x ∂y ∂x ∂y Γ # $ # $ ) ( ∂ ∂w∗ ∂w∗ ∂w∗ ∂w∗ ∂ − Nxx w − Nxy w dΩ + Nxy + Nyy ∂x ∂y ∂y ∂x ∂y Ω ∂x ∗ ∗ ∗ ∂Nxx ∂u ∂Nxy ∂w ∂Nxy ∂w ∂Nyy ∂w + w dΩ + + + ∂x ∂x ∂x ∂y ∂y ∂x ∂y ∂y Ω

Id =

+

Ω

w∗ Nxx

+

,

w Nxx

∂ 2 w∗ ∂ 2 w∗ ∂ 2 w∗ + Nyy + 2Nxy dΩ, ∂x2 ∂x∂y ∂y 2

(10)

The first integral on the right hand side of equation (10) is a boundary integral that can be expressed in terms of boundary tractions tn e ts . The second integral can be transformed into boundary integral using the first Green identity. The third integral vanishes because equilibrium equations (1). Thus, Id is finally expressed as a sum of a boundary integral and a domain integral in the following way: Id = It + Idw ,

(11)

where

It =

Γ

#

w ∗ tn

∂w ∂w + ts ∂n ∂s

$

#

− w tn

∂w∗ ∂w∗ + ts ∂n ∂s

$

dΓ

(12)

and +

Idw =

,

w Nxx

Ω

∂ 2 w∗ ∂ 2 w∗ ∂ 2 w∗ + 2Nxy dΩ + Nyy 2 ∂x ∂x∂y ∂y 2

(13)

So, substituting equations (11), (12) and (13) into equation (6), we have:

Kw(Q) + +

Γ

Nc i=1

Rc∗i (Q, P )wci (P ) =

+

Vn∗ (Q, P )w(P ) − m∗n (Q, P )

Γ

+λ

Nc i=1

∗ Rci (P )wci (Q, P )

Vn (P )w∗ (Q, P ) − mn (P )

Ω

∗ wNij w,ij dΩ +

! Γ

∂w(P ) dΓ(P ) ∂n

∂w∗ (Q, P ) dΓ(P ) ∂n

"

ti w∗ w,i − ti ww,i∗ dΓ .

(14)

A second integral equation is necessary in order to obtain the thin plate buckling boundary element formulation. This equation is obtained by the derivative of equation (14) in respect to the normal direction at the source point Q. This equation is given by:

K

∂w (Q) + ∂m +

Γ

Nc ∂Rc∗i i=1

+

+

Γ

∂m

∂Vn∗ ∂Mn∗ ∂w(P ) (Q, P )w(P ) − (Q, P ) dΓ(P ) ∂m ∂m ∂n (Q, P )wci (P ) =

Nc i=1

Rci (P )

∗ ∂wci (Q, P ) ∂m

,

∂w∗ (Q, P ) ∂ 2 w∗ − mn (P ) (Q, P ) dΓ(P ) Vn (P ) ∂m ∂n∂m

157 +

+λ

-

∗ ∂w,ij ∂w,i∗ ∂w∗ dΩ + w,i − ti w wNij ti ∂m ∂m ∂m Ω Γ

.

,

dΓ ,

(15)

∂() is the derivative with respect to the direction of the outward vector m that is normal to where ∂m the boundary Γ at the source point Q. To avoid the introduction of the derivative of the transversal displacement as unknown in equations (14) and (15), the following approximation is used:

w,i =

N NE

φj,i w(j)

(16)

j=1

where φj stands for shape functions used in the approximation of physical variables (w, ∂w/∂n, Vn , mn ), N N E stands for the number of nodes in the physical elements, and w(j) stands for the nodal value of w at node j. Domain integrals arise in the formulation owing to the contribution of in-plane stresses to the out of plane direction. The domain integral of equations (14) and (15) are given in the form:

I=

Ω

wv ∗ dΩ

(17)

where ∗ v ∗ = Nij w,ij

(18)

for equation (14), and v ∗ = Nij

∗ ∂w,ij ∂m

(19)

for equation (15). The domain can be discretized into cells as proposed by [1]. However, in this case, the boundary element method loses its main feature that is the boundary only discretization. In order to transform these integrals into boundary integrals, consider that the transversal displacement w is approximated over the domain Ω as a sum of M products between approximation functions fm and unknown coefficients γm , that is: w(P ) =

M

γ m fm

(20)

m=1

for approximation functions based on pure radial basis function, or w(P ) =

M

γm fm + ax + by + c

(21)

m=1

with M m=1

γ m xm =

M

γ m ym =

m=1

M

γm = 0

(22)

m=1

for approximation functions based on radial basis function combined with augmentation functions. Now, considering that the body force is approximated, for simplicity, by equation (20), the domain integral (17) can be written as:

I= or

Ω

w(P )v ∗ (Q, P )dΩ =

M m=1

γm

Ω

fm v ∗ (Q, P )dΩ,

(23)

158


I=

M

γm

Ω

m=1

fm v ∗ (Q, P )ρdρdθ,

(24)

or M

I=

γm

θ

m=1

r 0

fm v ∗ (Q, P )ρdρdθ,

(25)

where r is the value of ρ in a point of the boundary Γ. Defining Fm (Q) as the following integral:

Fm (Q) =

r

fm v ∗ (Q, P )ρdρ,

0

(26)

we can write: I=

M

γm

θ

m=1

Fm (Q)dθ.

(27)

Considering an infinitesimal angle dθ (Figure 1), the relation between the arch length rdθ and the infinitesimal boundary length dΓ, can be written as: J

dΓ 2

α

I r dθ 2

K K J

r

n α

rdθ r

I dθ Q

dΓ

Ω

Γ

Figure 1: Geometric relation for the domain transformation.

cos α =

r dθ 2 dΓ 2

,

(28)

or cos α dΓ, (29) r where α is the angle between unity vectors r and n. Using the inner product properties of the unity vectors n and r, showed in Figure 1, we can write: dθ =

dθ =

n.r dΓ. r

(30)

159

Substituting equation (30) into equation (27), the domain integral (17) can be written as a boundary integral given by: M

I=

m=1

γm

Γ

Fm (Q) n.rdΓ, r

(31)

or, in a matrix form:

I=

* Γ

*

F1 (Q) r n.rdΓ

Γ

F2 (Q) r n.rdΓ

*

...

Γ

FM (Q) n.rdΓ r

⎧ ⎪ γ ⎪ ⎪ 1 ⎨ γ2 ⎪

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

.. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ γ ⎪ ⎭ M

.

(32) To compute γm , it is necessary to consider the transversal displacement w in M points of the domain and of the boundary. In the case of this work, these points are the boundary nodes and some internal points. Thus, equation (20) can be written as: w = Fγ,

(33)

γ = F−1 b.

(34)

and γ can be computed as:

Substituting (34) into equation (32), we have: I=

* Γ

F1 (Q) r n.rdΓ

* Γ

F2 (Q) r n.rdΓ

...

* Γ

FM (Q) n.rdΓ r

F−1 b. (35)

Writing equation (35) for all source points, i.e., all boundary nodes and internal points, we have the following matrix equation: I = RF−1 w = Sw,

(36)

−1

where S = RF , I is a vector that contains the value of I in all source points Q, and R is a matrix that contains the value of integrals of equation (35) when this equation is written for all source points Q. 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 with coordinate (x, y) is computed by: Nij (x, y) = f (x, y)1×M F−1 M ×M Nij M ×M .

(37)

where Nij M ×M is a matrix that contains the value of Nij computed at all boundary nodes and internal points and M is the sum of the number of boundary nodes with the number of internal points.

2.2

Matrix Equations

Considering all body forces that appears in equations (3), (14), and (15), the vector P for these equations are given by: ⎡

p1 Sp1 bb Sbi Sp2 bi Sp2 ii p1 Sp1 cb Sci

⎢ Sp2 ⎢ P = ⎢ bb ⎣ Sp1 ib

Sp1 bc Sp2 bc Sp1 ic Sp1 cc

⎤

⎧ ⎫ ⎥⎪ ⎨ wb ⎪ ⎬ ⎥ wi ⎥ ⎦⎪ ⎩ w ⎪ ⎭ c

(38)

160


where the superscript index of matrix S stands for the type of equation that is being used, i.e., p1 stands for the first plate equation, given by equation (14) and p2 stands for the second plate equation, given by equation (15). In matrix S, the first subscript index stands for the location of the source points (b if source points are at a smooth part of the boundary, i if they are in the domain and c if they are at corners). The second subscript index shows where are the body forces that are multiplied by terms of the matrix S. For the second index, the same letters of the first subscript index are used with the same meaning. The right hand side vector has nodal values of w that stands for displacement in the transversal direction. Subscript indices in the right hand side vector indicate the location of nodes where displacements are computed (smooth boundary, internal point, or corner). Finally, if the boundary Γ is discretized in boundary elements and equations (14) and (15) is written for all source points, the following matrix equation can be obtained: ⎡ ⎢ ⎢ ⎢ ⎣

p1 Hp1 bb 0 Hbc p2 Hp2 0 H bb bc p1 Hib I Hp1 ic p1 Hp1 cb 0 Hcc

⎤

⎧ ⎥⎪ ⎨ vb ⎥ wi ⎥ ⎦⎪ ⎩ w

c

⎡

⎫ ⎪ ⎬

⎢ ⎢ =⎢ ⎪ ⎭ ⎣

p1 Sp1 bb Sbi Sp2 bi Sp2 ii p1 Sp1 cb Sci

⎢ Sp2 ⎢ +λ ⎢ bb ⎣ Sp1 ib

⎡

Sp1 bc Sp2 bc Sp1 ic Sp1 cc

Gp1 bb Gp2 bb Gp1 ib Gp1 cb ⎤

Gp1 bc Gp2 bc Gp1 ic Gp1 cc

⎤ ⎥ ⎥ ⎥ ⎦

pb pc

2

⎫ ⎧ ⎬ ⎥⎪ ⎨ wb ⎪ ⎥ wi ⎥ ⎦⎪ ⎭ ⎩ w ⎪ c

(39) where H and G are influence matrices of the BEM; the vector v contains transversal displacements and rotations of the nodes (not only transversal displacement as vector w). Vector p contain boundary node reactions of plate equation. Domain integrals due to qi ’s are transformed exactly into boundary integrals using the procedure presented in [19]. It is worth noting that terms that come from equation (12) are included in matrix H. Equation (39) can be written in a more concise form as: Hv = Gp + λSw.

(40)

Finally, columns of zero can be introduced into matrix S so that equation (40) can be rewritten as: ¯ Hv = Gp + λSv

(41)

¯ is the matrix S with columns of zero in positions that are multiplied by the rotation, i.e., by where S the derivative of vector w with respect to the normal to the boundary direction.

3

Approximation functions

Two approximation functions are used in this work. The first is a radial basis function that has been used extensively in the DRM and is given by: fm1 = 1 + R.

(42)

The second is the well known thin plate spline: fm3 = R2 log(R),

(43)

used with the augmentation function given by equations (21) and (22). It has been shown in some works from literature that this approximation function can give excellent results for many different formulations [see Partridge (2000), and Goldberg , Chen, and Bowman (1999)].

161

4

Eigenvalue problem

The only loads considered in the linear buckling equations are those related to the in-plane stress Nij and tractions ti that are multiplied by the critical load factor λ. All the known values of w, ∂w/∂n, Mn , Vn , wci , Rci (boundary conditions) are set to zero. Dividing the boundary into Γ1 e Γ2 (Figure 2), equation (41) can be written as: 3 Γ1: u3 = ∂u ∂n = 0

Ω

Γ2: Vn = Mn=0 Figure 2: Domain with constrained and free degrees of freedom. +

H11 H12 H21 H22

,

=λ

+

v1 v2

2

+

−

¯ 11 S ¯ 12 S ¯ 21 S ¯ 22 S

G11 G12 G21 G22 ,

v1 v2

2

,

p1 p2

,

2

(44)

where Γ1 stands for the part of the boundary where displacements or rotations are zero and Γ2 stands for the part of the boundary where bending moment or tractions are zero. Indices 1 and 2 stand for boundaries Γ1 and Γ2 , respectively. As v1 = 0 and p2 = 0, equation (44) can be written as: ¯ 12 v2 , H12 v2 − G11 p1 = λS ¯ 22 v2 H22 v2 − G21 p1 = λS

(45)

ˆ 2, ˆ 2 = λSv Hv

(46)

ˆ = H22 − G21 G−1 H12 , H 11 ¯ ˆ = S ¯ 22 − G21 G−1 S S 11 12 .

(47)

or,

ˆ e M, ˆ are given by: where, H

The matrix equation (46) can be rewritten as an eigen vector problem 1 v2 , λ

(48)

ˆ ˆ −1 S. A=H

(49)

Av2 = where,

Provided that A is non-symmetric, eigenvalues and eigenvectors of equation (48) can be found using standard numerical procedures for non symmetric matrices.

162

5


Numerical results

In order to verify the accuracy of the proposed method, a comparison with existing results in the literature on buckling of square unperforated and perforated plates has been performed. The results of Purbolaksono and Aliabadi [10] and El-Sawy and Nazmy [13] are used for assess the accuracy of the proposed method. The numerical results are presented in terms of the dimensionless buckling parameter Kcr , defined by [15] as: Kcr =

Ncr a2 π2D

(50)

where, Ncr is the critical load and a is the edge length of the square plates. Initially, the proposed formulation is applied to the analysis of buckling problems of unperforated square plates subjected to compression loads (figure 3) with different boundary conditions: all edges clamped (CCCC); all edges simply supported (SSSS); two edges clamped and two edges simply supported (CSCS); two edges simply supported, one edge free and one edge clamped (FSCS); two edges simply supported and two edges clamped (CSCS) and SCSC); two edges simply supported and two edges free (FSFS); and one edge free and three edges simply supported (FSSS). The square unperforated plate is discretized into 28 quadratic discontinuous boundary elements of equal length and 49 domain points, and the ratio between length a and thickness h of the square plate is a/h = 100. After, the formulation is applied to the analysis of buckling problems of two types of simply supported perforated square plates subjected to compression loads: square and rectangular perforations. The ratio between length a and thickness h of the square plate is a/h = 100, and the ratio between the edge length of the hole and the edge length of the plate is b/a and c/a (figure 4). The mesh used has 48 quadratic discontinuous boundary elements, 28 elements of equal length at the outer boundary and 20 elements of equal length at the inner boundary, with domain points distributed according to the aspect ratio of the hole.

Figure 3: Unperforated buckling model

Figure 4: Perforated buckling model

Critical load parameters Kcr obtained by the radial integration method (RIM) with the boundary element formulation using different number of domain points are shown in Tables 1, 2 and 3, for unperforated plates and square and rectangular perforations, respectively.

6

Discussion of the results

As it can be seen in Table 1 the comparison between the results obtained with the boundary element formulation using the radial integration method (RIM) for unperforated square thin plates, are in good

163

Table 1: Buckling coefficients Kcr for unperforated square plate Case 1 2 3 4 5 6

Boundary conditions CCCC SSSS CSCS SCSC FSCS FSSS

Kcr (FEM)

Kcr (DRM)

Kcr (RIM)

Kcr (Analytical)

10.39 4.01 7.80 6.88 1.72 1.42

10.14 4.00 7.68 6.78 1.71 1.43

10.12 4.02 7.79 6.79 1.74 1.46

10.07 4.00 7.69 6.74 1.70 1.44

Table 2: Buckling coefficients Kcr for a square thin plate with square perforation b/a 0 0.10 0.20 0.30 0.40 0.50 0.60 0.70

c/a 0 0.10 0.20 0.30 0.40 0.50 0.60 0.70

Kcr (FEM) 4.00 3.80 3.45 3.20 3.00 2.92 2.87 2.85

Kcr (BEM) 4.00 3.76 3.40 3.15 2.91 2.75 2.65 2.60

Error (%) 0.00 1.00 1.45 1.50 3.00 5.80 7.70 8.70

agreement with the results obtained by other workers using boundary and finite element methods and the analytical solution, as reported in Purbolaksono and Aliabadi [10]. Figure 5 shows the variation of buckling coefficient (Kcr ) with a normalized square hole dimension (b/a) for a square thin plate, of dimension a and with a central square hole of dimension b. The comparison between the results obtained with the proposed formulation are in good agreement with the results obtained by other workers using the finite element method [13]. It is shown that for an increasing size of perforation the proposed method consistently predict lower Kcr values.

Figure 5: Comparison between BEM and FEM [13] Figure 6 shows the variation of buckling coefficient Kcr with a normalized rectangular hole dimension (c/a) for a square thin plate, of dimension a, with a central rectangular hole of width 0.25 ∗ a and variable dimension y. As it can be seen, with a rectangular perforation from the aspect ratio 0.25 ∗ a, an increase in the major dimension of the perforation causes an increase in the buckling coefficient. This is because the ”side plates” increase in aspect ratio and become dominant, as reported in [14].

164


Table 3: Buckling coefficients Kcr for a square thin plate with rectangular perforation. b/a 0 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

c/a 0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

Kcr (FEM) 4.00 3.50 3.35 3.30 3.30 3.35 3.40 3.60 3.82 4.23 4.75 5.60

Kcr (BEM) 4.00 3.57 3.43 3.38 3.35 3.41 3.50 3.65 3.92 4.35 4.88 5.90

Error (%) 0.00 2.00 2.40 1.15 0.90 1.80 2.94 1.39 2.62 2.84 2.74 5.36

6

Buckling Coefficient K

5.5

5

4.5

BEM FEM [13]

4

3.5

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Normalized rectangular hole dimension (c/a)

Figure 6: Comparison between BEM and FEM [13]

7

Conclusions

This paper presented a boundary element formulation with boundary only discretization for the stability analysis of plates with non-uniform stress field (perforated plates). Domain integrals are transformed into boundary integrals by the radial integration method. The proposed formulation presents some advantages over cells and dual reciprocity. It is a formulation without domain integration and no particular solution is necessary. So, it is easier to implement than the dual reciprocity boundary element method and there is no need to discretize the domain, as in cell formulations. In order to validate this method, the formulation was applied for an unperforated plate and square plate with a square and rectangular holes. The results were compared with literature, showing good agreement.

Acknowledgement The first author would like to thank the Coordination of Improvement of Higher Education Personnel(CAPES), Brazil and State University of Campinas (UNICAMP), Brazil, for financial support for

165

this work. The authors wish to thank Professors P. M. Baiz and M. H. Aliabadi, from Imperial College of London, which contributed to the current state of the boundary element method and specifically to the development of this paper.

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] J. Lin, R. C. Duffield, and H. Shih. Buckling analysis of elastic plates by boundary element method. Engineering Analysis with Boundary Element, 23:131–137, 1999. [4] P. M. Baiz and M. H. Aliabadi. Local buckling of thin-walled structures by the boundary element method. Engineering Analysis with Boundary Elements, 33:302–313, 2009. [5] P. M. Baiz and M. H. Aliabadi. Linear buckling of shear deformable shallow shells by the boundary domain element method. CMES - Computer Modeling in Engineering and Sciences, 13:19–34, 2006. [6] P. M. Baiz and M. H. Aliabadi. Buckling analysis of shear deformable shallow shells by dual reciprocity boundary element method. Engineering Analysis with Boundary Elements, 31:361– 372, 2007. [7] P. M. Baiz and M. H. Aliabadi. Post buckling analysis of shear deformable shallow shells by the boundary element method. International Journal for Numerical Methods in Engineering, 84:379–433, 2010. [8] P. H. Wen, M. H. Aliabadi, and A. Young. A post buckling analysis of reissner plates by the boundary element method. Journal of Strain Analysis for Engineering Design, 41:239–252, 2006. [9] J. Purbolaksono and M. H. Aliabadi. Stability of euler’s method for evaluating large deformation of shear deformable plates by dual reciprocity boundary element method. Engineering Analysis With Boundary Elements, 34:819–823, 2010. [10] J. Purbolaksono and M. H. Aliabadi. Application of drbem for evaluating buckling problems of perforated thin plates. European journal of scientific research, 31:398–408, 2009. [11] S. Shimizu. Tension buckling of plate having a hole. Thin-Walled structures., 45:827–833, 2007. [12] C. J. Brown, A. L. Yettram, and M. Burnett. Stability of plates with rectangular holes. Journal of structural engineering., 113:1111–1116, 1986. [13] EL-SAWY Khaled M. and NAZMY Aly S. Effect of aspect ratio on the elastic buckling of uniaxially loaded plates with eccentric holes. Thin-walled structures, 39(12):983–998, 2001. eng. [14] T. M. Shakerley and C. J. Brown. Elastic buckling of plates with eccentrically positioned rectangular perforations. Int. journal mech. sci., 38:825–838, 1996. [15] S. Timoshenko and J. M. Gere. Theory of Elastic Stability. McGraw-Hill, New York, second edition, 1961. [16] M. H. Aliabadi. Boundary element method, the application in solids and structures. John Wiley and Sons Ltd, New York, 2002.

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[17] S. Syngellakis and M. Kang. A boundary element solution of the plate buckling problem. Engineering Analysis, 4:75–81, 1987. [18] A. Elzein. Plate stability by boundary element method. Springer-Verlag, Berlin, 1991. [19] 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.

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Combining Analytic Preconditioner and Fast Multipole Method for the 3-D Helmholtz equation M. Darbas1 , E. Darrigrand2 , Y. Lafranche2 1

LAMFA UMR CNRS 7352, Université de Picardie, Amiens, France, e-mail: [email protected]

2

IRMAR UMR CNRS 6625, Université de Rennes 1, Rennes, France, e-mails: [email protected], [email protected]

This work was funded in part by the ANR project MicroWave.

Keywords: integral equations, analytic preconditioner, On-Surface Radiation Condition, Fast Multipole Method, high-frequency scattering, Helmholtz equation.

Abstract. The numerical resolution of 3-D acoustic or electromagnetic scattering problems at high frequency remains to be a challenging issue. In the configuration of sound-hard acoustic problems solved using a Combined Field Integral Equation (CFIE), we propose a combination of an OSRC preconditioning technique and a Fast Multipole Method which leads to a fast and efficient algorithm independently of both a frequency increase and a mesh refinement. The OSRC-preconditioned CFIE exhibits very interesting spectral properties even for trapping domains. Moreover, this analytic preconditioner shows highly-desirable advantages: sparse structure, ease of implementation and low additional computational cost. In this paper, we investigate the numerical behavior of the eigenvalues of the related integral operators, CFIE and OSRC-preconditioned CFIE, in order to illustrate the influence of the proposed preconditioner. A large variety of tests validates the effectiveness of the method and justifies the interest of such a combination.

Introduction We present a detailed numerical study of an iterative solution to 3-D sound-hard acoustic scattering problems at high frequency considering Combined Field Integral Equations. We propose a combination of an OSRC preconditioning technique and a Fast Multipole Method (FMM) which leads to an efficient and robust algorithm independently of both a frequency increase and a mesh refinement. To validate its effectiveness, we have first applied the resolution algorithm to various and significant test-cases using a GMRES solver in the context of Helmholtz equation. The same scheme will be applied to Maxwell’s equations.

1

OSRC preconditioner and Fast Multipole Method

Let Ω− ⊂ R3 be a bounded domain representing an impenetrable body with Lipschitz continuous boundary Γ := ∂Ω− . We denote by Ω+ := R3 \ Ω− the exterior domain of propagation. We consider the scattering of an incident time-harmonic acoustic wave uinc of wavenumber k by the obstacle Ω− . The scattered field u+ satisfies the Helmholtz exterior boundary-value problem ⎧ Δu+ + k 2 u+ = 0, in D (Ω+ ), ⎪ ⎪ ⎨ + inc in H −1/2 (Γ), ∂n u|Γ = g = −∂n u|Γ , $ # (1) ⎪ x ⎪ ⎩ lim |x| ∇u+ · − iku+ = 0. |x| |x|→+∞ Several integral equations have been derived for solving this scattering problem (see for instance [3]). We consider the following CFIE (Combined Field Integral Equation): inc 1/2 (Γ) solution to Find ϕ = −(u+ |Γ + u|Γ ) ∈ H # $ $ # i i I − α∂n uinc (2) + M + αD ϕ = −(1 − α) uinc (1 − α) |Γ , on Γ, k 2 k |Γ

168


where α ∈ R\{0, 1} is a coupling parameter. I is the identity operator, and M and D the first and second traces of the double-layer potential respectively. The CFIE (2) is uniquely solvable in H 1/2 (Γ) for any frequency k > 0. This equation is a first-kind integral equation and does not provide an interesting spectral behavior. It involves the first-order, strongly singular and non-compact operator D. A preconditioning strategy consists in introducing an efficient approximation V3 of the Neumann-to-Dirichlet map, derived from On-Surface Radiation ! "−1/2 Condition (OSRC) methods (see [1] and the references therein): V3 = 1 I + Δ2Γ . The operator ΔΓ is ik

kε

the Laplace-Beltrami operator over the surface Γ and the parameter kε = k + iε is complex-valued (ε ∈ R∗ ). The parameter ε can be optimized in the case of a spherical obstacle [1]. The preconditioner V3 has a sparse structure, is very easy to implement and implies a low additional computational cost. This choice leads to the well-posed second-kind Fredhlom integral equation $ # I inc 3 + M − V3 D ϕ = −uinc (3) |Γ + V ∂n u|Γ , on Γ. 2 The resolution of the equation involves different numerical ingredients. The preconditioning operator V3 is computed by a Padé paraxial approximation of the square-root operator with a rotating branch-cut technique [9]. This leads to the resolution of sparse systems which are solved using a sparse direct solver such that the impact of the preconditioner is nearly negligible compared to the overall computational cost. The difficulties related to the integral operators are handled thanks to a Fast Multipole Method (e.g. [2], [8], [6]). The integral equations were solved using MUMPS (MUltifrontal Massively Parallel sparse direct Solver – http://mumps.enseeiht.fr/) and a single-level Fast Multipole Method.

2

Eigenvalues investigation

A thorough study of the eigenvalues behavior of the integral operators involved in (2) and (3) has been realized to illustrate the impact of the OSRC preconditioning technique on the spectrum of the CFIE operator according to the wavenumber k, the number of discretization points per wavelength nλ and the order of the Padé approximation Np considered for the localization of the square-root operator [5]. To this aim, we used Gmsh [7] for the meshes, and the libraries M E´ LINA ++ (http://anum-maths.univ-rennes1.fr/melina/ – finite element library) and ARPACK++ (http://www.ime.unicamp.br/∼chico/arpack++/ – “Implicit Restarted Arnoldi Method”) for the eigenvalues investigation. When the scatterer is the unit sphere, an explicit expression of the eigenvalues of the CFIE operators is known. We consider the sphere with an incident wave of direction ξinc = (0, 0, −1). In Fig. 1, we can observe that the numerical eigenvalues of the OSRC-preconditioned operator are well clustered at a point near to (1, 0) which is the accumulation point of the analytical ones. This is not the case for the CFIE. We observe a dispersion of the eigenvalues in the elliptic part.

$

! "# !

$ !

Figure 1: Unit sphere: distribution of the eigenvalues, k = 11.85, nλ = 10 Other examples were considered (trapping domains, industrial oriented objects). Fig. 2-4 give the distribution of the eigenvalues √ for a submarine of length 43m and for a spherical cavity, each of them with the incident direction ξ inc = −( 3/2, 0, 1/2). The clustering of the eigenvalues is very good even if some isolated eigenvalues might be close to 0 in the case of the trapping domain.


169

"

Y

Z

X

Figure 2: Submarine: mesh; distribution of the eigenvalues, k = 2.5, nλ = 10

A · X ·

+ O · B

Z Y

2D C-shape contour

X

Figure 3: Spherical cavity: mesh and 2D contour

%! $

Figure 4: Spherical cavity: distribution of the eigenvalues, k = 5.8, nλ = 10; resonant frequencies As expected, this impact on the eigenvalues induces a consequent increase of the speed of convergence of the GMRES solver when we use the preconditioner at high frequency combined to a single-level Fast Multipole Method for the calculation involving the integral operators D and M (see Fig. 5-6). Submarine, n=10

Submarine, k=2.5 500

CFIE CFIE+FMM CFIE+OSRC+FMM

300 200 100 0 0

GMRES iterations

GMRES iterations

400

300 200 100 0

2

4

6 k

8

CFIE+FMM CFIE+OSRC+FMM

400

10

20 n

Figure 5: Submarine: condition number vs. k or nλ

30

170

Eds: A Sellier & M H Aliabadi Sphere with cavity, n=10

300

CFIE CFIE+FMM CFIE+OSRC+FMM

400 300 200 100 0

GMRES iterations

GMRES iterations

500

Sphere with cavity, k=7 CFIE CFIE+FMM CFIE+OSRC+FMM

250 200 150 100 50 0

5

10

15 k

20

5

10

15

n

20

25

30

Figure 6: Spherical cavity: condition number vs. k or nλ

3

Conclusion

The combination of the OSRC preconditioner and the Fast Multipole Method leads to an impressive acceleration of the convergence of the iterative solver. Either for academical objects or more industrial test-cases, the scheme proves to be very efficient. It can be extended to the iterative resolution of the Maxwell exterior problem using the strategy developed in [4]. The OSRC preconditioning approach has already been successfully applied to the electromagnetism problem with impedance condition [10]. In a future work, we aim to carry out the study of the spectral behavior of the CFIE and OSRC-preconditioned CFIE operators for Maxwell exterior problem with perfectly conducting condition, and the study of the contribution of the FMM.

References [1] X. Antoine, M. Darbas and Y.Y. Lu, An Improved Surface Radiation Condition for High-Frequency Acoustics Scattering Problems, Comput. Meth. Appl. Mech. Eng. 195 (33-36) (2006), pp. 4060-4074. [2] R. Coifman, V. Rokhlin, and S. Wandzura, The Fast Multipole Method for the Wave Equation: A Pedestrian Prescription, IEEE Antennas and Propagation Magazine, 35(3) (1993), pp. 7-12. [3] D. L. Colton and R. Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics, John Wiley and Sons Inc., 1983. [4] M. Darbas, Generalized CFIE for the Iterative Solution of 3-D Maxwell Equations, Appl. Math. Lett. 19 (2006), pp.834-839. [5] M. Darbas, E. Darrigrand and Y. Lafranche, Combining OSRC preconditioning and Fast Multipole Method for the Helmholtz equation, J Comput. Phys., 236 (2013), pp. 289-316. [6] E. Darve, The Fast Multipole Method: Numerical Implementation, J. Comput. Phys., 160 (1) (2000), pp. 195-240. [7] C. Geuzaine and J.-F. Remacle, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering, 79, Issue 11 (2009), pp. 1309-1331. [8] S. Koc, J.M. Song, and W.C. Chew, Error Analysis for the Numerical Evaluation of the Diagonal Forms of the Scalar Spherical Addition Theorem, IEEE Trans. on Antennas and Propag., 45 (3) (1997), pp. 533-543. [9] F.A. Milinazzo, C.A. Zala and G.H. Brooke, Rational square-root approximations for parabolic equation algorithms, J. Acoust. Soc. Am., 101(2) (1997), pp. 760-766. [10] S. Pernet, A well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition, ESAIM: M2AN 44 (2010), pp. 781-801.


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Nonlinear BEM Formulation based on Tangent Operator Applied to Cohesive Crack Growth Modelling Hugo Luiz Oliveira1 and Edson Denner Leonel2 University of São Paulo, School of Engineering of São Carlos, Department of Structural Engineering. Av. Trabalhador São Carlense, 400. São Carlos-SP, Brazil. 13.566-590. 1 2 [email protected] [email protected] Keywords: Tangent operator, Dipoles of stresses, Cohesive crack model.

Abstract. Fracture mechanics and crack propagation problems have been widely studied by the scientific community in recent years, because crack growth phenomenon can explain the failure of structures. In order to model accurately the structural behaviour of complex engineering structures, including complex geometries and boundary conditions, numerical techniques are required. In this regard, the boundary element method (BEM) has been widely used to solve complex engineering problems, especially those where its mesh dimension reduction includes advantages on the modelling. This paper addresses the analysis of crack propagation in quasi-brittle materials using an alternative BEM formulation. In this type of problem, the damaged zone ahead the crack tip is modelled based on the fictitious crack model. Therefore, the residual resistance of the damaged zone is represented by cohesive stresses, which tends to close the crack faces. The alternative BEM formulation proposed aims at modelling the cohesive stresses using the domain term of the direct integral representation. This term is modified, in order to be non null only at the fictitious crack path. As a result of the domain term manipulation, appears a dipole of stresses, which will govern the cohesive stresses. It leads to a nonlinear formulation, where the cohesive stresses are determined according the crack opening displacements. In this paper, the nonlinear problem is solved using a tangent operator, which includes the nonlinear laws into the algebraic BEM equations. This operator is derived considering linear, bi-linear and exponential cohesive laws. The results achieved by the proposed formulation are compared with experimental and numerical ones, in order to validate and prove its robustness and accuracy.

Introduction The collapse of structures caused by discontinuities’ growth can only be accurately analysed if the discontinuities are properly modelled by the theory adopted. In this regard, fracture mechanics is recognized as a robust and accurate theory for modelling crack growth phenomenon and consequently the collapse of structures due to crack (discontinuity) propagation. This subject has received large attention by the scientific community in recent years [1,2]. Especially in the context of the development of numerical formulations and mathematical techniques applicable for modelling crack growth processes in complexes systems and bodies [3]. Due to the large advances in the field of science and engineering, the need to analyse larger and more complex structures in numerical form has become a challenger issue. Therefore, the development of numerical tools and more efficient algorithms regarding the computational cost and accuracy of results is quite essential. In this context, this paper presents an alternative BEM formulation based on dipoles of stresses applied to model cohesive crack propagation. The proposed formulation aims at modelling the cohesive stresses using the domain term of the direct BEM integral representation. This term is modified, in order to be non null only at the fictitious crack path. As a result of the domain term manipulation, appears a dipole of stresses, which will govern the cohesive stresses. This formulation solves the crack growth problem using three algebraic equations per source point positioned at the crack path. By comparing it with the classical dual BEM, that uses four algebraic equations per crack source point, it is a great advantage. The proposed formulation is nonlinear, since the cohesive stresses are determined according the crack opening displacement values. In this work, the nonlinear problem is solved using a tangent operator. This type of operator includes the nonlinear laws into the algebraic BEM equations, leading a faster and accurate solution. The tangent operator was derived for linear, bi-linear and exponential cohesive laws, which are the main contribution of this paper. The results achieved by the proposed nonlinear formulation are compared with experimental and numerical ones, in order to validate and prove its robustness and accuracy.

172


BEM Integral Equations Considering an elastic body with domain : and boundary * subjected to an initial stress field V 0jk acting into the domain, the following integral representation can be written, as presented by [4].

clki uki ³ plk* uk d * *

³u

* lk

*

* pk d * ³ V 0jk H ljk d :0

(1)

:0

* where ulk* , plk* and H lkj are the well known fundamental solutions for displacements, tractions and strains,

respectively [4]. : 0 is the initial stress region, which for the problem studied represents the FPZ. The initial stress C

region is a wide thin region limited by a boundary * , where *

C

C

C

*1 *2 as presented in Fig. (1).

Figure 1. Initial stress region.

Figure 2. Global and local coordinates.

In order to develop the cohesive fracture formulation, the initial stress integral shown in Eq. (1) has to be conveniently manipulated. Initially, it can be integrated over the initial stress region leading to:

³V

³uV

H d :0

0 * jk ljk

* lj

:0

*

C

Kk d * ³ ulj*V 0jk ,k d :0

0 jk

(2)

:0

C

in which K k is the normal vector to boundary * and the term V 0jkKk represents the tractions at the initial stress C

region boundaries, p . Therefore, the first integral on right hand side of Eq. (2) can be rewritten as: 01 j

³ u f , S p S d* * lj

*

01 j

³ u f , S p S d *

C

* lj

C

01 j

C 1

³ u f , S p S d *

* lj

C *1

01 j

C 2

(3)

C *2

C

C

where S and S are field points positioned at boundaries *1 and * 2 , respectively. Assuming that the width of initial stress region is too small compared to its length, the kernels defined at S and S can be rewritten for the middle path of the region, S . Therefore, these kernels can be redefined using Taylor series expansion as follows:

ulj* f , S

ulj* f , S

wulj* f , S w x1

a and ulj* f , S

ulj* f , S

wulj* f , S w x1 C

Considering the expansion terms presented by Eq. (4) and assuming that d *1 can be rewritten as:

³ u f , S p S d* * lj

*

01 j

C

C

³ *

wulj* f , S w x1

p 01 j S 2ad *

a

(4) C

d * and d *2

d * , Eq.(3)

(5)

The domain integral presented on right hand side of Eq. (2), which includes stress derivatives, must also be C C treated. This integral term can be transformed into integrals written on boundaries *1 and * 2 . In this regard, the width of the initial stress region is assumed as thin enough if compared to its length. Therefore, the stress’ variation along direction x1 is null. Therefore:

V 0jk ,k

wV 0jk wxk

wV 0jk w x m w x m wxk

wV 0jk w x 2 w x 2 wxk

wV 0jk w x2

tk

w V 0jk tk w x2

(6)

in which tk contains the outward normal components along x 2 , Fig. (2). Based on these assumptions, the domain integral presented on right hand side of Eq. (2) can be rewritten as:


³ ulj*V 0jk ,k d :0

w V 0jk tk

³ ulj*

:0

w x2

:0

d :0

³ ulj*

173

w V 0jk tk

x2

w x2

2ad x 2

(7)

Equation (7) can be integrated over direction x 2 leading to:

³ ulj*

w V 0jk tk w x2

x2

2ad x

1 x2 wulj* 0 ª¬ulj* V 0jk tk 2a º¼ 2 ³ V jk tk 2ad x 2 x2 x2 w x 2

2

(8)

As the width of the initial stress zone is small, the first term on right hand side of Eq. (8) becomes null. C

d * and V 0jk tk

Considering that d x 2

wulj*

³ ulj*V 0jk ,k d :0

³ wx

:0

*

, Eq. (8) can be rewritten as: p02 j

p 02 j 2ad *

(9)

2

where p 02 j indicates the tractions aligned to direction x 2 . Therefore, based on results presented in Eq. (5) and Eq. (9), the domain integral shown in Eq. (1) can be transformed as:

³V

H d :0

0 * jk ljk

:0

³

wulj* f , S w x1

*

p 01 j S 2ad * ³ *

wulj* w x2

p 02 j 2ad *

wulj*

³ wx *

p 0j k 2ad *

(10)

k

In order to avoid the local aspect of this deduction, Eq. (10) can be finally rewritten as:

³V

H d :0

0 * jk ljk

:0

wulj*

³ wx *

wulj* wxm 0 k p j 2ad * m w xk *

³ wx

p 0j k 2ad *

k

(11)

Until now, a domain integral defined over any thin zone was transformed into a line integral. Based on this formulation, stresses and displacements analyses for domains where nonlinear behaviours are assumed inside particular narrow regions can be performed. For instance, crack growth analysis in which cohesive stresses are assumed over a finite strip. Although a numerical algorithm based on the definition of a thin finite process zone could be derived, as shown above, the nonlinear zone must be wide enough to guarantee initial stress finite values. When the thickness goes to zero, infinite initial stresses are required due to the nature of the problem. Thus, in order to write a proper integral term for which the strip thickness is assumed as zero, a new tensor, denominated dipole, must be defined. Therefore:

wxm 0 k p j 2a w xk

q mj

(12)

This new variable is given by finite values when the initial stress goes to infinity. Thus, considering this new variable and the discussion presented above, Eq. (1) can be redefined as:

clki uki ³ plk* uk d * *

* lk

*

pk d * ³ Gmlj q mj d *

(13)

*

wulj* wxm . This new kernel is defined by:

where Gmlj

Gmlj

³u

ulj* ,m

1 ^3 4X r,mGlj r, jGlm r,lG jm 2r, jr,l r,m ` 8S G 1 X r

(14)

In order to derive the stress integral representation, a similar development can be followed. In this regard, Eq. (13) has to be derived and the Hooke´s law applied. This procedure leads to:

V im ³ Simk f , S uk S d * *

³ D f , S p S d * ³ G f , S q S d * g V p imk

*

lj im

k

l j

lj im

lj

(15)

*

lj in which gim V lj p is a free term that appears due to the singularity of the problem. The kernel Gimlj is given by:

lj Gim

1 4S 1 X r 2

1 2X G ijG lm G jmG il G jlG im 2 1 2X r,m r,lG ij r,i r,lG mj r, j r,lG im ½ ° ° ® ¾ G G G 2 r r r r r r 8 r r r r ° ° , j , m il , j ,i ml , m ,i jl ,i , j , m ,l ¯ ¿

(16)

Other parameters widely important for modelling fracture mechanics problems are the crack opening displacements. Regarding the proposed formulation, it can be obtained from Eq. (13), which has to be applied for

174


collocation points taken over the dipole line, where discontinuities of displacements and stresses will appear. This equation has to be written for points symmetrically positioned at S and S , as indicated in Fig. (1). As a result, different displacement representations are achieved for these points. By subtracting them, the crack opening displacements are obtained as follow:

'w1 ½ ® ¾ ¯'w2 ¿

ª 1 2X « 2G 1 X « « 0 « ¬

º 0» 1 q ½ » ® 11 ¾ 1 » ¯q2 ¿ » G¼

(17)

It is worth to stress that the presence of dipoles lead to a displacement discontinuity. On Eq. (18), 'w1 represents the crack opening displacement (mode I) and 'w2 the crack sliding displacement (mode II).

Algebraic Equations In previous section, the integral representations with additional terms for modelling stress and displacement discontinuities were derived. The boundary discretization leads to the known matrices H and G, which take into account boundary displacements and tractions. The remaining term can also be written into its algebraic form if the variable q mj is assumed to be approximate by standard shape functions along the crack line. After discretizing the whole crack path by elements and performing properly the integrals over them, Eq. (13) and Eq. (15) assume the following algebraic representations: (18) HU GP KQ (19) V H 'U G' P K 'Q in which K and K ' include the integral kernels due to dipoles Q . The boundary conditions can be applied on Eq. (18). As usual in BEM formulations, all unknown boundary values are stored in a vector X and all known in a vector F. Therefore: (20) AX BF KQ X M RQ where matrices A and B result from columns change between matrices H and G. M and R are defined as M A1BF and R A1K respectively. Similarly, the boundary conditions can be applied on Eq. (19). Performing this step together with the result presented by Eq. (20) one obtains: (21) V A' X B' F K 'Q V N SQ in which A’ and B’ result from columns change between matrices H’ and G’. The terms N and S are equal to N B' F A' M and S K ' A' R , respectively. The integrals that lead the algebraic equations Eq. (20) and Eq. (21) were evaluated by Gauss–Legendre numerical scheme accomplished with a sub-element technique. Based on these procedures, Eq. (13) and Eq. (15) are transformed into algebraic representations with very low integration error. It is worth to mention that due to lj the singularity present in the kernel Gim , only discontinuous elements have to be used at the crack path.

Solution Technique The cohesive crack propagation problem can be solved using Eq. (20) and Eq. (21). The first one leads the solution of boundary values whereas the second equation allows the determination of stresses and displacements at crack surfaces. It is worth to stress that boundary values are determined as long as all dipoles’ values in equilibrium configuration be achieved. The dipoles are calculated in the context of nonlinear solutions, which are performed in incremental form. In this regard, the exceeding stresses are reapplied on the structure and a dipoles variation is determined, as presented in Eq. (21). As a result of this dipoles variation, the crack opening displacements variation is determined, Eq. (17), which leads a new stress state of equilibrium. This last one is compared with the stresses due to the external loading. The differences between these two stresses’ state are reapplied until the residual stresses vector norm be smaller than a specified tolerance, indicating the convergence. This classical procedure is known as constant operator, since all relevant matrices are kept constant during the iterative process. However, this procedure may need too many iterations to achieve the convergence. In order to reduce the amount of required iterations, other numerical procedures can be adopted. The tangent operator is one of them.


175

In order to derive the tangent operator for dipoles’ formulation, Eq. (21) has to be rewritten into its incremental form. Thus: (22) Y 'Qn V n 'Qn 'Nn S 'Qn in which the subscript n indicates the current load step. The nonlinear problem is solved considering a NewtonRaphson scheme, thus, prevision and correction steps are used. In order to determine the corrections required for achieving the equilibrium condition, the equation above has to be represented using a Taylor series. Therefore:

Y 'Qni 1

Y 'Qni

wY 'Qni w'Qni

G'Q

i n

o G'Qni

(23)

in which o represents high order terms and the superscript i indicates the current iteration. At the equilibrium condition, Eq. (23) must be null. In this regard, the equation above can be rewritten for this condition considering only the first two terms of Taylor expansion. Thus:

Y 'Q i n

wY 'Qni w'Qni

G'Q

i n

ª wY 'Qni « i «¬ w'Qn

0 G'Q

i n

º»

»¼

1

Y 'Qni

(24)

Therefore, it is possible to calculate the dipoles’ variation based on the stress residuum of the current iteration and on the derivative of Y . The term that relates the derivative of Y with respect of Q is denominated tangent operator. It can be explicited as:

wY 'Qni w'Qni

w ª 'V ni Qni º¼ ¬ S w'Qni

(25)

It is worth to mention that V and Q are described on global coordinates, whereas 'w is described on local coordinates. Therefore, in order to determine the crack opening displacements, both V and Q must be described into local coordinates, as shown in Fig. (2). The transformation of coordinates, global to local and vice-versa, is performed considering the rotation matrix presented below: T

ª cos 2 T sen 2T « 1 « sen 2T cos 2T « 2 « sen 2 T sen 2T ¬

sen 2 T º » 1 sen 2T » » 2 cos 2 T »¼

(26)

Thus, the stresses and dipoles can be described into local coordinates as: i

i

(27) 'V n T 'V ni and 'Qn T 'Qni where the over bar indicates variables on local coordinate. Therefore, the derivate term positioned at the right hand side of Eq. (25) can be evaluated on local coordinates applying the chain rule as:

w'V ni w'Qni

i

T 1

w'V n w'Qni

w'V ni w'Qni

i

T 1

i

w'V n w'wni w'Q n w'wni w'Qi w'Qni n

(28)

The first derivate term presented on right hand side of Eq. (28) is obtained according the cohesive criteria adopted, which relates cohesive stresses to crack opening displacements. The second derivate term of this equation is achieved using the result presented on Eq. (17). Then: T

ª 1 2X º (29) 0 0» « i »¼ w'Q n «¬ 2G 1 X Finally, the last derivate term positioned at the right hand side of Eq. (28) is determined using the relations presented on Eq. (27). Thus: w'wni

i

w'Q n w'Qni

T

Therefore, Eq. (28) can be rewritten as:

(30)

176

w'V ni w'Qni


ª w'V criteria T 1 « ¬ w'w

º 0 0» ¼

T

ª 1 2X º w'V ni 0 0» T « w'Qni «¬ 2G 1 X »¼

1 2X w'V criteria R 2G 1 X w'w

in which R is given by: ª cos 4 T 2 cos3 T sen T cos 2 T sen 2 T º « » 3 R « cos T sen T 2 cos 2 T sen 2 T cos T sen3 T » 4 «cos 2 T sen 2 T 2 cos T sen3 T » sen T ¬ ¼

(31)

(32)

Finally, based on the results presented on Eq. (31) the tangent operator considering the dipole’s formulation assumes the following form:

wY 'Qni w'Q

i n

S

1 2X w'V criteria R 2G 1 X w'w

(33)

The tolerance to stop the iterative process within an increment of load is applied on the variation of the crack opening displacement corrections, i.e., wi wi 1 d tolerance . Moreover, it is worth to remark that the total crack opening displacement is always required to compute the local tangent operator for the next iteration.

Application The four point bending beam considered in this example is shown in Fig. (3). The geometry is given by its length of 675mm, height of 150mm and central notch of 75mm deep. The material properties were taken from [5], who have performed the laboratory test: tensile strength V tc 3.0 MPa , Young’s modulus E 37.000 MPa , Poisson ratio X

0.20 and fracture energy G f 69 N m . For the present analysis three cohesive laws were used: linear, bi-linear and exponential. The load was applied, for all cases, into 24 increments and the adopted tolerance within each increment was equal to 104 , based on the norm of non-equilibrated stresses.

Figure 3. Four point bending beam. Dimensions in mm.

Figure 4. Load x Displacement curves.


177

The load x displacement curves achieved in this application are shown in Fig. (4), where the results obtained by the proposed formulation are compared with experimental [5] and numerical [3], available in literature. In this figure, the symbol TO indicates the curves constructed using tangent operator. The other numerical curves were obtained using constant operator approach. According this figure, it can be observed that the proposed formulation is capable to represent the structural nonlinear behaviour introduced by FPZ. However, it leads to more rigid responses when compared with experimental and numerical results. It may be explained due to snapback behaviour observed on experimental test. This nonlinear behaviour can be modelled using arc-length algorithm. However, this solution technique is not the focus of this paper. In order to emphasize the large difference of performance observed between constant and tangent operators, Table 1 shows a comparative between the amounts of iterations required for convergence considering this two solution techniques. It can be observed that, at least, 68% of iteration reduction is observed when tangent operator is used. Table 1 – Comparative of iterations

Cohesive Law Linear Bilinear Exponential

Constant Operator (iterations) 2174 2637 4979

Tangent Operator (iterations) 687 721 740

Percentage of Reduction (%) 68.40 72.66 85.14

Conclusion In this paper, the crack growth process in quasi-brittle materials has been studied. This complex structural problem can be modelled solving a nonlinear system of equations, which appears due to the dependency between crack opening displacement and cohesive stresses along the crack path. In order to simulate this nonlinear structural problem, BEM has shown to be an accurate and efficient alternative. An alternative BEM formulation, based on initial stresses field, is proposed in order to simulate the nonlinear effects caused by damaged zones present in narrow regions into the domains. It is worth to stress that the proposed formulation uses three algebraic equations to describe the crack mechanical behaviour whereas the classical dual BEM requires four. Two iterative schemes have been applied to solve the nonlinear problem. The first one applies a constant operator, where all relevant matrices are kept constant along the iterative process. The second approach is developed by using a tangent operator. In this case, the derivate set of nonlinear equations is used and the problem is faster solved. The tangent operator has shown to be faster than the constant one. The use of tangent operator has shown to be always recommended to analyze crack propagation problems, particularly for the cases where the after pick region is reached.

Acknowledgements Sponsorship of this research project by the São Paulo State Foundation for Research (FAPESP), project number 2011/07771-7 is greatly appreciated.

References [1] Moes, N; Dolbow, J; Belytschko, T. A finite element method for crack growth without remeshing. Int. J. Numer Meth Engng. 46:131-150, 1999. [2] Leonel, E.D; Venturini, W.S; Chateauneuf, A. A BEM model applied to failure analysis of multi-fractured structures. Eng Fail Anal. 18:1538–1549, 2011. [3] Leonel, E.D; Venturini, W.S. Non-linear boundary element formulation with tangent operator to analyse crack propagation in quasi-brittle materials. Eng Anal Boundary Elem. 34:122–129, 2010. [4] Brebbia, C.A.; Dominguez, J. Boundary Elements: An introductory course. Second edition, WIT Press, Southampton, 1992. [5]Galvez, J.C; Elices, M; Guinea, G.V; Planas, J. Mixed mode fracture of concrete under proportional and nonproportional loading. Int J of Fracture. 94: 267-284, 1998.

178


Treatment of Singularities in Fundamental Solutions of Orthotropic Thick Plates A. P. Santana, E. L. Albuquerque, L. S. Campos and D. I. G. Costa Federal Institute of Maranho - IFMA Department of Mechanics and Materials So Luiz, MA, Brazil [email protected] University of Brazilia - UnB Faculty of Technology 70910-900, Brasilia, Bsb, Brazil [email protected], [email protected],[email protected]

Keywords: Boundary element method, thick plates and orthotropic.

Abstract. This work presents a proposal to treat hyper and strong singularities of orthotropic thick plate fundamental solutions and their derivatives. Provided that these fundamental solutions do not have a closed form, analytical procedures can not be used. In this work, a simple quadrature rule is proposed where strong and hypersingular integrals are treated in Cauchy and Hadamard sense, respectively. Derivatives of fundamental solutions are used in boundary integral equations to compute moments at internal points. Results are computed for a simple supported plate and show good agreement with literature.

1

Introduction

Fundamental solutions are the starting point to a boundary element formulation. Although necessary, they are not easily obtained. Fundamental solutions that take into account the effect of the transverse shear deformation through the thickness of a orthotropic plate can be derived using the Hörmander operator and the Radon transform. However, to the best of author’s knowledge, it is not possible to obtain a closed form fundamental solution. Besides being numerical, orthotropic shear deformable fundamental solutions are given by singular integrals (Wang and Huang [3]). The way how these singularities are treated is a key point in the performance of the boundary element code. Provided there is no closed form for fundamental solutions, the use of numerical methods is mandatory.. Reis et al. [1] presented a procedure to treat strong singular integrals in fundamental solutions of shear deformable orthotropic plates. Since hyper singularities were not treated, only displacement on internal points were computed. In the present work, the treatment of strong and hyper singularities is reached through the use of a numerical quadrature rule where few points are necessary to get accurate results. The formulation is applied to the computation of moments at internal points of a simply supported plate. The obtained results show good agreement with literature.

2

Mindlin Plate Theory

The Mindlin’s theory assumes displacement distribution through the thickness. Using the assumptions of the classical theory, he removed the hypothesis of the transverse shear deformation equal zero in the mid


179

plane, but considering that distortion variation is null, [3]. Thus:

∂ γ13 ∂ x3 ∂ γ23 ∂ x3

= 0.

(1)

= 0.

(2)

So, equations of equilibrium for the plate are given by: Mαβ ,β − Qα = 0.

(3)

Qα ,α + q = 0.

(4)

where q is the distributed transverse load per unit of area in the x3 direction. The bending moments Mαβ and she the shear forces Qα for orthotropic plates are expressed in terms of the rotations and the lateral displacement as: Mαβ = Dαβ (wα ,β + wβ ,α ) +Cαβ wγ ,γ . (5) Qα = Cα (wα + w3,α ).

(6)

where no summation is assumed in eq (5) and eq (6) with respect to the indices α and β . Material parameters Dαβ , Cαβ , and Cα are shown at [1, 3], w is the transversal displacement.

3

Differential Equations of Equilibrium The differential equation of equilibrium is given by [2]: Li jU j + bi = 0.

(7)

where bi represents the body force and Li j are Navier differential operators, see [3]. Forces and displacements at the boundary, Γ, can be expressed as: Pα = Mαβ nβ

P3 = Qα nα

Mn = Mαβ nα nβ

Mns = Mαβ tα nβ

ψn = ψα nα

ψS = ψα nα

(8)

where nα is the normal vector.

4

Moment Boundary Integral Equations

Bending moments at any internal point ζ can be computed by: Mαβ (ζ ) =

Γ

∗ Uαβ k (ζ , x)pk dΓ(x) −

Γ

∗ Pαβ k (ζ , x)uk (x)dΓ(x)

+q where kernels Ui∗jk , Pi∗jk and Wi∗jk are found in [3].

Γ

∗ Wαβ (ζ , x)dΓ(x)

(9)

180


5

Fundamental Solution

Fundamental solutions of orthotropic thick plates taking into account the transverse shear deformation are a set of particular solutions of the differential eq (7) under a unit concentrated load, i.e., the solutions satisfy the following inhomogeneous differential equations: Liadj jUk∗j (ζ , x) = −δ (ζ , x)δki .

(10)

in which δ (ζ , x) denotes the Dirac delta function, ζ represents the source point, x is a field point and Liadj j is the adjoint operator. Following Hörmander’s operator method, the solutions of eq (10) can be written as: j Uk∗j (ζ , x) =co Lad jk φ (ζ , x).

(11)

j ad j where φ (ζ , x) is a unknown scalar function and co Lad jk is the cofactor matrix of the operator L jk (see Ref. [3]). By substituting eq (11) into (10), we obtain the following equation: + , ∂4 ∂4 2 ∇2k D1 Dk 4 + (D1 D2 − D21 μyx − 2D1 Dk μyx ) 2 2 + ∂ x1 ∂ x1 ∂ x2 + ∂4 ∂2 ∂4 +D2 Dk 4 −C1C2 D1 2 + 2(2Dk + D1 μyx ) 2 2 ∂ x2 ∂ x1 ∂ x1 ∂ x2 ,2 ∂4 Φ(ζ , x) = −δ (ζ , x) +D2 4 (12) ∂ x2

The derivation of fundamental solutions (10) is reduced to that of eq (12). As soon as the solution of eq (12) is obtained, substituting it into eq (11) and by differentiation we can get the solutions of eq (10). Eq (12) is a sixth order partial differential equation. Using the plane wave decomposition method (Radon transform), the partial differential eq (12) can be reduced to an ordinary differential equation, which simplifies the treatment of the problem. We first expand δ (ζ , x) into a plane wave (see, for example, [3]):

δ (ζ , x) = −

1 4π 2

2π 0

| ω1 (x − ζ ) + ω2 (y − η ) |−2 d θ ,

(13)

in which (ω1 , ω2 ) are the coordinates of a point on the unit circle, i.e., ω1 = cos(θ ), ω2 = sin(θ ), (x, y) and (ζ , η ) are coordinates of field point and source point, respectively. Similarly, φ (ζ , x) can be written as: Φ(ζ , x) =

2π 0

ϕ (ρ )d θ ,

where ρ = ω1 (x − ζ ) + ω2 (y − η ), ϕ (ρ ) is a function depending only on ρ . By substituting eq (13) and eq (14) into eq (12), and considering differential relationship we obtain the following equation: # 2 $ d d4 1 2 − p ϕ (ρ ) = 2 2 | ρ |−2 , dρ 4 dρ 2 4π a

(14)

∂ ∂ xα

= ωα ddρ ,

(15)

in which 2 − 2D D μ )ω 4 ω 2 +C D D ω 2 ω 4 +C D D ω 4 ω 2 +C D D ω 6 a2 = C1 D1 Dk ω16 +C1 (D1 D2 − D21 μyx 1 k yx 1 2 k 1 2 2 1 k 1 2 2 2 k 2 1 2


181

2 − 2D D μ )ω 2 ω 4 , +C2 (D1 D2 − D21 μyx 1 k yx 1 2 4 = C1C2 [D1 ω1 + 2(2Dk + D1 μyx )ω12 ω22 + D2 ω24 ], p2 = b2 /a2 . The solution of eq (12) is now reduced to solve the ordinary differential eq (15). After four times integration of eq (15) and leaving out the constants of integration, we obtain:

b2

d 2 ϕ (ρ ) 1 − p2 ϕ (ρ ) = − 2 2 p2 ln | ρ | . dρ 2 8π a

(16)

The solution of eq (16) is given by:

ϕ (ρ ) =

1 [p2 ρ 2 ln | ρ | +2ln | ρ | +3 8π 2 p4 a2 +exp(pρ ) −exp(−pρ )

∞ exp(−pσ ) ρ

σ

ρ exp(pσ ) −∞

σ

dσ

d σ ].

(17)

Substituting eq (17) into eq (14) and integrating, we can obtain the function Φ(ζ , x). The generalized displacement and boundary tractions can be expressed in the following forms: Ui∗j (ζ , x) =

2π

Pi∗j (ζ , x) =

0

U˜ i∗j (ρ )d θ ,

2π 0

P˜i∗j (ρ )d θ .

(18) (19)

From eq (5), (6) and (8), we can easily obtain the kernels as follows: d4ϕ d2ϕ U˜ i∗j (ρ ) = aαβ 4 −C1C2 ωα ωβ 2 dρ dρ

(20)

d3ϕ dϕ U˜ 3∗α (ρ ) = −Uα∗ 3 (ρ ) = fα 3 −C1C2 ωα dρ dρ

(21)

dϕ 4 d2ϕ ∗ (ρ ) = α1 4 − β1 2 +C1C2 ϕ U˜ 33 dρ dρ

(22)

where repeated indices α and β do not mean summations. The constants aαβ , fα , α1 and β1 are shown at [3]. d5ϕ ∗ (ρ ) = [Dβ γ (aαβ ωγ + aαγ ωβ )nγ +Cβ γ dα nγ ] 5 P˜αβ dρ d3ϕ −[2Dβ γ C1C2 ωα ωβ ωγ nγ +C1C2Cβ γ ωα nγ ] 3 dρ d4ϕ P˜3∗α (ρ ) = [Dαγ ( fα ωγ + fγ ωα ) +Cαγ g]nγ 4 dρ D2 ϕ −[2Dαγ ωα ωγ +Cαγ ]nγ C1C2 4 dρ

(23)

(24)

182


6

d4ϕ P˜α∗ 3 (ρ ) = Cγ (aαγ − fα ωγ )nγ 4 dρ

(25)

d3ϕ d5ϕ ∗ P˜33 (ρ ) = Cγ ( fγ − β1 ωγ )nγ 3 + α1Cγ ωγ nγ 5 dρ dρ

(26)

dα = aαξ ωξ

(27)

g = f ξ ωξ

(28)

Numerical Treatment of the Fundamental Solution

Singularity order for moments computation are bigger than for displacements. As can been see on the eq (20) and (26), fundamental solutions Ui∗j and Pi∗j are function of of derivatives of ϕ up to fifth order. The fifth order derivative of ϕ is of order 1/ρ , i.e., it is strongly singular. For moment computation, as can be seen in Equation (9), we used the eq (5) and (6) and derivative order increases to d 6 ϕ /d ρ 6 . 6.1 The fundamental solution Ui∗j : As can be seen from eq (18), kernels given by eq (20), (21) and (22) have weak singularities that are treated using a symmetric integration and Telles transformation. In order to carry out the integration, we first determine the value θ0 which have ρ = 0. The value θ0 is determined using the following expression: . x−ξ (29) θ0 = tan−1 − y−η The interval of integration is given by:

θ0 , θ0 + 2π

(30)

Due to this symmetry, only half of the interval needs to be integrated, i.e., integration is carried out in the interval [θ0 , θ0 + π ] and multiplied by two. These kernels are, at most, weak singular due to the presence of derivative d 4 ϕ (ρ )/d ρ 4 . So, Telles transformation is enough to obtain an accurate integration. In this sense, Gauss quadrature was employed to compute integral of eq (18) and then Telles transformation to concentrate integration points near singularities. 6.2 The fundamental solution Pi∗j : Kernels given by eq (25) and (25) are strong singular. Of this form, again we can used the Gauss quadrature and Telles transformations to compute integral of eq (19). Singularities on kernels of eq (24) and (26) are treated using the quadrature proposed by [5]. 6.3 Derivatives of fundamental solutions Ui∗jk and Pi∗jk : Kernels Ui∗jk and Pi∗jk contain d 5 ϕ (ρ )/d ρ 5 and d 6 ϕ (ρ )/d ρ 6 , respectively. So, it is strongly and hypersingular, respectively. In this case the quadrature Gauss and quadrature proposed by [5] with 18 integration points was employed..

7

Numerical Results

Consider a square orthotropic plate, simply supported, under uniformly distributed load with amplitude q = −2, 07 × 106 MPa. The plate have the following properties: Ey = 0, 6895 × 1010 MPa, Ex = 2 × Ey , Gxy = Gxz =Gyz = 2652 × 106 Pa and νxy = 0, 3. The edges of plate are a = 254 mm wide and h = 25, 4


183

Table 1: Moments in the center of the plate. Number of elements 5 7 9 11 [4]

Mxx (N/m) 10114 10003 9964,7 9950,8 9540

Errors 6,0168% 4,853% 4,45% 4,3 %

mm thick. Constant boundary elements with equal length were used in the discretization. This problem is equivalent to the problem proposed by Sladek et al. [4]. Results for moments are given in Table 1. As can be seen in Table 1, results are in good agreement with [4]. A difference of 4,3% was obtained for the moment at the central point in x direction.

8

Conclusions

This paper proposes a new procedure to treat strong and hyper singular integrals presented in orthotropic shear deformable fundamental solutions. A simple quadrature was used and results were obtained with few integration points. To assess the treatment of singularities, the boundary element formulation for the computation of moments at internal points of orthotropic plates was implemented. Results has shown good agreement with literature. Acknowledgment. The authors would like to thank the State of Maranho Research Foundation (FAPEMA) and the Federal Institute of Maranhao (IFMA) for the financial support of this work.

References [1] A. Reis, E. L. Albuquerque, and L. Palermo Jr. The boundary element method applied to orthotropic shear deformable plates. Engineering Analysis with Boundary Elements. 37 (2013), 738746. [2] Y. F. Rashed, Boundary method formulations for thick plates, Southampton, Boston, 1999.

Topics in Engineering, WITpress

[3] J. Wang and M. Huang, Boundary element method for orthotropic thick plates, Sinica, vol. 7, pp. 258-266, 1991.

Acta Mechanica

[4] 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. [5] L. S. Campos and E. L. Albuquerque Quadrature Gauss to treated singularities of type 1/r and 1/r2 , University of Brazilia, College of Technology, 2013.

184


ANALYSIS OF FOLDED PLATES BY THE BOUNDARY ELEMENT METHOD K. R. P. Sousaa , E. L. Albuquerqueb , D. I. G. Costab , S. Hoefelc a Federal

˜ Institute of Maranhao ˜ Luis, MA, Brazil Sao

[email protected] b University

of Bras´ılia - UnB Bras´ılia, DF, Brazil [email protected]

c Federal

University of Piaui - UFPI Teresina, PI, Brazil

Keywords: Boundary Element Method, Plates.

Abstract. In this paper a boundary element formulation is developed for the analysis of structures formed by three-dimensional association of plates. Initially, boundary element formulations for plane elasticity and classical plates are associated in one plane structure with four degrees of freedom per node given by normal, tangential and transverse displacements and normal rotation. Then, the formulation is extended in order to allow the 3D assembling of these structures. Each plate is defined as a subregion. All plates are combined taking into account displacement compatibilities and traction equilibrium conditions. The numerical treatment is carried out by the direct boundary element method. A numerical example is presented and its results are in good agreement with results obtained by the finite element method.

1 INTRODUCTION A structure formed by the assembling of folded thin elastic plates is one of the most useful components of architectural or mechanical structures. Due to the practical importance of the subject, numerous researchers have carried out investigations within the last few decades. As a result, a considerable number of works is available in literature. Onate [2] e Zhang [1] used plane elements containing out-of-plane (flexural) and in-plane (stretching) degrees of freedom assembled to represent the folded plate finite element model. Palermo [3] presented a direct boundary element formulation to perform analyses of isotropic folded plate model. The stretching and bending were taken into account in each plane element, which was treated as a subregion in the boundary element method. Equations for each subregion were combined together considering the displacement compatibility and equilibrium conditions. This article presents boundary element formulation for the association of folded plates in 3D space. Each individual plate has four degrees of freedom per node, three displacement and one rotation, and is assumed as a subregion. The boundary element method is applied to the discretization of the obtained integral equations, in which the boundary is divided into boundary elements. Under the assumption of small deformation and small strain, the in plane motion and the out-of-plane motion are not coupled. Therefore, such a mixed state of stresses can be obtained by superposing the equations for both in-plane and out-of-plane motions. Then, these discretized equations for each plate component in the local coordinate


185

system are transformed into those in the global coordinate system. Then, the resulting equations are assembled in such a way that the compatibility and equilibrium conditions on the connected edges as well as the boundary conditions are satised. Finally, the entire system of equations for the whole plate structures composed of plate components can be obtained. Equations obtained to each subregion can be combined together taking into account the displacement compatibility and equilibrium conditions, forming a final folded plate model. The assembly matrix and numerical examples are presented and their results show good agreement when compared to results obtained by the FEM.

2

Integral equations for plane stress problems

The reciprocal relation between a static fundamental state in a infinity domain, with displacement given by u and tractions given by t defined over a body of domain Ω with boundary Γ, is given by the following integral equation (Aliabadi [4]):

ci j u j +

ti∗j u j dΓ =

Γ

Γ

u∗i j t j dΓ +

Ω

u∗i j b j dΓ,

(1)

where ci j is a constant, u j is the displacement vector, t j is traction vector and b is body forces. u∗i j and ti∗j are know as displacement and traction fundamental solutions, respectively.

3

Integral equations for bending problems

Using Betti theorem, an integral equation for a thin plate under transversal load g(p) can be written as (see Aliabadi [4]):

Kw +

Nc Nc ∂w dΓ + ∑ R∗ci wci = ∑ Rci w∗ci − ∂n Γ i=1 i=1 ∗ ∂ w gw∗ dΩg , Vn w∗ − mn − dΓ + ∂n Γ Ωg Vn∗ w − m∗n

(2)

where n is he outward unit normal vector to the boundary Γ, mn , Vn are respectively the normal bending moment and the Kirchhoffs equivalent shear force on the boundary Γ, Rc is thin plate reactions of corners, wci is the deflexion of corners and the symbol * stands for the fundamental solutions and Nc are the total number of boundary corners. The constant K is introduced in order to consider that the Dirac delta function can be applied in the domain, in the boundary, or outside the domain. If the Dirac delta function is applied in a point where the boundary is smooth, than K = 1/2. In order to have an equal number of equations and unknown variables, it is necessary to write an integral equation corresponding to the derivative of displacement w in relation to Cartesian coordinate system fixed in the source point, i.e., the point where the Dirac delta of the fundamental state is applied. The axis directions of this coordinate system are coincident with normal and tangent to the boundary directions in the source point. For a particular case where the source point is placed in a point where the boundary is smooth, the boundary equation is given by: Nc ∂ R∗ Nc ∂ w∗ci 1 ∂w ∂ Vn∗ ∂ m∗n ∂ w ci + w− wci = ∑ Rci + dΓ + ∑ 2 ∂m ∂m ∂n ∂ m ∂m Γ ∂m i=1 i=1 ) ( ∂ w∗ ∂ ∂ w∗ ∂ w∗ − mn dΩg . g (3) Vn dΓ + ∂m ∂m ∂m Γ Ωg ∂ m

186


Equations (2) and (3) can be discretized to find the system of equations from which the boundary values can be found. Domain integrals which come from linearly and uniformly distributed loads were transformed into a boundary integrals by exact radial transformation presented by Albuquerque [5].

4

Transformation Matrix

In the plane elasticity integral equations and plate bending integral equations, the displacement at any point are described in terms of its relative position with respect to the other points in the problem. The relations between force and displacement are preserved and does not depend on the coordinate system chosen to construct the matrices. Thus, each subregion may have its own local coordinate system. However, in order to make forces and displacements of several subregions compatible and to solve them simultaneously, they need to be described in the same global coordinate system. Consider Fig. 1a, a subregion described in a local coordinate system. It can be represented in a global system by the following equation: ⎧ ⎫ ⎡ ⎪ u1 ⎪ ⎪ ⎪ ⎪ ⎨ w ⎪ ⎬ ⎢ ⎢ =⎢ ⎪ ⎣ u2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ θ

n1 m1 l1 n2 m2 l2 n3 m3 l3 0 0 0

0 0 0 1

⎫ ⎤⎧ ⎪ u1 ⎪ ⎪ ⎪ ⎪ ⎬ ⎥ ⎨ w ⎪ ⎥ , ⎥ ⎪ ⎦ ⎪ u2 ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ θ

(4)

or simply: {u} = [T]{u }, where T is the matrix which takes local coordinates into global coordinates. In other words, if u is the displacement vector written in local coordinates, then {u} = [T]{u } is the displacement vector in global coordinates. y P3

Γ21 Γ12

P2

P4

θ w

u2 Q

x

Γ1

u1 z

Ω1 Interface

P1

Figure 1: a) Coordinate system ;

5

Γ2 Ω2

b)Association of the two subregion.

Compatibilization of bending moments and rotations

Bending moments and rotations in the normal direction appear in the association of the subregions. Those moments are not modified by the application of the transformation matrix T. The local positive moments


187

have opposed orientation in the subregions interface. The same thing occurs with rotations θ . Figure 2 shows the association of two subregions on the xz-plane. Bending moments Mn have opposite directions in accordance with the vector s, which is determined by the boundary orientation. It is necessary to compatibilize rotations and moments on each subregion interface. For this reason a positive direction has been adopted for each element.

z Mn

Mn s

s Mn

Mn

2

Mn

Mn

1

Mn

x

Mn

Figure 2: Compatibilization of bending moments and rotations The integral equations of each subregion can be obtained in an independent way and later coupled by equilibrium relationships and compatibilities of displacements and rotations. Considering the problem shown in Figure 1-b, traction equilibrium and displacement compatibility conditions over interface boundary Γ12 are given by (Palermo [3]):

u1i = u2i ,

θ 1 = −θ 2

ti1 + ti2 + ti = 0 Vn1 +Vn2 +Vn = 0 Mn1 + Mn2 + Mn = 0

6

(5)

Numerical Results

Consider a C-beam-clamped at both ends. The structure is formed by the association of three subregions with two interface (Figure 3). The structure is under a uniformly distributed load in subregion 2 equal q = −90 MPa. The material properties used in this analysis are: E = 2.1 GPa and ν = 0.3, and thickness h = 0.008 m. 4 Figure 4 shows the total displacement (ut = u21 + u22 + w2 ) of the edge A using 30, 40 and 70 constant elements per edge. The results obtained are compared whit results obtained by ANSYS, using the SHELL63 (3888 element). The relative errors obtained for the maximum deflexion along edge A are: 9.4%, 7.3% and 4.7%, respectively. Figure 5 shows the deformed shape of the C-beam.

7

Conclusions

This paper analyzed the behavior of structures formed by three-dimensional association of isotropic plates. The subregions association was done using displacement compatibility and equilibrium equations, as well as the compatibilization of bending moments and rotations on the interface. The formulation of 3-D association used coordinate transformation for each subregion. Equations in the local coordinate system

188


y

interface 1

interface 2

2m

edge A

z

2m

8m

x

Figure 3: Geometry of the structure. 0

−0.5

−1

Displacements

−1.5

−2

−2.5

−3

−3.5 SHELL63 BEM (20 c) BEM (40 c) BEM (70 c)

−4

−4.5

0

1

2

3

4

5

6

7

8

Nodes

Figure 4: Displacement obtained on the A edge using constant elements. were thus translated into a final system of equations in the global coordinate system. After imposing the boundary conditions, the system of equations can be solved. Based on the analysis of the obtained results, we can see that the use of constant elements provided a good agreement with results obtained by the FEM.

References [1] S. H. Zhang and L. P. R. Lyons, A thin walled box beam finite element for curved bridge analysis Computers & Structures, 18 6 (1984) 1035-1046. [2] E. Onate and B. Suarez, A unified approach for the analysis of bridges plates and axisymmetric shells using the Mindlin Strip Element, Computers & Structures 17 3-9 (1983). [3] L. Palermo The analysis of thin walled structure as an assemblage of plane elements with the boundary element method So Carlos, USP,(1989). (PhD. Thesis in Portuguese).


189

Total displacement

4

3.5

2.5 3

2 1.5

2.5

z

1 0.5 0

2

−0.5 8 1.5 7 5

6 4

5

1

3 4

2 3

1

0.5

0

2 −1 1 −2 y

0

0

−3 x

Figure 5: Total displacement (in mm) [4] M. H. Aliabadi, The boundary element method in engineering, vol. 2, (2002), New Jersey: Wiley. [5] E.L. Albuquerque, P. Sollero, W. Venturini and M.H. Aliabadi. Boundary element analysis of anisotropic Kirchhoff plates, International Journal of Solids and Structures, Vol. 43, pp. 4029–4046, (2006).

190


The Boundary Element Method applied to the Analysis of Fluid Extraction from a Reservoir A. B. Dias Jr1 , E. L. Albuquerque1 , E. Fortaleza1 1

University of Brasilia - Darcy Ribeiro Campus, North Wing - Brasilia - Brazil [email protected]

Keywords: Boundary Element Method, Oil Reservoir, Laplace Equation

Abstract. This work aims to study the pumping of oil in porous media using the boundary element method. Provided the reservoir may also contain gas, a formulation is implemented in which the shape of the free surface, the contact surface defined by the interface between oil and gas, is unknown in the problem. Oil flow in porous media is governed by Darcy differential equation and can be modeled by the laplacian of a potential function. The deformation of the free surface is obtained by an equation which relates the change in its form with the potential gradient. Linear continuous boundary elements are used and the pump is simulated as a vortex represented by a negative Dirac delta. Introduction. The main objective of this article is to study the pumping of oil out of a tank without gas production. It is well known that the excessive production of gas is a common problem in oil reservoirs, due to the existence of a layer of gas above the oil zone. This excessive production is related to a phenomenon known as gas cone. The gas cone is characterized by invasion of gas into the zone where the oil is extracted. This invasion occurs when the balance between the forces of the system is broken. Therefore, the gas enters the well and begins to be produced, because it has greater mobility than the oil. In this article we will address the flow of a single fluid in a porous medium. Therefore, this problem will have a deformable free surface, i.e., the shape of the free surface varies with time to represent the conical shape of the problem. The representation of the cone gas is illustrated in FIG. 1. Such a situation would represent the behavior of the oil while it is being pumped. The Boundary Element Method (BEM) will be used to simulate this flow.

Figure 1: Oil reservoir


191

The continuous extraction of immiscible fluids and fluid with free surface has been studied by different numerical methods. According [4], early work studied the critical and sub critical conditions of flow rate for stationary conditions. The model used by [5] considered layer flow confined below an impermeable boundary. The solution of the integral equation for this nonlinear model was solved numerically. With this model, the critical flow rate was calculated for every position of the sink. [2] used the linear stability theory and the boundary element method for modeling the recirculation of the flow in wells, that is the configuration used in the vicinity of ground water layers. Numerical simulations show that for this arrangement, there is a critical rate of pumping, whose value can be determined. [3] considered the problem of the cone of the water for vertical wells. They calculated a potential function of the oil zone assuming a horizontal radial flow. Furthermore, it was neglected the presence of water cone. In other works, such as [1], a small perturbation method and the Boundary Element Method was applied, assuming an approximated form for the suction pressure wells in an unconfined oil zone. [6] used the integral equation of the Boundary Element Method to solve numerically problems with axisymmetric geometries.

Figure 2: Oil reservoir Boundary Element Equations. This work will address the simulation of pumping from a oil reservoir. For this, consider the flow of one fluid in a porous medium shown in FIG. 2. Considering Darcy equation, this problem can be analyzed through the laplacian of a potential function given by [4]: ∇2 φ = 0

(1)

A fluid flow velocity is given by the derivative of this potential function. The boundary integral equation of the problem can be written as: φ (xd )c =

5 S

⎧ ⎨ 1,

where c=

⎩

θint 2π ,

0,

φ

∂φ∗ ds − ∂n

5 S

φ∗

∂φ ds ∂n

se (xd , yd ) ∈ domain if (xd , yd ) ∈ boundary / boundary or domain if (xd , yd ) ∈

(2)

(3)

The fundamental solution is given by: ∇2 φ ∗ = δ (x − x ) φ∗ =

1 ln(R) 2πk

(4) (5)

192


Figure 3: Discretization of reservoir The equation used to update deformed surface, along the time is given by: ∂φ∗ 1 ∂φ∗ ∂ η 2 −0.5 =− cos(l) = [1 + ( ) ] ∂t cos(l) ∂ n ∂x Where cos(l) = [1 + (

∂ η 2 −0.5 ) ] ∂x

(6)

(7)

NUMERICAL RESULTS In order to assess the implemented formulation, the extraction of oil from a reservoir was modeled. The oil reservoir was considered a square of area equal to 1 m2 (H = 1 m and w = 1 m). The boundary conditions were adopted according to [4], and their values are presented below in table 1: Table 1: Boundary Conditions Potential Flow Line 1 unknown 0 Line 2 1 unknown Line 3 Y unknown Line 4 1 unknown Each side was discretized into 16 linear boundary elements. The point where the pump was placed, called the sink, FIG. 2, is at x = 0.5 m and y = 0 m. The mesh used in the problem is shown in FIG. 3. The grid of points within the reservoir are the internal points. The boundary conditions for the potential are applied in lines 2, 3 and 4. The boundary conditions for the stream is applied in line 1. The power of pump is q˙ = -2.3. There is no concern with the correctness of the physical quantities of the problem. The pumping problem was analysed during a period of 2 s. The pump is turned on at time t = 0 s and turned off at time t = 1 s. The time was discretized into 100 time steps. FIG. 4 shows the position of the mid point of the free surface during the period of analysis. Note that the mid point reached a position around 0.87 m at time t = 1 s and then, as the pump was turned off, the free surface went back to its initial position. FIG. 5 shows the flux through lines 2 and 4. The magnitude of the flux is high when the pump is turned on (from 0 to 1 s) and low when the pump is turned off. Figure 5 shows the reservoir at instant t = 0 s and t = 1 s (just before the pump is turned off).


Figure 4: Displacement of the mid point of the free surface

Figure 5: Flux in the oil reservoir

193

194


Figure 6: Deformation of the free surface Conclusions. This paper presented a formulation of the boundary element method to simulate the extraction of oil from reservoir. The oil was considered as a Darcy fluid so that the Laplace equation can be applied in the analysis. The shape of the interface between liquid and gas has been considered as unknown and was calculated for each time step. The results were satisfactory, well modeling the phenomenon of cone formulation of gas. We can conclude that the Boundary Element Method is well suited for modeling moving boundaries problems. Acknowledgment. The authors are thank to Chevron for the financial support os this project.

References [1] J.R. Blake and M. Muskat. Coning in oil reservoirs. Math.Sci., 13:36–47, 1988. [2] T.R. MacDonald and P.K. Kitanidis. Modeling the free surface of an unconfined aquifer near a recirculation well. Ground Water, 31:774–780, 1993. [3] M. Muskat and R.B. Wyckoff. An approximate theory of water coning in oil production. Ground Water, 31:114–163, 1935. [4] D.A.Barray Zhang, H. and G.C. Hocking. Analysis of continuos and pulsed pumping of a phreatic aquifer. Advances in Water Resources, 22:623–632, 2002. [5] H. Zhang and G.C. Hocking. Withdrawal of layered fluid through a line sink in a poruos medium. Australian Mathematical Society, 38:240–254, 1996. [6] H. Zhang and G.C. Hocking. Flow in an oil reservoir of finite depth caused by a point sink above an oil-water interface. J. Hydraul. Eng., 32:365–376, 1997.


195

APPLICATION OF THE BOUNDARY ELEMENT METHOD TO ANALYSIS OF CREEP WITH PLASTICITY E. Pineda León 1, M.H. Aliabadi2, A. Rodríguez-Castellanos3 and J. Zapata1 1

Escuela Superior de Ingeniería y Arquitectura, Instituto Politécnico Nacional, México D. F., e-mail: [email protected] 2

Department of Aeronautical Engineering, Imperial College London, South Kensington campus, London SW72AZ 3

Instituto Mexicano del Petróleo, México D F, e-mail: [email protected]; [email protected]

Keywords: Creep, Plasticity, Boundary element method, nonlinear behaviour, plates . Abstract. This work presents a formulation to make a combined analysis of plasticity and creep in 2D plates using the Boundary Elements Method. This new approach is developed to combine the constitutive equation for time hardening creep and the constitutive equation for plasticity, which is based on the Von Misses criterion and the Prandtl-Reuss flow. The implementation of creep strain in the formulation is achieved through domain integrals. The creep phenomenon takes place in the domain which is discretized into quadratic quadrilateral continuous and discontinuous cells. The creep analysis is applied to metals with a power law creep for the secondary creep stage. Results obtained for three models studied are compared to those published in the literature. The obtained results are in good agreement and evinced that the Boundary Element Method could be a suitable tool to deal with combined non linear problems. 1. INTRODUCTION Most of the materials used in engineering have sophisticated material properties which may depend on stress, time and temperature. In order to model the complex behavior of such materials, stress analysis techniques are developed. These techniques are necessary to solve the elastic problem but also go further to model the nonelastic phenomenon such plasticity and creep. Without doubt FEM is computationally efficient and during many years has reached such popularity that a very wide range of linear and non-linear engineering problems have been solved with this powerful numerical method [1]. In many branches of science and engineering, the Boundary Element Method (BEM) has become a powerful tool for the solution of boundary value problems. The integral formulation is the foundation of the method and has been used to solve linear and nonlinear problems in finite and infinite regions. One of the first successful applications of the BEM to nonlinear problems in solid mechanics was focused to elastoplastic flow for work- hardening materials, for both anisotropic and compressible behavior, by Swedlow and Cruse [2]. This was followed by the numerical implementation of the boundary-integral technique for planar problems of elasticity and elasto-plasticity by Riccardella [3]. Kumar and Mukherjee [4] who presented the boundary integral equation analysis of time-dependent inelastic deformation of arbitrarily shaped three-dimensional metallic bodies subjected to arbitrary mechanical and thermal loading histories. Examples of creep of thick-walled spheres, long thick-walled cylinders and rotating discs were also discussed. Another formulation for plasticity based on initial stress is due to Banerjee and Mustoe [5]. Mukherjee [6] showed an indepth treatment of problems in nonlinear solid mechanics together with several interesting fracture mechanics applications using the Boundary Element Method. Also formulations for three-dimensional, two dimensional, axi-symmetric elastoplasticity as well as viscoplasticity and bending of plates, were covered. Telles [7] showed an extensive research on the elastoplasticy, viscoplasticity and creep of structural components and civil engineering structures using BEM. His work represented a wide range of benchmark problems, where important comparisons between BEM and FEM were done. Timedependent solution was obtained by the Euler step procedure highlighting for the selection of the time step

196


length. Brebbia et al. [8] also included elastoplasticy, viscoplasticity and wave propagation problems using BEM formulations. A broad range of time-dependent material non-linearity problems including

creep and plasticity are due to Chandenduang [9] and Aliabadi [10]. Some researches on creep continuum damage problems with plastic effects and crack propagation damage problems using BEM are by [11, 12]. Creeping analysis with variable temperature in plates applying the boundary element method was recently presented by Pineda et al. [13], here BEM showed good agreement between BEM results and experimental ones.

2 COMBINED PLASTICITY AND CREEP For the combined plasticity and creep analysis, the creep strain will be included to the total strain modeled above for plasticity only. Now the total strain rate consists of the elastic, plastic and creep strain rates as follow:

H where

H

is the total strain rate and

H e H p H c ,

H e , H p

and

H c

(1)

are the elastic, plastic and creep strain rate, respectively.

The constitutive equation for time hardening creep analysis can be presented as follows:

H c

3 ( n 1) mBV eq Sij t ( m1) , 2

where B, m and n are the material constants which dependent on the temperature.

(2)

V eq is the equivalent stress,

S ij is the deviatory stress and t is the time. The constitutive equation for plasticity based on von Mises yield criterion and Prandtl-Reuss flow rule is

H

p

3 ª SijHij º ( 2 ) V eq Sij , « Hc » 2« ¬1 3 P » ¼

(3)

where H´ is the plastic hardening modulus and P is the shear modulus. It is assumed that the plasticity and creep analysis are separable since elasto-plasticity is a time-independent process and creep is a time-dependent process. Therefore to combine the plastic and creep analysis superposition is used. 3 DISPLACEMENT BOUNDARY INTEGRAL FORMULATION The Somigliana’s equation rate can be obtained by neglecting the body forces, substituting the Dirac delta function property and the fundamental fields (displacements, tractions and stresses) into the equilibrium equations and traction definition to give ui

³ uc t d* ³ t c u *

ij j

*

ij

j d*

³ V c H :

ij

a ij d:

The above equation computes the displacement in any internal point of the domain

(4)

:

values of the boundary displacements and tractions as well as the anelastic deformation problems

H

a ij

0

once that we know the

Hija

. For linear elastic

, but in this work this deformation will be considered.

:c . In order to obtain a solution for the points on the boundary it is necessary to apply the definition of the limit to Somigliana’s equation when x o xc like in The equation (4) is for any internal point within the domain


elasticity, see Aliabadi [10]. Here

xc

197

is any point on the boundary

*

and

x

represent any point in the

domain :c This leads to the following boundary Integral representation of the boundary displacements when the initial strain approach for the solution of elastoplastic problems is used

cij u j ³ tijc u j d* *

³ uc t d* ³ *

ij j

:

³

*

On the left hand side of the equation (5), the integral equation

c H ajk d: V ijk

.

(5)

stands for the Cauchy principal value integral. In this

cij is called the jump term which depends on the geometry.

4 NUMERICAL IMPLEMENTATION The numerical expression for the displacement boundary equation (5) can be written as follows: N el § · cu ¦ ¨¨ ³ TId* ¸¸u n n 1©* ¹

N el

§

·

n 1

©*

¹

N el

§

n 1

© :N

¦ ¨¨ ³ UId* ¸¸t ¦ ¨¨ ³ V

2.92 8π 2 27δx2 δx2

(9)

2

We have chosen λ = 3/δx in order for the scheme to be stable. The error introduced by the terms being of order λδt, we also need to ensure the following accuracy criterion : 8π 2 δt < 1. (10) 27δx2 Of course, we normally would not use this kind of method to solve Kuramoto–Sivashinsky equation, since the highorder terms are linear and can therefore be treated implicitly. However, this was a way of demonstrating that our method can be used for higher order than second-order equations. Figure 1 shows a linear instability observed when λ is smaller than its critical value given by (9). This kind of short wavelength instability is typical of explicit schemes applied on stiff sets of PDEs, when the time-step is not small enough. We can also compare the long-time behaviour of Kuramoto-Sivashinsky equation, when varying λ. The solution of this equation is known to be very sensitive to small perturbations. Therefore the solutions for different values of λ on a long timescale cannot be directly compared. However, it is interesting to see how this parameter affects the solution and to note that the solutions are the same at short-time. Figure 2 shows two computations : the left-one without the stabilizing terms, using a time-step small enough for the computation to be stable, the right-one with the stabilizing terms, using a time-step hundred times larger. Although the computations look very different after t = 50, they present the same kind of features before this time.


277 3

100 80 60 40 20 0 -20 -40 -60 -80 -100 0

1

2

3

4

5

FIG. 1. Computation with linear instability, using λ = 2.5/δx2 , which is smaller than the critical value necessary for stability (9). We have used N = 512 points (δx 0.196) and δt = 0.01.

(a)

(b)

FIG. 2. The horizontal axis is space (x), the vertical axis is time (t). A regular grid with 512 points has been used in x. (a) Computation without the stabilizing terms. The time-step has been chosen such that an explicit scheme is stable : δt = 10−4 . This computation can be compared to the result presented in [2], although their scheme is fourth-order in time and spectrally accurate. (b) Computation with the stabilizing terms and a timestep hundred times bigger : δt = 10−2 . Since the solution is extremely sensitive to small perturbations, a direct comparison at long time between the two computations is not significant.

IV.

HELE–SHAW FLOW

When using boundary integral methods, with inviscid or Stokes flows, the interface is described using marker points, labeled with a parameter α. Since there is no fixed space grid in this case, the damping operator has to be chosen slightly differently. A natural choice is a diffusion operator of both x and y coordinates along the monotonic parameter α.

278

Eds: A Sellier & M H Aliabadi 4

Let’s consider a boundary integral method for which the interface is described using : {x(α, t), y(α, t)}, where α is a marker index, for instance α(j) = j/(N − 1), with N the number of points on the surface. We choose the following damping operator : 6 6 ∂ 2 x 66 xj−1 − 2xj + xj+1 ∂ 2 y 66 yj−1 − 2yj + yj+1 D[x]j = λ λ , D[y]j = λ λ . (11) 6 2 2 2 ∂α j δα ∂α 6j δα2 The choice of λ is again crucial to ensure the stability of the method. Thanks to a Von Neumann stability analysis, it can be shown [1] that the equivalent of stability criterion (6) – in the specific context of interfacial flow with surface tension in a Hele-Shaw cell, reads : λ>

4πS δα2 , 9 δs3min

(12)

where δsmin is the minimum distance between two successive points, and does depend on time. We have tested our method on the same Hele–Shaw flow as the one described in [4]. Using the velocity field described in the next section, our code is already stable for λ = 0.35 Sδα2 /δs3min . We have chosen the same configuration as the one used to produce figure 4 in [4], i.e. an interface in a vertical Hele–Shaw cell, separating two viscous fluids with the same dynamic viscosity. The complex velocity of marker points labeled with α is given by the Birkhoff–Rott integral : γ(α , t) 1 +∞ − dα , (13) w(α) = u(α) − iv(α) = 2iπ −∞ z(α, t) − z(α , t) where z(α, t) = x + iy. If the surface and γ are periodic with period 1, this equation can also be written : u(α) − iv(α) =

1 1 − γ(α , t) cot(π(z(α, t) − z(α , t)))dα , 2i 0

(14)

When there is no viscosity jump across the interface, the vortex sheet strength γ is given by : γ = Sκα − Ryα ,

(15)

where κ is the mean curvature of the interface : κ(α) =

xα yαα − yα xαα , s3α

where

sα = (x2α + yα2 )1/2 ,

(16)

S is the non-dimensional surface tension coefficient and R is the non dimensional gravity force. The initial condition we use is the same as the one used in [4] : x(α, 0) = α,

y(α, 0) = 0.01(cos(2πα) − sin(6πα))

To compute the complex Lagrangian velocity of the interface (14), we use the spectrally accurate alternate point discretization [5]. κα and yα are computed at each timestep using second-order centered finite differences. Since the evolution of a surface is only related to its normal velocity, we can freely choose the tangential velocity of the marker points, in order to keep a reasonable distribution of points and avoid point clustering. The marker points are advected according to : ∂X(α) = U n + T s, ∂t

(17)

where X(α) = (x, y) is the position vector, n = (−yα /sα , xα /sα ) and s = (xα /sα , yα /sα ) are the normal and tangential unit vectors, and U = (u, v) · n is the normal velocity, and T is the tangential velocity. We have chosen the same expression for the tangential velocity T as the one described in appendix 2 of [4], in order to avoid point clustering. Figure 3 shows a comparison between our computation and the result of [4]. The agreement is very good : the only quantitative difference that can be observed is visible on the bottom-right zoom panel.


279 5

2

2

2

1

1

1

0

0

0

-1

-1

-1

-2

-2 -0.5

0

0.5 t=0

1

1.5

-2 -0.5

0

0.5 1 t = 0.04

1.5

-0.5

0

0.5 1 t = 0.06

1.5

1.7 2

2

1

1

1.65

0

0

-1

-1

-2

-2

1.6

1.55

-0.5

0

0.5 t = 0.08

(a)

1

1.5

-0.5

0

0.5 t = 0.1

1

1.5

1.5 0.75

0.8

0.85

t = 0.1

(b)

FIG. 3. Figure 4 in [4] (a) compared to our numerical result (b), for the same physical parameters : S = 0.1, R = −50. In our computation, we have used N = 2048 points, δt = 3.125 × 10−5 and λ = 0.35 Sδα2 /δs3min , where δsmin is the minimum distance between two successive points on the surface, and is a function of time.

V.

CONCLUSION

This method can be used for PDEs on a fixed space grid, as well as with boundary integral methods for which the interface is described using marker points. The major advantage of such a method is that the damping terms added to the right-hand-side do not require a deep knowledge of the stiff terms in the equations. Indeed, we just need to know the scaling of these terms, often related to surface tension, in terms of δsmin . Then λ can be chosen in agreement with the stability criterion (9,12).

[1] L. Duchemin, J. Eggers, In preparation (2013). [2] A.-K. Kassam, L. Trefethen, Fourth-order time-stepping for stiff pdes, SIAM J. Sci. Comput. 26 (2005) 1214–1233. [3] J. Jim Douglas, T. Dupont, Alternating-direction Galerkin methods on rectangles. In Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970), Academic Press, Academic Press, New York, 1971. [4] T. Hou, J. Lowengrub, M. Shelley, Removing the stiffness from interfacial flows with surface tension, J. of Comp. Physics 114 (1994) 312–338. [5] M. Shelley, A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method, J. of Fluid Mechanics 244 (1992) 493–526.

280


Application of the Time-Domain Boundary Element Method to Analysis of Flow-Acoustic Interaction in Expansion Chamber Silencer Models Mikael A. Langthjem1 , Masami Nakano2

1

Faculty of Engineering, Yamagata University, Jonan, Yonezawa, Yamagata 992-8510, Japan [email protected]

2

Institute of Fluid Science, Tohoku University, Katahira, Aoba-ku, Sendai, Miyagi 980-8577, Japan [email protected]

Keywords: Self-sustained flow oscillations, Acoustic resonance, Hole tone, Feedback cycle, Discrete vortex method, Boundary Element Method Abstract. This paper is concerned with a mathematical model of a simple axisymmetric silencer model consisting of an expansion chamber followed by a tailpipe. The unstable shear layer is modeled via a discrete vortex approach, based on axisymmetric vortex rings. The aeroacoustic model, which is described in the present short paper, is based on the Powell-Howe theory of vortex sound. The boundary integrals, which represent the scattering by the cavity and the tailpipe, are discretized via the boundary element method. Introduction. Expansion chambers are often used in connection with silencers in engine exhaust systems, with the aim of attenuating the energy flow. The gas flow through the chamber may however generate selfexcited oscillations, thus becoming a sound generator rather than a sound attenuator. Similar geometries and thus similar problems may be found in, for example, solid propellant rocket motors and heat exchangers. A related problem is that of flow past a rectangular cavity. This, too, has connections to a number of practical applications, such as the sunroof in an automobile, weapon and landing gear bays of aircraft, and musical instruments. Analytically, the two-dimensionality makes the problem attractive; accordingly is has been extensively studied and a large number of articles are available [1]. The aim of the present work is to contribute to the understanding of the interaction between oscillations of the flow field and the acoustic field. By oscillations of the flow field we mean the self-sustained oscillations of the jet shear layer. It is unstable and rolls up into a large, coherent vortex (a ’smoke-ring’) which is convected downstream with the flow. It cannot pass through the hole in the downstream plate but hits the plate, where it creates a pressure disturbance. The disturbance is thrown back (with the speed of sound) to the upstream plate, where it disturbs the shear layer. This initiates the roll-up of a new coherent vortex. In this way an acoustic feedback loop is formed, making up one type of flow-acoustic interaction. These so-called hole-tone feedback oscillations may interact with the acoustic axial and radial eigenoscillations of the cavity and the tailpipe. It is these interactions that we seek to understand. In the present paper we study the configuration shown in Fig. 1. This is the hole-tone feedback system equipped with a tailpipe. A closed expansion chamber will be considered in a later publication. The unstable shear layer is modeled via a discrete vortex approach, based on axisymmetric vortex rings. The aeroacoustic model is based on the Powell-Howe theory of vortex sound [1, 2]. The boundary integrals, which represent the scattering by the cavity and the tailpipe, are discretized via the boundary element method.


281

Figure 1: The hole-tone feedback system with a tailpipe. The arrow indicates the direction of the flow. The present paper concentrates on the aeroacoustic analysis. A description of the flow analysis (discrete vortex method) has been given in earlier papers [3, 4]. The geometry of the problem facilitates the use of cylindrical polar coordinates (r, θ, z), with the fluid flowing in the positive z-direction. Although it is possible that non-axisymmetric modes may be excited, we will, at this stage, consider only the axisymmetric modes (r, z). Aeroacoustic model. The starting point is taken in Howe’s equation for vortex sound at low Mach numbers [1, 2]. Let u denote the flow velocity, ω = ∇ × u the vorticity, c0 the speed of sound, and ρ the the fluid density. The sound pressure p(x, t) at the position x = (r, z) and time t is related to the vortex force (Lamb vector) L(x, t) = ω(x, t) × u(x, t) via the non-homogeneous wave equation # $ 1 ∂2 2 − ∇ p = ρ∇ · L, (1) 2 c0 ∂t2 ∂p = ∇p · n = 0 on the surfaces (n = normal vector), and p → 0 for |x| → ∞. with boundary conditions ∂n To solve (1) and (2) in an axisymmetric setting, use is made of the time-domain axisymmetric Green’s function G(t, τ ; r, z; r∗ , z∗ ) which is a solution to

−

δ(r − r∗ ) 1 ∂ 2 G ∂ 2 G 1 ∂G ∂ 2 G + δ(z − z∗ )δ(t − τ ), + + =− ∂r2 r ∂r ∂z 2 r c20 ∂t2

(2)

where the δ’s are Dirac delta functions. It can be shown that the solution is given by G(t, τ ; r, z; r∗ , z∗ ) =

c0 H(fn+ )H(fn− ) 4 , π f+ f− d

(3)

d

where fn+ = r + r∗ −

4

c20 (t − τ )2 − (z − z∗ )2 ,

fn− =

4

c20 (t − τ )2 − (z − z∗ )2 − |r − r∗ |,

(4)

fd− = c20 (t − τ )2 − (z − z∗ )2 − (r − r∗ )2 .

(5)

and fd+ = (r + r∗ )2 + (z − z∗ )2 − c20 (t − τ )2 ,

H(f ) is the Heaviside unit function which takes the value 1 when f > 0 and the value 0 when f < 0. By making use of the Green’s function the pressure p(x, t) at any point x = (r, z) can be determined as z∗2 ! ∂G ∂p∗ " p(t, r, z) = −ρ p∗ 2πr∗ dz∗ ∇y G · Lr∗ dr∗ dz∗ + −G (6) ∂r ∂r∗ ∗ τ z∗ r∗ z∗1 2 r∗2 ! ∂G ∂p∗ " + p∗ 2πr∗ dr∗ dτ, −G ∂z∗ ∂z∗ r∗1 2

282


where, in the first term, ∇y = (∂/∂r∗ , ∂/∂z∗ ). This term represents the ‘source’ contribution ps from the vortex rings. The vorticity related to a single ring is given by ω j = Γj δ(r∗ − rj )δ(z∗ − zj )iθ , where iθ is a unit vector in the azimuthal direction of the cylindrical polar coordinate system (r, θ, z). Then, by making use of (3, 4), the first term in (14) takes the form c0 ∂ sgn(r, rj ) ps = ρ π ∂r j

t−d− j /c0

t−d+ j /c0

Γj (τ )vzj (τ ) rj ∂ 4 dτ − sgn(z, zj ) ∂z fd+ fd−

t−d− j /c0

t−d+ j /c0

2 Γj (τ )vrj (τ ) rj 4 dτ . fd+ fd−

(7)

The subscript ‘s’ stands for ‘source term’, and the summation over j refers to summation over all free vortex rings. Note that differentiation with respect to the source variables rj and zj have been converted into differentiation with respect to r and z. Here care should be taken with the signs related to rj and r and to zj and z; see (4). This is taken care of by the functions sgn(r, rj ) and sgn(z, zj ). The main contributions to the τ -integrations will be at the end point singularities. Hence the functions fd+ and fd− can be approximated as follows 7 7 8 8 τ − (t − d+ (t − d− fd− ≈ 2c0 d− (8) fd+ ≈ 2c0 d+ j j /c0 ) , j j /c0 ) − τ , where 1

2 2 2 d+ j = {(r + rj ) + (z − zj ) } ,

Let a = t −

d+ j /c0

and ab = t −

d− j /c0 .

1

2 2 2 d− j = {(r − rj ) + (z − zj ) } .

(9)

The integrals over τ in (7) then take the form b F (τ ) % Iτ (t) = , (τ − a)(b − τ ) a

(10)

which is a standard Gauss-Chebyshev integral. The corresponding integration formula is given by Iτ (t) =

I i=1

wi F (si ) + RI ,

si =

b+a b−a + ti , 2 2

ti = cos

(2i − 1)π , 2I

wi =

π , I

(11)

where RI is the reminder. Using just one point, i.e. taking I = 1, corresponds to assuming that the vortex strengths Γj (τ ) and the corresponding velocities vrj (τ, rj , zj ), vzj (τ, rj , zj ) are constant within the boundaries − of the integration over τ , and equal to their values at the mean retarded time t¯ = t−(d+ j +dj ))/2c0 . Applying this approximation, an evaluation of (7) gives + 2 2 rj r + r r − r 1 ρ 1 j j 4 − − 2 + Γj vrj (t¯)(z − zj ) + − 2 ps = − Γj vzj (t¯) (12) 2 2 − 4 (d+ (dj ) (d+ (dj ) j ) j ) d+ j j dj 2 2, r + rj r − rj 1 1 1 ∂ 1 ∂ ¯ ¯ (Γj vzj (t)) (Γj vrj (t)) (z − zj ) − + − + . + c0 ∂ t¯ c0 ∂ t¯ d+ d− d+ dj j j j The second and the third term of (14) make up the scattering contribution psc , due to the solid surfaces. We use the subscript ‘sc’ to refer to ‘scattered’, and the subscript asterisk in p∗ to refer to the surface pressure. The second term is for the horizontal sections (integration along the z axis) while the third term is for the vertical surfaces (integration along the r axis). By making use of the same kind of approximations as applied to the vortex source term ps these terms can be evaluated as + ( ( ) ), z∗2 r + r ∗ r − r∗ r π 1 ∂ ¯)) r + r∗ − r − r∗ % ∗ dz∗ p∗ (t¯) (p − ( t (13) + psc = δhc ∗ 2 2 − 2 c0 ∂ t¯ (d+ (d− d+ d− z∗1 ∗) ∗) ∗ ∗ d+ ∗ d∗ , + ( ( ) ) r∗2 1 r∗ (z − z∗ ) π 1 1 ∂ ¯)) 1 + 1 % (p − δvc + ( t + dr∗ p∗ (t¯) ∗ + − + − − 2 c0 ∂ t¯ (d∗ )2 (d∗ )2 d∗ d∗ r∗1 d+ ∗ d∗ z∗2 r∗2 r ∂p∗ (t¯) r ∂p∗ (t¯) % ∗ % ∗ +πδho dz∗ + πδvo dr∗ . + − ∂r∗ + − ∂z∗ z∗1 r∗1 d∗ d∗ d∗ d∗ 3


283

Here δhc is 1 on horizontal closed (i.e. physical) surfaces, and 0 otherwise; δvc is 1 on vertical closed surfaces, and 0 otherwise; δho is 1 on horizontal open (i.e. virtual, or control) surfaces, and 0 otherwise; and δvo is 1 on vertical open surfaces, and 0 otherwise. The total pressure at an observation point (r, z) is now given by ςp(t¯, r, z) = ps (t¯, r, z) + psc (t¯, r, z).

(14)

Here ς is equal to 1 when the observation point is in the acoustic medium and away from the solid boundaries, and equal to 12 when the observation point is located on a solid boundary. Discretization via a Galerkin-type boundary element method. Next we employ the boundary element methodology of dividing the surface into V elements, assuming that the pressure is constant within each element. The time dependence of the pressure is, within cosecutive time steps, interpolated via a cubic polynomial. Thus, the pressure (anywhere) on the boundary p∗ (t¯) can, at time step W , be expressed as p∗ (t¯, r∗ , z∗ ) =

V W

fv (r∗ , z∗ )gw (t¯)Pvw ,

(15)

1 for (r∗ , z∗ ) ∈ (rv , zv ) , 0 otherwise

(16)

v=1 w=1

(

where fv (r∗ , z∗ ) =

and gw (t¯) = g(t − wΔt), with ⎧ & t '2 1 & t '3 t ⎪ 1 + 11 ⎪ 6 Δt +& Δt' + 6& Δt' ⎪ 3 ⎪ 1 t 2 t ⎪ −1 t ⎨ 1 + 2 Δt − & Δt '2 21 & Δt '3 1 t t t g(t) = 1 − 2 Δt − Δt + 2 Δt ⎪ & t '2 1 & t '3 ⎪ t ⎪ ⎪ 1 − 11 − 6 Δt ⎪ 6 Δt + Δt ⎩ 0 otherwise.

for − Δt ≤ t < 0, for 0 ≤ t < Δt, for Δt ≤ t < 2Δt, for 2Δt ≤ t < 3Δt,

(17)

In the usual collocation type BEM (13) is evaluated at each of the V spatial control points in turn, to give V equations for the V unknown element pressures (at each time step). Here we employ the Galerkin method, where the ‘strong form’ of these equations are exchanged with a ‘weak form’. To this end, (13) is multiplied by the spatial shape function fu , followed by integration around the structure. Letting u run from 1 to V , we obtain a V × V equation system on the form A 0 pW = −

N save

Aw pW −w + fW ,

(18)

w=1

which is solved at each time step. Frictional attenuation in the pipe. The main interest of the present work lies in the interaction between acoustic resonance and self-sustained flow oscillations. In real acoustic resonances the pressure amplitude is kept in check (i ) by dissipative effects and (ii ) by nonlinear effects. The present problem is, as (1) shows, modeled as a problem in forced linear vibrations; and, as is well known from the theory of mechanical vibrations, at least a dissipation mechanism/model is needed to limit the pressure amplitude at resonance. Lighthill [5] gives an analysis of frictional attenuation of one-dimensional sound waves through a pipe, based on boundary layer theory. He shows that, when a pressure pulse has moved a distance z through the pipe, its strength (the pressure amplitude) has been reduced by the factor # $ ω z ! ν " r = exp −2 , (19) c0 d0 2ω

4

284


where ω is the cyclic frequency and ν the kinematic viscosity (of air). This factor, with z replaced by |zu − zv |, is included in (18). Acoustic feedback model. The acoustic particle velocity v(t, r, z) can be evaluated from the linearized Euler equation ∂v ρ = −∇p (20) ∂t once the pressure field p is known. In the present work the computation of v from (20) is based on the surface pressure pe on the boundary elements. The acoustic velocity field is superimposed onto the ‘hydrodynamic’ velocity field of the free vortex rings in the open domain between nozzle exit and end plate. Numerical examples. In the numerical examples to follow the diameter of both nozzle and end plate hole are d0 = 50 mm. The gap length between nozzle exit and the end plate is also 50 mm. The diameter of the end plate is 3d0 = 150 mm. The mean jet speed u0 = 10 m/s. The length of the pipe attached onto the end plate is l0 = 37.78d0 = 1889 mm. The pipe resonance frequencies are f0 = 90 n, n = 1, 2, · · · , where even values of n correspond to multiples of a whole wavelength. Figure 2 shows the appearance and location of free vortex rings in the vicinity of the end plate during one period of oscillation. The fundamental frequency f0 = 158 Hz, which is lower than for the case without a tailpipe [3, 4]. End plate

Nozzle

t=0

t=

1 8f0

t=

t=

3 8f0

t=

4 8f0

t=

5 8f0

t=

6 8f0

t=

7 8f0

t=

8 8f0

2 8f0

Figure 2: View of the vortex rings (cross sections) during one period of oscillation. Figure 3 shows the grid applied in the boundary element analysis. The left and right vertical end surfaces and the top horizontal surface are open; here the boundary condition p = 0 applies. To the solid surfaces, and also to the symmetry axis, the condition ∂p/∂n = 0 applies. 5


285

Symmetry axis

Figure 3: The grid applied in the bem. Figure 4 shows time series for the pressure signal at the position (r, z) = (0, 17 l0 ), measured from the tailpipe termination, in downstream direction. Parts (a)-(c) are for the case without acoustic feedback. Here, part (a) shows the source contribution ps only, part (b) the scattered (pipe resonance) contribution psc only, and part (c) the total pressure signal ps + psc . Parts (d)-(f) are similar to parts (a)-(c), but here acoustic feedback is included. Pressure p/p0 0.00025

0.00025

0

0

-0.00025

-0.00025

-0.0005 0

0.05

0.1

0.15

0.2

0.25

0.3

(a)

0.001

-0.0005 0

0.05

0.1

0.15

0.2

0.25

0.3

(d)

0

0.05

0.1

0.15

0.2

0.25

0.3

(e)

0

0.05

0.1

0.15

0.2

0.25

0.3

(f)

0.00025

0.0005 0

0

-0.0005 -0.001 0

0.05

0.1

0.15

0.2

0.25

0.3

(b)

0.001

-0.00025

0.00025

0.0005

0

0 -0.00025

-0.0005 -0.001 0

0.05

0.1

0.15

0.2

0.25

0.3

(c)

-0.0005

Time [s]

Figure 4: Sound pressure time series. The observation point is on the jet axis, 1/7 pipe-length downstream from the mouth of the pipe. Parts (a)-(c) are for the case without acoustic feedback, while parts (d)-(f) are for the case with acoustic feedback. (a, d) The source contribution ps only. (b, e) The scattered (pipe resonance) contribution psc only. (c, f) The total pressure signal ps + psc .

References [1] M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge University Press, Cambridge, 1998. [2] M. S. Howe, Theory of Vortex Sound, Cambridge University Press, Cambridge, 2003. [3] M.A. Langthjem, M. Nakano. Numerical study of the hole-tone feedback cycle based on an axisymmetric discrete vortex method and Curle’s equation. J. Sound Vibr. 2005, 288:133-176. [4] M.A. Langthjem, M. Nakano. A numerical simulation of the hole-tone feedback cycle based on an axisymmetric formulation. Fluid Dyn. Res. 2010, 42:1-26. [5] M. J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978. 6

286


On Explicit Expressions of 3D Elastostatic Green's Functions and Their Derivatives for Anisotropic Solids Longtao Xie1, Chuanzeng Zhang1, Yongping Wan2 and Zheng Zhong2 1

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

2

School of Aerospace Engineering and Applied Mechanics, Tongji University, 200092 Shanghai, P.R. China [email protected]; [email protected]

Keywords: Green’s functions, derivatives of Green’s functions, 3D elastostatics anisotropic solids

Abstract. A new mathematical construction of elastostatic Green’s functions and their derivatives for threedimensional (3D), homogeneous, anisotropic and linear elastic solids is presented in this paper. The new fractional form expressions of Green’s functions and their derivatives are valid for both non-degenerate and degenerate cases. Numerical examples show that they are convenient for an efficient and accurate computation of Green’s functions and their derivatives. Introduction Green’s functions or fundamental solutions are the kernel functions in boundary integral equations (BIEs) and they are required in boundary element method (BEM). Therefore, the efficient and accurate computation of Green’s functions plays an important role in solving BIEs especially in BEM. Commonly, Green’s functions in terms of contour or line integrals are reduced from triple integrals by applying the Fourier-transform to the governing partial differential equations [1, 2]. Green’s functions in terms of contour integrals can also be obtained by Randon-transform [3]. After using Cauchy’s residue theorem, Green’s functions become an explicit form in terms of Stroh eigenvalues pv v 1, 2, 3 , which are the roots of a sextic polynomial equation with positive imaginary part. These expressions are non-unique in the sense that they have different forms for non-degenerate case p1 z p2 z p3 and degenerate cases

p1 p2 and p1 p2 p3 . Although these expressions are theoretically correct, their computation in practice may have significant numerical difficulties in nearly degenerate cases, when any two of the eigenvalues are nearly equal. By rewriting these expressions for non-degenerate cases, Ting and Lee [4] derived alternative expressions for Green’s functions which remain valid in both degenerate and nearly degenerate cases. For the derivatives of the Green’s functions, the contour integral expressions were obtained by Barnett [5]. Mura represented these expressions in another form [6]. Following Barnett’s work, Lee gave explicit expressions in non-degenerate cases by means of Cauchy’s residue theorem [7]. Explicit expressions for the first derivatives of the Green’s functions in degenerate cases were given by Buroni et al. [8]. Like the Green’s functions themselves, the explicit expressions of their derivatives may still have some problems in the numerical evaluation. Recently, efforts have been made to overcome this numerical difficulty. By differentiating the Green’s functions of Ting and Lee [4], Lee [9] and Shiah and Tan [10] gave expressions for the derivatives of the Green’s functions in another form. However, since the expressions of Lee [9] still have different forms for non-degenerate and degenerate cases, significant errors may still remain in their numerical computation, especially in nearly degenerate cases. The main purpose of this paper is to derive exact expressions like Ting and Lee’s [4] for both first and second derivatives of 3D elastostatic anisotropic Green’s functions. Based on the contour integral expressions of Mura [6], new explicit algebraic expressions for Green’s functions and their derivatives are proposed. Numerical examples for transversely isotropic materials are presented and discussed. Problem Statement


287

The governing equations of 3D elastostatic Green’s functions for homogeneous, anisotropic and linear elastic solids are Cijkl Gkm,lj x G imG x 0 , (1)

where Cijkl is the elasticity tensor, G im is the Kronecker delta, and G x is the 3D Dirac-delta function

which is zero everywhere except at point x 0 . By applying either Fourier-transform or Radon-transform followed by some mathematical manipulations, the Green's functions Gij x can be expressed in a contour integral form as

Gx where, ξ is a parameter vector, Kik ξ

1

(2) K 1 ξ dI , 8S 2 x ³s1 Cijkl[ j[l , x is the distance from observation point x to the origin,

and S 1 is the unit circle on the oblique plane perpendicular to x . Furthermore, Green’s functions can be also obtained in an integral form as

G x

f N p 1 dp , 4S 2 x ³f D p

(3)

where N p and D p are the co-factor matrix and the determinant of K ξ , respectively. Here ξ m pn , where m and n are two mutually orthogonal unit vectors on the oblique plane perpendicular to x. It is well known in linear elasticity that D p has 6 roots. Three complex roots with a positive imaginary part are known as Stroh eigenvalues ( pv , v 1,2,3 ), and the other three roots pv are the conjugates of the

three complex roots pv . Therefore, D p can be written as

D p D v

3 1

p pv p pv ,

(4)

where D is the coefficient of p in D p , and the over bar denotes the complex conjugate. By applying Cauchy’s residue theorem to eq (3) we obtain for the Green’s functions 6

G x

i

N p

3

¦ 2S x v 1

D pv pv

3

p

n 1, n z v

v pn pv pn

(5)

for non-degenerate cases, when the three Stroh eigenvalues are distinct ( p1 z p2 z p3 ). However, for degenerate cases ( p1 p2 or p1 p2 p3 ), the Green’s functions have different expressions. The corresponding expressions in those two cases have been presented by Buroni et al. [8]. Indeed, all the above given expressions are mathematically correct. However, considerable errors may appear in the numerical evaluation when any two Stroh eigenvalues are nearly equal. Fig. 1 shows the results * G11 and T11* V11 from the work of Buroni et al. [8]. It can be seen from Fig. for the Green’s functions U11 1that numerical errors arise in nearly degenerate case by using eq (5) for Green’s functions. The numerical errors of the corresponding stress components resulted from the errors in the first derivatives of Green’s functions are even much more significant.

* Figure 1: Green’s functions U11

G11 and T11* V11 from the work of Buroni et al. [8]

288


After a careful analysis, we find that the numerical errors stem from the factors like pv pn in the denominators, which cause numerical errors when pv and pn are very close. Ting and Lee [4] presented a solution to this problem. By rewriting eq (5), they arrived at alternative expressions without factors like pv pn in the denominators. Besides higher accuracy, another advantage of Ting and Lee’s expressions [4] is that they are unified for degenerate and non-degenerate cases. Although their expressions were derived from non-degenerate cases, they remain valid also in degenerate cases. Ting and Lee [4] solved the numerical problem in evaluating Green’s functions. By differentiating Ting and Lee’s expressions, Lee [9] tried to solve the problem in the evaluation of the first and second derivatives of the Green’s functions. However, since the new expressions of the derivatives of the Green’s functions still include factors like pv pn in the denominators, numerical errors may still remain. So, an accurate evaluation of the derivatives of Green’s functions is still demanding, which is the main focus of this paper. New Expressions of Green’s Functions In this section, new expressions of Green’s functions are derived from eq (3). As Ting and Lee [4] pointed out that the co-factors, elements in the matrix N , are polynomials in p and the highest order is 4. Let 4

N p where N n n

¦ p N , n

n

(6)

n 0

0,1, 2, 3, 4 are independent of p . Substituting (4) and (6) into (3), we obtain 4

1

Gx

§

¦¨ N ³ x ¨

4DS

2

In

³

n 0

n

f

©

· dp ¸ . p pv p pv ¸¹ 1 pn

f

3 v

(7)

By defining f

pn

f

p p p p 3

v

v 1

dp ,

(8)

0,1, 2, 3, 4 .

(9)

v

eq (7) becomes 4

1

Gx

4DS

¦ I N n x n

2

n

n 0

By virtue of two special integrals f

1

f

v 1 p pv p pv

Un

³

Vn

³

3

dp

(10)

dp ,

(11)

and

I n n 0,1, 2, 3, 4 are expressed as

f

p

f

p p p p 3

v 1

v

v

I 0 U 3 , I1 V3 , I 2 U 2 p3 p3 I1 p3 p3 I 0 , I 3 V2 p3 p3 I 2 p3 p3 I1 ,

(12a) (12b) (12c)

U1 p3 p3 p2 p2 I 3 p3 p2 p3 p2 p3 p3 p3 p2 p3 p2 p2 p2 I 2 (12d) p3 p3 p2 p3 p3 p2 p2 p2 p3 p2 p2 p3 I1 p2 p2 p3 p3 I 0 . It is easy to obtain explicit expressions from eqs (10) and (11) by using Cauchy’s residue theorem directly. We call these expressions residual form (RF) expressions. Since RF expressions may cause numerical errors in practical calculations with computer, a reformulation of the expressions, which have no factors like pv pn in the denominators, is needed. We find out that after the summation of the terms in the RF expressions followed by a reduction of the fractions, new expressions have only factors like pv pn in the I4


289

denominators. Unfortunately, we cannot prove this yet in a strict mathematical way for general U n and Vn . However, when n is not larger than 9, it is confident that the RF expressions of U n and Vn are transformed

into fractional form (FF) expressions which have no factors like pv pn in the denominators. In fact, for elastostatic Green’s functions and their derivatives, n is not larger than 9. So we can conclude that the Green’s functions and their derivatives should have closed-form expressions in fractional form which have no factors like pv pn in the denominators. New Expressions of First and Second Derivatives of Green’s Functions In this section, fractional form expressions of the derivatives of Green’s functions are derived from those in contour integral form given by Mura [6]. Because of the size limitation of the paper, the details are not presented here. However the key idea is similar to that as described in the previous section. The main difficulty now is how to deal with the counterpart of eq (8), namely

I n,

³

pn

f

f

p p p p 3

2

v 1

2

v

0,1,,10

(13)

dp n 0,1,,16

(14)

dp

n

v

for the first derivatives of Green’s functions, and

I n3

³

pn

f

p p p p 3

f

3

3

v

v 1

v

for the second derivatives of Green’s functions. Since it is not so easy to derive similar expressions like eq (12) from eqs (13) and (14) directly, we consider instead of eqs (13) and (14)

I n,c

f

pn

f

p p p p

³

6

v 1

v

dp n 0,1,,10

(15)

dp n 0,1,,16 .

(16)

v

and

I n3c

f

pn

f

p p p p

³

9

v 1

v

v

The following steps are similar to that for evaluating Green’s functions as presented in the previous section. It should be mentioned here that in the final expressions, p4 and p7 , p5 and p8 and p6 and p9 should be replaced by p1 , p2 and p3 , respectively. Numerical Examples Although the fractional form expressions exist, it is too complex to write them out directly, especially when n becomes large. Therefore, new explicit fractional form expressions of Green’s functions and their derivatives are not listed in this paper, but the correctness, the efficiency and the accuracy of these expressions are checked. For simplicity in programming, fractional form expressions are evaluated from the residual form expressions with the help of MATHEMATICA in symbolic manipulations. By now, only Green’s functions and their first derivatives are numerically evaluated and compared with the analytical solutions. The numerical evaluation of the second derivatives and non-symbolic programming are in progress. To show the accuracy of the fractional form expressions, we consider an infinite, homogeneous, transversely isotropic and linear elastic body subjected to a unit point force normal to the plane of isotropy. The Green’s functions and the corresponding stresses are computed and compared with Lee’s analytical expressions for transversely isotropic materials [9]. The corresponding stresses are obtained by using V ij Cijkl Gk 3,l . (17) The used material constants are [9] C11 88 , C12 72 , C13

40 , C33

24 , C44 16 , (unit: 10 7 N/m 2 ).

(18)

290


The observation point is chosen as x

^sin I cosT , sin I sinT , cos I`. The results for the component of the

Green’s functions G11 and the stress component V 11 are shown in Figs. 2 and 3 for T

S / 4 with varying I . These figures show the correctness and the high accuracy of the present evaluation method by using

fractional form expressions of the Green’s functions and their derivatives.

Figure 2: G11 for a transversely isotropic material

Figure 3: V 11 for a transversely isotropic material Conclusions Fractional form expressions of Green’s functions and their derivatives are presented in this paper. The given expressions are unified in the sense that they are valid for both non-degenerate and degenerate cases. The numerical results for a transversely isotropic material show that the results from the fractional form expressions agree very well with the analytical results, which verifies the correctness and the high accuracy of the present fractional form expressions. Acknowledgment The work is supported by the China Scholarship Council (CSC) (Project No. 2011626148), which is gratefully acknowledged.


291

References [1] I. Fredholm Acta Mathematica, 23, 1 (1900). [2] I. M. Lifshitz and L. N. Rozenzweig Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, 17, 783 (1947). [3] H. L. Dunn International Journal of Engineering Science, 32, 119-131 (1994). [4] T. C. T. Ting and Ven-Gen Lee The Quarterly Journal of Mechanics and Applied Mathematics, 50, 407-426 (1997). [5] D. M. Barnett Physica Status Solidi (b), 49, 741-748 (1972). [6] Toshio Mura Micromechnics of Defects in Solids, Martinus Nijhof Publishers (1987). [7]

Ven-Gen Lee Mechanics Research Communications, 30, 241-249 (2003).

[8]

Federico C. Buroni, Jhonny E. Ortiz and Andrés Sáez International Journal for Numerical Methods in Engineering, 86, 1125-1143 (2011).

[9] Ven-Gen Lee International Journal of Solids and Structures, 46, 3471-3479 (2009). [10] Y. C. Shiah and C. L. Tan Computer Modeling in Engineering & Sciences, 78(2), 95-108 (2011).

292


QUADRATIC PROGRAMING FOR MINIMIZATION OF THE TOTAL POTENTIAL ENERGY TO SOLVE CONTACT PROBLEMS USING THE COLLOCATION BEM C. G. Panagiotopoulos, V. Mantiˇc, I.G. Garc´ıa, E. Graciani Group of Elasticity and Strength of Materials, Department of Continuum Mechanics, School of Engineering, University of Seville, Seville, ES-41092, Spain Keywords: Unilateral frictionless contact problems, adhesive contact, boundary element method, stiffness matrix, conjugate gradients.

Abstract. A unified approach for solving problems of frictionless as well as adhesive contact is presented. This approach is based on energetic principles and leads to a minimization problem of the total potential energy [1]. Appropriate boundary integral forms of the elastic energy are defined and the quadratic form of the problem is constructed [2]. The problem is solved by utilizing the collocation boundary element method (BEM) [3, 4]. Two algorithms for the solution of the quadratic problem obtained are proposed, both being variants of the well-known conjugate gradient algorithm [5], while the distinction between them consists of the explicit construction or not of the problem matrix. This matrix has a similar physical meaning as the stiffness matrix [6, 7] commonly used in the context of the finite element method (FEM) [8]. Both symmetric and non-symmetric formulations of this matrix are considered and discussed. The present procedure, in addition to its own interest, can also be extended to problems where dissipative phenomena take place such as friction, damage and plasticity [9, 10]. It seems also promising as a possible framework for the solution of contact problems of transient elastodynamics by BEM [11]. The present computational implementation is briefly described and numerical solutions of a frictionless contact problem are given and compared to a FEM solution. Introduction Contact problems are often present in engineering applications. Contact between deformable bodies is a complex and inherently non-linear problem. It is essentially a boundary phenomenon which has a strong effect on stress and displacement fields in the vicinity of the boundary of the contacting bodies. A simple model for the description of contact is the Signorini contact condition expressing the non-penetrability of the bodies in contact. In the context of problems of interface crack growth [9, 10], an energy based procedure together with its implementation in the collocation BEM has recently been developed. The acronym EC-BEM, standing for Energetic approach for the solution of elastic Contact problems by BEM, has been coined for this framework. In such kind of applications, a contact problem is always inherent in the procedure, hence it is important to present in detail the theoretical background as well as details of the numerical implementation for problems of unilateral (or Signorini) and adhesive frictionless contact. In the unilateral contact no other material is present at the interface between elastic domains. For this case the Signorini condition models the exact non-penetrability of the bodies in contact by the following conditions [12]: • The relative normal displacement of the interface cannot be larger than the distance between the bodies, • Only compressive normal tractions can be transmitted by contact, • Stresses can be transmitted only in the presence of contact. In the adhesive contact the common interface is represented by a continuous distribution of springs, similar to the Winkler spring model, with distinct normal and tangential elastic stiffnesses of values ranging from zero to infinity.


293

Theoretical background Let N be a finite number of mutually disjoint Lipschitz subdomains Ωi ⊂ Rd with the boundary ∂Ωi =Γi . Let Γij be the part of the boundary of Ωi possibly in contact with Ωj . A possible contact with a rigid obstacle is also considered on some parts of the outer boundary Γi0 . The rest of the outer part of the boundary ∂Ωi is the union of two disjoint subsets ΓiD and ΓiN . A time-dependent boundary displacement uiD (x, t) is imposed on the Dirichlet part of the boundary ΓiD , while on the boundary ΓiN i prescribed tractions piN (x, t) are imposed. Therefore, any admissible displacement ui (x, t) : ∪N i=1 Ω → d i i R has to be equal to a prescribed displacement uD (x, t) on ΓD . Thus, the Signorini contact can be developed on the surface ΓC = 0≤i 0 and, f (r) = i

! eikr − 1 + ikr − k3 r4

k2 r2 2

" .

(14)

In this case, we can’t obtain an explicit expression of the 1-D integrals in (13) but one obtains an accurate evaluation using standard numerical integration methods. It is also possible to express IG as a linear combination of 1-D integrals when the triangles are in general position in the same plane or in secant planes. When the triangles are in parallel planes, we can obtain only a reduction of one dimension using formula (5), since further reductions lead to hypergeometric functions thus the evaluation of the sum of the power series is not possible. Actually, the expression obtained is too intricate to be of practical interest. Numerical results:

For S = T , we denote

ref , the evaluation of (13) using enough Gauss-Legendre integration points to ensure a good accuracy • by IG (300 points on [0, 1]).


321

• by IG (N ), the evaluation of the 4-D integral (12) by the method presented in [5] by Sauter and Schwab. We use N Gauss-Legendre points on [0, 1]. We obtain 3 regular integrals over [0, 1]4 thus the total number of integration points is 3N 4 . • by I3G (M ), the evaluation of (13) using M Gauss-Legendre points on [0, 1] thus the total number of integration points is 3M . Finally, we introduce the relative differences : 6 6 6IG (N ) − I ref 6 6 ref 6 G E(N ) = 6I 6 G

and

6 6 63 ref 6 6 6IG (M ) − IG 3 6 ref 6 E(M )= . 6I 6 G

1 3 for E(M ). We observe Figure 3 shows these two relative differences according to 3N1 4 for E(N ) and to 3M that it takes fewer points with our method to achieve the same accuracy. Furthermore, we have to point out that similar results are available with linear basis functions.

10−2

ref = −0.0252 − 0.0097i IG

3N 4 = 48

3M = 6

Relative difference

10−4 10−6 10−8 3M = 48

10−10 10−12 10−14 10−16

10−5

E(N ) 3 ) E(M

10−4

10−3

Figure 3: E(N ) according to

4

1 and 1 3M 3N 4

1 3N 4

10−2

10−1

3 and E(M ) according to

100

1 3M .

Evaluation of finite-parts of integrals

˚ a point inside the triangle S defined in the Figure 1. Using formulas (6), we can explicit the finiteLet x ∈ S, * dy part of divergent integrals such as I3 = = x−y 3 . We introduce T an equilateral triangle of gravity center x such that x is an distance of each side of T . By applying formula (6), we obtain 3 3 dsy dsy 1 I3 ( ) = dy = − gi (x) + . (15) 3 3 3 S\T x − y αi x − y βi x − y i=1

i=1

It is equivalent to use T or a circle of center X and radius as exclusion area for this integrand and also to − → n→ replace x by z = x + − S , with nS the normal to the plane of S . As √ √3 dsy ds 3 , (16) = = √ 3 3/2 − 3 ( 2 + s2 ) βi x − y the only divergent terms come from the integrals on βj , j = 1, 3 and finally, the finite-part of I3 ( ) is , s+ + 3 i −σi (X) s % I3 = pf I3 ( ) = − gi (x) gi (x)2 gi (x)2 + s2 s− −σ (X) i=1 i

i

(17)

322


A2

α3 A1

β1 α2

β2

x

T β3

A3

α1

S

Figure 4: Exclusion zone for singular point X using triangle T . Remark: If X = Ai is a vertex of S, we obtain this simple expression: + I3 = pf I3 ( ) = −gi (x)

5

s % gi (x)2 gi (x)2 + s2

, s+ i −σi (X) .

(18)

s− i −σi (X)

Conclusion

The method presented provides explicit formulas to evaluate singular and nearly singular integrals arising in Galerkin BEM. We also apply the method for the explicit evaluation of the finite-part of a hypersingular integral. We can reduce the integral of the singular part of the Green kernel in a linear combination of 1-D regular integrals, as well as the integral of the whole Green kernel. Results obtained We have provided some results concerning the single layer potential but the results in this paper are only a small part of all those available which allow to take into account the various geometric situations and other integrands such as the singular part of the gradient of the Green kernel (double layer potential). We derived formulas for 3-D Helmholtz and Maxwell equations (EFIE and MFIE). Results for the double layer potential with triangles in parallel planes can be found in [7]. This method applies as well as to 2-D, to linear densities (see [6]) or even volume integral equations.

References [1] D.E. Cormac and D. Rosen, Singular and near singular integrals in the BEM: A global approach, SIAM Journal on Applied Mathematics, 1993. [2] M. Lenoir, Influence coefficients for variational integral equations, C. R. Acad. Sci., 2006. [3] L. Scuderi, A new smoothing strategy for computing nearly singular integrals in 3D Galerkin BEM, Journal of Computational and Applied Mathematics, 2009. [4] S. Nintcheu Fata, Semi-analytic treatment of nearly-singular Galerkin surface integrals, Applied Numerical Mathematics, 2010. [5] S. Sauter and C. Schwab, Boundary Element Methods, Springer-Verlag Berlin, 2010. [6] M. Lenoir and N. Salles, Exact evaluation of singular and near-singular integrals in Galerkin BEM, Proceedings ECCOMAS 2012, 2012 [7] M. Lenoir and N. Salles, Evaluation of 3-D Singular and Nearly Singular Integrals in Galerkin BEM for Thin Layers, SIAM Journal on Scientific Computing, 2012.


323

Density results and the method of fundamental solutions for Cauchy data reconstruction. Carlos J.S. Alves1 and Nuno F.M. Martins2 1

CEMAT-IST and Department of Mathematics, Instituto Superior Técnico, TULisbon, Avenida Rovisco Pais, 1096 Lisboa, Portugal, [email protected]. 2

CEMAT-IST and Department of Mathematics, Faculdade de Ciências e Tecnologia, Univ. Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal, [email protected]

Keywords: Cauchy data reconstruction, Laplace and Helmholtz equations, method of fundamental solutions.

Abstract. In this work we address the problem that consists in fitting Cauchy boundary data using the method of fundamental solutions (MFS). We study some density properties in order to justify the MFS fitting for full and partial Cauchy data. We focus on fitting partial data and in particular we will present some theoretical and numerical results concerning the reconstruction of data in an inaccessible part of the boundary. Overview The reconstruction of Cauchy boundary data is an inverse ill-posed problem with many applications in non intrusive evaluation problems. The problem here addressed consists in, given a pair of data (g,gn) solve the Cauchy problem

' k 2 u ° ®u g ½ ° ¾ ¯ w nu g n ¿

0 in : at 6 *

w:

where ∂nu denotes the normal derivative of u and the normal vector points outwards with respect to the regular (2D or 3D) domain Ω (that, for simplicity will be assumed to be simply connected). The part of the boundary Σ is assumed to be a relatively open set on the whole boundary Γ. In the following we shall consider the wave number k≥0 to be constant. Notice that, for k=0 we have a Cauchy problem for the Laplacian whereas for k>0 we are considering a Helmholtz problem. One of the most direct approaches for the numerical solution of the above Cauchy problem is the MFS. This meshfree boundary method has been mostly considered as a numerical method for direct boundary value problems (see the first papers by Krupradze [1] and Bogomolny [2]). Concerning inverse problems, the method was firstly applied in a decomposition method for acoustic scattering (see [3]) and has been mainly used for Cauchy data fitting (eg. [4] and [5]). For a survey on the subject, see [6]. Recall that a fundamental solution, Φk , is a response to a Dirac delta centered at the origin, that is, ' k 2 Ik G

and that the point source function ) ky is defined by ) k x y . For the Helmholtz equation ( k > 0) we consider

324


) ky ( x)

i (1) ° 4 H 0 k x y in 2D ° ® ik x y ° e in 3D °¯ 4S x y

) ky ( x)

1 °° 2S log x y in 2D . ® 1 ° in 3D °¯ 4S x y

and for k=0 (Laplace equation),

The method of fundamental solutions for the Cauchy problem consists in taking the approximation

u x

¦D ) x y k

j

j

j

where the source points are placed in the exterior of Ω. Thus, the above function satisfies the PDE 2 0 in Ω. In order to solve numerically the Cauchy problem using u , we compute the

' k u

coefficients D j by imposing the two boundary conditions, ¦ D j )k ( x y j ) g x ° j . ® k °¦ D j w n) ( x y j ) gn x ¯ j

(1)

Notice that the Cauchy problem may not have a solution. Even if such solution exists, the above linear system may not be solvable. Therefore, in order to deal with the ill-posed nature of Cauchy problem we usually compute the coefficients by considering some sort of regularization method. In turn, we obtain a function that provides a fitting for the given Cauchy data. An important theoretical question is, given a pair of Cauchy data (on appropriate functional spaces), can we obtain a good fitting using fundamental solutions basis functions ? It turns out that the question is related to the location where the sources are placed. Usually, the source points are placed at fictitious boundaries located outside the domain of interest Ω. The number of such boundaries is related to the number of boundary conditions. In this paper we study this density problem, for two situations: 1. The case where Σ = Γ (full boundary data) and 2. For partial boundary data, meaning that the Cauchy is available at part of the boundary. Density results for full boundary data. Consider the subspace of H 1 (:) defined by

H ' k 2

û H (:) : ' k u 0` 1

2

and notice that the trace and normal trace of u H ' k 2 belongs to H 1/2 * and H 1/2 * respectively. Let H * by

H 1/2 * u H 1/2 * be the data space and consider the linear map / * : H ' k 2 o H * defined

/* u

u

|*

, w nu|* .


325

Notice that, due to Holmgren’s lemma, the above map is injective. The Cauchy problem can now be recast as, given (g,gn) on HΓ solve the linear problem / * u g , gn . The range of / * is the so called set of compatible Cauchy data and is a proper subspace of H * , that is,

R / * z H * . Consider now the boundary set of fundamental solutions S*

^ )

`

| , w n ) ky |* : y *

k y *

where * is the boundary of a regular domain containing Ω (notice that there are other possible choices for artificial curves). Clearly, span S* R / * and system (1) is solvable if and only if

g , gn span S* .

We have: Proposition. If - k2 is not an eigenvalue for the Laplace-Dirichlet problem in Ω, the set of compatible Cauchy data is closed in H * and Proposition. Under the same above no resonance assumptions, the subspace span S* is dense in

R /* .

The first result means that for some pair of H * data there is no nearby data coming from H ' k 2 , In particular, it makes no sense to consider the MFS fitting for general data in H * taking only source points in the exterior of Ω. However, for compatible data, the second proposition states that the MFS provides a good fitting by taking sources located only in the exterior of Ω. One way to deal with the lack of density in the data space is to consider also interior source points, located at an extra interior artificial curve. This approach leads to a data fitting taking fundamental basis functions satisfying the PDE only at a subset of the whole domain Ω.

Density results for partial boundary data. Consider a decomposition of the boundary *

.

6 3

.

* \ 6

where 6 and * \ 6 are relatively open (non empty) subsets of * with common boundary 3 . As above, we define the space of partial data by H 6 H 1/ 2 6 u H 1/ 2 6 (see [7] for the definition of these trace spaces) and the map / 6 : H ' k 2 o H 6 . Again from Holmgren’s lemma, the map / 6 is injective hence, given g 6 , g 6 n R / 6 there exists an unique u H ' k 2 and

g

* \6

/ 6 u g 6 , g 6n ° ® g * \6 , g * \6 n °¯ / * \ 6 u

The pair g * \ 6 , g * \ 6 n will be referred as the missing data.

, g * \ 6 n R / * \ 6 such that

326


Proposition. Under the no resonance assumptions, the set of compatible data, R / 6 , is dense in H 6 . Notice that the set of full compatible data R / * is closed in the data space H * but is not dense (otherwise, all the data would be compatible). On the other hand, the set of partial compatible data R / 6 is dense but is not closed in the data space H 6 . Moreover, we have the following density result concerning fundamental solutions basis functions.

^

`

Proposition. Assuming the no resonance assumption, the subspace span ) ky |6 , w n ) ky |6 : y * is dense in H 6 . Therefore, we can obtain a good fitting of partial data taking fundamental solutions with source points located only in the exterior of Ω. An iterative method for Cauchy data reconstruction. In the classical MFS approximation for the Cauchy problem described at the beginning of the paper, the H 6 pairs of data are fitted simultaneously. However, in practice, one of the data is imposed (usually assumed with negligible error) and the other measured (usually contaminated with noise). In such situations, a simultaneous fitting of both data may lead to poor reconstruction results. Assume, for instance, that the Dirichlet datum g 6 is imposed and the Neumann datum g n6 is measured. Assume further that g 6 , gn6 R / 6 and define the linear map

\ g 6 : H 1/2 * \ 6 o H 1/2 6 , h

w nu |6 6

where u H ' k 2 is determined by the boundary conditions u |6 g and u |* \ 6 h . It follows that there

exists an unique u H ' k 2 satisfying \ g 6 u |* \ 6

g n6 and in particular, the missing boundary data is

/ * \ 6 u . Such solution can be obtained by solving the equation \ g6 h

gn6 .

Linearizing, we get the equation on the update hu \ 'g 6 h hu

gn6 \ g 6 h .

Numerical examples In order to illustrate the proposed methods we present two numerical simulations. First case, concerns Cauchy data reconstruction for the domain presented in Fig. 1 using a direct MFS approach. The partial data was obtained at 60 observation points (green dots) and the source points are represented by the red dots. In Fig. 2 we present several reconstruction results, taking observations at different parts of the boundary (the corresponding location is represented by the green dots on the x axis).


327

Fig. 1 Domain geometry and location of source and observations points.

Fig. 2 Cauchy data reconstruction. Plots (a) and (c) concerns Dirichlet datum and plots (b) and (d) Neumann datum.

Second simulation concerns a comparison between a direct MFS data reconstruction (Fig. 3) and the proposed iterative method (Fig. 4).

Fig. 3 Data reconstruction using a direct MFS approach

328


Fig. 4 Data reconstruction using an iterative approach

References [1] V. D. Kupradze and M. A. Aleksidze The method of functional equations for the approximate solution of certain boundary value problems, USSR Computational Mathematics and Mathematical Physics,4, 82-126 (1964). [2] A. Bogomolny Fundamental solutions method for boundary value problems, SIAM, J. Numer. Anal. 22, 644-669 (1985). [3] A. Kirsch and R. Kress On an integral equation of the first kind in inverse acoustic scattering, In: Inverse Problems (Cannon and Hornung, eds.), International Series of Numerical Mathematics, Birkhauser-Verlag Basel, 77, 93-102, (1986). [4] C. J. S. Alves and N. F. M. Martins The direct method of fundamental solutions and the inverse Kirsch Kress method for the reconstruction of elastic inclusions or cavities, J. Integral Equations and Applications, 21 (2), 153-178, (2009). [5] N. F. M. Martins An iterative shape reconstruction of source functions in a potential problem using the MFS, Inverse Problems in Science and Engineering, 20(8), 1175-1193, (2012). [6] A. Karageorghis, D. Lesnic and L. Marin A survey of applications of the MFS to inverse problems, Inverse Problems in Science and Engineering, 19 (3), 309-336, (2011). [7] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, (2000).


329

Elastoplastic Dynamic Analysis of Beam-Foundation Systems Employing BEM Andreas E. Kampitsis1 and Evangelos J. Sapountzakis2 1

School of Civil Engineering, National Technical University of Athens (NTUA) Zografou Campus, GR-157 80, Athens, Greece. e-mail: [email protected]

2

School of Civil Engineering, National Technical University of Athens (NTUA) Zografou Campus, GR-157 80, Athens, Greece. e-mail: [email protected]

Keywords: Dynamic Analysis, Beam on Nonlinear Foundation, Inelastic Analysis, Distributed Plasticity, Boundary Element Method

Abstract. In this investigation a Boundary Element Method (BEM) is developed for the elastoplastic dynamic analysis of an Euler-Bernoulli beam of simply or multiply connected constant cross section having at least one axis of symmetry, resting on inelastic foundation. Introduction Beam-foundation systems which are subjected to dynamic loading often exhibit inelastic material behavior either concerning the structural’s element or the foundation. Moreover, design of beams based on elastic analysis are most likely to be extremely conservative not only due to significant difference between initial yield and full plastification in a cross section, but also due to the unaccounted for yet significant reserves of strength that are not mobilized in redundant members until after inelastic redistribution takes place. Thus, material nonlinearity is important for investigating the ultimate strength of a beam that resists bending loading, while distributed plasticity models are acknowledged in the literature [1-3] to capture more rigorously material nonlinearities than cross sectional stress resultant approaches [4] or lumped plasticity idealizations [5,6]. Contrary to the good amount of attention in the literature concerning the elastic dynamic analysis of beams on elastic foundation, very little work has been done on the corresponding inelastic dynamic problems. In this investigation a Boundary Element Method (BEM) is developed for the elastoplastic dynamic analysis of an Euler-Bernoulli beam of simply or multiply connected constant cross section having at least one axis of symmetry, resting on inelastic foundation. The beam is subjected to arbitrarily distributed or concentrated dynamic bending loading along its length, while its edges are subjected to the most general boundary conditions. A displacement based formulation is developed and inelastic redistribution is modeled through a distributed plasticity model exploiting material constitutive laws and numerical integration over the cross sections. An incremental - iterative time discretization scheme is adopted to restore global equilibrium along with an efficient iterative process to integrate the inelastic rate equations [7]. The arising boundary value problem is solved employing the boundary element method [8]. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows. i. A BEM approach is employed for the dynamic elastoplastic analysis of a beam-foundation system. ii. The formulation is a displacement based one taking into account inelastic redistribution along the beam axis by exploiting material constitutive laws and numerical integration over the cross sections (distributed plasticity approach). iii. The inelasticity of the soil medium is taken into account. iv. An incremental - iterative time discretization scheme is adopted to restore global equilibrium of the beam. Integration of the inelastic rate equations is performed for each monitoring station with an efficient iterative process and stress resultants are obtained employing incremental strains. v. The beam is supported by the most general nonlinear boundary conditions including elastic support or restrain, while its cross section is an arbitrarily monosymmetric one. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.

330


Statement of the problem Let us consider a prismatic beam of length l (Fig. 1) of arbitrary constant cross–section having at least one axis of symmetry (z-axis), occupying the two dimensional multiply connected region : of the y,z

plane bounded by the * j j 1,2,...,K boundary curves, which are piecewise smooth, i.e. they may have

a finite number of corners. In Fig. 1 Cyz is the principal bending coordinate system through the cross section’s centroid. The normal stress-strain relationship for the material is assumed to be elastic-plasticstrain hardening with initial modulus of elasticity and yield stress E and V Y 0 , respectively (Fig. 1). The beam is resting on nonlinear inelastic tensionless Winkler type foundation and thus the foundation reaction is expressed as if p f ! 0 kw w ® 0 if p f d 0 ¯

p f

where kw

(1)

kw w,wy is the Winkler nonlinear inelastic functions depending on the yielding displacement

and the current one (Fig. 1).

pz(x,t)

Dynamic Load

Pz(x,t)

σ

my(x,t)

Εt

σy0 x

O

Normal stress-strain relationship

l

z,w

Ρ

Γ1

y,v Γ2

ε

kwt

Ρy

S C (Ω) z,w

Ε

(C: Center of gravity S: Shear center) Γk

O

kw

Force-Displacement relationship

δ

Fig 1: Prismatic beam resting on an inelastic foundation subjected to dynamic bending loading with an arbitrary cross-section having at least one axis of symmetry, occupying the two dimensional region : . The beam is subjected to the combined action of arbitrarily distributed or concentrated time dependent transverse loading pz pz x,t and bending moment my my x,t acting in the z direction (Fig. 1). Under the action of the aforementioned loading, the displacement field of the beam is given as u( x,z,t ) u x,t zT y x,t

(2a)

w x,t w x,t

(2b)

where u , w are the axial and transverse beam displacement components with respect to the Cyz system of axes; u x,t , w x,t are the corresponding components of the centroid C and T y x,t is the angle of rotation due to bending of the cross-section with respect to its centroid. Employing the strain-displacement relations considering small deflections and adopting the Euler-Bernoulli assumption the following strain components are obtained


H xx

J xz

z

331

d 2w

(3a)

dx 2

0 Ty

dw dx

(3b)

Considering strains to be small, employing the Cauchy stress tensor and assuming an isotropic and homogeneous material without exhibiting any damage during its plastification, the normal stress rate is defined in terms of the corresponding strain one as dV xx

el E d H xx

(4)

where d denotes infinitesimal incremental quantities over time (rates), the superscript el denotes the elastic part of the strain component and E

E 1 Q . If the plane stress hypothesis is undertaken 1 Q 1 2Q

E holds , while E is frequently considered instead of E* ( E | E ) in beam formulations. 1 Q 2 This last consideration has been followed throughout the paper, while any other reasonable expression of E* could also be used without any difficulty in many beam formulations.

then E

As long as the material remains elastic or elastic unloading occurs ( d H xx

el ) the stress rate is given d H xx

el pl with respect to the strain one from eqn. (4). If plastic flow occurs then dH xx dH xx , where the dH xx superscript pl denotes the plastic part of the strain component. The Von Mises yielding criterion is considered ignoring the influence of shear stresses and the yield condition is satisfied when the normal stress is equated with the yield stress of the material, that is

f

pl V xx V Y H eq

0

(5)

pl where V Y is the yield stress of the material and H eq is the equivalent plastic strain, the rate of which is pl defined in [9] and is given as d H eq

h is defined as h

d O ( d O is the proportionality factor). Moreover, the plastic modulus

pl dV Y d H eq

or dV Y

hd O and can be estimated from a tension test as

h Et E E Et . The stress rate is given with respect to the total strain one through eqn. (3) and the strain components as dV xx

pl Ed H xx Ed H xx

(6)

Equations of global equilibrium On the basis of Hamilton’s principle, the variations of the Lagrangian equation defined as

G ³t 2 U K Wext dt 0 t

(7)

1

and expressed as a function of the stress resultants acting on the cross section of the beam provide the governing equations and the boundary conditions of the beam. In eqn.(7) G denotes variation of quantities, V is the volume and l is the length of the beam, while U , K , Wext are the strain energy, the kinetic energy and the external load work, respectively given as

332


GU

³V V xxGH xx dV

1 2 ³ U G w dV 2 V

GK

G Wext

³l

p G w p G w m GT dx z

f

y

y

(8a,b,c)

The stress resultant corresponding to the internal bending moment of the beam is defined as

³ V xx zd :

SM y

(9)

:

After substituting eqn. (9) into eqn. (7) and conducting some algebraic manipulations, the global equilibrium equations of the beam is obtained as

d 2 SM y dx

2

U$ $w p f x

pz x

dm y x dx

(10)

along with its corresponding boundary conditions

D1

dSM y dx

E1SM y E2

D 2 w D3

dw dx

E3

(11a,b)

at the beam ends x 0,l , together with the initial conditions w x,0 w0 x

w x,0 w0 x

(12a,b)

where w0 x and w0 x are prescribed functions, while D i , Ei ( i 1,2,3 ) are functions specified at the beam ends. The boundary conditions (11) are the most general ones for the problem at hand, including also the elastic support. It is apparent that all types of the conventional boundary conditions (clamped, simply supported, free or guided edge) may be derived from eqns. (11) by specifying appropriately the functions D i and Ei (e.g. for a clamped edge it is D 2 E2 1 , D1 D3 E1 E3 0 ). Using the definition of eq. (9), elastoplastic constitutive equations at the beam level are derived by crosssectional integration of eq. (6) SM y

EI y wcc SM ypl

(13)

where I y is the moment of inertia with respect to the principle bending axis y and SM ypl is the plastic quantity defined as SM ypl

E ³ 'H pl zd :

(14)

:

Integral Representations – Numerical Solution According to the precedent analysis, the dynamic inelastic problem of an Euler-Bernoulli beams resting on resting on inelastic foundation, reduces to establishing the displacement component w x,t having continuous derivatives up to the fourth order with respect to x and up to the second order with respect to t, satisfying the initial boundary value problem described by the governing differential equation (10) along the beam, the boundary conditions (eqs. (11)) and the initial conditions (eqs (12)) at the beam ends x 0,l . This initial boundary value problem is solved employing the BEM [8], as this is developed in [10] for the solution of a fourth order differential equation with constant coefficients, after some modifications.


333

Numerical Example A rectangular cross section ( h 0.60m , b 0.30m ) pinned–fixed beam of length l 6.0m resting on a Winkler foundation with initial stiffness kw 20MPa and yielding force PwY 100kN / m has been studied. For the conducted analysis 20 linear longitudinal elements, 400 boundary elements, 72 quadrilateral cells and a 3 u 3 Gauss integration scheme for each cell, have been employed. The beam is subjected to a dynamic uniformly distributed loading acting at the first half of the beam’s length and following the time function presented in Fig. 2(a). The beam’s material is assumed to follow elastoplasticstrain hardening law with E 32318.4MPa , V Y0 20MN / m2 and Et 650MPa . In Fig. 2(b) the time history of the deflection w l /2,t of the midpoint of the beam subjected to the aforementioned dynamic loading is presented, performing either elastic or inelastic analysis ignoring the foundation reaction. Moreover, in Table 1 the static deflection of the midpoint of the beam is presented for different load stages and material properties taking into account or ignoring the elastic-plastic Winkler

foundation reaction, respectively. 0.015

1.5 Imposed Load vs Time

0.01

Displacement (m)

1

Load Factor

0.5

0

-0.5

Displacement Time History

0.005

0

-0.005

-0.01

-1

Elastic Material Strain Hardening Material

-0.015

-1.5

0

Time (sec)

0.02

0.04 Time (sec)

0.06

0.08

(a) (b) Fig. 2 Dynamic load vs. time function (a). Time history of the midpoint deflection w l /2,t of the beam ignoring the foundation reaction (b).

Concluding Remarks

In this investigation a Boundary Element Method (BEM) is developed for the elastoplastic dynamic analysis of an Euler-Bernoulli beam of simply or multiply connected constant cross section having at least one axis of symmetry, resting on inelastic foundation. The main conclusions that can be drawn from this investigation are a. The numerical technique presented in this investigation is well suited for computer aided analysis of prismatic beams of arbitrary simply or multiply connected cross section having at least one axis of symmetry, supported by the most general boundary conditions and subjected to the action of arbitrarily distributed or concentrated vertical loading. b. The inelastic analysis and the soil nonlinearity are of paramount importance for the dynamic response of the beam-foundation system. c. Accurate results are obtained using a relatively small number of nodal points across the longitudinal axis.

334


d. A small number of cells (fibers) is required in order to achieve satisfactory convergence. e. The developed procedure retains most of the advantages of a BEM solution even though domain discretization is required. Perfectly Plastic Winkler Foundation pz /w l / 2

350 420 480

Elastic – Et

E

0.495 0.594 0.679

Perfectly Plastic – Et

0

0.499 0.801 -

Strain Hardening – Et 0.512 0.780 2.186

Ignoring Foundation Reaction pz /w l / 2

150 290 325

Elastic – Et 0.344 0.666 0.746

E

Perfectly Plastic – Et 0.344 1.685 -

0

Strain Hardening – Et 0.345 0.952 2.176

Table 1: Midpoint deflection w l / 2 (cm) of the beam for different types of beam and foundation material properties

Acknowledgements

The work of this paper was conducted from the “DARE” project, financially supported by a European Research Council (ERC) Advanced Grant under the “Ideas” Programme in Support of Frontier Research [Grant Agreement 228254]. References [1]

[2] [3] [4] [5] [6] [7]

P. Nukala and D. White A mixed finite element for three-dimensional nonlinear analysis of steel frames, Computer Methods in Applied Mechanics and Engineering, 193, 2507-2545 (2004). L. The and M. Clarke Plastic-zone analysis of 3D steel frames using beam elements, Journal of Structural Engineering, 125, 1328-1337, 1999. A. Saritas and FC. Filippou Frame Element for Metallic Shear-Yielding Members under Cyclic Loading, J. Struct. Engrg, 135, 1115-1123 (2009). MR. Attalla, GG. Deierlein and W. McGuire Spread of Plasticity: Quasi-Plastic-Hinge Approach, J. Struct. Engrg 120, 2451-2473 (1994). JG. Orbison, W. McGuire and JF. Abel Yield surface applications in nonlinear steel frame analysis, Computer Methods in Applied Mechanics and Engineering, 33, 557-573 (1982). C. Ngo-Huu, S. Kim and J. Oh Nonlinear analysis of space steel frames using fiber plastic hinge concept, Engineering Structures, 29, 649-657, 2007. EA de Souza Neto, D Peri and DRJ Owen Computational Methods For Plasticity Theory and Applications, John Wiley and Sons (2008)

JT. Katsikadelis Boundary Elements: Theory and Applications, Amsterdam-London, United Kingdom, Elsevier (2002). [9] MA. Crisfield Non-linear Finite Element Analysis of Solids and Structures Vol. 1 Essentials. John Wiley and Sons, New York, USA (1991). [10] EJ. Sapountzakis Solution of non-uniform torsion of bars by an integral equation method, Computers and Structures, 77, 659-667 (2000).

[8]


335

Diffraction of In-Plane (P, SV) and Anti-Plane (SH) Waves in a Half - Space with Cylindrical Tunnels S. Parvanova 1, P. Dineva 2, G. Manolis3, F. Wuttke 4 1

Assoc. Prof., Dept. of Civil Engineering, UACEG, 1046 Sofia, Bulgaria, [email protected] 2 3

Prof., Institute Mech., BAS, 1113 Sofia, Bulgaria, [email protected]

Prof., Dept. of Civil Engineering, Aristotle University, 54124 Thessaloniki, Greece, [email protected] 4

Prof., Geomechanical Modeling, Bauhaus University, 99421 Weimar, Germany, [email protected]

Keywords: lined tunnels, stress concentration factor (SCF), scattered waves, elastodynamics, boundary element method (BEM)

Abstract. The diffraction wave field and dynamic SCFs in a half-plane with arbitrary free-surface relief containing two lined tunnels is studied by BEM based on sub-structuring approach. The numerical model is developed by use of the integral representation formula based on the elastodynamic fundamental solutions. The influence of the free surface relief on the dynamic SCFs, excited by P, SV or SH waves around the tunnel’s wall is examined. Also sensitivity of the wave signal along the free surface to the type and characteristics of the incident wave, to the geometrical configuration and elastic properties of the tunnels and to the type and geometry of the free surface relief is evaluated. Introduction The main aim of the study is to demonstrate the BEM computational potential for solution of the scattering, diffraction and dynamic stress concentration anti-plane and in-plane problems for frequency dependent seismic response of soil-tunnels system. Free surface relief, dynamic interaction between lined tunnels, soil-tunnel interaction and other key factors are taken into consideration. Although the advantages of the BEM for solution of the dynamic problems in the discussed field are well-known, there is a lack of BEM results for a laterally inhomogeneous half-plane with sub-surface underground structures as tunnels. The tunnels are modeled as cavities in most of the papers, and to the authors’ best knowledge there are no results for lined tunnels considering all components as rock or soil, external shot-crete ring layer and prefabricated concrete internal ring layer. Problem statement and its BEM solution Elastic isotropic half-plane with free surface relief of arbitrary shape is subjected to incident time-harmonic SH-, P- or SV- wave. Without any restriction concerning the geometrical shape of the free-surface relief we consider first a half-plane with free surface relief presented by two semi-elliptic hills with semi-axes a1 and a2 , see Figure 1. Cartesian co-ordinate system Ox1 x2 x3 is inserted and the anti-plane and in-plane wave motion in the and under incident angle plane x3 0 is studied. The plane wave propagates with a described frequency with respect to axis Ox1 . Two infinite cylindrical tunnels are embedded in the half-plane. Three types of the tunnel’s structure are considered and correspondingly three boundary-value problems are formulated and solved. The first type concerns unlined tunnel without any cover or ring, located directly in a stable rock and it is modeled as a cavity. The second and third types are presented in Figure 1 and they consider lined tunnels with walls with one or two ring layers. Free field is defined as the SH-, P- and SV-time-harmonic wave propagation in elastic half-plane with flat free surface and without any type of heterogeneities. In the considered cases of anti-plane and plane strain state, the only nonzero field quantities are: (a) for anti-plane wave motion-displacement u3 and

336


stresses

13 , 23 ;

(b) for in-plane wave motion-displacements u1 , u2 and stresses

the observer point x1 , x2 and frequency

11

,

22

,

12

, all depending on

. Since the response is also time harmonic, the common multiplier

exp(i t)) is suppressed in the following. Next, the governing equation of motion is given by

where

ij

Cijkl uk ,l , Cijkl

ij kl

ik

jl

il

jk

,

ij

is the Kronecker delta symbol,

2 ij , j

ui

0,

is the material density,

and μ are Lame constants, subscript commas denote partial differentiation, and the summation convention over repeated indices is implied. The boundary conditions are now as follows: traction-free conditions hold at all internal boundaries Γ1 of the tunnels, the equilibrium and compatibility conditions are posed along the interface between soil and tunnel walls Γ3, and between ring layers Γ2 of the tunnel walls. The strategy here is to compute the seismic wave field along the free surface of the half-plane and to evaluate the SCF at the tunnel interfaces. The BEM based on an efficient sub-structural approach is used as computational tool. The BEM description of the formulated problem is as follows: U ij* x, y ,

cij u j x,

t j y,

Tij* x, y ,

dS

S

where S

ff

1l

u j y,

(1)

dS

S

1r

2l

2r

3l

3r

, cij is a jump term depending on the surface geometry at the

collocation point, x and y are field and source points, respectively, and u x,

, t x,

are the displacement

and traction vectors at the boundaries. Furthermore, U ij* and Tij* are the displacement fundamental solution of the

a2

governing equation in elastodynamics and its corresponding traction.

x2

t3=0 а1 ρ0, μ0

Free surface (Γff)

а1

interface 1 (Γ1)

h Ω2

x1

Ω1

2d

R r1 r2

SH wave θ

interface 2 (Γ2) interface 3 (Γ3)

ρte, μte

ρti, μti

Ωh

Figure 1: Problem geometry Validation study A Matlab [1] software code has been developed to solve the BEM with sub-structuring capabilities. In order to verify its accuracy, several test examples are solved and the results are compared against available analytical 2r r / CS and/or numerical results. We first introduced a dimensionless frequencies defined as kr r CS 2 r , where is the wave length and r is the radius of the inhomogeneity, in case of and tunnels r R is the radius of the external ring, since in case of unlined tunnels R is cavity radius. The BEM mesh employs quadratic elements, which number can be decided upon following the verification procedure given below. Half space with cylindrical inclusions and single wall tunnels subjected to time-harmonic SH wave The first validation example is a homogeneous isotropic half-plane with a flat free surface containing one and nine circular inclusions of a unit radius, r, solved numerically by Dravinski and Yu in [2]. The depth of burial, with respect to the circle center, for the first upper row of all the inclusions, is h=2. The multiple inclusions of


337

number 9 are arranged equally spaced in a three-by-three array, the distance between every two adjacent inclusions in horizontal and vertical direction is d=3. Mechanical parameters of the inclusions are: shear modulus μ=1/6, density ρ=2/3, those of the half space in consistent units are: shear modulus μ=1, density ρ=1. The length of the free surface is S=20 for a single inclusion and S=26 for 9 inclusions. It is chosen so that the distance between center of the right most inclusions and the right end of the surface is S/2=10r. The numerical model consists of 50 quadratic boundary elements (BE) for the free surface and 12 BE for each of the inclusion interfaces.

2.5 2

|u3/u0|

3

|u3/u0|

3

9 incl. present single incl. present 9 inclusion [2] single incl. [2]

4

2

1.5 1

1

0.5

x1/r

0 -10

-8

-6

-4

-2

0

x1/r

0 2

4

6

8

10

-10

-8

-6

-4

-2

0

2

4

6

8

10

Figure 2: Comparison of the free surface response for single and nine inclusions in a half-plane for incident angle θ=0 (left panel) and θ=π/2 (right panel) and dimensionless frequency η=0.5 Figure 2 plots the surface displacement amplitude u3

Re u3

2

Im(u3 )2 versus distance x1 of the flat

free surface, at a fixed value of normalized frequency η 2r λ 0.5 , where is the SH-wave length and r is inclusion radius. The angle of incidence, , is equal to 0 and π/2. Apparently, the current BEM results are indistinguishable from those that were obtained in [2]. 0.6

φ

single tunnel s/r=5 s/r=3 s/r=2.5

present results

0.4

3

Balendra & al. [3]

single tunnel s/r=5 s/r=3 s/r=2.5

|u3/u0|

| σφz|r /u0

2

1

0.2

φ [degrees]

φ [degrees] 0

0

-180

-90

0

90

180

-180

-90

0

90

180

Figure 3: Displacement amplitudes along the circumference of the concrete tunnel (left panel); Stress amplitudes σφz at the inner surface of the left tunnel (right panel) Two parallel single wall tunnels of circular cross section under incident SH-waves in a half-plane with flat free surface are considered as the second validation example. A closed form solution by the method of wave function expansion was proposed in [3] and the image technique was used to represent the wave signal along the free surface. The mechanical and geometrical data of the numerical model are as follows: the half-plane is represented by loess at natural moisture with density ρ0=1640 kg/m3 and shear modulus μ0=0.111 GPa; the tunnels’ wall is made of concrete with density ρ1=2410 kg/m3 and shear modulus μ1=8.4 GPa [3]. For radius, r, of the outer tunnel contour the wall thickness is 0.1r, the depth of both tunnels with respect to the center of the circles is 2.5r and the distance, s, between both the centers is variable. The displacement amplitudes along the circumference of the left tunnel and shear stress amplitudes, σφz, at the inner surface, for frequency ω= 200 rad/s,

338


incident angle θ=π/3, and radius r=1 m, are depicted in figure 3.Apparently a very close agreement is observed with the results obtained in [3]. Half space with cylindrical cavity and canyon subjected to time-harmonic P- and SV- waves Scattering of plane harmonic P- and SV- waves, of normal incidence, by a circular cavity in an isotropic half plane is considered and compared with results obtained in [4]. The dimensionless frequency normalized with respect to the shear SV wave is η=1. The radius of the cavity is r=1, the depth of embedment is 2, the Poisson’s ratio is ν=1/3. The length of dizcretisized free surface is 40. This problem was solved in [4], applying the direct BEM with linear BE. Their mesh consists of 928 linear BE, 128 of which are used for modeling of the cavity. In order to reproduce their solution the BEM mesh used herein comprises of 80 quadratic BE (161 nodes) for representation of the free surface and 24 BE (48 nodes) for modeling of the hole. The displacement amplitudes of the free surface for vertical P- and SV- incidence are shown in Figure 4. 3.5

present

a)

2.5

1

2

1.5

0.5

-20 3

-15

-10

-5

0

5

1

0.5

x1/r

0

x1/r

0 10

15

20

-20

1

c)

-15

-10

-5

0

5

10

15

20

10

15

20

d)

0.8

2.5 2

|u2/u0|

|u1/u0|

Yu & Dravinski [4]

b)

3

|u2/u0|

|u1/u0|

1.5

1.5

0.6

0.4 0.2

1

x1/r

0.5 -20

-15

-10

-5

0

5

x1/r

0 10

15

20

-20

-15

-10

-5

0

5

Figure 4: Displacement amplitudes along the free surface of plane strain cavity model under: a), b) normal P- wave; c), d) SV- wave 3 3 u1 present u1 - Alvarez-Rubio et al. [5] a) b) u2 present u2 - Alvarez-Rubio et al. [5] 2

|u1/u0| |u2/u0|

|u1/u0| |u2/u0|

2

1

x1/r

0 -1

-0.8 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

0.8

1

x1/r

0 1

-1

-0.8 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

0.8

1

Figure 5: Displacement amplitudes along the free surface half-plane with semi-circular canyon under: a) normal SV- wave; b) normal P- wave


339

The last validation example is a half-plane with a semi-circular canyon of radius r swept by P- and SV waves, solved numerically by direct BEM in ref. [5]. This example, being one of the most popular topographic irregularities, has been investigated by many researchers applying different analytical and numerical techniques. The isotropic elastic half-plane is characterized by Poisson’s ratio 1/3. The BE mesh in present modeling is chosen accordingly ref. [5], namely the distance between two adjacent nodes of the mesh is equal to λ/15, where λ is SV-wavelength. The comparison of the horizontal and vertical displacement amplitudes of the free surface for vertical incident P- and SV- waves is given in Figure 5. Parametric study and conclusions A parametric study involving two cylindrical tunnels in the half-plane with free surface relief in the form of two elliptic hills with semi-axes a1 and a 2 (Fig. 1) is presented. The shape of the hills changes from circular 11 25 m; a2 5 m , and then vertically elongated as a1 a2 7.5m , to horizontally elongated ellipses as a1 11.25 9 m and their 11.25 5 m (see Fig. 1). In all cases, the depth of tunnel embedment is h 9m ellipses as a1 5 m; a2 11.25 center-to-center distance is 2 R 2d 15m . The material properties of the half-plane 18 518 MPa; 0 1746 kg / m 3 , while we look at an SH-wave with normal incidence, i.e., are 0 18.518 kr r CS respect to axis Ox1. The normalized frequency of the wave is defined as

/ 2 with 2 r ,

with CS 0 0 . As far as BEM modeling details go, we employ 79 quadratic BE along the free-surface and 48 for each of the two tunnel liners. Three basic types of tunnel construction have been considered: (a) unlined cylindrical circular tunnel (i.e., cavity) of radius R 6 m ; (b) lined tunnel with liner wall consisting of a single ring layer of shotcrete of material properties t 9,130 MPa; t 2,200 kg / m3 (see Fig. 1), where R 6 m, r1 r2 5.60 giving a wall thickness of t=40 cm; (c) lined tunnel with liner wall comprising two ring layers (see Fig. 1): an external layer as in case (b); an internal prefabricated concrete layer with material properties ti 12,295 12 295 MPa; ti 2 2,500 500 kg / m3 and thickness ti r2 r1 60 cm .

(a)

(b)

4.5

6

holes

1 wall

4

|u3|

|u3|

5

holes

5

1 wall

2 walls

3.5

2 walls

4

3 2.5

3

2

2

1.5 1

1

0.5

Ω

0 0

0.5

1

1.5

Ω

0 2

2.5

3

0

0.5

1

1.5

2

2.5

3

Figure 6: Displacement amplitude versus frequency for free surface relief with two semi-circular hills at receivers: (a) R1; (b) R2 Figs. 6-8 plot the frequency dependence of the displacement amplitude at different receiver points such as R1:( 10,7.5) 10,7. 7 5)) m for circular hills; R1:( 14,5) m for horizontally elongated elliptic hills; R1:( 6.25,11.25) 6.25,11.2 m 6.25,11.2 for vertically elongated hills, all located at the top of the left hill; and R 2 :(0,0) m located in the middle between two hills in all cases, see Fig. 1. Figures 6-8 compare the displacement amplitude versus frequency variation for all three type of tunnel’s structure, namely cavity, single liner and composite liner. The following conclusions can now be summarized: (a) site effects are most clearly demonstrated in the case of horizontally elongated hills along the free surface, see Figs. 7a,b; (b) the influence of tunnel construction type is also clearly visible, with marked differences observed between the cavity case and all other lined tunnel cases; (c) the displacement amplitude at any given receiver location strongly depends on the ratio of the tunnel

340


radius R and SH-wave length , keeping in mind that these two quantities filter into the definition of the non2 R . dimensional frequency (a)

(b)

3.5 3 2.5

holes

9

1 wall

8

2 walls

7

|u3|

10

|u3|

4

holes 1 wall 2 walls

6

2

5

1.5

4 3

1

2

0.5

1

Ω

0 0

0.5

1

Ω

0

1.5

2

0

3

2.5

0.5

1

1.5

2

2.5

3

Figure 7: Displacement amplitude versus non-dimensional frequency for free surface relief with two semi-elliptic horizontally elongated hills at two different receivers: (a) R1; (b) R2

|u3|

6

holes

5

2 walls

3

2

2

1

1

Ω 0

0.5

1

1.5

1 wall 2 walls

4

3

0

holes

5

1 wall

4

(b)

|u3|

(a)

6

Ω

0

2

2.5

3

0

0.5

1

1.5

2

2.5

3

Figure 8: Displacement amplitude versus non-dimensional frequency for free surface relief with two semi-elliptic vertically elongated hills at two different receivers: (a) R1; (b) R2 Acknowledgement: The second author and fourth author wish to acknowledge support provided through the DFG Grant No. DFG-Wu 496/5-1.

References [1] MATLAB, The Language of Technical Computing, Version 7.7. The MathWorks, Inc., Natick, Massachusetts (2008). [2] Dravinski, M. and Yu, Ch., Scattering of plane harmonic SH waves by multiple inclusions, Geophys. J. Int., 186, 1331–1346 (2011). [3] Balendra, T., Thambiratnam, D., Kohs, C. G. and Lee, S., Dynamic response of twin circular tunnels due to incident SH-waves, Earthquake engineering and structural dynamics, 12, 181-201 (1984). [4] Yu, Ch., and Dravinski, M., Scattering of plane harmonic P, SV or Rayleigh waves by a completely embedded corrugated cavity, Geophys. J. Int., 178, 479–487 (2009). [5] Alvarez-Rubio, S., Sanchez-Sesma, F. J., Benito, J. J., Alarcon, E., The direct boundary element method: 2D site effects assessment on laterally varying layered media (methodology), Soil Dynamics and Earthquake Engineering, 24, 167–180 (2004).


341

DRBEM solution of Liquid Metal MHD Flow in a Staggered Double Lid-Driven Cavity M. Tezer-Sezgin

a,b

and B. Pekmen

a,c

a

b

Institute of Applied Mathematics, Middle East Technical University, Department of Mathematics, Middle East Technical University, email: [email protected], c Department of Mathematics, Atılım University, email: [email protected], Ankara, Turkey

Keywords : MHD, Current Density, DRBEM Abstract. In this study, dual reciprocity boundary element method (DRBEM) is applied for solving the unsteady flow of a viscous, incompressible, electrically conducting fluid in a double lid-driven staggered cavity under the effect of an externally applied magnetic field. Magnetohydrodynamic (MHD) flow in a staggered double lid-driven cavity is a challenging problem showing that the symmetric flow pattern, and the well known MHD characteristics still hold accordingly with the movements of the lids for certain values of Reynolds number. MHD equations are coupled with the temperature effects (heat transfer in the cavity) by means of the Boussinessq approximation. 2D full MHD equations are solved in terms of stream function, vorticity, current density and energy equations by using DRBEM with implicit backward Euler time integration scheme. The velocity and the induced magnetic field components are also obtained through the relations with stream function and current density (with a relation to magnetic potential) by using the coordinate matrix. DRBEM translates all the differential equations defined in the problem domain to boundary integral equations defined on the cavity walls. It is also possible to treat all the terms together with nonlinearities as the right hand side functions in Poisson’s equations. This way, fundamental solution of Laplace equation is used in DRBEM formulation. Numerical results are obtained using linear boundary elements which give small sized discretized systems of equations due to the boundary only nature of DRBEM. This makes the whole procedure computationally effective and cheap. The results are given for several values of problem parameters as Reynolds number (Re), Prandtl number (P r), magnetic Reynolds number (Rem), Hartmann number (Ha) and Rayleigh number (Ra). An increase in Re unifies the two primary eddies in the flow into one main circulation at the center, and four secondary eddies are developed at the inner corners of the cavity. As a well known characteristic in MHD flow, as Ha increases Hartmann layers are formed at the perpendicular walls to the magnetic field. Heat transfer shows conduction dominated effect for large values of Ra. An increase in Rem causes two new cells close to the upper right and lower left corners for magnetic potential for a certain Ra due to the convection dominance in current density, and the movement of the lids in the opposite directions. Introduction A concise introduction about magnetohydrodynamics (MHD) is a difficult job due to the wise content of it. Therefore, lots of information about the physics and numerical approaches may be found in the books [1, 6]. MHD has wide range of applications in industry as the design of MHD generators, pumps, accelerators, nuclear reactors, crystal growth, etc. Numerically, buoyancy-driven MHD flow neglecting the induced magnetic field is solved by the domain discretization methods as the differential quadrature method (DQM), and penalty finite element method with bi-quadratic rectangular elements

342


in [3, 5], respectively. The use of the numerical techniques such as DQM and the discrete singular convolution (DSC) for the solution of the staggered double lid driven cavity requires division to subdomains before discretization [4, 8]. However, dual reciprocity boundary element method provides one to discretize only the boundary and to use arbitrary interior points. Physical problem is modelled with conservations of mass and momentum together with Maxwell’s equations through Ohm’s law, and conservation of energy. The governing non-dimensional, unsteady partial differential equations in terms of stream function (ψ)-temperature (T )-vector potential (A)-current density (j)-vorticity (w) are [2] ∇2 ψ = −w ∇2 T = P rRe

#

∂T ∂T ∂T +u +v ∂t ∂x ∂y

$

∇2 A = −Remj $ # $ # ∂j ∂j ∂w ∂w ∂j +u +v − Bx + By ∇2 j = Rem ∂t ∂x ∂y ∂x ∂y # $ # $ ∂u ∂v ∂Bx ∂By ∂Bx ∂v + + + −2 ∂x ∂x ∂y ∂y ∂y ∂x $ $ # # ∂w ∂w Ra ∂T ∂w ∂j ∂j +u +v − Ha2 Bx + By − ∇2 w = Re ∂t ∂x ∂y ∂x ∂y P rRe ∂x

(1)

where u = ∂ψ/∂y, v = −∂ψ/∂x and Bx = ∂A/∂y, By = −∂A/∂x, and ψ, T, A, j and w are the stream function, temperature, magnetic potential, current density and vorticity, respectively. The dimensionless parameters are in turn P r, Re, Rem, Ha and Ra, the Prandtl, Reynolds, Magnetic Reynolds, Hartmann, and Rayleigh numbers. y Th = 0.5 u=1 hz hz

g B hz hz u = −1 Tc = −0.5

x

Initially, w, j and T are taken as zero. The configuration of the problem is visualized in Fig.1. Stream function and velocity component v are all zero on the walls. u = 1 on the top wall and u = −1 on the bottom wall. The jagged walls are adiabatic (∂T /∂n = 0), the top wall is the hot wall Th = 0.5 and the bottom wall is the cold wall Tc = −0.5. Magnetic potential is A = −x on the walls due to the y-component of external magnetic field B0 = (0, 1).

Figure 1: Configuration of the problem.

DRBEM Application Eqs.(1) are rewritten as coupled Poisson equations ∇2 ϕ = b,

(2)

where ϕ denotes either ψ, T, A, j or w, and vector b contains all the terms when the diffusion terms are left on the left hand sides of each equation in (1).


343

In DRBEM, for a right hand side function b the following approximation is proposed [7] b≈

N +L

αj fj

j=1

where αj ’s are sets of initially unknown coefficients, the fj ’s are approximating functions, N is the number of boundary nodes and L is the number of interior points. The radial basis functions fj ’s are 2 + . . . + r n where i and j usually chosen as polynomials of radial distance rij as fij = 1 + rij + rij ij correspond to the source(fixed) and the field(variable) points, respectively. Furthermore, the fj ’s are related to particular solutions u ˆj ’s with the Poisson equation ∇2 u ˆj = fj and N +L N +L αj (∇2 u ˆ j ) ⇒ ∇2 ϕ = αj (∇2 u ˆj ). (3) b= j=1

j=1

Multiplying both sides by the fundamental solution of Laplace equation u∗ = grating over the domain, N +L (∇2 ϕ)u∗ dΩ = αj (∇2 u ˆj )u∗ dΩ, Ω

j=1

1 2π

ln

&1' r

and inte(4)

Ω

is obtained. Once the Green’s second identity is used, all the domain integrals will be transformed to the integrals on the boundary. The discretization of these boundary integrals using linear boundary elements, corresponding to stream function, temperature, magnetic potential, current density and vorticity equations, results in matrix-vector equations as ! " ˆ − GQ ˆ α, (5) Hϕ − Gϕq = H U ˆ and Q ˆ are constructed from u ˆj where the vectors ϕq define normal derivatives of ψ, T, A, j or w. U ∂u ˆ and then qˆj = ∂nj columnwise, and are matrices of size (N + L) × (N + L). The vector α is employed as α = F −1 b, (6) where F is the (N + L) × (N + L) coordinate matrix containing radial basis functions fj ’s as columns evaluated at N + L points. H and G are BEM matrices containing the boundary integrals of u∗ and q ∗ = ∂u∗ /∂n evaluated at the boundary elements, respectively. With the help of coordinate matrix F for evaluating the derivatives in b and the backward-Euler formula in time derivatives, the iteration with respect to time is carried between the system of equations for ψ, T, A, j and w as ψ m+1 − Gψqm+1 = −Swm $ # P rRe P rRe S − P rReSM T m+1 − GTqm+1 = − ST m H− Δt Δt

(7) (8)

− = −RemSj (9) HA # $ ' & Rem Rem H− S − RemSM j m+1 − Gjqm+1 = − Sj m − S {Bx }m+1 Dx + {By }m+1 Dy wm d d Δt Δt & ' & ' − 2S Dx {Bx }m+1 Dx v m+1 + Dy um+1 + Dy {v}m+1 Dy Bxm+1 + Dx Bym+1 (10) d d $ # Re Re S − ReSM wm+1 − Gwqm+1 = − Swm H− Δt Δt ' m+1 & Ra 2 m+1 m+1 SDx T m+1 − Ha S {Bx }d Dx + {By }d Dy j − (11) P rRe ! " ∂F −1 ∂F −1 ˆ − GQ ˆ F −1 , M = {u}m+1 Dx + {v}m+1 Dy , and F , Dy = F , S = HU where Dx = d d ∂x ∂y m+1 m+1 m+1 m+1 the vectors {u}d , {v}d , {Bx }d , {By }d enter into the system as diagonal matrices of size m+1

GAm+1 q

m

344


(N + L) × (N + L). Gaussian elimination with partial pivoting is used to solve the arranged system of equations of the form Ax = b. Systems of equations (7)-(11) are solved iteratively in the time direction to steady-state with a prescribed tolerance 10−4 . Unknown boundary conditions for current density j and vorticity w are obtained from relationships j=

' 1 1 & (∇ × B) = Dx Bym+1 − Dy Bxm+1 Rem Rem

w = ∇ × u = Dx v m+1 − Dy um+1 .

(12)

(13)

Numerical Results The radial basis function f = 1 + r + r 2 and 16-point Gaussian quadrature in the computation of integrals for construction of H and G matrices are used. In general, 140 boundary elements and 956 interior nodes are utilized by taking Δt = 0.1. As Re increases (Fig.2a-Fig.2b), twin primary eddies in the flow pattern turn out to be one main circulation at the center. The convection dominated effect increases in isotherms. Large values of Rem (Fig.2b-Fig.2c) cause the new cells in both magnetic potential and current density close to the upper right, and lower left corners due to the weakened dominance of diffusion terms. As Ha increases (Fig.2b-Fig.2d), the center of the streamlines rotate in counter-clockwise direction trying to form Hartmann and boundary layers. Isotherms do not change much. Current density and vorticity contours have the similar behavior which is the clustering through the moving walls and stagnancy at the center. Magnetic potential stays the same since Rem is not changed. Isotherms almost become perpendicular to the vertical walls which is an aspect of the increase in buoyancy effect as Ra gets larger (Fig.2b-Fig.2e). Counter rotating cells (counter-clockwise center cell and clockwise top and bottom cells) emerge in streamlines while the effect of moving lids becomes much more visible at the top and bottom cells. Finally, the increase in the dominance of convection for large P r in energy equation causes the isotherms to circulate inside the cavity (Fig.2b-Fig.2f). 6 1 ∂T 66 dx using The average Nusselt number at the hot top wall is computed from N ut = 6 0.4 ∂y y=1.4 Composite Simpson’s rule. As Ha increases, convection dominated effect in N ut is pronounced up to Ha ≤ 100 in Fig.3a (Rem = 10, Re = 400, P r = 0.1, Ra = 103 ). This may be due to the opposite motions of top and bottom lids. In addition, once the vorticity equation is solved for large values of Ha ≥ 100, a relaxation parameter 0 < γ < 1 is used as wm+1 ← γwm+1 + (1 − γ)wm . The increase in Rem (Ha = 10, Re = 400, P r = 0.1, Ra = 103 ) points to the decrease in mean Nusselt number through the hot top wall (Fig.3b) which means that the convection decreases with the increase in magnetic Reynolds number. Conclusion As Ha increases, boundary layer formation starts close to inner corner in the direction of the applied magnetic field. An increase in average Nusselt number is observed up to Ha ≤ 100. Ra increase causes three counter rotating cells in the cavity. Perturbation on magnetic potential lines emerging left down and right up corner eddies, and the decrease in convective heat transfer occurs with the increase in Rem.


ψ

T

345

A

j

(a) Re = 100, Rem = 10, Ha = 10, Ra = 103 , P r = 0.1

(b) Re = 400, Rem = 10, Ha = 10, Ra = 103 , P r = 0.1

(c) Re = 400, Rem = 250, Ha = 10, Ra = 103 , P r = 0.1

(d) Re = 400, Rem = 10, Ha = 100, Ra = 103 , P r = 0.1

(e) Re = 400, Rem = 10, Ha = 10, Ra = 105 , P r = 0.1

(f) Re = 400, Rem = 10, Ha = 10, Ra = 103 , P r = 1

Figure 2: Observations.

w

346


2.3

1.6

2.2

1.55

2.1

1.5

2

N ut

N ut

1.45 1.9

1.4 1.8 1.35

1.7 1.6

1.3

1.5

1.25

0

100

200

300

400

500

0

100

Ha

(a) N ut versus Ha

200

300

400

500

Rem

(b) N ut versus Rem

Figure 3: N ut at the hot top wall is observed w.r.t Ha and Rem.

References [1] P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge ; New York : Cambridge University Press, 2001. [2] K. S. Kang, D. E. Keyes, Implicit symmetrized streamfunction formulations of magnetohydrodynamics, Int. J. Numer. Meth. Fluids, 58 (2008) 1201-1222. [3] D. C. Lo, High-resolution simulations of magnetohydrodynamic free convection in an enclosure with a transverse magnetic field using a velocity-vorticity formulation, Int. J. Commun. Heat Mass Transfer, 37 (2010) 514-523. [4] S. H. Meraji, A. Ghaheri, P. Malekzadeh, An efficient algorithm based on the differential quadrature method for solving Navier-Stokes equations, Int. J. Numer. Methods Fluids 2012, 71:4 (2013) 422-445. [5] M. Sathiyamoorthy, A. Chamkha, Effect of magnetic field on natural convection flow in liquid gallium filled square cavity for linearly heated side wall(s), Int. J. Therm. Sci, 49 (2010) 1856-1865. [6] H. Ozoe, Magnetic Convection, Imperial College Press, 2005. [7] P.W. Partridge, C.A. Brebbia, L.C. Wrobel, The dual reciprocity boundary element method. Southampton, London, Computational Mechanics Publications, Elsevier Science, 1992. [8] Y.C. Zhou, B.S. Patnaik, D.C. Wan, G.W. Wei, DSC solution for flow in a staggered double lid driven cavity, Int. J. Numer. Meth. Engng 57 (2003) 211-234. Acknowledgement This study is supported by the project grant BAP-07-05-2013-001 of METU.


DRBEM Solution for the Incompressible MHD Equations in terms of Magnetic Potential B. Pekmen

a,c

and M. Tezer-Sezgin

a,b

a

b

Institute of Applied Mathematics, Middle East Technical University, Department of Mathematics, Middle East Technical University, e-mail: [email protected], c Department of Mathematics, Atılım University, e-mail: [email protected], Ankara, Turkey

Keywords : MHD, Magnetic Potential, DRBEM Abstract. This paper presents a dual reciprocity boundary element method (DRBEM) formulation for the solution of the incompressible magnetohydrodynamic (MHD) flow equations in a lid-driven square cavity. The governing equations are the coupled system of Navier-Stokes equations and Maxwell’s equations of electromagnetics through Ohm’s law. We are concerned with a stream function-vorticity-magnetic potential formulation of the full MHD equations in 2D. The velocity field and the induced magnetic field can be determined in turn through the relations with stream function and magnetic potential, respectively. In the DRBEM formulation, all the terms apart from the Laplace term (including nonlinearities) are treated as inhomogeneities, which make possible to use fundamental solution of Laplace equation. DRBEM transforms partial differential equations for stream function, vorticity and magnetic potential to the corresponding boundary integral equations which are discretized using linear boundary elements. Considerably small number of boundary nodes together with arbitrarily selected interior nodes are used to obtain numerical results at a cheap computational cost. The numerical solutions for the lid-driven cavity problem in the presence of a magnetic field are visualized for several values of Reynolds (Re), Hartmann (Ha) and magnetic Reynolds number (Rem). The results exhibit the well-known characteristics of the MHD flow. These are the shift of the core region of the flow and the development of the main vortex in vorticity through the center of the cavity as Re increases. An increase in Ha causes Hartmann layers for the flow at the bottom and top walls. Higher values of Rem result in circulation of the magnetic potential at the center of the cavity. Problem Definition Magnetohydrodynamics (MHD) is a branch of science dealing with the interaction between the electrically conducting fluids and electromagnetic forces. MHD has crucial applications such as in MHD generators, plasma confinement, fusion reactors, and designing cooling systems with liquid metals. Apart from physical investigations, numerical approaches are also developed on this subject. Stabilized Finite Element Methods (FEM) are applied to solve the incompressible MHD equations in two dimension in [1, 2]. A different stabilized FEM application is also examined in 3D MHD by Salah et al.[3]. The coupled equations in terms of velocity and magnetic field for unsteady MHD flow through a rectangular pipe is solved by finite volume spectral element method in [4]. The governing non-dimensional unsteady partial differential equations in terms of stream function-


magnetic potential-vorticity (ψ − A − w) are [5] ∇2 ψ = −w ∂A ∂A ∂A 1 ∇2 A = +u +v Rem ∂t ∂x ∂y ∂w ∂w ∂w 1 2 ∇ w= +u +v Re ∂t ∂x ∂y ∂ ∂By ∂ ∂By ∂Bx ∂Bx Ha2 Bx − + By − − ReRem ∂x ∂x ∂y ∂y ∂x ∂y

(1)

where u = ∂ψ/∂y, v = −∂ψ/∂x and Bx = ∂A/∂y, By = −∂A/∂x, and ψ, A and w are the stream function, magnetic potential and vorticity, respectively. The dimensionless parameters are Re, Rem and Ha as Reynolds, Magnetic Reynolds and Hartmann numbers, respectively. Initially, w is taken as zero in the unit square cavity. Upper lid moves with a constant velocity u = 1, while u and v are zero on all other boundaries. Stream function ψ, then may be taken as zero on the walls. Magnetic Potential is taken as A = −x through the relation By = −∂A/∂x with B0 = (0, 1) which is the +y-directed external magnetic field. DRBEM Application Eqs.(1) are rewritten as coupled Poisson equations ∇2 ψ = b1 (w) ∇2 A = b2 (x, y, t, u, v, Ax , Ay , At )

(2)

∇2 w = b3 (x, y, t, u, v, wx , wy , wt , Bx , By ). In DRBEM, an approximation for the source term b (here b1 , b2 or b3 ) is proposed as [6] b≈

N +L

αj fj

(3)

j=1

where N is the number of boundary nodes, L is the number of internal collocation points, αj ’s are sets of initially unknown coefficients, and the fj ’s are approximating functions. The radial basis functions 2 + . . . + r n where i fj ’s are usually chosen as polynomials of radial distance rij as fij = 1 + rij + rij ij and j correspond to the source(fixed) and the field(variable) points, respectively. For each source node i, the following integral equation is obtained by applying DRBEM ci ϕi +

Γ

ϕ

∂u∗ dΓ − ∂n

Γ

N +L ∂ϕ ∗ ∂u∗ u dΓ = dΓ − qˆj u∗ dΓ αj ci u îj + u ˆj ∂n ∂n Γ Γ

(4)

j=1

1 where ϕ denotes either ψ, A or w, u∗ = 2π ln 1r is the fundamental solution of Laplace equation, ci = 0.5 if the boundary Γ is straight line, and ci = 1 when node i is inside. The relation between the particular solution u ˆj and the approximating function fj is ∂u ˆj 1 ∂ r = ∇2 u ˆj = fj , j = 1, 2, . . . , N + L. (5) r ∂r ∂r Matrix-vector equations resulting from the discretization of these boundary integrals using linear boundary elements corresponding to stream function, magnetic potential and vorticity equations may be expressed as

ˆ − GQ ˆ α, (6) Hϕ − Gϕq = H U


where H and G are BEM matrices containing the boundary integrals of u∗ and q ∗ = ∂u∗ /∂n evaluated at the nodes, respectively, the vectors ϕq = ∂ϕ/∂n contain the known and unknown information at the ˆ and Q ˆ are constructed from u ˆj /∂n nodes about normal derivatives of ψ, A or w. U ˆj and then qˆj = ∂ u columnwise, and are matrices of size (N + L) × (N + L). The vector α is deduced from the Eq.(3) as α = F −1 b,

(7)

where F is the (N + L) × (N + L) coordinate matrix containing radial basis functions fj ’s as columns evaluated at N + L points. By using coordinate matrix for evaluating the derivatives in b and the backward-Euler formula for time derivatives, the iteration with respect to time is carried between the system of equations for ψ, A and w as Hψ m+1 − Gψqm+1 = −Swm

(8)

um+1 = Dy ψ m+1 , v m+1 = −Dx ψ m+1 Rem Rem S − RemSM Am+1 − GAm+1 SAm =− H− q Δt Δt

(9)

where

Bxm+1

m+1

Bym+1

(10)

m+1

= Dy A , = −Dx A Re Re S − ReSM wm+1 − Gwqm+1 = − Swm − H− Δt Δt

Ha2 m+1 S {Bx }d Dx (Dx By − Dy Bx ) + {By }m+1 Dy (Dx By − Dy Bx ) d Rem

− GQ F −1 , S = HU

Dx =

M = {u}m+1 Dx + {v}m+1 Dy , d d

∂F −1 F , ∂x

Dy =

(11)

(12)

∂F −1 F ∂y

and {u}d , {v}d , {Bx }d , {By }d enter into the system as diagonal matrices of size (N + L) × (N + L). Once the shuffling is done and the systems of the form Ax = b are obtained, they are solved by Gaussian elimination with partial pivoting. The iteration is performed as • Eq.(8) is solved using the values of vorticity at the m-th time level. Then the velocity components are computed as (13) um+1 = Dy ψ m+1 , v m+1 = −Dx ψ m+1 , inserting their boundary conditions. • The magnetic potential equation Eq.(10) is solved. Then, it is used to obtain the magnetic induction components as Bxm+1 = Dy Am+1 ,

Bym+1 = −Dx Am+1

(14)

inserting the concerned boundary conditions for them. • Vorticity boundary conditions are found by using the definition of vorticity with the help of coordinate matrix F w = ∇ × u = Dx v m+1 − Dy um+1 . (15) Inserting boundary conditions for w, vorticity equation (12) is solved in (m + 1)-th time level. • Iteration continues until the criterion 3 m+1 − cm ck m+1 k ∞ < = 1e − 4 c k=1

k

∞

is satisfied where ck stands for ψ, A and w, respectively.

(16)


Numerical Results The results are performed using f = 1 + r + r 2 radial basis functions in F matrix. Further, 16−point Gaussian quadrature is used for the integrals in H and G matrices. In general, 120 linear boundary elements and 841 interior points are taken in the computations. Steady-state flow pattern and magnetic potential contours are visualized in terms of streamlines, vorticity contours and magnetic potential lines depicting the effects of Re, Rem or Ha. The center of streamlines which is in the direction of moving lid for small Re numbers shifts through the center of the cavity forming new eddies at the lower corners of the cavity as Re increases. The circulation of vorticity is pronounced for large values of Re. These are the expected behaviors for lid-driven cavity flow. As can be seen from Fig.1, magnetic potential lines are not affected much with the variation of Re. The variation in magnetic Reynolds number causes the magnetic potential lines to circulate inside the cavity due to the dominance of convection terms in magnetic potential equation as Rem gets larger. Not much alteration occurs in streamlines and vorticity (Fig.2). Vorticity becomes stagnant at the center clustering through the walls as Ha increases (Fig.3). Thin boundary layers and Hartmann layers, respectively, on perpendicular and parallel walls to the direction of B0 , are well observed with an increase in Ha in streamlines. The magnitude of the velocity of the fluid decreases in the presence of high magnetic field intensity due to the retarding effect of Lorentz force. This can be observed from the centerline velocity components as Ha increases in Fig.4. Magnetic potential lines become perpendicular to the top and bottom walls pointing to the decrease in convection terms of magnetic potential equation due to the decrease in velocities. Since the reaction term dominates in vorticity transport equation for large values of Ha, a relaxation parameter 0 < γ = 0.1 < 1 is used as wm+1 = γwm+1 + (1 − γ)wm to accelerate the convergence. Conclusion As intensity of the external magnetic field increases (Ha increases), velocity of the fluid decreases. High magnetic Reynolds number causes the magnetic potential to circulate all around the cavity. Acknowledgement This study is carried under the project grant BAP-07-05-2013-001 of METU. References [1] S. H. Aydin, A. I. Neslit¨ urk, M. Tezer-Sezgin, Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations, Int. J. Numer. Methods Fluids 62 (2010) 188-210. [2] J.F. Gerbeau, A stabilized finite element method for the incompressible magnetohydrodynamic equations, Numer. Math. 87 (2000) 83-111. [3] N. B. Salah, A. Soulaimani, W. G. Habashi, M. Fortin, A conservative stabilized finite element method for the magneto-hydrodynamic equations, Int. J. Numer. Methods Fluids 29 (1999) 535554. [4] F. Shakeri, M. Dehghan, A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations, Appl. Numer. Math. 61 (2011) 1-23. [5] H. Ozoe, Magnetic Convection, Imperial College Press, 2005. [6] Partridge PW, Brebbia CA, Wrobel LC, The dual reciprocity boundary element method. Southampton, London, Computational Mechanics Publications, Elsevier Science, 1992.


Vorticity

Magnetic Potential

Re = 1500

Re = 500

Re = 100

Streamlines

Figure 1: Rem = 100, Ha = 10, Δt = 0.25, Re = 100, 500, 1500.

Vorticity

Magnetic Potential

Rem = 500

Rem = 100

Rem = 1

Streamlines

Figure 2: Re = 100, Ha = 10, Δt = 0.1, Rem = 1, 100, 500.


Vorticity

Magnetic Potential

Ha = 100

Ha = 50

Ha = 5

Streamlines

Figure 3: Re = Rem = 100, Ha = 5, 50, 100, Δt = 0.5, 0.2, 0.1.

1

0.2

0.9

0.15

0.8

0.1 Re=100 Rem=100

0.7

0.05 Velocity v

0.6 y

Re=100 Rem=100

0.5 0.4

0 −0.05 −0.1

0.3 Ha=5 Ha=25 Ha=50 Ha=100

0.2 0.1 0 −0.4

−0.2

0

0.2 0.4 Velocity u

0.6

0.8

Ha=5 Ha=25 Ha=50 Ha=100

−0.15 −0.2 −0.25

0

0.2

0.4

0.6

0.8

x

Figure 4: Velocity profiles at mid-sections of the cavity with various Ha.

1

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Numerical solution of 2d steady-state thermoelastic problems through a new and simple meshless Local Boundary Integral Equation (LBIE) method in combination with the Boundary Element Method (BEM) T. Gortsas1, S.V. Tsinopoulos2, E.J. Sellountos3, D.Polyzos1 1

Department of Mechanical Engineering & Aeronautics, University of Patras, Patras, Greece, 2

Department of Mechanical Engineering, Technical Research Institute, Patras, Greece. 3

Instituto Superior Tecnico CEMAT, Lisbon, Portugal

Keywords: Thermoelasticity, BEM, meshless LBIE method.

Abstract. A new and very simple meshless Local Boundary Integral Equation (LBIE) method is effectively combined with the Boundary Element Method (BEM) to solve steady state thermoelastic problems. The two dimensional integral equation valid for the uncoupled steady state thermoelasticity is employed for the BEM, while for the meshless LBIE representation the elastostatic fundamental solution is utilized and the thermal loading is considered as body force. The interpolation of the parameters involved in the BEM is accomplished through the use of quadratic line elements, while in LBIE method randomly distributed points without any connectivity requirement cover the analyzed domain and radial basis Functions (RBFs) are employed for the meshless interpolation of displacements and temperature. Since both methods conclude to a final system of linear equations expressed in terms of nodal displacement-temperature and tractions, their combination is accomplished directly with no further transformations as it happens in other combinations of domain methods with the BEM. Representative examples are provided in order to illustrate the achieved accuracy of the proposed here hybrid meshless LBIE/BEM formulation.

Introduction. The majority of engineering problems in solid and structures can be considered as thermoelastic problems. Thermal stresses induced in high temperature engines, fracture and fatigue processes, interfaces, geothermal systems, pressure vessels etc., are the main concern of a numerical analysis. An accurate and robust numerical method that has been used for the solution of transient is the Boundary Element Method (BEM).The main advantages it demonstrates against other well-known numerical methods such as the Finite element method (FEM) and the Finite Differences Method (FDM), is its high accuracy and the dimensionality reduction of the problem by one, which results in a boundary only discretization of the analyzed domain [1] However, for thermoelastic problems where internal thermal sources are included, volume integrals are inserted in the integral representation of the problem rendering the BEM a domain and not a boundary only discretization method. The application of the Dual Reciprocity BEM (DR-BEM) [2] or the equivalent Particular Integrals BEM (PI-BEM) [3, 4] which transform volume integrals to boundary ones, suffer from convergence problems and questions such as which is the best Radial Basis Function (RBF) for the approximation of displacements and how many interior collocation points are necessary for acceptable accuracy [5]. The Local Boundary Integral Equation (LBIE) method proposed by Zhu et al [6] is a meshless method that employs the same integral equations with the BEM and seems to circumvent most of the aforementioned problems without any sacrifice in accuracy. It is characterized as meshless method because the interpolation is accomplished through randomly distributed points covering the domain of interest and characterized by


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no-connectivity requirements. The combination of the advantages of both BEM and LBIE method and the fact that both utilize the same parameters, hybrid BEM/LBIE method seems to be an excellent alternative to FEM/BEM hybrid formulations. After the pioneering work of Zhu et al [6], meshless LBIE method has received considerable attention due to its accuracy as integral equation method and its flexibility of avoiding any kind of mesh [7,8,9]. Very recently Sellountos et al [10] proposed a stable, accurate and very simple meshless LBIE method for solving elastostatic problems, which utilizes Local Radial Basis Functions (LRBF) for the interpolation of elastic fields and its extension to three dimensions is straightforward. In the present work, that methodology is applied for two dimensional steady-state thermoelastic problems. Integral Representation and LBIEs for steady-state thermoelastic problem. Consider two thermoelastic materials of volume and with the same material properties and in contact to each other as it is shown in Fig 1. The steady state thermoelastic partial differential equations fulfilled for each domain are written as follows:

1 2T (1) Q k m 2u(1)

P

u(1) -

1- 2Q

1 2T (2) Q k m 2u(2)

0 2P (1 v) aT (1) 1- 2n

0

P 1- 2Q

u(2) -

2P (1 Q ) aT (2) 1- 2Q

½ °° (1) ¾x V 0° °¿ ½ ° ° (2) ¾x V ° 0 ° ¿

(1)

(2)

Figure 1. Coupling between two thermoelastic volumes with the same material properties.

where T , Q, u indicate temperature, internal thermal sources and displacement vector, respectively, while

P ,Q , D , N stand for the shear modulus, the Poisson ratio, the coefficient of linear thermal expansion and the

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thermal conductivity, respectively, all the same for both domains. The symbols , 2 represent the gradient and Laplace operators. At the global boundary S (1) S (2)

S (6) prescribed boundary conditions concerning the pairs T , q and t, u are satisfied, while at the interface S ( i ) continuity conditions are considered. The scalar q

k w nT represents thermal flux with w n meaning differentiation with respect to

the direction of outward normal to the boundary and vector t indicates field. The solution of Eqs (2) admits an integral representation of the form:

cT (x) ³ q* (x, y )T (y )dSy S

³ T * (x, y)q(y)dSy

(3)

S

cui (x) ³ tij* (x, y )u j (y )dSy ³ Pi* (x, y)T (y)dSy S

³u

ij

S

S

*

(x, y)t j (y)dSy ³ Qi* (x, y)q(y)dSy (4) S

* * * * * * Where all the kernels q , T , Pi , Qi , uij and tij are explicitly described in [1], while the constant c is equal

to 1 for x V (2) and equal to

1 for x S (2) S (3) S (4) S (i ) , except the corner points. 2

In the domain V (1) , we consider a group at randomly distributed and without any connectivity requirement points as shown in Fig. 1. The points at the global boundary of V (1) correspond to the nodes of a BEM mesh with quadratic elements. For each internal or boundary point x we assume a local circular domain, centered at x called support domain of x .The solution of Eqs. (1), for each support domain, is represented by the LBIEs:

cT (x)

³

*w:

ª¬w nT * (x, y ) w nT c (x, y ) º¼ T (y )dSy

³ ª¬T *

*

(x, y ) T c (x, y ) º¼ q(y ) dSy (5)

³ ª¬T * (x, y ) T c (x, y ) º¼ Q(y ) dSy :

cui (x)

³

*w:

ª¬tij * (x, y ) tij c (x, y ) º¼ u j (y )dSy

³ ª¬u

*

ij

*

(x, y ) uij c (x, y ) º¼ t j (y ) dS y

2P (1 Q ) D ³ ª¬w j uij * (x, y ) w j uij c (x, y ) º¼ T (y )dVy 1 2Q :

(6)

where T * is the fundamental solution of Laplace equation and T c , u c , t c are companion solutions illustrated in [6] and [8]. Finally, Γ represents the portion of the global boundary (Fig.1) when it is intersected by the support domain of the point x . Numerical Implementation and numerical results. For the domain V (2) , the standard BEM formulation described in [1] is applied. The global boundary of V (2) is discretized into quadratic elements and collocating integral equations (3) and (4) at all nodes, we obtain the following system of algebraic equations:

ª u(2) º ªu(21) º ª¬ H (2) º¼ « (2) » ª¬ H (21) º¼ « (21 » ¬T ¼ ¬T ¼

ª t (2) º ª t (21) º ª¬G (2) º¼ « (2) » ª¬G (21) º¼ « (21) » ¬q ¼ ¬q ¼

(7)


355

where (u(2) , t (2) , T(2) , q(2) ) and (u(21) , t (21) , T(21) , q(21) ) are vectors containing all displacements, tractions, temperatures and fluxes defined at all nodes of the external boundary S (2) S (3) S (4) and the interface, respectively. The LBIE method described in [10] is applied for the domain V (1) . According to this method on the global boundary S (1) S (i ) S (5) S (6) , displacements, tractions, temperatures and fluxes are treated as independent variables. On the local domains the LBIEs (5) and (6) are applied and the local boundaries are discretized into quadratic elements. The nodal values of displacements and temperatures are interpolated via a local RBF scheme illustrated in [10]. Thus the following system of algebraic equations is obtained:

ª u(1) º ª u(12) º ª¬ H (1) º¼ « (1) » ª¬ H (12) º¼ « (12) » ¬T ¼ ¬T ¼

ª t (1) º ª t (12) º ª¬G (1) º¼ « (1) » ª¬G (12) º¼ « (12) » ¬q ¼ ¬q ¼

(8)

where u(1) , T(1) are vectors containing displacements and temperatures defined at all internal and boundary points except those of interface S ( i ) which are represented by the vectors u(12) and T(12) . The vectors

(t (1) , q(1) ) and (t (12) , q(12) ) are comprised of nodal values of tractions and fluxes for the nodal points lying at the global boundary S (1) S (i ) S (5) S (6) and the interface, respectively. Applying the continuity conditions at S ( i ) and the boundary conditions at the problem we conclude to a system of the form:

> A@ x

b

(9)

with vectors x, b containing all the unknown and known parameters of the problem, respectively. The system (9) can be solved by a standard LU decomposition procedure. In order to demonstrate the accuracy of the just described BEM/LBIE method the following problem is solved. A hollow cylinder subjected to a thermal gradient is considered (Fig. 2). The internal and external radii of the cylinder are a 0.4m and b 1m .On internal and external surfaces constant temperatures are applied. The material properties are, Poisson ratio v 0.25 , coefficient of thermal expansion h 1E-5 /qC and thermal conductivity k= 1Wm1K 1 .Because of the symmetry of the problem, only one quarter of the cylinder was modeled. Two domains, one for each method, have been considered and depicted in Fig.2. For the LBIE treatment of the problem 120 totally points have been used while for the discretization of the global boundary of the cylinder 20 quadratic elements have been considered. The results have been normalized by dividing by P hT1 . The numerical results for radial displacements are presented in Table. 1 and are compared to those presented in [1].As it is apparent the numerical results are in excellent agreement with the corresponding analytical ones.

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Figure. 2 Hollow cylinder subjected to a temperature gradient. r 0.4 0.55 0.70 0.85 1.00

Numerical 0.323 0.345 0.452 0.610 0.805

Exact 0.322 0.346 0.453 0.611 0.806

Table 1.Radial displacement of the hollow cylinder. Conclusions A hybrid BEM LBIE method for solving 2D steady state thermoelastic problems has been proposed. The thermoelastic BEM formulation is the same with that described in Aliabadi (2002), while the LBIE methodology is based on the resent work of Sellountos et al(2012) properly modified for solving 2D thermoelastic problems. Acknowledgements This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Framework (NSRF) – Research Funding Program: ARCHIMEDES III. References [1] M.H.Aliabadi. The Boundary Element Method, Vol2 : Application in solids and Structures, Wiley, (2002). [2] D.Nardini and C.Brebbia A new approach to free vibration analysis using boundary elements. In:Brebbia CA (ed) Boundary Methods in Engineering, Springer, Berlin, Vol 50, 313-326(1982). [3] D.Polyzos, G.Dassios and D.E.Beskos On The Equivalence of Dual Reciprocity and Particular Integrals Approaches in the BEM, Boundary Elements Communications, Vol.5, 285-288 (1994). [4] S.Ahmad and P.K.Banerjee Free vibration analysis by BEM using particular integrals, Journal of Engineering Mechanics, ASCE, Vol. 112, 682-695 (1986).


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[5] J.Agnantiaris, D.Polyzos and D.Beskos Some Studies on Dual Reciprocity BEM for Elastodynamic Analysis,Computational Mechanics, Vol 17, 270-277 (1996). [6] T.Zhu, J.D.Zhang and S.N.Atluri A local boundary integral equation (LBIE) method in computational mechanics and a meshless discretization approach, Computational Mechanics, Vol. 21, 223– 235 (1998). [7] S.N.Atluri The meshless method (MLPG) for domain & BIE discretizations, Tech Science Press (2004). [8] E.J.Sellountos and D.Polyzos A MLPG (LBIE) method for solving frequency domain elastic problems. CMES: Computer Modelling in Engineering & Sciences, vol. 4, 619– 636 (2003). [9] E.J.Sellountos and D.Polyzos A MLPG (LBIE) approach in combination with BEM. Computer Methods in Applied Mechanics and Engineering, vol. 194, 859–875 (2005a). [10] E.J.Sellountos, D.Polyzos and S.N.Atluri A new and Simple Meshless LBIE-RBF Numerical Scheme in Linear Elasticity, CMES: Computer Modelling in Engineering & Sciences, 89(6), 513-551 (2012).

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A detailed Boundary Element analysis of the flow field outside a growing immiscible viscous fingering within a Hele-Shaw cell H. Power1, D. Stevens2 and A. Cliffe2 1

Faculty of Engineering, Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, UK, [email protected]

2

Faculty of Engineering, Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, UK, [email protected] 3

School of Mathematical Sciences, University of Nottingham, Nottingham, UK, [email protected]

Keywords: Viscous fingering, direct Boundary Element Method, field variable evaluation

Abstract. A BEM model for a viscous fingering problem in a Hele-Shaw cell is presented. The major point of interest in the study of this type of unstable problems is the evolution of the fluids interface where the entire nonlinear dynamic occurs. In this work besides considering the interface evolution, we will look at the corresponding flow field at each fluid region. To evaluate the flow field it is necessary to find the pressure field and its gradient inside the fluid domain. Since the reconstruction is valid only for nodes within the liquid domain, and the topology of the interface surface is typically complex, with its position constantly changing in time, it is useful to have a simple numerical scheme to determine whether or not a given location lies within the liquid or gas domain. The theory of harmonic surface potentials allow us to find such scheme, which generally is not a simple numerical task and usually very computational costly. Introduction. The onset and evolution of instabilities that occur in the displacement of the interface between two immiscible fluids with different viscosities is known as immiscible viscous fingering. Fingering instability appears when a fluid of higher mobility (lower viscosity) displaces a fluid of lower mobility (higher viscosity); in this case, the interface between the two fluids is unstable to perturbations of certain wavelengths, resulting in the evolution of long fingers of the less viscous fluid which penetrate into the more viscous fluid. As the fingers grow, their tips can also become unstable (tip splitting) leading to the formation of new fingers. This splitting pattern can be repeated successively as the evolution progresses, resulting in a complex interface patterns corresponding to nonlinear interfacial dynamics. The mathematical formulation of this type of interfacial problem, is described by the Darcy flow approximation (potential flow) at each side of the interface, which are matched at the sharp interface according to the continuity of normal velocity, and pressure force balanced by surface tension forces. Under these conditions the viscous instability competes with the stabilizing force of the surface tension at the interface as it is deformed by the finger (for more details see Homsy [1]). During the evolution of viscous fingering, the interface between the two fluids experiences large deformations, and the correct determination of its shape is of paramount importance. Consequently, the use of the boundary integral equation formulation of the problem is an attractive numerical technique for its solution, and has been successfully implemented previously to study fingering problems without the effect of dissolution; see, among others, DeGregoria and Schwartz [2], Tosaka and Sugino [3], Power [4], Zhao et al. [5], Hadavinia et al. [6] and Li et al [7]. The major point of interest in the study of this type of unstable problems is the evolution of the fluids interface where the entire nonlinear dynamic occurs. For this reason little attention has been given in the literature to the corresponding flow field at each fluid region, and most of the published articles in this area only focus their analysis to the moving interface, even those based on explicit analytical solutions of the governing partial differential equations, Howison [8]. In this work a BEM model of the problem is presented, where the pressure field at the unbounded viscous fluid domain is expressed as the potential field due to the injected flow plus a perturbation term . In our formulation


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we have the possibility of include more than one source point. To evaluate the flow field it is necessary to find the pressure field and its gradient (Darcy velocity) inside the fluid domain. Since the reconstruction is valid only for nodes within the liquid domain, and the topology of the interface surface is typically complex, with its position constantly changing in time, it is useful to have a simple numerical scheme to determine whether or not a given point lies within the liquid or gas domain. The theory of harmonic surface potentials allow us to find such scheme, which generally is not a simple numerical task and usually very computational costly. Boundary Element formulation of the problem. A BEM model for single-phase flow in a Hele-Shaw cell is presented, where it is considered that the viscosity of the less viscous fluid, the gas phase in our case, is so small that it may be ignored. Taking into account the asymptotic decaying condition at infinity of the perturbed field, it is possible to write the Green’s integral representation formula, direct integral equation formulation, for the perturbed pressure at a field point only in terms of the fluid interface , as (see Power and Wrobel [9]): (1) is the fundamental solution to the Laplace equation, and are the normal derivatives here and , respectively. of In the above formulation, it is necessary to impose the no-flux condition of across the fluid interface , i.e. (2) It is easy to verify that by imposing the above non-flux condition, the above pressure field vanishes at infinity. By evaluating the above pressure field at the gas/liquid interface the following integral equation of the first kind for the unknown normal derivative of is obtained: (3) (reference pressure) is an unknown constant value to be determined, subject to the non-flux condition (2), Here and required to guarantee the solvability of the integral equation (3); see Jaswon and Symm [10]. In this work, we implemented the cubic B-splines BEM numerical scheme previously suggested by Cabral et al. [11] and Zhao et al. [5], where both the geometry and surface densities are approximated by cubic B-spline functions. From our numerical results, we have verified that this type of numerical scheme shows an order convergence, with h as the characteristic element size. Zhao et al. [5] study the evolution of a single-phase radial viscous fingering similar to one considered here, but for only one point of injection and without the effect of dissolution. Their formulation is based on a direct integral equation approach, similar to the one used here, but they do not represent the pressure field in terms of the injected flow plus a perturbation, and consequently they cannot evaluate asymptotically the external surface integrals at infinity, requiring the use of an auxiliary far field surface enclosing the problem domain. The existence of such external surface makes the computation of long-time simulations more difficult due to the additional computational cost in the evaluation of corresponding external surface integrals and possible interaction between the moving interface and the auxiliary external surface.

Evaluation of the external flow field. To evaluate the flow field it is necessary to find the pressure field and its gradient (Darcy velocity) inside the fluid domain. The value of the perturbed pressure field, pˆ , at any location within the liquid domain, exterior to the growing bubble, can be reconstructed from the integral representational formula (1) with the obtained values of pˆ and qˆ at the moving surface interface , found from the solution of the integral equation (3) with prescribed boundary condition. Since the reconstruction is valid only for nodes within the liquid domain, and the topology of the interface surface is typically complex, with its position constantly changing in time, it is useful to have a simple numerical scheme to determine whether or not a candidate location x lies within the liquid or gas domain. The theory of harmonic surface potentials allow us to find such scheme, which generally is not a simple numerical task and usually very computational costly. By examining the behaviour of the double-layer potential with a constant density, unit density,

360


(4) at any point in the domain, inside or outside the moving bubble, it is possible to define such algorithm. Using the well-known jump property of the double layer potential, it follows that locations within the gas domain, inside the bubble, will return a value , whereas locations in the fluid domain will return . It is interesting to observe that no additional numerical coding is required since the matrix descretization of the double layer potential is already defined for the solution of equation (3). After defining if a point lies within the flow domain, external point, the pressure field can be directly found from equation (1) and the corresponding Darcy felicity field by taking the gradient of equation (1). In the evaluation of both external pressure and velocity all the integrals are regula since the selected point always lies outside the fluids interface. Numerical examples To verify the performance of the present B-spline BEM formulation, a numerical example has been carried out to follow the evolution of the six-tongue radial finger, i.e. a single well problem. For this validation example, the initial interface is given by a deformed circle having the shape of a waving curve with six small crests distributed symmetrically, and defined by the expression , with and d . For this case ( ), the surface tension is the mobility is taken to be ( ) and a single source is present at the origin with volume flux per unit length . of Initially, the fluid interface is divided into 72 boundary elements, but this number is dynamically increased as time develops, in order to maintain a target element size. The obtained numerical result of the fingering evolution can be seen in figure 1. This result is an exact reproduction of the result reported previously by Power [4] of the same problem, but using a different integral equation formulation (indirect approach). Besides, the present solution is consistent with the BEM numerical results reported by Hadavinia et al. [6], the analytical solution of the N fingers problem given by Howison [8]. However, the Howison solution is not constructive since it is given in terms of a generalised series of complex harmonic functions that can represent different cases by choosing different sets of the constant values of the series; i.e. it cannot be used to predict the evolution of a given initial condition.

Figure 1 – Evolution of the gas bubble, showing classical viscous fingering features. Bubble surface is plotted at intervals of Δt = 50.0 s


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To evaluate the exterior flow field a set of very dense uniform distribution of points, , inside and outside of the growing bubble is utilised to reconstruct the pressure field, and in each of them the above double layer potential with a unit density is evaluated to find if the corresponding point is inside or outside the bubble. This is achieved by a simple matrix vector multiplication. At any time of the evolution, points with are in the liquid phase and are therefore used in the reconstruction of the flow field. Figures 2a) and b) show the predicted perturbed pressure distribution and the magnitude of the total flow , corresponding to the last interface profile in velocity outside the growing bubble at the time Figure 1. As can be observed, the perturbed pressure field induces a localised pressure gradient against the moving interface at the base of the fingers, for both the first and second split, while at the finger tips the perturbed field leads to a localized pressure gradient in the direction of the moving finger. The perturbed pressure gradient against the moving interface induces a velocity field that opposes the injected velocity, delaying the growth of the finger base, while the pressure grading in the direction of the moving finger at its tip, which is the same direction than the injected velocity, increases its displacement speed. It is clear from Figure 2 b) that the tips of the fingers are growing most rapidly, with the inner region at the finger base almost stagnant. To find the velocity field it is necessary to obtain the gradient of the perturbed pressure at each of the evaluation points inside the fluid domain, i.e. outside of the integration surface , which can be found directly from the gradient of the integral representation formula (1), that needs only the evaluation of regular surface integrals. Figure 3 shows velocity vectors for the total flow field close to a developing finger, also at the time , in the top right quadrant of Figure 1. Close to the bubble surface the flow direction is roughly normal to the surface, and further from the surface the flow is radial, directed away from the source, i.e. given only by the injected velocity field. Once again the slow flowing fluid is visible around the stagnant finger bases.

a) Disturbed pressure distribution

b) Total velocity magnitude | u i |

Figure 2 – a) Predicted perturbed pressure distribution ( ) and b) magnitude of the total flow velocity (cm/s) outside the growing bubble at the time . The proposed numerical method allows for the inclusion of multiple sources. The enforced mass-flux will be zero for closed surfaces not containing a source, and non-zero for those containing sources. The following example validates this implementation by examining a model with two sources of equal strength. Figure 4 shows the bubble growth resulting from two sources of equal strength, in the absence of dissolution. Physical parameters are: M=0.05 , and Qtot=5 , i.e. 2.5 per source. The dipole effect is clearly visible here; the two bubbles repel each other, and finger growth is restricted to the outer sides.

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Figure 3 – Velocity vectors for the total flow field close to the a developing finger

Figure 4: Injection of two sources of equal strength; no gas dissolution. Surface is plotted at intervals of Δt=50 s. Figures 5a) and b) show the predicted perturbed pressure distribution outside the growing bubble at times and . As can be observed, the perturbed pressure field induces a localised pressure gradient against the moving interface at the base of the fingers, for both the first and second split, while at the finger tips the perturbed field leads to a localized pressure gradient in the direction of the moving finger. The perturbed pressure gradient against the moving interface induces a velocity field that opposes the injected velocity, delaying the growth of the finger base, while the pressure grading in the direction of the moving finger at its tip, which is the same direction than the injected velocity, increases its displacement speed.


a) t = 100

363

b) t = 300

) for the case of the injection of two Figure 5 - Predicted perturbed pressure distribution ( sources of equal strength. Acknowledgement: The present work has been partially supported by the European Commission projects MUSTANG (Project Reference 227286) and PANACEA (Project Reference 282900), seventh frame work programs. References [1] Homsy G.M. (1987), Viscous fingering in porous media, Annual Review of Fluid Mechanics, 19, 271311. [2] De Gregoria A.J. and Schwartz L.W., A boundary-integral method for two-phase displacement in HeleShaw, J. Fluid Mech., 164, 439-453, (1986). [3] Tosaka N. and Sugino R., Boundary element analysis of moving boundary in Laplacian growth, in Computational Modeling of Free and Moving Boundary Problems II, Computational Mechanics Publication, Southampton, (1993). [4] Power H. (1994) "The evolution of radial fingers at the interface between two viscous liquids", Engineering Analysis with Boundary Elements, Vol. 14, No. 4, 297-304. [5] Zhao K.X.H., Wrobel L.C. and Power H., Numerical simulation of viscous fingering using B-spline boundary elements, Computational Modeling of Free and Moving Boundary Problems III, Computational Mechanics Publication, Southampton, (1995). [6] Hadavinia H., Advani S.G. and Ferrer R.T. (1995), The evolution of radial fingering in a Hele-Shaw cell using continuous Overhauser boundary element method, Engineering Analysis with Boundary Elements, Vol. 16, 183-195. [7] Li S., Lowengrub J.S. and Leo P.H., (2007), A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell, Journal of Computational Physics, Vol. 225, 554-567. [8] Howison S.D., Fingering in Hele-Shaw cells, Journal of Fluid Mecjanics, 167, 439-453, (1986). [9] Power H. and Wrobel L.C., Boundary Integral Methods in Fluid Mechanics, Computational Mechanics Publications, (1995). [10] Jaswon M.A. and Symm G.T., Integral equation methods in potential theory and elasticity, Academic Press, New York, (1977). [11] Cabral J.J.S.P., Wrobel L.C. and Brebbia C.A., A BEM formulation using B-splines: I-uniform blending functions, Engineering Analysis with Boundary Elements, 7, 136-144, (1990).

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A solution procedure to 3D integral nonlocal elasticity: Coupling isotropic-BEM with strong form local radial point interpolation. Richard Kouitat Njiwa 1, Ngadia Taha Niane 2, Martin Schwartz 3 Université de Lorraine, IJL - Dpt N2EV - UMR 7198 CNRS. Parc de Saurupt, CS 14234, 54042 Nancy Cedex France 1

[email protected],

PSA Peugeot Citroën 18, rue des Fauvelles, 92250 La Garenne Colombes 2

[email protected], Aer-Alcen 106, rue du Lieutenant Petit Leroy, 94550 Chevilly-Larue 3

[email protected]

Keywords: Non local elasticity, Isotropic BEM, Meshfree strong form.

Abstract. In order to address some problems that lead to unphysical results in the framework of classical continuum mechanics, nonlocal theories are increasingly adopted. The nonlocal elasticity is a potential candidate for crack as well as size effect problems. In this work, we present a simple strategy to solve a 3dimensional nonlocal integral elasticity problem. The approach combines the conventional elastostatic boundary element method with a local radial point interpolation method. The effectiveness of the method is proved on some examples using the original integral model due to Eringen and the enhanced version proposed by Polizzotto.

Introduction Classical elasticity, either linear or not, has proven very efficient and useful in many engineering fields. Unfortunately this theory leads to questionable results in fields of fracture mechanics, small size objects where the size effect is prominent [1, 2]. According to Eringen, the local theory of elasticity is unsatisfactory whenever the influences of events arising at the microscopic level or at the microstructure are significant. Nonlocal elasticity is a way to include internal length scale in the material description. Its consideration appeared in the late sixties for example in the paper by Kroner [3]. It was noted that in the framework of classical elasticity, internal forces in the body are of contact type (zero range) whereas cohesive forces are known to be of finite range. They proposed enriched theories allowing a refine description of elastic behavior of the material. Following, improved formulations of nonlocal elasticity were proposed. A version of the nonlocal constitutive relation involves an integral term of the strain over the entire volume of the solid. Consequently the stress at a given point in the solid depends on the strain at all points in the solid. Following this approach, substantial works have been done to address crack problems, dislocations problems, problems with concentrated loads, contact problems, and carbon nanotubes (see e.g. [4-9]). With respect to continuum boundary value problems, Polizzotto pointed out a shortcoming of the integral formulation due to Eringen. Indeed, regarding a solid of finite extent, the latter does not provide a uniform stress under a uniform strain field. He then proposed [10] an enhanced constitutive law which alleviates this shortcoming. The model has already been applied to 2D crack problem by Jackiewicz and Holka [11]. As pointed out by Pisano et al. [12], regarding nonlocal elasticity of integral type, an issue that should be investigated thoroughly is related with the numerical solution of structural problems. These authors investigate the application of the finite element method to a 2D problem on the basis of the nonlocal finite element method proposed by Polizzotto [10]. The present work proposed a solution method of 3D nonlocal elastic problems. The proposed strategy couples the conventional boundary element method (BEM) for elastostatic problems with the local radial


365

point collocation method. The main steps of the approach are given in Section 2 below. In Section 3, the effectiveness of the method is demonstrated on examples of tension and shear loading of a unit cube and torsion of a cylindrical specimen.

Problem statement In the absence of body forces, the equilibrium of a solid occupying a domain : is described by:

V ij , j

0

(1)

Equation (1) must be supplemented by properly defined boundary conditions. This work is limited to the case of isotropic nonlocal elastic materials. The relation between the stress tensor V ij and the strain tensor

H ij as proposed by Polizzotto [11] reads: V ij x Rkl H :f

Cijkl Rkl H with

>1 J ( x)@H kl x ³ g x, x ' H kl x ' dV x '

(2)

:f

The function J(x) is such that

is called the domain of influence of the kernel g.

0 d J x

³ g x, x' dV x'

d 1 . The attenuation function g(x,x’) is positive and has the dimension of

:f

(length)-3. It reaches its maximum when x= x’ and decays to zero at large distance from x. It satisfies the normalization condition over an infinite domain

³ g x, x' dV x'

1 . The Eringen’s model is recovered

:f

when : f = : the function J(x) is equal to 1. In this work, we use g x, x ' f

exp x ' x l /(8S l 3 )

where l is the internal length characteristic of the material under study. Having selected the value of l, a spherical region : f with radius R is defined such that g x, x' dV x' | 1 . For the selected g kernel R is

³

:f

taken to be 9 times the value of l.

Solution Method Obviously the solution of the considered problem can be obtained by a specialized finite element method. It is also tempting to use one of the recently introduced numerical methods known as meshless or meshfree methods. Detailed of some of them can be found in the papers by Belytschko et al.[13] ; Atluri et al.[14] ; Liu et al.[15] ; Liu GR et al [16]. Most of these methods use either a local weak form or a local integral equation of the field equations. Each of them has its own shortcomings. In this work, we present an alternative to the weak-strong form point collocation method. First, we rewrite the stress as V ij x Cijkl H kl x Cijkl Qkl H with Qkl H Rkl H H kl . Assume that the displacement field is the sum of a complementary term ( u C ) and a particular solution ( u P ) with: w wx j w wx j

C

H klC 0

C

H klP

ijkl

ijkl

w wx j

(4)

C

ijkl

Qkl H 0

(5)

Equation (4) is similar to the standard Navier equation. Its approximation by the boundary element method is well documented and leads to a system of equations of the form:

> H @ û c ` >G@ ^t c `

(6)

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Let the domain : and its boundary * be represented by properly scattered nodes. A kinematical field at any point in the domain is interpolated by the local radial point interpolation. After some algebraic manipulations, one obtains vkh ( x ) > R1 R2 . . . Rn @> Fa @^vk / L ` > P1 P2 . . . Pm @> Fb @^vk / L ` h

which is written more compactly as: vk ( x )

>) ( x )@^v `

(7)

k/L

Ri ( r ) is the selected radial basis functions, n is the number of nodes in the neighbourhood of point x and m is the number of monomials Pj(x) in the basis of the selected augmented polynomial degree.

^z ` z /L

1 1

z21

z31 . . . z1N

z2N

z3N . Adopting interpolation (7) for all the kinematical fields in

equation (5), for each internal collocation center, one obtains:

ªQu º û/pL ` >Qu @û/ L ` ¬ p¼

^0`

(8)

Collecting equations (8) for all internal collocation points and assuming that the particular solution is zero at all centers located on the boundary, it comes

û ` >QG@û` P

(9)

The traction at the boundary point x is expressed, according to the proposed partition as:

ti ( x) V ij x n j ( x) CijklH klC x n j ( x) CijklH klP x n j ( x) Cijkl Qkl H n j ( x) or more compactly ti ( x) tiC ( x) tiP ( x) G ti ( x) . Using the interpolation (7), the last two terms of the right hand side of the above relation, are written for all boundary nodes in the matrix form:

^t ` p

ª Kt º û P ` and ^G t` ª KG t º û` ¬ ¼ ¬ p¼

(10)

Consider the partition of the displacement field and taking into account relations 9 and 10, equation (6) is rewritten as > H @ û` >G @ ^t` > H @ u p >G @ t p G t . After some algebraic manipulation it comes:

>H~ @û` >G@^t`

^0`

^ `

^

`

Numerical examples In this work, the generalized radial basis functions Ri r

r

2

c 2 are adopted. r is the Euclidian q

distance between the field point x and the center xi, c and q are known as shape parameters. Let us point out that the approach uses two distinct domain of influence. The first one is related to the constitutive relation and is selected smaller than the spherical region of influence for the local point interpolation. The effectiveness of the approach is demonstrated on the examples of tension load and torsion load. In the following, the stress-strain relation given by equation (2) is called the Eringen’s model when the function J(x) is taken equal to one at all points. Otherwise, it is called the Polizzotto’s model. The homogeneous isotropic solid has a Young modulus E = 220GPa and a Poisson ratio ν = 0.3. The characteristic internal length l is selected to be 10 μm. The latter value is chosen arbitrarily. Rigorously, it must be selected as a representative length of the inner microstructure of the material. Unit cube under uniform tension A scheme of the considered specimen is given in Figure (1a). The unit cube is subjected to uniform displacement u in the z direction on the upper face (z = 0.5). The lower face (z = - 0.5) is displacement constrained in the z direction and traction free in the tangential direction. The other faces of the cube are free of traction. This type of Dirichlet loading is selected since it allows a simple comparison between the two selected models. For the presented results, 27 internal collocation centers are used and the boundary of the cube is subdivided into 24 nine-node quadrilateral elements, i.e. four elements per face. The results presented


367

in table 1 below are obtained with the radial basis shape parameters q = 1.03 and c = d0 (minimal distance between two collocation centers)

Figure 1a : Case of uniform traction loading

Figure 1b : Cylindrical bar under torsion

Figure 1: Geometrical definition of the specimen: a/ simple tension, b/ torsion

point

0

field

Analytical ( V zz E u )

Eringen model

Polizzotto model

εyy (×10-3)

-0.1363636

-0.1363634

-0.1363635

εzz (×10-3 )

0.4545454

0.4545446

0.454550

0 0

point

0.5

field

Eringen model

Polizzotto model

εyy (×10-3

-0.1363638

-0.1363634

εzz (×10-3 )

0.4545453

0.4545451

σzz (MPa)

49.53003

99.99994

0.25 σzz (MPa)

100

99.05985

99.99989

0.25

Table 1: Uniform tensile loading of a unit cube: numerical results of strain and stress values at an internal point and a boundary point. Influence of the number of internal collocation points The results presented in table 1 are practically undisturbed when the radial basis shape parameters are varied from 0.5 to 15 for q and, 10-4d0 to d0 for c. In Figures 2, the evolutions of the stress σzz, calculated by the Eringen’s model, in the top surface (a) and the middle plane perpendicular to the loading direction are shown. As can be observed, the calculated values are expected ones except at those points near the solid boundary. Near and at the solid boundaries, the calculated stresses are lower than the expected value. There is no solid boundary effect when using the improved model of Polizzotto.

Figure 2: Evolution of σzz (Eringen model): (a) On the top of the cube ; (b) In the plane Oxy

368


Case of a cylindrical bar under torsion A cylindrical bar with radius 1mm and height 10 mm is considered (see Figure 1b). In order to simulate torsion, the tangential displacement at the upper and lower face of the specimen is specified. More specifically, let H be the half height of the cylinder and αt be the twist angle per unit length. At the upper face ux = -αtHy and uy = αtHx while at the lower ux = αtHy and uy = -αtHx. The lateral surface of the cylinder is free of traction.The simulation uses 80 nine-node quadrilateral boundary elements and 100 internal nodes. The variations of the stress σθz along a diameter of the top surface and a cross section of the cylinder are presented in Figures 3 for internal length 1, 10 and 20 μm. Once more, the effect of truncation of the non-locality domain of influence is evidenced and the higher the internal length the smaller the stress value. The general conclusions are similar to those of the loading of the unit cube. When using the Polizzotto’s model the results are closed to the classical solutions with a difference less than 2%. For this loading also, the quality of the results is not affected by the parameter c.

Figure 3: (a) Evolution of σθz between (-1;0;10) and (1;0;10) ; (b) Evolution of σθz between (-1;0;7.5) and (1;0;7.5) Circular crack in opening mode I Consider the case of a penny shaped crack centered in a bar with squared shape cross section whose upper and lower surfaces are subjected to uniform tensile stress σ0. The diameter (2c) of the crack is half the size (2a) of the cross section. Due to symmetry, only the upper half of the domain is modeled. All faces of the resulting cube are discretized into four quadrilaterals elements, except the lower face for which a particular mesh is required as represented in the figure below (see Figure 4).

Figure 4: Discretization of the lower face of the domain and evolution of the stress tzz at the front edge


369

As can be observed, the stress is finite at the crack front. The maximum value is located slightly ahead of the crack front at approximately one internal characteristic length. It seems possible to define cracking criteria based on a critical value of an equivalent stress.

Conclusion This paper addressed the problem of the numerical solution of an elastostatic problem with non local integral constitutive relation. Consider classical homogeneous isotropic elastostatic the boundary element method generally leads to highly accurate results. In the case of nonlocal integral elasticity a fundamental solution is not yet available and the method seems inapplicable. The present work proposes a promising BEM based solution procedure. It couples the conventional approach for isotropic elastostatic with the local radial point interpolation of a strong form differential equation. The overall approach is simple to implement. Its effectiveness and accuracy has been demonstrated on the example of loading of a unit cube and torsion of a cylindrical bar. The application of the method to fracture mechanics problems seems also promising. A deep investigation in this last case as well as the applicability to nanomaterials will be the subject of a future work.

References [1] A. Cemal Eringen Res Mech 21, 313-342 (1987). [2] LJ Sudak J. Appl Phys 94 7281-7 (2003). [3] E. Kröner Int J Solids Struct 3 731-742 (1967). [4] A.C. Eringen, B.S. Kim Mech Res Commun 1 233-237 (1974). [5] A.C. Eringen, C.G. Speziale, B.S. Kim J Mech Phys Solids 25 339-355 (1977). [6] A.C. Eringen Int J Eng Sci 15 177-183 (1977). [7] M. Lazar, G.A. Maugin, E.C. Aifantis Int J Solids Struct 43, 1404-1421 (2006). [8] Y.Z. Povstenko, I. Kubik Int J Eng Sci 43 457-471 (2005). [9] H. Heireche, A. Tounsi, A. Benzair, M. Maachon, E.A. Adda Beida Physica E 40, 2791-2799 (2008). [10] C. Polizzotto Int J Solids Struct 38 7359-7380 (2001). [11] J. Jackiewicz, H. Holka Eng Fract Mech 75, 461-474 (2008). [12] A.A. Pisano, A. Sofi, P. Fuschi Int J Solids Struct 46, 3836-3849 (2009). [13] T. Belytschko, Y.Y. Lu, L. Gu Int J Numer Methods Eng. 37, 229-56, (1994). [14] S.N. Atluri, T. Zhu Comput Mech 22, 117-27, (1998). [15] W.K Liu, S. Jun, S. Li, J. Andee, T. Belytschko Int J Numer Methods Eng 38, 1655-79, (1995). [16] G.R. Liu, Y.T. Gu J Sound Vib 246, 29-46, 2001.

370


Solutions for Free Vibration Analysis of Thick Square Plates by the Boundary Element Method W.L.A. Pereira 1,a, V.J. Karam 2,b, J.A.M. Carrer 3,c, W.J. Mansur 1,d 2

1 Department of Civil Engineering, Federal University of Rio de Janeiro, Brazil Department of Civil Engineering, State University of Norte Fluminense Darcy Ribeiro, Brazil 3 Department of Mathematics, Federal University of Paraná, Brazil a

[email protected], b [email protected], c [email protected], d [email protected]

Keywords: Boundary element method, thick square plates, free vibration, very large floating structure

Abstract. In this work, the BEM is employed to obtain the solutions for the free vibration analysis of thick square plates with two edges simply supported and the other two edges free. A formulation based on Reissner’s theory is used here, which includes the contribution of the additional translational inertia terms to the integral equation of displacements and internal forces. The boundary element method formulation used to discretize the space employs static fundamental solution. This problem type is very important in the hydroelastic analysis of very large floating structures (VLFS) which are commonly modeled as plates with free edges. To verify the accuracy this formulation some analyses are presented at the end of the paper. Introduction A vast literature exists for the free vibrations of square plates, but there are still some hypotheses either not introduced or not sufficiently tested that need to be studied. Plates, in general, are three dimensional structures used in many areas of applications as in the civil engineering; aerospace, nuclear, and marine industries, among others. In practical applications, two-dimensional theories may be considered in the analysis of plates. There three theories most used to study plates. The Kirchhoff’s theory is well described in [1, 2], does not take into account the effect of shear deformation and rotatory inertia, and is limited to thin plates. Another plate theory was proposed by Reissner [3-5] in the decade of 1940, and considers the effect shear deformation and requires three boundary conditions, instead of two, as established by the Kirchhoff’s theory. The Mindlin’s theory was proposed in two papers [6, 7]; to account for shear deformation based on a proposed displacement field through the plate thickness, and incorporated the effect of rotatory inertia. Nowadays, we are facing a population growth and a corresponding expansion of urban centers as is the case of Japan. For solve this problem type, engineers have proposed the construction of very large floating structures (VLFS) to long coastlines. In literature, VLFS may be classified under two broad categories, namely the pontoon-type and the semi-submersible type. The pontoon-type VLFS may be modeled as plates with free edges [8], thus using boundary conditions adequate in the formulation a set of equations is obtained being possible to compute the modal shapes and the stress-resultants. The Mindlin plate theory was used in [9] where the hydroelastic analysis of pontoon-type circular VLFS was performed. Watanabe et al. [10] presented a literature survey of the research on hydroelastic analysis of VLFS. Chen et al. [11] presented a review of hydroelastic theories for global response of marine structures. Recently, two theories of shear deformable plate vibrations that account for the influence of the transverse normal stress components were presented by Batista [12], being one of them based in the Mindlin theory and the other in the Reissner theory. Moreover, the transverse normal stress components are included in the boundary element method (BEM) for free vibration analysis of thick elastic plates [13, 14]. This work attempts to present accurate numerical results for the free vibration analysis of thick square plates with two edges simply supported and the other two edges free. For this, the BEM is employed to obtain the solutions for the free vibration analysis of thick square plates. A formulation based on Reissner’s theory is used here, which includes the contribution of the additional translational inertia terms to the integral equation of displacements and internal forces. The boundary element method formulation used to discretize the space employs static fundamental solution. This problem type is very important in the hydroelastic analysis of very large floating structures (VLFS) which are commonly modeled as plates with free edges. In this work, an indicial notation is used, where Latin subscripts vary from 1 to 3, while the Greek subscripts range is from 1 to 2.


371

Basic equations Consider a plate of thickness h made from homogeneous and isotropic elastic material with the modulus of elasticity E , Poisson’s ratio Q and mass density U . The equilibrium equations governing its free vibrations based on Reissner’s theory in a Cartesian coordinate system are given by Uh 3 (1a) ID M DE ,E QD 12 (1b) QD ,D Uhw where overdots indicated corresponds to differentiation with respect to time t . M DE and QD are the moments per unit length and the shear forces per unit length, respectively. The other variables involved in the problem are the generalized displacements which are the rotations ID and the vertical deflection w . Resultant moments and shear forces are written as [14] ª º 1 2Q GDE with mDE M DE mDE kUh w (2a) F J ,J G DE » D(1 Q ) «2 FDE 2 (1 Q ) ¬ ¼ 5 (2b) Gh ID w,D 6 in which k Q / 6(1 Q )O2 , where O2 10 / h 2 is the characteristic parameter of Reissner’s equations; G DE is QD

the Kronecker delta; D Eh3 /12(1 Q 2 ) is the bending stiffness; G E / 2(1 Q ) is the shear modulus. The expressions of the generalized deformations for the linear theory, in terms of the generalized displacements that appear in eq. (2) are given by Flexural strains components: NDE 2FDE ID ,E IE ,D (3a) Transverse shear strains components: J D

ID w,D

(3b)

For simplicity, the generalized displacements ID and w will be written generically as u i . Integral equations Let : be the domain, which is represented by the midsurface of the plate, and * be its contour, thus for present problem the following initial conditions in the domain are considered: Initial displacements: ui (x, t 0) ui 0 (4a) (4b) Initial velocity: ui (x, t 0) ui 0 The prescribe boundary conditions on * for the three generalized directions of the plate are defined by (5a) ui ui at *u (5b) pi pi at *p where * is the total boundary of the plate so that * *u *p , and pi are the generalized surface forces, defined as pD M DE nE

(6a)

p3

(6b)

QE nE

in which nE being the direction cosines of the outward normal on * . According to Pereira et al. [14], the integral equation for displacements can be written for three generic directions as Uh 3 cij ([ ) u j ([ , t ) p j (x, t ) uij* ([ , x) pij* ([ , x) u j (x, t ) d*(x) uD (x, t )ui*D ([ , x) d:(x) 12 : *

³>

³

@

³

³

Uh u3 (x, t ) ui*3 ([ , x) d:(x) kUh u3 (x, t ) ui*D ,D ([ , x) d:(x) :

(7)

:

The above equation holds for internal points with cij

G ij and for boundary points, with cij

G ij / 2 at

smooth boundaries. Moreover, [ and x are source point and field point, respectively. Note that the last term in eq. (7) also refers to the translational inertia term, being an additional part of the integral equations for free vibration analysis of thick elastic plate. The integral equations for moments and shear forces at an internal point [ are written as [15]

372


³ p (x, t )uDE ([ , x) d*(x) ³ u

M DE ([ , t )

*

k

k

*

[ , x) d*(x)

* k ( x, t ) pDEk (

Uh 3

*

³

12

³ uK (x, t ) uDEK ([ , x) d:(x) *

:

³

* * 3 (x, t ) zDE Uh u3 (x, t ) uDE ([ , x) d:(x) 3 ([ , x) d:(x) kUh u :

(8a)

:

and QE ([ , t )

³ p (x, t )u E ([ , x) d*(x) ³ u k

* 3 k

*

[ , x) d*(x)

* k ( x, t ) p3 Ek (

*

³

Uh 3 12

³ uK (x, t ) u EK ([ , x) d:(x) * 3

:

³

Uh u3 (x, t ) u3*E 3 ([ , x) d:(x) kUh u3 (x, t ) z3*E ([ , x) d:(x) :

(8b)

:

Where the tensors that appear with the asterisk in eqs. (7) and (8) represent the static fundamental solution, and were presented in Refs. [16, 17]. Numerical procedure Consider the boundary * discretized by isoparametric quadratic straight one-dimensional elements, each one denoted by * j , and the domain : discretized by constant triangular three nodes cell, the domain of a cell l being denoted by : l , according to in Fig. 1.

*j

:l :

x2

* x1

Figure 1: Domain discretized with boundary elements and internal cells. BEM guidelines consider boundary element and internal cell approximations as follows. Thus, displacements U ( j ) and surface forces P ( j ) within an element j are computed from its nodal values, U (n ) and P (n ) , according to the following approximations (9a) U ( j ) NU ( n) and P ( j ) NP ( n) while the translational inertia U (l ) at internal points are approximated by (l ) NU ( m) U (9b) By substituting (9) into (7) and from these resulting equations, we write the equations for all boundary nodes and for all cell nodes, with nodes being collocation points [ . Then, the following system of equations is formed: ª H bb 0 º U b ½ ªG bb 0 º P b ½ ª0 M bd º 0 ½ (10) « db »® ¾ « »® ¾ « »® d ¾ I ¼ ¯U d ¿ ¬G db 0 ¼ ¯ 0 ¿ ¬0 M dd ¼ ¯U ¿ ¬H The superscripts b and d in the above matrix equation correspond to the boundary and domain, respectively. Moreover, the first superscript corresponds to the source point, while the second concerns to the field point. The integrals over the boundary elements and internal cells are computed numerically, using Gaussian quadrature. In the case of singular integrals, special procedures can be used for the integrals in the boundary [18, 19], while that the finite part numerical quadrature is utilized for the integrals in the domain [20]. On the hypothesis of harmonic response, the displacement field is expressed as (11) u u~(x) sin(Z t ) where the tilde indicates amplitude and Z is the angular frequency of plate. Thus, substitution of equation (11) into equation (10) yields to the following eigenvalue problem: ~ ~ ~~ ~ ~ B X d (1 / Z 2 ) X d with B M dd A* A1 M bd (12) ~ B is a real matrix that is commonly sparse, non-symmetric and non-positive definite. This form, to evaluate the angular frequencies is necessary to use an iterative algorithm to solve the eigenvalue problem [21].


373

Numerical example In this section one example is presented to verify the influence of the additional translational inertia term in the formulation. For this, some analyses are made to various thicknesses of plates, where are considered four relations h / a equal to 0.001 , 0.01 , 0.1 and 0.2 . Consider a square plate with two edges simply supported and the other two edges free, with side length a 1.0 , mass density U 1.0 , modulus of elasticity E 1.0 and Poisson ratio Q 0.3 . Figure 2 shows the present problem that is discretized by 32 boundary elements mesh and 256 internal cells.

y

(a)

(b)

x Figure 2: Problem for analysis: (a) Square plate with two edges simply supported and the other two edges free, (b) Mesh with 32 boundary elements and 256 internal cells. In this work, the following symbolism SFSF will identify a square plate with the edges x 0 , y 0 , x a , y a having simply supported, free, simply supported, and free boundary conditions, respectively. Thus, boundary conditions must be satisfied for a simply supported edge, M x I y w 0 , and for a free edge, M xy

My

Qy

0.

To analysis of results, the non-dimensional frequency parameter is defined as follows: (13) ' Z a 2 Uh / D Table 1 presents the lowest eight frequency parameters with different thickness-side ratios h / a . The superscript ‘a’ indicates the consideration of additional translational inertia terms in the present formulation. It should be observed that for small values of the thickness-side ratio, the additional terms have no influence. While that for the others thickness-side ratios of plate, the responses already show some differences. Table 1: Comparison of frequency parameters ' for square plate with SFSF boundary conditions. h/a 0.001

0.01

0.1

0.2

Method Exact CPT [22] 2-D Ritz [23] Exact FSDT [24] Present Presenta [24-31] Present Presenta 3-D DQM [32] 3-D Ritz [33] Exact FSDT [24] Present Presenta 3-D DQM [32] 3-D Ritz [33] Exact FSDT [24] Present Presenta

Frequency parameters

'1

'2

'3

'4

'5

'6

'7

'8

9.6314 9.6327 9.6311 9.6428 9.6428 9.6270 9.6314 9.6314 9.4460 9.4462 9.4458 9.4484 9.4510 9.0011 9.0010 8.9997 8.9905 8.9995

16.1348 16.1368 16.1313 16.1623 16.1624 16.0971 16.1360 16.1360 15.3998 15.3995 15.4054 15.4194 15.4239 14.1229 14.1224 14.1341 14.1241 14.1372

36.7256 36.7248 36.7161 36.9377 36.9377 36.6112 36.8520 36.8523 33.9113 33.9129 33.9160 34.0643 34.0887 29.2632 29.2634 29.2558 29.2971 29.3592

38.9450 38.9455 38.9433 39.1610 39.1610 38.9043 39.0071 39.0076 36.4369 36.4376 36.4246 36.4670 36.5066 31.4724 31.4722 31.4338 31.3656 31.4672

46.7381 46.7326 46.7317 47.1038 47.1038 46.6393 46.8268 46.8274 42.8868 42.8874 42.8870 42.9758 43.0217 36.1734 36.1731 36.1646 36.1153 36.2206

70.7401 70.7355 70.7222 71.4096 71.4096 70.4846 71.0390 71.0403 62.3347 62.3374 62.3304 62.6232 62.7154 49.9345 49.9353 49.8953 49.9188 50.1042

75.2834 75.2853 75.2692 76.1792 76.1793 75.0554 76.0039 76.0054 66.4101 66.4096 66.3720 66.9421 67.0492 52.9119 52.8012 52.9793 53.2043

87.9867 87.9875 87.9819 89.2934 89.2934 87.8151 88.3262 88.3288 76.9730 76.9042 77.1451 77.3167 60.1064 60.0090 60.3416

374


Conclusions In this work, the BEM is employed to obtain the solutions for the free vibration analysis of thick square plates with two edges simply supported and the other two edges free. A formulation based on Reissner’s theory is used here, which includes the contribution of the additional translational inertia terms to the integral equation of displacements and internal forces. The boundary element method formulation used to discretize the space employs static fundamental solution. The numerical simulations carried out with the additional term considered by the present formulation modified results in comparison with results obtained without this term and its contribution in the analyses carried out here was more relevant for relation h / a 0.20 . This problem type is very important in the hydroelastic analysis of very large floating structures (VLFS) which are commonly modeled as plates with edges. Acknowledgments The authors are grateful for the financial support from CNPq and special grateful for the partnership Capes-FAPERJ, Pos-Doctorate Support Program in Rio de Janeiro–2009, registered under nº. 10/2009. References [1] I.H. Shames, C.L. Dym. Energy and finite element methods in structural mechanics. US: McGraw-Hill (1985). [2] S.P. Timoshenko, S. Woinowsky-Krieger. Theory of plates and shells. New York: McGraw-Hill (1959). [3] E. Reissner. On the theory of bending of elastic plates. Journal of Mathematics and Physics, 23: 184-191 (1944). [4] E. Reissner. The effect of transverse-shear deformation on the bending of elastic plates. Journal of Applied Mechanics, 12: 69-77 (1945). [5] E. Reissner. On bending of elastic plates. Quarterly of Applied Mathematics, 5: 55-68 (1947). [6] R.D. Mindlin. Influence of rotatory inertial and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics, 18: 1031-1036 (1951). [7] R.D. Mindlin, A. Schacknow, H. Deresiewicz. Flexural vibrations of rectangular plates. Journal of Applied Mechanics, 23: 430-436 (1956). [8] C.M. Wang, Y.C. Wang, E. Watanabe, T. Utsunomiya, Y. Xiang. Obtaining accurate modal stressresultants in freely vibrating plates that model VLSF. In: Proceedings of the Eleventh International Offshore and Polar Engineering Conference, Stavanger, Norway (2001). [9] E. Watanabe, T. Utsunomiya, C.M. Wang, Y. Xiang. Hydroelastic analysis of pontoon-type circular VLSF. In: Proceedings of the Thirteenth International Offshore and Polar Engineering Conference, Honolulu, Hawaii (2003). [10] E. Watanabe, T. Utsunomiya, C.M. Wang. Hydroelastic analysis of pontoon-type VLSF: a literature survey. Engineering Structures, 26: 245-256 (2004). [11] X.J. Chen, Y.S. Wu, W.C. Cui, J.J. Jensen. Review of hydroelasticity theories for global response of marine structures. Ocean Engineering, 33: 439-457 (2006). [12] M. Batista. Refined Mindlin-Reissner theory of forced vibrations of shear deformable plates. Engineering Structures, 33: 265-272 (2011). [13] W.L.A. Pereira, V.J. Karam, J.A.M. Carrer, W.J. Mansur. A dynamic formulation for the analysis of thick elastic plates by the boundary element method. Engineering Analysis with Boundary Elements, 36: 1138-1150 (2012). [14] W.L.A. Pereira, W.J. Mansur, V.J. Karam, J.A.M. Carrer. A formulation for free vibration analysis of thick elastic plates by the boundary element method. In: Proceedings of the XXXII Ibero-Latin American congress of computational methods and engineering (XXXII CILAMCE), Brazil (2011). [15] W.L.A. Pereira. A general formulation for dynamic analysis of thick plates by the boundary element method. D.Sc. thesis (in Portuguese), RJ, Brazil: COPPE/UFRJ (2009). [16] F. van der Weeën. Application of the boundary integral equation method to Reissner’s plate model. International Journal for Numerical Methods in Engineering, 18: 1-10 (1982). [17] V.J. Karam, J.C.F. Telles. On boundary elements for Reissner’s plate theory. Engineering Analysis, 5: 21-27 (1988). [18] J.C.F. Telles. A self-adaptive coordinate transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical Methods in Engineering, 24: 959-973 (1987). [19] V.J. Karam. Plate bending analysis by the BEM including physical nonlinearity. D.Sc. thesis (in Portuguese), RJ, Brazil: COPPE/UFRJ (1992).


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[20] H.R. Kutt. Quadrature formulae for finite part integrals. Report Wisk 178, The National Research Institute for Mathematical Sciences, Pretoria (1975). [21] B.T. Smith, J.M. Boyle, J.J. Dongarra, B.S. Garbow, Y. Ikebe, V.C. Klema, C.B. Moler. Matrix Eigensystem Routines. EISPACK Guide, Berlin, Springer-Verlag (1976). [22] A.W. Leissa. The free vibration of rectangular plates. Journal of Sound and Vibration, 31: 257-293 (1973). [23] K.M. Liew, K.C. Hung, M.K. Lim. Vibration of Mindlin plates using boundary characteristic orthogonal polynomials. Journal of Sound and Vibration, 182: 77-90 (1995). [24] S. Hosseini-Hashemi, M. Arsanjani. Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates. International Journal of Solids and Structures, 42: 819-853 (2005). [25] Y.K. Cheung, D. Zhou. Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions. Computers and Structures, 78: 757-768 (2000). [26] Y.K. Cheung, D. Zhou. Vibration of tapered Mindlin plates in terms of static Timoshenko beam functions. Journal of Sound and Vibration, 260: 693-709 (2003). [27] Y.K. Cheung, S. Chakrabarti. Free vibration of thick layered rectangular plates by a finite layer method. Journal of Sound and Vibration, 21: 277-284 (1972). [28] O.L. Roufael, D.J. Dawe. Vibration analysis of rectangular Mindlin plates by the finite strip method. Computers and Structures, 12: 833-842 (1980). [29] D.J. Dawe, S. Wang. Vibration of shear-deformable beams, plates using spline representations of deflection, shear strains. International Journal of Mechanical Sciences, 36: 469-481 (1994). [30] D.J. Gorman. Accurate free vibration analysis of clamped Mindlin plates using the method of superposition. Journal of Sound and Vibration, 198: 341-353 (1996). [31] J.L. Doong. Vibration and stability of an initially stressed thick plate according to a higher order deformation theory. Journal of Sound and Vibration, 113: 425-440 (1987). [32] K.M. Liew, K.C. Hung, M.K. Lim. A continuum three-dimensional vibration analysis of thick rectangular plates. International Journal of Solids and Structures, 30: 3357-3379 (1993). [33] M. Malik, C.W. Bert. Three-dimensional elasticity solutions for free vibrations of rectangular plates by the differential quadrature method. International Journal of Solids and Structures, 35: 299-318 (1998).

376


Numerical Simulation of Heat and Mass Transport Sevin Gümgüm Department of Mathematics, I˙zmir University of Economics, I˙zmir, Turkey e-mail:[email protected]. Keywords: DRBEM, mixed convection, Heat Transport, Mass Transport

Abstract. This study presents the numerical investigation of unsteady mixed convection in a square lid-driven cavity considering the eects of heat and mass diusion. The Dual Boundary Element Method (DRBEM) is used to solve the governing equations and the time derivative is approximated by Finite Dierence Scheme (FDM). Numerical solutions are obtained for several values of Lewis number (Le), and buoyancy ratio parameter (N). Introduction Combined buoyancy eects of thermal and mass diusion occurs at many transport processes in engineering applications such as spray and ash drying, combustion of atomized liquid fuels, cyclone evaporation and drying and dehydration operations in chemical and food processing plants, crystal growth, material and separation processes [1]. These processes are generally governed by the combined eect of natural and forced convection. Numerical simulation of combined thermal and mass transport in a square lid-driven cavity is studied by Al-Amiri et al [1]. The transport equations are solved using the Galerkin weighted residual method. They observed that heat and mass transfer characteristics are enhanced for low values of Richardson number. In another study Teamah and El-Maghlany [2] investigated the mixed convection in a rectangular cavity under the combined buoyancy of heat and mass diusion. The governing equations are solved by Finite Volume Method (FVM). They observed that both heat and mass transfer increased as the Richardson number is decreased. They also stated that an increase in the Lewis number increases the mass transfer, but no signicant eect on the heat transfer. In this study, DRBEM is used to discretize the spatial derivatives in the stream functionvorticity form of the Navier-Stokes equations, energy and the concentration equations. DRBEM idea is applied to the Laplace operator in each equation by using the fundamental solution of Laplace equation and keeping all the other terms as nonhomogeneity. The resulting matrices contain integrals of logarithmic function or its normal derivative. The DRBEM reduces all calculations to the evaluation of the boundary integrals only. This fact might be advantageous in geometrically involved situations that are frequently encountered in uid ow problems. DRBEM application yields a system of initial value problems in time which are solved by the nite dierence scheme. Formulation of the Problem The nondimensional form of the governing equations of unsteady mixed convection ow of


377

heat and mass transfer is given as in [1] ∇2 ψ = −ω ! ∂T ∂ω ∂ω ∂C " ∂ω 1 2 ∇ω= +u +v − Ri +N Re ∂t ∂x ∂y ∂x ∂x

∂T ∂T ∂T 1 ∇2 T = +u +v ReP r ∂t ∂x ∂y ∂C ∂C ∂C 1 ∇2 C = +u +v ReSc ∂t ∂x ∂y

where (x, y) ∈ Ω ⊂ R2 , t > 0. Ri, Re, P r and N are the Richardson number, Reynolds , u = − ∂ψ and number, Prandtl number, and buoyancy ratio number, respectively. u = ∂ψ ∂y ∂x ∂v − ∂u ) . The initial and the boundary conditions are taken as ω = ( ∂x ∂y ω=T =C=0

when t = 0

y=0:

0 ≤ x ≤ 1,

u = v = 0,

y=1:

0 ≤ x ≤ 1,

u = 1,

x = 0, 1 :

0 ≤ y ≤ 1,

u = v = 0,

T = 1,

C=1

v = 0,

T = 0, ∂T /∂x = 0,

C=0 ∂C/∂x = 0 .

The unknown vorticity boundary conditions are derived from its denition using DRBEM coordinate matrix. The buoyancy ratio number N is the ratio of the buoyancy forces due to GR

C , where GrC the concentration gradients to temperature gradients and dened as N = GRT is the Grashof number due to mass diusion, and GrT is the Grashof number due to thermal diusion. N > 0 represents aiding ow and N < 0 presents opposing ows [1].

Numerical Solution The equations in (1) are weighted through the domain Ω as in [3], by the fundamental 1

1

ln of Laplace equation in two dimensions in which r is the distance between solution u∗ = 2π r the source and the xed points. Applying Green's second identity, we have the following integral equations for each source point i:

ci ψ i +

Γ

(ψq ∗ ψ − ψ ∗ ψq )dΓ =

1 ci ωi + Re

Γ

Ω

(ωq ∗ ω − ω ∗ ωq )dΓ =

Ω

(

! " ∂ω ∂ω ∂T ∂ω +u +v − Ri ω ∗ dΩ ∂t ∂x ∂y ∂x + N ∂C ∂x

∂T ∂T ∗ ∂T 1 (Tq ∗ T − T ∗ Tq )dΓ = ( +u +v )T dΩ ReP r ∂t ∂x ∂y Γ Ω ∂C ∂C ∗ 1 ∂C 1 ci Ci + +u +v )N dΩ (Cq ∗ C − C ∗ Cq )dΓ = ( ReSc ReSc Γ ∂x ∂y Ω ∂t 1 ci Ti + ReP r

(−ω)ψ ∗ dΩ

!

378


where the subscript q indicates the normal derivative of the related function and ci = θi/2π with the internal angle θi at the source point i. One can expand the nonhomogeneties in each equation in terms of the radial basis functions fj which are are linked to the particular solutions of each equation with the Laplace operator. Then, substituting these expansions in Eq. (3) and the application of Green's second identity to the right hand sides will result in matrix vector equations for each unknown ψ, ω, T and C . Hψ − Gψq = (H ψˆ − Gψˆq )α 1 (Hω − Gωq ) = (H ω ˆ − Gωˆq )¯ α Re "

1 (HT − GTq ) = (H Tˆ − GTˆq )˜ α ReP r

1 (HC − GCq ) = (H Cˆ − GCˆq )˘ α ReSc where αj , α¯j , α˘j and α˜j are undetermined coecients, and G and H are (M + L) × (M + L) matrices dened as in [3]. The matrices ψˆ, ω ˆ , Tˆ and Cˆ are constructed by taking the

corresponding particular solutions as columns. Evaluation of the right hand sides of each equation in (4) at all boundary and interior (M + L) points gives Hψ − Gψq = (H ψˆ − Gψˆq )F −1 {−ω} (

) ∂ω ∂ω ! ∂T ∂C " ∂ω +u + v Ri +N ∂t ∂x ∂y ∂x ∂x ( ) ∂T ∂T 1 ∂T (HT − GTq ) = (H Tˆ − GTˆq )F −1 +u +v ReP r ∂t ∂x ∂y ( ) ∂C ∂C 1 ∂C −1 ˆ ˆ (HC − GCq ) = (H C − GCq )F +u +v ReSc ∂t ∂x ∂y 1 (Hω − Gωq ) = (H ω ˆ − Gωˆq )F −1 Re

#

where F is the (M + L) × (M + L) matrix containing coordinate functions fj 's as columns. ∂F −1 = F R, where R denotes Space derivatives in Eqs. (5) are approximated with F as ∂R ∂x ∂x ω , ψ , T , and C . Substituting convection terms back into Eq. (5), and nally rearranging, we end up with the following system of ordinary dierential equations for ω, T and C respectively ˜ + Gω ˜ q − SF = 0 ω˙ − Hω ˜t T + G˜t Tq = 0 T˙ − H

$

C˙ − Hñ C + Gñ Cq − SF 1 = 0

and a linear system of equations for ψ Hψ − Gψq = −Sω.

For the derivatives of ω, T and C in Eq. (6) implicit central dierences are used assuming the previous two time level solutions are known.


379

Results and Discussion The no-slip boundary conditions of the velocities are assumed. The vertical walls are insulated, while the horizontal walls are isothermally heated or cooled. Reynolds number and Richardson number are xed at 100 and 0.01, respectively. The Prandtl number is taken as 1. The Lewis number is dened as Le = Sc/P r. Computations are carried out for N = −100, −25, 25, 100 with the time increment Δt = 0.8, and the number of linear boundary elements is M = 100. The eect of varying N on the streamlines and isotherms are presented in Fig. 1. When N = 0, the mass (concentration) transport phenomenon vanishes and the problem reduces to a pure thermal convection problem. The results show that as N increases thermal boundary layer formation starts along the heated bottom wall, and higher temperature and mass gradients in the vertical direction are observed. The buoyancy forces enhances energy transport since it acts in the same direction of the liddriven wall. On the other hand as N decreases the eect of the buoyancy force weakens. Thus, the main vortex breaks into two prime vortices. Fig. 2 presents the eect of Lewis number on the concentration, isotherms, streamlines and vorticity. For Le = 5 and 50, 100 and 120 linear boundary elements are used, respectively with time step Δt = 0.8. As the Lewis number increases thinner concentration boundary layers are observed along the bottom wall. It is also observed that the eect of Lewis number on the other function (streamlines, isotherms and vorticity) is insignicant. This can be attributed to the fact that the combined buoyancy eects are dominated by the lid-driven wall. In Fig. 3 the variation of the temperature and u-velocity prole at the mid-plane of the cavity is shown for Le = 1 and varying values of N . The u-velocity takes higher values when N is positive.This is because the buoyancy force is assisting the ow. Conclusion The mixed convective heat and mass transport in a lid-driven cavity is analyzed by DRBEM. The results illustrate that an increase in the Lewis number causes thinner mass boundary layers at the bottom wall of the cavity, but does not have any signicant eect on the other functions. Positive buoyancy force enhances the heat transfer inside the cavity. Negative buoyancy forces slow the main vortex and breaks up into two vortices.

References [1] A. M. Al-Amiri, K. M. Khanafer and I. Pop Numerical simulation of combined thermal and mass transport in a square lid-driven cavity, 46, 662-671 (2007). [2] M. A. Teamah and W. M. El-Maghlany Numerical simualtion of double-diusive mixed convective ow in rectangular enclosure with insulated moving lid, 49, 1625-1638 (2010). [3] P. W. Partridge, C. A. Brebbia and L. C. Wrobel The Dual Reciprocity Boundary Element Method, Comp. Mech. Pub., Southampton and Elsevier Sci., London, 1992.

380


STREAM FUNCTION 1

1

0.145 0.343

0.9

−0.08

0.8

STREAM FUNCTION

TEMPERATURE 1

−0.02 −0.04 −0.06

0.9

TEMPERATURE

0.7

1

−0.02 −0.04 −0.06

0.9 0.0459

0.8

−0.1

0.7

0.151

0.9 0.346

−0.08

0.8

0.8

0.7

0.7 −0.07

0.6

0.6

−0.09

0.5 0.4

0.6

−0.05

−0.03

0.3

0.5

0.4

0.4

0.3

0.1 0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

0.6

0.8

1

0.2

0.1

0.1

−0.02

0

0

1

0

0.2

0.4

0.6

0.8

0.0403

0.7

0.7

0.7

−0.13

0.6

0.6 −0.11

0.639

0

0.834

0.2

0.4

0.6

0.8

1

0.5 0.4

−0.08

0.4

0.3

−0.06

0.3

0.161

0.2

0.1

0.1

0

0

1

−0.06 −0.04

0.6

−0.02

0.0675

0.5

−0.007

0.536 0.255

0.4 0.3

0.2 0.44 0.64

0

0.349

0.7

0.442

0.723

0.24

0.74

0.2

0.4

0.6

0.8

1

0

0.63

0.2 −0.007 −0.005 −0.0035 −0.004−0.003

0.1 0.84

0.94 0.8

0.8

0.3

−0.04 0.2

0.6

1

−0.03 −0.05

0.9

0.6

0.54

0.5

0.4

0.4

0.443

−0.07 0.8

0.2

1

0.9

0.8

0

0.01

1

0.14 0.34

0.9

−0.09

0.8

0.5

0.248

0.931

N = −25

0.01 −0.03 −0.05 −0.07

1

0.3

−0.01

0.2

N = 25

0.9

0.736

−0.005

0.442 0.64 0.838

0.937

0.5

−0.03

0.4

0.244

0.739 0.2

−0.05

0.3

−0.01

0.1

0.0531

0.541

0.5

0.2

0

0.6

0.541

−0.07

0

0.2

0.4

0.6

0.8

N = 100

1

0

0.817

0.911

0.1 0

0.2

0.4

0.6

0.8

1

N = −100

Figure 1: Eect of buoyancy force N on the streamlines and isotherms for Re = 100, Ri = 0.01, Le = 1

STREAM FUNCTION 1

0.8

1

0.147

0.9

−0.0871

0.6

0.6

−0.0782 −0.0605

0.5

1 0.9

0.355

−4 −2

3

−5

1

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

25 3

−3 −1

0.0485

0.7

−

0.18

0.9

0.344

0.8

0.7

VORTICITY

CONCENTRATION

TEMPERATURE 0.0162 −0.0339 −0.0516 −0.0694

1 0.9

4

0.541 0.5

−0.0428

0.531

0.4

0.4

0.4

1

0.4

−0.0251 0.3

0.3

0.2

0.2

−0.00735

0.1 0

0.2

0.4

0.6

0.8

1

0

0

0.707

0.64 0.837

0.2

0.4

0.6

0.3 2 0.0915

0.2

0.443

0.1 0.935

0

0.3

0.246

0.738

0.1

0.8

1

0

0.2

0.2 0.267 0.443 0.619 0.795 0.971

1.15 0.883 1.06 0

0.4

0.6

0.8

0.1 1 1

0

0

2

0.2

0.4

0.6

0.8

1

Le = 5

0.0152 −0.0336 −0.052 −0.0703

1 0.9

1

1 −0.000501 0.0245 0.037

0.146

0.9

0.343

0.8

0.7

0.6

−0.0795

0.5

−0.0612

0.4

−0.0428−0.0244

0.54

0.5 0.4

0.3

0.3

0.244

0.737 −0.00601

0.2

0.2

0.1

0.442 0.639 0.836

0.1 0.934

0

0

0.2

0.4

0.6

0.8

1 4 0.9

0.8

0.8

1

0

0

0.2

0.4

0.6

0.8

−4 −2

1

−5 −3 −1

0.0473

0.7

0.6

0.0−−−

0.012

0.9

3

−0.0887

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1 1

0

0.0621 0.125 0.15 0.162 0.137 0.112 0.0871 0

0.2

0.4

0.6

0.0370.0495 0.0746 0.0996 0.8

3 24

5

1

2

0.1 1 1

0

0

2 0.2

0.4

0.6

0.8

1

Le = 50

Figure 2: Eect of Lewis number Re = 100, Ri = 0.01, N = 1

Le

on the streamlines, isotherms, concentration and vorticity for


1

1 N=100 N=50 N=25 N=−25 N=−50 N=−100

0.9 0.8 0.7

0.9 0.8 0.7

0.6

0.6

0.5

0.5

Y

Y

381

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.2

0.4 0.6 TEMPERATURE

0.8

1

0 −0.4

N=−100 N=−50 N=−25 N=100 N=50 N=25 −0.2

0

0.2

0.4

0.6

0.8

1

U

Figure 3: Eect of N on the temperature and u-velocity along the mid-plane of the cavity for Re = 100, Ri = 0.01, Le = 1

382


The Method of Fundamental Solutions of MHD Pipe Flow in an Exterior Region S. Han Aydın

∗

Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey Keywords: Fundamental solutions method, MHD pipe flow, Conducting exterior medium.

Abstract. In this study, an application of the fundamental solutions method is presented for the numerical solution of the magnetohydrodynamic (MHD) flow through a circular pipe under the influence of a transverse magnetic field when the outside medium is also electrically conducting. The coupled equations are transformed into homogenous modified Helmholtz equations, and Laplace equation for 1 the interior and exterior mediums, respectively. The fundamental solutions K0 (r) for the modified 2π 1 log(r) for the Laplace equation are used for the evaluation of the corHelmholtz equation and 2π responding Dirichlet and Neumann types coupled boundary conditions in terms of the fundamental solutions. Proposed numerical scheme is computationally very cheap and efficient corresponding to other numerical methods since there is no need for mesh discretization or integral evaluation. Computations are carried for several values of Reynolds number Re, magnetic pressure Rh of the fluid, and magnetic Reynolds numbers Rm1 and Rm2 of the fluid and the outside medium, respectively. Obtained numerical solutions display well known characteristics of the MHD flow.

Introduction The objective of this work is to present MHD pipe flow of circular cross-section under the influence of a transverse magnetic field numerically by using fundamental solutions method when the outside medium is also electrically conducting. It is already known that there are many applications of MHD pipe flow such as the design of the cooling systems with liquid metals for nuclear reactors, electromagnetic pumps, MHD generators, and flowmeters measuring blood pressure, etc. The exact solution of the problem can be obtained only for some special cases [1, 2]. The fundamental solutions method (FSM) is a mesh-free, very efficient and computationally cheap numerical technique. Therefore there are many papers used FSM for the solution of many physical and engineering problems (see [3, 4, 5] and references therein). Generally, the method is applicable for homogeneous differential equations where it’s fundamental solutions is known. There are also some special studies for the non-homogeneous equations [3, 5]. FSM is based on the selection of the points on the boundary of the domain called source points and selection of the nodes called collocation points on the artificial boundary which is just behind the real boundary of the problem domain. The convergence of the method is already proved for the Laplace [4] and modified Helmholtz equations [5]. ∗

Electronic address: [email protected]; 1


383

Figure 1: Problem definition of the circular pipe We have considered the general MHD problem inside of a pipe when the surrounding medium is also electrically conducting, and has small magnetization compared to the fluid inside the pipe. The flow is assumed as incompressible and viscous, and the fluid inside the pipe is electrically conducting. The axis of the pipe is coincident with the z -axis, and the y -axis is parallel with the magnetic induction at infinity. The externally applied magnetic field with an intensity B0 is assumed to be in y -direction. The outside medium has also electrical conductivity which is small compared to fluid. However, the continuity of both induced magnetic fields inside and outside of the pipe is maintained through the conducting wall of the pipe (see Fig.1). Therefore, this problem is a generalization of the special cases of MHD flow through a pipe. Thus, we consider in this paper, the most general case of MHD flow in a pipe which has electrically conducting wall with the same conductivity of outside infinite medium.

Mathematical Model The mathematical modelling of the MHD flow considered above is expressed with the partial differential equation in non-dimensional form in inside the pipe and its exterior as [6, 7]

∇2 V (x, y) + ReRh

∂B (x, y) = −1 ∂y

∂V (x, y) = ∇ B(x, y) + Rm1 ∂y 2

∇2 Bext (x, y) =

in Ωin

(1)

0 0

in Ωext

(2)

with the no-slip condition on the pipe wall

V =0

on ∂Ωin = Γ

(3)

and continuity conditions for the induced magnetic fields

B(x, y) = Bext (x, y) , 1 ∂B(x, y) 1 ∂Bext (x, y) , = Rm1 ∂n Rm2 ∂n Bext (x, y) = 0 lim 2

x2 +y →∞

on Γ on Γ

(4) (5) (6)

where n and n are unit outward normals on Γ for the regions Ωin and Ωext , respectively. Rm1 and Rm2 are the magnetic Reynolds numbers inside the pipe and in external medium. In order to apply

2

384


FSM, eq(1) need to be transformed to the form where the fundamental solution is known. √ Let’s start with by denoting V1 = V and B1 = ReRh ReRhRm1 is the Hartmann M B , where M = number of the fluid. Then the eq(1) is written as [1]

∇2 V 1 + M

∂B1 = −1 ∂y

∇ 2 B1 + M

∂V1 = 0. ∂y

in Ωin

(7)

Then, applying the transformations

U1 = V1 + B1

,

U2 = V1 − B1

1 My

,

W2 = U 2 −

W1 = U 1 +

u1 = W1 eky

,

1 My

(8)

u2 = W2 e−ky

sequently as done in [7], eq(7) are reduced to two homogeneous modified Helmholtz equations as

∇2 u 1 − k 2 u 1 = 0 in Ωin

(9)

∇2 u2 − k 2 u2 = 0 where k = M/2 and the boundary conditions (4)-(6) in terms of new variables are [7]

e−ky u1 + eky u2 = 0 2ReRh 2 Bext + y e−ky u1 − eky u2 = M M 2 ∂y ∂u2 2M ∂Bext ∂u1 + − eky = . e−ky ∂n ∂n Rm2 ∂n M ∂n

(10) (11) (12)

The Method of Fundamental Solutions In order to solve eqs(9) and (2) with FSM, define the unknowns u1 , u2 and Bext in terms of the fundamental solutions of the modified Helmholtz and Laplace equations as

u1 =

N 1 k=1

2π

cu1 k K0 (|x, yk |), u2 =

N 1 k=1

2π

cu2 k K0 (|x, yk |), and Bext

N 1 k=1

2π

ext ln(|x, yk |) cB k

Bext u2 where cu1 are the unknown constants to be determined, x is the source point, yk ’s are k , ck and ck the collation points and |x, yk | is the distance between x and yk . Selecting N number of source points (xi ) on the boundary, gives the following linear algebraic equations as

e−kyi e−kyi

N

kyi cu1 k K0 (|xi , yk |) + e

k=1 N

k=1

ky cu1 k K0 (|xi , yk |) − e

N k=1 N k=1

cu2 k K0 (|xi , yk |) = 0 cu2 k K0 (|xi , yk |) =

3

N 2ReRh 4π cBext ln(|xi , yk |) + yi M k=1 k M

(13) (14)

Advances in Boundary Element Techniques XIV e−kyi

N k=1

cu1 k

N

∂K0 (|xi , yk |) ∂K0 (|xi , yk |) cu2 − ekyi k ∂n ∂n k=1

385 = −

2M Rm2

N k=1

ext cB k

∂ ln(|xi , yk |) 4π ∂yi + (.15) ∂n M ∂n

Bext u2 After calculating the unknown coefficients cu1 , unknowns u1 , u2 and Bext , and so using k , ck and ck back-transformations the original unknowns V, B and Bext can be calculated at any point on the domain.

Numerical Results We consider a long pipe of circle cross-section defined by {(x, y) : x2 + y 2 ≤ 1}. On the boundary we have taken N = 64 discretization points and same number of corresponding collocation points. The behaviors of the velocity of the fluid and inside and outside induced currents (induced magnetic fields) are visualized in terms of contour plots for very high values of magnetic Reynolds numbers Rm1 , Reynolds number Re and magnetic pressure number Rh of the fluid. The accuracy of the FSM is displayed in terms of the maximum values of the velocity and induced currents by comparing with the Reference solution [7] in Table 1. Fig. (2) shows equal velocity and induced current lines, respectively, for increasing values of Rm1 = 10, 100, 500 when Re = 1, Rm2 = 1 and Rh = 10. A boundary layer formation which is a well known behavior of the MHD flow is seen for the large values of the Hartmann number (M ). Also, flow becomes stagnant and reaches its maximum velocity value at the center of the pipe. The similar behavior is also seen for the different values of the Reynolds number Re = 1, 10, 100 from Fig. (3). The effect of magnetic pressure of the fluid on the velocity and induced current of the fluid and outside induced current is visualized in Fig. (4). Induced current of the fluid tries to close itself inside the pipe due to the high magnetic pressure as in the case of insulated pipe wall and boundary layer formation behaviour is also observed as Rh getting large.

Parameters Rm1 Re Rh 10 1 10 100 1 10 500 1 10 10 10 10 10 100 10 10 1 5 10 1 20

Velocity FSM Ref [7] 0.053 0.056 0.023 0.024 0.0105 0.011 0.009 0.008 0.0013 0.0011 0.0808 0.087 0.033 0.034

Induced Current FSM Ref 0.065 0.064 0.084 0.083 0.092 0.091 0.008 0.008 0.0009 0.0009 0.1 0.1 0.037 0.037

Table 1: Comparison of maximum values of the velocity and induced current

Summary We consider a fundamental solution method approach for the approximate solution of the MHD flow through a circular pipe under the influence of a transverse magnetic field when the outside medium is also electrically conducting. Coupled equations with coupled boundary conditions for the pipe region are transformed to the modified Helmholtz equations. Using the fundamental solutions of the modified

4

Eds: A Sellier & M H Aliabadi V

V

V

0.053 0.042 0.033 0.022 0.011 0.001

0.023 0.018 0.014 0.009 0.005 0.001

0.0105 0.0084 0.0063 0.0042 0.0021 0.0001

0

0 X

Y

0

Y

Y

386

0

0 X

0 X

(a) Velocity B

-3

0 X

-3

0 X

0

-3

3

0.092 0.035 0.005 0 -0.005 -0.035 -0.092

3

Y

0

-3

3

B

0.084 0.045 0.009 0 -0.009 -0.045 -0.084

3

Y

Y

0

-3

B

0.065 0.043 0.021 0 -0.021 -0.043 -0.065

3

-3

0 X

3

(b) Induced currents

V

V

V

0.053 0.042 0.033 0.022 0.011 0.001

0.0099 0.0079 0.0059 0.0041 0.0021 0.0001

0.0013 0.0011 0.0008 0.0005 0.0003 0.0001

0

0 X

Y

0

Y

Y

Figure 2: Velocity of the fluid and induced currents for Rm1 = 10(left), Rm1 = 100(center) and Rm1 = 500(right) and Rm2 = 1, Re = 1, Rh = 10

0

0 X

0 X

(a) Velocity B

-3

0 X

3

-3

0 X

3

0.0009 0.0006 0.0003 0 -0.0003 -0.0006 -0.0009

3

Y

0

-3

B

0.0088 0.0058 0.0029 0 -0.0029 -0.0058 -0.0088

3

Y

Y

0

-3

B

0.065 0.043 0.021 0 -0.021 -0.043 -0.065

3

0

-3

-3

0 X

3


Figure 3: Velocity of the fluid and induced currents for Re = 1(left), Re = 10(center) and Re = 100(right) and Rm1 = 10, Rm2 = 1, Rh = 10

5

387

V

V

V

0.0808 0.0646 0.0485 0.0323 0.0162 0.0001

0.053 0.042 0.033 0.022 0.011 0.001

0.033 0.026 0.021 0.013 0.007 0.001

0

0 X

Y

0

Y

Y


0

0 X

0 X

(a) Velocity B

-3

0 X

3

-3

0 X

3

0.037 0.024 0.012 0 -0.012 -0.024 -0.037

3

Y

0

-3

B

0.065 0.043 0.021 0 -0.021 -0.043 -0.065

3

Y

Y

0

-3

B

0.108 0.072 0.036 0 -0.036 -0.072 -0.108

3

0

-3

-3

0 X

3


Figure 4: Velocity of the fluid and induced currents for Rh = 5(left), Rh = 10(center) and Rh = 20(right) and Rm1 = 10, Rm2 = 1, Re = 1 Helmholtz equation and Laplace equation, unknowns are expressed in terms of the linear combinations of the fundamental solution with constant coefficients which are calculated from the solution of the corresponding linear system of equations. Obtained solutions shows the accuracy and efficiency of the proposed numerical scheme for the high values of the Hartmann number.

References [1] L. Drago¸s Magnetofluid Dynamics, Abacus Pres (1975). [2] J.A. Shercliff J. Fluid Mech, 1(6), 644–666 (1956). [3] M.A. Golberg Engrg. Analy. Bound. Elem., 16 205–213 (1995). [4] M. Katsurada, H. Okamoto Computers Math. Applic 31(1) 123–137 (1996). [5] X. Li Appl. Math. Comput. 159 113–125 (2004). [6] A. Carabineanu, E. Lungu Int J Numer Methods Eng, 68(2), 173–191 (2006). [7] M. Tezer-Sezgin, S. Han Aydın Computing, DOI 10.1007/s00607-012-0270-4.

6

388


Quadratic B-Splines in the Analog Equation Method for the Nonuniform Torsional Problem of Bars E.J.Sapountzakis1 and I.N.Tsiptsis2 1,2

School of Civil Engineering, National Technical University, Zografou Campus,GR-157 80 Athens, Greece, emails: [email protected], [email protected]

Keywords: analog equation method, boundary element method, nonuniform torsion, b-splines

Abstract. In this paper, the Analog Equation Method (AEM), a boundary element based method, is employed for the nonuniform torsional problem of bars of arbitrary constant cross section, considering a quadratic b-spline approximation for the fictitious loads of a substitute problem. The fictitious loads are established using a BEM-based technique and the solution of the original problem is obtained from the integral representation of the solution of the substitute problem. The bar is subjected to arbitrarily distributed twisting moments along its length, while its edges are subjected to the most general torsional (twisting and warping) boundary conditions. The problem is numerically solved introducing a quadratic b-spline function for the fictitious load in the integral representations of the aforementioned technique. Numerical results are worked out to illustrate the method, designate its efficiency, accuracy and less computational cost while verify its integrity comparing with the results of analytical solutions. 1. Introduction Boundary Element Methods (BEM) [1], which implement integral equations, are the most contemporary numerical methods for solving boundary value problems. A BEM approach uses in-line elements for discretisation, instead of area elements used in Finite Element Methods or Finite Differential Methods resulting to a small number of elements required to achieve high accuracy. Remodelling to reflect design changes becomes simpler. However, BEM, such as other numerical methods, is not free of drawbacks. Particularly, application of BEM requires the so-called fundamental solution. A promising technique that overcomes these drawbacks is the Analog Equation Method, developed by Katsikadelis [2,3]. AEM constitutes a numerical method for solving linear and nonlinear boundary value problems (elliptical, parabolic and hyperbolic) with linear or nonlinear boundary conditions. This method is based on BEM while improves it and eliminates its drawbacks. According to AEM, the real problem, which is described by a differential operator not reversed in practice, is transformed to an equal problem which is described by a linear differential operator of the same order with known fundamental solution and integral representation. In the substitute problem, the geometry of the space under consideration and boundary conditions are preserved while the non-homogenous terms of the linear operator stand for fictitious loads. Fictitious loads are computed through the numerical implementation of AEM which leads to a system of linear or nonlinear algebraic equations. In this paper, AEM is presented in a general form for one-dimension boundary problems described by fourth-order differential equations. The problem of nonuniform torsion of a homogeneous isotropic bar is reduced to solving the fourth-order differential equation with respect to the angle of twist of the cross section. The bar is subjected to an arbitrarily distributed twisting moment while its edges are restrained by the most general linear torsional boundary conditions. The essential features and novel aspects of the present formulation of AEM compared with previous ones are summarized as follows. The method used is based on quadratic b-splines, that is, piecewise quadratic polynomials with C1 continuity (lowest-degree polynomial representing a planar curve), and the collocation discretisation methodology with the points of a uniform partition being the collocation points. B-splines have been only sparsely used in finite element analyses (FEM) and boundary element methods (BEM). However, lately integrated computer aided design (CAD) and finite element analyses (FEA) using b-splines gained greater insight with the introduction of NURBS (NonUniform Rational B-splines) by Hughes et al. [4]. Thus, an introduction of b-splines in a BEM-based numerical technique is a natural starting point for the introduction of Isogeometric Analysis in the numerical solution of generalized beam theories with BEM. The most important property of b-splines in general is that both continuity and local controllability can be achieved by their use. Local controllability in simple words is the ability of b-splines to change only a portion of a curve when a single point is moved. The introduction of the quadratic b-spline to replace the approximation of fictitious loads with constant values and its integration to the expressions of the AEM technique improves accuracy and reduces nodal points required for


389

discretisation. Unknown values of the problem are reduced, too. The dimensions of matrices used for the numerical implementation of AEM become smaller and less algebraic equations are required to compute fictitious loads. The employed b-spline is a special class of b-splines called uniform quadratic b-spline. As the name implies, parametric quadratic polynomials are used on a uniform knot sequence, which is called the knot vector, composed of successive integers equally spaced (linear elements of the same length used for discretisation). Three control points have been used to represent the b-spline which is the minimum number that can be used for a quadratic b-spline [5]. The computation of fictitious loads at collocation points depends now on the calculation of the three control points. 2. Statement of the problem-Integral Representations Consider a prismatic bar of length l with a cross section of arbitrary shape, occupying the two dimensional multiply connected region : of the y, z plane bounded by the K+1 curves *0 ,*1 ,*2 ,...,* K as shown in Fig.1.

z

y

tz

n

0 ty

Ε, G

t

mt

mt x

θ(x)

ZS

x

a=0

Y

Ο x

b=L *

y y

Y

K 1 j 1

ΓΚ

Ρ

S

yS

v

ω

S

C

w ω

z

zS

L

Ρc

Z z

s

0

*j

Γ2

Γ1 (Ω) ΓΚ+1

(a) (b) Fig.1. Prismatic bar subjected to a twisting moment (a) with a cross section of arbitrary shape occupying the two dimensional region : (b). When the bar is subjected to the arbitrarily distributed twisting moment mt governed by the following boundary value problem according to [6]. d 4θ x ( x ) d 2θ x ( x ) GI t mt 4 dx dx 2 dT E2 M b D1θx ( x) D2 M t D3 , E1 dx

along the beam

ECM

E3

at the beam ends z 0,l

mt ( x) its angle of twist is

(1) (2a,b)

where E , G are the modulus of elasticity and the shear modulus of the isotropic material of the bar; CM and I t are the warping and torsional constants of the beam’s cross section, respectively. Moreover, dθx ( x) / dx denotes the rate of change of the angle of twist θx ( x) and it can be regarded as the torsional curvature, while M t is the twisting moment and M b is the warping moment due to the torsional curvature at the boundary of the beam. The boundary conditions (2a,b) are the most general linear torsional boundary conditions for the beam problem including also the elastic support. It is apparent that all types of the conventional torsional boundary conditions (clamped, simply supported, free or guided edge) can be derived from these equations by specifying appropriately the functions ai and E i (e.g. for a clamped edge it is a1 E1 1 , a2 a3 E2 E3 0 ). The solution of the boundary value problem given from eqs (1), (2a,b), which represents the nonuniform torsion of bars, presumes the evaluation of the warping and torsion constants CM and I t , respectively, which are given as

390


³M

CM

:

P2 M

d:

It

§ 2 wM P wM P · y z 2 y M z M ¸d : ¨ : wz wy ¹ ©

³

(3a,b)

where MMP ( y, z ) is the primary warping function with respect to the shear center M of the cross section of the bar, which can be established by solving independently the Neumann problem 2MMP

0 in : ,

wIMP wn

zny ynz on *

(4),(5)

where 2 w 2 / wy 2 w 2 / wz 2 is the Laplace operator; w / wn denotes the directional derivative normal to the boundary * and n y , nz the direction cosines. 3. Numerical Solution The evaluation of the angle of twist θx ( x) is accomplished using AEM [3]. According to this method, for the function θx ( x) , which is four times continuously differentiable along the beam and three times continuously differentiable at the beam ends z 0,l the following relation is valid d 4T x ( x) dx 4

(6)

q ( x)

where q( x) is the fictitious load. The fundamental solution of eq (6), also known as the fundamental solution of flexural beam, is a partial solution of the following differential equation

d 4T x ( x,[ ) dx 4

G( x [ )

(7)

where θx ( x, [ ) and its derivatives are given as follows d 3T x ( x, [ ) 1 sgn U 2 dx3 2 d T x ( x, [ ) 1 / 2 ( x, [ ) l (1 U ) 2 dx 2 d T x ( x, [ ) 1 2 /3 ( x, [ ) l U ( U 2) 4 dx 1 3 3 2 / 4 ( x, [ ) θx ( x, [ ) l 2 U 3 U 12

/1 ( x, [ )

with U

(8a) (8b) (8c)

(8d)

r / l , r x [ , x, [ points of the beam. Thus, the following representation of the angle of twist is obtained 1

T x ([ )

ª º d 3T ( x ) d 2T ( x ) dT ( x ) ³0 / 4 ( x,[ )q( x)dx «¬/ 4 ( x,[ ) dxx 3 / 3 ( x, [ ) dxx 2 / 2 ( x, [ ) dxx /1( x, [ )T x ( x) »¼ 0 1

(9)

Eq (9) implies that if q( x) and all boundary values (θx ( x), θx' ( x), θx'' ( x), θx''' ( x)) at the bar ends 0, l are known, θx ([ ) can be calculated at each internal point of the bar. Differentiating eq (9), the expressions for the derivatives of θx ([ ) can be derived as


391 1

dT x ([ ) d[

1 ª d 3T x ( x ) d 2T x ( x ) dT ( x ) º ³ / 3 ( x, [ )q( x )dx « / 3 ( x, [ ) / 2 ( x, [ ) / 1 ( x, [ ) x » 3 0 dx dx 2 dx ¼0 ¬

(10)

1

d 2T x ([ ) d[ 2

³/

d 3T x ([ ) d 3[

1 ª d 3T x ( x) º ³ /1 ( x, [ )q( x)dx « /1 ( x, [ ) » 0 dx3 ¼ 0 ¬

1

0

2

ª d 3T x ( x) d 2T x ( x) º /1 ( x, [ ) ( x, [ )q( x)dx « / 2 ( x, [ ) » 3 dx dx 2 ¼ 0 ¬

(11)

1

(12)

The introduction of a b-spline in the above mentioned expressions can now be done by substituting q( x) with the polynomial representation of a quadratic b-spline with a uniform knot vector. According to [5], the ith b-spline basis functions of p-degree denoted by Ni , p ([) is defined as § 1 if [i d [ [i 1 · ° ½ ° Ni , 0 ([) ®¨ ¸¾ , p 0 0 otherwise ° ¹° ¯© ¿ [ [i [i p 1 [ Ni , p ([) Ni , p 1([) Ni 1, p 1([), p t 1 [i p [i [i p 1 [i 1

(13) (14)

These basis functions are piecewise polynomials which form a basis for the vector space and multiplied by the control points give the representation of the b-spline curve. Considering the interval [0,1], which contains the bar element with length equal to unity, with ξ ϵ [0,1] and substituting to eq (14) the following basis functions are derived for the quadratic b-spline N0,2 ([) N1,2 ([) N 2,2 ([)

[ 0 00

[ 0 1 0

[ 0 1 0

N0,1 N1,1

1 [ N1,1 1 0

2 °§ (1 [ ) ®¨ °© 0 ¯

1[ N 2,1 1 0

§ 2[ (1 [ ) ° ®¨ 0 °© ¯

1 [ N3,1 1 1

§ [ 2 ° ®¨ ° ¯© 0

N 2,1

if 0 d [ 1· ½ ° ¸¾ otherwise ¹ ¿ ° ½ if 0 d [ 1· ° ¸¾ otherwise ¹ ¿ °

½ if 0 d [ 1· ° ¸¾ otherwise ¹ ° ¿

(15) (16) (17)

where N 0,1 , N1,1 , N 2,1 and N 3,1 calculated by eqs (13), (14). Thus, the quadratic b-spline curve is defined by C ([ )

n

¦N

i ,2

([ ) Pi

(18)

i 0

where Pi are the control points P0 , P1 and P2 . Substituting eqs (15)-(17) to eq (18), we derive the expression for the fictitious load q( x) q( x)

P0 2 xP0 x2 P0 2 xP1 2 x2 P1 x 2 P2

(19)

The length of the bar is considered equal to unity in order to simplify this initial approach of fictitious load using a quadratic b-spline in AEM, make the comparison with the AEM using constant values of fictitious loads easier and the results more obvious. Three equidistant collocation points have been used Collocation and control points are shown in Fig.2. Now q( x) can be substituted in eqs (9)-(12) in order to produce the matrices [A], [A’], [A’’], [A’’’], [F] and consequently [B], [B’], [B’’] and [B’’’] of AEM. The integrals entering these matrices are computed analytically. Matrices [E] and [C] used for the derivation of [B, B’, B’’, B’’’] are the same as in the original application of the method. Thus, inputs of matrices [A], [A’], [A’’], [A’’’] and [F] are now functions of the

392


three control points. The same case is for [B], [B’], [B’’] and [B’’’] matrices. Then, eq (1) yields the following linear system of equations ( E[CM ][ A0 ] G[ I t ][ B '']) > P0

P1

P2 @

7

{mt } G[ I t ]{R ''}

(20) which gives the values of the control points P0 , P1 and P2 instead of the values of q( x) at collocation points as in the original AEM. The diagonal matrix [A0] contains the values of basis functions N 0,1 , N1,2

Fig.2. Bar element, representation of fictitious load q(x), control and collocation points.

and N 2,2 for X=Xi1, Xi2 and Xi3. Matrices [A, A’, A’’, A’’’], [F] and [B, B’, B’’, B’’’] as they have been formed after substitution of b-splines in the relevant integrals are 3X3, 8X3 and 3x3 respectively. The vector of control points substitutes the fictitious load vector of the original AEM. Matrices [CM] and [It] are diagonal with 2X2 dimensions and their values depend on the cross section geometry and primary warping function. Then, the values of the control points are used in order to compute (θx ( x), θx' ( x), θx'' ( x), θx''' ( x)) at midpoint of each element using the following expressions {4} [ B]> P0 {4 ''} [ B '']> P0

P2 @ {R}, {4 '} [ B ']> P0 7

P1 P1

P1

P2 @ {R ''}, {4 '''} [ B ''']> P0 7

P2 @ {R '} 7

P1

(21a,b)

P2 @ {R '''} 7

(21c,d)

4. Numerical Example In order to evaluate warping and torsion constants, given in eqs (3), a computer program has been written, while the ability of pre- and post-processing of input and output data has been employed. This program employs constant elements to approximate line integrals. Another program in Matlab language is used in order to derive [A, A’, A’’, A’’’], [F] and [B, B’, B’’, B’’’] matrices. Control Points P and Angle of twist θx ( x) (and its derivatives) have been calculated according to eqs (20),(21) using Matlab software tool. A clamped steel beam of length L=1m, of rolled doubly symmetric I-section IPE-200 ( It 6,846cm4 , CM 12746cm6 , max ISP 47,50cm2 , according to [7]), loaded along the length by a uniform twisting moment mt 1kNm / m has been studied in order to examine the advantages attained by the use of a quadratic b-spline as the fictitious loads in AEM technique (instead of constant values in the original AEM) in terms of accuracy and computational cost. The modulus of elasticity and Poisson ratio of the beam are E 2.1E8kPa and v 0.3, respectively. As mentioned in section 2, the values of coefficients in boundary conditions for clamped edge given in eq (2) are a1 E1 1 , a2 a3 E2 E3 0 . Thus, vectors {D} , {R} , {R '} , {R ''} and {R '''} are equal to zero. q(x)(rad/m4)/ θ(x)(rad) q(1/4)=d4θ/dx4(1/4) q(1/2)=d4θ/dx4(1/2) θ(1/4) θ(1/2)

AEM original (1)

AEM b-splines (2)

Analytical Solution (3)

Error % (1)-(3)

Error % (2)-(3)

0.1849 0.1681 0.000315 0.000519

0.3656 0.3432 0.000559 0.0009485

0.3657 0.3433 0.000523 0.000925

49.40 51.03 39.65 43.90

0.027 0.029 6.440 2.478

Table 1. Values of fictitious loads at positions of collocation points and angles of twist for the bar element of IPE-200 cross section. The following three cases studied: 1) three discretization elements using original AEM technique with constant values of fictitious load, 2) three collocation points for the numerical implementation of AEM technique with quadratic b-spline fictitious load and 3) the analytical solution of the fourth order differential equation with the aid of Maple programming.


393

5. Concluding Remarks The main conclusions that can be drawn from this investigation are the following mentioned. An improvement in the accuracy of the results is achieved by around 50% as it can be observed in Table 1 comparing the values of fictitious loads at the positions of collocation points at Xi1-Xi3 and Xi2 between the aforementioned AEM technique and the original one. Comparing the values of the angles of twist, the difference showed around 40%. The analytical solution regarding the values of fictitious loads and angles of twist at collocation points is used to verify the improvement in the results. Comparing the results between the AEM with the b-spline as the fictitious load and the analytical solution, a difference of 2,5-6,4% in the angles of twist can be observed. Regarding the Fig.3. Parabolic Curve of fictitious fictitious load, results are more positive with values closely load for the bar element of IPE-200 related. As shown in Fig.3, the curve formed to represent the cross section as obtained from the fictitious load given by the analytical solution is a parabola. analytical solution. This means that highly accurate results can be obtained using a quadratic b-spline curve. In addition to this, the number of the unknowns is now restricted to the number of the control points which depend on the order of the b-spline used and a smaller number of collocation points is required in order to analyse the beam comparing to the original AEM technique where the length of the elements should be small (thus their number should be high) especially when there are concentrated loads. Thus, the computational cost is much less by using a quadratic b-spline. However, further investigation should be done in order to examine the use of higher order b-splines in AEM or refinement methods for the quadratic b-splines in order to increase accuracy, the optimum positions of collocation points used, how changes in the position of control points could affect the results (sensitivity of the method) and other aspects that could arise through the application in practical examples.

6. References [1] J.T. Katsikadelis, Boundary Elements:Theory and Applications, Elsevier (2002). [2] J.T. Katsikadelis, The Analog Equation Method: A Powerful BEM-Based Solution Technique for Solving Linear and Non-linear Engineering Problems, Boundary Element Method XVI, 167-182, (1994). [3] 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, 1338, (2002). [4] T. Hughes, J. Cottrell and Y. Bazilevs, Isogeometric analysis: Toward integration of CAD and FEA, Wiley (2009). [5] L. Piegel and W. Tiller, The NURBS Book, Berlin, Heidelberg: Springer (1997). [6] E.J. Sapountzakis and V.G. Mokos, Warping Shear Stresses in Nonuniform Torsion by BEM, Computational Mechanics, 30(2), 131-142, (2003). [7] M. Kraus and R. Kindmann, St. Venant’s Torsion Constant of Hot Rolled Steel Profiles and Position of the Shear Centre, NSCC (2009).

Arbitrary Stokes flow about a fixed or freely-suspended slip particle A. Sellier LadHyx. Ecole polytechnique, 91128 Palaiseau Cédex, France e-mail: [email protected] Keywords: Stokes flow, Ambient flow, Navier slip condition, Boundary-integral equation, Green tensor.

Abstract. The rigid-body migration of a slip and arbitrary-shaped solid particle freely suspended in a prescribed and arbitrary ambient Stokes flow is determined after calculating the hydrodynamic force and torque exerted on the particle when it either experiences a given rigid-body in a quiescent liquid or is held fixed in the ambient Stokes flow. The adopted procedure extends a recent work [14] and consists in inverting at the most seven problems involving coupled and regularized boundary-integral equations on the particle boundary. In addition to the numerical implementation, preliminary computations for spheroidal slip particles are presented. Introduction Not surprisingly, the flow of a fluid past a solid surface is strongly sensitive to the boundary conditions it satisfies on this surface. Such conditions are dictated by the surface properties and ability to let the liquid flow tangent to it. Although the usual no-slip condition (equal fluid and surface velocities at the surface) is valid for most surfaces, it sometimes breaks down for surfaces (such as hydrophobic ones) allowing a tangent slip. In most cases, one then adopts the celebrated Navier [1] slip condition in which (see (1)) the surface ability to let the fluid flow tangent to it is characterized by its so-called surface slip length λ ≥ 0. For a Newtonian fluid with uniform viscosity μ, velocity u and stress tensor σ flowing past the slip solid surface S moving at the velocity v and having unit normal n directed into the liquid, the Navier slip condition reads u = v + λ{σ .n − (n.σ .n)n}/μ on S.

(1)

Many applications require to determine the flow about a solid slip particle P, with smooth surface S having given slip length λ, when P either experiences a prescribed righid-body motion in a quiescent liquid or is held fixed in a given ambient flow.

μ, ρ

(ua , pa )

S

P

n O •

D λ

U Ω

Figure 1. A solid slip particle P experiencing a rigid-body migration (U, Ω) and immersed in a prescribed arbitrary ambient Stokes flow (ua , pa ). Such a tremendously-involved problem, depicted in Fig. 1, fortunately becomes more tractable whenever inertial effects are negligible (for instance, for micro-scaled particle and/or a sufficiently viscous liquid) since the flow is then governed by the linear Stokes equations (instead of the non-linear NavierStokes ones). Under the Low-Reynolds-Number flow assumption, one can quote the following works:

(i) The case of a slip particle experiencing a prescribed rigid-body migration in a quiescent liquid. It has been addressed using quite different approaches for either spherical [2], weakly or slightly nonspherical [3-6], toroidal [7,8], spheroidal [9,10], axi-symmetric [11-13] and also recently arbitrary-shaped [14] slip particles. (ii) The case of a slip particle held fixed in a given and arbitrary ambient Stokes flow. In contrast to previous case (i), the available literature solely deals with a spherical slip particle held fixed either in a linear [15] or an arbitrary [16] Stokes flow. Note that [16] actually nicely extends to the case of a slip sphere (lambda > 0) the famous Faxen relations [17] derived for a no-slip sphere (λ = 0). To the author’s very best knowledge, there is currently no work dealing with the challenging case (ii) of a slip and arbitrary-shaped solid particle held fixed or freely-suspended in a given and arbitrary ambient Stokes flow. The present paper introduces a suitable method to solve this issue. As in [14], it rests on the treatment of a few boundary-integral equations on the slip particle surface. Governing equations and resulting problems and linear system This section presents the general addressed problem and a trick to determine the force and torque exterted on a slip particle held fixed in a given arbitrary ambient Stokes flow and also its incurred rigid-body when it is freely-suspended in this flow. Governing equations and basic issues As shown in Fig. 1, we consider a solid slip particle P immersed in a Newtonian and unbounded liquid with uniform density ρ and viscosity μ. This particle has attached point O and smooth slip surface S with unit normal n directed into the liquid domain D. It is embedded in a steady and arbitrary Stokes flow with prescribed velocity field ua , pressure field pa and stress tensor σa obeying μ∇2 u∞ = ∇p∞ and ∇.u∞ = 0 in IR3 .

(2)

The particle rigid-body motion has translational velocity U (the velocity of O) and angular velocity Ω. In addition, the disturbed flow about P has velocity ua + u and pressure pa + p in the liquid domain D. Inertial effects are negligible, i. e. the particle length scale a and the typical magnitude V > 0 of the velocity u satisfy Re = ρV a/μ 1. The flow (u, p) with stress tensor σ then obeys μ∇2 u = ∇p and ∇.u = 0 in D, (u, p) → (0, 0) at ∞,

(3)

u(M ) − λ{σ.n − (n.σ.n)n}/μ = −ua (M ) + λ{σa.n − (n.σa.n)n}/μ + U + Ω ∧ OM on S (4) where (4) is the Navier [1] slip condition and λ ≥ 0 the particle surface slip length. Since (ua , pa ) is a Stokes flow inside the particle (recall (2)) it exerts zero force and torque on it. Consequently, the force F and torque Γ (about O) experienced by the particle read

F=

S

[σa + σ].ndS =

S

σ.ndS, Γ =

S

x ∧ [σa + σ].ndS =

S

x ∧ σ.ndS.

(5)

In practice one is interested in determining the flow (u, p) about the slip and arbitrary-shaped particle, the resulting force F and torque Γ and if not provided the particle rigid-body motion (U, Ω). By superposition, one may then confine the analysis to three different Cases: Case 1: particle experiencing a prescribed rigid-body migration (U, Ω) in a quiescent liquid. The resulting flow, obtained for ua = 0 and pa = 0, exerts on the moving particle hydrodynamic force Fh and torque Γh given by (5). By superposition, one gets Fh = −μ{A.U + B.Ω} and Γh = −μ{C.U + D.Ω} with second-rank resistance tensors A, B, C and D depending upon the particle geometry and slip length λ (in addition, tensors B and C are transposed). Case 2: particle held fixed in the arbitrary ambient Stokes flow (ua , pa ). Here, U = Ω = 0 and one seeks the flow (u, p) and the force and torque it exerts on the fixed particle given by (5) and further denoted by Fa and Γa , respectively.

Case 3: particle freely-suspended in the Stokes flow (ua , pa ). The particle has negligible inertia and is thus force-free and torque-free. Its motion (U, Ω) is then first obtained by solving the relations Fa = μ{A.U + B.Ω}, Γa = μ{C.U + D.Ω}.

(6)

Once this is done, combining the previous solutions to Case 1 and Case 2 finally provides (if necessary) the required flow about the moving and freely-suspended slip particle. First approach to obtain the force Fa and torque Γa We use Cartesian coordinates (O, x1 , x2 , x3 ) and introduce, for i = 1, 2, 3, the auxiliary Stokes flows (i) (i) (i) (i) (ut , pt ) or (ur , pr ) solution to (3)-(4) for (ua , pa ) = (0, 0) and (U, Ω) = (ei , 0) or (U, Ω) = (0, ei ), (i) (i) respectively. These flows exert on the particle surface S tractions σt .n and σ r .n. In solving the Case 2 we apply the usual reciprocal identity [18] for each auxiliary flow and the unknown flow (u, p) about the particle therefore arriving at the following key relations

λ (i) (i) (σa.n).[σ t .n − (n.σ t .n)n]dS (i = 1, 2, 3), μ S S λ (σa.n).[σ r(i) .n − (n.σ r(i) .n)n]dS (i = 1, 2, 3). Γa .ei = − ua .σ (i) r .ndS + μ S S

Fa .ei = −

(i)

ua .σ t .ndS +

(7) (8)

Recently [14] solved the Case 1 by appealing to a boundary formulation which provides on the particle (i) (i) surface S the tractions σ t .n and σ r .n. By virtue of (7)-(8), it is then sufficient to solve the Case 1 (by the treatment developed in [14]) to determine the force Fa and torque Fa prevailing in Case 2. Another procedure, given later, is however possible. Analytical results for a sphere Analytical results [2, 16] have been obtained for a slip sphere with center O and radius a and read 1 + 2λ/a Ω ]U, Γh = −8πμa3 [ ], (9) 1 + 3λ/a 1 + 3λ/a 1 + 2λ/a 1 4πμa3 ]ua (O) + πμa3 [ ]∇2 ua (O), Γa = [ ](∇ ∧ ua )(O). (10) Fa = 6πμa[ 1 + 3λ/a 1 + 3λ/a 1 + 3λ/a

Fh = −6πμa[

By virtue of (6), the migration of a sphere freely suspended in a Stokes flow (ua , pa ) is given by U = ua (O) + [

a2 1 ]∇2 ua (O), Ω = (∇ ∧ ua )(O). 6(1 + 2λ/a) 2

(11)

The results (9)-(11) provide nice benchmark tests for the treatment proposed in the next sections. Advocated boundary-integral approach This key section gives the relevant boundary-integral equations which permit one to solve the Case 1 and obtain the required vectors Fa and Γa in Case 2. Boundary-integral equations for the auxiliary Stokes flows The Stokes flow for a particle migration (U, Ω) in a quiescent liquid (Case 1) exerts on the particle surface S a traction σ.n. We then introduce on S the quantity d and vector d tangent to S such that d = n.σ.n/μ, d = [σ.n − (n.σ.n)n]/μ.

(12)

As established in [14], those unknown surface quantities d and d satisfy the boundary problem d.n = 0 and Li [d, d] = [U + Ω ∧ OM].ei for i = 1, 2, 3 and x on S

(13)

where the linear regularized boundary operators Li read (summing over indices k and l in (14)) 8πLi [d, d] = −8πλdi (x) −

+λ

S

S

Gki (y, x)dk (y)dS(y) −

S

Gki (y, x)nk (y)d(y)dS(y)

[dk (y) − dk (x)]Tkil (y, x)nl (y)dS(y)

(14)

with di = d.ei and, denoting by δ the Kronecker delta symbol, the definitions [(y − x).ei ][(y − x).ej ] δij + , |x − y| |x − y|3 6[(y − x).ei ][(y − x).ej ][(y − x).ek ] . Tijk (y, x) = − |x − y|5

Gij (y, x) =

(15) (16)

(i)

(i)

Adequately selecting (U, Ω) then provides the required tractions σt .n and σ r .n on the particle boundary by numerically inverting the problem (13). Boundary-integral equations for Case 2 Let us now consider the Stokes flow (u, p) with stress tensor σ about the particle held fixed in the ambient Stokes flow (Case 2). It then obeys the boundary condition u − λ{σ.n − (n.σ.n)n}/μ = −ua + λ{σa.n − (n.σa.n)n}/μ on S. Introducing this time on the particle surface S the quantity

d

and vector

d

(17)

as

d = n.[σa + σ].n/μ, d = {[σa + σ].n − (n.[σa + σ].n)n}/μ.

(18)

it is possible to prove that (d , d ) fulfills the following boundary problem d .n = 0 and Li [d , d ] = −ua .ei for i = 1, 2, 3 and x on S.

(19)

Hence, a second approach to compute the force Fa and the torque Fa consists in obtaining the vector [σa +σ].n = d n+d on the particle surface S by solving (19) and then employing the relations (6). Numerical method and preliminary results for spheroids This section briefly presents the implemented numerical treatment and a few results for spheroidal slip particles immersed in pure linear or quadratic ambient shear flows. Numerical implementation Since it is detailed in [14], we briefly present the boundary element technique employed to numerically invert the regularized boundary-integral (13) or (19). We use on the particle surface S a N −node mesh made of 6-node curvilinear and triangular boundary elements. At each nodal point, where the unit normal n and two unit vectors t1 and t2 tangent to the particule surface S such that t1 .t2 = 0 are calculated, one then (for instance for the problem (13)) ends up with three unknown quantities: d and also dt1 such dt2 that d = dt1 t1 + dt2 t2 . This choice ensures the property d.n in (13) while the discretized boundary-integral equations result in a linear system. This system with 3N × 3N non-symmetric and dense matrix A is solved by LU factorization algorithm. Results for spherical and spheroidal slip particles We consider spheroidal slip particles, with surface having equation (x1 /a)2 + (x2 /a)2 + (x3 /b)2 = 1, immersed in linear or quadratic shear flows (ua , pa ) = (ks x3 e1 , 0) or (ua , pa ) = (kq x23 e1 , 2μx1 ) shear flows. For such flows symmetry easily show that U = U1 e1 , Ω = Ω2 e2 and also that Fh = −6πμaf1 U1 e1 , Γh = −8πμa3 c2 Ω2 e2 ,

Γa = 4πμa3 ks cs e2 , Fa = U = 0, ws = Ω2 /ks = cs /(2c2 ) for linear shear,

Fa = 2πμa3 kq fq e1 , Γa = Ω = 0, uq = U1 /(kq a2 ) = fq /(3f1 ) for quadratic shear

(20) (21) (22)

with dimensionless coefficients f1 , fq , torque factors c2 , cs and translational and angular velocities uq and ωs . For a sphere (b = a) the analytical results (9)-(11) give f1 = (1 + 2λ/a)/(1 + 3λ/a), fq = cs = c2 = 3uq = (1 + 3λ/a)−1 and ω = 1/2. As shown in Table 1, our computations converge to those results as the number N of nodes put on the sphere boundary increases (the relations (7)-(8) were employed but values obtained after solving (19) are also reported for comparisons). Table 1: Computed coefficients cs , fq , ωs and uq for a sphere with radius a versus the number N of nodal points for λ/a = 0.5, 2. The values computed after solving (19) for d instead of using (7)-(8) are given in the columns with overlined symbols. N 74 242 1058 exact 74 242 1058 exact

λ/a 0.5 0.5 0.5 0.5 2 2 2 2

cs 0.39758 0.39992 0.40001 0.4 0.14355 0.14280 0.14285 0.14286

fq 0.39973 0.39972 0.39997 0.4 0.15021 0.14303 0.14285 0.14286

ωs 0.49926 0.50006 0.50002 0.5 0.50588 0.50005 0.49999 0.5

uq 0.16596 0.16651 0.16665 0.16667 0.06950 0.06670 0.06666 0.06667

cs 0.39685 0.39977 0.39999 0.4 0.14146 0.14276 0.14285 0.14286

fq 0.38526 0.39584 0.39967 0.4 0.13915 0.13927 0.14252 0.14286

ωs 0.498343 0.499864 0.499996 0.5 0.49852 0.49992 0.5 0.5

uq 0.15995 0.16490 0.16652 0.16667 0.06438 0.06494 0.06651 0.06667

Note that for a sphere ws does not depend upon λ/a. This is not the case any more for spheroidal particles as illustrated in Table 2 for two oblate (b/a = 0.8) and prolate (b/a = 1.2) spheroids. Note that ωs increases or decreases as λ/a increase for the prolate or oblate spheroid, respectively. Table 2: Computed coefficients cs , fq , ωs and uq for one oblate (ob) spheroid with b = 0.8a and one prolate (pro) spheroid with b = 1.2a. λ/a 0 0.2 0.5 0.8

cs (ob) 0.61483 0.32776 0.14987 0.06602

fq (ob) 0.58828 0.31031 0.13647 0.05357

ωs (ob) 0.39016 0.33912 0.23998 0.14092

uq (ob) 0.21338 0.13029 0.06326 0.02611

cs (pro) 1.49872 1.02335 0.74875 0.62052

fq (pro) 1.55486 1.06734 0.78707 0.65709

ωs (pro) 0.59034 0.62285 0.68911 0.75785

uq (pro) 0.48039 0.37307 0.29857 0.25967

Other results will be reported and discussed at the oral presentation. Conclusions A new boundary approach has been proposed to accurately computed the force and torque applied on a slip solid particle held fixed in an arbitrary Stokes flow and, if necessary, the resulting rigid-body migration of a freely-suspended particle. The task is reduced to the treatment of at the most seven boundary problems involving regularized boundary-integral equations on the particle boundary. The adopted boundary element implementation is benchmarked against the analytical results established elsewhere for a spherical particle and preliminary results for slip spheroids are given. 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). [2] A. B. Basset A treatise on Hydrodynamics, Vol. 2. Dover, New York, (1961).

[3] D. Palaniappan Creeping flow about a slighty deformed sphere. Z. Angew. Math. Phys., 45, 832–838 (1994). [4] H. Ramkissoon Slip flow past an approximate spheroid. Acta Mech., 123, 227–233 (1997). [5] S. Senchenko and H. J. Keh Slipping Stokes flow around a slightly deformed sphere. Physics of Fluids, 18, 088104. (2006). [6] Y. C. Chang and H. J. Keh Translation and rotation of slightly deformed colloidal spheres experiencing slip. Journal of Colloid and Interface Science, 330, 201–210 (2009). [7] M. M. R. Williams A closed torus in Stokes flow with slip boundary condition. Q. J. Mech. Appl. Math., 40, 235–243 (1987). [8] S. K. Loyalka Rotation of a closed torus in the slip regime. J. Aerosol. Sci., 25, 371–379 (1996). [9] S. Deo and S. Datta Slip flow past a prolate spheroid. Indian J. Pure Appl. Math., 33, 903–909 (2002). [10] H. J. Keh and Y. C. Chang Slow motion of a slip spheroid along its axis of revolution. International Journal of Multiphase Flow, 34, 713–722 (2008). [11] S. K. Loyalka and J. L. Griffin Rotation of non-spherical axi-symmetric particles in the slip regime. J. Aerosol. Sci., 27, 509–525 (1994). [12] H. J. Keh and C. C. Huang Slow motion of axisymmetric slip particles along their axes of revolution. Keh, H. J. & Huang, C. H. 2004 Slow motion of axisymmetric slip particles along International Journal of Engineering Science, 42, 1621–1644 (2004). [13] Y. C. Chang and H. J. Keh Theoretical study of the creeing motion of axially and fore-and-aft symmetric slip particles in an arbitrary directions. European Journal of Mechanics B/Fluids, 30, 236–244 (2011). [14] A. Sellier Stokes flow about a slip arbitrary-shaped particle. CMES, 87 (2), 157–176 (2012). [15] B. U. Felderhof Force density induced on a sphere in linear hydrodynamics II. Moving sphere, mixed boundary conditions. Physica., 84A, 569–576 (1976). [16] H. J. Keh and S. H. Chen The motion of a slip spherical particle in an arbitrary Stokes flow. European Journal of Mechanics B/Fluids, 15, 791–807 (1996). [17] H. Faxen Die Bewegung einer starren Kugel langs der Achse eines mit zaeher Fluessigkeit gefuellten Rohres. Arkiv for Matematik, Astronomi och Fysik,, 17 (27), 1–28 (1922-1923). [18] J. Happel, H. Brenner Low Reynolds number hydrodynamics, Martinus Nijhoff, (1973).

Gravity-driven migration of bubbles and/or solid particles near a free surface M. Guémas1 ,2 , A. Sellier1 and F. Pigeonneau2 Ecole polytechnique, 91128 Palaiseau Cédex, France 2 Surface du Verre et Interfaces, UMR125 CNRS St Gobain, 39 quai Lucien Lefranc, BP 135, 93303 Aubervilliers, Cedex, France e-mail: [email protected] e-mail: [email protected] 1 LadHyx.

Keywords: Bubble, free surface, surface tension, Stokes flow, Boundary-integral equation, film drainage.

Abstract We investigate the challenging problem of bubble(s) and rigid particle(s) interacting near a free surface. The time-dependent bubble(s) and free surface shapes are determined for a large range of Bond number by solving the creeping flow induced by the bubble(s) and the particle(s) motion. This works extends the boundary-integral formulation handled in a recent work solely dealing with bubble(s) ascending toward a free surface. 1. Introduction The gravity-driven motion of bubble(s) interacting with solid particle(s) in a viscous liquid in presence of a free surface is of high interest in applications such as geophysics, chemistry, glass process, . . . This task is quite involved due to the interactions occurring between the different solid and evolving surfaces. The axisymmetric gravity-driven migration of bubble(s) ascending toward a free surface has been numerically investigated either for bubble with equal surface tension in [4] or unequal surface tension in [1]. In contrast, this work considers, still for axisymmetric geometry, the more-involved case of cluster made of both bubble(s) and solid particle(s). This problem is solved adopting regularized and carefully-selected boundary-integral equations enforced on the entire liquid domain. 2. Challenging time-dependent problem 2.1 Assumptions and relevant axisymmetric quasi-steady Stokes flow We consider a cluster made of M ≥ 0 bubbles Bm and/or N ≥ 0 solid particles Pn with M +N ≥ 1 immersed in a Newtonian fluid with uniform density ρ and viscosity μ. This liquid is bounded by a free surface and both the cluster and the liquid are subject to the uniform gravity g = −ge3 (with g > 0). The bubble Bm , the solid particle Pn and the free surface have smooth and time-dependent surfaces Sm (t) with uniform surface tension γn , Σn (t) and S0 (t) with uniform surface tension γ0 , respectively. As illustrated in Figure 1 for M = N = 1, all surfaces S0 (t), Sm (t) and /or Σn (t) admit unit normal vector n directed into the liquid domain D(t) and the same axis of revolution (O, e3 ) (axisymmetric problem). As the cluster migrates under the gravity, the shapes of the bubble(s) and free surface evolve in time. At initial time, each bubble is spherical with typical radius a and the free surface is the z = 0 plane. At any time t, the pressure p0 above the disturbed free surface S0 (t) and pm inside the disturbed bubble Bm are assumed to be constant. Each solid particle Pn with uniform density ρn has, for symmetry reasons, time-dependent velocity U (n) (t)e3 . In addition, the liquid flow has pressure p + ρg.x (here x = OM with O denoting the origin of our Cartesian coordinates) velocity u and stress tensor σ. We denote by a the bubble(s) and solid particles typical length scale and by V the typical magnitude of velocities u and U (n) (t). Assuming that Re = ρV a/μ 1, inertial effects are negligible


401

z

γ0

S0 (t)

n γ1

D(t)

x

B1 S1 (t)

n

g = −ge3

P1 n

Σ1 (t)

Figure 1: One bubble B1 and a solid sphere P1 moving near a free surface S0 (t). and the flow (u, p) obeys the following quasi-steady creeping flow equations and boundary conditions ∇ · u = 0 and μ∇2 u = gradp in D(t), (u, p) → (0, 0) as |x| → ∞,

(1)

σ · n = (ρg · x − pm + γm ∇S · n) n on Sm (t) for m = 0, ..., M,

(2)

u = U (n) (t)e3 on Σn (t) for n = 1, ..., N

(3)

where H = [∇S · n]/2 is the local average curvature. Assuming bubbles with constant volume, one supplements (1)-(3) with the relations 1 u · n dS = 0 on Sm for m=0,...,M. (4) Sm (t)

For N ≥ 1 the velocities U (n) (t) are unknown. By symmetry, each solid particle Pn is torquefree. In addition, each Pn with negligeable inertia is force-free. This latter property results in the additionnal conditions e3 · σ · ndS = (ρn − ρ)Vn g for n = 1, ..., N (5) Σn (t)

where ρn and Vn designate the uniform density and volume of the particle Pn . The material surface(s) Sm (t) have velocity V. Since there is no mass transfer across the surfaces Sm (t), one has (6) V · n = u · n on Sm for m = 0, ..., M. 2.2 Proposed tracking algorithm for the time-dependent entire liquid boundary We compute the time-dependent shape of the free surface, the bubble(s) and particle(s) surface(s) by running at each time t the following steps : 1

Note that (4) indeed also holds for m = 0 because u is divergence-free and u → 0.

i

Step 1: From the knowledge at time t of the liquid domain D(t), one first computes the quantity ∇S · n on each surface Sm (t). Step 2: One then solves at time t the relations (1)-(5) to get the unknown velocities U (n) (t) and the fluid velocity u on each surface Sm (t). Step 3: The liquid boundary D(t + dt) at time t + dt is obtained by moving between times t and t + dt each surfaces Sm by exploiting the relation (6) and each solid surface Σn at the velocity U (n) (t)e3 . One should note that for such a procedure the following issues are of the utmost importance: (i) To accurately compute the local average curvature (σ · n)/2 on each surface Sm in Step 1. (ii) To efficiently and accurately solve the Stokes problem (1)-(5) in Step 2. (iii) To adequately select a time step at in Step 3. This work introduces a suitable treatment to cope with the previous issue (ii). 3. Advocated method This section presents a new procedure to appropriately solve the problem (1)-(5). 3.1 Auxiliary Stokes flows for cluster involving at least one solid particle. As soon as N ≥ 1, each velocity U (n) occuring in (1)-(5) is unknown. Fortunately, it is possible to determine U (1) (t), · · · , U (N ) (t) prior to obtain the liquid flow (u, p) ! The trick consists in introducing, for n = 1, · · · N , auxiliary Stokes flows (u(n) , p(n) ) obtained without stress on each Sm and when each solid surface Σq is motionless for q = m with the surface Σn of Pq which translates at the velocity e3 . In other words, the flow (u(n) , p(n) ) with stress tensor σ (n) satisfies (1) and the following boundary conditions u(n) = δnq e3 on Σq for q = 1, · · · , N σ

(n)

(7)

· n = 0 on Sm for m = 0, · · · , M.

In addition, one supplements (1), (7)-(8) with the additionnal conditions u(n) · n dS = 0 on Sm for m = 0, ..., M.

(8)

(9)

Sm (t)

Denoting by ∂D the liquid boundary, the reciprocal identity [2] for the flows (u, p) and (u(n) , p(n) ) reads u(n) · σ · n dS = u · σ (n) · n dS. (10) ∂D

∂D

Enforcing the relations (5) by exploiting the aformentionned identity (10) and the boundary conditions (2)-(3) and (7)-(8), one then arrives at the N -equation linear system " ! e3 · σ (n) · n dS U (q) (t) = (ρ − ρn )Vn g q≥1

Σq

+

m≥0 Sm

u(n) · (ρg · x − pm + γm ∇S · n) n dS

for n = 1, · · · , N.

(11)

Furthemore, the pressure pm is uniform in the bubble Bm which leads, in conjection with (5), to " ! e3 · σ(n) · n dS U (q) (t) = (ρ − ρn )Vn g q≥1

Σq

+

m≥0 Sm

u(n) · (ρg · x + γm ∇S · n) n dS for n = 1, · · · , N.

(12)

It is possible (and here admitted) to prove, invoking the energy dissipation in Stokes flow, that (12) is well-posed (i. e. presents a non-singular matrix). Note that, one solely needs to evaluate the surface quantities u(n) on each Sm and σ (n) ·n on each Σq to obtain the translational velocity U (q) (t)e3 of the particle Pq . As shown in the next subsection, those required key surface quantities u(n) and σ(n) · n are calculated by inverting relevant boundary-integral equations on the entire liquid boundary ∂D. 3.2 Relevant boundary-integral equations 3.2.1 Three-dimensionnal formulation For a Stokes flow (u, p) with stress tensor σ obeying (1) with prescribed values of the stress σ · n on each Sm and of the velocity u on each Σn , one has the key coupled regularized boundary-integral equations (see for instance [5]), μ[u(x) − u(x0 )] · T(x, x0 ) · n(x)dS − G(x, x0 ) · σ · n(x)dS −8μπu(x0 ) + m≥0 Sm

=

m≥0 Sm

n≥1 Σn

G(x, x0 ) · σ · n(x)dS

for x0 on Sm

(13)

and m≥0

Sm

μ[u(x)−u(x0 )] · T(x, x0 ) · n(x)dS − = +8μπu(x0 ) +

m≥0 Sm

n≥1 Σn

G(x, x0 ) · σ · n(x)dS

G(x, x0 ) · σ · n(x)dS

for x0 on Σn

(14)

where the second-rank tensor G and third-rank stress tensor T are defined as [3] I (x − x0 ) ⊗ (x − x0 ) + ; |x − x0 | |x − x0 |3 (x − x0 ) ⊗ (x − x0 ) ⊗ (x − x0 ) T (x, x0 ) = −6 |x − x0 |5 .

G(x, x0 ) =

(15) (16)

with I the identity tensor. Clearly, solving (13)-(14) permits one to get the unknown vectors u on Sm and σ · n on Σn from the knowledge of u on Σn and σ · n on Sm . 3.2.2 Axisymmetric formulation Since we restrict the analysis to the% axisymmetric configuration depicted in Fig.1, we adopt cylindrical coordinates (r, φ, z) with r = x2 + y 2 , z = x3 and φ the azimuthal angle in the range [0, 2π]. We set u = ur er + uz ez = uα eα (with α = r, z), f = σ · n = fr er + fz ez = fα eα and n = nr e + nz ez = nα eα and introduce the traces Ln of Σn and Lm of Sm in the φ = 0 half plane.

Integrating over φ the equations (13)-(14), then yields the equivalent coupled boundary equations μ[uβ (x) − uβ (x0 )] Cαβ (x, x0 )dl − Bαβ (x, x0 )fβ nβ (x)dl −8πuα (x0 ) + m≥0 Lm

=

m≥0 Lm

and

m≥0 L

n≥1 Ln


μ[uβ (x) − uβ (x0 )]Cαβ (x, x0 )dl − = 8πuα (x0 ) +

m≥0 Lm

for x0 on Lm

n≥1 Ln

(17)



for x0 on Ln

(18)

for α = r, z, the differential arc length dl in the φ = 0 plane and the so-called single-layer and doublelayer 2 × 2 square matrices Bαβ (x, x0 ) and Cαβ (x, x0 ) given in Pozrikidis [5]. Note that a summation over β = r, z holds in (17)-(18). 3.2.3 Resulting boundary-integral equations for the axisymmetric Stokes flow problem Dealing with our axisymmetric problem (1)-(4), we first evaluate for each axisymmetric flow (n) (n) (u(n) , p(n) ) the needed vectors u(n) = uα eα on each Sm and σ (n) = fβ eβ on each Σq . We per(n)

(n)

form this calculation by inverting (17)-(18) for uz = δnq and ur = 0 on each Σq and σ(n) · n = 0 on each Sm . Once both the velocity and stress vectors are known on each surfaces Σq and Sm , one then obtains each velocity U (q) (t) by solving the linear system (12). Finally, we gain the required velocity u = uα eα on each Sm by inverting one more time (17)-(18) using the boundary conditions (2)-(3), i. e. μ[uβ (x) − uβ (x0 )] Cαβ (x, x0 )dl − Bαβ (x, x0 )fβ nβ (x)dl −8πuα (x0 ) + m≥0 Lm

=

m≥0 Lm

and

m≥0 Lm

n≥1 Ln


μ[uβ (x) − uβ (x0 )]Cαβ (x, x0 )dl − +

m≥0

Lm

n≥1 Ln

for x0 on Lm

(19)

Bαβ (x, x0 )fβ nβ (x)dl = +8πU (n) (t)e3 (x0 )


for x0 on Ln

(20)

for α = r, z and γm uniform on each surface Sm . In summary, our approach consists, for N ≥ 1 solid particle(s), in inverting N + 1 boundary-integral equations (17)-(18). 4. Numerical method The coupled boundary-integral equation (17)-(18) are numerically inverted by appealing to the following key steps (see for further details [4, 1]): (i) First, the entire contour L = Ln ∪ Ln is divided into Ne curved boundary elements with the L0 truncated free surface. Each boundary element has Nc collocation points spread with a uniform distribution. An isoparametric approximation is used for the components u and f = σ · n on each boundary element.

For the nodes located on Lm , the vectors U and Fd collect the unknown and prescribed components of u and f . In a similar fashion, F and Ud are the vectors associated with the unknown and given values of f and u at the nodes of the solid contours Ln . Finally, once the coupled boundary-integral equations (17)-(18) are discretized, these vectors satisfy indeed the 2Ne Nc -equation linear system U + C · U − B1 · F = −B2 · Fd

for x0 on ∪m≥0 Lm ,

(21)

C · U − B1 · F = −Ud + B2 · Fd

for x0 on ∪n≥1 Ln .

(22)

The matrices B1 , B2 and C involve integrations of the quantities Bαβ and Cαβ introduced in §3.2.2 over the entire contours ∪m≥0 Lm and ∪n≥1 Ln . . (ii) One finds the solution (U,F) of (21)-(22) by Gaussian elimination. (iii) The shape of each surface Sm and the position of each Σn if N ≥ 1 is tracked in time using the boundary condition (6) and solving the equation dx/dt = u(x, t) for each nodal point. A RungeKutta-Fehlberg method performs this task using a time-step selected by controlling the errors for the second and third-order schemes. Furthermore, as the distance between two surfaces tends to zero, the adjusted time step is then very small and the computations is stopped. 5. Conclusions Preliminary numerical results will be exposed at the oral presentation for a cluster made of one bubble and one spherical solid particle. Furthermore, this particular case will be dicussed and compared with the two-bubbles configurations studied in [4, 1]

References [1] M. Guémas, F. Pigeonneau, and A. Sellier. Gravity-driven migration of one bubble near a free surface: surface tension effects. In M. H. Aliabadi P. Prochazca, editor, Advances in Boundary Element & Meshless Techniques XIII, 2012. [2] J. Happel and H. Brenner. Low Reynolds number hydrodynamics. Martinus Nijhoff Publishers, The Hague, 1983. [3] S. Kim and S. J. Karrila. Microhydrodynamics. Principles and selected applications. ButterworthHeinemann, Boston, 1991. [4] F. Pigeonneau and A. Sellier. Low-reynolds-number gravity-driven migration and deformation of bubbles near a free surface. Phys. Fluids, 23:092302, 2011. [5] C. Pozrikidis. Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, Cambridge, 1992.

406


Elastoplastic analysis of structures using the NNRPIM S.F. Moreira1, J. Belinha2, L.M.J.S. Dinis 3 and R.M. Natal Jorge 4 1

Instituto de Engenharia Mecânica, Rua Dr. Roberto Frias, s/n 4200-465 Porto – Portugal, [email protected]

2

Instituto de Engenharia Mecânica, Rua Dr. Roberto Frias, s/n 4200-465 Porto – Portugal, [email protected]

3

Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s/n 4200-465 Porto – Portugal, [email protected]

4

Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s/n 4200-465 Porto – Portugal, [email protected]

Keywords: Meshless Methods, Natural Neighbours, Radial Point Interpolators, Natural Neighbour Radial Point Interpolator Method (NNRPIM), Elastoplasticity

Abstract. In this work the Natural Neighbour Radial Point Interpolation Method (NNRPIM), an innovative meshless method, is used to study several solid mechanics benchmark examples considering elastoplasticity. The NNRPIM uses the natural neighbour mathematical concept to impose the nodal connectivity and to determine the integration mesh. The interpolation functions are obtained with the radial point interpolators. The NNRPIM interpolation functions possess the delta Kronecker property, which permits to impose directly the essential and the natural boundary conditions. In the present analysis were considered isotropic materials, with an isotropic hardening, assuming the von Mises yield criterion. The nonlinear solution algorithm used was the Newton Raphson Method. The results obtained are very close to the results obtained using the Finite Element Method, taking into account exactly the same conditions. It is also possible to observe that the NNRPIM stress field distribution is smooth and accurate. Introduction A fundamental aspect of engineering is the aspiration to design artifacts exploiting materials to a maximum performance under working conditions. In order to optimize the material usage, it is required to consider in the design the non-linear material properties associated within the working environments [1]. Despite being the Finite Element Method (FEM) one of the most frequently used methods in stress analysis in both industry and science [2] this work proposes the use of an innovative meshless method to the elastoplastic analysis of structures, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). When compared with the FEM, meshless methods have some drawbacks, such are the complexity in the numerical implementation and programming and the higher computational cost. However the advantages are overwhelming. In the FEM distorted or low quality meshes lead to higher errors, the domain dynamic discontinuities (crack path opening) represent a numerical problem and the element mesh rigidity does not permit to easily remesh the problem domain. To overcome this limitation, an extended finite element method (XFEM) as appeared, however, the XFEM presents drawback, which is the lack of smoothness of the resulting derivatives. And so, neither XFEM nor FEM can deal with distorted meshes very well [3, 4]. The smooth particle hydrodynamics (SPH) method, [4, 5], was one of the first meshless methods to be developed and initially was used for modeling astrophysical phenomena. Libersky and Petschek [6] extended this method to solve solid mechanics problems. This method was based on a strong form, as well as the SPH corrected versions that meanwhile emerged [4]. The strong form requires strong continuity on the dependent field variables (generally displacements) and the variable functions should be differentiable up to the order of the partial differential equations. Besides, the strong form is very difficult to obtain for practical engineering problems. Alternatively, a weak form, which can be obtained using energy principles or weighted residual methods, requires a weaker continuity 1


407

on the adopted functions [7, 8]. Developed in 1994, the element free Galerkin (EFG) [9], was one of the first meshless methods based on a global weak form. The EFGM is based on the moving least-squares approximants (MLS) of Lancaster and Salkauskas [10], which satisfies the patch test. One year later, the reproducing kernel particle method (RKPM) was developed [11]. The meshless local Petrov-Galerkin (MLPG) method [12], is one of the most popular meshless method based on local weak forms, in which the local weak forms are generated on overlapping subdomains. However, the aforesaid methods employ approximation functions, which do not satisfy the Kronecker delta property. Consequently the treatment of the essential and natural boundary conditions is not as straightforward as in other numerical methods presenting interpolation functions [3, 4, 13]. In order to overcome this drawback, in the last few years some meshless methods that use interpolation functions were developed, among which are the Point Interpolation Method (PIM) [14, 15], the Natural Neighbour Finite Element Method (NNFEM) [16, 17] and the Meshless Finite Element Method (MFEM) [18]. The numerical method used in this work, the Natural Neighbour Radial Point Interpolation Method (NNRPIM), combines the radial point interpolators (RPI) with the natural neighbours geometric concept. Natural Neighbour Radial Point Interpolation Method (NNRPIM) The NNRPIM uses mathematic concepts, such as the Voronoï Diagrams and the Delaunay tessellation, to construct the influence-cells – the basic structure of the nodal connectivity in the NNRPIM – and the background integration mesh – totally dependent on the nodal mesh. Unlike the FEM, where geometrical restrictions on elements are imposed for the convergence of the method, in the NNRPIM there are no such restrictions, which permits a random node distribution for the discretized problem. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed with the Radial Point Interpolators (RPI). The NNRPIM interpolation functions possess the delta Kronecker property – which facilitates the enforcement of boundary conditions, since these can be directly imposed – and its construction is simple and its derivatives are easily obtained. The radial basis function (RBF) used in the RPI is the multiquadric RBF [19, 20]. The natural neighbour concept allows to impose the nodal connectivity [21], which can be determined by the application of a useful mathematical tool, the Voronoï diagrams. The Voronoï diagrams are constituted by a set of Voronoï cells, and to each one of these cells is associated one and only one node. In addition all points within each Voronoï cell domain are closer to the Voronoï cell associated node than any other node of the discretized domain. The influence-cells, being formed by a set of nodes in the neighbourhood of a point of interest [3, 22], permit to establish the nodal connectivity. The influence-cells also enable the construction of a background mesh for integration purposes completely dependent on the nodal mesh. Several types of influence-cells can be used. In this work it is used the second degree influence-cell, which is constituted by the first natural neighbours of the interest point x I and the natural neighbours of interest point x I first natural neighbours. The second degree influence-cell permits to achieve a higher nodal connectivity and to obtain more accurate results [3]. In order to construct the integration mesh, each Voronoï cell is subdivided in sub-cells using the Delaunay triangulation. The quadrature points are applied in each one of the sub-cells, being distributed by the GaussLegendre integration procedure. In this work only two dimensional cases were studied, therefore it was used one quadrature point in each sub-cell. The advantage of this integration scheme is that it permits to construct a numerical integration mesh total dependent of the nodal mesh. In this work it is used the radial point interpolators (RPI) function [23]. Thus, consider a function u( xI ) , defined in the domain : and discretized by a set of N nodes, assuming that only the nodes that belong to the influence-cell of the interest point x I have effect on the function u( xI ) . It is also assumed that the function u( xI ) pass throw all of the nodes of the influence-cell using a radial basis function (RBF). Thereby, considering the value of the function in a interest point x I , it follows the presented in equation (1) , where Ri ( xI ) is the radial basis function (RBF), n is the number of nodes inside of the influence-cell of the point of interest x I and ai ( xI ) and bi ( xI ) are non-constant coefficients of Ri ( xI ) and pi ( xI ) ,

2

408


respectively. The monomials of the polynomial basis are defined by pi ( xI ) and m is the basis monomial number.

u xI

½ ¦R x a x ¦p x b x ^R x , p x ` ®¯b ¾¿ n

m

T

i

I

i

I

j

i 1

I

j

I

a

T

I

I

(1)

j 1

In the radial basis function (RBF) used, the variable is the Euclidean norm between the point of interest x I and the neighbour node,

( xI xi )2

rij

(2)

In this work it is used the multiquadric radial basis function, initially proposed Hardy [24], presented in equation (3), where c and p are shape parameters that require an optimization in order to maximize the performance of the method. R(rij ) (rij 2 c 2 ) p (3) In previous NNRPIM works [3, 20] were obtained the following optimized c and p parameters: c 0.0001 and p 0.9999 . The polynomial basis added to the RBF was a constant basis, defined as presented in equation (4).

xT

{x, y}; pT ( x) {1}; m 1,

(4)

However, in order to obtain a unique solution, the polynomial basis has to satisfy an extra requirement [3], n

¦ p ( x )a ( x ) j

i

i

i

0,

j 1, 2,..., m

(5)

i 1

Thus, a new equation matrix can be written,

us ½ a ½ ® ¾ G® ¾ ¯0¿ ¯b ¿

(6)

Applying equation (6) in equation (1), is obtained

u ½ u( xI ) {RT ( xI ), pT ( xI )}G 1 ® s ¾ M ( xI )us ¯0¿

(7)

where M ( x ) is the interpolation function defined by

M ( xI ) {RT ( xI ), pT ( xI )}G 1 {M1 ( xI ), M2 ( xI ),..., Mn ( xI )}

(8)

Elastoplasticity In this work it is only considered bilinear elastoplastic material assumptions. In order to describe the elastoplastic material behaviour, it is necessary – in addition to the elastic stress-strain relations – to fulfil three requirements [2, 8]: 1. A yield criterion: based on a function that defines when the yielding can occur, taking into account the values of the stresses; 2. A flow rule: which relates the plastic strain increments to the current stresses and the stress increments outside the yield function; 3. A hardening rule: which specifies how the yield function is modified during plastic flow; 3


409

The characterization of the yield for a given material is idealized by a yield surface, which split the space of stresses in an elastic and a plastic domain. The yield occurs only if the stress satisfies the general yield criterion, usually stated in the form presented in equation (9).

F V , H p , N

f V , H p , N V Y N 0

(9)

where f V , H p , N is the yield function, which is dependent of the stress state V , the plastic strain H p and

of one hardening parameter N . The yield stress associated to the material is represented by V Y N . Considering a material yield stress independent of the hardening parameter N and a material displaying isotropic properties it is possible to simplify equation (9),

F V

f V V Y

0

(10)

The yield criterion used in this work for isotropic materials is the von Mises yield criterion, commonly applied on metals,

F V

2 2 1ª V xx V yy V yy V zz V xx V zz 2 6 V yz2 V zx2 V xy2 º¼» V Y N 0 2 ¬«

(11)

Once reached the plastic material domain, the material behaviour will become conditioned by the value of the variation of the yield function f with respect to the stress state, V , as it is stated in equation (12), where wf / wV is the f gradient, and thereby an orthogonal vector to the yield surface for a considered stress state V . T

df

§ wf · ¨ ¸ dV © wV ¹

(12)

When the yield function f presents a value equal to the yield stress of the material, V Y , we are in the threshold of the elasticity and in the beginning of the plastic behaviour. The hardening rule describes how the yield surface changes as the result of plastic deformation. In this work, we only have considered isotropic hardening, which means that the yield surface expands uniformly during plastic flow. The nonlinear solution algorithm used in this work was the Newton Raphson Method. Numerical Example In this work several benchmark examples considering elastoplastic materials were studied using the meshless method proposed, the NNRPIM. The example presented in this manuscript is the elastoplastic analysis of an infinite plate with a circular hole. Due to the double symmetry of the infinite plate, the problem can be simplified as represented in Figure 1(a). The material properties are also presented in Figure 1(a). The problem domain was discretized with 1654 nodes, Figure 1(b). The elastoplastic results obtained with the NNRPIM regarding the punctual displacements evolution with the load increment in control points: A, B and C (Figure 1(a)) were compared with the results obtained with commercial FEM software. The comparison shows that the NNRPIM solution is very close with the FEM solution. Regarding the stress distribution, it is possible to observe in Figure 2 that the stress fields distributions obtained, for distinct load levels, present a smooth variation. With Figure 2 it is also possible to visualise the evolution of the stress along the plate.

4

(a)

(b)

Figure 1 (a) Geometric Model; (b) Nodal mesh.

Figure 2 Stress field distribution (σxx) obtained for the infinite plate with a circular hole, using the NNRPIM.

Conclusions The most relevant conclusions of this work can be summarized as follows: i. The NNRPIM solution is always very close to the FEM solution. ii. The obtained variables fields are smooth and accurate. iii. The nodal connectivity, the integration mesh and the interpolation functions are defined based uniquely on the nodal mesh. Due to that, the NNRPIM can be seen as a truly meshless method. References

[1] [2] [3]

[4]

[5]

[6]

J. Bonet and R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis: Cambridge University Press, 1997. K.-J. Bathe, Finite Element Procedures in Engineering Analysis. New Jersey: PrenticeHall, Inc., 1982. L. M. J. S. Dinis, R. M. Natal Jorge, and J. Belinha, "Analysis of 3D solids using the natural neighbour radial point interpolation method," Computer Methods in Applied Mechanics and Engineering, vol. 196, pp. 2009-2028, 2007. V. P. Nguyen, T. Rabczuk, S. Bordas, and M. Duflot, "Meshless methods: A review and computer implementation aspects," Mathematics and Computers in Simulation, vol. 79, pp. 763–813, 2008. P. W. Randles and L. D. Libersky, "Smoothed Particle Hydrodynamics: Some recent improvements and applications " Computer methods in applied mechanics and engineering, vol. 139, pp. 375-408, 1996. L. D. Libersky and A. G. Petschek, "Smoothed particle hydro-dynamics with strength of materials," in The Next Free Lagrange Conference, 1991, pp. 248-257

5

Advances in Boundary Element Techniques XIV [7] [8]

[9] [10] [11]

[12] [13]

[14]

[15]

[16] [17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

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G. R. Liu and S. S. Quek, The Finite Element Method. A Pratical Course: ButterworthHeinemann, 2003. J. A. O. P. Belinha, "The Natural Neighbour Radial Point Interpolation Method," PhD, Departamento de Engenharia Mecânica, Faculdade de Engenharia da Universidade do Porto, Porto, 2010. T. Belytschko, Y. Y. Lu, and L. Gu, "Element-free Galerkin methods," International Journal for Numeric Methods in Engineering, vol. 37, pp. 229–256, 1994. P. Lancaster and K. Salkauskas, "Surfaces Generation by Moving Least Squares Methods," Mathematics of Computation, vol. 37, pp. 141-158, 1981. W. K. Liu, S. Jun, S. Li, J. Adee, and T. Belytschko, "Reproducing kernel particle methods for structural dynamics," International Journal for Numeric Methods in Engineering, vol. 38, pp. 1655-1679, 1995. Z. T. Atluri SN, "A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics," Computational Mechanics, vol. 22(2), pp. 117–27, 1998. K. M. Liew, X. Zhao, and A. J. M. Ferreira, "A review of meshless methods for laminated and functionally graded plates and shells," Composite Structures, vol. 93, pp. 2031–2041, 2011. G.R. Liu and Y.T. Gu, "A Point Interpolation Method for Two-Dimensional Solids," International Journal for Numeric Methods in Engineering, vol. vol. 50, pp. 937-951, 2001. J.G. Wang, G.R. Liu, and Y.G. Wu, "A Point Interpolation Method for Simulating Dissipation Process of Consolidation," Computational Methods in Applied Mechanics and Engineering, vol. 190, pp. 5907-5922, 2001. J. Braun and M. Sambridge, "A numerical method for solving partial differential equations on highly irregular evolving grids," Nature, vol. vol. 376, pp. 655-660, 1995. N. Sukumar, B. Moran, A.Yu Semenov, and V.V. Belikov, "Natural neighbour Galerkin methods," International Journal for Numeric Methods in Engineering, vol. 50, pp. 1-27, 2001. R. Sergio, S. Idelsohn, E. Oñate, N. Calvo, and F. Del Pin, "The Meshless Finite Element Method," International Journal for Numeric Methods in Engineering, vol. vol. 58, pp. 893912, 2003. S. Moreira, J. Belinha, L. M. J. S. Dinis, and R. M. N. Jorge, "The enriched natural neighbour radial point interpolation method for the analysis of crack tip stress fields," presented at the 1st International Conference of the International Journal of Structural Integrity, Faculty of Engineering, University of Porto, Portugal, 2012. S. Moreira, J. Belinha, L. M. J. S. Dinis, and R. M. N. Jorge. (2013) Análise de vigas laminadas utilizando o Natural Neighbour Radial Point Interpolation Method (NNRPIM). Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería. Dinis L.M.J.S., Jorge R.M.N., and Belinha J., "A Natural Neighbour Meshless Method with a 3D Shell-Like Approach in the Dynamic Analysis of Thin 3D Structures," ThinWalled Structures, p. DOI: 10.1016/j.tws.2010.09.023, 2011. L. M. J. S. Dinis, R. M. Natal Jorge, and J. Belinha, "Extensão do "Natural Neighbour Radial Point Interpolation Method" à análise de laminados compósitos," presented at the CMNE/CILAMCE 2007, Porto, 2007. J.G. Wang and G.R. Liu, "A Point Interpolation Meshless Method based on Radial Basis Functions," International Journal for Numeric Methods in Engineering, vol. 54, pp. 16231648, 2002. R. L. Hardy, "Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988," Computers & Mathematics with Applications, vol. 19, pp. 163-208, 1990. 6

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A simple BEM based solution procedure for some extended continuum mechanical problems: application to a microdilatation medium Nicolas Thurieau1, Richard Kouitat Njiwa2, M’Barek Taghite3 Université de Lorraine, Institut Jean Lamour - Dpt N2EV - UMR 7198 CNRS. Parc de Saurupt, CS 14234, 54042 Nancy Cedex France 1

[email protected], [email protected], 3 [email protected]

Keywords: Micromorphic, Microdilatation, Isotropic BEM, Meshfree strong form.

Abstract. Extended continuum mechanical approaches are nowadays increasingly adopted for the modelling of various types of materials including foams, porous solids, and biological tissues. As for the conventional mechanical field there is a need of numerical results in order to help the understanding of the material response. The present work is concerned with the simplest micromorphic medium (microdilatation). It is shown that accurate numerical results can be obtained for some problems by combining the conventional isotropic boundary elements method with local radial point interpolation.

Introduction A detailed understanding of the microscopic mechanisms is invaluable for optimizing the properties of new materials and a better understanding of the behavior materials. Microstructural information is taken into account either implicitly or explicitly in the description of the continuum. In recent years, there has been a growing interest in theories of extended continuum mechanics. One can mention the theory of micromorphic media of Eringen and Suhubi [1] which is intended to capture the impact of the microstructure on the overall response of the material. In the full theory, a material point of a micromorphic medium has twelve degrees of freedom: the three traditional components of displacement and the nine components of a micro-deformation tensor. Usually it is specialized depending on the particular microscopic aspect to be highlighted. In the simplest micromorphic theory, a material point can only dilate or contract. This theory is known as microdilatation theory and has already been applied to model foam as well as some porous media [2]. The literature on microdilatation media is not dense. Indeed it is not easy to understand and qualify they response using conventional mechanical tests. We believe that numerical experiments are extremely useful in this field. Following, we have taken the party to investigate the response of a 3D microdilatation medium to various types of loading. The work presented is essentially numerical and based on a specifically developed simulation tool. First of all, we present the governing equations of a microdilatation medium. Then, we describe the adopted numerical method called “local point interpolation – boundary element method” (LPIBEM). Finally, we discuss our results on some numerical examples.

Governing Equations In the theory of microdilatation medium, the material point is attached to a triad of directors that can only stretch. The material point has four degrees of freedom: the three components of the displacement vector and the scalar microdilatation. The field equations of such a medium under quasi-static evolution are: ߪ௝௜ǡ௝ ൌ Ͳ (1) (2) ‫ݏ‬௞ǡ௞ െ ‫ ݌‬ൌ Ͳ ߪ௜௝ is the stress tensor, the vector ‫ݏ‬௞ and the scalar ‫ ݌‬are known as the microstress vector and microstress function respectively. The latter can be viewed as an internal pressure.


413

When the considered solid is homogeneous and isotropic the constitutive relations are as follows (3) ߪ௜௝ ൌ ߣ‫ݑ‬௥ǡ௥ ߜ௜௝ ൅ ߤ൫‫ݑ‬௜ǡ௝ ൅ ‫ݑ‬௝ǡ௜ ൯ ൅ ߟ߰ߜ௜௝ (4) ‫ݏ‬௞ ൌ ܽ߰ǡ௞ ‫ ݌‬ൌ ߟ‫ݑ‬௥ǡ௥ ൅ ܾ߰ (5) ߰ is the micro-stretch function and ‫ݑ‬௜ the macroscopic displacement vector. ߣ and ߤ are the Lamé coefficients and ߟ, ܽ and ܾ are constitutive parameters. With ݊௝ the outward normal vector on the boundary, the macro and micro traction are given respectively by: (6) ‫ݐ‬௜ ൌ ߪ௝௜ ݊௝ (7) ‫ ݏ‬ൌ ‫ݏ‬௜ ݊௜ ൌ ܽ߰ǡ௜ ݊௜ From thermodynamic considerations, it has been established that, the constitutive constants must fulfil the following conditions (see e.g. [3]): ܾሺ͵ߣ ൅ ʹߤሻ െ ͵ߟ ଶ ൐ Ͳ, ߤ ൐ Ͳ, ܽ ൐ Ͳ and ܾ ൐ Ͳ.

Solution Method In the case of linear problems with well-established analytical fundamental solution, the boundary element method has already proved very efficient. When the fundamental solution of the field equations does not exist, the BEM loses its main appeal (reduction of the problem dimension by one) as traditional volume cell are needed in the so called field boundary element method. In order to alleviate this shortcoming, a number of strategies have been proposed. One can mention the dual reciprocity method (DRM) and the radial integration method (RIM) that allow to convert volume integral into surface ones. In recent years, a large number of researchers have invested in the development of the so called meshless or meshfree methods. Among the various meshless approaches, the local point interpolation method is very attractive because it is simple to implement. The accuracy of this approach deteriorates in presence of Neumann type boundary conditions which are practically inevitable when solving solid mechanic problems. Liu et al [4] have proposed to circumvent this difficulty by adopting the so called weak-strong form local point interpolation method. In a recent paper Kouitat [5] proposed a strategy to take advantage of the BEM method and the local point interpolation method. The LPI-BEM has proved efficient in the case of anisotropic elasticity [5], piezoelectric solid [6] and nonlocal elasticity [7]. It is adopted in this work and its main steps are presented below. First, assume that the kinematical primary variables are the sum of a complementary term and a particular term. That is ‫ ݑ‬ൌ ‫ ܪݑ‬൅ ‫ ܲݑ‬and ߰ ൌ ߰‫ ܪ‬൅ ߰ܲ . The complementary fields satisfy the following homogeneous equations: ு ு ு ቀߣ‫ݑ‬௥ǡ௥ ߜ௜௝ ൅ ߤ൫‫ݑ‬௜ǡ௝ ൅ ‫ݑ‬௝ǡ௜ (8) ൯ቁ ൌ Ͳ ǡ௝

ு ൌͲ ܽ߰ǡ௞௞

(9)

Accordingly, the particular fields solve: ௉ ௉ ௉ ቀߣ‫ݑ‬௥ǡ௥ ߜ௜௝ ൅ ߤ൫‫ݑ‬௜ǡ௝ ൅ ‫ݑ‬௝ǡ௜ ൯ቁ ൅ ߟ߰ǡ௜ ൌ Ͳ

(10)

௉ ܽ߰ǡ௞௞ െ ܾ߰ െ ߟ‫ݑ‬௥ǡ௥ ൌ Ͳ

(11)

ǡ௝

The introduced partition affects Neumann type boundary conditions. Indeed, the macrotraction and microtraction can be written as: ‫ݐ‬௜ ൌ ‫ݐ‬௜ு ൅ ‫ݐ‬௜௉ ൅ ߜ‫ݐ‬௜ (12) (13) ‫ ݏ‬ൌ ‫ݏ‬ு ൅ ‫ݏ‬௉ ு ு ௉ ௉ ு ௉ where ‫ݐ‬௜ு ൌ ቀߣ‫ݑ‬௥ǡ௥ ߜ௜௝ ൅ ߤ൫‫ݑ‬௜ǡ௝ ൅ ‫ݑ‬௝ǡ௜ ߜ௜௝ ൅ ߤ൫‫ݑ‬௜ǡ௝ ൅ ‫ݑ‬௝ǡ௜ ൯ቁ ݊௝ , ‫ݐ‬௜௉ ൌ ቀߣ‫ݑ‬௥ǡ௥ ൯ቁ ݊௝ , ߜ‫ݐ‬௜ ൌ ߟ߰݊௜ , ‫ ݏ‬ு ൌ ܽ߰ǡ௜ு ݊௜ , ‫ ݏ‬௉ ൌ ܽ߰ǡ௜௉ ݊௜ .

414


Equations (8) and (9) are similar to those of the classical small strain elastostatic and the well-known potential problem respectively. Applying the usual boundary element method, systems of equations of the following form are obtained: ሾ‫ܪ‬ሿሼ‫ݑ‬ு ሽ ൌ ሾ‫ܩ‬ሿሼ‫ ݐ‬ு ሽ (14) ഥ ሿሼ߰ ு ሽ ൌ ሾ‫ܩ‬ҧ ሿሼ‫ ݏ‬ு ሽ ሾ‫ܪ‬ (15) Let us now consider the solution of equations (10-11) by a local radial point collocation method. In this method [4], a field

N

Z x is approximated as: w x

M

¦ R r a ¦ P x b i

i

i 1

N

with the constraint condition:

¦ P x a j

i

i

j

j

.

j 1

0 j 1 m .

i 1

Ri r is the radial basis functions, N the number of nodes in the neighbourhood (support domain) of point x and M is the number of monomial terms in the polynomial basis ܲ௝ ሺ‫ݔ‬ሻ. Coefficients ai and b j can be determined by enforcing the approximation to be satisfied at the N nodes in the support domain. After some algebraic manipulation the interpolation is written in the compact form: Z x >)x @^Z/ L `. Using this relation, for each internal collocation point, equations (10) and (11) read:

~ @û P `/ L K > )ˆ @^\ `/ L ^0` >B @T >C @>B @>) T ~ T ˆ ` ^\ P ` b ^) ˆ `T ^\ ` K ^`T > ) @û`/ L a ^` ^`^) /L /L § w © wx1

û`/ L z

u

1 1

u12

z1, z2 , z3 T

u31 . . . u1N by B( z )

ª z1 «0 « ¬« 0

T

w · ¸ , ^\ `/ L wx3 ¸¹

w wx2

In these equations, ^` ¨¨

u2N

\

(16)

0

1

u3N , and matrix T

0

0

z2

z3

z2 0

0 z3

z1 0

0 z1

(17)

\ 2 \ 3 . . . \ N 2 \ N 1 \ N , T

B

is given in terms of a vector

T

0º z3 » . Matrix C is the usual Voigt representation of the » z2 ¼»

isotropic elastic constants and N the number of points in the influence domain of point x. Now, consider all internal collocation points and if the particular integrals are selected such that u P 0 and \ P 0 at all boundary points, relations (16, 17) lead to systems of equations that can be written in the compact form: ሼ‫ݑ‬௉ ሽ ൌ ሾ‫ܣ‬ሿሼ߰ሽ ሼ߰ ௉ ሽ ൌ ሾ‫ܤ‬ሿሼ߰ሽ ൅ ሾ‫ܥ‬ሿሼ‫ݑ‬ሽ

(18) (19)

After some simple algebraic calculations, one obtains:

>H~ @û` >G@^t` >F~@^\ ` >Hˆ @ ^\ ` >G @ ^ s` >F @ û`

(20)

Remarkably, the final equations contain boundary primary variables and internal kinematical unknowns as in traditional BEM. Boundary conditions can be taken into account as usual and the resulting system of equations solved by a standard direct solver.


415

Numerical examples ௤

In this work the multi-quadrics radial basis functions are adopted: ܴ௜ ሺ‫ݎ‬ሻ ൌ ൫‫ݎ‬௜ଶ ൅ ܿ ଶ ൯ where ‫ݎ‬௜ ൌ ԡ‫ ݔ‬െ ‫ݔ‬௜ ԡ, ܿ and ‫ ݍ‬are known as shape parameters. The shape parameter ܿ is taken proportional to a minimum distance ݀଴ defined as the maximum value among the minimum distances in the x, y and z directions between collocation points. More specifically we set ܿ ൌ ߚ݀଴ . The material parameters (cf. eq (3-5)) used for the results presented in this work are: μ= 0.85 GPa, O = 3.4 GPa, a = 26 kN, b = 26 GPa.

Example 1: Loading of a unit cube This academic case is considered in order to assess the effectiveness of the proposed approach. It is well known that the shape parameters of the multi-quadrics radial basis functions affect the solution accuracy. Accordingly, numerical results are always given with associated optimal values of the shape parameters. As a first case, let us consider the unidirectional loading of a unit cube in the third direction. In this case, a uniform traction is applied on the top surface of the specimen. The microdilatation is uniform in the specimen leading to zero microstress vector [8]. The analytical solution of the problem is available and reads: ߟ ߰ൌ ଶ ߪ ͵ߟ െ ܾሺ͵ߣ ൅ ʹߤሻ ଷଷ ͳ ߣ ߟଶ ቈ െ ቉ߪ ߝଵଵ ൌ ߝଶଶ ൌ െ ͵ߣ ൅ ʹߤ ʹߤ ሺ͵ߣ ൅ ʹߤሻܾ െ ͵ߟଶ ଷଷ ଶ ͳ ߣ൅ߤ ߟ ቈ ൅ ቉ߪ ߝଷଷ ൌ ሺ͵ߣ ൅ ʹߤሻܾ െ ͵ߟ ଶ ଷଷ ͵ߣ ൅ ʹߤ ߤ Note that in this case, the solution is not affected by the parameter ܽ since the microstress is identically zero. For the results presented in this part, the boundary of the unit cube is subdivided into 24 nine-node elements. 27 internal collocation centres are used. For the multi-quadrics shape parameters q = 1.03 and E = 10-3, the obtained numerical solution are practically analytical ones. These results are undisturbed over a wide range of variation of these parameters (q between 0.5 and 1.5, E between 10-4 and 10-1). In all figures in this work, the undeformed geometry of the specimen is drawn with dashed lines and the deformed shape is obtained by magnifying the displacement 50 times. The effect of the particle microdilatation is highlighted in figure 1, where the results are compared to those of the pure elastic specimen with the same material parameters. Figure 1A compares the transverse displacements within the plane (x=0). It is observe that, due to microdilatation, the lateral contraction of the specimen is about 27.3 % lower than what would have been obtained in a pure elastic material. If the material were purely elastic, we would obtain a lower axial displacement. Figure 1B shows that this is not because the axial displacement is larger (about 11%). A1

A2

B1

B2

Figure 1: Transverse displacement (A) and displacement in the loading direction (B) in a unit cube under unidirectional tension load (10-2 GPa): (1) elastic material, (2) microdilatation (K = -8 GPa).

416


This is a particular feature of a microdilatation medium. These results are undisturbed when changing the load level, the number of internal collocation points and the number of boundary elements. Then, the numerical approach is considered effective and accurate for this type of loading.

Example 2: Loading of a tubular specimen This case is considered in order to assess the ability of the proposed approach to deal with a slightly more complex geometry. A tubular specimen with height 0.75mm has an external radius of 0.75 mm and an internal radius of 0.25 mm. The results presented are obtained with 240 regularly spaced internal collocation centres. The boundary of the specimen is subdivided into 80 nine-node elements. In the case of a unidirectional loading in the z-direction (applied traction or imposed displacement), the obtained numerical results are practically analytical ones (same expressions as those in the preceding section but in cylindrical coordinates). For the following cases, results are shown on a half-plane due to the symmetry. Assume now that the microtraction free outer lateral boundary is imposed a given radial displacement (0.0027 mm). The other surfaces of the specimen are free of traction and microtraction. The numerical results of the radial displacement (for various values of K) are compared in figures 2 I) to the purely elastic solution. (a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

I)

II)

Figure 2: Radial displacement I) and axial displacement II) in the plane x=0: (a) elastic case, (b) K = 2GPa, (c) K = -8GPa, (d) K = -10.15GPa. It is observed that the higher the absolute value of K, the smaller the radial displacement of the inner surface. Note that the radial displacement of the pure elastic case is higher than all the preceding ones. This is in agreement with results of the preceding section. Consider the axial displacement (see Fig. 2 II)), there is an elongation of the purely elastic specimen. The amount of extension decreases with K down to approximately -8 GPa. As a surprising effect, the specimen shortens for much higher absolute values of K. This is another particular feature of the model, which can simulate material with negative Poisson ratio (also called auxetic materials).


417

As a final example, let us consider that the tubular specimen is subjected simultaneously to tension loading (10-2 GPa) in the axial direction and to a radial displacement (0.0027 mm) on the outer lateral surface. As can be observed in figure 3a, if the material is purely elastic, the overall response is dominated by the imposed displacement. The specimen shortens. As the absolute value of the coupling parameter K increases, the amplitude of shortening of the specimen decreases. As in the former case for higher absolute values of K the specimen elongates. Imposed traction

(a)

(b)

(c)

(d)

Imposed displacement

Imposed traction

Loading

Figure 3: Radial displacement in the plane x=0: (a) elastic case, (b) K = -2GPa, (c) K = -8GPa, (d) K = -10.15GPa.

Conclusion In this work a boundary element based method namely the LPI-BEM is used for the numerical solution of 3D microdilatation media. The approach uses a partition of the primary field variables and then couples conventional boundary element method with a strong form point collocation method. The obtained and presented results prove effectiveness and accuracy of the method. This alternative to other mesh reduction approach seems promising for a wide range of problems. The results reveal that the model can capture particular material response such as those of auxetic materials. We believe that many types of microstructural effect can be represented in this simple way. Following, such simple approach can be applied to the modelling of living tissue in order to highlight the effect of microscopic damage.

References [1] A. C. Eringen et E. S. Suhubi, « Nonlinear theory of simple micro-elastic solids—I », International Journal of Engineering Science, vol. 2, no 2, p. 189̻203, mai 1964. [2] H. Ramézani, H. Steeb, et J. Jeong, « Analytical and numerical studies on Penalized Micro-Dilatation (PMD) theory: Macro-micro link concept », European Journal of Mechanics - A/Solids, vol. 34, p. 130̻ 148, juill. 2012. [3] D. Iesan et A. Pompei, « On the equilibrium theory of microstretch elastic solids », International Journal of Engineering Science, vol. 33, no 3, p. 399̻410, févr. 1995. [4] G. R. Liu et Y. T. Gu, « A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids », Journal of Sound and Vibration, vol. 246, no 1, p. 29̻46, sept. 2001. [5] R. Kouitat Njiwa, « Isotropic-BEM coupled with a local point interpolation method for the solution of 3D-anisotropic elasticity problems », Engineering Analysis with Boundary Elements, vol. 35, no 4, p. 611̻615, avr. 2011. [6] N. Thurieau, R. Kouitat Njiwa, et M. Taghite, « A simple solution procedure to 3D-piezoelectric problems: Isotropic BEM coupled with a point collocation method », Engineering Analysis with Boundary Elements, vol. 36, no 11, p. 1513̻1521, nov. 2012. [7] M. Schwartz, N. T. Niane, et R. Kouitat Njiwa, « A simple solution method to 3D integral nonlocal elasticity: Isotropic-BEM coupled with strong form local radial point interpolation », Engineering Analysis with Boundary Elements, vol. 36, no 4, p. 606̻612, avr. 2012. [8] N. Kirchner et P. Steinmann, « Mechanics of extended continua: modeling and simulation of elastic microstretch materials », Computational Mechanics, vol. 40, no 4, p. 651̻666, sept. 2007.

418


Efficient BEM Stress Analysis of 3D Generally Anisotropic Elastic Solids With Stress Concentrations and Cracks Y.C. Shiah1, C.L. Tan2* and Y.H. Chen3 1. 2 3

Department of Aeronautics and Astronautics National Cheng Kung University, Tainan 701, Taiwan, R.O.C. Department of Mechanical & Aerospace Engineering Carleton University, Ottawa, Canada K1S 5B6 Department of Computer Science & Information Management Providence University, Taichung, Taiwan, R.O.C. (* Corresponding author; Email: [email protected])

Keywords: Anisotropic elasticity, Green’s function, Fourier series, stress concentrations, fracture mechanics.

Abstract. The present authors have recently proposed an efficient, alternative approach to numerically evaluate the fundamental solution and its derivatives for 3D general anisotropic elasticity. It is based on a double Fourier series representation of the exact, explicit form of the Green’s function derived by Ting and Lee [5]. This paper reports on the successful implementation of the fundamental solution and its derivatives based on this Fourier series scheme in the boundary element method (BEM) for 3D general anisotropic elastostatics. Some numerical examples of stress concentration problems and a crack problem are presented to demonstrate the veracity of the implementation. The results of the BEM analysis of these problems show excellent agreement with those obtained using the commercial finite element code ANSYS and with known analytical solutions in all cases. Introduction The boundary element method (BEM) is well established as an efficient computational tool for threedimensional (3D) linear elastic stress analysis of isotropic bodies. It is, however, significantly less so for treating 3D generally anisotropic elastic solids. The primary reason lies in the relatively slow progress made over the years for an efficient means to numerically evaluate the fundamental solution (or Green’s function) and its derivatives for this class of problems. These quantities are necessary items in the development of the boundary integral equation (BIE) which is the analytical basis of the BEM. The Green’s function, U(x), for a 3D anisotropic medium as first derived by Lifschitz and Rosenzweig [1] is expressed as a line integral around a unit circle with the integrand containing the Christoffel matrix defined in terms of elastic constants. Since then, there have been numerous efforts to reformulate it into simpler or more explicit analytical forms, including those carried out in the context of BEM development (see, e.g. [2-12]). In the pioneering work of Wilson and Cruse [4], a large database of the fundamental solution is generated in advance for a given material from direct numerical computations of the Lifschitz and Rozentsweig’s solution, and an interpolation scheme is used in their BEM implementation. The efficiency and accuracy of this scheme have been called into question for highly anisotropic materials and various other schemes to this end have been proposed for use in BEM, see, e.g., [8, 11,12]. Of significance to note here is that Ting and Lee [5] have derived a fully algebraic, explicit form of the 3D anisotropic Green's function, expressed in terms of Stroh’s eigenvalues. Lee [13, 14] further showed how its derivatives could be obtained; the complete explicit expressions for the derivatives for general anisotropy were, however, derived and presented only by the present authors and their co-workers in [15, 17]. They were also implemented in Tan et al [16] to analyse some benchmark problems by the BEM. In their attempts to develop less elaborate forms of this U(x) and its derivatives to facilitate efficient numerical evaluation of these quantities, the present lead authors have very recently [18] proposed that advantage can be taken of the periodic nature of the spherical angles if U(x) is expressed in spherical coordinates. This allows the Green’s function to be represented by a double Fourier series and its derivatives can also be obtained in a straightforward manner by direct differentiation of the series. Not only are the resulting formulations significantly more concise, a very important advantage is that the evaluation of the coefficients of the Fourier series is performed only once, regardless of the number of field points involved in the BEM analysis. This makes the scheme very efficient indeed without any sacrifice in accuracy. To further enhance


419

the computational efficiency, the authors [19] reformulated the scheme by organising and simplifying the terms, and taking advantage of some of the characteristics of the Fourier series, so that less number of terms needs to be summed. This has implications for the efficiency of the numerical algorithm in the BEM analysis, noting too that slightly more refined meshes are typically needed when treating, for example, 3D stress concentration and cracked problems of anisotropic bodies than of isotropic ones. The reformulated Fourier series scheme for U(x) and its derivatives have been implemented into an existing BEM code for 3D general anisotropic elasticity. Some examples involving stress concentrations and a crack problem is presented in this paper to demonstrate this and the accuracy of the solutions obtained. Before this, a review of the approach is in order. Boundary integral equation and fundamental solutions of 3D anisotropic elastic bodies The boundary integral equation (BIE) that relates the displacements, ui, to the tractions, ti, on the surface S of the domain can be expressed in indicial notation as Cij ( P ) u i( P ) ³ u i( Q ) T ij( P , Q ) dS S

³ t ( Q )U i

ij

(1)

( P , Q ) dS

S

In Eq. (1), Uij(P,Q) U(x) and Tij(P,Q) represent the fundamental solutions of displacements and tractions, respectively, in the xi-direction at the field point Q due to a unit load in the xj-direction at P in a homogeneous infinite body; also, Cij(P) depends on the geometry of the surface at P. The numerical evaluation of U(x) for generally anisotropic materials proposed by Ting and Lee [5] has been discussed by Shiah et al [15]. Nevertheless, m x3 it is useful to first provide a brief review. Q Field Pt. With reference to Fig.1, let n and m be two mutually perpendicular n unit vectors on the oblique plane at Q normal to the position vector x; r I the vectors [n, m, x/r] forms a right-angle triad. By considering a spherical coordinate system as shown, the explicit form of the Green’s P x2 function can be expressed as Source Pt. T 4 1 1 n ( ) U( x )= (2) ¦ qn ˆ , 4S r n= 0 x1

where r represents the radial distance between the source point P and the field point Q; qn , ˆ ( n ) , and are given by [15] -1 ° ° 2 E1 E 2 E 3 ® ° 1 qn = ° ¯ 2 E1 E 2 E 3

ª ° 3 º ½° ptn « Re ®¦ ¾ -G n 2 » p -p p -p 1 t= ° ° t t+1 t t+ 2 ¿ ¬« ¯ ¼» ° 3 ptn- 2 pt 1 pt 2 ½° Re ®¦ ¾ °¯ t=1 pt -pt+1 pt -pt+2 °¿

Fig.1: Vectors definitions in the spherical coordinate system

for n= 0, 1, 2,

for n= 3, 4,

* (i( n)1)( j1)(i 2)( j 2) * ((in)1)( j 2)(i 2)( j1) , (i, j 1, 2, 3) ,

*ˆ ij( n )

N ik =Cijks m j ms , m (sinT , cosT , 0) .

(3a) (3b)

(3c)

In Eq. (3a), pi, are the Stroh’s eigenvalues; they the roots of the sextic equation, obtained by setting = 0; and Ei are the positive imaginary parts of pi. In Eq.(3b), are defined as follows:

* (4) pqrs

N pqN rs , * (3) V pqN rs +N pqVrs , * (2) pqrs pqrs

* (1) pqrs

V pqWrs +VrsW pq , * (0) pqrs

where W, V are given by

W pqWrs ,

N pqWrs +N rsWpq +V pqVrs ,

(4)

420


Wik =Cijks n j ns , Vik =(Cijks +Ckjis )n j ms .

(5)

In the above equations, Cijks are the stiffness coefficients of the anisotropic material. It has already been shown [15, 16] that the direct computation of Eq.(2) for U(x) is relatively straightforward and very efficient indeed. Lee [13] and Shiah et al [15] have also obtained the analytical expressions for the derivatives of U(x). They are in terms of some very high order tensors and although direct to evaluate, is found to be the best form for computations. This is re-examined by Lee [14], who showed that the very high order tensors can be avoided by differentiating U(x) with respect to spherical coordinates as an intermediate step, separating the terms associated with the radial distance, and then using the usual chain rule. This approached was followed in Shiah et al [17], where explicit expressions for the 1st and 2nd derivatives of U(x) in general anisotropy are obtained and implemented in the BEM. Although relatively more efficient to compute, their implementation is somewhat tedious because of their lengthy forms. As an alternative approach for the numerical evaluation of U(x), the present authors very recently [18] proposed a Fourier series representation of U(x) and its derivatives in terms of the spherical coordinates. This scheme yields significantly more concise expressions that can be easily implemented in BEM programming and was shown to be computationally very much more efficient for the evaluation of the Green’s function and its derivatives. The Green’s function can be expressed in the spherical coordinates as U uv ( r, T , I )

H uv (T , I ) , 4S r

u, v

1, 2, 3 .

(7)

By virtue of its periodical nature, one may further rewrite the H uv in Eq.(7) into a Fourier series, viz.

H uv (T , I )

D

D

m

n

¦D ¦D O

( m ,n ) uv

e

i mT n I

,

(8)

where D is an appropriately large integer for convergence of the series and the unknown Fourier coefficients Ouv( m,n) are given by

Ouv( m,n)

1 4S 2

³ S ³ S H T ,I e S

S

uv

i mT nI

dT dI .

(9)

Equation (8) can be numerically determined by, for example the Gaussian quadrature scheme as follows,

Ouv( m,n )

1 k k ¦¦ wp wq fuv( m,n ) S [ p ,S [q , 4 p 1q1

(10)

where f uv( m,n ) represents the integrand in Eq.(8); k is the number of the Gauss abscissa, p, and wp is the corresponding weight. It should be noted that the computation of the Fourier coefficients is performed only once, irrespective of the number of nodes and elements in the BEM mesh; the CPU time for this evaluation is trivial indeed in a complete BEM analysis. To reduce the number of terms truly required in the series for ( m, n ) ( m, n ) ( m,n ) into its real part Ruv and imaginary part I uv as the computations, Tan et al [19] separated Ouv follows,

Ouv(m,n) Ruv(m,n) iIuv(m,n) . and expressed the Green’s function as

(11)


D D ª R uv( m ,n ) cos mT Iuv( m ,n ) sin mT cos nI º ½ ° ¦¦ « »° ( m ,n ) ( m ,n ) 1 °° m 1 n 1 «¬ Rˆ uv sin mT Iûv cos mT sin nI »¼ °° ® ¾, 2S r ° D § R (0,m ) cos mI I (0,m ) sin mI · Ruv(0,0) ° uv uv ¸¸ ° ¦ ¨¨ ( m ,0) ° ( m ,0) 2 ¯° m 1 © Ruv cos m T I uv sin mT ¹ ¿°

U uv

where Ruv

( m ,n )

( m ,n ) , Rˆ uv , Iuv

( m ,n )

421

(12)

( m ,n ) , and Iûv are given by

R uv( m ,n ) I ( m ,n ) uv

Ruv( m ,n ) Ruv( m , n ) , Rˆ uv( m ,n ) I ( m ,n ) I ( m , n ) , Iˆ( m ,n ) uv

uv

Ruv( m ,n ) Ruv( m , n ) ,

(13)

I uv( m ,n ) I uv( m , n ) .

uv

Since no operations of complex numbers are involved and the number of terms in the series is minimized, the numerical computations are very efficient indeed. The derivatives of Eq. (7) can be obtained in a straightforward manner by simply differentiating the Fourier series of U uv as follows, U uv ,l

wU uv wr wU uv wT wU uv wI . wr wxl wT wxl wI wxl

(14)

This can be shown [19] to result in the following form: D ªD D Ruv(0,0) º ½ m ( m ,n ) m ( m ,n ) ° Zl (T , I ) « ¦¦ * uv (T ) cos nI * uv (T ) sin nI ¦ J uv (T ) J uv (I ) ° 2 »¼ ° m 1 ¬m 1 n 1 ° D °° 1 °° ªD D º m ( m ,n ) ( m ,n ) ® Z c(T , I ) « ¦¦ m * uv (T ) cos nI *ˆ uv (T )sin nI ¦ m Juv (T ) » ¾ , (15) 2S r 2 ° l m 1 ¬m 1 n 1 ¼ ° D ° ° ªD D º ( m ,n ) m ( m ,n ) ° Zlcc(T , I ) « ¦¦ n * uv (T )sin nI * uv (T ) cos nI ¦ m Jûv (I ) » ° m 1 ¬m 1 n 1 ¼ ¯° ¿°

U uv ,l

where ( m ,n )

*uv (T ) * ( m ,n ) (T ) uv

m

( m ,n )

R uv( m,n ) cos mT Iuv( m ,n ) sin mT , *uv (T ) R ( m,n ) sin mT I ( m ,n ) cos mT , *ˆ ( m,n ) (T ) uv

uv

uv

m

Rûv( m,n ) sin mT Iûv( m ,n ) cos mT , Rˆ ( m ,n ) cos mT Iˆ( m ,n ) sin mT , uv

uv

J uv (T )

Ruv( m ,0) cos m T I uv( m ,0) sin m T ,

Juvm (T ) Zl (T , I ) Zl (T , I ) Zl (T , I )

Ruv( m ,0) sin m T I uv( m ,0) cos m T , Jûvm (I ) Ruv(0,m ) sin m I I uv(0,m ) cos m I , sin I cos T , Zlc(T , I ) sin T / sin I , Zlcc(T , I ) cos I cos T for l 1, for l 2, sin I sin T , Zlc(T , I ) cos T / sin I , Zlcc(T , I ) cos I sin T cos I ,

Zlc(T , I ) 0,

J uv (I )

Ruv(0,m ) cos mI I uv(0,m ) sin mI ,

Zlcc(T , I ) sin I

for l

(16)

3.

Compared to the previous exact analytical forms, the above is even simpler to implement into an existing BEM computer code. However, it should be noted that in the above expressions, there is numerical singularity when I =0 or S. This is due to the multi-valued definition of T when I =0 or S. The details of this are discussed in [19] and the problem may be easily resolved by re-definition of the coordinates. Once the 1st-order derivatives of U(x) are computed, the fundamental solution of tractions can be determined by Tij

Cikmn U mj ,n U nj ,m N k / 2 ,

(17)

422


where N k denotes the components of the unit outward normal vector at the field point. In a similar manner, the derivatives of higher orders may also be derived, but they are not of concern for the study in this paper. Numerical examples Three examples are presented here to demonstrate the successful implementation of the Fourier series scheme for the computation of the fundamental solution and its derivatives in BEM for 3D anisotropic elasticity. The BEM code employs the standard quadratic isoparametric element formulation. For the numerical calculations of U(x) and its derivatives, the quantity D for the Fourier series was set to be 16, and the number of Gauss points, k, was deliberately set to 64, noting that the latter has very little influence on the overall CPU time of the BEM analysis as the Fourier coefficients are only computed once for a given material system. The first example, shown in Fig. 2, is a sphere containing a solid spherical rigid inclusion (R2/R1=2). It is subjected to uniform hydrostatic tensile P on its outer surface while the innner surface is fully constrained to the rigid inclusion. An alpha-quartz single crystal is chosen as the material, whose stiffness coefficients, denoted by C* , are given by [20]:

C

*

§ 87 .6 6.07 13 .3 17 .3 ¨ ¨ 6.07 87 .6 13 .3 17 .3 ¨ 13 .3 13 .3 106 .8 0 .0 ¨ 0 .0 57 .2 ¨ 17 .3 17 .3 ¨ 0 .0 0 .0 0 .0 0 .0 ¨ ¨ 0 .0 0 .0 0 .0 0 .0 ©

0 .0 0 .0 0 .0 0 .0 57 .2 17 .3

0 .0 · ¸ 0 .0 ¸ 0.0 ¸ GPa. (18) ¸ 0 .0 ¸ 17 .3 ¸ ¸ 40 .765 ¸¹

For demonstrating the capability of the approach in dealing with general anisotropy, the material principal axes are arbitrarily rotated with respect to the x1-, x2- and x3-axis by -200, 550, and 1450, respectively, with positive rotation angles being defines in the conterclockwise direction. This results in the material stiffness matrix defined in the global Cartesian coordinates as §102.22 10.90 1.16 3.39 7.76 20.08 · ¨ 10.90 114.23 4.39 13.87 3.57 11.79 ¸ ¨ ¸ ¨ 1.16 4.39 120.20 0.19 17.52 0.01 ¸ C ¨ ¸ GPa. (19) ¨ 3.39 13.87 0.19 37.77 2.88 1.42 ¸ ¨ 7.76 3.57 17.52 2.88 37.95 0.19 ¸ ¨ ¸ © 20.08 11.79 0.01 1.42 0.19 52.12 ¹

x3 I

V0

R2 R1

x2 T

x1

Fig.2: A sphere with a solid inclusionExample 1

Fig.3: Meshes used in the BEM and ANSYS- Example 1

For the BEM analysis, 240 quadratic isoparametric elements with 644 nodes were employed. For verification, the problem was also analysed by ANSYS, using 61440 SOLID186 elements with 70092 nodes. Figure 3 shows the mesh discretisations of the BEM full model and a partial (one-eighth) ANSYS FEM model for the sake of clarity. The computed variations of the normalised hoop and meridional stress, VTT/V0 and VII/V0, at the inner and outer radius around the equator in the x1-x2 plane are shown in Figure 4. It can be seen that the BEM results are in excellent agreement with those obtained from the FEM analysis by ANSYS. As expected, the fluctuations of the stress concentrations are more evident on the inner surface. Both analyses were carried out and clocked on a PC-based computer equipped with quad-core Intel CPU; the runs recorded 43.96 seconds and 27 minutes for the BEM and ANSYS operations, respectively.


VTT/V0

VII/ V0

BEM ANSYS

1.2

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

-0.2

-0.2 45

90

r=R1 r=R2

1.0

0.8

0

BEM ANSYS

1.2

r=R1 r=R2

1.0

423

135

180

225

270

315

360

0

T (Degrees)

45

90

135

180

225

270

315

360

T (Degrees)

Figure 4: Normalized stresses along the equator on the x1-x2 plane – Example 1 The second example considered is an ellipsoidal cavity in a cylindrical bar, as shown in Figure 5. Three values of the aspect ratio, R2/R1, as defined in the figure, are considered, namely, 1.0, 2.0, and 3.0. The radius of the cylindrical bar R0=10 R1 is taken and its length is 2R0. The bar is subjected to axial tension V0 at one end and is fully constrained at the other. The same material properties as in the previous example are used again for the analysis. Also shown in Fig.5 is the BEM full model, where 216 quadratic elements were employed. For verification, the problem was also analysed by ANSYS, using 216934 SOLID187 elements. Due to the relatively large dimension of the cylinder as compared with the size of the ellipsoid, this case actually approximates an infinite anisotropic domain with an ellipsoidal cavity when it is subjected to a remote uniaxial tension. The variations of calculated stress concentration factor (SCF), defined by V33/V0, around the surface of the ellipsoidal cavity at x3=0 are shown in Fig.6, where again, excellent agreement between the two sets of results from the BEM and FEM analyses is achieved. The cpu times recorded for the BEM and ANSY analysis were 32.65 seconds and 26 minutes, respectively. BEM ANSYS

V33/V0

o

R2/R1= 1.0

2.6

R2/R1= 2.0

2.4

R0 x3

R1 R2

R2/R1= 3.0

2.2 2.0

x2 x1

1.8

2R0

T

1.6 1.4 1.2 1.0 0

45

90

135

180

225

270

315

360

T (Degrees)

Fig. 5: An ellipsoidal cavity in a circular bar in tension and BEM mesh– Example 2

Figure 6: Stress concentration on the surface of the ellipsoidal cavity; x3=0

The last example treated here is a penny-shaped crack in an infinite transversely isotropic medium; it is a special case of anisotropy. It has also been has been studied using BEM by Tan el al [21], and the exact solutions for the stress intensity factors (SIFs) under different load conditions are available. For the material, a graphite-epoxy composite with the following stiffness coefficients [22] is considered:

424


C11 = 13.92 MPa, C12 = 6.92 MPa, C13 = 6.44 MPa, C33 = 160.7 MPa, C44 = 7.07 MPa.

(20)

Two load cases are treated, namely, (i) remote uniform tension of 33= o and (ii) remote shear stress 23= o, applied at the top and bottom faces of the cube modelled. For the BEM analysis, the body is modelled as a cube with side lengths ten times the diameter, 2a, of the crack which lies on the mid-plane; this is shown in Fig.7. Although sub-region interface exterior surface advantage can be taken of the planes of symmetry, the full problem was modelled as the mesh will be used for more general case of loading and anisotropy in other studies. Figure 7: BEM mesh for Example 3; 216 Special O(r-1/2) traction-singular crack-front elements are elements, 604 nodes employed and the SIFs are obtained using the computed traction coefficients at the crack-front nodes using the well-established “traction-formula” [23], as follows:

(KI )A

(t3* ) A S l , ( K II ) A

(t1* ) A S l , ( K III ) A

(t2* ) A S l ,

(21)

In Eq. (21), (ti* ) A are the traction coefficients computed at the crack-front node A of the traction-singular element and l is the width of this element. The width of the crack-front element for this case was set to be l/a = 0.15. For case (i), the exact solution for the normalized stress intensity factor, K I / V o S a , is 0.637. The BEM result computed using the traction formula is 0.638, with an error less than 1%. For load case (ii), the exact normalized SIFs for the material properties used in the analysis are [ K II / W o S a 0.8115sin T , K III / W o S a 0.4617 cos T ] ; the angle here is the angular position measured from the x1-axis. Table 1 lists all computed SIFs of case (ii) for various angular positions Table 1: Computed normalised SIFs at various angular positions of a penny-shaped crack - Example 3

T

K II / W o S a

(Deg.) Exact

BEM

K III / W o S a

|Error %|

Exact

BEM

|Error %|

0

0.0000

0.0000

N/A

0.4617

0.4566

1.10

15

0.2100

0.2097

0.03

0.4459

0.4422

0.83

30

0.4057

0.4042

0.15

0.3999

0.3954

1.12

45

0.5738

0.5740

0.02

0.3264

0.3237

0.83

60

0.7016

0.7004

0.17

0.2309

0.2284

1.08

75

0.7838

0.7842

0.05

0.1194

0.1185

0.75

90

0.8115

0.8087

0.35

0.0000

0.0000

N/A

It can be seen that the SIFs calculated by the BEM using the Fourier approach agree with the exact solutions with the maximum error of 1.12%. Conclusions The efficient evaluation of the fundamental solutions is critical to the success of the BEM as a numerical tool for treating three-dimensional generally anisotropic bodies. Very recently, the lead authors [18] presented an efficient scheme to compute the fundamental solutions, where the Green’s function and


its derivatives are represented by a Fourier series; a modification to the scheme was also very recently [19] developed to reduce the number of calculations required in the series summation. This revised Fourier series scheme for the evaluation of the Green’s function has been implemented into a BEM code for threedimensional anisotropic elasticity. The success of this implementation has been demonstrated in this paper by two examples in stress concentrations where the results are compared with those obtained by FEM. Excellent agreement of the results have been obtained while requiring significantly less computational effort. A fracture problem has also been presented and the BEM results when compared with the exact analytical solutions again showed very good agreement indeed. Acknowledgement The authors gratefully acknowledge the financial support from the National Science and Engineering Research Council of Canada and the National Science Council of Taiwan (NSC 99-2221-E-006-259-MY3). References [1] Lifshitz, I.M., Rozenzweig, L.N., Zh. Eksp. Teor. Fiz. 17, pp.783-791 (1947). [2] Synge, J.L., The Hypercircle in Mathematical Physics. Cambridge University Press, Cambridge (1957) [3] Barnett, D.M., Phy. Stat. Solid (b) 49, 41-748 (1972). [4] Wilson, R.B., Cruse, T.A., Int. J. Numer. Methods Engng. 12, 1383-1397 (1978). [5] Ting, T.C.T., Lee, V.G., Q. J. Mech. Appl. Math. 50, 407-426 (1997). [6] Nakamura, G., Tanuma, K., Q. J. Mech. Appl. Math. 50, 179-194 (1997). [7] Wang, C.Y., J. Engng. Math. 32, 41-52 (1997). [8] Sales, M.A., Gray, L.J., Comp. & Struct. 69, 247-254 (1998). [9] Pan, E., Yuan, F.G., Int. J. Numer. Methods Engng. 48, 211-237 (2000). [10] Tonon, F., Pan, E., Amadei, B., Comp. & Struct. 79, 469-482 (2001). [11] Phan, P.V., Gray, L.J., Kaplan, T., Comm. Numer. Methods Engng. 20, 335-341 (2004). [12] Wang, C.Y., Denda, M., Int. J. Solids Struct. 44, 7073-7091 (2007). [13] Lee,V.G., Mech. Res. Comm. 30, 241–249 (2003). [14] Lee,V.G., Int. J. Solids Struct. 46, 1471-1479 (2009). [15] Shiah, Y.C., Tan, C.L., Lee, V.G., CMES-Comp. Modeling Eng. & Sci. 34(3), 205-226 (2008). [16] Tan, C.L., Shiah, Y.C., Lin, C.W., CMES-Comp. Modeling Eng. & Sci. 41, 195-214 (2009). [17] Shiah, Y.C., Tan, C.L., Lee, R.F., CMES-Comp. Modeling Eng. & Sci. 69, 167-197 (2010). [18] Shiah, Y.C., Tan, C.L., Wang, C.Y., Engng. Analysis Boundary Elem. 36, 1746-1755 (2012). [19] Tan, C.L., Shiah, Y.C., Wang, C.Y., International Journal of Solids and Structures (2013). (Submitted) [20] Huntington, H.B., The Elastic Constants of Crystal. Academic Press, New York (1958). [21] Tan, C.L., Shiah, Y.C., Armitage, J.R., Hsia, W.C., Advances in Boundary Element Techniques XI, 468-473, Eds. C.H. Zhang, M.H. Aliabadi, M. Schanz, Berlin, Germany, 12-14 July (2010). [22] Saez, A., Ariza, M.P., Dominguez, J., Engng. Analysis Boundary Elem., 20,287-298 (1997) [23] Luichi, M.L., Rizzuti, S., Int. J. Num. Methods Engng., 24, 2253-2271 (1987)

425

426


Optimization of Cathodic Protection Systems Combining Genetic Algorithms and the Method of Fundamental Solutions W. J. Santos1, J. A. F. Santiago2 and J. C. F. Telles3 Department of Civil Engineering, COPPE/UFRJ. Rio de Janeiro – Brazil 1

[email protected], 2 [email protected], 3 [email protected]

Keywords: Optimization, Gauss-Newton method, cathodic protection, MFS, BEM.

Abstract. The method of fundamental solutions (MFS) is used for the solution of Laplace's equation, with nonlinear boundary conditions, aiming at analyzing cathodic protection systems. The metallic surfaces, in contact with the electrolyte, to be protected, are characterized by a nonlinear relationship between the electrochemical potential and current density, called polarization curve. Thus, the calculation of the intensities of the virtual sources entails a nonlinear least squares problem. In this work, it is proposed the Gauss-Newton (GN) method to minimize the resulting nonlinear objective function, whose design variables are the coefficients of the linear superposition of fundamental solutions and the positions of the virtual sources located outside the problem domain. First, an example is presented to validate the standard MFS formulation as applied to the simulation of cathodic protection systems, comparing the results with a direct boundary element (BEM) solution procedure. Second, a MFS methodology is presented, coupled with genetic algorithms (GAs), for the optimization of anode positioning and their respective current intensity values. 1. Introduction For evaluation of the design of cathodic protection systems, several numerical methods have been used to predict current and potential distributions on metallic structures. The BEM is one of the most appropriate technique to solve problems involving CP systems, mainly to solve large problems and considering homogeneous conductive medium. Several applications of BEM to study CP systems have been reported in the literature [1,2]. The method of fundamental solutions (MFS) belongs to the class of meshless methods and is a technique which can also be applied to CP problems, even though not many references can be found in the literature. Just like BEM, MFS is applicable when a fundamental solution of the differential equation in question is known, with the advantage of not requiring any integration procedure or specific treatment for the singularities of the fundamental solution. In the MFS, the approximate solution of the problem is represented in the form of a linear superposition of fundamental solutions with singular points located outside the domain of the problem. These singular points are called virtual sources and form a “pseudo-boundary” having no common points with the actual boundary of the region. The basic ideas for the formulation of the MFS were first proposed by Kupradze and Aleksidze [3]. The MFS has successfully been applied for solving several problems. For example, acoustic problems [4] and crack problems [5]. In paper [6] a formulation using a genetic algorithm (GA) was proposed with the MFS to simulate cathodic protection systems. In order to satisfy the corrosion protection criterion, the minimization of an objective function using, for example, GAs and a penalty method for handling constraints can be adopted. This type optimization has been successfully performed with the BEM [7,8]. The purpose of the present contribution is to present a formulation using a GA and the MFS to determine the optimum location and the optimum current intensity of the anodes inserted in the electrolyte. Here, an initial two-dimensional simulation is carried out to compare the standard MFS formulation with the direct boundary element (BEM) solution procedure [9] prior to assessing the complete optimization procedure to solve a test example.


427

2. The MFS for CP simulations In order to design the CP system, one needs to know the expected current density or the electrochemical potential over the metallic surfaces. The mathematical model of the problem, within the conducting domain π (electrolyte), is based on a Poisson equation for the electrochemical potential: (1) ݇ߘ ଶ ߶ ሺ࢞ሻ ൌ ܾሺ࢞ሻǡ ࢞ ‫ א‬πǡ where ܾ is a known function representing the anodes as external sources and ݇ is the conductivity of the electrolyte. Over the metal surfaces in direct contact with the electrolyte, the boundary conditions for eq (1) are of the following form ݅ ሺ࢞ሻ ൌ ‫ ܨ‬ሺ߶ሻǡ ࢞ ‫߁ א‬ǡ (2) where ߁ is the boundary of π, ݅ ሺ࢞ሻ is the current density in the outward normal direction ࢔ and ‫ ܨ‬ሺ߶ሻ is a nonlinear function of the ߶. The general solution ሺ߶௚ ሻ of eq (1) is given by adding a particular solution ሺ߶௣ ሻ to the solution of the associated homogeneous equationሺ߶௛ ሻ, subjected the corresponding homogeneous boundary conditions. One technique for obtaining a particular solution of Poisson's equation is based on the Newton potential [10], which is given by the integral

߶௣ ሺ࢞ሻ ൌ ‫׬‬π ‫ ܩ‬ሺࣈǡ ࢞ሻܾሺࣈሻ ݀πǤ

(3)

The function ‫ ܩ‬ሺࣈǡ ࢞ሻ is a fundamental solution of Laplace's equation given by ଵ

ଵ

ሺ૆ǡ ‫ܠ‬ሻ ൌ ቀୖቁ ǡ ଶ஠୩ where ܴ is the Euclidean distance between point ࣈ and the field point ࢞.

(4)

Considering an impressed current anode, the external sources can be treated as point sources and the term ܾሺࣈሻ becomes equal to ௡೛ೞ ௣௦ ܾሺࣈሻ ൌ ෌௠ୀଵ ܲ൫࢞௣௦ (5) ௠ ൯ߜ൫࢞௠ ǡ ࣈ൯ǡ ௣௦ ௣௦ where ࢞ represent the coordinates of the point sources, ܲሺ࢞ ሻ is the intensity of the source, ߜ is the Dirac delta "function" and ݊௣௦ is the number of point sources inserted in the electrolyte. Therefore, the particular solution can be written in the following form ߶௣ ሺ࢞ሻ ൌ

ଵ ଶ஠୩

௡

೛ೞ ௣௦ ௣௦ ෌௠ୀଵ ܲ൫࢞௠ ൯ ‫ܩ‬൫࢞௠ ǡ ࢞൯ǡ

(6)

௣௦ where now ܴ is the Euclidean distance between point ࢞௠ and the point ࢞.

In addition, from Ohm's law, the particular solution for a current density is equal to ݅௣ ሺ࢞ሻ ൌ ݇

డథ೛ డ௡

ଵ

ൌ െ ଶగ ෍

௡೛ೞ ௠ୀଵ

௣௦

ଵ డோ

ܲ൫࢞௠ ൯ ோ డ௡ .

(7)

The approximate solution of the associated homogeneous problem by MFS is represented in the form of a linear superposition of fundamental solutions with singular pointsሺ࢞௦௣ ሻ located outside the domain of the problem. Thus, the electrochemical potential can be considered as the following summation ௡ೞ೛

߶௛ ሺ࢞ሻ ൌ ෍

௝ୀଵ

‫ܩ‬ሺ࢞ǡ ࢞௝௦௣ ሻ ܿ௝ ǡ

(8)

428


with ݊௦௣ being the total number of virtual sources and the coefficients ܿ௝ present in the approximate solution are the unknown source intensities. డீ Similarly, defining ‫ ܪ‬ൌ ݇ , the homogeneous solution for the current density ሺ݅௛ ሻ is given as డ௡ ௡ೞ೛

݅௛ ሺ࢞ሻ ൌ ෍

௝ୀଵ

‫ܪ‬ሺ࢞ǡ ࢞௝௦௣ ሻ ܿ௝ Ǥ

(9)

The idea of MFS is to determine the coefficients ܿ௝ Ԣ‫ ݏ‬by imposing satisfaction of the boundary condition at certain collocation points. In the present work, the polarization curve is given by the expression [11]: ݅ ൌ ‫ ܨ‬ሺ߶ሻ ൌ ݁

ഝశలవయǤవభ ഁభ

ଵ

െ ൤௜ ൅ ݁

ഝశఱమభǤల ഁమ

భ

ିଵ

൨

െ݁

షሺഝశళబళǤఱళሻ ഁయ

ି

ǡ

(10)

with ߶ and ݅ having units ܸ݉ and ߤ‫ܣ‬Ȁܿ݉ ଶ, respectively, and ߚଵ ǡ ߚଶ ǡ ߚଷ e ݅ଵ are given constant parameters: ߚଵ ൌ ʹͶܸ݉, ߚଶ ൌ ʹ͵ǤͶ͹ܸ݉,ߚଷ ൌ ͷͷܸ݉ e ݅ଵ ൌ ͺ͸ǤͲ͸ߤ‫ܣ‬Ȁܿ݉ ଶ. The conductivity of the electrolyte is equal to ݇ ൌ ͲǤͲͶ͹ͻπିଵ ܿ݉ ିଵ and the critical value of the electrochemical potential is ߶௖ ൌ െͺͷͲܸ݉ሺ‫ݏݒ‬Ǥ ሻ. The general solution of the problem must satisfy eq (10), i.e., ݅௚ ൌ ݅௣ ൅ ݅௛ ൌ ‫ܨ‬൫߶௣ ൅ ߶௛ ൯ ൌ ‫ܨ‬൫߶௚ ൯.

(11)

The relationship given by eq (11) generates a of nonlinear least square problem in which the design variables are the coefficients ܿ௝ and the positions of the virtual sources. Here, the Gauss-Newton (GN) method is proposed to solve the nonlinear least square problem [12]. In least square problems, the objective function ݂ has the following special form ଵ

௡

೎೛ ଶ ‫ݎ‬௝ ሺࢉǡ ࢞ ௦௣ ሻ, ݂ሺࢉǡ ࢞ ௦௣ ሻ ൌ ଶ σ௝ୀଵ (12) where ݊௖௣ is the number of collocation points, ࢉ is a vector containing the coefficients ܿ௝ Ԣ‫ ݏ‬and each ‫ݎ‬௝ is a smooth function referred to as a residual given by

௝

௝

‫ݎ‬௝ ൌ ݅௚ െ ‫ܨ‬൫߶௚ ൯ǡ ݆ ൌ ͳǡ ǥ ǡ ݊௖௣ Ǥ

(13)

Considering the arrangement of the virtual sources on a circular contour, it is only necessary to search the radius of this circle. Thus, design variables of eq (12) are ࢉ and a radius ߩ. Furthermore, a vector ࢓ containing the design variables ࢉ and ߩ is also considered in this work, i.e., ࢓ ൌ ሺܿଵ ǡ ܿଶ ǡ ǥ ǡ ܿ௡ೞ೛ ǡ ߩሻ. A necessary condition for a vector ࢓‫ כ‬to be a minimum of ݂ሺ࢓ሻ is that ‫ ݂׏‬ሺ࢓‫ כ‬ሻ ൌ Ͳ. Assuming that ݂ሺ࢓ሻ is twice continuously differentiable and a Taylor series approximation, the gradient in the vicinity of ሺ࢓૙ ሻ can be given as ‫݂׏‬൫࢓૙ ൅ ο࢓൯ ൎ ‫݂׏‬൫࢓૙൯ ൅ ‫׏‬ଶ ݂൫࢓૙ ൯ο࢓.

(14)

Setting the approximate gradient (14) equal to zero yields ‫׏‬ଶ ݂൫࢓૙ ൯ο࢓ ൎ െ‫݂׏‬൫࢓૙ ൯. (15) The solution of eq (15) for successive solution steps leads to Newton's method for minimizing ݂ with ࢓૙ acting as the initial solution. Considering the individual components ‫ݎ‬௝ into a residual vector ࢘ሺ࢓ሻ, the gradient of ݂ሺ࢓ሻ can be written as


429

௡

೎೛ ‫ݎ‬௝ ሺ࢓ሻ ‫ݎ׏‬௝ ሺ࢓ሻ ൌ ࡶሺ࢓ሻࢀ ࢘ሺ࢓ሻ, ‫݂׏‬ሺ࢓ ሻ ൌ σ௝ୀଵ

(16)

where each ‫ݎ׏‬௝ , ݆ ൌ ͳǡ ǥ ǡ ݊௖௣ is the gradient of ‫ݎ‬௝ and ࡶሺ࢓ሻ is the Jacobian matrix. Applying the product rule for derivatives, the Hessian of ݂ can then be expressed as follows: ௡

೎೛ ‫ݎ‬௝ ሺ࢓ሻ ‫׏‬ଶ ‫ݎ‬௝ ሺ࢓ሻ. ‫׏‬ଶ ݂ ሺ࢓ ሻ ൌ ࡶሺ࢓ሻࢀ ࡶሺ࢓ሻ ൅ σ௝ୀଵ

(17)

In the Gauss-Newton method, the second summation term in eq (17) is ignored and the Hessian is estimated by first term of eq (17): ‫׏‬ଶ ݂ ሺ࢓ ሻ ൎ ࡶሺ࢓ሻࢀ ࡶሺ࢓ሻǤ

(18)

Using eq (18), the equations for successive iterations in the GN method become ࢀ

ࢀ

ࡶ൫࢓࢑ ൯ ࡶ൫࢓࢑ ൯ ο࢓࢑ ൌ െࡶ൫࢓࢑ ൯ ࢘൫࢓࢑ ൯Ǥ

(19)

Implementations of the Gauss̽Newton method usually perform a line search in the direction ο࢓࢑, requiring the step length ߙ ௞ Ǥ Each iteration of a line search method is given by ࢓࢑ା૚ ൌ ࢓࢑ ൅ ߙ ௞ ο࢓࢑ .

(20)

3. GA for optimization of CP In order to guarantee effective cathodic protection the electrochemical potential distribution over the interface metal electrolyte should be kept, as uniform as possible, close to the critical potential: ߶ ൑ ߶௖ Ǥ The optimum location and the optimum current intensity delivered by the anodes have to be determined in such a way as to satisfy the protection criterion. Hence, to satisfy the protection criterion with the minimum possible power input, it is necessary to minimize the following objective function: ܼ൫࢞௣௦ ǡ ܲሺ࢞௣௦ ሻ൯ ൌ ට

ଵ ௡೎೛

௡

೎೛ σ௜ୀଵ ሾ߶ ௜ െ ߶௖ ሿଶ ൅ ‫ ݒ‬ට

ଵ ௡೎೛‫כ‬

௡

೎೛ σ௜ୀଵ ݇௟ ଶ ሺ࢞௣௦ ǡ ܲ ሺ࢞௣௦ ሻሻ

(21)

where ܼ calculates the root mean square error (RMSE) between the electrochemical potential at each boundary node and the critical potential. Furthermore, ݊௖௣‫ כ‬is the number of collocation points that do not satisfy the protection criterion, the constant ‫ ݒ‬is a penalty number and the function ݇௟ is equal to: ݇௟ ൫࢞௣௦ ǡ ܲ ሺ࢞௣௦ ሻ൯ ൌ ൫߶ ௜ െ ߶௖ ൯ ή ‫ݑ‬൫߶ ௜ െ ߶௖ ൯ǡ where ‫ ݑ‬is the unit step function. Typical values of ‫ ݒ‬are within the range ͳͲଶ െ ͳͲହ .

(22)

In this paper, the minimization of eq (21) is achieved using a genetic algorithm. The adopted GA used for the minimization has a binary representation and is inspired by the algorithm presented in [13]. However, some characteristics have been included as the two-point crossover and the elitism. Furthermore, the probabilities of mutation and crossover can vary linearly over the generations. 4. Numerical results Example 1 presents an initial simulation with the sole purpose of testing the standard MFS formulation, comparing results with a direct boundary element (BEM) solution procedure. In this case, the anodes are fixed in different positions within the electrolyte. In example 2 the optimization model for cathodic protection systems using GA and the MFS is tested. The adopted GA design variables are the coordinates and the current intensity of the anodes. Two-dimensional tank representation has been assumed.

430


Example 1 In this example the cathodic protection of a two-dimensional rectangular storage metal tank using four anodes is analyzed. The dimensions of the structure are ͳͲͲܿ݉‫ݔ‬ͷͲܿ݉ and each anode has a current intensity ofെͺͶͲͲǤͲͲߤ‫ܣ‬. The boundary was represented with ͳʹͲ collocation points and ͸Ͳ virtual sources and therefore ʹͳ design variables for the Gauss-Newton method. The initial solution is considered equal to ሺͳͲǤͲǡ െͳͲǤͲǡ ǥ ǡͳͲǤͲǡ െͳͲǤͲሻ for the intensities of virtual sources and the radius starts equal to ͸͹ǤͲܿ݉. The step length ߙ ௞ was considered constant and equal to ͳǤͲ. The stopping criterion adopted for the GN method was the objective function of eq (12) reaching the tolerance of ͳͲିସ . In Fig. 1 are illustrated the virtual source arrangements for the initial radius and for the optimum radius. The optimum radius determined by GN method after ͹͹ iterations is ߩ ൌ ͸ͲǤʹͺͶͲ. Fig. 2 presents the potential distribution on the metal surface determined by MFS and by BEM. The RMSE between the potential values on the boundary determined by BEM and MFS is ͲǤͲͻͺͶͲ͵ܸ݉.

Figure 1: Virtual sources arrangement.

Figure 2: Potential distribution on the metal.

The potential distribution in the electrolyte using BEM is presented in Fig. 3 whereas Fig. 4 indicates the MFS solution. The RMSE between the potential values at internal points determined by BEM and MFS isͲǤͲͻͲͺͻ͸ܸ݉.

Figure 3: Potential in the electrolyte using BEM.

Figure 4: Potential in the electrolyte using MFS.

Example 2 In this example the tank geometry of problem 1 is repeated considering four anodes randomly distributed within the electrolyte. The anodes have coordinates ‫ݔ‬௔௣௦ and ‫ݕ‬௔௣௦ with ranges within ሾͷǤͲܿ݉ǡ ͻͷǤͲܿ݉ሿ and ሾͷǤͲܿ݉ǡ ͶͷǤͲܿ݉ሿ. The range of current intensity is ሾെͻͲͲͲǤͲߤ‫ܣ‬ǡ െ͹ͲͲͲǤͲߤ‫ܣ‬ሿ. In addition, the following GA values have been used: population sizeൌ ͵Ͳ, maximum number generationsൌ ͳͲͲ, required precision ൌ ͲǤͲͳ, initial crossover probabilityൌ ͹Ͳ, final crossover probabilityൌ ͷͲ, initial mutation probability ൌ ͲǤͷ, final mutation probabilityൌ ͵Ǥͷ and the penalty number ‫ ݒ‬ൌ ͳͲଶ . The parameters for the GN method and MFS are the same presented in example 1. The potential distribution over the metal surface after optimization is presented in Fig. 5. The optimum location determined by GA is presented in Fig. 6 and the optimum current intensity is equal toെ͹ͻ͸͹Ǥ͹Ͷߤ‫ܣ‬Ǥ


Figure 5: Boundary potential distribution.

431

Figure 6: Potential distribution in the electrolyte.

5. Conclusions The test performed using constant elements BEM and the proposed GN MFS indicates a good agreement between the electrochemical potential distribution on the metal surface and in the electrolyte. The results found confirm GA as a robust optimization procedure to determine the optimum locations and the minimum current intensity for the anodes inserted in the electrolyte. References [1] J. C. F. Telles, W. J. Mansur, L. C. Wrobel and M. G. Marinho Numerical Simulation of a Cathodically Protected Semisubmersible Platform using PROCAT System, Corrosion, 46, 513-518 (1990). [2] J. A. F. Santiago and J. C. F. Telles On Boundary Elements for Simulation of Cathodic Protection Systems with Dynamic Polarization Curves, International Journal for Numerical Methods in Engineering, 40, 2611-2622 (1997). [3] V. D. Kupradze and M. A. Aleksidze Aproximate method of solving certain boundary-value problems, Soobshch akad nauk Gruz SSR, 30, 529-536 (1963). [4] E. G. A. Costa, L. Godinho, J. A. F. Santiago, A. Pereira and C Dors Efficient numerical models for the prediction of acoustic wave propagation in the vicinity of a wedge coastal region, Engineering Analysis with Boundary Elements, 35, 855–867 (2011). [5] E. F. Fontes Jr., J. A. F. Santiago and J. C. F. Telles On a regularized method of fundamental solutions coupled with the numerical Green’s function procedure to solve embedded crack problems, Engineering Analysis with Boundary Elements, 37, 1–7 (2013). [6] W. J. Santos, J. A. F. Santiago and J. C. F Telles An Application of Genetic Algorithms and the Method of Fundamental Solutions to Simulate Cathodic Protection Systems, Computer Modeling in Engineering & Sciences, 87, 23-40 (2012). [7] S. Aoki and K. Amaya Optimization of cathodic protection system by BEM, Engineering Analysis with Boundary Elements, 19, 147–156 (1997). [8] L. C. Wrobel and P. Miltiadou Genetic algorithms for inverse cathodic protection problems, Engineering Analysis with Boundary Elements, 28, 267-27 (2004). [9] C. A. Brebbia., J. C. F. Telles and L. C. Wrobel Boundary Element Techniques: Theory and Applications in Engineering, Springer, Berlin (1984). [10] O. D. Kellog, Foundations of Potential Theory, Dover, New York (1954). [11] J. F. Yan, S. N. R. Pakalapati, T. V. Nguyen and R. E. White Mathematical Modelling of Cathodic Protection using the Boundary Element Method with Nonlinear Polarisation Curves, Journal of the Electrochemical Society, 139, 1932-1936 (1992). [12] J. Nocedal and S. J. Wright Numerical Optimization, Springer (2006). [13] Z. Michalewicz Genetic Algorithms + Data Structures = Evolution Programs, Spinger-Verlag (1996).

432


A level set-based topology optimization method using 3D BEM

T. Yamada1,a , S. Shichi2,b , T. Matsumoto2,c , T. Takahashi2,d , H. Isakari2,e 1 Department

of Mechanical Engineering and Science, Kyoto University Furo-cho, Chikusa-ku, Kyoto, 464-8603, Japan 2 Department of Mechanical Science and Engineering, Nagoya University Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan a [email protected], b [email protected], b [email protected], d [email protected], e [email protected]

Keywords: Optimum design, structural analysis, boundary element method, topology optimization, level

set method, sensitivity analysis Abstract The boundary element method (BEM) is used in the level set-based topology optimization procedure for three-dimensional elastostatics. The level set function is used to control the boundary of the optimizing solid. In the present study, the boundary mesh is generated at every iterative step of updating the distribution of the level set function. Thus, the boundary condition can be specified explicitly also for the varying boundary. The effectiveness of the present approach is demonstrated through some numerical examples for maximum stiffness design optimization problems. Introduction Boundary element method has been considered as a useful numerical method to apply for shape sensitivity analyses and shape optimization of continuum media [1, 2, 3]. However, because of its heavy computation cost and storage requirement, it has not been applied effectively to industrial applications. Recent development of fast BEM algorithms has changed this situation and it is now ready to use not only for shape optimization problems but also for topology optimization problems. As the shape controlling method for topology optimization problems, level set-based approaches [4, 5] have been proposed and successfully used combined with the finite element method. In this approach, a socalled level set function is used to represent the boundary of the domain that changes its topology during the optimization process. By using the level set function, one can define a definite boundary of the domain compared with other topology optimization methods such as homogenization and density methods. Wei, et al. [6] and Yamada, et al. [7] proposed approaches that use the distribution of the level set function as the design variable. In particular, Yamada’s approach is powerful in controlling the obtained topology from complicated to simpler ones by giving a parameter of Tikhonov’s regularization [8]. Also, the distribution of the level set function is updated by solving a type of reaction diffusion equation, whose source term corresponds to the topological sensitivity of the objective functional, for the level set function. This method is very strong and could give the same optimum topology no matter what the initially assumed topologies would be. The drawback of this conventional approach is that, although the shape of the designing object is clearly defined with the levelset function, the entire design space is always used for the repeated numerical analyses by giving small values for the material properties of those of the elements corresponding to exterior part of the domain [5]. Since the boundary condition cannot be specified explicitly for the varying boundary using this method, topology optimizations with continuous loading on the newly generated boundary are not tractable in the conventional approach. In this study, we present a topology optimization approach in which the boundary elements are generated at every iteration step accompanying a shape change. BEM mesh generation is rather easy by employing the background voxels for solving the reaction-diffusion equation of the level set function. This approach is applied to some maximum stiffness design problems for three-dimensional solids.


433

Boundary element method for elastostatics We use the boundary element method for calculating the displacements and stresses that are used to evaluate the objective functional and its topological sensitivities required in the updating process of the level set function. Somiglina’s identity, which is the integral representation of the displacement at a point in the solid, is given as

u j (y) +

Γ

ti∗j (x, y)ui (x)dΓ(x) =

Γ

u∗i j (x, y)ti (x)dΓ(x),

y ∈ Ω,

(1)

where Ω is the domain, Γ the boundary of the solid, ui and ti are the i-th component of displacement and traction vectors, respectively. u∗i j and ti∗j are the fundamental solution of the displacement and corresponding traction, and can be given as 1 (3 − 4ν )δi j + r,i r, j 16π G(1 − ν ) ∂r −1 ti∗j (x, y) = (1 − 2ν )δi j + 3r,i r, j + (1 − 2ν )(r,i n j − r, j ni ) 2 8π (1 − ν )r ∂ n

u∗i j (x, y) =

(2)

(3)

where r = |x − y|, G is the shear modulus, and ν is Poisson’s ratio. The repeated indices follow Einstein’s summation convention, and an index after a comma denotes the derivative with respect to the corresponding component of the coordinate. The boundary integral equation, a regularized version, is derived from Somiglina’s identity (1), as follows: Γ

& ' ti∗j (x, y) ui (x) − ui (y) dΓ(x) = u∗i j (x, y)ti (x) dΓ(x), Γ

y ∈ Γ.

(4)

After discretizing eq (4), we obtain a system of linear algebraic equations as [H] {u} = [G] {t} ,

(5)

which can be solved for the boundary unknown nodal values after the boundary conditions are applied. The displacement gradients at internal points can be calculated by the following integral representation obtained by differentiating Somiglina’s identity:

∂ u j (y) = ∂ yl

∂ u∗ (x, y) ij Γ

∂ yl

∂ t ∗ (x, y) ij

ti (x) dΓ(x) −

Γ

∂ yl

ui (x) dΓ(x),

y∈Ω

(6)

Topology optimization based on level set method We use level set function in order to represent the shape of the domain under optimization. The level set function, φ (x), is defined as a scalar function of point taking positive value for an internal point, zero for an boundary point, and negative value for a point out of the domain defined for the object. We consider a fixed design domain for which φ (x) is defined. The designed object is assumed to exist within D. The upper bound and lower bound of φ (x) are given as +1 and −1, respectively, for simplicity. Then, φ (x) can be written as ⎧ for ∀x ∈ Ω \ ∂ Ω ⎨ 0 < φ (x) ≤ 1 φ (x) = 0 for ∀x ∈ ∂ Ω , (7) ⎩ −1 ≤ φ (x) < 0 for ∀x ∈ D \ Ω where ∂ Ω = Γ. The objective functional defined with a boundary integral term can be written as

inf F = subject to

Γ Γ

f (u,t) dΓ ti vi dΓ −

G(χ (φ )) =

Ω

Ci jkl uk,l vi, j dΩ = 0,

D

χ (φ ) dD −Vmax ≤ 0,

(8)

434


where Ci jkl is the elastic tensor, vi is the adjoint displacement, χ is the Heaviside function, and Vmax is the allowable maximum volume in the fixed design space D. The second equality constraint is the weak form of the displacement field of the elastostatic problem under consideration. For a maximum stiffness design of a solid, we have Γ

f (u,t) dΓ =

Γt

uiti dΓ.

(9)

The augmented objected function with these constraints can be defined as

inf J =

Γt

f (u,t) dΓ +

Γ

ti vi dΓ −

Ω

Ci jkl uk,l vi, j dΩ + λ G +

D

τ (−sgn(φ ))∇2 φ dD,

(10)

where λ is Lagrange’s multiplier. The last integral of eq (10) is a regularization term defined to control the curvature of the level set function with a parameter τ . Now, we consider an adjoint problem defined as Ci jkl vk,li (x) = 0, in Ω, ∂f vi = − on Γu , ∂ ti ∂f on Γt . si = C jikl vk,l n j = ∂ ui

(11) (12) (13)

The adjoint problem has the same differential equation and the boundary condition pattern as the original linear elastostatic problem. Therefore, the boundary element method can be used to obtain their solutions as well. By taking the variation of J after an infinitesimal topology change, we obtain the topological sensitivity of J, as follows: dT J = Ai jkl u0k,l v0i, j + λ − τ ∇2 φ ,

(14)

where u0k,l and v0i, j are the gradients of the displacement and adjoint displacement at the point at which a topology change is assumed, respectively, and Ai jkl is a tensor related to the solution of the elastostatic problem with an infinitesimal spherical cavity [9]. Both the gradients in eq (14) can be calculated with eq (6) by using the boundary solutions obtained by BEM. In order to update the distribution of the level set function, we assume that the derivative of φ with respective to fictitious time, which is considered as a variable corresponding to the updating iteration of the level set function, is proportional to the above topological sensitivity dT J. Hence, we have & ' ∂φ = −K Ai jkl v0i, j u0i, j + λ − τ ∇2 φ . ∂t

(15)

Since eq (15) is an evolution equation with a source term, BEM is not suitable for solving it. However, when we consider a fixed design domain D of a simple rectangular solid geometry discretized into voxels, it can be solved very efficiently and easily by using FEM [7]. In Fig.1 is shown the flowchart of the present topology optimization procedure. An appropriate distribution of the level set function for the initial solid shape is given first, and the corresponding boundary elements are generated. Then, the displacement and traction of the original and adjoint problems are calculated by BEM, and the objective function is evaluated. When its value is not converged, the topological sensitivities are calculated subsequently, and the reaction diffusion equation (15) for the level set function is solved by FEM to update the distribution of the level set function, and we return to the BEM mesh generation process.


Fig. 1

Fig. 2

435

Flow chart of topology optimization procedure.

Fixed design domain and the boundary conditions for a cantilever model.

Numerical examples Maximum stiffness design of cantilever model We consider a cantilever model subjected to a uniform loading on a limited part of the boundary at the end as shown in Fig.2, and search for its optimum topology to maximize the stiffness under a volume constraint. The fixed design domain is assumed as a rectangular parallelepiped area of 0.8m × 0.6m × 0.04m. The distribution of the level set function is approximated at the nodes of the uniformly discretized hexahedral finite elements in the fixed design domain. The edge length of the hexahedron is equally 1.0×10−2 m. The maximum volume of the designed solid is assumed to be 40% of that of the fixed design domain. Also, the solid is assumed to be of an isotropic material and its Young’s modulus and Poisson’s ratio are 2.16 × 1010 Pa and 0.3, respectively. We try the optimization for four different regularization parameter values: τ = 1.0 × 10−3 (Case 1), 5.0 × 10−4 (Case 2), 1.0 × 10−4 (Case 3), and 5.0 × 10−5 (Case 4). We show in Fig.3 the obtained optimum shapes. As can be seen in Fig.3, BEM is effectively used to obtain the optimum topologies with definite boundaries. For a smaller value of the regularization parameter, more complicated topology is obtained. Hence, a finer mesh would be needed both for the fixed design domain and object boundary. Maximum stiffness design of solid subjected to pressure load on generated boundary Next we show an example [5] of the topology optimization of a solid subjected to a uniform normal pressure on the boundary that are generated in the process of topology optimization. In Fig.4 is shown the fixed design domain, a rectangular

436


Fig. 3

Optimal configurations for a cantilever model.

parallelpiped area of 0.5m × 0.5m × 0.3m, and is devided uniformly into hexahedral linear elements whose edge lengths are equally 1.0 × 10−2 m. The allowable maximum volume Vmax is assumed to be 8% of that of the fixed design domain. Also, Young’s modulus and Poisson’s ratio are assumed to be 2.16×1010 Pa and 0.3, respectively. The solid is fixed at five different points on the base and all the boundary except the base is subjected to a uniform normal pressure. The regularization parameter is given as τ = 1.0 × 10−4 . As can be seen in Fig.5, a similar optimum topology as that presented in [5] is obtained. The main advantage of the present approach is that we can keep giving a load explicitly on the boundaries newly generated through the optimization process. Concluding remarks A level set-based topology optimization method for three-dimensional solid using the boundary element method has been presented. The level set function is used to control the shape of the solid having different

Fig. 4

Fixed design domain and pressure boundary condition on the surface.


Fig. 5

437

Intial and optimal configuration of the solid subjected to pressure on the newly generated boundary.

topologies. The topological sensitivity of the objective functional with a regularization term is used for updating the distribution of the level set function. The boundary elements are generated for the iso-surface of the level set function every time when the level set function is updated. This procedure enables obtaining the definite boundary of the optimum topology and giving the boundary condition on its boundary through the optimization process. References

[1] J.H. Choi and B.M.Kwak Int. J. Numer. Methods Engng., 26, 1579–1595 (1988). [2] T.Burczynski, J.H.Kane and C.Balakrishna Int. J. Numer. Methods Engng., 38, 2839–2866 (1995). [3] R.Aithal and S.Saigal Engineering Analysis with Boundary Elements, 15, 115–120 (1995). [4] M.Y.Wang, X.Wang and D.Guo, Computer Methods in Applied Mechanics and Engineering, 192, 227– 246 (2003). [5] G.Allaire, F.Jouve and A.Toader Journal of Computational Physics, 194, 363–393 (2004). [6] P.Wei and M.Y.Wang Int. J. Numer. Methods Engng., 78, 379–402 (2008). [7] T.Yamada, K.Izui, S.Nishiwaki and A.Takezawa Computer Methods in Applied Mechanics and Engineering, 199, 2876–2891 (2010). [8] A.N.Tikhonov and V.Y.Arsenin Solution of Ill-posed Problems, Wiston and Sons (1997). [9] B.B.Guzina and M.Bonnet Q. Jl. Mech. Appl. Math. 57, 161–179 (2004).

438


Mesh-free solutions for bending of thin plates with variable bending stiffness V. Sladek1, J. Sladek1 and L. Sator1 1

Institute of Construction and Architecture, Slovak Academy of Sciences, 845 03 Bratislava, Slovakia, [email protected], [email protected], [email protected]

Keywords: Kirchhoff-Love theory, variable thickness, functionally graded materials, governing equations, exact solutions, decomposed formulation, meshless approximation.

Abstract. The paper concerns the stationary bending of thin elastic plates composed of functionally graded materials within the Kirchhoff-Love theory. Two gradations of the Young modulus are considered: (i) in-plane gradation, (ii) gradation across the plate thickness. It is discussed when the bending deformations and the in-plane deformations are decoupled in the treatment of FGM plates in the Kirchhoff-Love theory. The theoretical analysis is supplemented with numerical results.

1.Introduction Plates are defined as plane structural elements with a small thickness compared to the planar dimensions. The typical thickness to width ratio of a plate structure, h / L , is less than 0.1. The advantage of a plate theory consists in reduction of the full three-dimensional solid mechanics problem to a two-dimensional problem. The most widely accepted and used in engineering are the Kirchhoff–Love theory of plates (classical plate theory [1]) and the Mindlin–Reissner theory of plates (the first-order shear plate theory, or the third order theory by Reddy [2]). Having used the expression for the rotations of the straight lines normal to the mid-surface in terms of gradients of the deflection in the Kirchhoff–Love theory, the shear strain is vanishing in contrast to the Mindlin-Reissner theory, where it is constant across the plate thickness owing to introduction of rotations as new independent variable. The relative simplicity of the Kirchhoff–Love theory consists in the fact that all the solution can be expressed in terms of the deflection and its derivatives. On the other hand, in the Mindlin-Reissner theory, the inplane and deflection deformations are coupled and should be solved simultaneously. In this paper, we deal with the stationary bending of thin elastic plates in the Kirchhoff-Love theory and discuss problems when coupling between the bending and in-plane deformations is necessary. We consider functionally graded plates with two gradations of the Young modulus: (i) the in-plane gradation, (ii) the gradation across the plate thickness. It is shown that in the case of gradation across the plate thickness it is necessary to consider the coupling. Furthermore, the plane stress conditions are not satisfied. The theoretical conclusions are illustrated by simple numerical examples with showing the differences between the correct and incorrect formulations.

2. Governing equations in the Kirchhoff-Love theory Let us consider a straight plate structure occupying the 3D domain V

{ ( x1, x2 , x3 )

3

;x

( x1, x2 )

, x3 [ h / 2, h / 2]}

[ h / 2, h / 2] 2 .

According to the Kirchhoff hypothesis the 3D displacement field is expressed as vi (x, x3 ) GiD ¬ªuD (x) x3w,D (x) ¼º Gi3w(x) ,

(1)

where uD (x) are the in-plane displacements of the mid-surface : with D {1, 2} , and w(x) is the displacement of the mid-surface in the x3 -direction. Thus, the shear strain eD 3 is vanishing in the Kirchhoff-Love theory [1] in contrast to the Mindlin-Reissner theory [2], where it is constant across the plate thickness. According to the Hooke law, the 3D elastic stresses in the plate structure are given as

VDE (x, x3 )

E 1 ªWDE (x) x3PDE (x) º , VD 3 (x, x3 ) V 3D (x, x3 ) 0 ¼ 1 Q H ¬


439

E Q uD ,D (x) x3w,DD (x) , 1 Q H

V 33 (x, x3 )

(2)

with H : 1 FQ and H WDE : uD ,E (x) uE ,D (x) QGDE uJ ,J (x) , PDE : Hw,DE (x) QGDE w,JJ (x) , 2

(3)

where F 2 , E and Q is the Young modulus and the Poisson ratio, respectively. Since h L (where h and L are the thickness and a characteristic width of the plate, respectively), the variations on x3 [h / 2, h / 2] can be treated by using the average stresses across the thickness of the plate Tij (x) :

1 h /2 ³ V ij (x, x3 )dx3 h h /2

(4)

Then, assuming the material coefficients to be independent on x3 , we obtain TDE (x)

E ( x) 1 WDE (x) , 1 Q H

TD 3 (x) { 0 ,

T33 (x)

E ( x) Q uD ,D (x) | 0 , 1 Q H

(5)

where the last assessment is justified in plates subjected only to transversal loading. According to Eq. (5), the average stresses satisfy the plane stress conditions and therefore they are referred to as the generalized plane stresses [3,4]. Then, it is required to replace F 2 by F 1 in the plane stress semi-integral formulation with the force equilibrium and equilibrium of moments of forces yielding the sets uncoupled governing equations for inplane displacements and deflections

WDE ,E (x)

(1 Q 2 ) qD (x) , hE (x)

ª D(x) P,DE (x) º ¬ ¼,DE

h /2

qD (x) :

³

h /2

X D (x, x3 )dx3 ,

(6)

q(x) , q(x) : V 33 (x, h / 2) V 33 (x, h / 2)

h /2

³

h /2

X 3 ( x, x3 )dx3 , D(x) :

E ( x ) h3 ( x ) 12(1 Q 2 )

(7)

with q(x) being the surface density of transversal loading. Quite different situation appears when the Young modulus is dependent on the x3 -coordinate. For instance, E0 ª1 ] 1/ 2 x3 / h º , the plane stress conditions are not satisfied «¬ »¼ and the in-plane and deflection deformations are not further uncoupled, since the equilibrium equations yield p

assuming the power-law gradation E ( x3 )

ªWDE ,E (x) GLPDE ,E (x) º ¬ ¼

(1 Q ) H qD (x) , E0 hd p

Ld p ª º «12G 2 WDE ,ED (x) 1 ] f p PDE ,ED (x) » h ¬ ¼

with d p : 1

] p 1

, sp :

G:

h] s p Ld p

,

H q(x) , (1 Q ) D0

(8)

D0 :

E0 h3 12(1 Q 2 )

,

(9)

1 1 12 12 3 , fp : . p 2 2( p 1) p 3 p 2 p 1

If the medium is not graded in the transversal direction ( ] 0 ), the plane stress conditions are applicable, the interaction constant vanishes ( G 0 ) and the governing equations reduce to the uncoupled ones given by eqs. (6),(7) for homogeneous medium with E E0 . In view of eqs. (8) and (9), one can eliminate the in-plane strains from (9) with getting the equations 2 2 w

F 1 q(x) , D : (1 Q ) ª1 ] f p 12d p (GL / h)2 º , F : 1 (1 F )Q ¼ H¬ DD0

(10)

440


2uD

F Q uE ,ED H

2FGL 2 w,D H

,

(11)

where we have assumed qD (x) 0 . The set of governing equations should be supplemented by boundary conditions. If we are interested in the response to pure transversal loading, the in-plane tractions are vanishing on the boundary edge (with the necessity to eliminate the RBM), i.e. TDE (x)nE (x)

E0 d p nE (x) ª¬WDE (x) GLPDE (x) º¼ 1 Q

0

nE (x)WDE (x) GLnE (x)PDE (x)

(12)

The relevant physical boundary quantities for specification of a deflection problem are the deflection ( w ), normal slope ( ww / wn ), bending moment ( M ) and generalized shear force ( V ) which are given as M ( x)

V ( x)

(1 Q ) D0 H

Ld p ª º «12G 2 WDE (x) 1 ] f p PDE (x) » nD (x)nE (x) h ¬ ¼

DD0 PDE (x)nD (x)nE (x) F

DD0 ª2 w(x) Hw,DE (x)tD (x)tE (x) º , ¬ ¼ Ld p º (1 Q ) D0 ª wT (x) «12G 2 WDE ,E (x) 1 ] f p PDE ,E (x) » nD (x) H wt h ¬ ¼

DD0 nD (x)2 w,D (x) T ( x)

wT (x) wt

(14)

Ld p º (1 Q ) D0 ª «12G 2 WDE (x) 1 ] f p PDE (x) » nD (x)tE (x) H h ¬ ¼

DD0 H w,DE (x)nD (x)tE (x) , F

(13)

tE (x) H 3JE nJ (x) .

DD0 PDE (x)nD (x)tE (x) F

(15)

From eq.(10), one can see that w 1/ ( DD0 ) . Hence and from eqs. (13)-(15), we conclude that the boundary values of the bending moment, twisting moment and the generalized shear force are independent on the gradation factor ] in solutions of the boundary value problems.

3. Numerical examples Let us consider circular plate with circular hole subjected to uniform transversal loading in order to illustrate the behaviour of FGM thin elastic plates. For simplicity, we shall consider either the gradation along the radius of the plate or the gradation through the plate thickness. Then, the problem exhibits circular symmetry and the governing equations become the ordinary differential equations. In the case of transversal gradation, we have to deal with the ODE with constant coefficients and the exact solutions are available for all events of standard boundary conditions. Rather more complex is the case of radial gradation of the Young modulus, since the governing equation is the ODE with variable coefficients and the exact solutions for standard plate boundary value problems are available only for special case of power-law gradation of the bending stiffness D(r ) D0 (r / r0 ) p . Therefore we have developed also the numerical method for solution of considered problems in FGM thin plates. The method consists in decomposition of the fourth order differential equation into two Poisson equations for the deflection and a new field variable and collocation of the governing equations as well as the boundary conditions at nodal points with using the meshless approximations for field variables [5]. In the case of coupled in-plane and deflection deformations, the third differential equation is eq.(11) for the in-plane displacements, which is again the second order differential equation. In numerical examples, we shall assume FGM composed of two micro-constituents with power-law gradation of density of volume fraction for the material #2 in x3 -direction

Advances in Boundary Element Techniques XIV p

§ x h/ 2· v(2) ( x3 ) ¨ 3 ¸ , h © ¹

441

dV(2)

Sv(2) ( x3 )dx3

v(2) ( x3 )dx3

V

V

h

, V

Sh , S S r12 r02

and for the material #1 it is v(1) ( x3 ) 1 v(2) ( x3 ) . 1 h /2 ³ v(2) ( x3 )dx3 h h /2

V(2)

Then, the volume fraction for material #2 is F(2)

V

1 p 1

and the composed Young modulus in the FGM (according to the mixed rule reflecting the densities of volume fractions) is E ( x3 )

p ª E(2) §1 x · º 1 , E0 «1 ] ¨ 3 ¸ » , ] : E(1) h 2 © ¹ »¼ ¬«

E(1) v(1) ( x3 ) E(2) v(2) ( x3 )

E0 : E(1) .

Similarly in the case of power-law gradation of densities of volume fractions for two materials #2 in radial direction are p

§ r r0 · v(2) (r ) ¨ ¸ , © r1 r0 ¹

dV(2)

2S rhv(2) (r )dr

2r v(2) (r )dr

V

V

r12 r02

The volume fraction for material #2 is F(2)

V(2) V

2

r12

, v(1) (r ) 1 v(2) (r ) . r1

³ v(2) (r )rdr r02 r0

r r · 2 § r0 1 0¸ ¨ r1 r0 © p 1 p 2 ¹

and the composed Young modulus is

^

E (r ) E(1) v(1) (r ) E(2)v(2) (r ) E0 1 H ¬ª r r0 / r1 r0 ¼º

p

`,

H:

E(2) E(1)

1 ,

E0 : E(1) .

Fig.1 Distribution of densities of volume fractions and volume fractions for material # 2 in considered FGM The correct treatment of the FGM plate with transversal gradation of the Young modulus requires consideration of coupling of the in-plane and deflection deformations and the choice of F -parameter as F 2 . Fig. 2 shows percentage deviations of maximal deflections in plate with clamped both edges and obtained by using various formulations vs the gradation factor ] . It is seen that wmax (coupling, F 1) Awmax (coupling, F 2) with A being a constant independent on ] . On the other hand, the coupling effect is dependent on ] .

442


Fig.2 Comparison of maximal deflections in FGM plate with E ( x3 ) using formulations with ( G z 0 ) and without( G 0 ) coupling and two values of the F -parameter In Fig. 3, it can be seen the depressing effect of gradation of the Young modulus on the deflection of the FGM plate. Furthermore, it is seen the influence of incorrect treatments of the problem on the radial distribution of deflections.

Fig.3 Radial dependence of deflections FGM plate with E ( x3 ) using formulations with ( G z 0 ) and without ( G 0 ) coupling and two values of the F -parameter. The coloured lines correspond to results for two values of the gradation factor ] and the dashed line shows the deflection of the homogeneous plate.


443

Fig. 4 Radial distributions of deflections FGM plate with two radial gradations E (r ) and E1 (r ) Numerical results for the radial distributions of deflections in the FGM plate with two different radial gradations of the Young modulus (a) E(r ) : E(r0 ) E0 o E(r1) (1 H ) E0 ; (b) E1(r ) : E(r1) E0 o E(r0 ) (1 H )E0 and two different values of the exponent of power-law gradations are shown in Fig. 4. The inner edge is clamped, while the outer one is simply supported. It is seen that the effect of reduction of deflections is decreasing with increasing the value of the exponent p (i.e. with decreasing the volume fraction of the material with higher value of the Young modulus). Furthermore, it can be seen that the maximal deflections are shifted toward the edge with lower value of the Young modulus.

4. Conclusions The bending of thin elastic plates is studied with assuming the Kirchhoff hypothesis and allowing the spatial variation of the bending stiffness due to functional gradation of material coefficients and/or the plate thickness. It is shown that even within the Kirchhoff – Love theory the coupling between the in-plane and deflection deformations cannot be omitted, if the bending stiffness is variable across the plate thickness. On the other hand, in the case of FGM plates with continuous in-plane variation of either the material coefficients or the plate thickness, the in-plane and deflection deformations are decoupled completely, but the governing equations become the PDE with variable coefficients. The strong formulation is applied to numerical solution of boundary value problems with using the Point Interpolation Method for meshless approximation of field variables. Owing to high order derivatives in the governing equations, the decomposition method has been developed and applied in this paper too. The perfect coincidence of numerical results with exact ones has been achieved.

Acknowledgements This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0032-10. References [1] S.P. Timoshenko and S. Woinowsky-Krieger Theory of Plates and Shells, McGraw-Hill (1959). [2] J.N. Reddy Mechanics of Laminated Composite Plates: Theory and Analysis, CRC Press (1997). [3] J.R. Barber Elasticity, Springer Science + Business Media B.V. (2010). [4] A.I. Lurie Theory of Elasticity, Springer-Verlag, Berlin, 2005. [5] V. Sladek, J. Sladek and L. Sator Engineering Analysis with Boundary Elements, 37, 348–365 (2013).

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An SGBEM implementation with quadratic programming for solving contact problems with Coulomb fricton Roman Vodiˇcka1 , Vladislav Mantiˇc2 1

Technical University of Košice, Civil Engineering Faculty,Vysokoškolská 4, 042 00 Košice, Slovakia,

[email protected] 2

University of Seville, School of Engineering, Camino de los Descubrimientos s/n, 41092 Seville, Spain,

[email protected]

Keywords: symmetric Galerkin BEM, frictional contact, viscous regularization, energetic approach.

Abstract. A new numerical procedure for the solution of contact problems with Coulomb friction is developed and tested. The solution process follows a quasi-static evolution of the problem. The model proposed provides a generalized weak solution approximated by time-stepping and boundary element algorithms. The spatial discretization is obtained by the symmetric Galerkin BEM and leads to an application of algorithms of quadratic programming to cope with the optimality character of the approximated solution. Finally, the numerical procedure is applied to a simple receding contact problem. Introduction Numerical solution of contact problems with friction may be very challenging. There exist several approaches for the solution of contact problems by Boundary Element Method (BEM), see e.g. [2, 4] and references therein. The present work tries to enhance the energetic model of interface debonding proposed in [9] in order to cover also the frictional contact between the debonded parts of a specimen or structure. In the present work, the frictional law is regularized to cope with the energetic character of the model, see [11]. The regularization is proposed so that convex quadratic energy functionals are obtained and algorithms of quadratic programming [3] can successfully be applied. This includes a weakening of the Signorini contact conditions, which allows a small overlapping of solids in contact, and making the bulk domains viscoelastic, although the parameters characterizing the viscous response of the structure are small [6, 7]. In the following sections the proposed model is described, its numerical solution is outlined and an example is solved to assess the applicability of the approach to contact problems. Contact model For the sake of simplicity, only two-dimensional contact problems between two solids, Ω η (η = A, B), will be considered in the present work, Figure 1. The Signorini condition of unilateral contact&[u]n ≥0'on the contact zone Γc will govern the contact problem. Here, the relative normal displacement [u]n = uB −uA ·nA is defined at the contact zone. Similarly, the relative tangential displacement [u]s is also defined. The solution of the contact problem is based on the evolution of energies during the loading process: the elastic energy stored in the bulks and the energy dissipated due to friction and (possibly a very small amount) due to viscosity. From ˙ a physical point of view, frictional dissipation is given by the functional R(t; u)=

Γc

˙ s dΓ , where the rate ts · [u]

˙ s , and t denote the of change of the relative tangential displacement as a function of time [u]s is denoted by [u] contact traction vector and ts its tangential (shear) component. A problem arises from the mathematical point of view when the contact traction in the dissipation functional R is approximated, leading to the observation that it is better to express this traction vector in terms of displacements. Therefore, the tangential traction is related to the displacements in the following way: first, the Coulomb friction law |ts |≤μ|tn | with a constant friction coefficient μ ≥ 0 is considered providing a relation between normal and tangential tractions, tn and ts , and then the Signorini contact condition is penalized by defining tn =kg [u]− n such that a tiny overlapping of solids is allowed originating a pertinent contact compression

Advances in Boundary Element Techniques XIV wA

nA

sA ΓtA

ΓAu

Γc

ΩA

x2

445

ΩB

ΓtB

sB nB

ΓBu x1

O

wB

Figure 1: A model – contact of two subdomains ([u]− n denotes the negative part of the relative normal displacement). This penalization can also be justified by a presence of a very thin layer of the normal stiffness kg > 0 which is compressed in contact and stress-free out of contact. For the numerical solution, it is also advantageous to include a viscous effect in the bulk in order to regularize the frictional relations. The viscosity is considered by a simple linear Kelvin-Voigt model which provides the ˙ where D is the fourth-order tensor of viscosity parameters, stress tensor σ by the relation σ=C:ε(u)+D:ε(u), given in the present work as D=τR C, where τR ≥ 0 is a relaxation time parameter. Based on the above assumptions, a quasi-static viscoelastic evolution is governed by the following inclusion: ˙ −δu F (τ,u), ∂u E (τ, u) + ∂u˙ R(u; u)

(1)

where the symbol ∂ refers to partial subdifferential relying on the convexity of the energy functionals, see [8]. It includes the stored energy functional [5, 8] E (τ,u) =

ΩA

1 A A A ε :C :ε dΩ + 2

1 B B B ε :C :ε dΩ + 2

ΩB

Γc

1 & − '2 kg [u]n dΓ , 2

(2)

with the admissible displacements uη = wη (τ) on Γuη and the small strain tensor εη =ε(uη ), the potential energy of external forces (acting only along the boundary in the present work): F (τ,u) = −

Γt A

fA · uA dΓ −

Γt B

fB · uB dΓ .

(3)

and the dissipation potential ˙ = R(u; u)

Γc

˙ s |dΓ + τR −μkg [u]− n · | [u]

ΩA

1 A A A ε˙ :C :ε˙ dΩ + τR 2

ΩB

1 B B B ε˙ :C :ε˙ dΩ 2

(4)

where ε˙η =ε(u˙ η ) is the strain rate. Computer implementation The numerical procedure devised for solving the above problem considers time and spatial discretizations separately, as usual. The procedure is formulated in terms of the boundary data only, with the spatial discretization leading to the Symmetric Galerkin BEM (SGBEM). Time discretization The time-stepping scheme is defined by a fixed time stepsize τ0 such that τ k =kτ0 for k k−1 ˙ u −u k=1, . . . Tδ . The displacement rate is approximated by the finite difference u≈ , where uk denotes the τ0 k solution at the discrete time τ . The differentiation with respect to the displacement rate can be replaced by the k−1 ; u−uk−1 ). It means that ˙ differentiation with respect to u as well, in the sense that ∂u˙ R(uk−1 ; u)≈τ 0 ∂u R(u τ0 the inclusion (1) is approximated at discrete times τ k by the first order optimality condition for the functional H k (u) = E (kτ0 , u) + τ0 R(uk−1 ;

u−uk−1 ) + F (kτ,u). τ0

(5)

The optimality solution is denoted by uk . Substituting the previous time-step result into the dissipation potential due to friction makes the pertinent functional convex with respect to the unknown u, the optimality solution being thus unique and defining the minimum.

446


The simple viscosity model is chosen in order to exploit the reformulation of the viscoelastic problem in the bulk in terms of an elastic problem in the bulk, which is solved by the elastostatic SGBEM. Let us introduce a new variable v (a fictitious displacement), which will replace the admissible u in (5) for the time step k, as v = u + τR

u − uk−1 , τ0

and also vk = uk + τR

uk − uk−1 . τ0

(6)

Then, the functional H k is defined as ⎧ - − .2 τ0 kg τ0 ⎨ 1 1 1 τR k ε(vA ):CA :ε(vA )dΩ + ε(vB ):CB :ε(vB )dΩ + dΓ v+ uk−1 H (v)= τ0 +τR ⎩ 2 Ω A 2 ΩB 2 Γc τ0 +τR τ0 n − 6 6 1 τR 1 τR 6 6 + −μkg uk−1 · 6 v−uk−1 6 dΓ + ε(uA k−1 ):CA :ε(uA k−1 )dΩ + ε(uB k−1 ):CB :ε(uB k−1 )dΩ n 2 τ0 Ω A 2 τ0 Ω B s Γc # # $ $ ) τR τR − fA · vA + uA k−1 dΓ − fB · vB + uB k−1 dΓ , (7) τ0 τ0 Γt A Γt B for any admissible v satisfying the condition ' τR & η ˜ η (kτ) on Γuη . w (kτ) − wη ((k − 1)τ) = w vη = wη (kτ) + τ0

(8)

Let vk = argmin H k (v). It is worth observing that the viscosity in (7) is eliminated in the sense that the only energy term in the bulk associated to the unknown v is the elastic strain energy, uk−1 is known from the previous time step. In finding the minimum of H k (v), an iteration for v is rendered as a solution of an elastic BVP for unknown (fictitious) displacements v and the actual tractions t of the viscoelastic model. For the solution of these BVPs an SGBEM code is used, thus using in the minimization process only such solutions, it is convenient to change the bulk integrals with v in (7) to boundary based integrals

Ωη

ε(vη ):Cη :ε(vη )dΩ =

Γη

tη (vη ) · vη dΓ .

(9)

Once vk is obtained it can be transformed back to the original solution uk by the relation (6). Spatial discretization and SGBEM The role of the SGBEM in the present computational procedure is to provide a complete boundary-value solution from the given boundary data for each domain in order to calculate the elastic strain energy in these domains by using the boundary integral in (9). Thus, the SGBEM code calculates unknown tractions along Γc ∪Γu ,assuming the displacements at Γc to be known from the used minimization procedure, in the same way as proposed and tested in [9]. The integral equations solved by SGBEM are the Somigliana displacement and traction identities, written for each particular domain Ω η separately. The numerical solution is obtained by the piecewise linear approximations of the form (10) vη (x) = ∑ Nηψ p (x)vηp , tη (x) = ∑ Nηϕl (x)tηl , p

l

with nodal shape functions Nηψ p (x) and Nηϕl (x) and nodal values vηp and tηl . Let the subvectors of the nodal unknowns at the boundary parts Γuη , Γt η and Γc , respectively, be distinguished by the same subscripts u, t and c. Then, SGBEM leads to the symmetric square matrix of the following system of linear algebraic equations: ⎞⎛ η ⎞ ⎛ 1 η ⎞⎛ η⎞ ⎛ Uηut −Tηuc − 2 Muu −Tηuu −Uηuu Tηut −Uηuc tu g η ηT η ηT ηT ηT η η 1 ⎠ ⎝fη ⎠ . ⎝ Ttu (11) −Stt Ttc ⎠ ⎝vt ⎠ = ⎝ Suu Mtt −Ttt Stc 2 tηc vηc −Uηcu Tηct −Uηcc −Tηcu Uηct − 12 Mηcc −Tηcc The elements of the submatrices denoted with letters U, T and S are formed by double integrals including the singular integral kernels denoted by the same letter as is usual in SGBEM, see [10]. The square 2×2 submatrices, associated with the nodes l and p, of the mass matrices Mηrr (with r=u,t, or c), are formed by the integrals: (Mηrr )l p =

Γrη

Nηϕl (x)Nηψ p (x)dΓ .

(12)


447

Notes on the minimization algorithm Once all the boundary data are known, the energy of the state given by H k in (7) can be calculated using (9). It is worth to see how it is carried out in the present implementation. First, let us reconsider the absolute value term and the term with [·]− in H k . A classical trick of removing the unpleasant terms and replacing them by additional unknowns with restrictions is used [1]. Let the additional auxiliary unknowns be denoted as α and β and the following restrictions hold: α − [v]s ≥ − uk−1 , β ≥ 0, s (13) τR k−1 u β + [v]n ≥ − . α + [v]s ≥ uk−1 , s τ0 n For the discretization, the approximation formulas for both auxiliary parameters α, β given by pertinent boundary element mesh should be considered. In what follows, the same mesh and approximation as used in (10) for displacements on the boundary part ΓcA is considered. The approximation formulas can be written in the form α(x) = ∑ Nψm (x)αm , m

β (x) = ∑ Nψm (x)βm ,

(14)

m

A. where αm , βm are the nodal unknowns pertinent to the node xm Then, the discretized energy H k , from equation (7) (using (9) and approximations (10) and (14)) is expressed as

1 1 τ 0 + τR k H (v, α, β) = ∑ NAψ p (x)vAp · ∑ NAϕl (x)tAl dΓ + Γ B 2 ∑ NBψq (x)vBq · ∑ NBϕr (x)tBr dΓ τ0 ΓA 2 p q r l ⎡ .2 .− .⎤ k τ 1 0 g AB B k−1 A k−1 + ⎣ ∑ Nψ q (x)αq ⎦ dΓ ∑ Nψ q (x)βq − μkg ∑ Nn pq un q − un p Γc 2 τ0 + τR q q q −

∑ NAψ p (x)vAp · ∑ NAϕl (x)f Al dΓ −

Γt A p

l

∑ NBψq (x)vBq · ∑ NBϕr (x)f Br dΓ + V (uA k−1 , uB k−1 , f A k , f B k ),

Γt B q

(15)

r

AB =N B (xA ). The functional V includes all the data which are constant with respect to v. where N pq ψq p In minimization of the functional (15) with the restrictions (13), it may be useful to reformulate the problem in such a way that the restrictions change to bound constraints. The left-column and right-column restrictions in (13) provide linearly independent constraints which can respectively be written in a matrix form as ⎛ ⎞ ⎛ ⎞ # $ # $ $ α #A $ β #A ξ1 0 IA ⎝ B ⎠ 0 0 ⎝ B⎠ I I −NAB s vs ≥ vn ≥ , . (16) A AB A A AB A ξ3 ξ2 I Nn −I I Ns −I vAs vAn AB AB with the identity matrix IA , the matrices NAB n and Ns consisting of N pq and ξi corresponding to the right-hand sides in (13). Both inequalities are defined by full row-rank matrices. Thus, denoting arbitrary matrices whose columns span respectively the null-spaces of the left-hand-side matrices in (16) by Kα and Kβ , the following relations hold: ⎞ ⎛ ⎞ ⎛ ⎛ α⎞ & AB 'T & AB 'T .−1 # $ IA IA Kα α AB ' & ' & Ns Ns + 2IA −NAB y1 ⎜ s AB T ⎟ Ns ⎝vBs ⎠ = ⎝− NAB T + ⎝KBα ⎠ zα , (17a) & AB 'T & AB 'T ⎠ Ns s AB AB A y2 −Ns Ns Ns Ns + 2I vAs KAα IA −IA

with the restrictions applied only to yi :

# $ # $ y1 ξ ≥ 1 and y2 ξ2

⎞ ⎛ ⎞ ⎞ ⎛A β .−1 # $ # $ # $ I IA Kβ β A A I I ' & y3 y 0 ⎟ ⎜ ⎜ T 3 B B ⎟ z , with AB ⎝vs ⎠ = ⎝ 0 & ' + ≥ . ⎠ A ⎠ ⎝ K Nn T β β y4 y4 ξ3 NAB I NAB + 2IA A A n n vAs 0 −I Kβ ⎛

(17b)

448


Thus, there is the same number of bound constraints as provided by the more general restrictions (13). The discretized functional (15) can be expressed in a general matrix form as 1 H k (y) = yT Ay − bT y + c, 2

−∞ ≤ ylow ≤ y ≤ yup ≤ +∞.

(18)

The bounds ylow and yup are in fact determined by the constraints applied on y j ( j=1, 2, 3, 4) in (17) and, as indicated, some of them may be infinite, which means that no restrictions are applied for the pertinent components of y, in fact those corresponding to zα and zβ . The constrained minimizer of (18) is denoted by yk . The problem with standardly applied algorithms is that the matrix A might not necessarily be calculated in an explicit way. The terms which arise from the first two integrals in the right-hand side of (15) provide the energy and calculating the derivative with respect to the unknown v they provide a projected traction Mt with M defined as in (12). The projected traction can naturally be calculated from the SGBEM algorithm, represented by the product Ay in equation (18). Thus, each time the optimization algorithm requires a matrix-by-vector product actually a system from the SGBEM is solved. The influence matrices of the SGBEM, however, are calculated only once at the beginning of the solution process, as they are the same for all the iterations and all time steps, considering only small displacements. Numerical example The present formulation has been tested numerically by a computer code, which uses the SGBEM for finding elastic solution in each subdomain, and a conjugate gradient based method for constrained minimization leading to the contact solution. A typical example of receding contact is solved, the geometry and load configuration of the example are shown in Figure 2. The prescribed displacements (defining a kind of hard-device loading) w= (0, −w2 ) are

h

x2

x1

Γt

w

w Γu Γc

Γt

L

Figure 2: Receding contact geometry. increasing during the loading process. The incrementally prescribed loading is given by the relation wk2 =vτ k , k=1, 2, . . . 100 with v=1mm s−1 and τ k =kτ0 , τ0 =2×10−5 s. The plane strain state is considered. The dimensions of the aluminum layer are L=1500mm and h=125mm and elastic properties E=7×104 MPa and ν=0.35. The layer is loaded along the mid part of the top face, where the given length parameters are 7 = 16 L, w = 18 L. The time-relaxation parameter for the viscoelastic model is τR =1×10−3 s. The Coulomb friction coefficient is μ=0.4. The contact model requires the stiffness parameter to be set kg =1×107 MPa mm−1 . The computed scaled deformation of the layer is shown in Figure 3. It behaves as expected for a receding contact problem. The central part remains in contact during the loading process, the contact tractions are plotted in the bottom graph of Figure 3. Also the traction results of the numerical solution are in a good agreement with an expected stress distribution and contact zone length. Conclusions An energy based model for solving frictional contact problem has been considered. The model uses a regularization of standard contact conditions by allowing a small interpenetration and also adds a small amount of viscosity to make the solution more regular. The numerical implementation of spatial discretization via SGBEM has permitted the whole problem to be defined only by boundary and interface data. A simple 2D example has been used to validate the model.


−t1 [MPa]

0.5 0.4

750 850 950

1500

k=100

t1 t2

k=60

1.5 1.25 1

k=30

0.3

0.75

0.2

0.5

0.1

0.25

0 750

t2 [MPa]

0 0.6

449

0 800

850

x1 [mm]

900

950

1000

Figure 3: Deformation of the layer (top) and contact tractions (bottom). Acknowledgement The authors are grateful to Prof. Tomáš Roubíˇcek (Charles University in Prague) for fruitful discussions. R. V. acknowledges support from the grant VEGA 1/0201/11. V.M. acknowledges support from the Junta de Andalucía (TEP-4051), the Spanish Ministry of Science and Innovation (MAT2009–14022) and the Spanish Ministry of Economy and Competitiveness (MAT2012-37387). References [1] S. Bartels and M. Kružík. An efficient approach to the numerical solution of rate-independent problems with nonconvex energies. Multiscale Modeling & Simulation, 9(3):1276–1300, 2011. [2] A. Blázquez, R. Vodiˇcka, F. París, and V. Mantiˇc. Comparing the conventional displacement BIE and the BIE formulations of the first and the second kind in frictionless contact problems. Engng. Anal. Boundary Element, 26:815–826, 2002. [3] Z. Dostál. Optimal Quadratic Programming Algorithms, volume 23 of Springer Optimization and Its Applications. Springer, Berlin, 2009. [4] C. Eck, O. Steinbach, and W.L. Wendland. A symmetric boundary element method for contact problems with friction. Mathematics and Computers in Simulation, 50:43 – 61, 1999. [5] M. Koˇcvara, A. Mielke, and T. Roubíˇcek. A rate-independent approach to the delamination problem. Math. Mech. Solids, 11:423–447, 2006. [6] T. Roubíˇcek. Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J. Math. Anal., 45(1):101–126, 2013. [7] T. Roubíˇcek, C.G. Panagiotopoulos, and V. Mantiˇc. Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity. ZAMM – Z. Angew. Math. Mech., DOI 10.1002/zamm.201200239. [8] T. Roubíˇcek, M. Kružík, and J. Zeman. Delamination and adhesive contact models and their mathematical analysis and numerical treatment (Chapter 9). In V. Mantiˇc, editor, Mathematical Methods and Models in Composites, Imperial College Press, 2013. [9] R. Vodiˇcka and V. Mantiˇc. An SGBEM implementation of an energetic approach for mixed mode delamination. In P. Procházka and M.H. Aliabadi, editors, Advances in Boundary Element Techniques XIII, pages 319–324, EC ltd, Eastleigh, 2012. [10] R. Vodiˇcka, V. Mantiˇc, and F. París. Symmetric variational formulation of BIE for domain decomposition problems in elasticity – an SGBEM approach for nonconforming discretizations of curved interfaces. CMES – Comp. Model. Eng., 17(3):173–203, 2007. [11] R. Vodiˇcka, V. Mantiˇc, and T. Roubíˇcek. A quadratic programming approach for frictional contact problem — an SGBEM implementation (under preparation).

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Nonlocal elasticity analysis by local integral equation method P.H. Wen1,a, X.J. Huang1,b and M.H. Aliabadi2,c 1

School of Engineering and Material Sciences, Queen Mary, University of London, UK 2

Department of Aeronautics, Imperial College, London, UK

a

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

Keywords: Two dimensional nonlocal elasticity, Eringen’s model, local integral equation method, weak form, radial basis functions. Abstract. This paper developed an algorism for the two-dimensional nonlocal elasticity problems by using the local integral equation method (LIEM). The approach is based on the Eringen’s model with LIEM and the interpolation using the radial basis functions to obtain the numerical solutions for 2D problems. A weak form for a set of governing equations with a unit test function is transformed into the local integral equations. The meshless approximation technique with radial basis functions is employed for the implementation of displacements. A set of the local domain integrals is obtained in analytical form for the local elasticity and by using a standard integral scheme for the nonlocal elasticity. Two examples are given to demonstrate the convergence and accuracy of LIEM including a rectangular plate and disk subjected to a uniformly distributed displacement or tensile load. Introduction It is well known that the classical continuum theories like the linear theory of elasticity are intrinsically size independent. Although the development of classical theories of linear elasticity has been quite successful for solving most engineering problems at intrinsically size independent, the need for efficient and accurate numerical method is increasingly demanded for problems with nonlocal elasticity. Nowadays, it is believed that the physical nature of materials is discrete if atoms are regarded to be the basic constituents. Inter atomic forces are long range in character while quantum mechanics, molecular dynamics, and lattice dynamics are the fundamental theory and supporting methods of approach. According to the classical theories, the elastic strains and stresses are singular at the tips of crack (dislocation) and at the corner of the notch. The continuum damage mechanics has been established to fill the gap between the classical continuum mechanics and fracture mechanics [1-4]. A continuum model for micro-cracking in these materials leads inevitably to strain softening. It is causes a loss of positive definiteness of the elastic modulus matrix and results as an ill-posed boundary value problem [5-6]. The finite elements calculations using elasto-plastic models with yield limit degradation in the framework of the classical theory of plasticity give very different results for different discretization meshes [7]. In other words, the finite elements results are not independent with respect to the finite element mesh refinements and converge at infinite mesh refinement to a solution with zero energy dissipation during failure. To prevent such physically unrealistic behavior, the mathematical models with localization limiters that force the strain-softening region to have a certain minimum finite size has been proposed in [8, 9]. A nonlocal elastic model proposed by Erigen [1011] and reviewed by Altan [12] is based on the key idea that the long-range forces are adequately described by a constitutive relation. A theory of nonlocal elasticity of bi-Helmholtz type is suited based on the Erigen’s model by Lazar et al [13]. A comprehensive review on the nonlocal elasticity theory can be found in [14] by Pisano et al. It is no necessary to introduce in this paper. In recent years, the computational mechanics community has turned its attention to so-called mesh reduction methods. These mesh reduction methods (commonly referred to as Meshless or Meshfree) have received much interest since Nayroles et al [15] proposed the diffuse element method. Later, Belyschko et al [16] and Liu et al [17] proposed element-free Galerkin method and reproducing kernel particle methods, respectively. One key feature of these methods is that meshless methods do not need any grid and are hence meshless. More recently, a family of Meshless methods, based on the Local weak Petrov-Galerkin formulation (MLPGs) for arbitrary partial differential equations with moving least-square (MLS) approximation has been developed (Atluri [18]). 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 [19]. Both methods (MLPG and LBIE) are meshless, as no domain/boundary meshes are


451

required in these two approaches. However, Galerkin-base meshless methods, except MLGP presented by Atluri [18] still include several awkward implementation features such as numerical integrations in the local domain. In this paper, two-dimensional local boundary integral method (LIEM) is developed for the nonlocal elasticity theory with 2D Eringen’s model. With the use of radial basis functions, the analytical solutions for the domain integrals in the weak form are derived for local elasticity. For the nonlocal elasticity, as there no singularity in the integral kernels, the domain integrals are obtained numerical by standard integration scheme. To compare with a high accuracy solution, two-dimension problem of a disk subjected to inner pressure load is transformed to a one-dimension problem with a second order differential equation in terms of displacement, which is solved numerically using the point collocation method [20]. Three numerical examples demonstrate the accuracy and efficiency of LIEM.

Local integral equation method A nonlocal elastic model proposed by using nonlocal elasticity [10,11] is based on the key idea that the long-range forces are adequately described by a constitutive relation of the form, for two-dimension isotropic medium, as

V ij , j f i σ ( x)

0

[1 σ (x) [ 2 ³ D (x, x' , l )σ (x' )dV (x' )

(1)

V

σ {V 11 , V 22 , V 12 }T , ε {H 11 , H 22 , H 12 }T , σ

Dε, H ij

(u i , j u j ,i ) / 2

where [ 1 and [ 2 are portion factors and

support domain R of x

[1 [ 2

1 , V represents the volume of domain, f i body forces, D a nonlocal kernel defined as the influence coefficient, l the characteristic length or influence distance; x, x' are collocation and domain integration

node x Local integral domain Ωs with boundary Γs

Γ

Ω

Node in

support points and u i displacements; σ , σ and ε are domain yk vectors of nonlocal stress, local stress (classical stress) and strain; D denotes the elastic moduli matrix. An improvement of L nonlocal elasticity model considers the nonlocal elastic material as a two-phase elastic *s=*D+*T material which includes phase 1 material complying with local elasticity and phase 2 Figure 1. Arbitrary distributed node, support domain of x, local material with nonlocal elasticity. Two kinds of integral domain for weak formulation. boundary conditions are considered for nonlocal elasticity, namely, for nonlocal traction boundary: (1) traction boundary: V ij n j t i0 ; (2) displacement boundary: u i u i0 , in which ui0 and t i0 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 nonlocal integral equation approach, the weak form of differential equation over a local integral domain : s can be written, from Eq.(1), as

³ (V

ij , j

f i )u i* d:

0

(2)

:s

where u i* is test function. By use of the divergence theorem, Eq.(2) above can be rewritten in a symmetric weak form as

452

Eds: A Sellier & M H Aliabadi (3)

* * * ³ V ij n j ui d* ³ (V ij ui, j f i ui )d: 0

w: s

: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

ij

:s

ui*, j d: ³ t i ui* d* ³ t i ui* d* *D

Ls

³t

0 i

ui* d*

*T

³ f u d: i

* i

(4)

:s

in which, Ls indicates the other part of the local boundary inside the local integral domain : s ; *D is the intersection between the local boundary *s and the global displacement boundary; *T is a part of the traction boundary as shown in Figure 1.

Analytical forms of domain integrals Consider a unit test function, i.e. M i (x) 1 and the local domain is enclosed by several straight lines, therefore, the local integral equation (4) becomes

³V

ij

(x)n j (x)d*(x)

*s

L

L

[1 ¦ n lj ³ V ij (x)d*(x) [ 2 ¦ n lj ' l ³ D (x i , x' , l )V ij (x' )dV (x' ) l 1

*l

l 1

(5)

V

where L is number of straight line for the boundary of local domain, n ij and ' i are components of normal and length of segment i of the boundary of local domain. Suppose there are M nodes both in the domain and on the boundary, M M : M T M D , where M : indicates the number of nodes collocated in domain,

M T and M D are numbers of nodes on the traction/displacement boundaries and consider the radial basis function interpolation and relationship between stress and strain in Eq.(1), Eq.(5) becomes ª

º

[1 ¦ u1( k ) ¦ «¦ E ' F1il n1l PF2il n2l D ik ¦ E ' G1tl n1l PG2tl n2l E tk » K

L

k 1

T

t 1 ¼ ¬i 1 ªL º l l ³ «¦ E'Ik ',1 (x' )n1 (x l ) PIk ',2 (x' )n2 (x l ) D (x l , x' , l )' l »¼dV (x' ) V ¬l 1

l 1

K'

[2 ¦u

( k ') 1

k' 1

K

ª

º

[1 ¦ u 2( k ) ¦ «¦ QE ' F2il n1l PF1il n2l D ik ¦ QE ' G2tl n1l PG1tl n2l E tk » K

L

K

T

t 1 ¬i 1 ¼ K' L º ª [ 2 ¦ u 2( k ') ³ «¦ QE 'I k ', 2 (x' )n1l (x l ) PIk ',1 (x' )n2l (x l ) D (x l , x' , l )' l »dV (x' ) k' 1 ¼ V ¬l 1 k 1

l 1

K

L

ª

k 1

l 1

¬i

T

º

t 1

¼

0

(6a)

[1 ¦ u1( k ) ¦ «¦ QE ' F1il n2l PF2il n1l D ik ¦ QE ' G1tl n2l PG2tl n1l E tk » K

1

K'

ª

k' 1

V ¬l

1

º

K

[ 2 ¦ u1( k ') ³ «¦ QE 'I k ',1 (x' )n2l (x l ) PIk ', 2 (x' )n1l (x l ) D (x l , x' , l )' l »dV (x' ) L

K

L

ª

k 1

l 1

¬i

¼

T

º

t 1

¼

[1 ¦ u 2( k ) ¦ «¦ E ' F2il n2l PF1il n1l D ik ¦ E ' G2tl n2l PG1tl n1l E tk » ª

1

º

[ 2 ¦ u 2( k ') ³ «¦ PIk ',1 (x' )n1l (x l ) E 'Ik ', 2 (x' )n2l (x l ) D (x l , x' , l )' l »dV (x' ) 0 K'

k' 1

V

¬l

L

1

¼

(6b)

where k ' 1,2,...K ' are numbers of node in the support domain centred (x' ) at the local integral area 'V (x' ) . By using the interpolation of radial basis function, we have

Advances in Boundary Element Techniques XIV sl sl wR wP F jil ³ i ds, G jtl ³ t ds xj x w w j 0 0 sin E l , n2l

Consider n1l

453 (7)

cos E l , we have solutions in closed form

F1il

(r2 r1 ) cos E l [( x y1i ) sin E l ( xal 2 y 2i ) cos E l ] sin E l ln( d1 / d 2 )

F2il

(r2 r1 ) sin E l [( xal 2 y 2i ) cos E l ( xal 1 y1i ) sin E l ] cos E l ln( d1 / d 2 )

r1

c 2 ( xal 1 y1i ) 2 ( xal 2 y 2i ) 2

d1

( xal 1 y1i ) cos E l ( xal 2 y 2i ) sin E l r1

r2

d2

( xbl 1 y1i ) cos E l ( xbl 2 y 2i ) sin E l r2

G11l

c 2 ( xbl 1 y1i ) 2 ( xbl 2 y 2i ) 2 6 , one has G13l G16l 0 ; G12l s, G14l

2 xa1 s s 2 cos E i , G15l

G21l

G22l

xa1 s 0.5s 2 cos E i , G26l

l a1

If T

G24l

0, G23l

s, G25l

xa 2 s 0.5s 2 sin E i

2 xa 2 s s 2 sin E i

(9)

( xbi 1 xai 1 ) 2 ( xbi 2 xai 2 ) 2 .

s

(8)

For all domain integrals, four-point standard integral scheme is adopted in computation. Then Eqs (6a) and (6b) above are rewritten as K

L

ª

k 1

l 1

¬i

º

[1 ¦ u1( k ) ¦ «¦ E ' F1il n1l PF2il n2l D ik ¦ E ' G1tl n1l PG2tl n2l E tk » K'

[2 ¦u

( k ') 1

k' 1

K

T

¦¦¦ E 'I V

L

4

k ',1

L

ª

k 1

l 1

¬i

(x )n (x l ) PIk ', 2 (x )n (x l ) D (x l , x , l )' l w p 'Vq ' qp

l 1

' qp

q 1 p 1 l 1

K

¼

t 1

1

l 2

' qp

T

º

t 1

¼

[1 ¦ u 2( k ) ¦ «¦ QE ' F2il n1l PF1il n2l D ik ¦ QE ' G2tl n1l PG1tl n2l E tk » K

1

[ 2 ¦ u 2( k ') ¦¦¦ QE 'Ik ', 2 (x 'qp )n1l (x l ) PIk ',1 (x 'qp )n2l (x l ) D (x l , x 'qp , l )' l w p 'Vq K'

k' 1

V

L

4

0

(10a)

q 1 p 1 l 1

K

L

ª

k 1

l 1

¬i

T

º

t 1

¼

[1 ¦ u1( k ) ¦ «¦ QE ' F1il n2l PF2il n1l D ik ¦ QE ' G1tl n2l PG2tl n1l E tk » K

1

[ 2 ¦ u1( k ') ¦¦¦ QE 'Ik ',1 (x 'p )n2l (x l ) PIk ', 2 (x 'p )n1l (x l ) D (x l , x 'p , l )' l w p 'Vq K'

k' 1

V

L

4

q 1 p 1 l 1

K

L

ª

k 1

l 1

¬i

T

º

t 1

¼

[1 ¦ u 2( k ) ¦ «¦ E ' F2il n2l PF1il n1l D ik ¦ E ' G2tl n2l PG1tl n1l E tk » K

1

[ 2 ¦ u 2( k ') ¦¦¦ PIk ',1 (x 'p )n1l (x l ) E 'Ik ', 2 (x 'p )n2l (x l ) D (x l , x 'p , l )' l w p 'Vq K'

k' 1

V

4

L

0

(10b)

q 1 p 1 l 1

where V in the summation above indicates the number of total rectangular segments of whole integral domain using a background grid, x 'qp x 'q x 'p .

Numerical examples A square plate under tensile load First, we consider a square plate of side a 5 cm subjected to a uniformly distributed force along two sides t10 rV 0 as shown in Figure 2(a). Young's modulus is one unit and and Poisson ratio Q 0 . In this case, the accurate solution for one-dimension bar as observed by Li et al [20] can be used for comparison.

454

Eds: A Sellier & M H Aliabadi The dimensionless parameter [1 [ 2 0.5 and the characteristic length l / a 0.1 . Two types of uniform distribution of node are considered, i.e. the number of total node N1 u N 2 11u11 and 21u 21 respectively. To demonstrate the convergence of this method, the variation of the normalized local stress V 22 ( x1 ) EH 22 / V 0 at x2 a / 2 by both the local integral equation method (2D) and the finite integration method [20] (1D) are plotted in Figure 3. It is illustrated that the boundary effect for 2D nonlocal elasticity theory is significant as same as 1D problem. As expected, the results given by LIEM are convergent for two densities of node distribution and hence the solution for the fine grid of node is closer to that for one dimensional case too. Both normalized solutions tend to one unit at the middle of the plate/bar. However, these two solutions, i.e. two-dimension and one-dimension, should not be expected to be the same as the different influence functions D (x, x' , l ) . Consider the same geometry of the plate above fixed along the edge at x1 0 and a uniform distribution of displacement u10 0.001cm along the edge x1 5 [14] as shown in Figure 2(b). In this case, Young's

modulus E 2.1u 10 6 N/cm 2 , Poisson ratioQ 0.2 , characteristic length l 0.1 cm and parameter [1 0.5 . A uniform distribution of node ( 21u 21 ) is considered and all other free parameters are the same as in example above. Figure 4 shows the distribution of strain H 11 versus x1 at x2 0.019 cm and x2 2.519 cm respectively. The results by Pisano et al [14] using NL-FEM are presented in Figure 5 for comparison. x2

x2 V0

a (a)

V0

u10

x1

x1

a (b)

Figure 2. Square plate. (a) under tensile load; (b) uniform displacement.


455

A disk under internal pressure load

A disk under an internal pressure load is analysed. Poisson ratio Q 0.3 and dimensions a 1 cm and 2 cm. Due to the symmetry, one quarter of the disk is studied by using LIEM in this example. However, in the domain integral in the governor equation, the contributions to the strain from the whole disk must be taken into account as shown in Figure 5(a). The boundary condition is described as

b

t10 0 1

t

p a cos M , t 20 0, u

0 2

p a sin M for

0 for x2

0; t

0 2

x12 x22 0 1

0, u

1cm; t10

0 for x1

0, t 20

0 for

x12 x22

2cm;

(11)

0.

The node distribution is shown in Figure 5(b), here the total number of node N 1039 . Again, all free parameters c , D and r0 are selected as same as example above. The number of grid is selected as

(41u 41) . The normalised strains EH 11 ( x1 ,0) / pa and EH 22 ( x1 ,0) / pa along the axis x are plotted in Figures 6 and 7 for different parameters l ( [1 0.1) . One dimensional solution with high accuracy by using the point collocation method is reported in the same figures for comparison. Obviously good agreement between these solutions has been achieved. x2 ( x1 , x2 )

( x1 , x2 )

ra

pa

rb ( x1 , x2 )

x1

( x1 , x2 ) (a)

M

(b)

Figure 5. A disk under pressure load. (a): superposition for simplification; (b) nodal distribution in a quarter of disk and boundary conditions.

456


Conclusions In this paper, the formulation for the meshless local integral equation method is presented for the nonlocal elasticity analysis. Based on the Eringen’s model, a weak form for a set of governing equations with a unit test function is transformed into local integral equations. The meshless method is carried out by using the radial basis functions. Three numerical examples are presented to demonstrate the convergence and accuracy of the proposed method. It is concluded that the meshless local integral equation method is of high accuracy and suitable to deal with 2D nonlocal elasticity problems. As 2D nonlocal elasticity is linear, LIEM can be extended to dynamic case easily using the Laplace transform domain.

References [1] Sudak LJ. Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J Appl Phys, 94:7281–7, 2003. [2] Wang Q, Varadan VK. Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater Struct, 16: 178–90, 2008. [3] Filiz S, Aydogdu M. Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity. Comp Mater Sci, 49:619–27, 2010. [4] Hu YG, Liew KM, Wang Q, He XQ, Yakobson BI. Nonlocal shell model for elastic wave propagation in single and double walled carbon nanotubes. J Mech Phys Solids, 56:3475–85, 2008. [5] Bazant ZP. Instability, ductility and size effect in strain softening concrete. Journal of the Engineering Mechanics Division ASCE 1976;12: 331-344. [6] Sandler IS. Strain-softening for static and dynamic problems, in: Proc. Symp. On Constitutive Equations; Micro, Macro and Computational Aspects (ed. K.J. Willam), ASME, New York 1984; 217-231. [7] Bazant ZP, Lin FB. Non-local yield degradation. Int. J. Num. Meth. Engn 1988; 26: 1805-1823. [8] Bazant ZP, Belytschko TB, Chang TP. Continuum theory for strain-softening. J. Engrg. Mech. Div. ASCE 1984; 110: 1666-1692. [9] Sladek J, Sladek V, Bazant ZP. Non-local boundary integral formulation for softening damage. Int. J. Num. Meth. Engn 2003; 57: 103-116. [10] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys 1983; 54: 4703-4710. [11] Eringen AC. Theory of nonlocal elasticity and some applications. Res. Mech. 1987; 21: 313-342. [12] Altan SB. Existence in nonlocal elasticity. Archive Mechanics 1989; 41: 25-36. [13] Lazar M., Maugin G. A., Aifantis E.C., On a theory of nonlocal elasticity of bi-Helmholtz type and some applications, Int. J. Solids and Struct. 2006, 43, 1404-1421. [14] Pisano AA, Sofi, A., Fuschi, P., Nonlocal integral elasticity: 2D finite element based solutions, Int. J. Solids and Struct. 2009; 46, 3838-3849. [15] B. Nayroles, G. Touzot & P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics, 10, 307-318, 1992. [16] T. Belytschko, Y.Y. Lu & L. Gu, Element-free Galerkin method, Int. J. Numerical Methods in Engineering, 37, 229-256, 1994. [17] W.K. Liu, S. Jun & Y. Zhang, Reproducing kernel particle methods, Int. J. Numerical Methods in Engineering, 20, 1081-1106, 1995. [18] S.N. Atluri, The Meshless Method (MLPG) for Domain and BIE Discretizations, Forsyth, GA, USA, Tech Science Press, 2004. [19] V. Sladek, J. Sladek., Ch. Zhang, Comparative study of meshless approximations in local integral equation method, CMC: Computers, Materials, & Continua, 4, 177-188, 2006. [20] Li M., Hon, Y.C., Korakianitis, T., Wen P. H., Finite integration method for nonlocal elastic bar under static and dynamic loads, Engineering Analysis with Boundary Elements (to appear).


457

Boundary element analysis of polymer composites under frictional contact conditions L. Rodr´ıguez-Tembleque∗1 , F.C. Buroni∗ , R. Abascal and A. Sáez∗ Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los Descubrimientos s/n, E41092 Sevilla, SPAIN. 1 [email protected]

Keywords: Polymeric Composites, Anisotropic Friction, Contact Mechanics, Boundary Element Method.

Abstract. A boundary element based formulation is applied to study numerically the tribological behavior of fiber-reinforced plastics (FRP) under different frictional contact conditions. The Boundary Element Method (BEM) with an explicit approach for fundamental solutions evaluation is considered for computing the elastic influence coefficients. Contact constraints on the potential contact zone are enforced by a proposed contact operators over the augmented Lagrangian, which allow to consider an orthotropic law of friction. Furthermore, the formulation considers a micromechanics model for FRP that allows also to take into acount the fiber volume fraction. In these studies, it can be observed the influence of fiber volume fraction, fiber orientation and the sliding orientation on the contact variables. Introduction This work studies numerically the tribological behavior of fiber-reinforced plastics (FRP) under different frictional contact conditions. Although the FRP are highly demanded in the industry and they are widely applied in many machine components, a study of their tribological response has not been fully completed. Furthermore, there are not many numerical formulations that allow to analyze these polymer composites under different frictional contact conditions, taking into account the tribological characteristics of these materials. This paper presents a boundary element methodology to study FRP in contact problems. The formulation, based on previous works [1, 2, 3, 4], uses the Boundary Element Method (BEM) for computing the elastic influence coefficients, and contact operators over the augmented Lagrangian, to enforce anisotropic frictional contact constraints. The variation of fiber volume fraction has also a considerable influence on the contact variable. Micromechanics allows to estimate the mechanical properties of composite materials from the known values of the fiber and the matrix. There are different micromechanical approaches. The simplest approach is the rule of mixtures, but it fails to represent some of the properties with reasonable accuracy, so the modified and more accuracy micromechanical model proposed by Hopkins and Chamis [5] is considered in the formulation. The proposed methodology is applied to study a carbon FRP under orthotropic frictional contact. In these studies, it can be observed the influence of fiber volume fraction, fiber orientation and the sliding orientation on the normal and tangential contact compliance, as well as the contact traction distribution. Boundary integral equations The BEM formulation for an elastic anisotropic continuum Ω was presented in [6]. In order to implement the well known displacement boundary equation, we need an explicit scheme to evaluate the displacement and traction fundamental solution, U∗ and T∗ , respectively. The displacement fundamental solution for anisotropic media can be expressed as a singular term by a modulation function H: 1 U∗ (rˆ H(ˆ e) (1) e) = 4πr ∗ The authors would like to dedicate this work to the memory of Prof. Ram´ on Abascal Garc´ıa (1956-2013). We mourn his untimely death as we loose a great engineering educator, researcher and above all, a good person.

458

where r = x(Q) − x(P ) and ˆ e = (x(Q) − x(P ))/r, being · the Euclidic norm. H(ˆ e) is one of the three Barnett-Lothe tensors which is symmetric and positive-definite. The tensor H(ˆ e) can be evaluated as [7] 1 +∞ −1 Γ (p)dp (2) H(ˆ e) = π −∞ with Γ(p) = Q + (R + RT )p + Tp2 , expressed in terms of the parameter p, Qjk = Cijkl n în ˆ l , Rjk = Cijkl n îm ˆ l and Tjk = Cijkl m ˆ im ˆ l . In these expressions, n ˆ i and m ˆ i are the components of any two mutually orthogonal unit vectors such that {ˆ n, m, ˆ ˆ e} is a right-handed triad. Repeated index implies sum. The components of the traction fundamental solution follow easily from the derivative of the displacement fundamental solution. The derivative of the Green’s function may be expressed in a ˆ as similar way to equation (1), as a singular term by a modulation function which only depends on e ˜ ∗ (ˆ ∂U∗ (rˆ e) e) 1 ∂U = ∂xq 4πr2 ∂xq

(3)

where, according to Lee’s approach [8], the components of the modulation function are given by ˜ ∗ (ˆ ∂U Cpqrs ij e) = −ˆ el Hij + (Mlqiprj eˆs + Msliprj eˆq ) ∂xl π The Msliprj integrals (4) have the following representation in terms of the parameter p: +∞ Φijklmn (p) 1 Mijklmn = dp 2 |T| −∞ (p − p1 )2 (p − p2 )2 (p − p3 )2

(4)

(5)

where pα are the Stroh’s eigenvalues and corresponds to the three complex roots of the sixth-order polynomial equation |Γ(p)| = 0 with positive imaginary part [9]. In equation (5), Φijklmn (p) :=

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

(6)

în ˆ j + (ˆ ni m ˆj + m ˆ in ˆ j )p + m ˆ im ˆ j p2 , being has been introduced together with the definition of Bij := n ˆ jk the adjoint of Γjk , defined as Γpj Γ ˆ jk = |Γ(p)|δpk , where δpk is the Kronecker delta. Γ In order to provide an explicit boundary element formulation, the Cauchy’s residue theory for multiple poles is applied to evaluate the integrals in (2) and (5), so no integration is performed [6]. In addition, possible repeated Stroh’s eigenvalues are allowed in this formulation (see [6] for details). Recently, Buroni and S´ aez [10] have derived new unique and explicit expressions for the anisotropic fundamental solutions that may be used as an alternative evaluation scheme. It is worth to point out that others 3D anisotropic BEM formulations have also been recently proposed as, among others, those by Wang and Denda [11] or Shiah et al. [12]. Contact discrete variables and restrictions To consider the contact between two solids Ωα (α = 1, 2), we have to compute the contact gap for each pair I of nodes in contact (k)I = (kgo )I + (d2 )I − (d1 )I (7) or in a more compact for as:

k = Cg kgo + (C2 )T x2 − (C1 )T x1

(8)

according to [4], where k is the contact pairs gap vector and kgo is the initial geometrical gap and rigid body displacement vector. Contact restrictions for every contact pair I are summarized in: the Non-penetration condition, the Coulomb friction law and the Principle of maximum energy dissipation. The mathematical expressions for these contact restrictions, can be classified into two groups: normal and tangential.


459

• Normal direction: The unilateral contact conditions can be written, in the form of a complementarity relation, as: (kn )I ≥ 0 ;

(pn )I ≤ 0 ;

(pn )I (kn )I = 0

(9)

• Tangential direction: For tangential direction, the fulfilment of friction law and principle of maximum dissipation is guaranteed by: (pt )I μ ≤ |(pn )I | ; (kt )I = −λM−2 (pt )I /||(pt )I ||μ ; λ (pt )I μ − |(pn )I | = 0 (10) being λ ≥ 0. In the expressions above, || • ||μ denotes the elliptic norm @ (pte2 )I 2 (pte1 )I 2 + ||(pt )I ||μ = μ1 μ2 M=

μ1 0 0 μ2

(11)

(12)

and the coefficients μ1 and μ2 are the principal friction coefficients in the directions {e1 , e2 }. The contact restrictions (9) and (10) for every contact pair I can be expressed, according to [4], in a discrete form as: (pt )I − PEρ ( (p∗t )I ) = 0 (pn )I − PR− ( (p∗n )I ) = 0 (13) where augmented contact variables were defined in [2, 3, 4] as: (p∗t )I = (pt )I − rt M2 (kt )I and (p∗n )I = (pn )I + rn (kn )I , as well as the projection functions: PR− and PEρ , with ρ = |PR− ( (p∗n )I )|. BE contact discrete equations The boundary integral equations for a body Ω, can be written in a matrix form as: ˜ − Gp ˜ =F Hd

(14)

where the vector d represents the nodal displacements, and F contains the applied boundary conditions. These equations are well known and can be found in many books like [13] or [14]. Equation (14) can be written for contact problems as: Ax x + Ap pc = F, being (x)T = [(xq )T (dd )T ] the nodal unknowns vector that collects the external unknowns (xq ), and the contact nodal displacements (dc ). ˜ belonging to the conpc is the nodal contact tractions. Ap is constructed with the columns of G ˜ and G, ˜ corresponding to the tact nodal unknowns, and Ax = [Ax Ad ] with the columns matrices H exterior unknowns (Ax ), and the contact nodal displacements (Ad ). Considering a boundary element discretization for every solid Ωα (α = 1, 2), the resulting BEMBEM non-linear contact equations set can be expressed according with [4], as ⎡

A1x ⎢ 0 ⎢ ⎣ (C1 )T 0

0 A2x −(C2 )T 0

⎤⎧ 1 ˜1 A1p C 0 x ⎪ ⎪ ⎨ 2 2 2 ˜ x −Ap C 0 ⎥ ⎥ Λ 0 Cg ⎦ ⎪ ⎪ ⎩ k Pλ Pg

⎫ ⎪ ⎪ ⎬

⎧ ⎪ ⎪ ⎨

F1 F2 = ⎪ ⎪ Cg kgo ⎪ ⎭ ⎪ ⎩ 0

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(15)

The first two rows in the expression above represent the equilibrium of each solid Ωα (α = 1, 2). The third row is the contact kinematics equations and the last row express the nodal contact restrictions. ˜ 1 Λ and p2c = −C ˜ 2 Λ. Matrices Pλ Vector Λ represents the nodal contact tractions, so that: p1c = C and Pg are the non-linear terms obtained by assembling the matrices (Pλ )I and (Pg )I , associated to the I pair of nodes in contact. The values of the matrices depends on the I pair contact state:

460

• No-Contact: (Λ∗n )I ≥ 0 ⎤ ⎤ ⎡ 1 0 0 0 0 0 (Pλ )I = ⎣ 0 1 0 ⎦ , (Pg )I = ⎣ 0 0 0 ⎦ 0 0 1 I 0 0 0 I ⎡

• Contact-Adhesion: (Λ∗n )I < 0 ⎡ 0 0 (Pλ )I = ⎣ 0 0 0 0

and (Λ∗t )I μ < |(Λ∗n )I | ⎤ ⎡ ⎤ 0 −rt μ21 0 0 0 ⎦ , (Pg )I = ⎣ 0 −rt μ22 0 ⎦ 0 I 0 0 −rn I

• Contact-Slip: (Λ∗n )I < 0 and (Λ∗t )I μ ≥ |(Λ∗n )I | ⎤ ⎡ ⎡ ⎤ 1 0 ωt∗1 0 0 0 ∗ 0 ⎦ (Pλ )I = ⎣ 0 1 ωt2 ⎦ , (Pg )I = ⎣ 0 0 0 0 −rn 0 0 0 I I

(16)

(17)

(18)

being: (ω ∗t )I = (Λ∗t )I /(Λ∗t )I μ , and (Λ∗n )I and (Λ∗t )I the normal and tangential augmented variables components associated to the contact pair I: (Λ∗n )I = (Λn )I +rn (kn )I and (Λ∗t )I = (Λt )I +rt M2 (kt )I . Solution method To solve the system (15), Rz = F, the Generalized Newton Method with Line Search (GNMLS) can be applied over: Θ(z) = Rz − F = 0. This method can summarized in the following steps: (1) Start iteration, loop n, defining an arbitrary initial vector z (0) , and the positive scalars: q > 0, β ∈ (0, 1), σ ∈ (0, 1/2), and ε > 0. (2) Solve for Δz(n) , the system BΘ(z(n) , Δz(n) ) = −Θ(z(n) ), where BΘ(z(n) , Δz(n) ) is the function B-derivative. (n) (3) Obtain first integer m = 1, 2, ... that fulfills the following decreasing error condition: Ψ(z + α(n) Δz(n) ) ≤ 1 − 2σα(n) Ψ(z(n) ), with α(n) = β m q and Ψ(z(n) ) = 12 Θ(z(n) )2 .

(4) Actualize solution: z(n+1) = z(n) + α(n) Δz(n) . (5) If Ψ z(n+1) ≤ ε, the solution is achieved: z(n+1) , else compute new iteration (n ← n + 1). Application This example presents a contact problem between an steel sphere of radium: R = 50 mm and a FRP half-space (see Figure 1(a)). The sphere is subjected to a normal displacement g o,x3 = −0.02 mm and a tangential translational displacement of module: go,t = 0.001 mm, which forms an angle θ with axis x1 . The materials of the two contacting bodies are similar to the previous example, well as the orthotropic friction law: μ1 = 0.1 and μ2 = 0.2. For simplicity, due to the contact half-width (a) will be much less than the radius (R), the solids are approximated by elastic half-spaces, each one discretized using linear quadrilateral boundary elements. Figure 1(b) shows the meshes details, where the half-space characteristic dimension is L = 1.2 mm. The carbon FRP considered is IM7 Carbon/ 8551 − 7, whose mechanical properties of its fiber and matrix can be found in [15]. So the influence of fiber volume fraction V¯f can be studied: V¯f = {0.30, 0.45, 0.60, 0.75}. Figures 2 (a) and (b) shows the influence of the fiber volume fraction on the normal and tangential contact loads, for a fixed normal indentation and tangential translational displacement of module. Figures 2(a) and (b) show the normal load variations relative to the load for the fiber alignment parallel to the axe x1 (α1 = 0) and a volume fraction of 30 %. In Figure 2(a), for


461

(a)

(b)

Figure 1: (a) Sphere indentation over a FRP halfspace. (b) Boundary elements mesh details.

every fiber orientation, the normal load increases its value with V¯f , but the biggest increment occurs in the normal fiber orientation. Same behavior is observed in Figure 2(b) for the tangential load: its values increases with V¯f . Finally, the Figure 2(c) shows the influence of the fiber volume fraction on the orthotropic tangential contact compliance for a fixed fiber orientation (α 1 = 0o ). For every sliding direction θ, the tangential load increases in the same proportion with V¯f .

(a)

(b)

(c)

Figure 2: Influence of the fiber volume fraction on the normal (a) and tangential (b) contact compliance. (c) Influence of the fiber volume fraction on the orthotropic tangential contact compliance.

Conclusions This work presents a three-dimensional boundary element methodology which allows us to analyze fiber-reinforced polymer under contact conditions, taking into account both the mechanical and the tribological anisotropic characteristics. All these examples show the importance of considering, in the contact problems of FRP, the anisotropy and the micromechanics of the bulk, and the anisotropy of the surface properties. As it can be observed in the results, the contact variables, since contact traction distributions and contact compliances are clearly modified by the fiber orientation, the volume fraction or the sliding direction. So they have to be considered in the simulation. In other case, we could over- or underestimate contact magnitudes and their distribution over the contact zone.

462

Acknowledgments This work was co-funded by the DGICYT of Ministerio de Ciencia y Tecnolog´ıa, Spain, research projects DPI2010-19331 and DPI2010-21590-C02-02, which were co-funded by the European Regional Development Fund (ERDF) (Fondo Europeo de Desarrollo Regional, FEDER). References [1] Rodr´ıguez-Tembleque, L., Buroni, F.C., Abascal, R., S´ aez, A. 3D frictional contact of anisotropic solids using BEM. Eur. J. Mech. A. Solids. 2011; 30: 95–104. [2] Rodr´ıguez-Tembleque, L., Abascal, R., Aliabadi, M. H. Anisotropic wear framework for 3D contact and rolling problems. Comput. Meth. Appl. Mech. Eng. 2012; 241: 1–19. [3] Rodr´ıguez-Tembleque, L., Abascal, R., Aliabadi, M. H. Anisotropic fretting wear simulation using the boundary element method. CMES–Computer Modeling in Engineering and Sciences 2012; 87: 127–155. [4] Rodr´ıguez-Tembleque, L., Abascal, R. Fast FE-BEM algorithms for orthotropic frictional contact. Int. J. Numer. Methods Eng. 2013; 94: 687–707. [5] Hopkins, D. A., Chamis, C. C. A Unique Set of Micromechanics Equations for High Temperature Metal Matrix Composites. In: Testing Technology of Metal Matrix Composites, ASTM STP 964, American Society for Testing and Materials, Philadelphia, 1988; 159–176. [6] Buroni, F.C., Ortiz, J.E., S´ aez, A. Multiple pole residue approach for 3D BEM analysis of mathematical degenerate and non-degenerate materials. Int. J. Numer. Methods Eng. 2011; 86: 1125– 1143. [7] Ting, T.C.T., Lee, V.G. The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids Q. J. Mech. Appl. Math. 1997; 50:407–426. [8] Lee, V.G. Explicit expressions of derivatives of elastic Greens functions for general anisotropic materials. Mech. Res. Comm. 2003; 30: 241–249. [9] Ting, T.C.T. Anisotropic Elasticity, Oxford University Press, Oxford, 1996. [10] Buroni, F.C., S´ aez, A. Unique and explicit formulas for Green’s function in three-dimensional anisotropic linear elasticity. Journal of Applied Mechanics, In press. [11] Wang CY, Denda M. 3D BEM for general anisotropic elasticity. Int. J. Solids Struct. 2007; 44: 7073–7091. [12] Shiah, Y. C., Tan, C. L., Wang, C. Y. Efficient computation of the Green’s function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis. Eng. Anal. Boundary Elem. 2012; 36: 1746–1755. [13] Brebbia, C.A., Dominguez J. Boundary Elements: An Introductory Course (second edition). Computatinal Mechanics Publications. John Wiley & Sons, 1992. [14] Aliabadi, M.H. The Boundary Element Method Vol2: Applications in Solids and Structures. John Wiley & Sons, 2002. [15] Kaddour, A. S., MJ Hinton, M. J.. Input data for test cases used in benchmarking triaxial failure theories of composites. J. Compos. Mater. 2012; 54: 2295–2312.


463

Efficient FFT–MFS algorithms for boundary value problems in two-dimensional linear thermoelasticity Andreas Karageorghis1 and Liviu Marin2,3

of Mathematics and Statistics, University of Cyprus/Panepist mio KÔprou, P.O. Box 20537, 1678 Nicosia/LeukwsÐa, Cyprus/KÔproc, [email protected] 2 Institute of Solid Mechanics, Romanian Academy, 15 Constantin Mille, P.O. Box 1-863, 010141 Bucharest, Romania, [email protected] 3 Centre for Continuum Mechanics, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei, 010014 Bucharest, Romania

1 Department

Keywords: Linear thermoelasticity; Method of fundamental solutions (MFS); Fast Fourier transforms (FFTs). Abstract. We propose efficient algorithms for the numerical solution of boundary value problems in planar linear thermoelasticity which combine fast Fourier transforms (FFTs) with the recentlyproposed method of fundamental solutions (MFS)–method of particular solutions (MPS), see e.g. [1, 2]. Mathematical Formulation Consider an isotropic solid which occupies a two-dimensional domain Ω bounded by a smooth curve ∂Ω and is characterised by the thermal conductivity, κ, the coefficient of linear thermal expansion, αT , Poisson’s ratio, ν, and the shear modulus, G, respectively. In the framework of linear isotropic thermoelasticity, the stress tensor, σ = [σij ]1≤i,j≤2 , is related to the strain tensor, = [ij ]1≤i,j≤2 , by means of the constitutive law of thermoelasticity, namely ν tr ((x)) I − γ T(x) I , x ∈ Ω . (1) σ(x) = 2G (x) + 1 − 2ν Here I = [δij ]1≤i,j≤2 is the identity matrix in R2×2 and the constants ν and γ are related to the shear modulus, Poisson’s ratio and the coefficient of linear thermal expansion by the following equations: ν plane strain state 1+ν ν= γ = 2G αT , (2) and 1 − 2ν ν /(1 + ν) plane stress state respectively, where αT = αT (plane strain state) and αT = αT (1 + ν) /(1 + 2ν) (plane stress state). In the absence of heat sources and body forces, the governing equations of isotropic linear thermoelasticity in terms of the temperature and the displacement vector are given by & ' −∇ · κ ∇T(x) = 0 , x ∈ Ω , (3a) −∇ · σ(x) ≡ L u(x) + γ ∇T(x) = 0 ,

x ∈ Ω,

(3b)

and they are subject to either Dirichlet boundary conditions 3 T(x) = T(x) ,

x ∈ ∂Ω ,

(3c)

3 (x) , u(x) = u

x ∈ ∂Ω ,

(3d)

or mixed boundary conditions 3 T(x) = T(x) ,

x ∈ Γ1 ,

q(x) = 3 q(x) ,

x ∈ Γ2 ,

(3e)

3 (x) , u(x) = u

x ∈ Γ1 ,

t(x) = 3 t(x) ,

x ∈ Γ2 .

(3f)

In eqs. (3a)–(3f) the boundary segments Γ1 , Γ2 ⊂ ∂Ω are & such that ' Γ1 ∩ Γ2 = ∅ and Γ1 ∪ Γ2 = ∂Ω, n(x) is the outward unit normal vector at Γ2 , q(x) ≡ − κ ∇T(x) · n(x) and t(x) ≡ σ(x) n(x) are the normal heat flux and the traction vector at x ∈ Γ2 , respectively, while L = (L1 , L2 )T is the partial differential operator associated with the Navier-Lamé system of isotropic linear elasticity, i.e. " ! & ' 2ν ∇ ∇ · u(x) , x ∈ Ω . (4) L u(x) ≡ −G ∇ · ∇u(x) + ∇u(x)T + 1 − 2ν

464

Solution Strategy The MFS–MPS approach for the solution of the two-dimensional thermoelasticity problem (3a)–(3b) with boundary conditions (3c)–(3d) or (3e)–(3f) relies on the existence of a particular solution of the non-homogeneous equilibrium equations (3b) which is justified by the result presented in [1, 2, 3]. Due to the structure of the problem investigated herein, see eqs. (3a)–(3f), we apply the following strategy which combines the MFS and the MPS, see also [1, 2, 3]: Step 1. Solve the thermal (3a) and (3c) [(3a) and (3e)] using the MFS to determine the unknown 6 problem 6 6 6 boundary data q6∂Ω [q6Γ1 and T6Γ2 ], as well as the temperature distribution in the domain T6Ω . Step 2. Determine a particular solution u(P) of the non-homogeneous equilibrium equations for two-dimensional isotropic linear elasticity (3b) in Ω, as well as the corresponding particular strain tensor (P) , stress tensor σ(P) and traction vector t(P) . Step 3. Solve the resulting homogeneous equilibrium equations in two-dimensional isotropic linear elasticity L u(H) (x) = 0 ,

x ∈ Ω,

(5a)

with either Dirichlet boundary conditions 3 (x) − u(P) (x) , u(H) (x) = u

x ∈ ∂Ω ,

(5b)

or mixed boundary conditions 3 (x) − u(P) (x) , x ∈ Γ1 , u(H) (x) = u t(H) (x) = 3 t(x) − t(P) (x) − γ T(x) n(x) , x ∈ Γ2 , using the MFS to determine the unknown boundary data, namely either 6 6 6 6 u(H) 6Γ2 , as well as u(H) 6Ω , (H) 6Ω and σ(H) 6Ω .

(5c) (5d) 6

t(H) 6

∂Ω

or

6

t(H) 6

Γ1

and

6 Step 4. By applying the superposition principle, determine the unknown boundary data, i.e. either t6∂Ω = 6 6 6 6 6 6 6 6 & ' & ' t(H) 6∂Ω + t(P) − γ T n 6∂Ω or t6Γ1 = t(H) 6Γ1 + t(P) − γ T n 6Γ1 and u6Γ2 = u(H) 6Γ2 + u(P) 6Γ2 , as 6 6 6 6 6 6 6 6 & '6 well as u6Ω = u(H) 6Ω + u(P) 6Ω , 6Ω = (H) 6Ω + (P) 6Ω and σ6Ω = σ(H) 6Ω + σ(P) − γ T I 6Ω . Method of Fundamental Solutions Step 1. Consider the fundamental solution, F, of the two-dimensional steady-state heat conduction equation (3a) in an isotropic homogeneous medium [4], namely 1 log |x − ξ| , x ∈ Ω , (6) 2πκ ' & ' & where x = x1 , x2 is a collocation point and ξ = ξ1 , ξ2 ∈ R2 \ Ω is a singularity or source point, and approximate the temperature in the solution domain by a linear combination of fundamental solutions N with respect to N singularities, ξ(n) n=1 , in the form F(x, ξ) = −

T(x) ≈ TN (a, ξ; x) =

N

' & an F x, ξ(n) ,

x ∈ Ω,

(7)

n=1

'T & where a = a1 , . . . , aN ∈ RN and ξ ∈ R2N is a vector containing the coordinates of the singularities (n) N ξ . Consequently, the normal heat flux on ∂Ω is approximated by n=1 q(x) ≈ qN (a, ξ; x) = −

N n=1

' & an κ ∇x F x, ξ(n) · n(x) ,

x ∈ ∂Ω .

(8)


465

(n) N

Next, we select N collocation points, x ⊂ ∂Ω, and collocate the boundary conditions (3c) or n=1 (3e) to obtain the following system of linear equations with respect to a ∈ RN : Aa = f .

(9)

Here A ∈ RN×N is the matrix whose elements are calculated from equations (7) and (8), while f ∈ RN is the right-hand side vector containing the corresponding discretised boundary data as given by equations (3c) or (3e). Step 2. The MFS approximation for the particular solution of the non-homogeneous equilibrium equations (4) in R2 is given by (P)

u(P) (x) ≈ uN (a, ξ; x) = − x∈

R2

\

N A

ξ

γ 8πκG

(n)

#

1 − 2ν 1−ν

$ N

& ' an x − ξ(n) log |x − ξ(n) | ,

n=1

(10)

.

n=1

From eq. (10), the corresponding particular traction vector on the boundary ∂Ω is approximated as # $ 1 − 2ν γ (P) t(P) (x) ≈ tN (a, ξ; x) = − 4πκ 1 − ν + , ' & N (11) " ' x − ξ(n) · n(x) & 1 ! (n) × x − ξ , x ∈ ∂Ω . log |x − ξ(n) | + ν n(x) + an (n) 2 1 − 2ν |x − ξ | n=1

Note that once the coefficients, a ∈ RN , corresponding to the thermal problem (3a) and (3c) [(3a) and (3e)] are retrieved by solving equation (9), the particular solutions for the boundary displacement vector on Γ2 boundary traction vector on Γ1 are expressed via eqs. (10) and (11), respectively. Step 3. For the Cauchy-Navier system associated with the two-dimensional isotropic linear elasticity, the fundamental solution matrix U = [Uij ]1≤i,j≤2 , for the displacement vector is given by [5] 1 Uij (x, ξ) = 8πG(1 − ν)

+

, xi − ξi xj − ξj − (3 − 4ν) log |x − ξ| δij + , |x − ξ| |x − ξ|

x ∈ Ω,

i, j = 1, 2 . (12)

By combining the definition of the traction vector, Hooke’s constitutive law, the kinematic relations and eq. (12), the fundamental solution matrix T = [Tij ]1≤i,j≤2 , for the traction vector in the case of two-dimensional isotropic linear elasticity is obtained as [5] 2G (1 − ν) ∂x1 U1k (x, ξ) + ν ∂x2 U2k (x, ξ) n1 (x) 1− 2ν + G ∂x2 U1k (x, ξ) + ∂x1 U2k (x, ξ) n2 (x) , x ∈ ∂Ω , k = 1, 2 ,

T1k (x, ξ) =

and

T2k (x, ξ) = G ∂x2 U1k (x, ξ) + ∂x1 U2k (x, ξ) n1 (x) 2G ν ∂x1 U1k (x, ξ) + (1 − ν) ∂x2 U2k (x, ξ) n2 (x) , + 1 − 2ν

x ∈ ∂Ω ,

k = 1, 2 ,

(13a)

(13b)

where ∂xk Uij denotes the derivative of the fundamental solution for the displacement vector (12) with respect to xk , k = 1, 6 2. Analogously to the MFS for the thermal problem, we approximate the displacement vector u(H) 6Ω by (H)

u(H) (x) ≈ uN (d, ξ; x) =

N n=1

U(x, ξ(n) ) dn ,

x ∈ Ω,

(14)

466

'T & && 'T & 'T 'T where dn = bn , cn ∈ R2 , n = 1, . . . , N, and d = d1 , . . . , dN ∈ R2N . In a similar manner, 6 (H) 6 the traction vector t is approximated by Ω (H)

t(H) (x) ≈ tN (d, ξ; x) =

N

T(x, η(n) ) dn ,

x ∈ ∂Ω .

(15)

n=1

By collocating the boundary conditions (3d) or (3f) at system of linear equations with respect to d ∈ R2N :

x(n)

N n=1

⊂ ∂Ω, one obtains the following

Bd = g.

(16)

R2N×2N

3 − C a is is the matrix whose elements are calculated from eqs. (14) and (15), g ≡ g Here B ∈ 3 ∈ R2N is the right-hand side vector containing the corresponding the modified right-hand side, where g boundary data as given by eqs. (3d) or (3f), while the elements of the matrix C ∈ R2N×N are determined from those elements of the matrices that approximate u(P) and t(P) − γ T n. Step determined the coefficients d ∈ R2N , the approximations for the unknown boundary 6 4. Having 6 6 data t6∂Ω (u6Γ2 and t6Γ1 ) are obtained via the superposition principle and eqs. (10), (14) and (15). Dirichlet Problems in Circular Domains There are considerable computational savings in the proposed approach when the domain Ω is, for example, a disk and we consider the Dirichlet problem (3a)–(3d). In particular, if Ω is the disk N Ω = x ∈ R2 : |x| < ρ , we choose the collocation points x(n) n=1 such that # $ & (n) (n) ' 2(n − 1)π 2(n − 1)π , sin x1 , x2 = ρ cos , n = 1, . . . , N , N N N and the singularities ξ (m) m=1 such that # $ & (m) (m) ' 2(m − 1 + α)π 2(m − 1 + α)π = R cos , sin ξ1 , ξ 2 , m = 1, . . . , N , N N B where R > ρ and the positions of the sources differ by an angle (2πα) N from the positions of the boundary points and 0 ≤ α < 1. B& ' Laplacian System. The matrix A = 1 2πκ log |x(n) − ξ(m) | in eq. (9) is known [6] to 1≤n,m≤N

be circulant [7]. If we define the matrix U ∈ CN×N by 1 ! −2πi(k−1)(−1)/N "N e , U=√ k,=1 N

(17)

we premultiply system (9) by U to obtain U A U∗ U a = U f

or

?=? Da f

(18) ' & ? = U a and the matrix D = U A U ∗ is diagonal. The elements of a ?= ? where ? f = U f, a a1 , ? a2 , . . . , ? aN B ∗ ?. can be easily calculated from ? an = f?n Dnn , n = 1, . . . , N, and then a can be recovered from a = U a ? are carried out efficiently using Fast Note that the operations ? f = U f , D = U A U ∗ and a = U ∗ a Fourier Transforms (FFTs) (for details, see [6]). Cauchy-Navier System. The matrix B in eq. (16) is not circulant but with some manipulations can become so [8]. More specifically, we may write system (16) as ⎡ ⎤⎛ ⎞ ⎛ ⎞ B11 B12 · · · B1N d1 g1 ⎢ B21 B22 · · · B2N ⎥ ⎜ d2 ⎟ ⎜ g2 ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟ (19) ⎢ .. .. .. ⎥ ⎜ .. ⎟ = ⎜ .. ⎟ , .. ⎣ . . . . ⎦⎝ . ⎠ ⎝ . ⎠ BN1 BN2 · · · BNN dN gN


467

where Bnm =

& ' & ' U11 x(n) , ξ (m) U12 x(n) , ξ (m) & (n) (m) ' & (n) (m) ' , U22 x , ξ U21 x , ξ

# dm =

bm cm

-

$ ,

gn =

(P)

31 (x(n) ) − u1 (x(n) ) u (P) u2 (x(n) ) − u2 (x(n) ) 3

. .

Following [8], we next define the matrices Si ∈ R2×2 , i = 1, . . . , N, by 1 Δx(i) Δy(i) , Si = ri Δy(i) −Δx(i) % (i) (i) (i) (i) where Δx(i) = x1 − ξ1 , Δy(i) = x2 − ξ2 and ri = (Δx(i) )2 + (Δy(i) )2 . Si Si = I, i = 1, . . . , N, system (19) can be re-written as ⎤⎛ ⎞ ⎛ ⎡ S1 B11 S1 S1 B12 S2 · · · S1 B1N SN S1 d1 S 1 g1 ⎢ S2 B21 S1 S2 B22 S2 · · · S2 B2N SN ⎥ ⎜ S2 d2 ⎟ ⎜ S2 g2 ⎥⎜ ⎟ ⎜ ⎢ ⎢ ⎥ ⎜ .. ⎟ = ⎜ .. .. .. .. .. ⎣ ⎦⎝ . ⎠ ⎝ . . . . . SN BN1 S1 SN BN2 S2 · · · and rearranged as

+

SN BNN SN

3 11 B 3 12 B 3 21 B 3 22 B

SN dN

Using the property ⎞ ⎟ ⎟ ⎟, ⎠

S N gN

,# $ # $ 3 31 g b = , 32 g 3 c

3 ij ∈ RN×N , i, j = 1, 2, are defined by where the matrices B ' ' ⎤ & ⎡ & S1 B11 S1 ij · · · S1 B1N SN ij ⎥ .. .. .. 3 ij = ⎢ B ⎦, ⎣ . . . ' ' & & SN BNN SN ij SN BN1 S1 ij · · ·

(20)

(21)

i, j = 1, 2,

and ' & 'T & 3 = S1 b 1 , S 2 b 2 , . . . , S N b N T , g 31 = S1 g1 (x(1) ), S2 g1 (x(2) ), . . . , SN g1 (x(N) ) , b 'T & & 'T 3 32 = S1 g2 (x(1) ), S2 g2 (x(2) ), . . . , SN g2 (x(N) ) . c = S1 c1 , S2 c2 , . . . , SN cN , g

(22)

3 ij , i, j = 1, 2, can be shown to be circulant [8]. Hence we re-write (21) as Each of the matrices B , + # $ # $ 3 12 3 11 B 3 31 g B b ∗ (I ⊗ U (I ⊗ U ) 3 ) (I ⊗ U ) = (I ⊗ U ) , (23) 3 22 32 g 3 c B21 B where U is defined by eq. (17). System (23) can be written as # $ # $ ? 31 Ug D11 D12 b = , 32 D21 D22 Ug ? c

(24)

? = U b, 3 ? where b c = U3 c and the matrices Dij = U Bij U ∗ , i, j = 1, 2, are diagonal. Thus solving 3 and system (24) is equivalent to solving N systems of order 2, which yield the unknown vectors U b 'T & 3 and 3 3i = 3 U3 c. The vectors b c can then be obtained by pre-multiplication by U ∗ . By defining d bi , 3ci , i = 1, . . . , N, the vectors di , i = 1, . . . , N, in eq. (19), and hence the solution, can be recovered from 3 i , i = 1, . . . , N. the relations di = Si d Matrix Decomposition Algorithm. Combining the results mentioned above, we obtain the following matrix decomposition algorithm [9]: Step 1. Calculate the vector ? f = U f in eq. (18). Step 2. Evaluate the elements of the diagonal matrix D = U A U ∗ .

468

B Step 3. Evaluate ? ai = ?fi Dii , i = 1, . . . , N, in eq. (18). ?. Step 4. Recover a = U ∗ a Step 5. Use the MFS approximation (7) to modify the boundary conditions as shown in eq. (5b). Step 6. Calculate the matrices Si Bij Sj , i, j = 1, . . . , N, and the vectors Si gi , i = 1, . . . , N, in eq. (20). Step 7. Rearrange system (20) to obtain system (21). 31 and g 32 from eq. (22). Step 8. Calculate g 31 and U g 32 . Step 9. Compute U g Step 10. Evaluate the elements of the diagonal matrices Dij = U Bij U ∗ , i, j = 1, 2. ? and ? Step 11. Evaluate b c by solving N (2 × 2)−systems in eq. (24). 3 = U∗ b ? and 3 Step 12. Compute b c = U∗ ? c. 'T 'T & & 3 i , i = 1, . . . , N. 3i = 3 bi , 3ci , i = 1, . . . , N, and then evaluate bi , ci = di = Si d Step 13. Define d Note that FFTs can be used in Steps 1, 2, 4, 9, 10 and 12 leading to an algorithm of total cost O(N log N). Although not presented herein, it is worth mentioning that similar conclusions can be drawn for linear thermoelasticity mixed boundary value problems in annular domains; for details, we refer the reader to [3]. Conclusions In this study, we have proposed efficient FFT–based algorithms for the numerical solution of certain problems in two-dimensional thermoelasticity by employing the MFS–MPS approach of [1, 2]. These algorithms are applicable to problems in domains possessing radial symmetry, such as disks and annuli, for both Dirichlet and Neumann boundary conditions. The MFS matrices arising in such situations possess circulant or block-circulant structures. Consequently, the solution of the resulting systems can be carried out efficiently by using FFTs. Acknowledgements. The financial support received from the Romanian National Authority for Scientific Research (CNCS–UEFISCDI), project number PN–II–ID–PCE–2011–3–0521, is gratefully acknowledged.

References

[1] L. Marin and A. Karageorghis, Boundary Elements and other Mesh Reduction Methods XXXIV (BEM/MRM 2012) (C.A. Brebbia and D. Poljak, eds.), WIT Press, Southampton (2012). [2] L. Marin and A. Karageorghis, Engineering Analysis with Boundary Elements (2013); in press, doi: 10.1016/j.enganabound.2013.04.002. [3] A. Karageorghis and L. Marin, Journal of Scientific Computing (2013); in press, doi: 10.1007/s10915-012-9664-x. [4] G. Fairweather and A. Karageorghis, Advances in Computational Mathematics 9, 69–95 (1998). [5] M.H. Aliabadi, The Boundary Element Method. Volume 2: Applications in Solids and Structures, John Wiley & Sons, London (2002). [6] Y.–S. Smyrlis and A. Karageorghis, Journal of Scientific Computing 16, 341–371 (2001). [7] P.J. Davis, Circulant Matrices, John Wiley & Sons, New York-Chichester-Brisbane (1979). [8] A. Karageorghis, Y.–S. Smyrlis, and T. Tsangaris, Numerical Algorithms 43, 123–149 (2006). [9] B. Bialecki, G. Fairweather, and A. Karageorghis, Numerical Algorithms 56, 253–295 (2011).


469

Coupling boundary integral and shell finite element methods to study the fluid structure interactions of a microcapsule in a simple shear flow. Claire Dupont1,2, Anne-Virginie Salsac2, Dominique Barthès-Biesel2, Marina Vidrascu3, Patrick Le Tallec1 1

Laboratoire de Mécanique des Solides (UMR CNRS 7649), Ecole Polytechnique, emails: (dupont, letallec)@lms.polytechnique.fr

2

Laboratoire de Biomécanique et Bioingénierie (UMR CNRS 7338), Université de Technologie de Compiègne, emails: (a.salsac, dbb)@utc.fr 3

Equipe-projet REO, INRIA Rocquencourt - LJLL UMR 7958 UPMC, email: [email protected]

Keywords: fluid-structure interaction, shell element, finite element method, boundary integral method Abstract. We simulate the motion of an initially spherical capsule in a simple shear flow in order to determine the influence of the bending resistance on the formation of wrinkles on the membrane. The fluid structure interactions are obtained numerically coupling a boundary integral method to solve for the Stokes equation with a nonlinear finite element method for the capsule wall mechanics. The capsule wall is discretized with MITC linear triangular shell finite elements. We find that, at low flow strength, buckling occurs in the central region of the capsule. The number of wrinkles on the membrane decreases with the bending stiffness and, above a critical value, wrinkles no longer form. For thickness to radius ratios below 5%, the bending stiffness does not have any significant effect on the overall capsule motion and deformation. The mean capsule shape is identical whether the wall is modeled as a shell or a two-dimensional membrane, which shows that the dynamics of thin capsules is mainly governed by shear elasticity and membrane effects. Introduction Bioartificial capsules consisting of an internal liquid droplet enclosed by a thin hyperelastic wall have numerous applications in bioengineering and pharmaceutics, where they are used as vectors for drug targeting or the development of artificial organs. Their membrane may undergo large deformations due to the hydrodynamic stresses exerted by the flows of the internal and suspending fluids. Their motion and deformation are therefore solution of a complex problem of fluid-structure interaction in the viscous flow regime. It is important to predict their behaviour in order to avoid/provoke the membrane rupture depending on the applications. Different numerical methods have been considered to simulate the dynamics of a capsule in an external flow. Previous studies have solved the membrane equilibrium either locally using spectral elements [1] or bicubic B-splines [2], or globally implementing a finite element method [3]. To solve the low Reynolds number equations, the boundary integral method [1,2,3,4] is the technique the most classically used as compared to the lattice Boltzmann method [5] and spectral method [6]. Since the velocity field at any position within the fluid domain is given by surface integrals calculated on the geometric boundaries, it allows reducing the dimension of the problem by one and avoids re-meshing the fluid domain at each time step. It also allows a Lagrangian tracking of the membrane position with high accuracy. The boundary integral – finite element (BI-FE) coupling method, initially proposed by [3], has been shown to be stable in the presence of in-plane compression. It has thus enabled the study of little explored cases, such as the dynamics of a capsule in a pore with a square cross-section [7] or the motion of an ellipsoidal capsule in a simple shear flow, when its revolution axis is initially placed off the shear plane [8]. So far, the capsule wall has typically been considered to be infinitely thin with negligible bending stiffness and modeled as a 2D hyperelastic material. We presently introduce a boundary integral – shell finite element method to take into account the bending stiffness of the capsule wall and study its influence on the motion and deformation of a spherical capsule in an external shear flow. An MITC (mixed interpolation of the tensorial component) shell finite element method is used to model both membrane and bending effects [9]. After a brief outline of the problem at stake, we detail the fluid-structure interaction numerical method and validate it on the classical test of an initially spherical capsule in simple shear flow. We then investigate the influence of the bending resistance on the capsule motion and on the wrinkle formation.

470

Problem Statement We consider an initially spherical microcapsule (radius ℓ) consisting of a liquid droplet enclosed by a thin membrane. The membrane is a three-dimensional material of thickness h, shear modulus G, Poisson coefficient ν and bending modulus κ. It is modeled as a midsurface shell defined by the material properties (1) Gs hG, Qs Q where Gs is the surface shear modulus and νs the surface Poisson coefficient. The capsule is suspended in a simple shear flow in the (e1,e2) plane: (2) v f Jx2 e1 , where J is the shear rate. The inner and outer fluids are supposed to be Newtonian and to have the same viscosity μ and density ρ. Owing to the small capsule size, the inner and outer flow Reynolds numbers Re U" 2J / P are infinitely small. The dynamics of the microcapsule is mostly governed by the capillary and bending numbers

Ca

PJ" Gs

and

B

1 N . " Gs

(3)

The capillary number compares the viscous to the shear elastic forces and the bending number the bending to the shear elastic forces. The latter can also be considered as the ratio of the membrane thickness h to the sphere radius ℓ. Numerical Method: General Principal For the first time, the BI-FE method is enriched with shell finite elements to account for the capsule bending stiffness. The capsule wall is discretized with MITC (Mixed Interpolation Tensorial Components) triangular shell finite elements, the nodes being located on the midsurface. The first step consists in computing the displacement field of the capsule membrane material points, as well as the Green-Lagrange strain tensor. The tension tensor is obtained assuming the capsule to follow the Hooke’s law. The membrane equilibrium equation, expressed in its weak form, is solved using a finite element method to deduce the viscous load on the membrane. The velocity of the membrane nodes is then obtained solving the Stokes equations in the internal and external fluids with a boundary integral formulation. The new position of the capsule membrane points is finally calculated integrating the velocity with an explicit Euler integration scheme. Membrane Mechanics In this subsection, we briefly describe the shell kinematics based on [9] and [10] as well as the mechanical problem solved. The surface tensor components will be designed with Greek indices and the 3D tensor components with Latin indices. We adopt the Einstein summation convention on repeated indices. The capsule wall is represented as a shell of midsurface S and thickness h. At each instant of time, the midsurface is defined by the 2D chart ߮ሺߦଵ ǡ ߦ ଶ ሻ,which takes values in the bounded open subset ߱ ‫ א‬Թଶ . It is convenient to define the local covariant base (a1, a2, a3) following the midsurface deformation. The two base vectors (a1, a2) are tangent to the midsurface ܽఈ ൌ ߮ǡఈ ǡ ߙ ൌ ͳǡʹ (4) where the notation Ǥǡఈ denotes the partial derivative with respect toߦ ఈ . The third vector a3 is the unit normal vector n of the capsule midsurface S. The contravariant base (a1, a2, a3) is defined by ܽ ఈ Ǥ ܽఉ ൌ ߜఉఈ , with ߜఉఈ the Kronecker tensor and a3 = a3= n. The three-dimensional position within the capsule wall is given by ߮ଷ஽ ሺߦଵ ǡ ߦ ଶ ǡ ߦ ଷ ሻ ൌ ߮ሺߦଵ ǡ ߦ ଶ ሻ ൅ ߦ ଷ ܽଷ ሺߦଵ ǡ ߦ ଶ ሻǡ (5) ଵ ଶ ଷ for ሺߦ ǡ ߦ ǡ ߦ ሻ in the reference domain ௛൫క భ ǡక మ ൯ ௛൫క భ ǡక మ ൯

߱ଷ஽ ൌ ቄሺߦଵ ǡ ߦ ଶ ǡ ߦ ଷ ሻ ‫ א‬Թଷ Ȁሺߦଵ ǡ ߦ ଶ ሻ ‫ א‬ɘǡ ߦ ଷ ‫ א‬ቃെ (6) ǡ ቂቅǤ ଶ ଶ We define the 3D covariant base vector (g1, g2, g3) such that ଷ஽ ݃ఈ ൌ ߮ǡఈ ൌ ܽఈ ൅ ߦ ଷ ܽǡఈ and ݃ଷ ൌ ܽଷ . (7) 1 2 3 The 3D contravariant base (g , g , g ) is likewise defined by ݃௠ Ǥ ݃௡ ൌ ߜ௡௠ . The components of the 3D metrics tensor are


471

݃ఈఉ ൌ ݃ఈ Ǥ ݃ఉ ǡ ݃ఈଷ ൌ Ͳǡ and݃ଷଷ ൌ ͳǤ (8) We assume that the displacement satisfies the Reissner-Mindlin kinematical assumption, i.e. the material line orthogonal to the midsurface remains straight and unstretched during deformation. The displacement is then expressed by ‫ݑ‬ଷ஽ ሺߦ ଵ ǡ ߦ ଶ ǡ ߦ ଷ ሻ ൌ ‫ݑ‬ሺߦଵ ǡ ߦ ଶ ሻ ൅ ߦ ଷ ߠఒ ሺߦଵ ǡ ߦ ଶ ሻܽ ఒ ሺߦଵ ǡ ߦ ଶ ሻǤ The first term u represents the global infinitesimal displacement of a line perpendicular to the midsurface at the coordinatesሺߦଵ ǡ ߦ ଶ ሻ. The second term is the displacement due to the rotation of this line. The deformation of the membrane is computed from the displacement. The expression of the nonlinear 3D Green-Lagrange strain tensor is ଵ ݁௜௝ ൌ ଶ ቀ݃௜ Ǥ ‫ݑ‬ǡ௝ଷ஽ ൅ ݃௝ Ǥ ‫ݑ‬ǡ௜ଷ஽ ൅ ‫ݑ‬ǡ௜ଷ஽ Ǥ ‫ݑ‬ǡ௝ଷ஽ ቁ i, j = 1, 2, 3. (10) The second Piola-Kirchhoff tension tensor ߑis then obtained from ߑ ൌ

డ௪ሺ௨యವ ሻ డ௘

ǡ

(11)

where the strain energy function takes the form ଵ

‫ݓ‬൫‫ݑ‬ଷ஽ ൯ ൌ ଶ ‫׬‬ఠయವ ൣ‫ ܥ‬ఈఉఒஜ ݁ఈఉ ൫‫ݑ‬ଷ஽ ൯݁ఒஜ ൫‫ݑ‬ଷ஽ ൯ ൅ ‫ ܦ‬ఈఒ ݁ఈଷ ൫‫ݑ‬ଷ஽ ൯݁ఒଷ ൫‫ݑ‬ଷ஽ ൯൧ ݀߱

α, β, λ, μ = 1,2.

(12)

with ‫ ܥ‬ఈఉఒஜ ൌ ‫ܩ‬௦ ቀ݃ఈఉ ݃ఒஜ ൅ ݃ఈஜ ݃ఉఒ ൅

ଶఔೞ ݃ఈఉ ݃ఒஜ ቁǡ ଵିఔೞ

(13)

(14) ‫ ܦ‬ఈఒ ൌ Ͷ‫ܩ‬௦ ݃ఈఒ for the Hooke's law. Tensions, which are forces per unit length of the deformed midsurface, are obtained integrating the stresses across the wall thickness. Knowing the internal tension tensor, the unknown viscous load exerted by the fluid on the membrane can be calculated solving the wall equilibrium ‫׏‬ୱ Ǥ ܶ ൅ ‫ ݍ‬ൌ ͲǤ (15) The operator ‫׏‬ୱ is the surface gradient and ܶ the Cauchy tension tensor such that ଵ

ܶ ൌ ௃ ‫ܨ‬Ǥ ߑǤ ‫் ܨ‬

(16)

with J the Jacobian and ‫ܨ‬the deformation gradient with respect to the reference configuration. The local equilibrium (eq. (15)) is then written in a weak form using the virtual work principle and solved by means of the finite element method. Let V be the Sobolev space H1. For any virtual displacement field ‫ݑ‬ොଷ஽ ‫ܸ א‬, the internal and external virtual work balance requires ‫׬‬ఠయವ ‫ݑ‬ොଷ஽ Ǥ‫ ߱݀ ݍ‬ൌ ‫׬‬ఠయವ ߑ ‫݁ߜ ׷‬Ƹ ݀߱ where ߜ݁Ƹ ൌ ݁൫ܷ

ଷ஽

൅ ‫ݑ‬ො

ଷ஽

(17)

൯ െ ݁൫ܷ ଷ஽ ൯ with U3D the displacement in the reference configuration. The

equation is solved to compute the viscous load q. Internal and External Flow Dynamics Knowing the load q, the velocity of the points can be expressed as an integral equation over the deformed capsule surface S using the boundary integral method: ଵ

ூ

‫߮׊‬଴ଷ஽ ‫߱ א‬ଷ஽ ǡ ‫ ݒ‬ቀ߮଴ଷ஽ ቁ ൌ ‫ ݒ‬ஶ ቀ߮଴ଷ஽ ቁ െ ଼గఓ ‫׬‬ௌ ቀԡ௥ԡ ൅ where ‫ݒ‬

ஶ

௥ٔ௥ ቁǤ ‫ ݍ‬ቀ߮ᇱଷ஽ ቁ ݀ܵ ቀ߮ᇱଷ஽ ቁǡ ԡ௥ԡయ ଷ஽

is the undisturbed flow velocity and I is the identity vector. The vector ‫ ݎ‬ൌ ߮

െ߮

(18) ᇱଷ஽

is the

distance vector between the point ߮ଷ஽ , where the velocity vector is calculated, and the point ߮ ᇱଷ஽ that describes the midsurface S in the integral. The displacement is related to the velocity v of the wall through the kinematic condition: ‫ݒ‬ሺߦଵ ǡ ߦ ଶ ǡ ߦ ଷ ሻ ൌ ‫ݑ‬ǡ௧ଷ஽ ሺߦଵ ǡ ߦ ଶ ǡ ߦ ଷ ሻ (19) where Ǥǡ௧ is the time derivative. An explicit Euler method is then used to integrate the velocity over time and obtain the new position of the wall points.

472

Discretization The capsule wall is discretized using linear triangular shell finite elements. We use the mixed interpolation of the tensorial component approach, which can handle the modeling of objects with wall thicknesses much smaller than their characteristic size, a situation that is prone to locking phenomena [9,10,11]. The MITC approach is based on a mixed formulation that interpolates strains and displacements separately and connects both interpolations at specific tying points. In the following, we will show results for MITC3 linear elements consisting of three nodes (one at each vertex) with 5 degrees of freedom by node. The mesh of the spherical capsule is generated by inscribing an icosahedron (regular polyhedron with 20 triangular faces) in a sphere. The elements are subdivided sequentially until the desired number of elements is reached [3]. We denote η the mesh size. Validation Before coupling the finite element method to the fluid solver, we have validated the mechanical behavior of the shell finite elements in large static deformations by simulating the inflation of a capsule under an internal pressure p. During the inflation, the wall is subjected to an isotropic traction characterized by the stretch ratio ߣ ൌ ͳ ൅ ߙ. The bending resistance plays no role in this test case. We have plotted the evolution of the pressure p according to the inflation factor α (Fig.1). The numerical results are in agreement with the analytic results. The error remains small in all cases. It increases with increasing wall thickness h and decreasing number of elements. Among the simulated cases, it is therefore maximum when the capsule is modeled with ‫ ܤ‬ൌ ͲǤͳand 1280 MITC3 elements. Even in this case, it is only equal to 0.49%, which validates the shell finite element method. 5 4 3 pℓ/Gs 2

Analytic B = 0.01 B = 0.05 B = 0.1

1 0

0

0.1

0.2

0.3 0.4 0.5 α Fig.1: Non-dimensional pressure as a function of the inflation factor α for bending numbers ‫ ܤ‬ൌ ͲǤͲͳǡ ͲǤͲͷand ͲǤͳ. The capsule wall is discretized with 5120 MITC3 elements. We have then studied the convergence of the numerical procedure by simulating the motion of an initially spherical capsule in a simple shear flow at ‫ ܽܥ‬ൌ ͲǤ͸. The numerical procedure converges linearly with ߛሶ ο‫ݐ‬ and quadratically with η. In the following, all the results are provided for 5120 MITC3 elements and a time step ߛሶ ο‫ ݐ‬ൌ ͳͲିଷ. The characteristic mesh size is then ߟ ൌ ͷ ൈ ͳͲିଶ. Capsule Dynamics in a Simple Shear Flow We consider the dynamics of an initially spherical capsule in a simple shear flow at ‫ ܽܥ‬ൌ ͲǤ͸. For all the values of the bending number (‫)Ͳ ് ܤ‬, the capsule is elongated in the straining direction at the steady state, while the vorticity of the flow induces the rotation of the wall around the steady deformed shape. The larger the capillary number, the more elongated the capsule (Fig. 2). This motion, called tank-treading, is exactly the same as that observed when the wall of the capsule is modeled with a membrane model (‫ ܤ‬ൌ Ͳ) [2,3]. The deformed shape can be approximated by its ellipsoid of inertia. We define L1 and L2 the lengths of the two principal axes of the ellipsoid of inertia in the shear plane. The deformation of the capsule in the shear plane can measured by the Taylor parameter D12:

Advances in Boundary Element Techniques XIV ‫ܦ‬ଵଶ ൌ

473

௅భ ି௅మ ௅భ ା௅మ

(23)

To determine the influence of the bending resistance on the average shape in the shear plane, we compare the ∞ Taylor parameter at steady state ‫ܦ‬ଵଶ for several capillary numbers and two values of bending number(Fig. 3). For a given capillary number, the capsule has the same average shape in the shear plane as the one predicted when the capsule wall is modeled as a two-dimensional membrane (without bending resistance). ௤Ǥ௡

e2

ீೞ

Ͳ

e1 െ

(b)

(a)

௤Ǥ௡ ீೞ

Fig.2: Steady deformed shape for a capsule with ‫ ܤ‬ൌ ͲǤͲͲͷ at (a) ‫ ܽܥ‬ൌ ͲǤͳ and (b) ‫ ܽܥ‬ൌ ͲǤ͸ǤThe color scale corresponds to the normal load ‫ݍ‬Ǥ݊ , where the maximum value is equal to ‫ݍ‬Ǥ ݊Ȁ‫ܩ‬௦ ൌ ͲǤͷ at ‫ ܽܥ‬ൌ ͲǤͳand 2.5 at ‫ ܽܥ‬ൌ ͲǤ͸. When the capsule wall is modeled with a membrane model (‫ ܤ‬ൌ Ͳሻ, wrinkles appear in the central region of the capsule for capillary numbers below a critical value CaL (Fig. 4a). They result from the presence of compressive tensions in the equatorial area and are in the straining direction. They persist at steady state. Wrinkles are observed at exactly the same location with the shell model (Fig. 4b,c) for low values of ‫ܤ‬. When the bending number is increased, the wrinkle wavelength decreases: it is due to the increase in bending stiffness (Fig. 3b,c). For ‫ ܤ‬ൌ ͲǤͲͷ, the wrinkles no longer form (Fig. 4d). There is therefore a critical bending number, above which the capsule wall is too stiff for buckling to occur. For capillary numbers above CaL, the capsule is more elongated by the flow (Fig. 2b): the tensions at the equator become positive and the wrinkles disappear. 0.5 0.4 0.3 D12∞ 0.2 B=0

0.1

B = 0.005 B = 0.05

0 0

0.2

0.4

Ca

0.6

0.8

1

ஶ as a function of Ca for bending numbers ‫ ܤ‬ൌ Ͳǡ ͲǤͲͲͷ and ͲǤͲͷ , for an initially Fig. 3: Values of ‫ܦ‬ଵଶ spherical capsule subjected to a simple shear flow.

474

(a)

(b)

(c)

(d)

Fig. 4: Steady-state profiles of capsules subjected to a simple shear flow (‫ ܽܥ‬ൌ ͲǤͳ): bending number ‫ ܤ‬ൌ Ͳ (membrane model) (a), ‫ ܤ‬ൌ ͲǤͲͲͷ (b), ‫ ܤ‬ൌ ͲǤͲͳ(c) and ‫ ܤ‬ൌ ͲǤͲͷ (d). The color scale is the same as in as Fig.2. The maximum value is equal to ‫ݍ‬Ǥ ݊Ȁ‫ܩ‬௦ ൌ ͲǤͷ.

Conclusion We have simulated the motion of a capsule in a simple shear flow using a boundary integral - MITC shell finite element coupling strategy. The numerical method is stable and free of locking phenomenon. We have shown that the motion and deformation of a thin membrane capsule is marginally influenced by the bending stiffness. The latter controls the amplitude and wavelength of the wrinkles that appear at low capillary number in the straining direction. However, the average deformed shape that the capsule assumes, as it tanktreads, remains identical to that predicted by a two-dimensional membrane model. References [1] W. R. Dodson, P. Dimitrakopoulos Spindles, cups, and bifurcation for capsules in Stokes flow. Phys. Rev. Lett., 101 (20), 208102 (2008). [2] É. Lac, D. Barthès-Biesel, N. A. Pelekasis, J. Tsamopoulos Spherical capsules in three-dimensional unbounded Stokes flow : effect of the membrane constitutive law and onset of buckling, J. Fluid Mech., 516, 303–334 (2004). [3] J. Walter, A.-V. Salsac, D. Barthès-Biesel, P. Le Tallec Coupling of finite element and boundary integral methods for a capsule in a Stokes flow, Int. J. Num. Meth. Engng, 83, 829–850 (2010). [4] C. Pozrikidis Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow, J. Fluid Mech., 297, 123–152 (1995). [5] Y. Sui, H. T. Low, Y. T. Chew, P. Roy Tank-treading, swinging, and tumbling of liquid-filled elastic capsules in shear flow. Phys. Rev. E, 77 (1), 016310 (2008). [6] S. Kessler, R. Finken, U. Seifert Swinging and tumbling of elastic capsules in shear flow. J. Fluid Mech., 605, 207–226 (2008). [7] X.-Q. Hu, A.-V. Salsac, D. Barthès-Biesel Fow of a spherical capsule in a pore with circular or square cross-section, J. Fluid Mech., 705, 176–194 (2012). [8] C. Dupont, A.-V. Salsac, D. Barthès-Biesel Off plane motion of a prolate capsule in shear flow, J. Fluid Mech., 721, 180-198 (2013). [9] D. Chapelle, K. J. Bathe The Finite Element Analysis of Shells – Fundamentals, Computation Fluid and Solid Mechanics. Springer (2003). [10] I. Paris Suarez Robustesse des éléments finis triangulaires de coque, phD Thesis, Université Pierre et Marie Curie (Paris VI) (2006). [11] P.-S. Lee, K.-J. Bathe Development of MITC isotropic triangular shell finite elements, Comp. Struct, 82, 945–962 (2004).


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⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝

0 0 0 0 e31 0 0 h31 C11 C12 C13 0 C12 C22 C23 0 0 0 0 0 e31 0 0 h31 C13 C23 C33 0 0 0 0 0 e33 0 0 h33 0 0 0 C44 0 0 0 e15 0 0 h15 0 0 0 0 0 C44 0 e15 0 0 h15 0 0 0 0 0 0 0 0 0 0 0 0 0 C66 0 0 0 0 e15 0 −11 0 0 −β11 0 0 0 0 0 −11 0 0 −β11 0 0 0 0 e15 0 0 0 0 0 −33 0 0 −β33 e31 e31 e33 0 0 0 0 h15 0 −β11 0 0 −γ11 0 0 0 0 0 −β11 0 0 −γ11 0 0 0 0 h15 0 0 0 0 0 −β33 0 0 −γ33 h31 h31 h33

⎞⎛ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

ε1 ε2 ε3 2ε23 2ε13 2ε12 −E1 −E2 −E3 −H1 −H2 −H3

σij

Cij

Di

eij hij

βij

Bi

ij γij εij Ei

Hi

" #$ ! % &

!

C66 = 0.5(C11 − C12 )

CiJKl uK,il + ρω 2 δJK uK + bJ = 0

CiJKl () * % + ! ω %! δJK , * uK ⎧ ⎨ uk , ,- ' . uK = ϕ, ,-( ⎩ φ, ,-/,

%

uk

'

ρ

%

bJ

.

ϕ φ

" % . % / 0%

!

'

∂2Λ ∂b2 ∂b1 + − ∂x1 ∂x2 ∂x23 ∂b3 ∂2Γ ∂2Δ ∂2Ψ ∂2Φ −ω 2 Γ = a5 ∇2 Γ + a2 2 + a3 2 + d2 ∇2 Ψ + d3 2 + m2 ∇2 Φ + m3 2 + ∂x3 ∂x3 ∂x3 ∂x3 ∂x3 ∂b1 ∂b2 ∂2Δ −ω 2 Δ = a3 ∇2 Γ + a1 ∇2 Δ + a5 2 + d1 ∇2 Ψ + m1 ∇2 Φ + + ∂x1 ∂x2 ∂x3 −ω 2 Λ = a4 ∇2 Λ + a5

∂2Γ

∂2Δ

∂2Ψ

∂2Φ

∂b4 + d1 2 + d4 ∇2 Ψ + d5 2 + z4 ∇2 Φ + z5 2 + ∂x3 ∂x23 ∂x3 ∂x3 ∂x3 ∂b5 ∂2Γ ∂2Δ ∂2Ψ ∂2Φ 0 = m2 ∇2 Γ + m3 2 + m1 2 + z4 ∇2 Ψ + z5 2 + m4 ∇2 Φ + m5 2 + ∂x3 ∂x3 ∂x3 ∂x3 ∂x3 0 = d2 ∇2 Γ + d3

∇2 =

∂2 ∂x21

+

Λ=

( / 1 2 3

∂2 + ∂x22

∂u2 ∂u1 − , ∂x1 ∂x2

Δ=

∂u1 ∂u2 + , ∂x1 ∂x2

Γ=

∂u3 , ∂x3

Ψ=

∂ϕ , ∂x3

Φ=

∂φ ∂x3

4

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠


477

C11 C33 C13 + C44 C11 − C12 C44 , a2 = , a3 = , a4 = , a5 = a1 = ρ ρ ρ 2ρ ρ e15 + e31 e15 e33 −11 −33 , d2 = , d3 = , d4 = , d5 = d1 = ρ ρ ρ ρ ρ h15 + h31 h15 h33 −γ11 −γ33 m1 = , m2 = , m3 = , m4 = , m5 = ρ ρ ρ ρ ρ −β11 −β33 , z5 = z4 = ρ ρ ! ! " Λ # ∞ Λ= Λ exp(iα.x) dα $% −∞

# " " α = (α1 , α2 , α3 ) # " " & !

! " Δ Γ Ψ Φ ' u1 u2 u3 ϕ φ

α1 Δ − α2 Λ α1 Δ + α2 Λ Γ Ψ Φ , u2 = , u3 = , ϕ= , φ= , $$ iα3 iα3 iα3 i(α21 + α22 ) i(α21 + α22 ) (

)* # !

! +# ),* &-. / 0. )1* 23 4 ! uK " x !

⎞ ⎛ NH NG 4π 2 ⎝ CHr fKH (α(r) ) CGr fKG (α(p) ) (r) (p) % % exp(iα .x) + exp(iα .x)⎠ uK ≈ $5 |x| |KGp ||∇G(p) | |KHr ||∇H (r) | r=1 p=1 u1 =

# fHr 6 NH H = 0 #

x α(r) .>06 ∇H (r) H = 0 α(r) 6 KHr 2

" H = 0 KHr = κH1 κH2 κH1 κH2 " H = 06 CHr = exp( 14 πi(sgn κH1 + sgn κH2 )) " fGr NG ∇G(r) KGr CGr G = 0 4 G = 0

G(α1 , α2 , α3 ) = a4 (α21 + α22 ) + a5 α23 − ω 2 = 0

$1

# #

" 7

#

" H = 0 # 8 8 "

#

9

#

⎡ ⎢ ⎢ ⎣

⎤ a5 (α21 + α22 ) + a2 α23 − ω 2 a3 α23 d2 (α21 + α22 ) + d3 α23 m2 (α21 + α22 ) + m3 α23 2 2 2 2 2 2 2 2 2 2 ⎥ m1 (α1 + α2 ) d1 (α1 + α2 ) a3 (α1 + α2 ) a1 (α1 + α2 ) + a5 α3 − ω ⎥ ... d2 (α21 + α22 ) + d3 α23 d1 α23 d4 (α21 + α22 ) + d5 α23 z4 (α21 + α22 ) + z5 α23 ⎦ 2 2 2 2 2 2 2 2 2 2 m2 (α1 + α2 ) + m3 α3 m1 α3 z4 (α1 + α2 ) + z5 α3 m4 (α1 + α2 ) + m5 α3 ⎛ ⎞ ∂b3 ⎛ ⎞ Γ ⎜ ∂b1 ∂x3 ∂b2 ⎟ ⎜ Δ ⎟ ⎜ ∂x + ∂x ⎟ 2 ⎟ ≡ Ax = b ⎟ ⎜ 1 ... ⎜ ⎟ ⎝ Ψ ⎠=⎜ ∂b4 ⎝ ⎠ ∂x3 Φ ∂b5 ∂x3

$

478

x3 ≡ x3 x1 x x1 − x2

α(r) 2 = 0 x2 ! H(α1 , α2 , α3 ) = det(A) = 0

x3 ≡ x3

x x2

θ x2 φ

x1

x1

" #

# $ %&' ()! *

+! %,' G = 0 α(p) - H

- . - . H− H+ */ %' - . α(p) H+ !

α(r) # 0 %&' α(r) 0

- H = 0 0

α1 = R(θ)sin(θ), α3 = R(θ)cos(θ) 1 .1/2 % B(θ) ± B(θ)2 − 4A(θ)C(θ) 4 2 2 4 H = A(θ)R(θ) − ω B(θ)R(θ) + C(θ)ω = 0 ⇒ R± (θ) = ω 2 2A(θ)

R± (θ) 0 H± A(θ) B(θ) C(θ) 3- 1 θ α

. - R(θ)(cosθex1 − sinθex3 ) +

dR(θ) (sinθex1 + cosθex3 ) = 0 dθ

4

ex1 ex3 x

0

" C11 = C33 = 283 5$ C12 = C13 = 121 5$ C44 = 80.8 5$ α(r)


479

CL =

%

C33 /ρ

r

x3

θ

x3

! −3

1.5

−3

Real part

x 10

1.5

Imaginary part

x 10

Analytical solution Far field solution 1 1

0.5

44 33

C U

C44U33

0.5

0

0 −0.5

−0.5 −1

−1.5 10

15

20

25 ω r/C

30

35

−1 10

40

15

20

L

−3

35

40

"

U33

θ = 30

−3

Real part

x 10

30

L

3

25 ω r/C

3

Imaginary part

x 10

2

1

1

44 33

2

C U

C44U33

Analytical solution Far field solution

0

0

−1

−1

−2

−2

−3 10

15

20

25 ω r/C

30

35

40

−3 10

15

20

L

25 ω r/C

30

35

40

L

"

U33

θ = 60

# $ % &''

(

# )*+ ,-+ . ,

480

!"# !$ % &' % (#) % % *+, -% &' .% . / 0, 1% 20, 3% 4 / 2 0, % 5 )% 4 6 ) ## % , $7$$8-, -% &9' 1% 20 :% / )0% 3 ;# ) 6 *!< =# # % , 9 >?7989, % &$' *!

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