Advances in Boundary Element Techniques IX

ISBN 978-0-9547783-5-4 Publish by EC Ltd, United Kingdom

Advances in Boundary Element Techniques IX

This book is a collection of the edited papers presented at the 9th International Conference on Boundary Element Techniques (BeTeq 2008), held in Seville, Spain, during 911 July 2008

ECltd


Edited by R Abascal and M H Aliabadi

ECltd

Advances In Boundary Element Techniques IX

Advances In Boundary Element Techniques IX

Edited by R Abascal M H Aliabadi

EC

ltd

Published by EC, Ltd., UK Copyright © 2008, Published by EC Ltd, P. O. Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

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

ISBN: 978-0-954778354

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

International Conference on Boundary Element Techniques IX 9-11 July 2008, Seville, Spain Honorary Chairmen: Professor Enrique Alarcón Professor José Domínguez Escuela Técnica Superior de Ingenieros Industriales Escuela Técnica Superior de Ingenieros Universidad Politécnica de Madrid Universidad de Sevilla Organising Committee: Professor Ramón Abascal Escuela Técnica Superior de Ingenieros Universidad de Sevilla

Professor Ferri M.H. Aliabadi Department of Aeronautics Imperial College London

Local Organising Committee: Ramón Abascal Escuela Técnica Superior de Ingenieros Grupo de Estructuras Universidad de Sevilla M. Pilar Ariza Escuela Técnica Superior de Ingenieros Grupo de Estructuras Universidad de Sevilla J. Manuel Galán Escuela Técnica Superior de Ingenieros Dpto. Ingeniería del Diseño Universidad de Sevilla

Andrés Sáez Escuela Técnica Superior de Ingenieros Grupo de Estructuras Universidad de Sevilla J. Ángel González Escuela Técnica Superior de Ingenieros Dpto. Ingeniería del Diseño Universidad de Sevilla Luís Rodríguez-Tembleque Escuela Técnica Superior de Ingenieros Grupo de Estructuras Universidad de Sevilla

International Scientific Advisory Committee: Abe, K. (Japan) Baker, G. (USA) Blázquez, A. (Spain) Carpentieri, B. (Austria) Cisilino, A. (Argentina) Dabnichki, P. (UK) Davies, A. (UK) Denda, M. (USA) Sapountzakis, E. J. (Greece) Fedelinski, P. (Poland) Frangi, A. (Italy) Gatmiri, B. (France) Gallego, R. (Spain) Gospodinov, G. (Bulgaria) Gumerov, N. (USA) Hirose, S. (Japan) Kinnas, S. (USA) Mallardo, V. (Italy) Mansur, W. J. (Brazil) Mantic, V. (Spain) Marin, L. (UK) Matsumoto, T. (Japan) Mattheij, R. M. M (The Netherlands) Mesquita, E. (Brazil)

Millazo, A. (Italy) Minutolo, V. (Italy) Mohamad Ibrahim, M. N. (Malaysia) Ochiai, Y. (Japan) Pérez Gavilán, J. J. (Mexico) Prochazka, P. (Czech Republic) Polyzos, D. (Greece) Sáez, A. (Spain) Seok Soon Lee (Korea) Sellier, A. (France) Salvadori, A. (Italy) Schneider, S (France) Sladek, J (Slovakia) Sollero, P. (Brazil) Song, C (Australia) Taigbenu, A (South Africa) Tan, C. L. (Canada) Tanaka, M. (Japan) Telles, J. C. F. (Brazil) Venturini, W. S. (Brazil) Wen, P. H. (UK) Wrobel, L. C. (UK) Yao, Z. (China) Zhang, Ch. (Germany)

PREFACE The Conferences in Boundary Element Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques in both boundary element methods (BEM) and boundary integral equation techniques (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), and Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the Escuela Técnica Superior de Ingenieros of the Universidad de Sevilla, Spain, during 9-11th July 2008. Research papers received from 18 countries 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 Keynote Lectures were given by P. H. Wen, R. Gallego, O. Maeso, A. Sáez and V. Mantic. The organizers are indebted to the University of Seville and to the Escuela Técnica Superior de Ingenieros for their support of the meeting. The organizers would also like to express their appreciation to the International Scientific Advisory Board for their assistance in supporting and promoting the objectives of the meeting and for their assistance in the form of reviews of the submitted papers.

Editors July 2008

Advances in Boundary Element Techniques

Contents Characterization of CNT-reinforced composites via 3D continuum-mechanics-based BEM F.C. Araújo, L.J. Gray

1

A 2-D hypersingular BEM for transient analysis of cracked magneto electro elastic solids R. Rojas-Díaz, F. García-Sánchez, A Sáez, Ch. Zhang

7

A boundary element analysis of symmetric laminate composite shallow shells E.L. Albuquerque, M.H. Aliabadi

15

A Simple Numerical Procedure to Improve the Accuracy of the Boundary Element Method C.F. Loeffler, L.C. Wrobel

21

A boundary element formulation for 3D wear simulation in rolling-contact problems L. Rodríguez-Tembleque, R. Abascal, M.H. Aliabadi

27

A CQM based symmetric Galerkin boundary element formulation for semi-infinite domains L. Kielhorn, M. Schanz

33

An efficient method for boundary element analysis of thermoelastic problems without domain discretization R. Fazeli and M. R. Hematiyan

41

Analysis of piezoelectric active patches performances by boundary element techniques A. Alaimo, A.Milazzo, C. Orlando

49

Analytical integrations in 3D BEM elastodynamics A. Temponi, A. Salvadori, E. Bosco,E, A. Carini, J.Alsalaet

55

BEM analysis of an interface crack with friction E. Graciani, V. Mantiþ, F. París

61

BEM-FEM coupling in antiplane time-harmonic elastodynamics using localized Lagrange multipliers J.M. Galán and R. Abascal

67

BEM-FEM coupling in contact problems J. A. González, K. C. Park, C.A. Felippa, R. Abascal

73

BEM-FEM coupling model for the analysis of soil-pile-structure interaction in the frequency domain L. A. Padrón, J.J. Aznárez y Orlando Maeso

79

EM-FEM coupling with non-conforming interfaces for static and dynamic problems T. Ruberg, M. Schanz, G. Beer

85

Cartesian transformation method for evaluation of regular and singular domain integrals in boundary element and other mesh reduction methods M. R. Hematiyan, M. Mohammadi, R. Fazeli, A. Khosravifard

93

Design of noise barriers with boundary elements and genetic algorithms O. Maeso , D. Greiner , J.J. Aznárez , G. Winter

101

Determination of the elastic response of materials with defects in three dimensional domains using BEM and topological Derivative I.H. Faris, R. Gallego

107

Dual boundary elements for analysis of non-shear reinforced concrete beams S. Parvanova, G. Gospodinov

113

Evaluation of Green’s functions for 3D anisotropic elastic solids Y.C. Shiah, C.L. Tan, V.G. Lee, Y.H. Chen

119

Experimental measurement error in the inverse identification problem in a viscoelastic layer A.E.Martínez-Castro, R.Gallego

125

Computation of stresses on the boundary of laminate composites plates by the boundary element method A.R.Gouvea, E.L.Albuquerque, L.Palermo Jr., P.Sollero

131

General 3-D dynamic fracture mechanics problems in transversely isotropic solids P. Ruiz, M.P. Ariza, J. Domínguez

137

Identification of blood perfusion parameters using an inverse DRBEM/genetic algorithm P.W. Partridge, L.C. Wrobel

143

Improved solution algorithm for the BEM analysis of multiple semi permeable piezoelectric cracks in 2-D M. Denda

149

BEM model for the free-molecule flow in MEMS A. Frangi

155

Laplace domain two dimensional fundamental solutions to dynamic unsaturated poroelasticity I. Ashayeri, M. Kamalian, M. Kazem Jafari

163

Meshless convection-diffusion analysis by triple-reciprocity boundary element method Y. Ochiai and S. Takeda

171

Modeling of inhomogeneities and reinforcements in elasto-plastic problems with the BEM K. Riederer, C. Duenser and G. Beer

179

Modelling of changing geometries for the excavation process in tunnelling with the BEM C. Duenser, G. Beer

185

Multi-grain orthotropic material analysis by BEM and its application Dong-Eun Kim, Sang-Hun Lee, Il-Jung Jeong and Seok-Soon Lee

191

Potential problem for functionally graded materials: two-dimensional study Z. Sharif Khodaei, J. Zeman

201

Scalar wave equation by the domain boundary element method with non-homogeneous initial Conditions J. A. M. Carrer, W. J. Mansur

209

Shear deformation effect in plates stiffened by parallel beams by BEM E.J. Sapountzakis and V.G. Mokos

215

Slow viscous migration of a solid particle in a rigid and motionless cavity: boundary-element techniques against a new and boundary integral-free approach A. Sellier

223

High-Order spectral elements for the integral equations of time-harmonic Maxwell problems E. Demaldent, D. Levadoux, G. Cohen

229

Advances in Boundary Element Techniques

A BE-topology optimization method enhanced by topological derivatives K.Abe, T.Fujiu, K.Koro

235

Simulation of transient flows in highly heterogeneous media with the flux Green element method A. E. Taigbenu

241

A procedure for 2-D elastostatic analysis of functionally graded materials by the boundary element-free method L. S. Miers, J. C. F. Telles

247

A comparative study of two time-domain BEM for 2D dynamic crack problems in anisotropic solids M. Wünsche, Ch. Zhang, F. García-Sánchez, A. Sáez

255

Transient thermal bending problem by local integral equation method J. Sladek, V. Sladek and P.H. Wen

263

BEM for thin vortex layers G. Baker, N. Golubeva

269

A boundary element method for simultaneous Poisson’s equations T. Matsumoto, T. Takahashi, T. Naito, S. Taniguchi

275

Scaled boundary finite-element analysis of dynamic stress intensity factors and T-stress C. Song

281

The BEM applied to an inverse procedure for the determination of the acoustic pressure distribution of a radiating body J.A. Menoni, E.Mesquita, E.R. Carvalho

289

Error controlled analysis of non-linear problems using the boundary element method K.Thoni, G. Beer

295

On the use of boundary element methods for inverse problems of damage detection in structures A.B. Jorge, P.S. Lopes and M.E. Lopes

301

Algebraic preconditioning techniques for large-scale boundary integral equations in electromagnetism: a short survey B. Carpentieri

309

Solution of nonlinear reaction-diffusion equation by using dual reciprocity boundary element method with finite difference or least squares method G.Meral, M.Tezer-Sezgin

317

A Galerkin boundary element method with the Laplace transform for a heat conduction interface problem R.Vodicka Assembled plate structures by the boundary element method D. D. Monnerat, J. A. F. Santiago, J. C. F. Telles

331

Development of a time-domain fast multipole BEM based on the operational quadrature method in 2-D elastodynamics T. Saito, S. Hirose, T.Fukui

339

Characteristic matrix in the bending plate analysis by SBEM T. Panzeca, F. Cucco, M. Salerno

347

323

Computational aspects in thermoelasticity by the symmetric boundary element method T. Panzeca, S. Terravecchia, L. Zito

355

Natural convection flow of micropolar fluids in a square cavity by DRBEM S.Gümgüm, M.Tezer-Sezgin

363

Two-dimensional Thermo-Poro-mechanic fundamental solution for unsaturated soils P.Maghoul, B. Gatmiri, D.Duhamel

371

A three-step MDBEM for nonhomogeneous elastic solids X.W. Gao, J. Hong Ch. Zhang

381

DBEM for fracture analysis of stiffened curved panels (Plates and Shallow Shells Assemblies) P.M.Baiz, M.H.Aliabadi

389

An incremental technique to evaluate the stress intensity factors by the element-free method P.H. Wen, M.H. Aliabadi

395

Stress analysis of composite laminated plates by the boundary element method F.L. Torsani, A.R. Gouveia, E.L. Albuquerque, P. Sollero

401

Boundary element formulation for dynamic analysis of cracked sheets repaired with anisotropic patches M. Mauler, P. Sollero, E.L. Alburquerque Homogenization of nonlinear composites using Hashin-Shtrikman principles and BEM P. Prochazka, Z. Sharif Khodaei

407

413

Transient dynamic analysis of interface cracks in 2-D anisotropic elastic solids by a time-domain BEM S. Beyer, Ch. Zhang, S. Hirose, J. Sladek, V. Sladek

419

Modelling of topographic irregularities for seismic site response F. J. Cara, B. Benito, I. Del Rey, E. Alarcón

427

A fast 3D BEM for anisotropic elasticity based on hierarchical matrices I. Benedetti, A. Milazzo, M.H. Aliabadi

433

A BEM approach in nonlinear acoustics V. Mallardo, M.H. Aliabadi

439

Time convoluted dynamic kernels for 3D saturated porelastic media with incompressible constituents M. Jiryaei Sharahi, M. Kamalian, M.K. Jafari

445

Stability analysis of composite plates by the boundary element method E.L.Alburquerque, P.M.Baiz, M.H. Aliabadi

455

BEM model of mode I crack propagation along a weak interface applied to the interlaminar fracture toughness test of composites L. Távara, V. Manti, E. Graciani, J. Cañas, F. París

461


1

Characterization of CNT-reinforced composites via 3D continuum-mechanics-based BEM F.C. Araújo1,2 and L.J. Gray2 1

Dept Civil Eng, UFOP, Ouro Preto, MG, Brazil; [email protected] 2 Oak Ridge National Laboratory, TN, U.S.A; [email protected]

Keywords: CNT-based composites, 3D standard BE formulations, special quadratures for singular and quasi-singular integrals, subregion-by-subregion technique.

Abstract. A continuum-mechanics-based Boundary Element Method (BEM) has been applied to estimate mechanical properties of carbon-nanotube-reinforced composites. The analyses have been carried out for three-dimensional representative volume elements (RVEs) of the composite. To model the thin-walled carbon nanotubes (CNTs), special integration procedures for calculating nearly-strongly-singular integrals have been designed, and a generic BEM substructuring algorithm allows modeling complex CNT-reinforced polymers, containing any number of nanotubes of any shape (straight or curved). Moreover, this substructuring algorithm, based on Krylov solvers, makes the independent generation, assembly, and storage of the many parts of the complete BEM model possible. Thus, significant memory and CPU-time reductions are achieved by avoiding an explicit global system of equations. Further CPU-time reduction is obtained by employing a matrix-copy option for repeated fiber reinforcement. Long and short fibers arranged according to square-packed and hexagonal-packed patterns can be considered to idealize the composite microstructure. Single-cell and multi-cell RVEs can be employed for evaluating the macroscopic material constants. Introduction In recent years, carbon nanotubes (CNTs) have been extensively employed to design advanced materials with improved physical properties. For example, as reported in [1] and [2], an excellent gain in the mechanical and thermal characteristics of polymeric matrices is achieved by spreading in the samples just a small portion of CNTs (about 1% of their weight). As a matter of fact, nanomaterials have developed an important role in many engineering fields as in electronics, sensors, computing, etc. Particularly in structural engineering, they have been combined with polymeric matrices to manufacture fiber-reinforced light-weight composites, actually a new generation of composites. Available formulations for treating nanosysstems and their validity in the light of the different scales, going from atomistic to continuum models, are discussed in [3]. Considering the dimensions of nanosystems, molecular dynamics (MD) formulations should be employed to describe their physical behavior appropriately. Nevertheless, still constrained by limited computer power (even for present-day computers) to apply MD-based simulations to practical problems, attempts to assure the use of continuum-mechanics (CM) formulations to nanosystems have been made in the computational and experimental level [4-8]. Of course, because of their geometric similarity to hollow cylindrical tubes (single-walled or multi-walled), most of the CM-models for CNTs bases upon shell elements [4-7]. In [9], Chen and Liu apply a CM-based strategy to characterize CNT-reinforced composites. There, a 3D quadratic solid (brick) finite element is employed to model representative volume elements containing a single CNT (single-unit-cell RVEs), and a 2D quadratic 8-node finite element, to model multi-unit-cell RVEs (containing many CNTs). In the present work, single-unit and multi-unit cells are also considered to characterize CNT-reinforced composites, but a 3D standard boundary element technique is applied to solve the RVEs. In general, the strategy applied here comprises a robust subregion-by-subregion (SBS) technique, necessary for modeling heterogeneous materials, and efficient integration procedures, needed to evaluate the singular and nearly-singular integrals coming up in the matrix-coefficient calculations. Details of the SBS technique, based on Krylov solvers, are given in [10]. It accounts for the micromechanical modeling of composites consisting of a large number of fibers spread in a polymeric matrix. Moreover, to efficiently model composites containing geometrically and physically identical substructures, such as a number of

2

Eds: R Abascal and M H Aliabadi

identical fibers, a matrix-copy option is included. Thus, matrices for repeated subregions are immediately obtained simply by copy and rotation transformations. Here, just the diagonal-preconditioned biconjugate gradient solver (J-BiCG) is applied along with the SBS technique. In addition, discontinuous boundary elements [10-11] are employed to make the modeling of complex composites considerably easier. As mentioned above, the other pillar of the strategy is the special integration schemes for coping with nearly-singular integrals resulting from either thin-walled domains or discontinuous boundary elements. In the applications here, for weakly-singular and nearly-weakly-singular integrals, numerical quadratures that combine triangle and polynomial coordinate transformations [10,12] are employed, and for the nearlystrongly-singular integrals, the line-integral approach detailed in Araújo and Gray [13], which uses closed expressions for the strongly-singular line integrals involved, is adopted. For verifying the robustness of the strategy, 3D simulations of CNT-based RVEs derived from squarepacked fiber arrays are considered. Possible future developments are also commented upon. 3D modeling of CNT-based RVEs Provided that continuum mechanics models satisfactorily describe the mechanical behavior of CNTs, a 3D standard boundary element technique is employed to model general CNT-based composites consisting of a number of CNT fibers scattered in a material matrix (see Fig. 1). For that, the subregion-by-subregion (SBS) technique described in [10], essential to cope with heterogeneous materials, is applied. Here, structured matrix-vector products (SMVP) [10] and the special matrix-copy option for repeated substructures are considered.

Figure 1: Fiber-reinforced composite The SBS technique is similar to the element-by-element (EBE) technique, widely applied to solve largeorder engineering problems with finite elements (FEM). Its main idea is to use an iterative solver, which allows working with smaller parts of the system of equations without explicitly assembling the global matrix. Yet, in the BE-SBS technique, contrary to EBE techniques, no data-structure optimization is further needed to reduce CPU time and memory, as coefficients belonging to edges shared by different subregions do not overlap. For n s subregions, after introducing the boundary conditions, the BE global system of equations can be written as i 1

¦ H m 1

n

im u mi

G im p im A ii x i

¦ H

m i 1

im u im

G im p mi B ii y i

, i 1, n s ,

(1)

where H ij and G ij denote the usual BE matrices obtained for source points pertaining to subregion : i and associated respectively with the boundary vectors u ij and p ij at *ij . Note that if i z j , *ij corresponds to the


3

interface between : i and : j ; *ii is the outer boundary of : i . The SBS algorithm bases then on separately storing and manipulating the subsystems in Eq. (1). During the solution by means of some Krylov-type solver, the fiber-matrix interfacial conditions (here supposed to be perfectly bonded) are taken into account. In addition, for the i-th subregion, the following data structure for matrices H i and G i is adopted: block 1

block 3

block 2

(2)

Hi

[ H i1

H i , i 1

A ii

H i ,i 1

H in ]

Gi

[ G i1

G i , i 1

B ii

G i ,i 1

G in ]

.

As shown by Araújo and Gray [14], using structured matrix-vector products (SMVP) increases the solver efficiency. Moreover, using discontinuous boundary elements considerably simplifies the modeling of complex CNT-based composites [10], and additional efficiency is also brought about by a "matrix-copy" option, which accounts for promptly obtaining the coefficient matrix for repeated substructures by copying and rotating a previously assembled one. Thus, matrix-assembly CPU time is considerably reduced for composites containing many physically and geometrically identical reinforcement elements (as usual). Characterizing CNT-based composites The material characterization of composites takes place on the micromechanical level, where the many parts compounding it (polymer and fibers) are directly modeled. In this paper, square-packed arrays are considered to idealize the smearing of fibers inside the matrix material (Fig. 2), and the BE SBS technique discussed above is applied to analyze the corresponding 3D RVEs. The macroscopic material parameters are then calculated solving the RVEs for proper loading cases [15]. 3 CNT-based fiber matrix

2

unit cell

Figure 2: Square-packed array For measuring constants E1 , X12 , and X13 , one considers the specimen under stretching (or shortening) in the 1 principal material direction (fiber direction), and boundary conditions on the lateral surfaces perpendicular to the 2 and 3 directions, so as to simulate the surrounding medium (Fig. 3). The following expressions are employed [13]: E1

§ l1 ¨ G (1) © 1

V 1(1) ¨

· ¸, ¸ ¹

Q 12 Q 13

§ G (1) ¨ 2 ¨ l2 ©

·§ l1 ¸¨ ¸¨ G (1) ¹© 1

· ¸ , ¸ ¹

(3)

4

Eds: R Abascal and M H Aliabadi 1

3

1

G1(1) CNT

traction-free lateral boundary

CNT

G t(1)

l1

3 traction-free lateral boundary

constrained lateral boundary

G 1(1)

constrained lateral boundary

G 3( 2)

G t(1) l3

G 2(2)

CNT

G 2( 2)

CNT

2

2

2

2

l2

l2

l2

(a) 1st analysis (b) 2nd analysis Figure 3: Boundary conditions in strain state 1

l2

G 3( 2)

(a) 1st analysis (b) 2nd analysis Figure 4: Boundary conditions in strain state 2

where V 1(1) denotes the average stress in the 1 direction of the RVE. For evaluating the constants E 2 , X 23 and X 21 , strain state 2 in Fig. 4, in which V 1( 2)

E2

§ G 2( 2) · ¸ ¨ l2 ¸ ¹ ©

V 2( 2) ¨

1

Q 23

,

§ G ( 2) ¨ 3 ¨ l3 ©

0 , is considered. It turns out

·§ l ¸¨ 2 ¸¨ G ( 2) ¹© 2

· ¸ , ¸ ¹

§E · 2 ¸ , Ë ¸ © 1 ¹

Q 21 Q 12 ¨

(4)

where V 2( 2) is the average stress in the 2 direction of the RVE in 2nd analysis [13].

Applications and discussions The BE SBS technique has been applied to evaluate engineering constants for square-packed CNT-based composites. In the numerical tests, a long CNT through single-unit-cell and 2 u 2 -unit-cell RVEs (Fig. 5 and Fig. 6) is considered. For comparison purposes, the same physical constants used by Chen and Liu [9] are adopted here: E CNT 1,000 nN 2 GPa and Q CNT 0.30 for the CNT, and E m 100 nN 2 GPa and nm

Qm

nm

for the polymer matrix. The cylindrical cross section of the CNT fibers has outer radius r0 5.0 nm and inner radius ri 4.6 nm . The RVE in Fig. 5 has dimensions l1 10 nm , and l 2 l 3 20 nm , and in Fig. 6, l1 10 nm , and l 2 l 3 40 nm .Discontinuous boundary elements, when needed, are generated by shifting the nodes interior to the elements a distance of d 0.10 (measured in natural coordinates). In both analyses, an 8-node quadrilateral boundary element is adopted, and the tolerance for the iterative solver (J-BiCG) taken as ] 10 6 . 0.30

3

1

3

2

2

1

Figure 5: BE model for single-unit-cell RVE for long CNT square-packed arrays

Figure 6: BE model for 2 u 2 -unit-cell RVE for long CNT square-packed arrays


5

The BE model in Fig. 5 has two subregions: one for the matrix material, one for the CNT. Both the matrix material and the CNT are modeled with 64 boundary elements, resulting in a total of 1,824 degrees of freedom for the global system. The 2u 2 -unit-cell RVE in Fig. 6 gives rise to a global system with 6,990 equations, the matrix material having been modeled with 224 elements (666 nodes), and each CNT, again with 64 elements (192 nodes). For the CNTs, the BE matrix is only assembled for one of them and then copied for the others. Discontinuous boundary elements are placed at the polymer-CNT interfaces in both models. The engineering constants estimated using these RVEs are shown in Tab. 1. Table 1: Engineering constants (long CNT, square-packed array) single-unit-cell RVE 2u 2 -unit-cell RVE Chen & Liu (3D FE) BE SBS BE SBS E1 /E m 1.3255 1.3227 1.3225 E2 /Em , E3 /Em 0.8492 0.8323 0.8319 Q 12 ,Q 13 0.3000 0.2974 0.2975 Q 23 0.3799 0.3757 0.3597 For the single-unit-cell RVE, good agreement with the values obtained by Chen and Liu [9] using refined 3D FE models is achieved. It should also be noted that the nit n values, where nit is the number of iterations for the solver and n the system order (n=1824), indicate good solver performance for both loading cases; these numbers were 0.28 for strain state 1 and 0.25 for strain state 2. The sparsity of the global matrices is 29% in both cases. For the 2 u 2 -unit-cell RVE, no considerable changes are observed in the material constant values (see Tab. 1), which indicates that, for the determination of elastic constants, single-unit-cell-based RVEs satisfactorily represent the composite material. Moreover, these calculations highlight the computational efficiency of the matrix-copy option in case of repeated substructures ( 2u 2 -cell RVE). The nit n values once again indicate good solver performance; they are 0.21 in the strain state case 1 and 0.19 in the strain state case 2. The global matrix sparsity for this model (with 5 subregions) is 57%. Conclusions A 3D linear elasticity boundary-element formulation based upon a robust subregion-by-subregion (SBS) technique has been developed and proven to be very convenient to analyze representative volume elements of CNT-reinforced composites. First, the efficiency of the line-integral approach (see reference [13] for details) should be stressed: besides allowing modeling thin-walled domains, it also accounts for the reliable use of discontinuous boundary elements, so providing easier modeling of complex coupled domains by means of the SBS algorithm. The matrix-copy option also increases the efficiency of the BE SBS algorithm, avoiding the repeated calculation of matrices for identical substructures. This significantly reduces the total matrix-assembly time for fiber-/particle-reinforced composites, while the J-BiCG iterative solver has shown very good performance in general: for a stopping criterion of ] 10 6 , nit n 0.20 for the larger model (6,990 equations). In addition, more efficient Krylov solvers, such as the BiCGSTAB(l), can be applied. As a final comment regarding efficiency, note that the SBS approach effectively exploits the sparsity of the global system, usually high for many coupled models. The computed effective material constants from the SBS algorithm compared very well with results from refined FE models [9]. Finally, a boundary element formulation is well suited to the study of composites. The determination of the effective elastic constants is dependent upon the surface stress solution, and as tractions are directly obtained from solving the boundary integral equations, this evaluation is straightforward. Moreover, for complex composites, surface meshes are simpler to generate than volume discretizations. By the way, the formulation is also adequate to simulate the matrix-fiber delamination. Nonlinear interfacial constitutive laws can be readily implemented in the code. Of course, a parallel version of the SBS algorithm is straightforward.

6


Acknowledgement This research was sponsored by the Office of Advanced Scientific Computing Research, U.S. Department of Energy under contract DE-AC05-00OR22725 with UT-Battelle, LLC, the Brazilian Research Council, CNPq, and by the U.S. Department of Energy Higher Education Research Experiences (HERE) Program, administered by the Oak Ridge Institute for Science and Education, ORISE. References [1] D.Qian, E.C.Dickey, R.Andrews, and T.Rantell Applied Physics Letter, 76, 2868-2870 (2000). [2] MJ.Biercuk, M.C.Llaguno, M.Radosavljevic, J.K.Hyun, A.T.Johnson, and J.E.Fischer Appl. Phys. Lett., 80 (15), 2767–2769 (2002). [3] N.M.Ghoniem and K.Cho Computer Mod. Eng. Sci., 3,147-174 (2002). [4] X.Q.He, S.Kitipornchai, and K.M.Liew J. Mech. Phys. Solids, 53, 303-326 (2005):. [5] S. Kitipornchai, X.Q.He, and K.M.Liew J. Appl. Phys., 97, 114318 (2005). [6] A.Pantano, D.M.Parks, and M.C.Boyce J. Mech. Phys. Solids, 52, 789-821 (2004). [7] C.M.Wang, Y.Q.Ma, Y.Y.Zhang, and K.K.Ang, Journal of Applied Physics, 99, 114317 (2006). [8]

Y.Chen, B.L.Dorgan Jr, D.N.McIlroy, and D.E.Aston Journal of Applied Physics, 100, 104301 (2006).

[9] X.L.Chen and Y.J.Liu Computational Materials Science, 29, 1–11 (2004). [10] F.C.Araújo, K.I.Silva and J.C.F.,Telles Int. J. Numer. Methods Engrg.,68, 448-472 (2006). [11] F.C.Araújo, K.I.Silva, and J.C.F.Telles Commun. Numer. Meth. Engng., 23, 771–785 (2007). [12] X.L.Chen and Y.J.Liu Eng. Anal. Boundary Elements, 29, 513-523 (2005). [13] F.C.Araújo and L.J.Gray Computer Modeling in Engineering & Science, .812(1), 1-19 (2008). [14] F.C.Araújo and L.J.Gray Computational Mechanics (in press, 2008). [15] M.W.Hyer Stress Analysis of Fiber-Reinforced Composite Materials, McGraw-Hill (1998).


15

A boundary element analysis of symmetric laminate composite shallow shells E. L. Albuquerque1 and M. H. Aliabadi2 1

Faculty of Mechanical Engineering, State University of Campinas Campinas, Brazil, [email protected] Currently at Imperial College London as an academic visitor. 1

Department of Aeronautics, Imperial College London London, UK, [email protected]

Keywords: Laminated composites, shallow shells, radial integration method. Abstract. This paper presents a boundary element formulation for the analysis of symmetric laminate composite shallow shells where only the boundary is discretized. Classical plate bending and plane elasticity formulations are coupled and effects of curvature are treated as body forces. Body forces are written as a sum of approximation functions multiplied by coefficients. Domain integrals which arise in the formulation are transformed into boundary integrals by the radial integration method. The accuracy of the proposed formulation is assessed by comparison with results from literature. Introduction Nowadays, the demand for construction of advanced aerospace, automotive, and marine structures has increased the interest in composite laminated shells. Some requirements, as for example high strength-to-weight ratio, good resistance to corrosion, as well as long fatigue life, cannot be obtained with the use of metallic or any other engineering materials except composites. Other requirements, as aerodynamic profile and good stealth characteristics demand curved structures or shell like structures. Although the large majority of papers about the numerical analysis of composite shells are related to the finite element method, there are few works in literature that present boundary element formulations applied to orthotropic or even anisotropic shells [1, 2, 3]. However, all these works involve complicated fundamental solutions that need to be computed numerically. An alternative approach to the these previous formulations is the coupling of plate bending and plane elasticity formulations, as proposed by Zhang and Atluri [4] who derived a formulation for static and dynamic analysis of isotropic classical shallow shells. Domain integrals were computed by the domain discretization into cells. Dirgantara and Aliabadi [5] extended this approach to the analysis of shear deformable isotropic shallow shells. For the best of authors knowledge, there is no paper in the literature for anisotropic shells using the coupling of plate bending and plane elasticity formulations. In this paper, a boundary element formulation for anisotropic shallow shells with no domain discretization is presented. Classical plate bending and plane elasticity formulations are coupled and effects of curvature are treated as body forces. The domain integrals due to body forces are transformed into boundary integrals using the radial integration method. Numerical results are presented to assess the accuracy of the method. Displacements computed using the proposed formulation are in good agreement with results available in literature. Boundary integral equations Consider a shallow shell of an anisotropic elastic material with the mid-surface being described by z = z(x1 , x2 ). The base-plane of the shell is defined in a domain Ω in the plane x1 , x2 whose boundary is given by Γ. Using the equilibrium equation of shallow shells, the reciprocity relation, and the Green theorem, it is possible to derive integral equations that can be divided in terms of plane elasticity and plate bending formulations as shown by Zhang and Atluri [4] for isotropic shallow shells. These equations

16


are coupled by the domain integrals that arise in both of them. Integral equations for the plane elasticity formulation are given by:

cij uj +

Γ

t∗ik (Q, P )uk (P )dΓ(P ) =

Γ

+

Ω

u∗ik (Q, P )tk (P )dΓ(P ) Cκkj u3 u∗ik,j (Q, P )dΩ,

(1)

where i, j, k = 1, 2; uk is the displacement in directions x1 and x2 , ti = Nij nj , Nij are membrane forces applied in the shell; u3 stands for the displacement in the normal direction of the shell surface; κ depends on the curvature radii Rij of the shallow shell; kij are the inverse of curvature radii. P is the field point; Q is the source point; and asterisks denote fundamental solutions. The anisotropic plane elasticity fundamental solutions can be found, for example, in [6]. The constant cij is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. The integral equation for the plate bending formulation is given by:

Ku3 (Q) +

=

Γ

Nc i=1

Rci (P )u∗3ci (Q, P ) +

+

Γ

+

Γ

+

Ω

Vn∗ (Q, P )w(P ) − m∗n (Q, P ) Ω

N

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

q3 (P )u∗3 (Q, P ) dΩ

Vn (P )u∗3 (Q, P ) − mn (P )

∂u∗3 (Q, P ) dΓ(P ) ∂n

Cκnj ui (P )u∗3 (Q, P ) dΓ(P ) +

Ω

C

κij ∗ u (Q, P )u3 (P ) dΩ ρij 3

[Cκij (P )u∗3 (Q, P )],j ui (P ) dΩ,

(2)

where ∂() ∂n is the derivative in the direction of the outward vector n that is normal to the boundary Γ; mn and Vn are, respectively, the normal bending moment and the Kirchhoff equivalent shear force on the boundary Γ; Rc is the thin-plate reaction of corners; u∗3ci is the transverse displacement of corners; q3 is the domain force in the x3 direction; The constant K is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. As in the previous equation, an asterisk denotes a fundamental solution. Fundamental solutions for anisotropic thin plates can be found, for example, in [7]. As can be seen, domain integrals arise in the formulation owing to the curvature of the shell. In order to transform these integrals into boundary integrals, consider that a body force b is approximated over the domain Ω as a sum of M products between approximation functions fm and unknown coefficients γm , that is: b(P ) =

M m=1

γm fm .

(3)

The approximation function used in this work is: fm1 = 1 + R,

(4)

Equation (3) can be written in a matrix form, considering all source points, as: b = Fγ

(5)


17

Thus, γ can be computed as: γ = F−1 b

(6)

Body forces of integral equations (1) and (2) depend on the displacements. So, using equation (6) and following the procedure presented by Albuquerque et al. [8], domain integrals that come from these body forces can be transformed into boundary integrals. Then, by discretization of these boundary integrals, a matrix equation can be obtained. Finally, after applying boundary conditions, this matrix equation is transformed in a linear system that can be solved to find the unknowns of the shell problem. Numerical results In order to assess the accuracy of the proposed formulation, consider a square spherical shallow shell, as shown in Figure 1. The geometry and material properties of the shell are as follow: length of the base edge of the shell a =0.254 m, thickness h = 0.0127 m, curvature radii R1 = R2 = R = 2.54 m (R12 = R21 = 0), elastic moduli E2 = 6.895 GPa and E1 = 2E1 , Poisson ratio ν12 = 0.3, and shear modulus G12 = E2 /[2(1 − ν12 )]. The shell is under a uniformly distributed load in the transversal direction (internal pressure) q3 = 2.07 MPa (q1 = q2 = 0).

a

x2

a

R x1 R h Figure 1: Square spherical shallow shell. This problem was analysed considering two types of boundary conditions, i.e., clamped and simplysupported. Three meshes were used. Mesh 1 has 12 constant boundary elements and 9 internal points, mesh 2 has 20 constant boundary elements and 25 internal points, and mesh 3 has 28 constant boundary elements and 49 internal points. Mesh 3 is shown in Figure 2. All meshes have elements of equal length and uniformly distributed internal points. Figures 3 and 4 show results for the clamped and simply-supported boundary conditions, respectively, together with meshless results obtained by Sladek et al. [9] for the same problems but considering the shell as shear deformable. As it can be seen, the results for the clamped boundary conditions are in good agreement with the meshless results while for the simply-supported boundary conditions they are slightly lower.

18


Figure 2: Mesh and internal points for the square shallow spherical shell.

0.7

0.6

0.5

0.4

0.3 Sladek et al. (2007) Mesh 1 Mesh 2 Mesh 3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

Figure 3: Transversal displacement for the spherical shell with clamped edge.


19

0.7

0.6

0.5

0.4

0.3 Sladek et al. (2007) Mesh 1 Mesh 2 Mesh 3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

Figure 4: Transversal displacement for the spherical shell with simply-supported edge.

Conclusions This paper presented a boundary element formulation for the analysis of symmetric laminated composite shallow shells where domain integrals are transformed into boundary integrals by the radial integration method. As the radial integration method doesn’t demand particular solutions, it is easier to implement than the dual reciprocity boundary element method. Results obtained with the proposed formulation are in good agreement with results presented in literature. Acknowledgment The first author would like to thank the CNPq (The National Council for Scientific and Technological Development, Brazil), AFOSR (Air Force Office of Scientific Research, USA), and FAPESP (the State of S˜ ao Paulo Research Foundation, Brazil) for financial support for this work.

References [1] W. Jianguo. The fundamental solutions of orthotropic shallow shells. Acta Mechanica, 94:113–121, 1992. [2] P. Lu and O. Mahrenholtz. The fundamental solution for the theory of orthotropic shallow shells involving shear deformation. Int. J. of Solids and Structures, 31:913–923, 1994. [3] J. Wang and K. Schweizerhof. The fundamental solution of moderately thick laminated anisotropic shallow shells. Int. J. Engng. Sci., 33:995–1004, 1995. [4] J. D. Zhang and S. N. Atluri. A boundary/interior element method for quasi-static and transient response analysis of shallow shells. Computers and Structures, 24:213–223, 1986. [5] T. Dirgantara and M. H. Aliabadi. A new boundary element formulation for shear deformable shells analysis. Int. J. for Numerical Methods in Engn., 45:1257–1275, 1999. [6] P. Sollero and M. H. Aliabadi. Fracture mechanics analysis of anisotropic plates by the boundary element method. Int. J. of Fracture, 64:269–284, 1993.

20


[7] E. L. Albuquerque, P. Sollero, W. Venturini, and M. H. Aliabadi. Boundary element analysis of anisotropic kirchhoff plates. International Journal of Solids and Structures, 43:4029–4046, 2006. [8] E. L. Albuquerque, P. Sollero, and W. P. Paiva. The radial integration method applied to dynamic problems of anisotropic plates. Communications in Numerical Methods in Engineering, 23:805–818, 2007. [9] J. Sladek, V. Sladek, J. Krivacek, and M. H. Aliabadi. Local boundary integral equations for orthotropic shallow shells. International Journal of Solids and Structures, 44:2285–2303, 2007.


21

A Simple Numerical Procedure to Improve the Accuracy of the Boundary Element Method Loeffler, C.F. 1 and Wrobel, L.C. 2 1

2

Mechanical Engineering Department, Federal University of Espírito Santo Av Fernando Ferrari, 514, Vitória, ES – CEP 29075-910 – Brazil [email protected]

School of Engineering and Design, Brunel University, Uxbridge UB8 3PH – UK [email protected]

Keywords: Boundary Element Method, recursive calculations, accuracy

Abstract. This paper presents a simple procedure to improve the accuracy of numerical calculations with the Boundary Element Method. In the BEM, values at internal points are usually determined through the application of an integral equation after all the boundary values are calculated. In this work, it is shown that the same idea can be used to improve the accuracy of the boundary results. The aim of this numerical procedure is to correct the boundary values by recalculating them through the use of a boundary integral equation, based on the boundary values calculated previously. The procedure is applied to the solution of the Laplace equation. Numerical results are compared with analytical ones to show the improvements obtained with the proposed technique. Boundary Integral Equation Let u(x) be a scalar potential in a two-dimensional domain : . Considering mathematical fundamentals of the Theory of Integral Equations [1], an integral equation equivalent to the Laplace equation may be written as:

u([)

*

*

³ u ([; X)q(X)d* ³ q ([; X)u(X)d* *

(1)

*

in which q(X) is the normal derivative of u(X) and ī defines the boundary. The two-dimensional fundamental solutions are of the form: u*(ȟ;X)=(-1/2S) ln r([;X)

(2)

q*(ȟ;X)=(-1/2Sr) r,i ni

(3)

where r(ȟ;X) is the Euclidean distance between the source point ȟ and the field point X, and ni is the external unit normal vector at the field point. When the source point is positioned on the boundary, u*(ȟ;X) and q*(ȟ;X) present singularities. The mathematical procedure applied to solve this problem considers the domain ȍ augmented by a circular sector of radius İ centered at ȟ and then takes the limit as İĺ0 [2]. Considering this extended boundary, the integral equation (1) may be rewritten as:

22


u([)

°

*

½° q* ([; X)u(X)d* ³ q* ([; X)u(X)d* ¾ *H ¯**H ¿°

³ u ([; X)q(X)d* lim ®° ³ Ho 0

*

(4)

The third integral in eq (4) must be studied considering an extra potential field u(ȟ), that is:

lim ³ q* ([; X)u(X)d* Ho0

*H

lim{ ³ q* ([; X)[u(X) u([)]d* ³ q* ([; X)u([)d*} Ho 0

*H

(5)

*H

Provided u(X) is a continuous function, a Taylor series expansion of first order may hold in the first term on the right-hand side of eq (5). Taking into consideration that, in a circular sector, rŁİ and d* H HdT , results in: lim{ ³ q* ([; X)[u(X) u([)]}d* Ho0

*H

lim{ ³ Ho0

*H

1 [u,i ([)'x i ] }d* 2S H

1 u,i ([)n i HdT} 0 2 S T

lim{³ Ho0

(6)

If u(x) satisfies the Holder condition, the second integral on the right-hand side of eq (4) exists in the Cauchy Principal Value (CPV) sense [2]. The second term on the right-hand side of eq (5) can be easily integrated, but it is left here in general form because of its dependence on the internal angle between adjacent elements [2]. Thus, the integral equation for source points located on the boundary is given by: u([)[1 lim{ ³ q* ([; X)d*}] c([)u([) Ho0

*

*

³ u ([; X)q(X)d* ³ q ([; X)u(X)d* *

*H

(7)

*

Following standard BEM discretisation and integration procedures [2], eq (7) is collocated at a number N of source points, generating the system of equations: (8)

HU–GQ=0 Recursive Boundary Element Procedure

The solution of the BEM system (8) provides the missing potential or its normal derivative at all N boundary nodes. Regarding internal unknowns, it is possible to determine their values by directly using the integral eq (1) for source points inside the domain. The accuracy for the internal variables is usually better than for the boundary ones. The discrete form of eq (1), for a boundary point and considering smooth boundaries, can be written as: N

(0.5)u([i )

N

¦ Q ³ I u *([ ; X)d* ¦ U ³ I q *([ ; X)d* e k

e 1

i

e k

k

*e

e 1

i

k

(9)

*e

where Ik are the interpolation functions and Q ek and U ek are the nodal values at the boundary. The potential values and their spatial derivatives in the xj direction (or normal and tangential directions) can be also calculated. However, the integral equations for the spatial derivatives are hyper-singular [3-5]. Taking the Cartesian derivatives of eq (1) and using the Leibniz rule gives:


u,i ([) q i ([)

* i

³ q, *

23

([; X)u(X)d* ³ u,*i ([; X)q(X)d*

(10)

*

where: u,*i ([; X)

q*i

q,*i ([; X) p*i

(1/ 2Sr)r,i

(11)

(1/ 2Sr 2 )[2r,i r, j n j rj,i n j ]

(12)

Considering, as previously, an augmented sector of radius İ and applying a constant potential field u(ȟ) in the complete domain for convenience, eq (10) can be rewritten as: q i ( [)

³

** H

lim{ Ho0

³

** H

p*i ([; X)[u(X) u([)]d* ³ p*i ([; X)[u(X) u([)]d* *H

(13)

q*i ([; X)q(X)d* ³ q*i ([; X)q(X)d*} *H

On the augmented boundary * H the expression of pi* (ȟ;X) can be simplified as: p*i ([; X)

(1/ 2SH 2 )n i

(14)

in which rŁİ and r,i = ri/İ . Taking into consideration the last equation and a first-order Taylor expansion, the second integral on the right-hand side of eq (13) results in: lim{ ³ p*i ([; X)[u(X) u([)]d*} lim{ ³ (1/ 2SH 2 )n i [u, j 'x j ]HdT} lim{ ³ (1/ 2SH)n i [u, j n j ]HdT} Ho 0

Ho 0

*H

Ho 0

TH

(15)

TH

Except for the negative sign, the same result can be found by analysing the last integral on the righthand side of eq (13). The complete expression is dependent on the internal angle between neighboring element [3-5], represented here simply as s(ȟ). Considering smooth boundaries, especially when constant boundary elements are used, the result of eq (15) is 0.5qi. For the remaining integrals, the two limits exist when considered together in a CPV sense [5]. Thus, taking into consideration the former procedures, eq (13) can finally be written as: s([)q i ([) CPV{³ p*i ([; X)[u(X) u([)]d* ³ q*i ([; X)q(X)d*} *

(16)

*

Once the boundary values are calculated, eq (16) can be discretised and applied to points ȟi on the boundary: N

N

s([ j )q i ([ j ) CPV{¦ Qek ³ Ik q*i ([ j ; X)d* ¦ U ek ³ Ik p*i ([ j ; X)d*} e 1

*e

e 1

(17)

*e

Numerical Simulations All the following numerical simulations consider a semi-circular domain, with zero normal flux along the curved boundary and a prescribed temperature along the horizontal boundary. The temperature values are discontinuous at r=0, according to Fig.1. The BEM discretisation has 22

24


constant elements. The geometric nodes (extreme points) of the boundary elements along the circular boundary were taken as new points for the application of the recursive BEM (RBI).

Y

T1=1

X T0=0

Fig. 1: Semi-circular domain subjected to a discontinuous temperature field Table 1 presents analytical and numerical results for the temperature at nodal points. Although a small number of boundary elements is used the numerical results display very good accuracy, with the largest errors appearing near the corners.

I (o )

X

Y

NUMERICAL

10 30 50 70 85 95 110 130 150 170

.38794E+01 .34115E+01 .25321E+01 .13473E+01 .34730E+00 -.34730E+00 -13473E+01 -.25321E+01 -34112E+01 -.38791E+01

.68405E+00 .19696E+01 .30176E+01 .37016E+01 .39696E+01 .39696E+01 .37016E+01 .30176E+01 .19696E+01 .68405E+00

.5175E-01 .1644E+00 .2764E+00 .3880E+00 .4719E+00 .5281E+00 .6120E+00 .7236E+00 .8355E+00 .9483E+00

ANALYTICAL .5555E-01 .1667E+00 .2778E+00 .3889E+00 .4722E+00 .5277E+00 .6111E+00 .7222E+00 .8333E+00 .9444E+00

ERROR (%) 6.85 1.38 0.50 0.23 0.06 0.08 0.15 0.19 0.26 0.41

Table 1 - Temperatures at nodal points calculated directly on the circular boundary Table 2 shows normal derivatives on the horizontal boundary. Results have the same value but opposite sign for negative values of the coordinate x. Comparing with the temperature results, it can be seen that there is a decrease in numerical accuracy.

X .350E+01 .250E+01 .175E+01 .125E+01 .750E+00 .250E+00

Y .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00

NUMÉRICAL .9631E-01 .1251E+00 .1819E+00 .2502E+00 .3592E+00 .1557E+01

ANALYTICAL .9095E-01 .1273E+00 .1819E+00 .2546E+00 .4244E+00 .1273E+01

ERROR (%) 5.89 1.73 0.00 1.73 15.36 22.31

Table 2 - Normal fluxes at nodal points calculated directly on the horizontal boundary


25

Table 3 contains temperature results on the circular boundary obtained with the recursive use of the BEM. The comparison of results in this table with results presented in Table 1 shows a similar level of accuracy for both cases. In principle, the RBI does not appear to improve significantly the accuracy of the original temperature results in this case. It must be noted that the points selected for the RBI procedure connect two straight boundary elements, being necessary to determine the correct angle between them to determine s(ȟ).

X . 37588E+01 .30642E+01 .20000E+01 .69460E+00 .00000E+00 -.69460E+00 -.20000E+01 -.30642E+01 -.37582E+01

Y .13681E+01 .25711E+01 .34641E+01 .39392E+01 .40000E+01 .39392E+01 .34641E+01 .25712E+01 .13681E+01

NUMERICAL .1081E+00 .2204E+00 .3323E+00 .4437E+00 .5002E+00 .5545E+00 .6676E+00 .7794E+00 .8922E+00

ANALYTICAL .1111E+00 .2222E+00 .3333E+00 .4444E+00 .5000E+00 .5555E+00 .6667E+00 .7778E+00 .8889E+00

ERROR (%) 2.70 0.81 0.30 0.16 0.00 0.18 0.14 0.20 0.37

Table 3 - Temperatures at nodal points calculated recursively on the circular boundary Table 4 presents the numerical results for the normal flux at points located at the horizontal boundary, calculated recursively by the RBI procedure. In this case, the performance of the RBI procedure was more effective, significantly increasing the accuracy of the results in comparison to Table 2. X .300E+01 .200E+01 .150E+01 .100E+01

Y .000E+00 .000E+00 .000E+00 .000E+00

NUMERICAL .1056E+00 .1587E+00 .2119E+00 .3181E+00

ANALYTICAL .1061E+00 .1591E+00 .2122E+00 .3183E+00

ERROR (%) 0.47 0.25 0.14 0.06

Table 4 - Normal fluxes at nodal points calculated recursively on the horizontal boundary Table 5 shows a further comparison of the results obtained directly with 25 linear boundary elements [6] and recursively with the previous discretisation of 22 constant elements. Again, the RBI procedure significantly improves the accuracy of the normal fluxes, particularly at points located close to the singularity. X

Y

.800E+00 .450E+00 .170E+00 .090E+00

.000E+00 .000E+00 .000E+00 .000E+00

NUMERICAL RECURSIVE .3977E+00 .7072E+00 .1872E+01 .3537E+01

NUMERICAL LINEAR BEM .3689E+00 .6732E+00 .1472E+01 .5975E+01

ANALYTICAL SOLUTION .3979E+00 .7073E+00 .1819E+01 .3537E+01

Table 5 – Comparison of normal fluxes calculated directly with linear boundary elements and recursively using constant elements

26


Conclusions It is known that the use of a boundary integral equation to determine internal values produces numerical results with better accuracy than at boundary points. This behaviour is justified by the error minimization imposed by the discretisation procedure. However, the use of this strategy to recalculate boundary values has not been previously discussed in the BEM literature. Following preliminary tests, the recursive procedure has been shown to improve the solution especially for potential derivatives. In the example presented, the RBI displayed a superior performance to a standard BEM calculation, particularly close to discontinuities and singularities. The computational cost is not high because no new system of equations needs to be solved. The singularities in the flux calculations can be removed analytically or numerically, e.g. by using Kutt quadrature points [7]. References [1] F.B. Hildebrand, Methods of Applied Mathematics, Dover, New York (1992). [2] C.A. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary Element Techniques, Springer-Verlag, Berlin (1984). [3] J.C.F. Telles, A.A. Prado, Hyper-singular Formulation for 2-D Potential Problems, Advanced Formulations in Boundary Element Methods, Vol. 6, Elsevier, London (1993). [4] W.J. Mansur, P. Fleury Jr, J.P.S. Azevedo, A Vector Approach to the Hyper-singular BEM Formulation for Laplace´s Equation in 2D, International Journal of BEM Communication 8, pp 239-250 (1997). [5] L.C. Wrobel and M.H. Aliabadi, The Boundary Element Method, Wiley, Chichester (2002). [6] A.L. Halbritter, J.C.F. Telles, W.J.Mansur, Boundary Element Application to Field Problems, Conference on Analysis, Design and Construction of Structures of Nuclear Central Stations (in Portuguese), Porto Alegre, Brazil, pp 707-724 (1978). [7] H.R. Kutt, On the Numerical Evaluation of Finite Part Integrals Involving an Algebraic Singularity, Special Report WISK 179, National Research Institute for Mathematical Sciences, Pretoria (1975).


27

A Boundary Element Formulation for 3D Wear Simulation in Rolling-Contact problems L. Rodr´ıguez-Tembleque1∗ , R. Abascal2 , M.H. Aliabadi3 1,2

Departamento de Mecnica de los Medios Continuos, Escuela Técnica Superior de Ingenieros, Camino de los descubrimientos s/n, 41092 Sevilla, SPAIN 1 [email protected], 2 [email protected] 3 Department of Aeronautics, Faculty of Engineering, Imperial College, South kensington Campus, London SW7 2AZ, UK [email protected]

Keywords: Boundary Element Method, Rolling-Contact Mechanics, Wear Simulation.

Abstract. The present work shows a new methodology for wear simulation in rolling-contact 3D problems. The formulation is based on the Boundary Element Method (BEM) for computing the elastic influence coefficients, and on projection functions over the augmented Lagrangian of tractions for rolling-contact restrictions fulfilment. The loss of material on the bodies’ surface is modeled using the Archard’s linear wear law. The methodology is applied to simulate a disc-on-disc dry wear test. Results will show the solids profiles evolution in a disc-on-disc rolling-contact problem when the wear is taking place. Introduction Wear phenomenon is presented in the mechanical surface interaction between solids in rolling-contact. From an engineering point of view, wear estimation is a very interesting mechanical topic because allow to predict the life of a mechanical component, to select proper materials, and to have an optimum design for durability. Wear prediction also allow to plan maintenance operations more precisely, which reduces costs as a result of engine or machine immobilization. In spite of the importance of this phenomenon, there are no many works in the literature related with the numerical simulation of wear in rolling contact problems. Among these works should be mentioned Olofsson and Andersson [1], who simulate mild wear in boundary lubricated rollers thrust bearings using Archard’s wear law, and Jendel’s work [2], where computes train wheel profiles wear compared with field measurements. In both cases the wear depth is computed neglecting the elastic slip. Telliskivi et al. [3]-[4] simulates wear in rolling problems using a semi-winkler model. This work presents a new methodology for wear simulation in rolling-contact 3D problems. The formulation is based on the Boundary Element method (BEM) for computing the elastic influence coefficients, and on projection functions over the augmented Lagrangian of tractions for rolling-contact restrictions fulfilment. The material loss of the bodies is modelled using the Archard’s [5] linear wear law which is one of the most used models for engineering applications. Boundary element method equations ¯ using the BEM, could be written respectively, as: The elastic equations for the domains Ω and Ω ¯ ¯ u − G¯ ¯p = b Hu − Gp = b ; H¯

(1)

¯ contains the applied boundary where the vector u (¯ u) represents the nodal displacements, and b (b) conditions. This equations are well known and can be found in books like [6] or [7]. The Equations (1) have to be regrouped to study a rolling contact problem as ¯ ¯ qx ¯ uu ¯ pp ¯q + A ¯c − A ¯c = b Aq xq + Au uc − Ap pc = b ; A

(2)

28


where uc = [uTn uTt ]T (¯ uc ) are the nodal displacements on the potential contact zone (Ac ), pc = pc ) are the nodal contact tractions, xq (¯ xq ) are the unknown nodal displacements or [pTn pTt ]T (¯ ¯ q ) are the columns of H (H) ¯ and G (G) ¯ matrices, depending on the tractions outside Ac , and Aq (A outside Ac boundary conditions. The discrete equilibrium equation (proposed by Signorini) for the contact tractions of each body, could be expressed in the following way: pc (3) pc = −T¯ where T is a Boolean matrix which establishes the relations between the tractions nodes components of each domain. The couple of nodes which are going to be in contact have the same but opposite tractions. The normal separation for each pair in contact, and the tangential difference: (ut − ut ), can be written as follows: uc − uc + kog (4) kc = T¯ In the expressions above, kc = [kTn kTt ]T represents the contact pair separation, and kog = [kTng 0T ]T ; (kng = Xn − Xn ) is the geometric normal contact pair separation. Equations (2)-(4), can be rewritten in the following way: xq + Ru uc + Rp pc + Rk kc = F Rq (5) x ¯q where the matrices Rq , Ru , Rp and Rk are defined as: Aq 0 Au −Ap ; R ; R = = Rq = u p ¯ u TT ¯ pT ¯T ¯q A A 0 A Rk =

0 ¯T ¯ uT A

; F=

b ˜ b

(6)

˜=b ¯+A ¯ u TT kog and b Rolling-Contact equations The rolling 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. • 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

(7)

• Tangential direction: For tangential direction, the fulfilment of friction law and the principle of maximum dissipation is guaranteed by: (pt )I ≤ µ |(pn )I | ; (st )I = −λ(pt )I

; λ ≥ 0 ; (st )I ((pt )I − µ |(pn )I |) = 0

(8)

In the expression above, st stores the tangential slip velocities of every contact pair I, (st )I , which can be written as [8] ˆ (kt ) ≈ (¯ ˆ kt c)I + D c)I + row I D (9) (st )I = (¯ I where

⎡ ˆ =⎣ D

vt,1

∂ ∂ Xt1

+ vt,2 0

∂ ∂ Xt2

⎤

0 vt,1

∂ ∂ Xt1

+ vt,2

⎦ ∂ ∂ Xt2

(10)


29

So, considering all the nodes in the contact surface, the tangential slip velocity expression is c + D kt st = ¯

(11)

being kt and st the tangential slip, and the slip velocities vectors, respectively, and D an algebraic matrix operator, as [8]. Using the projection defined in [8, 9], over the augmented tractions, the rolling contact restrictions (7-8) can be written as (pn )I = Pn [(pn )I + rn (kn )I ] (12) (pt )I = Pt [(pt )I − rt (st )I ] Wear modelling One of the most used model for engineering applications is the linear wear equation, the Archard’s wear law [5]. This law considers that the wear arises from the adhesive forces set up when atoms come into intimate contact. This kind of wear is known as a Delamination Wear and it occurs when the sliding speed remains at low levels, such as the surface heating can be neglected, and the applied load not exceed a limit where seizure takes place. The expression for the Archard’s wear law could be written as: W =κ P d (13) and shows that the volumetric wear of one body is proportional to the product between the applied load and the sliding tangential distance (d). The constant κ is defined as: κ = κ/Ho , where the constants κ and Ho are respectively, the non-dimensional wear coefficient and the hardness of the material in contact. For Archard [5], the constant κ represents the probability that a fragment will be formed at an adhesive joint resulting in a wear particle. In order to simulate numerically the worm on contact surfaces, we can write (13) in a differential equation form, for every contact pair I, as (pn )I (st )I (h˙ w )I = κ

(14)

using the nodal wear depth vector hw , the nodal normal pressure vector pn and the nodal tangential slip vector st . From the solid particle point of view which is travelling with velocity V through the contact zone, the wear depth after the rotation (k+1) can be approximated as )I = (∆h(k+1) w

NI

(k+1) ∆t κ ( p(k+1) )I ( st )I n

(15)

I {I:I ∈ Ac and X(I) is pararell to V direction}

where the super-index (k + 1) express the rotation which is taking place, NI is the number of contact pairs that are located in the same direction (parallel to V ), ∆t = τ /NI , and τ is the solid particle time of residence in Ac travelling with V velocity. The same scheme was used by Jendel in [2], but in this work, we consider the elastic slip velocity. Solution algorithm The quasi-static wear problem presented in equations (5,12,15) is solved as follows. Considering that ¯Tq uTc kTc sTt hTw ] are known for the rotation (k ), their values at all the problem variables, zT = [xTq x (k+1 ) rotation are computed using a Uzawa scheme with index (n): (n)

(I) Apply the rigid body rapprochement increment: ∆ko

30


(II) Initialize all the contact pair tractions and their wear depth: (n)

(k)

(n)

(k)

(n)

(16)

(pc )I = (pc )I , (hw )I = (hw )I , (∆hw )I = 0 (III) Solve the linear equations set:

Rq Ru

⎡ ⎤(n+1) xq ⎢ x 0 ¯q ⎥ (k) (n) ⎥ Rk ⎢ = F + (n) (n) ⎣ uc ⎦ ¯ u TT (∆hw + ∆ko ) + Rp pc A kc (n+1)

(IV) Compute the contact tractions: pc (n+1) for each pair I : increment: ∆hw

(18)

(n+1) (p(n+1) )I = Pn (p(n) )I n n )I + rn (kn

(19)

(n) (n+1) )I = Pt (pt )I − rt (st )I

(20)

)I = (¯ c)I + row

(n+1)

(pt (n+1)

)I =

, and the wear depth

ˆ k(n+1) D t

(n+1)

(st

(∆hw

(n+1)

, the tangential slip velocity: st

(17)

NI I {I:I ∈ Ac and X(I) is pararell to V direction}

I

(k+1) (k+1) ∆t κ ( pn )I ( st )I

(21)

(V) Compute the error function: (n+1)

Ψ(pc (n+1)

(a) If Ψ(pc

(n+1) (n) ) = pc − pc

(22)

) ≤ ε, the solution for the rotation (k+1 ) is reached: z(k+1) = z(n+1) .

In case the applied boundary condition is the external load i -component ((Q(k+1) )i ), before reaching the solution at rotation (k+1 ), the resultant loads applied on the contact zone (n+1) )i dA (Γc ) have to be computed: (Q(n+1) )i = Ac (pc (n)

(a.1) If |(Q(n+1) )i | > |(Q(k+1) )i | + εload , modify ∆ko

and return to (II).

(a.2) Otherwise, the solution for the rotation (k+1 ) is reached: z(k+1) = z(n+1) . (n)

(b) Otherwise, return to (III) doing pc convergence is reached.

(n+1)

= pc

(n)

(n+1)

and ∆hw = ∆hw

, and iterate until

After the solution at rotation (k+1 ) is reached (z(k+1) ), the solution for the next rotation is achieved evaluating: z(k) = z(k+1) and returning to (I).


(a)

31

(b)

(c) Figure 1: Disc-on-disc scheme (a), solids wear profiles evolution(b) and elastic half-space boundary elements mesh (c). Results The proposed methodology was applied to solve a disc-on-disc rolling contact simulation (see Figure 1a), considering wear, in 30000 cicles. The applied force was F = 300N , the discs geometric parameters were: R11 = 32.5mm, R1 = 125mm and R2 = 32.3mm, constant rotational speed: ω = 300rev/min, the Creep = 0.5%, the materials properties: G = 8 · 1010 P a (modulus of rigidity ) and ν = 0.25 (Poisson’s ratio), and the wear parameters: H = 2.5 · 109 P a (hardness) k = 5 · 10−3 (coefficient of wear) kw = k/H = 2 · 10−6 mm2 /N . Figure 1b shows the discs profiles evolution during the wear process, and in Figure 1c we can see the elastic-halfspace boundary elements mesh used. Another wear simulation has been done increasing only the creep value (Creep = 1.5%). The resulting discs profiles after 30000 rotations in both cases (Creep = 0.5% and Creep = 1.5%), are showed in Figure 2a together with the experimental results (Figure 2b) showed by Telliskivi in [3]. The results reveal a good enough agreement with the experimental results. Summary and conclusions This work presents new methodology for wear simulation in rolling-contact 3D problems using the BEM for computing the elastic influence coefficients, and the projection functions for rolling-contact restrictions fulfilment. The material loss of the bodies is modelled using the Archard’s linear wear law. The methodology is applied to simulate a disc-on-disc profile wear. The comparison with the

32


(a)

(b)

Figure 2: Wear depth evolution in two different creep cases: numerical results (left) and experimental results (right). experimental results has a good enough agreement, so the proposed methodology seems to be a powerful numerical tool for wear computing in 3D rolling contact problems. Acknowledgments This work was funded by the Conserjer´ıa de Innovaci´ on Ciencia y Empresa de la Junta de Andaluc´ıa, Spain, research project P05-TEP-00882, and by the Ministerio de Educaci´ on y Ciencia, Spain, research project DPI2006-04598. References [1] U. Olofsson, S. Andersson, and S. Bj¨ orklund. Simulation of mild wear in boundary lubricated spherical roller thrust bearings. Wear, 241(2):180–185, 2008. [2] T. Jendel. Prediction of wheel profile wear - comparisons with field measurements. Wear, 253(12):89–99, 2002. [3] T. Telliskivi. Simulation of wear in a rolling-sliding contact by a semi-winkler model and the archard’s wear law. Wear, 256(7-8):817–831, 2004. [4] T. Telliskivi and U. Olofsson. Wheel-rail wear simulation. Wear, 257(11):1145–1153, 2004. [5] J.F. Archard. Contact and rubbing of flat surfaces. J. Appl. Phys., 24:981–988, 2006. [6] C.A. Brebbia and J. Dom´ınguez. Boundary Elements: An Introductory Course (second edition). Computational Mechanics Publications, 1992. [7] M.H. Aliabadi. The Boundary Element Method Vol 2: Application in Solids and Structures. Wiley, London, 2002. [8] R. Abascal and L. Rodr´ıguez-Tembleque. Steady-state 3d rolling-contact using boundary elements. Comm. Num. Meth. Eng., 23(10):905–920, 2007. [9] R. Abascal and L. Rodr´ıguez-Tembleque. Fast 3d contact mechanics bem algorithm. In Selvadurai A.P.S., Tan C.L., and Aliabadi M.H., editors, Advances in Boundary Elements Techniques VI. EC, Ltd., UK, 2005.


33

A CQM based symmetric Galerkin Boundary Element Formulation for semi-infinite domains Lars Kielhorn1,a , Martin Schanz1,b 1

Institute for Applied Mechanics, Technikerstraße 4/II, 8010 Graz, Austria a

[email protected] , b [email protected] ,

Keywords: Time domain SGBEM, CQM, 3-d elastodynamics, hypersingular kernels, semi-infinite domains

Abstract. The present work deals with the problem of modelling wave propagation phenomena within a 3-d elastodynamic halfspace. While there exist several Boundary Element formulations based on the more common collocation method, the development of Symmetric Galerkin Boundary Element Methods in this field is at the very beginning and rather challenging. On that score the present work should be understood as a first step. Here, the time discretization of the underlying Boundary Integral Equations (BIEs) is done via the Convolution Quadrature Method (CQM) proposed by Lubich. After the time discretization, a variational formulation is established resulting in a Galerkin based method in space. Moreover, to obtain a symmetric Galerkin Boundary Element formulation the 2nd BIE is required. This BIE involves hypersingular kernel functions which must be treated carefully in the numerical implementation. Hence, a regularization based on integration by parts of the elastodynamic fundamental solution is presented which, finally, results in a Boundary Element formulation containing at least only weakly singular kernel functions. In Boundary Element Methods semi-infinite domains are commonly approximated in space by considering just a sufficiently large enough region. Unfortunately, applying this procedure to the symmetric formulation implies the evaluation of additional terms on the truncated surface’s boundary due to the regularization of the involved kernel functions. Therefore, a methodology based on infinite elements intended to overcome this drawback will be presented. The numerical tests done so far show that this approach might be capable of treating semi-infinite domains also within a symmetric Galerkin scheme. Initial boundary value problem for elastodynamics By definition of the Lamé-Navier operator L := −(λ + µ)∇∇ · −µ∇ · ∇

(1)

in terms of the Nabla operator ∇ and Lamé’s constants λ, µ the initial boundary value problem for some linear elastic solid occupying the domain Ω ⊂ R3 with its boundary Γ = ∂Ω reads as (Lu)(˜ x; t) +

∂2 u(˜ x; t) = b(˜ x; t) ∂t2 uΓ (y; t) = gD (y; t) t(y; t) = gN (y; t)

(˜ x; t) ∈ Ω × (0, ∞) (y; t) ∈ ΓD × (0, ∞) (y; t) ∈ ΓN × (0, ∞) .

(2)

˜ ∈ Ω and the time In (2), the unknown displacement field u(˜ x; t) depends on the location x t ∈ (0, ∞). Furthermore, uΓ and t are the boundary displacements and tractions for which the

34


Dirichlet data and Neumann data, gD and gN are prescribed on the boundary parts ΓD and x; t) is a given body force per ΓN , respectively. The body’s mass density is denoted by and b(˜ unit volume. For simplicity this body force is assumed to be absent in the following. Finally, ∂ ˜ ∈ Ω. homogeneous initial conditions are considered, i.e., u(˜ x; 0) = 0 and ∂t u(˜ x; 0) = 0 for all x Boundary integral equations. To obtain a boundary element formulation of the stated problem, first, an appropriate boundary integral representation of the given system (2) is introduced [1] t ˜ ; t − τ ) · (Ty u)(y; τ ) dsy dτ u(˜ x; t) = U(y − x 0 (3) tΓ ˜ ; t − τ )]T · u(y; τ ) dsy dτ ∀ x ˜ ∈ Ω, y ∈ Γ, t ∈ (0, T ) − [(Ty U)(y − x 0

Γ

˜ ; t−τ ). In (3), Ty = T (∂y , n(y)) denotes the stress containing the fundamental solution U(y − x operator based on Hooke’s law (Ty u)(y; t) = t(y; t) = σ(y; t) · n(y)

(4)

where σ(y; t) is the Cauchy stress tensor and n(y) is the outward normal vector. The first ˜ → x ∈ Γ onto the boundary integral equation is obtained by applying a limiting process Ω x representation formula (3). Using operator notation, this boundary integral equation reads for a sufficiently smooth boundary Γ (V ∗ t)(x; t) = (( 12 I + K) ∗ u)(x; t)

∀x ∈ Γ .

(5)

The introduced operators are the single layer operator V, the identity operator I, and the double layer operator K which are defined by t (V ∗ t)(x, t) = U(y − x, t − τ ) · t(y, τ ) dsy dτ (6a) 0 t Γ (I ∗ u)(x, t) = δ(y − x; t − τ )I · u(y; τ ) dsy dτ (6b) Γ 0 t (K ∗ u)(x, t) = lim (Ty U) (y − x, t − τ ) · u(y, τ ) dsy dτ . (6c) ε→0

Γ\Bε (x)

0

In these expressions, Bε (x) denotes a ball of radius ε centered at the point x. Note that the single layer operator (6a) involves a weakly singular integral and that the integration of the double layer operator (6c) has to be understood in the sense of a Cauchy principal value. Moreover, in the definition (6b) I denotes the identity matrix, and δ is the Delta-distribution. To obtain a symmetric formulation, additionally the second boundary integral formula is needed. The application of the traction operator Tx to the dynamic representation formula (3) ˜ → x ∈ Γ yields with a subsequent limit Ω x (D ∗ u)(x, t) = (( 12 I − K ) ∗ t)(x, t)

∀x ∈ Γ .

(7)

The newly introduced operators are the hypersingular operator D and the adjoint double layer operator K t (D ∗ u)(x, t) = − lim Tx (Ty U) )(y − x, t − τ ) · u(y, τ ) dsy dτ (8a) ε→0

0

t

(K ∗ t)(x, t) = lim

ε→0

0

Γ\Bε (x)

Γ\Bε (x)

(Tx U)(y − x, t − τ ) · t(y, τ ) dsy dτ .

(8b)

The application of the hypersingular operator has to be understood in the sense of a finite part.


35

Symmetric formulation. For the solution of the initial boundary value problem (2) the symmetric formulation as proposed in [3, 11] using both boundary integral equations (5) and (7) is considered. While the first integral equation (5) is used only on the Dirichlet part ΓD of the boundary the second one (7) is evaluated on the Neumann part ΓN (V ∗ t)(x; t) − (K ∗ u)(x; t) = ( 12 I ∗ gD )(x; t) (K ∗ t)(x; t) + (D ∗ u)(x; t) = ( 12 I ∗ gN )(x; t)

(x, t) ∈ ΓD × (0, ∞) (x, t) ∈ ΓN × (0, ∞) .

(9)

Further, the Cauchy data u, t are decomposed into ˜ +g ˜D u=u

Ñ . and t = ˜t + g

(10)

Ñ , of the given Dirichlet and ˜D and g In these decompositions, arbitrary but fixed extensions, g Neumann data, gD and gN , are introduced such that ˜D (x; t) = gD (x; t) g ˜ gN (x; t) = gN (x; t)

(x, t) ∈ ΓD × (0, ∞) (x, t) ∈ ΓN × (0, ∞)

(11)

˜D of the given Dirichlet datum has to be continuous due to holds. Note that the extension g regularity requirements [12]. Inserting the decompositions (10) into (9) leads to the symmetric formulation for the unknown ˜ , ˜t Cauchy data u ˜D − V ∗ g Ñ ˜ = ( 12 I + K) ∗ g V ∗ ˜t − K ∗ u ˜ 1 ˜ =( I −K)∗g Ñ − D ∗ g ˜D K ∗t+D∗u 2

(x, t) ∈ ΓD × (0, ∞) (x, t) ∈ ΓN × (0, ∞) .

Variational principles. Using the inner product f, g Γ = ˜ and ˜t such that formulation is introduced to find u

Γ

(12)

f (x)g(x) dsx a variational

˜D − V ∗ g Ñ , w ΓD ˜ , w ΓD = ( 12 I + K) ∗ g V ∗ ˜t, w ΓD − K ∗ u ˜ 1 ˜ , v ΓN = ( 2 I − K ) ∗ g Ñ − D ∗ g ˜D , v ΓN K ∗ t, v ΓN + D ∗ u

(13)

holds for all test functions w(x) and v(x). Note, as this Galerkin scheme is used only for the spatial integrations the test functions w(x) and v(x) exhibit no time dependency. Temporal discretization. The time discretization of (13) can be done in several ways. Here, the Convolution Quadrature Method (CQM) developed by Lubich [8] is chosen. Thereby, the time convolution integrals of the form t y(t) = f ∗ g = f (t − τ )g(τ ) dτ . (14) 0

are approximated by a quadrature rule which forms a discrete convolution ym = y(m ∆t) ≈

m

ωm−k (fˆ, ∆t)g(k ∆t) .

(15)

k=0

In (15), the quadrature weights ωm−k depend only on the time step size ∆t and the Laplace transform fˆ of the function f . Confer to [10] for more details about the application of the CQM in boundary element methods.

36


Finally, applying this time stepping scheme to (13) yields a semi-discrete variational form m m 1 ˆm−k u ˆ m−k g ˜Dk − Vˆm−k g ˜ k , w ΓD Ñk , w ΓD Vˆm−k ˜tk , w ΓD − K ( 2 Iˆ + K) (16) = ˆ m−k ˜tk , v Γ + D ˆ m−k u ˆ m−k g ˆ )m−k g ˜ Nk − D ˜Dk , v ΓN ˜ k , v ΓN K ( 12 Iˆ − K N k=0 k=0 where (·)m−k denotes the weight ωm−k ((·), ∆t) depending on the respective Laplace transformed integral operator. A Boundary Element formulation for semi-infinite domains ˜∈ The typical problem statement for an elastodynamic halfspace where the domain Ω = {˜ x|x R3 ∧ x˜3 < 0} as well as the boundary Γ = {y | y ∈ R3 ∧ y3 = 0} are unbounded reads as ((L +

∂2 )u)(˜ x; t) = 0 ∂t2 t(y; t) = g(y; t)

(˜ x, t) ∈ Ω × (0, ∞) (y, t) ∈ Γ × (0, ∞)

(17)

∂ with homogeneous initial conditions u(˜ x; 0) = 0 and ∂t u(˜ x; 0) = 0 ensuring that the solution of (17) fulfill the Sommerfeld radiation condition [5]. Since ΓD = {∅} and ΓN = Γ the semi-discrete variational form (16) reduces to m

ˆ m−k u ˜ k , v Γ = D

k=0

m k=0

ˆ )m−k gk , v Γ ( 12 Iˆ − K

(18)

which is an appropriate Galerkin formulation of (17) in terms of boundary integral equations. Regularization of the hypersingular operator. Concerning the numerical evaluation of the bilinear form in (18) the involved hypersingularity makes a direct evaluation rather impossible. Therefore, a regularization based on Stokes theorem is used to transfer the hypersingular bilinear form to a weak one. This regularization is given in detail in [7] and based itself mainly on the work of Han [6]. Additionally, the double layer potential is also transfered to a weak form using the same techniques as for the hypersingular bilinear form. Within this work it is sufficient to mention that the regularization process demands either a closed boundary surface Γ or vanishing integral kernels at infinity. Since the involved kernels fulfill the Sommerfeld radiation condition the last constraint is satisfied and the regularization holds also for the elastodynamic halfspace. Nevertheless, problems might occur on a discrete level. There, it is a common practice to model just a truncated part of the infinite geometry. Unfortunately, the emerging truncation’s borderline represents the surface’s boundary such that Γ is neither closed anymore nor that the integral kernels can be assumed to vanish. Therefore, it must be ensured that the discretized area is closed or modelling the infinite surface. Spatial discretization. Figure 1(a) illustrates the discretization approach of an unbounded domain. Thereby, the boundary Γ is represented in the computation by an approximation Γh which is the union of two sets of different geometrical elements f

Γh =

Ne =0

Ne i

τf ∪

m=0

i τm .

(19)


37

In (19), τ f denotes standard linear finite elements, e.g., surface triangles, and Nef is their total number. Additionally, the boundary’s far-field is represented by Nei infinite boundary elements τ i whose configuration is depicted in Fig. 1(b). For the concept of infinite elements refer to [2] and the references cited there. ˜ and g are approximated by the separation of variables with Further, the boundary functions u trial functions ϕi and ψj , which are defined with respect to the geometry partitioning (19), and time dependent coefficients ui and gj ˜ (x) ≈ u

N

ui (t)ϕi (x) and g(x) ≈

i=1

M

gj (t)ψj (x) .

(20)

j=1

In case of finite boundary elements the functions ϕi are chosen to be equivalent to those shape functions forming the geometry approximation. x2 x1

z0 x ˆ2

x0

x ˆ1 z1

(a) Halfspace with infinite elements

x1

(b) Infinite element

Figure 1: Discretized halfspace and infinite mapping ˆ = [ˆ By introducing local coordinates x x1 , xˆ2 ] ∈ [0, 1] × [0, 1) the mapping χτ : τˆ → τ from the reference element τˆ to an infinite element τ reads as xˆ2 x z x = χτ (ˆ (21) x) = φ(ˆ x1 ), 0 + φ(ˆ x1 ), α 0

x1 z1 1 − xˆ2 where x0 and x1 denote the two fixed vertex nodes, and z0 and z1 represent two direction vectors with the property zk , zk = 1. The scalar value α > 0 is a scaling factor. The function φ within the dot product ·, · describes the approximation for the finite extent and is given by φ(ˆ x1 ) = [1 − xˆ1 , xˆ1 ] . Since the integral kernels depend mostly on the distance r = |y − x| between two points y and x it is preferable to note that for infinite elements the asymptotic behavior of the distance is of order O((1 − xˆ2 )−1 ). Moreover, the transformation of the integral kernels to the reference element demands the computation of the Gram determinant which itself can be expressed via the Jacobi matrix ∂x 2 (22) x) = ∂∂x Jτ i (ˆ x ˆ1 ∂x ˆ2 =: J1 J2 = O(r) O(r ) and reads as

gτ i (ˆ J i ) = J1 , J1 J2 , J2 − J1 , J2 2 = O(r3 ) . x) = det(J τi τ

(23)

For some kernel function k(x(ˆ x), y(ˆ y)) = O(r−1 ) the integrand in a Galerkin scheme takes the form gτ (ˆ x)gτm (ˆ y) k(x(ˆ x), y(ˆ y)) ϕˆ (ˆ x)ϕˆm (ˆ y) dˆ x dˆ y. (24) I[ , m] = τ

τm

38


From (24), it is obvious that the trial function for an infinite element has to be of order O(r−3 ) to guarantee that the integral is finite. Therefore, the trial and test functions ϕi of an infinite element are chosen as ϕi (x) ◦ ϕ(ˆ ˆ x) = φ(ˆ x1 ) (1 − xˆ2 )3

(25)

where φ(ˆ x1 ) is identical to the function used for the geometry approximation (21). Finally, a comment concerning the singular integrals must be made. All integral operators used in the present work are weakly singular. They are treated completely numerical based on quadrature rules developed by Sauter and Erichsen [4]. Numerical examples Now, numerical results for the present Boundary Element formulation are given. The material data represents soil with Lamé’s constants λ = µ = 1.3627 · 108 N/m2 , and mass density = 1884 kg/m3 . The discretization of the infinite domain consists of 800 regular linear triangles and 80 infinite elements with a scaling factor of α = 1. The triangles occupy a total area of 20m × 20m. Moreover, at the mesh’s center an area of A = 2m2 is excited by a traction jump g = [0, 0, −1] H(t) N/m2 according to the unit step function H(t). The remaining surface is traction free.

(a) Vertical displacements u3

(b) Radial displacements u1

Figure 2: Vertical and radial displacements at the observation point x Figure 2 depicts the solution for an observation point x on the surface in 4m distance to the center of the loading. The first and the second numerical solution vary in the chosen time step size but reveal in general the same behavior. Compared to the analytical solution [9] both displacement solutions exhibit oscillations for larger times which are due to artificial reflections at the crossing of finite and infinite boundary elements. But beside these effects, both numerical solutions show approximately the characteristics of the analytical solution. Contrary, a computation without infinite elements titled as ’NOINF’ but with the same time step size as the first depicted numerical solution namely ∆t = 8.5 · 10−4 s yields a defective result for times larger than 0.034s. This is exactly the time the compression wave with the velocity c1 = λ+2µ/ = 465.8m/s has to travel from the center of loading to the truncated boundary and back to the observation point x . From this it can be stated that the infinite element approach for the treatment of semi-infinite domains by symmetric Boundary Element Methods is expedient but requires further research.


39

Conclusions A boundary element method for elastodynamics based on a Galerkin discretization in space and on the Convolution Quadrature Method in time was presented. To obtain a symmetric formulation, also the usage of the second boundary integral equation is required which demands the computation of hypersingular kernel functions. Therefore, a regularization of the elastodynamic hypersingular integral operator is used leading to a weakly singular bilinear form. Since the regularization is based on integration by parts it is not suitable for treating problems where the mesh of a halfspace is truncated. To overcome this drawback the concept of infinite elements was used and adopted to the present boundary element formulation. Unfortunately, the numerical results obtained so far are not completely satisfactory since they feature oscillations for larger times. To reduce these oscillations or, better, to eliminate them further investigations are essential. Nevertheless, the infinite element approach is advantageous compared to computations without any infinite elements as it reaches the static limit for larger times. References [1] J.D. Achenbach. Wave propagation in elastic solids. North-Holland, 2005. [2] P. Bettes. Infinite Elements. Penshaw Press, 1992. [3] M. Costabel and E. P. Stephan. Integral equations for transmission problems in linear elasticity. Journal of Integral Equations and Applications, 2:211–223, 1990. [4] S. Erichsen and S. A. Sauter. Efficient automatic quadrature in 3-d Galerkin BEM. Computer Methods in Applied Mechanics and Engineering, 157:215–224, 1998. [5] D. Givoli. Numerical Methods for Problems in Infinite Domains, volume 33 of Studies in applied mechanics. Elsevier, 1992. [6] H. Han. The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity. Numerische Mathematik, 68:269–281, 1994. [7] L. Kielhorn and M. Schanz. Convolution Quadrature Method based symmetric Galerkin Boundary Element Method for 3-d elastodynamics. International Journal for Numerical Methods in Engineering, 2008. (accepted). [8] C. Lubich. Convolution quadrature and discretized operational calculus I & II. Numerische Mathematik, 52:129–145 & 413–425, 1988. [9] C. L. Pekeris. The seismic surface pulse. Proceedings of the National Academy of Sciences of the United States of America, 41:469–480, 1955. [10] M. Schanz. Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach, volume 2 of Lecture Notes in Applied Mechanics. Springer-Verlag Berlin Heidelberg, 2001. [11] S. Sirtori. General stress analysis method by means of integral equations and boundary elements. Meccanica, 14:210–218, 1979. [12] O. Steinbach. Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer, 2008.


41

An efficient method for boundary element analysis of thermoelastic problems without domain discretization R. Fazeli1,a and M. R. Hematiyan1,b 1

Department of Mechanical Engineering, Shiraz University, Shiraz, Iran a

e-mail: [email protected], be-mail: [email protected]

Keywords: boundary element method, thermoelasticity, temperature rise, boundary integrals.

Abstract. In thermoelastic problems with a non-uniform temperature rise, a domain integral appears in the boundary element formulation of the problem. To save the main advantage of boundary element method as a boundary-only technique, a method is presented to solve thermoelastic problems without domain discretization. Temperature rise function is approximated by a function with an arbitrary distribution in a Cartesian direction and a piecewise quadratic variation in the other Cartesian direction. The domain integrals in 2D displacement and stress integral equations of thermoelasticity are then exactly transformed into boundary by Cartesian transformation method. Two 2D thermoelastic examples are presented to show the efficiency of the proposed method. Introduction To solve engineering problems by boundary element method (BEM), it is sufficient to discretize only the boundary of the problem. This is the main attractiveness of BEM in comparison with finite element method (FEM). High accuracy of BEM as a semi-analytic method is another characteristic of the method. On the other hand, presence of body forces in elastic problems or temperature rise in thermoelastic problems produce some domain integrals in the displacement and stress integral equations. Evaluating the mentioned domain integrals by boundary-only discretization is an important task in BEM. Various techniques have been developed for this purpose. Danson presented an approach for gravitational, centrifugal and thermoelastic forces based on the Galerkin vector and Gauss-Green theorem [1]. A different approach was presented by Henry and Banerjee based on using a particular-integral approach [2]. A simple and effective technique, called Radial Integration Method (RIM), was developed by Gao [3, 4]. This method is based on mathematical operations. It can evaluate any domain integral with boundary-only discretization. Recently, a method for exact transformation of a wide variety of BEM domain integrals into boundary has been presented by Hematiyan [5, 6]. In this method, domain integrals are transformed into boundary integrals simply by Green’s theorem in Cartesian coordinates system. It has been successfully employed for potential and elastostatic problems [5] and also for thermoelastic problems [6]. For very complicated cases where, the domain integral cannot be exactly transformed into the boundary, the integral can be evaluated by a meshless adaptive method [7]. The general form of the methods presented in [5-7] is called Cartesian Transformation Method [8]. In this paper, the displacement integral equation of thermoelasticity is employed to develop a new form of stress integral equation of thermoelasticity. The temperature rise function is approximated by a function with an arbitrary variation in one Cartesian direction and Piecewise-quadratic variation in the other direction. By this form of temperature rise function, it is possible to transform exactly all domain integrals into boundary integrals in both displacement and stress integral equations. The mentioned approximation for temperature rise function is accurate enough and unlike approximation methods with radial basis functions, it can be carried out with no need of system solving and therefore it is faster.

42


Integral equations of thermoelasticity Temperature distribution over an isotropic material can cause a volumetric expansion. The relationship between the total strain H ij and the stress V ij in thermoelasticity can be expressed as Q 1 (1) {V ij G ijV kk } G ijDT 2G 1 Q where, G is shear modulus, Q is Poisson’s ratio, T is temperature rise function and D is coefficient H ij

of linear thermal expansion. The first term in the right hand side of Eq. 1 represents the strain produced by actual stress, while the second term represents the thermal expansion effect. The stressstrain relation can be expressed as V ij

Q

2G{H ij

G ijH kk }

1 2Q

2G (1 Q ) DG ijT 1 2Q

(2)

The displacement integral equation of thermoelasticity for an isotropic material can be expressed as [9] ~ (3) Cij ( x )u j ( x ) ³ Tij ( x, y )u j ( y )dS ³ U ij ( x, y )t j ( y )dS k ³ U ik ,k ( x, z )T ( x )dV S

S

V

where, ~ k

2G (1 Q ) D 1 2Q

(4)

S and V in Eq. 3 represent the boundary and the domain of the problem respectively. Cij

0.5G ij for

smooth boundaries and Cij G ij for internal points. x , z and y are respectively source point, field point and boundary point. U ij and Tij are 2D fundamental solutions and are expressed as follow [9]: 1 Q (5) U ij {(3 4Q )G ij ln r r,i r, j } 8SG (1 Q ) 1 wr ½ ® [(1 2Q )G ij 2r,i r, j ] (1 2Q )( n j r,i ni r, j ) ¾ 4S (1 Q ) r ¯ wn ¿ where ri z i xi .

(6)

Tij

Differentiating Eq. 5 with respect to z j yields (1 2Q ) r,i 4S (1 Q )G r

U ij , j

(7)

The following relations are employed to derive Eq. 7: wr wn

r,i ni ,

r,i r,i

1,

r,ij

wf ( r ) wx j

1 G ij r,i r, j , r

wf ( r ) wr wr wx j

wf ( r ) r, j wr

Substituting Eq. 7 in Eq. 3, results in:

³ (U

Cij ( x )u j ( x )

S

ij

( x, y )t j ( y ) Tij ( x, y )u j ( y )) dS ³ 10) [2–4]. Reliable tools exist for the prediction of gas damping in the first two regimes (see e.g. [1,5–8]) but the situation is less defined at lower pressures. Even though numerical techniques of deterministic and statistical nature have been developed in the transition regime, their application to realistic 3D low-speed MEMS is still under investigation. On the contrary, the collisionless or free-molecule flow lends itself to the development of simpler numerical models. Based on the formal classification given above, the free-molecule flow represents the limiting case where the Knudsen number tends to infinity. In inertial MEMS this applies typically at pressures in the range of few mbars and below. The numerical technique typically employed for estimating the quality factor of MEMS at low pressure is the Test Particle Monte Carlo (TPMC). However, this approach is stochastic in nature which is a major drawback for the extremely low-speed applications at hand, often implying long runs in order to obtain reliable averages, while in the design and optimization phase of MEMS fast and agile tools are preferred. A different technique based on a Boundary Integral Equation (BIE) approach is very competitive with respect to the TPMC for the typical working conditions of inertial MEMS since it is fast, robust, of relatively simple implementation and does not suffer from the issue of statistical noise. In the last Section, the formulations analysed are benchmarked against data available for a real inertial MEMS.

156


Numerical analysis: theoretical basis It is assumed that the velocity of moving surfaces v w is small with respect to the average thermal molecular speed, hence it is v w = v w / 2 RT 1 exp(a)@ , 2S r

wT3* ( p, q, t ,W ) wxi

rr ,i 1 exp(a ) ½ ® E1 (a ) ln(a) C 1 ¾ , a 8S ¯ ¿

(27) (28)

(29)

(30)

1 exp(a) 1 exp(a) 1 wr {ni [ E1 (a) ln(a ) C 1 ] 2r ,i [1 ]} , a 8S wn a

(31)

where r ,i wr / wxi Numerical solutions are obtained using the interpolation functions for time and space. If constanttime interpolation and the time step (t F t f 1 ) are used, the time integral can be treated analytically. The time integrals for T f* ( P , q ,t , W ) and wT f* / wn are given as follows:

³

tF

tf

T1* ( p, q, t ,W )dW

* t F wT1 ( p, q, t ,W )

³t

wn

f

dW

tF * T2 ( p, q, t ,W )dW

³t

f

wT2* ( p, q, t , W ) dW wn

tF

³t

³

tF

tf

f

T3* ( p, q, t ,W )dW

1 E1 (a f ) , 4N S

1 wr exp( a f ) , 2N S r wn

r2 1 {E1 (a f ) [E1 (a f ) ln(a f ) C 1 exp(a f )]} , 16NS af

r wr 1 exp( a f ) [ E1 (a f )] , 8NS wn af

(32)

(33) (34) (35)

1 r4 1 {E1 (a f ) [4E1 (a f ) 4 ln(a f ) 4C 1 exp(a f )] 2 [2 E1 (a f ) af 256NS af 2 ln(a f ) 2C 2a f 3 3 exp( a f ) 5a f ]} , (36)

³

* t F wT3 ( p, q, t , W ) tf wn

where

3

dW

1 r wr 1 exp(a f ) a f E1 (a f ) [2 E1 (a f ) 2 ln(a f ) 2C 1 exp(a f )]} , (37) { 64NS wn af a 2f


af

175

r2 . 4N (t F t f )

(38)

In the same way, the time integrals for wT f* / wxi and w 2T f* ( P, q, t ,W ) / wxi wn in Eq.(16) are given as follows:

³

wT1* ( p, q, t ,W ) dW wxi

tF

tf

³

tF

tf

³

f

tF

tf

³

tF

tf

(39)

wT2* ( p, q, t ,W ) dW wxi

wr 1 exp( a f ) [ E1 (a f )] , 8NS wxi af

w 2T2* ( p, q, t ,W ) dW wxi wn

1 1 wr 1 ni {E1 ( a f ) [1 exp( a f )]} 2r ,i [1 exp( a f )] 8NS af wn a f

wxi

f

1 wr [ni 2r , i (1 a f )] exp( a f ) , 2SNr 2 wn

* t F wT3 ( p, q, t ,W )

³t

,


tF

³t

r ,i exp(a f ) 2N S r

dW

r

(40)

(41) ,

(42)

r 3r,i 1 exp(a f ) a f 1 { E1(a f ) [2E1(a f ) 2 ln(a f ) 2C 1 exp(a f )]} , (43) 64NS af a2f


2 E1 (a f ) 4 ln(a f ) 1 r2 E1 (a f ) [1 exp(a f )] ni { 64NS af af

wr 1 [1 exp(a f ) a f exp(a f )]} 2r ,i {E1 (a f ) 2 wn af

2 1 [1 exp(a f )] 2 [1 a f exp(a f )]} . af af

(44)

Assuming that functions T ( Q , W ) and wT ( Q , W ) wn remain constant over time in each time step, Eq.(15) can be written in matrix form. Replacing T ( Q , W ) and wT ( Q , W ) wn with vectors T㨒 and Q㨒, respectively, and discretizing Eq.(15), we obtain [3] F

F

¦ * H(6 H ¦ ) H(3 H $ ,

f 1

(45)

f 1

where B0 represents the total effect of the pseudo-initial temperature, heat generation term, and convection term. Adopting a constant time step throughout the analysis, the coefficients of the matrix at several time steps need only computed and stored once. In the conventional boundary element method using the time-dependent fundamental solution, the number of boundary integrals increases rapidly with increasing number of time steps. In this method, there is no such disadvantage, because the integration is considered within several time steps, taking the temperature at the end of the previous time step as the initial (pseudo-initial) value at the current time step [10]㧚If we use a constant length for the time steps, the integration in the present method is carried out once for the whole time step procedure. However, the iteration process is necessary for the convection term. 3. Numerical examples We analyzed the rectangular region shown in Fig. 1 to verify the efficiency of this method. The velocities Ax and Ay (m/s) are given by

Ax 40 y (1 y ) ® Ay 0 ¯

.

(46)

The coefficient of thermal diffusivity N is 1 m2s-1. As shown in Fig. 1, one part of the rectangular region is 10͠, the remainder of the region is 0͠ and the right side of the region is adiabatic. The internal points are used for interpolation. Figure 2 shows the temperature distributions at t=0.025, 0.1 and 0.5 s. Figure 3 shows the comparison of BEM result with one by the finite difference method (FDM).

176


These results using presented method are in good agreement with the values obtained using FDM. Next, the temperature distribution of L-shaped region with a round corner shown in Fig. 4 is obtained. When the flow is that of an ideal fluid and the flow rates at x 0 are Ax =10 (m/s) and Ay =0, the distributions of the flow rates are obtained using Laplace’s equation. The coefficient of thermal diffusivity N is 2 m2s-1. As shown in Fig. 4, one part of the region is 10͠ and the remainder is 0͠. Fig. 5 shows the temperatures distribution at t = 0.005 and 0.1 s. We obtained the temperature distributions for the convection-diffusion problem upon heat generation WN 1 for the region shown in Fig. 6. The homogeneous heat 100 K / m 2 is generated in the hatched region

O

in Fig. 6. The coefficients of convection term Ax and Ay (m/s) are given by Ax 0 . ® ¯ Ay 20 x(1 x)

(47)

The coefficient of thermal diffusivity N is 1 m2s-1. The temperature at the boundary is 0͠ as shown in Fig. 6. Fig. 7 shows the temperature distributions at t=0.005, 0.1, 0.2 and 2 s.

Fig.1

(a)

t=0.025 s

Rectangular region

(b) t=0.1 s Fig.2 Temperature distributions

Fig.3 Comparison of BEM an FDM results

(c)

t=0.5 s


Fig. 4 L-shaped region with round corner

(a) t=0.05 s (b) t=0.1 s Fig.5

Temperature distributions

Fig.6 Rectangular region with heat generation

177

178


(a) time=0.05 s

(b) time=0.2 s Fig. 7

(c) time=0.2 s

(d) time=2 s

Temperature distributions

4. Conclusion In the conventional boundary element method, internal cells are required to solve convection-diffusion problems using time-dependent fundamental solutions. However, domain integrals are converted into boundary integrals by the triple-reciprocity boundary element method, and internal cells thus become unnecessary. Although there is disadvantage of the conventional boundary element method, in which it takes a considerable amount of time to calculate as an arithmetic series with increasing number of time steps when using a time-dependent solution, it has been overcome using the triple-reciprocity boundary element method in which we regard the calculated temperature as the initial temperature. Even if this method uses internal points, the preparation of data is simpler than that for internal cells, and the numerical examples show the efficiency of this method. References [1] A. Brebbia, J. C. F. Tells and L. C. Wrobel, Boundary Element Techniques㧙Theory and Applications in Engineering, Berlin, Springer-Verlag, pp.47-107, (1984). [2] C. A. Brebbia and P. Skerget, Diffusion-Convection Problems Using Boundary Elements, Advances in Water Resources, Vol.7, pp.55-57(1984). [3] L. C. Wrobel and B. D. Figueiredo, Numerical Analysis of Convection-Diffusion Problems Using the Boundary Element Method, International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 1, pp.3-18 (1991). [4] S. J. DeSilva, C.L Chan, A. Chandra, and J. Lim, Boundary Element Method Analysis for the Transient Conduction-Convection in 2-D with Spatially Variable Convective Velocity, Applied Mathematical Modelling, Vol.22, pp.81-112 (1998) [5] L. C. Wrobel, The Boundary Element Method, Vol. 1, John Wiley & Sons, West Sussex, p.118 (2002). [6] H. S. Carslaw and J. C. Jaeger, Conduction in Heat in Solids, 2nd ed., Oxford Clarendon Press, 1986. [7] Y. Ochiai, Two-Dimensional Unsteady Heat Conduction Analysis with Heat Generation by Triple-Reciprocity BEM, International Journal for Numerical Methods in Engineering, Vol.51, No.2, pp.143-157 (2001). [8] Y.Ochiai, Multidimensional Numerical Integration for Meshless BEM, Engineering Analysis with Boundary Elements, Vol.27, No.3, pp.241-249 (2003). [9] . Ochiai and T. Kobayashi, Initial Strain Formulation without Internal Cells for Elastoplastic Analysis by Triple-Reciprocity BEM, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 1877-1892 (2001). [10] Y.Ochiai, V. Sladek and J. Sladek, Transient Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method, Engineering Analysis with Boundary Elements, Vol.30, pp.194-204 (2006).


179

Modeling of Inhomogeneities and Reinforcements in Elasto-Plastic Problems with the BEM Katharina Riederer1, a, Christian Duenser1,b and Gernot Beer1,c 1

Institute for Structural Analysis, Graz University of Technology, Lessingstrasse 25/II 8010 Graz a

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

Keywords: Internal Cells, Inclusions, Rock Bolts, Reinforcements, Tunneling, Boundary Element Method.

Abstract. The Boundary Element Method (BEM) is particular suitable to analyze problems involving infinite and semi-infinite spaces. However, the classical BEM can not deal with heterogeneous ground conditions and inclusions such as rock bolts. In this work a novel approach for the simulation of inclusions in 2- and 3- dimensional problems and in connection with elastoplastic material behavior is presented. Two kinds of inclusions are treated, common geological inhomogeneities and the reinforcement of the ground by rock bolts. Internal cells are required to consider the effects of the inclusions inside the domain (the domain-force-terms). To avoid the increase of the degrees of freedom in the system of equations, an iterative algorithm is used for the calculation. This iteration procedure is combined with the plasticity-algorithm. Introduction In underground engineering problems different kinds of inclusions plays an important role. This includes in particular geological inhomogeneities and the reinforcement of the ground by rock bolts (linear inclusions). Such a rock bolting technique is an often used way of supporting underground openings. In tunneling and underground mining, steel rod inserted in a hole drilled into the excavation surface to provide support to the roof or sides of the cavity. To analyze an underground structure containing several small inclusions with numerical methods is very time consuming, for the finite element method as well as for the boundary element method. Even for generating a model the effort is huge, in particular for a domain with many rock bolts and in three dimensional analyses. In this work a BEM approach is presented to generate and analyze elastic or plastic structures containing a large number of inclusions in an efficient way. The inclusions are discretized with cells (volume-cells or line-cells). Only a few numbers of line-cells are necessary to discretize one rock bolt and these cells are independent of the boundary-element mesh (see Fig. 1).

Fig. 1: Discretization of a tunnel involving rock bolts and geological inhomogeneities

180


By modeling inclusions in this way, the size of the system of equations does not increase with the number of cell nodes. The additional degrees of freedom are treated as an additional right-handside and the system is solved within an iterative algorithm. Especially for large scale problems and in elasto-plastic domains this iterative procedure is an efficient alternative. It avoids the increase of the system of equations and the iteration procedure can be combined within the iterations from the plasticity-algorithm. Solution Procedure The boundary integral equation for an elastic problem considering initial-stress inside the domain can be expressed as (see also [1], [2], [3], [6], [5]): c( P)u( P )

³ U( P, Q)t(Q) T( P, Q)u(Q) dS ³ E( P, Q)V S

0

(Q ) dV

(1)

V

Where u and t are the displacements and tractions, respectively on the boundary S, c is related to the boundary geometry, U and T are the fundamental solutions for the displacements and tractions, respectively. The last integral over the volume V contains the initial-stress V 0 and the strain fundamental solution E . P and Q denote the source point and field point, respectively, for nodes on the boundary and Q are the internal points where the initial-stress is acting on. In conventional BEM the initial-stress is known and the system of equations (Eq. 1) can be solved directly to obtain the unknown boundary quantities u and t . Here the effect of the inclusions is considered by their action-reaction stresses (initial-stresses V 0 ), which are unknown at the beginning. The problem can either be solved directly or iteratively. In the direct approach the unknown volume integral can be rearranged to the unknowns on the left-hand-side, in this case the system of equations increases with the number of degrees of freedom related to the cell nodes. The method proposed here shows an iterative approach, the unknown volume term is calculated iteratively by a right-hand-side adding to the system of equations. Therefore the system itself has no additional degrees of freedom based on the inclusions; it has only the size of the boundary-element-modell. In the first step of the calculation the unknown volume-term of Eq. 1 is neglected. An initial analysis is carried out assuming that no inclusions are inside the domain. After solving the system, the displacement- or the stress-field in the domain can be carried out via post-processing using Somigliana’s Identity. The stress on an interior point P is given by: V( P )

³ D( P, Q)t(Q) S( P, Q)u(Q) dS ³ W( P, Q)V S

0

(Q ) dV

(2)

V

Where the kernels D , S and W are the corresponding kernels for the stress computation. The stress in the rock-mass is calculated at all cell-nodes. Because of the different materials of the rock mass and the inclusion (geological inhomogeneities or rock bolts), the stress is different in both. The residual initial-stress increment is computed by:

ı 0

ı Inclusion ı Rock

DInclusion DRock İ

D

Inclusion

1 DRock I ı Rock

(3)

Where D Inclusion and D Rock are the constitutive matrices of the inclusion and of the rock mass, respectively; I is the identity matrix. In the next iteration step the residual initial-stress increment ı 0 is applied to the system as loading in Eq. 1. With this new loading the system is solved again and a new stress-field in the domain can be computed and from this the new residual initial-stress


181

follows. The iteration proceeds until the residual stress in the incremental step vanish. The sum of the residual stress increments of all iteration-steps gives the complete initial-stress which represents the effect of the inclusions. To consider elasto-plastic material behavior of the rock mass, a nonlinear algorithm is needed anyway, as discussed for example in [5] and [8]. The inclusion algorithm can be combined with the plasticity algorithm (Fig. 2) and thus produces not much additional expense in comparison to a system without inclusions.

Initial Analysis

> A@^x 0 ` ^b B ` ^b B ` Contributions due to the boundary conditions ^x` Unknown boundary values 0 Initial-stress is zero at the beginning

ı0

i=1 Evaluate the stress-increment at all cell nodes

ı i ( P )

Residuum for Plasticity:

ı 0 P i

f ı i , ı 0 i 1 , DEP

Residuum for Rock Bolts:

^ `

o b P i

ı 0 RB i

f ı i , ı 0 i 1 , ERB

^

o b RB i

Residuum for Inhomogeneities:

ı 0 I i

ı 0 i

f ı i , ı 0 i 1 , DI

^ `

o b I i

ı 0 P i ı 0 RB i ı 0 I i

Incremental Elastic Solution

> A@^x i ` ^b P i ` ^b RB i ` ^b I i ` Update results

xtot

i = i+1

no

¦ x

i

ıtot

¦ ı

i

ı0 tot

¦ ı

0 i

Check Convergence yes Solution

Fig. 2: Flow chart diagram for the iterative solution algorithm

`

182


Integral Formulation for Inhomogeneities and Rock Bolts To calculate the volume-term of the integral equations (Eq. 1, Eq. 2) different methods are possible. Here a cell-integration technique is used: the volume V is subdivided into a number of integration cells Vc and over these cells numerical integration is used. For geological inhomogeneities general volume-cells are used: C

³ E( P , Q ) V

0

(Q )dV

¦ ³ E( P , Q ) V

0

(Q )dV

(4)

c 1 Vc

V

To model linear inclusions like rock bolts, line-cells with a predefined cross-section-area are used instead of volume-cells (see also [7], [4], [9]). It is assumed that the rock bolts are in continuous contact with the ground (fully grouted) and that they are only able to carry axial loading. The three dimensional integration over the volume of the bolt is reduced to a one dimensional integration along its length by integrating over the cross-section-area of the bolt ABolt analytically. The variation of stress across the cross-section is assumed to be constant. The domain integral (Eq. 4) may be replaced by: C

³ E( P , Q ) V

V

0

(Q )dV

§

· · E( P, Q ) dA ¸ V 0 (Q )dL ¸ ¸ ¸ Lc © ABolt ¹ ¹ §

¦ ¨¨ T ³ ¨¨ ³ g

c 1

©

C

§

¦ ¨¨ ³ E ( P, Q)V c 1

© Lc

0

· (Q )dL ¸ ¸ ¹

(5)

is a modified Where V 0 is the initial-stress in axial direction of the bolt (a scalar value) and E fundamental solution for strain which is computed as follows: First the terms of the general fundamental solution E are computed by using the local coordinates of the bolt x , y , z (see Fig. 3).

Fig. 3: Local coordinate system of the bolt in 3 dimensions The fundamental solution for strain is now given in local coordinates E and the analytical integration over the cross-section-area can easily be done. However, to insert the kernel in the global expressed equations (Eq. 1), the entire terms have to be transformed to the global system, from Eq. 5 is with the geometrical transformation matrix Tg . The modified fundamental solution E computed by:


E

Tg

³

E( P, Q )dA

183

(6)

ABolt

Eq. 5 is evaluated numerically over all line-cells by using Gauss Quadrature. C

§

¦ ¨¨ ³ E ( P, Q)V c 1

© Lc

0

· (Q )dL ¸ ¸ ¹

C

NG

¦¦ E ( P, [

n

)V 0 ([ n ) J Wn

(7)

c 1 n 1

The integral (Eq. 5) becomes singular if the field point Q coincide with the source point P on the boundary or with P in the domain. In this case the numerical integration using the Gauss Quadrature does not give accurate results. Therefore the integral (Eq. 5) is carried out analytically over the hole bolt volume. The volume term of the stress equation (Eq. 2) is calculated analog to volume term of Eq. 1: The is computed by: modified fundamental solution W

W

TH

³

W ( P, Q ) dA

(8)

ABolt

Where W is the fundamental solution calculated in the local coordinate system, and TH is the strain transformation matrix. The numerical integration over the cell length can be done like for Eq. 7 and in the singular case the integral is carried out analytically too.

Numerical Applications The described method is applied in the boundary element program BEFE++ and verification examples for two- and three- dimensional problems are carried out. Here plane strain example is presented.

Fig. 4: left: Boundary element model, right: Comparison of the results The example shown here is a circular hole (radius = 10m) in an infinite domain which is reinforced by 24 rock bolts. The boundary of the hole is subjected to an internal tension of 15 MN/m². Figure 3 shows the mesh used for the analysis. It consists of 40 linear boundary elements

184


for the hole and two quadratic line-cells for each rock bolt. A finer mesh with 120 linear boundary elements and 4 cells per rock bolt was analyzed, too. For the analysis the following material properties were assumed: ERock=5000MN/m², Q=0.3, EBolt=400000MN/m². The Bolts have a crosssection of ABolt=0.007854m² and a length of l=10m. In Fig. 4 the results are compared with a finite element analysis using a very fine mesh. The displacement along the axis of the rock bolt is shown. It can be seen that the solutions compare well, even the coarse mesh gives very accurate results. Further examples that tread rock bolts in combination with plastic material behavior will be presented at the conference.

Acknowledgements The work presented here is supported by the Austrian science fund FWF (project number: L293N07) and by the European Commission, under its 6th framework program, within the integrated project TUNCONSTRUCT.

Literature References [1] M.H. Aliabadi: The Boundary Element Method. John Wiley & Sons, Volume 2, 2002. [2] P.K. Banerjee: The Boundary Element Methods in Engineering. McGRAW-HILL Book Company, 1981. [3] G. Beer: Programming the boundary element method: an introduction for engineers. John Wiley & Sons, 2001. [4] G. Beer: Numerical simulation in tunnelling. SpringerWienNewYork, 2003. [5] X.W. Gao, T. Davis: Boundary Element Programming in Mechanics. Cambridge University Press, 2002. [6] L. Gaul, M. Kögl, M. Wagner: Boundary Element Methods for Engineers and Scientists, Springer-Verlag Berlin Heidelberg, Deutschland, 2003. [7] K. Riederer, P.G.C. Prazeres, G. Beer: Numerical Modelling of Ground Support with the Boundary Element Method, ECCOMAS Thematic Conference on Computational Methods in Tunnelling, Austria, 2007. [8] K. Thöni, R. Hafizi, G. Beer: Efficient Calculation of Non-Linear Problems using the Boundary Element Method in Tunneling, ECCOMAS Thematic Conference on Computational Methods in Tunnelling, Austria, 2007. [9] Y.C. Wang, X.W. Gao: Practicable BEM Analysis of Frictional Bolts in Underground Opening. Journal of Structural Engineering, March 1998, 342-346.


185

Modelling of Changing Geometries for the Excavation Process in Tunnelling with the BEM Christian Duenser1,a and Gernot Beer1,b 1

Institute for Structural Analysis, Graz University of Technology, Lessingstrasse 25/II 8010 Graz, Austria a

[email protected], [email protected]

Keywords: Tunnelling, Sequential excavation, Boundary Element Method, Corners and Edges, Multiple Regions, Discontinuous Elements

Abstract. The modelling of the sequential excavation process in tunnelling is a complicated task with the Boundary Element Method (BEM), because it depends heavily on changing geometries and changing boundary conditions. One possibility to model the sequential excavation is to use the Multiple Region BEM (MRBEM). The problem of corners and edges has a big influence to the results of this modelling strategy. This will be pointed out and a solution with discontinuous elements will be shown. Another possibility to solve the excavation problem in tunnelling is to use a single region boundary element calculation in combination with internal result computation. Stresses in the interior domain are calculated to obtain the loading for subsequent analysis steps. With this method the problem of corners and edges is largely avoided, but the crucial point is the correct evaluation of the loading for the next excavation step. These methods are shown for tunnelling examples in 2D. Introduction The New Austrian Tunnelling Method (NATM) is characterised by a complicated sequence of excavation and installation of support systems. In this context the modelling of the sequential tunnel excavation in an infinite/semi-inifinite domain will be discussed. The numerical method which will be applied is the Boundary Element Method (BEM). Fig. 1 shows a typical excavation with the NATM. As a possible example the tunnel cross section is divided into two parts, top heading and bench. The volumes of material are excavated at different time and location. One way to model the sequential tunnel excavation with the BEM is to use multiple regions [1,2,3]. The technique of multiple regions is shown to be applied successfully for the modelling of the sequential excavation process [4,5]. Another possibility to simulate the sequential excavation is the use of a single region boundary element calculation for each step of excavation [6]. To obtain the loading of each subsequent excavation step stresses in the interior domain are calculated. These two solution strategies are explained in the following sections and an example in 2D is shown. Multiple Region Boundary Element Method (MRBEM) As in Fig. 1 shown the model consists of several finite regions which represent the volumes of excavation. These regions are embedded in an infinite region, which represents the infinite extent of the continuum. The excavation process is simulated by deactivating of regions from the model system. The geometry and boundary conditions are changing progressively from excavation step to step. For all regions stiffness matrices are calculated and assembled to a global system of equations, which is solved for the interface displacements.

186


Fig. 1 – Example for a staged excavation process in 3D (only half of the mesh shown) The stiffness matrix of a region is determined by using the boundary integral equation shown in Eq.1 c( P)u( P)

³ U( P, Q)t(Q)dS ³ T( P, Q)u(Q)dS S

(1)

S

Where u and t are the displacements and tractions on the boundary S, respectively, c is related to the boundary geometry, U and T are the fundamental solutions for displacements and tractions, respectively. In discretized form this equation is equivalent to E

N

¦¦ 'T

e ni

E

u en

e 1 n 1

N

¦¦ 'U

e ni

t en

(2)

e 1 n 1

'U eni and 'Tnie are the integrated kernel shape function products of the element e for a collocation point ( Pi ) at point i . u en and t en are vectors containing all displacements and tractions for the element. Collocating at all boundary nodes results in an equation system. In general the boundary conditions are mixed and the known and unknown values are separated to the right and left side of the equation and this gives the following: N °t c0 °½ B N ® N ¾ f0N ; ¯°u f 0 ¿°

N °t cn °½ B N ® N ¾ f nN ¯°u fn ¿°

for n 1, 2......Ndofc

(3)

Matrix B N is the assembled left hand side for region N , containing the integrated kernel shape N function products related to the unknown boundary conditions. t c0 is a part of the unknown vector representing the traction at the interface c and u Nf0 are the displacements due to the loading (excavation tractions) at the Neumann boundary. The second equation of Eq. 3 obtains the same left hand side as the first but the loading are unit displacements at the interface degrees of freedom. Applying unit displacements at all degrees of freedom at the interface results in the stiffness matrix K N of region N . The final solution t cN , the tractions at the interface nodes, and u Nf , the displacements at the Neumann boundary, can be expressed in terms of u cN , the displacements at interface nodes:


°t cN °½ ® N¾ ¯°u f ¿°

N N °t c0 °½ °K °½ N ® N ¾ ® N ¾ uc ¯°u f 0 ¿° °¯ A °¿

where

N °K °½ ® N¾ °¯ A °¿

187

ªt c1 , t c 2 , ..... t cNdofc ¼° º½ °¬ ® ¾ ¯° ¬ªu f 1 , u f 2 ,.... u fNdofc ¼º ¿°

(4)

Using the conditions of equilibrium and compatibility the stiffness matrices K N and the vectors t of all regions can be assembled into a global system of equation, which can be solved for the unknown interface displacements u c as shown in Eq. 5. N c0

t c 0 K uc

0

(5)

K is the assembled stiffness matrix related to the interface nodes only, t c 0 is the assembled vector of tractions at the interface due to the given boundary conditions at the Neumann boundary. The remaining unknowns, displacements at the Neumann boundary and tractions at the interface can be evaluated returning from system to region level, by using Eq. 4. The excavation is modelled by deactivating regions, this means removing stiffness matrices from the equation system [5]. This implies that the geometry of the system and the boundary conditions of some regions are changing, for instance from interface conditions to Neumann conditions. For these regions the stiffness matrices have to be updated each calculation step before the assembly process starts. A problem, which arises is the one of corners and edges [2,5,6,7]. In the following an example (shown in Fig. 2) is shown where this problem is pointed out. This example concerns an excavation in 2D and models a tunnel in longitudinal section. Of course this means an excavation of infinite extend out of plane, which is not a real tunnel excavation, but the problem of corners can be shown clearly. The example shown in Fig. 2 consists of 10 regions which are sequentially removed from the system. At the beginning most of the boundary elements belong to interface boundaries. The regions are of rectangular shape and exhibit geometrical corners. If adjacent elements of these corners belong to the interface, Dirichlet boundary conditions are applied for the evaluation of the stiffness matrices. The resultant tractions at these corners are multi-valued in reality, this means different on both sides of the corner node. If continuous elements are used the boundary integral equation is able to deliver only one distinct vector of tractions at the corner node, therefore the tractions are unique. As the resultant tractions at the interface belong to the loading of the subsequent load step erroneous results will be achieved. LC 1 A

LC 2

LC 3

LC 4

LC 5

LC 7

LC 8

LC 9

LC 10

B

LC 6

Fig. 2 – Example for a sequential excavation process in 2D In the diagram of Fig. 3 vertical displacements for load case 1 to 5 at the line AB (shown in Fig.2) are shown. The dashed lines show the results for calculations with continuous quadratic elements. It can be seen that the results, except for load case 1, are erroneous and theses errors accumulate from step to step. With the use of continuous elements the problem of corners and edges is neglected, especially at nodes of the interface where the traction is singular. Doing the same

188


calculations with discontinuous quadratic elements gives excellent results [2]. These results are validated with single region boundary element calculations as well as with finite element calculations. 0.03

LC1 Discont LC2 Discont LC3 Discont LC4 Discont LC5 Discont LC1 Cont LC2 Cont LC3 Cont LC4 Cont LC5 Cont

displacements uy [m]

0.025 0.02 0.015

LC5

0.01 0.005

LC2

LC3

LC4

LC1 0 0

3

6

9

12

15

chainage [m]

Fig. 3 – Vertical displacements for load case 1 to 5 along the line AB (shown in Fig. 2)

Single region boundary element method

The method discussed before requires a predefinition of the complete geometry of the tunnel excavation problem. From beginning the calculation has to deal with all the regions, interfaces, etc. which in subsequent load cases will be part of the excavation process. This means, that the size of the equation system is determined by the number of all degrees of freedom at the entire geometry and it nearly remains the same for every load step of calculation. Now only a single boundary element region is used to represent the actual excavation surface [6]. Stresses are calculated at points inside the domain, provided that the boundary conditions are known. For the respective load step excavation loads are determined with this approach. The algorithm is explained for the same example as in the previous section. Fig. 4 shows the excavation sequence for 10 load cases. The excavated parts of each step, where the loading is applied, are indicated by hatched areas. With this method it is necessary to discretize the excavated tunnel surface only. A single boundary element region is sufficient for the discretization of the respective load step which can be seen in Fig. 4. LC 1

LC 2

LC 3

LC 4

LC 5

LC 6

LC 7

LC 8

LC 9

LC 10

Fig. 4 – Example for a sequential excavation process in 2D The crucial part is the determination of excavation tractions for the current load step. This is explained next for load case LC4. In Fig. 5 the excavation steps from LC1 until LC4 are shown. The region to be excavated for LC4 (indicated by hatched areas) is shown in the sketches for each


189

of the previous load cases. The resultant displacement field and stress field for LC4 is the accumulation of incremental results of all previous load cases. This implies that the previous load cases have to be considered for the determination of the loading for LC4. LC 4

LC 3

LC 2

LC 1

Fig. 5 – Excavation of LC4 The excavation tractions are calculated with internal stress evaluation. The stress at an internal point is calculated by the following integral equation: ı ( Pi )

³ R( P, Q)t(Q)dS ³ S( P, Q)u(Q)dS S

(6)

S

where ı ( Pi ) is the stress at an internal Point Pi , R and S are the fundamental solutions for stress, t and u are the boundary traction and displacement values, respectively. The stress is evaluated at the same points for load case 1 to 3. For load case 1 and 2 all points are internal points as shown in Fig. 5. For these load cases there is no difficulty in the evaluation of the stress. For LC 3 some of the points of the excavated volume are boundary points. Because of the sharp corners at point A and B (shown in Fig. 6) the stress is infinite and a calculation directly at these points is not possible. To overcome this problem the stress is evaluated inside the adjacent element very near to the boundary, at an intrinsic coordinate of value [ 0,90 . The stress is extrapolated to the boundary according to the parabolic shape function of the element. This is shown in Fig. 6. A

theoretical stress distribution

load case LC3 Detail A

assumed stress distribution

B

t

[ = -0,90

t

load case LC4

t

Fig. 6 – Treatment of singular point The traction vector t is calculated by multiplying the stress ı with the outward normal vector n to the excavation surface. The resulting traction at the excavation surface for LC4 is the sum of tractions obtained by internal stress calculation for LC1 to LC 3 completed by the tractions due to the virgin stress field. Once the loading tractions are found the solution for the current load step is evaluated by a single region boundary element calculation using Eq. 3. The results of the vertical displacements along the crown of the tunnel are shown in Fig. 7. These results are compared with the reference solution. As can be seen in Fig. 7, it seems that some loading is lost from one load case to the other. The reason for this is the inaccurate evaluation of the tractions near to the singular points A and B, shown in Fig. 6.

190


0 -0.005

displacements uy [m]

LC1 NEW LC2 NEW LC3 NEW LC4 NEW LC5 NEW LC1 REF LC2 REF LC3 REF LC4 REF LC5 REF

LC1

-0.01

LC2

-0.015

LC3

-0.02 LC4 -0.025 LC5 -0.03 0

3

6

9

12

15

chainage [m]

Fig. 7 – Vertical displacements for load case 1 to 5 along at tunnel crown

Conclusion The main advantage of the single region approach against the conventional method of domain decomposition (MRBEM) is that the effort for the solution is much less. The simulation starts with a very small BEM mesh and this will extend from excavation step to step. Additional effort has to be spent for the correct evaluation of the stress distribution near to the boundary, where singular values of stress appear. Currently the implementation for 3D into the computer code BEFE++ is an ongoing task. A reasonable comparison of the efficiency to the conventional method of MRBEM only makes sense for a 3D example.

Acknowledgement The work reported here is supported by the European Commission, under its 6th framework program, within the integrated project TUNCONSTRUCT.

References [1] G. Beer: Programming the Boundary Element Method, John Wiley & Sons, 2001. [2] G. Beer, I. Smith, C. Duenser: The Boundary Element Method with Programming, Springer, Wien New York, 2008. [3] X.W. Gao and T.G. Davies: Boundary Element Programming in Mechanics, Camebridge University Press, 2002. [4] C. Duenser, G. Beer: Boundary element analysis of sequential tunnel advance. Proceedings of the ISRM regional symposium Eurock, 475-480, 2001. [5] C. Duenser: Simulation of sequential tunnel excavation with the Boundary Element Method, Verlag der Technischen Universität Graz, 2007. [6] C. Duenser, G. Beer: New algorithms for the simulation of the sequential tunnel excavaton with the boundary element method. Proceedings of ECCOMAS Thematic Conference on Computational Methods in Tunnelling, Vienna, 2007. [7] X.W. Gao and T.G. Davies: 3D multi-region BEM with corners and edges, Int. J. Solids Structures, 37, 2000, 1549-1560.


191

Multi-grain Orthotropic Material Analysis by BEM and its Application Dong-Eun Kim1, Sang-Hun Lee2a, Il-Jung Jeong3b and Seok-Soon Lee4c 1

Suite 903. Jeljon Tower 1 17-1 Jeonja, Bundang, Seongnam, Gyeonggi, 463-811, Korea [email protected] 2,3,4 Gyeongsang National University, School of Mechanical and Aerospace Engineering, Jinju, Gyeongnam, 660-701, Korea a b c [email protected], [email protected], [email protected]

Key Words : Multi-Grain, Orthotropic, Effective elastic modulus, Effective Poisson`s ratio, BEM

Abstract Most of the MEMS parts are made of multi-grain silicon wafer, which is the orthotropic material and its material direction is arbitrary. The stress analysis for the multi-grain is important factor in order to apply the MEMS parts to industrial applications. The finite element method (FEM) is commonly used by a stress analysis method but the boundary element method (BEM) is known as the result of the BEM is more accurate than that of the FEM since the fundamental solution are used. In this study, we derived the boundary integration equation for the orthotropic material by applying fundamental solutions with complex variables. The multi-region analysis procedure for the BEM is developed in order to apply the analysis of the multi-grain orthotropic material. The effective elastic modulus and its Poisson’s ratio are calculated by the BEM. The results of the present method are compared with those of the finite element method in order to verify the present procedure.

1.

Introduction

Reliability and safety of MEMS parts are an important factor in its application. The material properties are determined by the computational mechanics in order to reduce the testing efforts. Toonder, et al.(1). extracted the effective elastic modulus and Poisson’s ratio for an orthotropic material which has 30 grains and has different orientation in each grain with the FEM. Mullen, et al.(2) simulated the effective elastic modulus for a multi-grain thin layer by Monte-Carlo method, Yin(3) also extracted the elastic modulus by the same method. Chu(4) extracted the effective elastic modulus by applying Johnson-Mehl model. Choi(5) calculated the effective elastic modulus and Poisson’s ratio by applying the FEM to an simulated multi-grain structure. For the numerical simulation, the FEM is popular. But the BEM has advantage in numerical accuracy and approximate boundary only and was applied to analysis crack growth analysis by dual boundary element method.(6) In this study, we developed the program to simulate the effective elastic modulus and Poisson’s ratio for the multi-grain orthotropic material. For the simulation, we re-derived the fundamental solution for the orthotropic material(7,8) and applied the discontinuous element for the corner problem. We applied the developed program to simulate the multi-grain orthotropic material and determined the effective elastic modulus and Poisson’s ratio, which was compared with the results by the FEM.

192


2.

orthotropic material relation and BE formulation

By Gibson’s composite material principle,(9) the stress –strain relation in orthotropic material is given in Eq.(1). ª H11 º «H » « 22 » «H 33 » « » «H 23 » «H 31 » « » ¬«H12 ¼»

X21 / E2 ª 1/ E1 « X / E 1/ E2 « 12 1 « X13 / E1 X32 / E2 « 0 « 0 « 0 0 « 0 ¬« 0

X31 / E3

0

0

X32 / E3

0

0

1/ E3

0

0

0

1/ G22

0

0

0

1/ G31

0

0

0

0 º ªV 11 º 0 »» ««V 22 »» 0 » «V 33 » »« » 0 » «V 23 » 0 » «V 31 » »« » 1/ G12 ¼» ¬«V 12 ¼»

(1)

Since Two dimensional orthotropic material is V 3 W 23 W 31 0 , Eq.(1) can be reduced as following relation. ª H11 º «H »= « 22 » ¬« 2H12 ¼»

ª S11 «S « 12 ¬« 0

S12 S 22 0

0 º 0 »» S66 ¼»

ªV 11 º «V » « 22 » ¬«V 12 ¼»

(2)

where ıij and İij (i,j=1,2) are stress and strains, respectively, and the coefficients Sij are the elastic compliances of the material. These compliances can be written in terms of engineering constants as S11 S12

1 E1

1

, S 22

S 21

E2 v12 E1

(3)

v21 E2

, S66

1 G12

,

where Ek is Young’s modulus in xk direction,, G12 is shear modulus in x1-x2 plane and ȣij is Poisson’s ration. The characteristic equation is given as follows: S22 2S26 P (2S12 S66 )P 2 2S16 P S11P 4

0.

(4)

The solution of the characteristic equation can be denoted by: P1 D1 iE , P2 D 2 iE 2 , P3 P1 , P4 P2 .

(5)

Here, Di , Ei are real and i is imagery, Pi is the conjugate of the complex variable of Pi . Since the directions of the grain crystallization of the MEMS parts are determined randomly, the strains can be expressed in terms of stresses in non-principal coordinates of the laminates as: ª H11 º «H »= « 22 » ¬« 2H12 ¼»

where the by:

ª S11 « « S12 «0 ¬

Sij

S12 S 22 0

0 º » 0 » S66 »¼

ªV 11 º «V » , « 22 » ¬«V 12 ¼»

(6)

are the components of the transformed lamina compliance matrix which are defined


193

S11 V1 V2 cos 2J V3 cos 4J

(7)

S12

V4 V3 cos 4J

S22

V1 V2 cos 2J V3 cos 4J

S66

2(V1 V4 ) 4V3 cos 4J .

The invariants (V1,V2,V3,V4) are given by: V1 V2

V3 V4

1 (3S11 3S 22 2 S12 S66 ) 8 1 ( S11 S 22 ) 2 1 ( S11 S 22 2 S12 S 66 ) 8 1 ( S11 S22 6 S12 S66 ). 8

(8)

The boundary integral equation for anisotropic materials may be written as: (8) Cij ( zk0 )u j ( zk0 ) ³ Tij ( zk ,zk0 )u j ( z k )d *( zk ) *

³U

ij

( zk , zk0 )t j ( zk ) d *( zk ).

(9)

*

Cij is determined by the local boundary shape.

zk is the complex variable and can be written as:

zk zk0 [ x1' Pk x2' ](] ] 0 ). xi' is the derivative of xi and ] 0 is the position of

(10)

zk0. The displacement and traction

fundamental solution are given as follows: Uij

2Re[rj1 Ai1 ln( z1 z10 ) rj 2 Ai 2 ln( z2 z20 )]

Ti1 2n1 Re[P12 Ai1 /( z1 z10 ) P22 Ai 2 /( z2 z20 )] 2n2 Re[P1 Ai1 /( z1 z10 ) P2 Ai 2 /( z2 z20 )] Ti 2

(11)

2n1 Re[P1 Ai1 /( z1 z10 ) P2 Ai 2 /( z2 z20 )] 2n2 Re[ Ai1 /( z1 z10 ) Ai 2 /( z2 z20 )],

here nj is normal unit vector in x1-x2 coordinate, rij is give as: r1 j

S11P 2j S12 S16 P j

r2 j

S12 P j S 22 / P j S 26 ,

(12)

and Ajk are complex variables which are determined by following equation: 1 ª1 « « P1 P 1 « r r11 « 11 ¬« r21 r 21

1

P2 r12 r22

1 º ª Aj1 º »« » P 2 » « A j1 » » « » A r12 » « j2 » r 22 ¼» ¬« A j 2 ¼»

ª G j 2 / 2S i º « » « G j1 / 2S i » , « » 0 « » 0 ¬ ¼

where įij, i,j=1,2 represents the Dirac delta.

(13)

194


3.

multi-grain boundary element

For the multi-grain boundary element, the matrix equation for the region-k with interface surface denoted by I can be written as follows:(10) H kU Ik

G k TIk .

(13)

The relation between the traction TIk and the displacement U Ik in region-k can be are expressed: K 1U 1 TIk o ® 2 I2 ¯K U I

K kU Ik

TI1 ½ ¾. TI2 ¿

(14)

On the interface surface, the displacement must be the same between two regions and the traction can be given by adding individual value: U I1 U I2

TI1 TI2 .

U I , TI

(15)

We can write the relation between displacement and traction on the interface surface by combining eq.14 and eq.15 as following: ( K 1 K 2 )U I

TI .

(16)

4.

Discontinuous Boundary Element

The edge and corner problem in the boundary element method occurs at the point, where the normal vector is defined incorrectly. Aliabadi, et. al.(8) introduced the discontinuous element in order to remove the singularity in the crack tip. The nodes in this element are shifted on each end nodal point as: 2 ,] 3

]

0, ]

2 . 3

(17)

The coordinates can be written as: N dn (] ) x in ,

x i (] )

n

(18)

1, 2, 3,

here, N dn (] ) is the discontinuous shape function and xin is nodal coordinate which is shifted by eq.17. Since this element does not have common nodal points and the shape function in eq.18 is smooth enough within the element, the edge and corner problem does not occur. For this element, the integral of eq.9 can be represented by: 1

³ T ( z ,z ij

*

k

0 k

)u j ( z k )d *( zk ) u nj ³ Tij ( z (] ), z (] 0 )) N dn (] ) J (] )d ]

Fijn (] , ] 0 )u nj ,

(19)

1

where ujn denote the nodal displacement component and J is the Jacobian of the coordinate transformation and is given by J=L/2, where L represents the element length. Fijn can be given by


Fiju (] , ] 0 )

195

(20)

ª º 1 Nn 1 1 d Re « ' q j1 ( P1n1 n2 ) Ai1 + ' q j 2 ( P2 n1 n2 ) Ai 2 » l ³ d] . 0 ' ª¬ x1 P2 x2' º¼ «¬ ª¬ x1 P1 x2 º¼ »¼ 1 ] ]

Here, qij ª« P1 P2 º» . 1 1 ¬

¼

As the same method, the traction can be given by: 1

³U *

Here

ij

( zk ,zk0 )t j ( z k )d *( zk ) t nj ³ U ij ( z (] ), z (] 0 )) N dn (] ) J (] )d ]

Lnij

Lnij

is given as follows: (22)

2 Re ª¬ rj1 Ai1 ln(] ] 0 ) rj 2 Ai 2 ln(] ] 0 )].

5. 5.1

(21)

Lnij (] , ] 0 )t nj .

1

Application of multi-grain BEM

Stress analysis of multi-grain structure

Geometry shapes for the numerical analysis are shown in Fig. 1, which has 8 regions. For the verification of the present method, we apply the FEM method to analysis the same model as shown in Fig.4(a). We used Altair HyperMesh(11) for FEM modeling and ABAQUS V6.6/standard(12) for the analysis. For the material properties are given in Table 1 and the element and node numbers are shown in Table 2 for each region, respectively. For the analysis, boundary conditions are given in Fig.1(b) and the applied force is 10N/mm2.

(a)

(b)

Fig. 1 FEM mesh (a) and BEM element (b) and boundary condition for stress analysis

The tractions of two surface A-A’ and B-B’ are compared in Fig.5 and Fig.6. These Figures show that the present method is good agreement with the FEM results. Table 3 shows the tractions along the line A-A’ and line B-B’. In the Table 3, the dark raw shows that the traction has some gap. At the sharp corner along the line A-A’, the normal direction does not have a unique value. Therefore the traction has some gap on the sharp corner.

196


Table 1 Material properties of each region

Table 2 Element and node number of each region

Region No.

E1 (MPa)

E2 (MPa)

Q

G (MPa)

Region No.

1 2 3 4 5 6 7 8

198373.0 197844.3 209803.8 199086.3 197477.1 209231.4 199994.7 210000.0

204073.0 205489.8 197017.6 202683.5 206849.3 197077.1 201392.5 197000.0

0.2936 0.2908 0.3001 0.2962 0.2880 0.3005 0.2984 0.3000

78449.69 78200.41 77247.72 78641.28 77925.13 77385.28 78750.46 77200.00

1 2 3 4 5 6 7 8

FEM BEM

BE model

146 120 47 22 65 23 12 41

167 141 62 30 80 32 19 51

26 25 15 9 18 11 7 13

53 51 31 19 37 23 15 27

FEM BEM

10.2

9

10.0

Traction (MPa)

8

Traction (MPa)

FE model

Element No Node No Element No Node No

7

6

5

9.8

9.6

9.4

9.2

4 9.0

1

2

3

4

5

6

7

8

9

10

11

Node Count (from A to A') Fig. 2 Comparison of the result at A – A`

5.2

1

2

3

4

5

6

7

8

9

Node Count (from B' to B) Fig. 3 Comparison of the result at B – B`

Extract Effective Elastic Modulus and Poisson’s Ratio and Discussion

Effective elastic modulus and Poisson’s ratio was calculated by Choi(5), who used the FEM with the model as shown in Fig. 4. Node Count 1 2 3 4 5 6 7 8 9 10 11

Line A-A’ ABAQUS BEM 5.9787 5.9695 5.7770 5.7694 5.5766 5.5820 7.5716 7.3345 9.089 9.0863 6.9696 6.6760 4.2575 4.2457 6.3100 6.0357 7.8254 7.8115 7.2258 7.1247 6.409 6.4174

Line B-B’ ABAQUS BEM 9.1628 9.1522 9.5798 9.7723 10.0226 10.0001 10.0153 9.9780 9.9868 9.9714 9.9989 10.0834 10.0231 10.0107 10.0951 10.0607 10.1245 10.1184

Table 3 Comparison of traction by FEM and BEM(MPa)

Fig. 4 General model for the effective material constants extraction


197

Effective Poisson’s ration and elastic modulus are given by the relation between stress and strain based on the Hooke’s Law. These relations are expressed by a plane stress and plane strain condition respectively: Q effV

V 22 V , Eeff V 11

Q effH

V 22 H , Eeff V 11 V 22

here,

Q effV , EeffV

2

2

V 11 V 22 , V 11 H 11

and

(23)

(V 11 2V 22 )(V 11 V 22 ) , H 11 (V 11 V 22 )

Q effH , EeffH

(24)

are the effective Poisson’s ratio and elastic modulus in plane stress and

plane strain conditions, respectively. V 11 and V 22 is the nominal stresses in x and y direction, respectively, and H 11 is nominal strain in x-direction. The nominal stress and strain in unit thickness are give by : V 11

Fx , V 22 LV

Fy LH

, H 11

G LH

(25)

,

here, Fx and Fy are the reactions in x and y-direction, respectively, and ˡ is the displacement on the applied force surface. The geometries for the BEM and FEM are shown in Fig. 5, which were generated by the random grain generation process and we assume that the problem is a plane stress condition.

(a)

(b) Fig.5 BEM element(a) and FEM element(b)

The material properties in each region are given in Table 4, in which region number 7 has material properties with 0 rotational angle but the other region has the material properties by changing the rotational angle randomly.

198


Table 4 Material properties of each region Region no.

E1 (MPa)

E2(MPa)

Q

Table 5

G(MPa)

1

2

3

4

5

6

7

Section Left_side(MPa) Upper_side(MPa) Bottom_side (MPa) V (MPa) Eeff

Extracted result of each method ABAQUS

BEM

ty

tx

tx

ty

We used Altair HyperMesh for modeling the geometry and AQUS V6.6/ Standard for analysis. We calculated the nominal reaction in eq.23 on the boundary surface by the FEM and BEM and the effective elastic modulus and Poisson’s rations as show in Table 5. Table 5 shows that the results by the FEM and BEM are almost the same. This means that the present method is very accurate.

6.

Conclusion

The tractions analysis of the multi-grain structure shows that the present method is good agreement with the FEM results. The results of the effective elastic modulus and Poisson’s ratio for the multi-grain orthotropic material by the BEM show almost the same as those by the FEM. From these results, the present method is accurate to analysis the multi-grain orthotropic material that we can apply the present method to MEMS structures with multi-grains.

Acknowledgements The authors would like to thank to 2nd Stage Post-BK 21/ NURI project for the financial support of this work.

References (1) den Toonder, J. M., van Dommelen, J. A. W. and Baaijens, F. P. T., 1999, “The relation between single crystal elasticity and the effective elastic behavior of polycrystalline materials : theory, measurement and computation”, Modeling and Simulation in Materials Science and Engineering, Vol.7, pp.909-928. (2) Mullen, R. L., Ballarini, R., Yin, Y. and Heuer, A. H., 1997, “Monte Carlo simulation of effective elastic constants of polycrystalline thin films”, Acta Materialia, Vol.45, No.6, pp.22472255 (3) Yin, Y., 1997, “Monte Carlo Simulation of Effective Elastic Constants of Polycrystalline Thin Films”, M.Sc Thesis, Case Western Reserve University, Civil Engineering (4) Chu, Z., 2000, “Monte Carlo simulation of elastic properties of polycrystalline materials using


199

the Johnson-Mehl model”, M.Sc Thesis, Case Western Reserve University, Civil Engineering. (5)Choi. J. W., 2004, Statistical Approach to the Elastic Property Extraction and Planar Elastic Response of Polycrystalline Thin-Films, Ph.D. Dissertation, The Ohio State University Mechanical Engineering. (6) Portela, A., 1993, Dual Boundary Element Analysis of Crack Growth, Computational Mechanics Publications. (7)Rasskazove, A. O., “Calculation of a Multilayer Orthotropic Shallow Shell by the Method of Finite Elemets,” International Applied Mechanics, Vol. 14, No. 8, pp. 826-830, 1978 (8) Aliabadi. M. H., 2002, The Boundary Element method – Applications in Solids and Structures, John Willy & Sons LTD., Vol. 2. (9) Gibson. R. F., 1994, Principles of composite material mechanics, McGRAW-Hill, pp. 46-58. (10) Beer. G., 2001, Programming the Boundary Element Method – An Introduction for Engineers, John Willy & Sons LTD., pp. 247-285 (11) Altair Engineering, 2007, HyperMesh 8.0 User’s Guide. (12) ABAQUS, 2006, Analysis User’s Manuals, Version 6.6.


201

Potential problem for Functionally Graded Materials: two-Dimensional study Z. Sharif Khodaei1,a and J. Zeman2,b 1

Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague 6, Czech Republic,

2

Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague 6, Czech Republic, a

[email protected], b [email protected]

Keywords: Functionally graded materials, stochastic Hashin-Shtrikman variational principle, Potential problem, Green’s function, Boundary element method.

Abstract. Functionally graded materials (FGMs) are two-phase composites with continuously changing microstructure adapted to performance requirements. Compared to a layered system, an FGM avoids the discontinuity in material properties across the interface. FGMs have already been employed in many important areas, e.g., thermal barrier coatings for aerospace applications and implants for bio-medical applications. As analytical solutions for non-homogeneous materials are rare, there is a need for reliable and efficient numerical methods for solving problems in FGMs. Traditionally, the overall behavior of FGMs has been determined using local averaging techniques or a given smooth variation of material properties. Although these models are computationally efficient, their validity and accuracy remain questionable as their link with the underlying microstructure is not clear. In this paper, we propose a modeling strategy for the 2-D potential problem of FGMs systematically based on a realistic random microstructural model by the BEM. The overall response of FGMs is addressed in the framework of stochastic Hashin-Shtrikman variational principles. Introduction Due to the continuously and functionally varying volume composition of constituent particles, functionally graded materials (FGM) provide superior thermo-mechanical performance under given loading circumstances compared with classical laminated composite materials. In order to achieve the best performance, computing the temperature and overall stress/strain distributions in FGMs needs an appropriate estimate for properties of the graded layer, such as the thermal conductivity, coefficient of thermal expansion, the Young modulus, Poisson’s ratio and so on. A detailed description of the geometry of actual graded composite microstructure is usually not available, except for information on volume fraction distribution and approximate shape of the dispersed phase or phases. Therefore, the evaluation of thermo-mechanical response and local stresses in graded materials must rely on analysis of micromechanical models with idealized geometries. There are significant differences between the analytical models of macroscopically homogeneous composites and FGMs. It is well known that the response of macroscopically homogeneous systems can be described in terms of certain thermo-elastic moduli that are evaluated for a selected representative volume element, subjected to uniform overall thermo-mechanical fields. However, such representative volumes are not easily defined for systems with variable phase volume fractions, subjected to non-uniform overall fields. Heat conduction problems can be efficiently solved with the boundary element method. In this work a two-dimensional FGM plate subjected to steady state heat conduction is solved. The model used here is systematically derived from a fully penetrable sphere microstructural model introduced by [1]. The statistics of local fields then follow from re-formulation of the Hashin-Shtrikman (H-S) variational principles introduced, e.g., in [2,3] and summarized in the current context together with

202


the Galerkin scheme allowing to treat general bodies. The application of the Boundary Element Method (BEM) is covered in the spirit of [4], [5] and [6]. Microstructural model As already indicated in the introductory part, the morphological description adopted in this work is the one-dimensional case of a microstructural model studied by Quintanilla and Torquato [1]. A particular realization can be depicted as a collection on N fully penetrable spheres (fibres in 2D) of radius R randomly distributed within a plate (matrix material) of size H1 u H 2 , whose particle density obeys any specified variation in volume fraction, see Figure 1. The position of the i-th sphere is specified by the x and y coordinates of its reference point x which in our case coincides with the centre of the sphere. As individual spheres are allowed to freely overlap each other, the family of microstructures attainable by the model is very versatile. Hashin-Shtrikman variational principle Problem statement. Laplace’s equation of heat transfer 2u 0, is considered in a 2-D rectangular domain depicted in Figure 1 with a basic example of discretization of the boundary into elements and nodes.

Figure 1: Two-deimensional potential problem associated with realization D. Hashin-Shtrikman decomposition Following the seminal ideas of [7,8], the solution of the stochastic problem is sough as a superposition of two auxiliary problems, each characterized by constant coefficient of thermal conductivity k0. By introducing a comparison medium, with constant thermal conductivity coefficient and defining “polarization flux” W through the relation

q x; D

k

0

u x; D W x; D

(1)

and by introducing it into the steady state condition we obtain:

k 0 u x; D W x; D

0

(2)


203

Multiplying the above equation by the test function v(x) for the temperature, integrating it over the domain and using the Gauss divergence theorem, the following variational boundary value problem (BVP) is obtained (weak formulation of the Laplace’s equation):

³ k

0

u x; D v x W x; D v x

0

(3)

:

The unknown polarization flux is now a new variable to be determined from the critical point of the two-field Hashin-Shtrikman-Willis functional

u (x;D ), W (x;D )

arg

min

stat

v ( x )V T ( x;D )T D

U (v(x),T (x; D ); D )

(4)

and, hence the functional for this problem yields §1 0 · ¨ k u x; D v x T x; D v x ¸d : ©2 ¹ (5) 1 § 1 · ³ ¨ T x; D ª¬ kij x k 0 º¼ T x; D ¸d : : © 2 ¹ where Tdenotes an admissible polarization flux from the realization-dependent set 7(D). In order to determine the distribution minimizers for a given probability distribution p(D), we introduce the average energy functional: U (v(x), T x; D ; D ))

³

:

³ 3 v(x;D );D p D dD ,

3 (v(x); D ))

S

(6)

The minimization of the functional with respect to v can be performed using Green’s function technique. Therefore the decomposition of the temperature field u(x;D) is introduced. u x; D u 0 x u1 x; D ,

(7)

0

where u is the solution of the reference problem and u1 denotes the temperature field related to a structure loaded by test polarization flux T. Determination of u0 is a standard task, which can be solved by a suitable numerical technique. Furthermore u1 can be solved by introducing the Green’s function of the polarization problem satisfying k 0 x 2G 0,f x, y G y x 0,

(8)

When the body is loaded by polarization flux T, we get

k 0 u x; D T x;D

0,

(9)

Solving the above equation will result in the temperature field u1 x; D ³

wG 0 x, y wy

:

:

(10)

w G x, y 2

u x; D

T y ; D dy ³ ' 0 x , y T y ; D d y ,

0

T y ; D dy ³ * x , y T y ; D d y . : wxwy By exploiting the optimality properties of the minimizing temperature field u(x;D), after some steps described in e.g. [9,10], the Hashin-Shtrikman functional in Eq. (5) is restored solely in terms of the polarization flux 1

³

0

:

W x;D arg

stat

T x;D T D

H T x;D ; D

where H T x; D ; D is the “condensed” energy functional and is defined as

(11)

204


H T x; D ;D

min U v x ,T x;D ; D 3 0 u 0 x

v x V

(12) 1 1 T x;D ª¬N ij x k 0 º¼ T x;D dx : : 2 with 3 0 denoting the thermal potential energy of the reference structure. The stochastic problem then can be solved by repeating the previous equation in probabilistic framework. Taking the ensemble average of the Eq. (7) and Eq. (10) yields to ³³ T x;D * x, y T y;D dydx ³

u x u 0 x ³ ' 0 x, y W :

y dy

(13)

where the expectation W is a solution of the stochastic variational problem

W x arg

H T x; D ;D p D dD .

³

min

T x ;D T D uS

S

(14)

From the above equations, it is now possible to link the unknown temperature field and flux to the test polarization flux T, which becomes the primary variable to be solved. Due to limited knowledge of detailed statistical characterization of phase distribution, the previous variational principle can only be solved approximately. The following form of polarization flux is assumed:

W x; D

¦W x F x;D , r

r

r

T x; D

(15)

¦T x F x;D . r

r

r

where Wr and r denote a realization-dependent polarization stresses related to the r-th phase and Fr is the characteristic function which specifies the material distribution in case of binary heterogeneous materials. Replacing the stochastic polarization flux into the "condensed" HashinShtrikman functional Eq. (14) and after some manipulations detailed in e.g. [11,10], leads to the following variational principle

W 1 ( x),W 2 ( x)

arg 2

min

T1 x;D ,T 2 x;D

2

¦¦ ³ r 1 s 1

3 u0 x

³W

: :

r

(x) S rs (x, y )W s (y )* 0 (x, y )dxdy

(16)

1 1 2 ¦ W r x cr x W r (x) ª¬kr k 0 º¼ dx 2 r 1 ³:

Discretization. The condition in Eq. (16) presents an infinite system to be fulfilled. Two steps need to be taken for converting the equation to a finite-dimensional system: (i) to represent the reference flux field and the Green’s function related quantities and (ii) discretization of the phase polarization flux. Next by using the standard Galerkin procedure the Eq. (16) is reduced to a finite-dimensional format. To that end, the following discretization of the phase polarization flux is introduced

W r ( x ) | NW (x)d rW ,

T r (x) | NW (x)d rT

W

W

(17) T

where N is the matrix of (possibly continuous) shape functions; d r and d r denote the degrees-offreedom (DOFs) of trial and true polarization flux, the latter related to the discrete Green’s function. Introducing the approximation Eq. (17) into the variational statement Eq. (16) and using the arbitrariness of d rT leads to the system of linear equations 2

KWr d rW ¦ KWrs d sW s 1

with the individual terms given by (r, s = 1, 2)

0

(18)


205

1 1 cr (x)NW (x)T ª¬ kr k 0 º¼ NW (x)dx, 2 ³:

KWr

KWrs

¦ ³

2

s 1

³

: :

(19)

Srs x, y NW (x)T * 0 x, y NW (x)dxdy

Finally, once the approximate values of phase polarization flux are available, the elementary statistics of the temperature field follows from the discretized form of Eq. (13) 2

u x u 0 (x) ¦ r 1

³ ' x, y c (y) N (y)dy d W

0

:

r

W r

(20)

Note that additional information such as conditional statistics or higher-order moments can be extracted from the polarization fields in post-processing steps similar to Eq. (20); see [12], [13] and [14] for more details. BEM approximation of reference problem Starting point for the boundary element formulation is the weighted residual (or weak) statement of the differential equation. For Laplace’s equation 2u 0, is given by

³

:

2u (x; D )v(x)d :

(21)

with the Green’s function as the test function v(x) and boundary condition

u x u x

on *u ,

q x

on * q .

q x

(22)

Next step is to transform the differential operator to the boundary terms. This transformation is done by applying Green’s theorem twice to the weighted residual statement. Substituting for the Green’s function i.e. the fundamental solution u* as the test function and integrating it twice by parts, taking the load point ȟ to the boundary and accounting for the jump of the left-hand side integral as is described in [15], yields the boundary integral equation c(ȟ )u 0 (ȟ )

wu (x) 0,f wu (x) 0,f G ȟ, x d *u x ³ G ȟ, x d * q x *q wn wn wG 0,f ȟ, x wG 0,f ȟ, x u ( x) d * q x ³ u (x)d * u x ³ *q *u wn wn

³

*u

(23)

wu 0 (x) . wn Since one component of the pair u , q is always specified on *u and * q z 0 , the above equation

and q 0 x

determines the unknown boundary data (i.e. u on * q and q on *u ). We introduce a decomposition of the Green’s function into the discretization-independent infinite-body part and the discretization-dependent boundary contribution: G 0 ȟ, x | G 0,f ȟ, x G 0,h ȟ, x

(24)

0,

where G refers to the two-point GF for the corresponding homogeneous material and the second term G0,h is bounded and refers to the additional term due to property gradation; Expression of ¨0 is derived following an analogous procedure. We exploit the infinite-bodyboundary split ' 0 ȟ, x | ' 0,f ȟ, x ' 0,h ȟ, x

(25)

206


The first part ' 0,f ȟ, x is directly obtained from equation (10) for the reference problem. 0 The BEM-based approach is completed by approximating * function, in particular we get

* 0 ȟ, x | * 0,f ȟ, x * 0,h ȟ, x , * 0,f ȟ, x *0,h ȟ, x

w' 0,f ȟ, x

(26)

wx w' 0,h ȟ, x

wx Some examples of these functions are illustrated in Figure 2 illustrated below. Finally note that the previous procedure can be directly translated to multi-dimensional and/or vectorial case; see [6] for more details.

(a)

0 *11 function

(c)

* 021 function

(b)

(d)

0 *12 function

* 022 function

0 Figure 2: (a)-(d) Examples of * function

Acknowledgments. The financial support of this work provided by grants No.GACR 103/07/0304 and CZE MSM 684 077 0001, 3 of the Grant Agency of the Czech Republic is gratefully acknowledged. References [1] J. Quintanilla and S. Torquato. Microstructure functions for a model of statistically inhomogeneous random media. Physical Review E, volume 55, (1997), p.1558. [2] J. R. Willis. Bounds and self-consistent estimates for the overall properties of anisotropic composites. Journal of the Mechanics and Physics of Solids, volume 25, (1977), p.185.


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[3] J. R. Willis. Variational and related methods for the overall properties of composites. In Advances in Applied Mechanics, volume 21, (1981), p. 2–74. [4] Z. Bittnar and J. Šejnoha. Numerical methods in structural mechanics. ASCE Press and Thomas Telford, Ltd, New York and London, 1996. [5] P. Procházka and J. Šejnoha. Behavior of composites on bounded domain. Boundary Elements Communications, volume 7, (1996), p.6–8. [6] P. Procházka and J. Šejnoha. A BEM formulation for homogenization of composites with randomly distributed fibers. Engineering Analysis with Boundary Elements, volume 27, (2003) p.137. [7] S. Ghosh, K. Lee, and S. Moorthy. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method. International Journal of Solids and Structures, volume 32, (1995), p.27. [8] A. Sutradhar and G.H. Paulino. The simple boundary element method for transient heat conduction in functionally graded materials. Computer Methods in Applied Mechanics and Engineering, volume 193, (2004), p.4511. [9] M. Grujicic and Y. Zhang. Determination of effective elastic properties of functionally graded materials using Voronoi cell finite element method. Materials Science and Engineering AStructural Materials Properties Microstructure and Processing, volume 251, (1998), p.64. [10] Y. Tomota, K. Kuroki, Mori T., and Tamura I. Tensile deformation of two-ductile-phase alloys. Journal of Material and science engineering, volume 21, (1976), p.85. [11] V. Sládek, J. Sládek, and Ch. Zhang. Domain element local integral equation method for potential problems in anisotropic and functionally graded materials. Computational Mechanics, volume 37, (2005), p.78. [12] M. Lombardo, J. Zeman, G. Falsone, and M. Šejnoha. Random field models of heterogeneous media via microstructural quantification. In preparation, (2008). [13] R. Luciano and J.R. Willis. FE analysis of stress and strain fields in finite random composite bodies. Journal of the Mechanics and Physics of Solids, volume 53, (2005), p.1505. [14] R. Luciano and J.R.Willis. Hashin-Shtrikman based FE analysis of the elastic behavior of finite random composite bodies. International Journal of Fracture, volume 137, (2006), p.261. [15] C.A. Brebbia, J.C.F. Telles, and L.C. Wrobel, editors. Boundary Element Techniques (Springer Verlag, Berlin Heidlberg New York Tokyo, 1984).


209

Scalar Wave Equation by the Domain Boundary Element Method with Non-homogeneous Initial Conditions J. A. M. Carrer1 and W. J. Mansur2 1 PPGMNE: Programa de Pós-Graduação em Métodos Numéricos em Engenharia, Universidade Federal do Paraná, Caixa Postal 19011, CEP 81531-990, Curitiba, PR, Brasil email: [email protected] 2

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

Keywords: scalar wave equation, initial conditions, D-BEM

Abstract. This work is concerned with the development of a D-BEM approach to solve 2D scalar wave propagation problems. The process of time-marching is accomplished with the use of the Houbolt method, as usual. Special attention is devoted to the development of a procedure that allows for the computation of the initial conditions contributions. Two examples are presented and the D-BEM results are compared with the corresponding analytical solutions. Introduction The development of BEM formulations for solving time-dependent problems is a very attractive area of research. For this reason, a great number of formulations appeared during the last years, enriching the BEM literature concerning this matter. The reader is referred to Beskos [1] for a very complete discussion concerning dynamic analysis by the BEM,. Bearing in mind that the integral equations can be obtained by means of a weighted residuals statement, the solution of time-dependent problems can be accomplished with the use of time-dependent fundamental solutions: in this case, BEM formulations are denominated TD-BEM (TD meaning time-domain) , e.g. Mansur [2], Dominguez [3]. Alternatively, one can use static fundamental solutions instead of time-dependent fundamental solutions. In this case, the BEM basic integral equation presents a domain integral with the kernel constituted by the fundamental solution multiplied by the second order time derivative of the potential, thus generating the socalled D-BEM formulations, D means domain, e.g. Carrer and Mansur [4], Hatzigeorgiou and Beskos [5]. It is important to note that the time variable does not appear explicitly in the integral equations. As a consequence, in order to perform the time-marching, the choice of an adequate approximation to the acceleration plays an important role: this approximation seems to be, as long as the authors know, the Houbolt method [6]. Note that unless responses only at early times are required, D- -BEM formulation is not appropriate for infinite domain analyses. In spite of this drawback, D-BEM formulations provide reliable results, justifying the interest in its development. In this work a D-BEM formulation for the solution of 2-D scalar wave equation problems with nonhomogeneous initial conditions is presented. Two examples are included at the end of the article, with the aim of validating the proposed formulation. D-BEM Formulation The starting D-BEM equation is written as follows:

´ ´ 1´ c([) u([,t) = µ u*([,X) p(X,t) d*(X) µ p*([,X) u(X,t) d*(X) c2 µ µ ¶*

µ ¶*

µ ¶:

.. u*([,X) u(X,t) d:(X)

(1)

In eq(1), u(X,t), that is generically referred to as potential function, can represent the transversal displacements of a membrane, and p(X,t) represents its normal derivative (support reaction in a membrane). Besides, c is the wave propagation velocity, t is the time, * is the boundary and : is the domain of the

210


problem. As usual in BEM formulations, X and [ represent the field and source points, respectively. The fundamental solution, u*([;X), is given by: u*([,X) =

1 §1· ln 2S © r ¹

(2)

where r is the distance between X and [, and p*([;X) is the normal derivative of the fundamental solution, computed according to: p*([,X) =

du*([,X) du*([,X) dr = dn dr dn

(3)

In order to solve eq(1), boundary and domain discretization must be carried out and an approximation to the acceleration must be adopted. In the present work, the discretization of the boundary is accomplished by linear boundary elements; the discretization of the domain, by triangular linear cells. The reader is referred to Mansur [2] for further details concerning this matter. Once the spatial discretization has been carried out, the resulting matrices can be assembled thus generating an enlarged system of equations, written below: b ª Hbb 0 º ° un+1 « db » ® d ¬ H I ¼ °¯ un+1

½° ª Gbb º ¾ = « db » °¿ ¬ G ¼

{

b pn+1

}

bb bd 1 ªM M º 2 « db dd » c ¬M M ¼

u..bn+1 ½ ® ..d ¾ ¯ un+1 ¿

(4)

In eq(4), in order to simplify the notation, the subscript (n + 1) represents the time tn+1 = (n + 1)'t, where 't is the selected time interval. The superscripts b and d correspond to the boundary and to the domain (internal points), respectively. In the sub-matrices, the first superscript corresponds to the position of the source point and the second superscript, to the position of the field point. The identity matrix is related to the coefficients c([) = 1 of the internal points. From eq(4), the boundary unknowns are the values of u and p at * (as usual in BEM formulations) and the values of u at :. It is important to mention that the assemblage of such an enlarged system of equations is necessary because the domain integral relates boundary values to domain values. This remark is confirmed by matrix Mbd in eq(4). Time-marching Scheme. The Houbolt method [6] is obtained by cubic Lagrange interpolation of u = u(t) from time t(n 2) = (n 2)'t to time t(n + 1) = (n + 1)'t. Exact differentiation with respect to time gives the approximations to the velocity and acceleration below: . 1 ª 11un+1 18un + 9un1 2un2º¼ un+1 = 6't ¬

(5)

.. 1 un+1 = 2 ª¬2un+1 5un + 4un1 un2º¼ 't

(6)

After substituting (6) in (4), the time-marching scheme can start:

ª §©(c't)2Hbb + 2Mbb·¹ « § 2 db ¬ ©(c't) H + 2Mbb·¹

º ° ubn+1 »® §I + 2Mdd· °¯ udn+1 © ¹¼ 2Mbd

b

b

b

bb bd ° 5un + 4un1 un2 1 ªM M º »® 2« db dd c ¬ M M ¼ ° 5ud + 4ud ud ¯ n n1 n2

½° ª(c't)2Gbb º ¾=« » °¿ ¬(c't)2Gdb ¼

½° ¾ °¿

Eq(7) can be represented in a concise manner as:

b n+1

{p } (7)


211

H un+1 = G pn+1 + gn

(8)

The contributions of the previous instants of time are stored in vector gn. After the boundary conditions are imposed, a final system of equations, that arises from eq(8), can be solved. It is important to point out that an adequate choice of the time-step plays a fundamental role in the analysis. A dimensionless variable, say E't, was adopted in order to provide a measure of the time-step 't, see Mansur [2] and Carrer and Mansur [4]: E't =

c't "

(9)

where " is the length of the smallest element used in the boundary discretization. Initial Conditions Contributions. In the Houbolt method, the computation of the velocity and accelerations at time tn+1 = (n + 1)'t requires the knowledge of the values of u from time tn 2 = (n 2)'t to time tn+1 = (n + 1)'t. At the beginning of the analysis, i. e., at the beginning of the time-marching process, n = 0 and, consequently, the values u 2 and u 1 must be computed appropriately in order to provide a good start of the analysis. . For the determination of u 1, uo is computed by employing the forward and the backward finite difference formulae at t = 0 and assuming that the finite difference expressions are equal, that is: u1 u0 u0 u1 . uo = = 't 't

(10)

Solving eq(10) for u1, one has: u1 = 2u0 u1

(11)

. One can also assume that uo can be computed by employing a central finite difference formula, which gives: u1 u1 . uo = 2't

(12)

Solving eq(12) for u 1: . u1 = 2't u0 + u1

(13)

. From (11) and (13) one has u 1 as a function of u0 and u0: . u1 = u0 't u0

(14)

. Now, for the determination of u 2, initially u1 is computed by employing the forward and the backward finite difference formulae at t = 't and assuming that the finite difference expressions are equal, that is: u0 u1 u1 u2 . u1 = = 't 't

(15)

212

Eds: R Abascal and M H Aliabadi Solving eq(15) for u 2: u2 = 2u1 u0

(16)

. Substituting (14) in (16) one obtains the value of u 2 as a function of u0 and u0: . u2 = u0 2't u0

(17)

Examples In the following examples, the BEM results are always compared with the corresponding analytical solution, computed following the procedures described by Kreyszig [7]. Square Membrane under Prescribed Initial Displacement over the Entire Domain. This example deals with a square membrane defined over the domain 0 d x d a, 0 d y d a, subjected to the initial conditions given by eq(18), see Fig.1: uo(x,y) = U x(x a)y(y a);

vo(x,y) = 0

(18)

Boundary discretization employed 80 elements and the domain, 800 cells, see Fig. 2.

Figure 1. Square membrane under displacement field over the entire domain.

initial

Figure 2. Square membrane: boundary domain discretization and selected points.

and

The analytical solution to this problem, for the general case of a rectangular membrane with dimensions a and b, according to Kreyszig [8], is: 64Ua2b2 u(x,y,t) = 3 3 6 mnS

f

f

¦ ¦ sin §©mSxa·¹ sin §©nSyb·¹ cos §© O

mn

Sct·¹;

with Omn =

m2 n2 + a2 b2

(19)

m=1n=1

The results corresponding to the displacement at point A(a/2,a/2) and to the support reaction at node B(a,a/2) are shown in Fig. 3 and Fig. 4, respectively. The analysis was carried out with the time interval computed from E't = 3/10. Good agreement between D-BEM and analytical results was already expected, as in this example time jumps do not appear.


u/U

p/U

analytical D-BEM

1.5

213

analytical D-BEM

0.4

1.0 0.2 0.5

0.0

0.0

-0.5 -0.2 -1.0

-1.5

-0.4 0.0

4.0

8.0

12.0

16.0 ct/a

0.0

Figure 3. Square membrane under initial displacement field over the entire domain: displacement at point A(a/2,a/2).

4.0

8.0

12.0

16.0 ct/a

Figure 4. Square membrane under initial displacement field over the entire domain: support reaction at node B(a,a/2).

Square Membrane under Prescribed Initial Velocity over Part of the Domain. The square membrane shown in Fig. 5, with the initial condition vo = V prescribed over :o, is analysed here. y u=0

a u=0

a/5

:R

u=0

a/5

u=0 a

x

Figure 5. Square membrane under prescribed initial velocity over part of the domain. The analytical solution to this problem is given by, see Mansur [2] and Kreyszig [7]: f u(x,y,t) =

2V S3

f

¦ ¦ mnO1

mn

mSx nSy sin § a · sin § a · Gmn © ¹ © ¹

(20)

m=1n=1

where: c Omn =2

m2 n2 3nS 2nS 3mS 2mS + ; and Gmn = §cos § 5 · cos § 5 ·· §cos § 5 · cos § 5 ·· sin(2SOmnt) © © ¹ © ¹¹ © © ¹ © ¹¹ a2 a2

(21)

This example was analysed previously with the use of the TD-BEM formulation, e.g. Mansur [2]. In the D-BEM formulation presented here, care should be taken when considering the initial conditions contributions. Because linear interpolation is adopted to the initial conditions in the internal cells, the use of double-nodes in the boundary of :o became necessary in order to avoid the spreading of the initial conditions to the cells in the neighbouring of :o. Aiming at providing a good representation of the time jumps in the support reaction response, a more refined mesh, constituted

214


of 160 elements and 3200 cells, not shown here, was employed. The results corresponding to the displacement at point A(a/2,a/2) and to the support reaction at node B(a,a/2), are depicted in Fig. 6 and Fig. 7, respectively. These results were obtained by adopting E't = 3/10. The D-BEM responses are in good agreement with the analytical ones, although it must be pointed out that the TD-BEM formulation provides a better representation of the time jumps. Nevertheless, the good responses furnished by the D-BEM formulation demonstrate its applicability to the solution of this kind of problems. u/V

p/V 1.2

analytical D-BEM

1.0

analytical D-BEM 0.8

0.5 0.4

0.0

0.0

-0.4 -0.5 -0.8

-1.0

-1.2 0.0

0.8

1.6

2.4

3.2

4.0

4.8 ct/a

Figure 6. Square membrane under prescribed initial velocity over part of the domain: displacement at point A(a/2,a/2).

0.0

0.8

1.6

2.4

3.2

4.0

4.8 ct/a

Figure 7. Square membrane under prescribed initial velocity over part of the domain: support reaction at node B(a,a/2).

Conclusions The D-BEM formulation is a very promising approach and, consequently, it is expected that some research work concerning its development will be done in the next years. Aiming at contributing with the development of the D-BEM formulation, this work is concerned with the solution of 2-D scalar wave equation problems with non-homogeneous initial conditions; in other words, once the Houbolt method was adopted to approximate the acceleration, this work is concerned with finding general expressions to the terms u 2 and u 1 that appear at the beginning of the time-marching. The results presented here demonstrate the validity of the expressions found, and also encourage the extension of the present work to elastodynamics, as well as to the DR-BEM formulation. References [1] D.E. Beskos, Boundary Elements in Dynamic Analysis: Part II (1986-1996), Applied Mechanics Reviews, 50, 149-197 (1997). [2] W.J. Mansur, A Time-stepping Technique to Solve Wave Propagation Problems Using the Boundary Element Method, Ph.D. Thesis, University of Southampton, England, (1983). [3] J. Dominguez, Boundary Elements in Dynamics, Computational Mechanics Publications, Southampton and Boston, (1993). [4] J.A.M. Carrer and W.J. Mansur, Alternative Time-Marching Schemes for Elastodynamic Analysis with the Domain Boundary Element Method Formulation, Computational Mechanics, 34, 387-399 (2004). [5] G.D. Hatzigeorgiou and D.E. Beskos, Dynamic Elastoplastic Analysis of 3-D Structures by the Domain/Boundary Element Method, Computer & Structures, 80, 339-347 (2002). [6] J.C. Houbolt, A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft, Journal of the Aeronautical Sciences, 17, 540-550 (1950). [7] E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, Inc., 8th edition, (1999).


215

Shear Deformation Effect in Plates Stiffened by Parallel Beams by BEM E.J.Sapountzakis1 and V.G. Mokos2 1,2

School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece

Key words: Stiffened plate, reinforced plate with beams, bending, nonuniform torsion, warping, ribbed plate, slab-and-beam structure, shear deformation, Reissner’s theory

Abstract. In this paper a general solution for the analysis of shear deformable stiffened plates subjected to an arbitrary loading is presented. The analysis of the plate is based on Reissner's theory, while the analysis of the beams is performed employing the linearized second order theory taking into account shear deformation effect. Six boundary value problems are formulated and solved using the Analog Equation Method (AEM), a BEM based method. The solution of the aforementioned plate and beam problems, which are nonlinearly coupled, is achieved using iterative numerical methods. The adopted model permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. 1. Introduction Structural plate systems stiffened by beams are widely used in buildings, bridges, ships, aircrafts and machines. Stiffening of the plate is used to increase its load carrying capacity and to prevent buckling especially in case of in-plane loading. Moreover, for cases wherein the plate or the beams are not very “thin” or the stiffeners are closely spaced, the error incurred from the ignorance of the effect of shear deformation may be substantial, while the accuracy of a classical analysis decreases and the truthfulness of the results is lost with growing plate or beam thickness. The extensive use of the aforementioned plate structures necessitates a rigorous analysis. In this paper a general solution for the static analysis of plates stiffened by arbitrarily placed parallel beams of arbitrary doubly symmetric cross section subjected to an arbitrary loading is presented taking into account shear deformation effect in both the plate and the beams. The employed structural model is an improved one of that presented by Sapountzakis and Mokos in [1], so that a nonuniform distribution of the interface transverse shear force and the nonuniform torsional response of the beams are taken into account. According to this model, the stiffening beams are isolated again from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. The utilization of two interface lines for each beam enables the nonuniform torsional response of the beams to be taken into account as the angle of twist is indirectly equated with the corresponding plate slope. The unknown distribution of the aforementioned integrated tractions is established by applying continuity conditions in all directions at the two interface lines. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second-order effects. The analysis of the plate is based on Reissner's theory, which may be considered as the standard thick plate theory with which all others are compared, while the analysis of the beams is performed employing the linearized second order theory taking into account shear deformation effect. Six boundary value problems are formulated and solved using the Analog Equation Method (AEM) [2], a BEM based method. The effectiveness, the range of applications of the proposed method and the influence of shear deformation effect are illustrated by working out numerical examples with great practical interest. 2. Statement of the problem Consider a plate of homogeneous, isotropic and linearly elastic material with modulus of elasticity

E, shear modulus G and Poisson ratio P , having constant thickness h p and occupying the two dimensional multiply connected region : of the x, y plane bounded by the boundary * . The plate is stiffened by a set

216


of i 1,2,...,I arbitrarily placed parallel beams of arbitrary doubly symmetric cross section of homogeneous, isotropic and linearly elastic material with modulus of elasticity Ebi , shear modulus Gbi and Poisson ratio

Pbi , which may have either internal or boundary point supports. For the sake of convenience the x axis is taken parallel to the beams. The stiffened plate is subjected to the lateral load g g x, y . For the analysis of the aforementioned problem a global coordinate system Oxy for the analysis of the plate and local coordinate ones Oi xi y i corresponding to the centroid axes of each beam are employed. : : Middle Surface of the Plate

g

f ji

f ji

qizj

1

2

qiyj x,u p

qixj *

O

hp

y,v p

E, P

qixj

z,w p b if

4

: if : Interface

q izj qiyj i

C : Center of gravity S i: Shear Center

The solution of the problem at hand is approached by an improved model of that proposed in [1]. According to this model, the stiffening beams are isolated again from the plate by sections in its lower outer surface, taking into account the arising tractions at the fictitious interfaces (Fig.1). Integration of these tractions along each half of the width of the i-th beam results in line forces per unit length in all directions in two interface lines, which are denoted by qixj , qiyj and qizj j 1,2 encountering in this way the nonuniform distribution of the interface transverse shear forces qiy , which in the aforementioned model

in [1] was ignored. The aforementioned integrated tractions result in the loading of the ith beam as well as the additional loading of the hbi plate. Their distribution is unknown and can be Ebi , Pbi established by imposing displacement continuity i i z ,wb conditions in all directions along the two interface lines, enabling in this way the b if : Width of Interface nonuniform torsional response of the beams to be taken into account, which in the Fig.1 Isolation of the beams from the plate. aforementioned model in [1] was also ignored. On the base of the above considerations the response of the plate and of the beams may be described by the following boundary value problems. xi ,ubi O { Ci { S i y i ,vbi i

a. For the plate. The plate undergoes transverse deflection and inplane deformation. Thus, for the transverse deflection, according to Reissner’s theory, if \ \ x, y is a stress function satisfying the equation

k\ 2\

0,

k

10 / h 2p

in ȍ

(1)

the average rotations I px , I py of the plate with respect to the axes y, x, respectively, taking into account the effect of shear deformation can be written as

I px

ww p wx

6 Q px 5Gh p

I py

ww p wy

6 Q py 5Gh p

(2a,b)

where the second term in the right hand side of these equations represents the additional angle of rotation due to shear deformation. Moreover, the stress resultants are given as


Q px

Q py M px

M py

M pxy

217

ww p ww p w\ w 2 wp N x N xy wx wx wy wy ww p ww p w\ w 2 D wp N y N yx wy wy wx wx

D

§ w2 wp D ¨ 2 w p P 1 ¨ wy 2 © § w2 wp D ¨ 2 w p P 1 ¨ wx 2 ©

D 1 P

w2 wp wxwy

(3a) (3b)

· 2 wQ gP px ¸ k 1 P ¸ k wx ¹ · 2 wQ gP py ¸ k 1 P ¸ k wy ¹

1§ wQ px wQ py ¨¨ k © wy wx

(3c)

(3d)

· ¸¸ ¹

(3e)

and the equation of equilibrium employing the linearized second order theory can be written as

§ w 2 wp w 2 wp w 2 wp 2N xy Ny D4 w p ¨ N x w xw y ¨ w x2 w y2 ©

· ¸ ¸ ¹

g

h 2p 2 P 2 g 10 1 P

§ 2 § w wi pj w wi pj h 2p 2 P 2 i w mipxj w mipyj ¦ ¨ ¦ ¨ qizj q zj qixj qiyj wy wx wx wy 10 1 P i 1 ¨ j 1 ¨© © I

where w p

w p x, y is the transverse deflection of the plate; D

· ¸G ij y y j ¸ ¹

in ȍ

· ¸ ¸ ¹

(4)

Eh p 3 / 12( 1 v 2 ) is its flexural rigidity;

N x x, y , N y N y x, y , N xy N xy x, y are the membrane forces per unit length of the plate cross section and G ( y yi ) is the Dirac’s delta function in the y direction. The governing differential equations (1), (4) are also subjected to the pertinent boundary conditions of the problem, which are given as Nx

D p1w p D p2Q pn D p3

E p1I pn E p2 M pn

where a pl , E pl , J pl ( l

E p3

J p1I pt J p2 M pnt

J p3

on ī

(5a,b,c)

1,2,3 ) are given functions specified on the boundary * , Q pn , M pn , M pnt are

the shear force, the bending moment and the twisting moment along the boundary, respectively and I pn , I pt are the average rotations of the plate with respect to the axes t, n, respectively. Since linearized plate bending theory is considered, the components of the membrane forces N x , N y ,

N xy are given as

Nx

w vp · § w up P C ¨¨ ¸ w w y ¸¹ x ©

Ny

Eh p / 1 P 2 ; u p

§ w up w vp · C ¨¨ P ¸ w y ¸¹ © wx

C

1 P § w u p w vp · ¨ ¸ 2 ¨© w y w x ¸¹

(6a,b,c)

v p x, y are the displacement components of the middle qxj , qiyj (i=1,2,…I), (j=1,2). These displacement surface of the plate arising from the line body forces i components are established by solving independently the plane stress problem, which is described by the following boundary value problem (Navier’s equations of equilibrium) where C

u p x, y , v p

N xy

218


· 1 P w ªw up w vp º 1 I § 2 i i ¨ ¦ qxjG j y yi ¸ 0 ¦ « » ¸ 1 P w x ¬ w x w y ¼ Gh p i 1 ©¨ j 1 ¹ · 1 P w ªw up w vp º 1 I § 2 i i 2v p ¦ ¨ ¦ q G y yi ¸¸ 0 « » 1 P w y ¬ w x w y ¼ Gh p i 1 ©¨ j 1 yj j ¹ G p1u pn G p2 N n G p3 H p1u pt H p2 Nt H p3 2u p

in which G

(7a)

in :

(7b)

on *

(8a,b)

E / 2( 1 Q ) is the shear modulus of the plate; N n , Nt and u pn , u pt are the boundary

membrane forces and displacements in the normal and tangential directions to the boundary, respectively; G pl , H pl ( l 1,2,3 ) are functions specified on the boundary * . b. For each (i-th) beam. Each beam undergoes transverse deflection with respect to z i and y i axes, axial deformation along xi axis and nonuniform angle of twist along xi axis. Thus, for the transverse deflection with respect to z i axis the equation of equilibrium employing the linearized second order theory and taking into account shear deformation effect can be written as [27]

§ Nbi Ebi I iy ¨ 1 ¨ Gbi Azi ©

· w 4 wi b ¸ ¸ i4 ¹ wx

Ebi I iy 2 § w 2 qizj ¦¨ Gbi Azi j 1 ¨© wxi2

2

§

¦ ¨¨ qizj qixj

i

i Nbj

wqixj 3 i

w 2 wbi i2

j 1©

w 3 wbi 3qixj i3 wx

wwbi wx

wx

wx

i · wmbyj ¸ wx wxi ¹¸ w 2 qixj wwbi · ¸ wxi2 wxi ¸¹

w 2 wbi i2

in Li , i 1,2,...,I

(9)

subjected to the following boundary condtions i a1zi wbi a2zi Rbz

where wbi

a3zi

i i E1ziTby E 2zi M by

E3zi

at the beam ends xi

0, Li

(10a,b)

i is its moment of wbi xi is the transverse deflection of the i-th beam with respect to z i axis; I by

i inertia with respect to y i axis; Nbj

are the axial forces at the xi centroid axis arising from the

i Nbj xi

i line body forces qixj ; alzi , Elzi ( l 1,2,3 ) are coefficients specified at the boundary of the i-th beam; Tby , i i Rbz , M by are the slope, the reaction and the bending moment at the i-th beam ends, respectively given as

i Tby

i M by

i 2 § Ebi Iby Ni ¨ 1 bj ¦ Gbi Azi j 1 ¨© Gbi Azi

· w 3 wi w wi b b ¸ ¸ w xi3 w xi ¹ i · 2 i 2 § N w w bj i Ebi I by ¦ ¨¨ 1 i i ¸¸ i2b Gb Az ¹ w x j 1©

Similarly, the vbi

vbi xi

boundary value problem

i Rbz

i Ebi I by

2

§

¦ ¨¨ 1

j 1©

i · 3 i i · 2 § Nbj i w wb ¸ w wb ¦ ¨ Nbj ¸ w xi ¸¹ Gbi Azi ¸¹ w xi3 j 1 ¨©

(11a,b,c)

transverse deflection with respect to y i axis must satisfy the following


i § 2 § Nbi ·¸ w 4 vbi i ¨ qiyj qixj wvb Nbj Ebi I zi ¨ 1 ¦ i i i4 i ¨ ¸ ¨ G A x x w w j 1 b y ¹ © © i i 2 § w 2 qi wqixj w 2 vbi w 3 vbi E I yj b z ¦¨ 3qixj 3 i i i2 i3 wx wxi wxi2 Gb Ay j 1 ¨© wx

i a1yi vbi a2yi Rby

a3yi

219

i · wmbzj ¸ wx wxi ¸¹ w 2 qixj wvbi · ¸ wxi2 wxi ¸¹

w 2 vbi i2

i i E1yiTbz E 2yi M bz

in Li , i 1,2,...,I

E3yi

at the beam ends xi

(12)

0, Li

(13a,b)

i where Ibz is the moment of inertia of the i-th beam with respect to y i axis; alyi , Elyi ( l 1,2,3 ) are i i i , Rby , M bz are the slope, the reaction and the bending moment at coefficients specified at its boundary; Tbz

the i-th beam ends, respectively given as i · 3 i i Nbj Ei I i 2 § ¸ w vb w vb b bz ¦ ¨ 1 i i i i i3 Gb Ay j 1 ¨© Gb Ay ¸¹ w x w xi i · 2 i 2 § Nbj w v i i M bz Ebi Ibz ¦ ¨¨ 1 i i ¸¸ i2b Gb Ay ¹ w x j 1©

i Tbz

i Rby

i Ebi Ibz

2

§

¦ ¨¨ 1

j 1©

i · 3 i i · 2 § Nbj i w vb ¸ w vb ¦ ¨ Nbj ¸ i i ¸ i3 ¨ Gb Ay ¹ w x w xi ¸¹ j 1©

(14a,b,c)

In eqns (10), (12)-(14), (15), (17)-(19) Gbi Aiy , Gbi Azi are the shear rigidities of the Timoshenko’s beam theory, where

Aiy

N iy Ai

1 aiy

Ai

Azi

N zi Ai

1 aiz

Ai

(15a,b)

are the shear areas with respect to y , z axes, respectively with N iy , N zi the shear correction factors, aiy , aiz the shear deformation coefficients and Ai the cross section area of the i-th stiffening beam. Since linearized beam bending theory is considered the axial deformation ubi of the beam arising from the arbitrarily distributed axial forces qixj (i=1,2,…I), (j=1,2) is described by solving independently the boundary value problem

Ebi Abi

w 2ubi wx

i2

2

¦ qixj

in Li , i 1,2,...,I

(16)

j 1

J 1xi ubi J 2xi Nbi J 3xi

at the beam ends xi

0, Li

(17)

where Nbi is the axial reaction at the i-th beam ends given as

Nbi

2

¦ Nbji j 1

Ebi Abi

w ubi w xi

Finally, the nonuniform angle of twist with respect to boundary value problem

(18)

i

x shear center axis has to satisfy the following

220

i Ebi I bw

Eds: R Abascal and M H Aliabadi i w 4Tbx

wx

i Gbi I bx

i4

i i a1xiTbx a2xi M bx

a3xi

i where Tbx

i Tbx xi

i w 2Tbx

wx

i2

2

in Li , i 1,2,...,I

i ¦ mbxj

(19)

j 1

E1xi

i wTbx

i E 2xi M bw

w xi

E3xi

at the beam ends xi

0, Li

(20a,b)

is the variable angle of twist of the i-th beam along the xi shear center axis;

i i Gbi Ebi / 2( 1 Pbi ) is its shear modulus; I bw , Ibx are the warping and torsion constants of the i-th beam cross section, respectively given as

i I bw

³Ai M S P

2

with M SP y i ,z i

dAi

§

³Ai ¨¨ y

i I bx

©

i

2

zi

2

yi

wM SP wz i

zi

wM SP · i ¸dA wy i ¸¹

(21a,b)

the primary warping function with respect to the shear center S of the Ai beam cross

section; alxi , Elxi ( l 1,2,3 ) are coefficients specified at the boundary of the i-th beam;

i wTbx

w xi

denotes the

i rate of change of the angle of twist and it can be regarded as the torsional curvature; M bx is the twisting i moment and M bw is the warping moment due to the torsional curvature at the boundary of the i-th beam. Eqns. (1), (4), (7a), (7b), (9), (12), (16), (19) constitute a set of eight coupled partial differential equations i including fourteen unknowns, namely \ , w p , u p , v p , wbi , vbi , ubi , Tbx , qix1 , qiy1 , qiz1 , qix2 , qiy2 , qiz2 .

Six additional equations are required, which result from the displacement continuity conditions in the directions of z i , xi and y i local axes along the two interface lines of each (i-th) plate – beam interface. These conditions can be expressed as In the direction of z i local axis: bif i wip1 wbi Tbx 4 bif i i i Tbx w p2 wb 4 In the direction of xi local axis: u ip1 ubi u ip2 ubi

h p w wip1 hbi w wbi bif 2 wx 2 w xi 4 i i i h p w w p2 hb w wb bif 2 wx 2 w xi 4

w vbi w xi w vbi wx

i

i

f 1 wTw xbxi

+ ISiP

i

f 2 wTw xbxi

+ ISiP

along interface line 1 ( f ji 1 )

(22a)


(22b)


(22c)


(22d)

along interface line 1 ( f i 1 ) j

(22e)

In the direction of y i local axis:

h p w wip1

hi i b T bx 2 wy 2 h p w wip2 hbi i i i v p2 vb T bx 2 wy 2

vip1 vbi

where


(22f)


u ipj

h p w wipj 2

wx

3 Q px 2 G

vipj

221

h p w wipj 2

wy

3 Q py 2 G

(23a,b)

while

§ · 3 i 3 i 2 i i · wqi E i I i § wqi ¨ E i I i w wb b y ¨ zj N i w wb 2qi w wb zj wwb ¸ mi ¸ b y bj xj byj ¸ wxi Gbi Azi ¨ wxi3 Gbi Azi ¨© wxi wxi3 wxi2 wxi wxi ¸¹ © ¹ i § · 3 i i i § wqi 3 i 2 i i · w q wvbi w w w w v E I v v v 1 ¨ i i yj yj i b b z ¨ b 2qi b b ¸ mi ¸ Eb I z Nbj xj bzj ¸ wxi Gbi Aiy ¨ wxi3 Gbi Aiy ©¨ wxi wxi3 wxi2 wxi wxi ¸¹ © ¹

i Tby

i Tbz

wwbi

1

(24a)

(24b)

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

and ISiP

section at the point of the j-th interface line of the i-th plate – beam interface f . It is worth here noting that the coupling of the aforementioned equations is nonlinear due to the terms including the unknown qixj and

qiyj interface forces.

3. Numerical Solution The numerical solution of the boundary value problems described by eqns (4-5a,b,c), (7a,b-8a,b), (9-10a,b), (12-13a,b), (16-17) and (19-20a,b) will be accomplished employing the Analog Equation Method [2]. This method is applied for the aforementioned problems as this is presented in [1]. 4. Numerical example

(Beam V)

1.00m

1.00m

1.00m

1.00m

(Beam IV)

ly=9.00m j=2

a

FR (free) (C)

(Beam III)

1.00m

(Beam II)

j=1

1.00m

0.20m

10cm

SS

SS (simply supported)

lpy=9.00m

(Beam I)

1.00m

concrete C35 40 ( Eb 3.35 u 107 kPa , Pb 0.2 ) I-section beams symmetrically placed (Fig.2) forming a bridge deck has been studied.

five

(C) (A)

1.00m

20kN / m2 and stiffened by

3.00m

1.00m

uniform loading g

g=20kPa

hp=0.20m

1.00m

A concrete C20 25 ( E 2.9 u 107 kPa , P 0.0 ) rectangular plate with dimensions a u b 18.0 u 9.0 m subjected to eccentric

x 2.00m

0.20m

a

FR (A) y

lpx=18.00m

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

(b)

The plate along its small edges is simply supported according to its transverse boundary conditions, clamped according to its inplane ones, while the other two edges are free according to both its transverse and inplane boundary conditions. The beams at their edges are also simply supported according to their bending

222


boundary conditions and clamped according to their axial and torsional ones. In Table 1 the obtained deflections of the stiffened plate at its center and at the middle of the free edges A and C (Fig. 2) and in Fig.3 the obtained axial forces Nbi of each stiffening beam are presented taking into account or ignoring shear deformation effect. Table 1. Deflections w p mm of the stiffened plate.

Including Inplane Forces with s.d. without s.d. 1.1094E+00 8.9581E-01 –9.3156E-02 –6.6141E-02 9.6674E-01 7.1705E-01

Center Middle of the free edge A Middle of the free edge C

Ignoring Inplane Forces with s.d. without s.d. 7.2096E+00 7.1152E+00 2.5232E-01 1.8170E-01 7.3523E+00 7.2615E+00

FR

Beam I Beam II Beam III Beam IV Beam V

SS

SS FR

1000

Axial Force N (kN) of beam

Axial Force N (kN) of beam

1000

0

-1000

-2000

500 0 -500 -1000 -1500 -2000

-3000 -9

-7

-5

-3

-1

1

3

Beam length (m)

5

7

-9

9

(a)

-7

-5

-3

-1

1

3

Beam length (m)

5

7

9

(b)

Fig. 3 Axial forces of the stiffening beams taking into account (a) or ignoring (b) shear deformation effect. 5. Concluding remarks a. The proposed model permits the study of a stiffened plate subjected to an arbitrary loading, while both the number and the placement of the nonintersecting stiffening beams are also arbitrary. b. The proposed model permits the evaluation of both the longitudinal and the transverse inplane shear forces at the interfaces between the plate and the beams (estimation of shear connectors). c. The evaluated lateral deflections of the plate - beams system are found to exhibit considerable discrepancy from those of other models, which neglect inplane and axial forces and deformations. d. In some cases, the influence of the shear deformation effect is remarkable and should not be neglected. References [1] E.J.Sapountzakis and V.G.Mokos, Analysis of Plates Stiffened by Parallel Beams, International Journal for Numerical Methods in Engineering, 70, 1209-1240 (2007). [2] 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).


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229

High-Order Spectral Elements for the Integral Equations of Time-Harmonic Maxwell Problems 1

E. Demaldent1,a , D. Levadoux1,b , G. Cohen2,c Electromagnetism and Radar Department, ONERA, chemin de la Hunière, 91761 Palaiseau, FRANCE 2 INRIA Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay, FRANCE. a [email protected],b [email protected],c [email protected]

Keywords: integral equations, electromagnetic scattering, high-order method.

Abstract. We present a novel, high-order, Method-of-Moment (MoM) with interpolatory vector functions, on quadrilateral patches. The main advantage of this method is that the Hdiv conforming property is enforced, meanwhile it can be interpreted as a point-based scheme. Our approach consists in a specific choice of Degrees Of Freedom (DOF), made in order to fulfill a fast integral evaluation. We apply this method to Field Integral Equations (FIE) to solve time-harmonic electromagnetic scattering problems. We also discuss the hybrid Integral Equation - Finite Element Method (IE-FEM), for simulations performed on coated scatterers. Introduction Field integral representation methods allow an electromagnetic field to be expressed in terms of its Cauchy data: the electric and magnetic current densities. The discretization is then performed on the boundary of the scatterer. A drawback of this method is that one has to solve a full matrix system. Even if fast evaluation techniques can be used to speed up an iterative solver, reducing the number of unknowns is a challenge which still remains. It is well known that high-order schemes provide a significant reduction of the number of unknowns and yield more accurate solutions. As the Method-of-Moment (MoM) with Galerkin testing results in optimal convergence for scattering problems, one should try to expand MoM low-order basis functions into their higher-order form ([1]). Unfortunately, this choice increases dramatically the matrix fill time, thus rendering classical high-order MoM useless for large scatterers. The problem lies in an overpriced evaluation of the double integral, for every term of the impedance matrix. By contrast, high-order point-based discretizations, such as the Nyström method ([2][3]), excel with their low precomputation time. However, these schemes are commonly not Hdiv conforming, and line charges may appear on patch’boundaries with the Maxwell problem. Various authors have developped high-order MoM which can be interpreted as point-based methods too ([4][5][6][7]). The main idea consists in choosing specific Degrees Of Freedom (DOF), defined by Lagrange polynomials whose roots are some quadrature points (spectral element method). It results in fast integral evaluation since both testing and basis functions cancel on these. Nevertheless, the frequently used Gauss quadrature rule does not lead to edge functions, thus enforcing the Hdiv property becomes problematical. The purpose of our talk is to present such a Galerkin/point-based scheme, with Hdiv conforming elements. The key component, introduced in [8][9][10] for volumic methods, lies in the use of GaussLobatto quadrature rules which lead to edge functions. Therefore, the Hdiv conforming property is easily enforced. In the following sections, we first adapt the spectral element method to the first family of Nédélec Hdiv conforming elements ([11]). Next, we compare this formulation to other spectral schemes (L2 functions, Gauss quadrature rules) to state on the advantages/drawbacks of our method. We then describe slight modifications which lead to a robust discretizaton scheme, with fast matrix fill time and accurate solutions, even on scatterers with sharp edges. Finally, we expand our approach to the

230


hybrid Integral Equation - Finite Element Method (IE-FEM), with high-order anisotropic hexahedral elements, for simulations performed on coated scatterers. Hdiv Conforming Spectral Elements In this section, we adapt the spectral element method to the first Nédélec family of Hdiv conforming elements, on quadrilateral patches. We start with a brief description of the variationnal formulations we use. Then, we present our DOF and their associated quadrature rules, which yield to a spectral formulation. Let (Einc , Hinc ) be the incident field which illuminates an arbitrarily-shaped object Ω of boundary Γ (in R3 ), and Γh a mesh of Γ. This field induces electric and magnetic current densities on Γ, which radiate and generate a scattered field (Escat , Hscat ) in R3 \ Ω. For simplicity, we restrict the study to perfectly conducting (PEC) scatterers so that the magnetic current density vanishes. Hence, our unknown is the electric current density J. Let Jh be the approximation of J by a linear combination of N basis functions Φn : Jh (r) =

N

In Φn (r) ∀r ∈ Γh , where In , n = 1, . . . , N are the unknown coefficients.

(1)

n=1

Variationnal formulations of Magnetic and Electric Field Integral Equations (MFIE (2) and EFIE (3), respectively) are given by: 1 Φm (r) · Φn (r)dΓh + Φm (r) · n × ∇ G(r, r ) × Φn (r )dΓh dΓh 2 Γh Γh Γh = Φm (r) · n × Hinc dΓh , (2) Γh ¯ r )Φm (r) · Φn (r )dΓh dΓh = −jk Φm (r) · Einc dΓh , (3) G(r, Γh

Γh

Γh

−jkr−r

e ¯ r ) = [Idk 2 + ∇∇]G(r, r ), k is the wave number, and n is the , G(r, 4πr − r ¯ yields a hypersingular integral. Applying exterior normal to Γ. The Dyadic Green’s function G Stokes’ theorem, the use of Hdiv conforming functions allows us to handle the double gradient operator, and results in a weakly singular integral. We make use of this process when r and r are close to each other (near interactions). Therefore, we deal with divergence terms of basis and testing functions. Both EFIE and MFIE have some resonance frequencies. A common way to avoid this problem is to solve the Combined Field Integral Equation (CFIE). The CFIE is obtained by a linear combination of the two previous ones, and doesn’t present any resonance frequencies (for real frequencies): CFIE = αEFIE + (1 − α)MFIE, with α ∈]0, 1[. where G(r, r ) =

e We set Γh = N i=1 {Ki }, where Ki are quadrilaterals (with straight or curved edges). Let K be the = Ki . DFi is the Jacobian matrix unit square and Fi the conforming mapping of R3 , such that Fi (K) and {e1 , e2 } the canonical basis of Fi and Ji = det(DFi ). (u1 , u2 ) denotes the 2D-coordinates on K, of R2 . We have s.t. : r = Fi (u1 , u2 ). (4) ∀r ∈ Γh , ∃Ki ⊂ Γh , ∃(u1 , u2 ) ∈ K, Let Φ ∈ {Φn }N n=1 be defined on K (for simplicity, indexes i are suppressed in following equations).


231

As we use Lagrange polynomials to expand our vector function Φ, it reads: Φ|K (r) =

1 1 , u2 ) , Φ(u 1 , u2 ) = L1 (u1 )L2 (u2 )eµ DF(u1 , u2 ) · Φ(u |J (u1 , u2 )|

µ = 1, 2 ,

(5)

1 DF is the local Hdiv conforming isomorphism. |J | We introduce two sets of Lagrange polynomials complete to pth order (1D), and a space on K: where L1,2 are the Lagrange polynomials, and

• the set of Lagrange polynomials whose roots are the p+1 Gauss quadrature points: Gp = {gip }p+1 i=1 , • the set of Lagrange polynomials whose roots are the p + 1 Gauss-Lobatto quadrature points: GLp = {glip }p+1 i=1 , p = (GLp+1 ⊗ Gp ) e1 ⊕ (Gp ⊗ GLp+1 ) e2 . • U Our approximate space for the first Nédélec family of Hdiv conforming elements is: 1 p . U p = Φ ∈ Hdiv (Γ) ; ∀K ∈ Γh , Φ|K ∈ DF U |J |

(6)

The Gauss-Lobatto quadrature rule gives us DOF on the edges, shared with the neighbour elements, so that the Hdiv conforming property is easily implemented (notice that functions in U 0 correspond to Rooftop functions). p+1 Now let’s define our specific quadrature rules: {(ξ, ω)gl,p q }q=1 denotes the set of points and weights g,p th for the p order Gauss-Lobatto quadrature rule ((ξ, ω)q for the Gauss one).We have:

glip (ξqgl,p ) = δiq , gip (ξqg,p ) = δiq , 1 ≤ i, q ≤ p + 1.

(7)

p+1 Thus, choosing {(ξ, ω)gl,p+1 } ⊗ {(ξ, ω)g,p ⊗ Gp polynomials, and {(ξ, ω)g,p q q } within GL q }⊗ gl,p+1 p p+1 } within G ⊗ GL , leads to a fast evaluation of the impedance matrix: each double {(ξ, ω)q integral estimation requires only one pair of quadrature points (far interactions), instead of (p + 2)4 with classical p-order MoM.

Establishing a Robust Discretization Scheme with Spectral Elements Enforcing the Hdiv conforming property, with spectral elements, involves the use of Gauss-Lobatto quadrature rules in direction of current’s flow. However, these rules are not as accurate as Gauss ones: the Gauss quadrature rule with p + 1 points is exact for polynomials of degree 2p + 1, whereas the Gauss-Lobatto one is exact for 2p − 1 (1D). Hence, replacing (p + 2)2 Gauss points (classical p-order MoM) with (p+2)⊗(p+1) Gauss-Lobatto and Gauss points, respectively, provides an increase of the error estimate with small values of p (p = 0, 1, 2). When simulations are performed on scatterers with sharp edges, or on coarse meshes (less than 5 points per wavelength), higher-order approximations (p = 3, 4, 5...) also suffer from the error increase. In order to relieve the constraint on quadrature choice, one should make use of L2 functions with DOF associated to Gauss points. This set is characterized by (8) and (9). Nevertheless, line

232


charges may appear on patch’boundaries, thus yielding accurate solutions on scatterers with sharp edges becomes problematical too. (Gp+1 ⊗ Gp ) e1 ⊕ (Gp ⊗ Gp+1 ) e2 V p = , 1 p 2 p V = Φ ∈ L (Γ) ; ∀K ∈ Γh , Φ|K ∈ , DF V |J |

(8)

g,p g,p+1 {(ξ, ω)g,p+1 } ⊗ {(ξ, ω)g,p } quadrature points. q q } and {(ξ, ω)q } ⊗ {(ξ, ω)q

(9)

To improve precision, we have to incorporate the quadrature set (9) with the Hdiv conforming set of functions (6). This combination is called semi-spectral, since roots of polynomials in GLp+1 do not match their associated quadrature points anymore. Of course, it involves a slower pre-computation time: (p + 2)2 points are required to estimate each term instead of one. Obviously, there is a trade-off between speed and accuracy in the choice of quadrature rules, with Hdiv conforming functions. We choose to combine these rules, with regards to the geometry discretization: • the semi-spectral form is restricted to functions on sharp edges, or on coarse elements, • the spectral form is employed with remaining functions. As a result, matrix fill time is reduced and accuracy is ensured. Consequently, the number of unknowns can be minimized (4-5 points per wavelength), as illustrate in Fig.1.

Figure 1: Cobra cavity, 5GHz, polar Phi (θinc = 110, φinc = 20): |Re(Jh )| (top) and bistatic Radar Cross Section (bottom).


233

The Hybrid Integral Equation - Finite Element Method A few years ago, M. Duruflé and G. Cohen have developed spectral methods for FEM, applied to the time-harmonic Maxwell problem ([9][10]). We’ve employed the same procedure to define our set of Hdiv-conforming functions (6), thus expanding our study to IE-FEM (with quadrilateral and hexahedral elements) do not present any difficulty. However, when the coating is thin compared to the wavelength, the use of high-order hexahedrons leads to an over-discretized thickness. To overcome this problem, we’ve implemented high-order anisotropic hexahedral elements: p-order in tangential directions and r-order in normal direction, with r ≤ p (Fig. 2).

Figure 2: coated scatterer.

As an example, we’ve run simulations on a sphere (1GHz), coated with a uniform dielectric coating: RΓ0 = 0.564, RΓ = 0.594, εr = 4.0. The coating acts like a waveguide, thus yielding accurate results with low-order methods is difficult, in comparison with the anisotropic spectral method (Fig. 3).

Figure 3: Comparison between classical low-order functions and high-order spectral methods on the coated sphere.

Conclusion The spectral element method has been expanded to the first Nédélec family of Hdiv conforming functions. We’ve discussed the drawbacks of the method, and slight modifications have been introduced to ensure precision and fast pre-computation time. Finally, our study has been extended to IE-FEM, with spectral anisotropic finite elements, for simulations performed on coated scatterers. Significant decrease of the number of unknowns has been observed through numerical examples.

234


References [1] R.D. Graglia, D.R. Wilton, A.F. Peterson, High Order Interpolatory Vector Bases for Computational Electromagnetics, IEEE Trans. Ant. Prop., vol. 45, no. 3, pp. 329-342, March 1997. [2] L.F. Canino, J.J. Ottusch, M.A. Stalzer, J.L. Visher, S.M. Wandzura, Numerical Solution of the Helmholtz Equation in 2D and 3D Using a High-Order Nyström Discretization, J. of Computational Physics,vol.146,pp 627-663,1998 [3] S.D. Gedney, A. Zhu, C.-C. Lu, Study of Mixed-Order Basis Functions for the Locally-Corrected Nyström Method, IEEE. Trans. Ant. Prop., vol. 52, no. 11, pp. 2996-3004, 2004. [4] S.D. Gedney, High-Order Method-of-Moments Solution of the Scattering by Three-Dimensional PEC Bodies Using Quadrature-Based Point Matching, Microwave and Optical Technology Letters, vol. 29, no. 5, June 2001. [5] G. Liu, S.D. Gedney, High-Order Moment Method Solution for the Scattering Analysis of Penetrable Bodies, Electromagnetics, vol. 23, pp. 331-345, 2003. [6] G. Kang, J.M. Song, W.C. Chew, K. Donepudi, J.M. Jin, A Novel Grid-Robust Higher-Order Vector Basis Function ofr the Method of Moment, IEEE Trans. Ant. Prop. vol.49 ,pp. 908-915, 2001. [7] K. Donepudi, J.M. Jin, W.C. Chew, A Grid-Robust Higher-Order Multilevel Fast Multipole Algorithm for Analysis of 3-D Scatterers, Electromagnetics, vol. 23, pp. 315-330, 2003. [8] G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations, Springer-Verlag, 2004. [9] M. Durufle, Intégration numérique et e´ léments finis d’ordre e´ levé appliqués aux e´ quations de Maxwell en régime harmonique, PhD thesis, Universite Paris Dauphine, 2006. [10] G. Cohen, M. Durufle, Non Spurious Spectral-Like Element Methods for Maxwell’s Equations, J. of Computational Mathematics, vol. 25, no. 3, pp282-304, May 2007. [11] J.C. Nedelec, Mixed finite elements in R3 , Numer. Math., vol 35, no. 3, pp. 315-341, 1980.


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Simulation of transient flows in highly heterogeneous media with the flux 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: [email protected]

Keywords: Green element method; boundary element; potential flow; highly heterogeneous media. Abstract In many geological formations, rapidly varying medium properties are encountered. In certain instances the variability of the medium properties is of significant orders of magnitude to cause numerical difficulties to conventional numerical schemes because of the sharp discontinuity in the gradient of the potential or pressure. The flux-based Green element method (GEM), that implements the singular boundary integral equations in an element-by-element manner, is used to simulate transient flow in heterogeneous media where changes of several orders of magnitude of the hydraulic conductivity are encountered. The formulation is flexible to accommodate either type 1 heterogeneity in which there is spatial distribution of the medium properties or type 2 in which abrupt changes of medium properties exist at certain boundaries. Only examples of type 2 heterogeneity are addressed in this paper; those of type 1 have earlier been addressed. The current formulation is tested on two numerical examples, and it is observed that high accuracy is achieved with coarse discretization of the domain. 1. Introduction Previous Green element formulations have been applied to transient potential flow problems in which the medium parameters depend on the primary or dependent variable (nonlinear potential problem) and/or have a functional distribution with respect to space (type 1 heterogeneous problem) [1-4]. This paper addresses type 2 heterogeneous problems in which the medium parameters exhibit sharp discontinuities at certain parts of the medium and also referred to as zoned problems. Boundary element simulations of steady flows in zoned media have previously been solved [5]. Lorinczi et al. [6] had solved the steady state form of this problem with a flux-based GEM which retains only three unknowns (the primary variable, and fluxes in x and y directions) at the internal nodes. Although problems solved were handled with rectangular elements, the authors demonstrated how triangular elements can be incorporated into the formulation. Here another flux based GEM formulation is used in solving the heterogeneous problem with sharp discontinuities in the medium parameter. It is a more direct approach that retains the primary variable and all normal directional fluxes at the internal nodes. It applies to both regular and irregular grids, as demonstrated in previous work [7]. Whereas it generates a lot more degrees of unknowns at the internal nodes, its high accuracy using coarse grids makes it quite attractive. The compatibility equation at an internal node shared by elements with different conductivity values is derived and used in resolving the closure problem at such nodes. Its derivation enables the evaluation of the integral of the normal fluxes over a closed circular contour centred at the internal node, taking into consideration that the fluxes are continuous across boundaries with sharp discontinuity in medium properties. Two examples of transient potential flows in zoned media are solved with the current formulation. 2. Flow equation and flux GEM The equation that governs transient potential flow in a heterogeneous media is given by ( K h )

wh f wt

(1)

242


where is the 2-D gradient operator, h is the potential, K is the hydraulic conductivity which is homogeneous in each zone but exhibits sharp discontinuities at the inter-zonal boundaries of the flow domain, t is time and f is a forcing term. The usual boundary conditions of the Dirichlet, Neumann and Cauchy types are accommodated at different sections of the external boundaries of the medium, while h is prescribed everywhere in the domain at the initial time. Because K is homogeneous in each zone, Eq. (1) can be treated as a diffusion equation of the form K 2 h

wh f wt

(2)

Compatibility conditions for h and its normal flux are admitted at the inter-zonal boundaries. Applying Green’s theorem to Eq. (2) gives the integral equation commonly encountered in boundary element circles. § wh · OKhi ³ KhG n G q ds ³³ G ¨ f ¸ dA 0 * / © wt ¹

(3)

where G ln(r ri ) is the fundamental solution to the Laplacian operator, q Kh n , the subscript i denotes the source point ri ( xi , y i ) and O is the nodal angle at ri that is obtained from a Cauchy integration of the Dirac delta function about the source point. The integral equation is implemented in a subdomain or element that is used to discretize / , while the quantities h and q are interpolated by Lagrange-type interpolation functions. The discrete form of Eq. (3) in a subdomain or element is § dh j · f j ¸¸ Rij h j Lij q j Wij ¨¨ © dt ¹

(4)

0

where the indices take values equivalent to the number of nodes in the element, and Rij Lij Wij

§ · K ¨ ³ N j G (r , ri ) n ds G ijO ¸ e ©* ¹ ³ N j G (r , ri ) ds,

(5)

*e

³³ G ( r , ri ) N j dA.

/e

N j is the interpolation function with respect to node j of the element. All the element integrations

in Eq. (5) are evaluated analytically for the 4-node linear and 8-node quadratic rectangular elements, and as well as the 3-node linear and 6-node quadratic elements. Achieving the exact integrations by GEM is largely due to the fact that the source node is always located in the element over which integration proceeds. Approximating the time derivative with a difference expression dh / dt | (h ( 2 ) h (1) ) / 't (where 't t 2 t1 ), using the generalized finite difference scheme with weighting factor T [0,1] and aggregating the discrete element equations for all the elements used in discretizing the domain produces a matrix equation of the form Vij u (j2 )

where

u (j2 )

(6)

Si { {h 2j , q 2j }T

T

{h j (t 2 ), q j (t 2 )}

is a mixed vector of unknowns. The matrix Eq. (6) is

readily solved to produce the solution for h and q at all nodal points at the current time t 2 . Thus


243

far, the formulation has proceeded in the usual way with previous flux GEM applications [1-3]. For nodes located at the interface of two zones with abrupt changes in hydraulic conductivity K , the compatibility relations for the potential and the flux are applied [5]. What is new in this work is the treatment given to internal nodes located where more than two zones of different K values meet. The equation that is applicable for such nodes is presented in the subsequent section. 4. Compatibility equation at internal nodes where more than 2 zones meet This situation is illustrated for a general case where 5 zones meet. The number of generated integral equations falls short by one the number of unknowns, resulting in a closure anomaly. This anomaly is resolved by introducing an additional equation for the normal directional fluxes meeting at the node. At the same time it should be recognized that the K values for the 5 zones are different. We introduce a circle C centred at the internal node P of infinitesimally small radius R that defines a surface S . It is observed that the normal directional fluxes at the segments emanating from the internal nodes are tangential to the circle (Figure 1a). The integral of the normal directional fluxes around this circle is given by ³ qds C

Where M q

K h n ³ qds C

M

(7)

¦ ³ q ds j 1 Cj

for the illustrated case is 5. Using the expression for the normal directional flux, Kwh / wn , Eq. (7) becomes M

¦ K j ³ j 1

Cj

wh ds wn

(8)

Noting that the normal n on the segments of the elements is also the tangential direction s on the circle C , Eq. (8) becomes ³ qds C

M

¦ K j ³ j 1

Cj

wh ds ws

M

¦ K j ³ dh j 1

(9)

Cj

Since h is continuous at the node, then ³ qds C

M

¦ K j ³ dh j 1

(10)

0

Cj

The numerical implementation of Eq. (10) follows: ³ qds C

M E j 1

R ¦ ³ q dE

(11)

0

j 1 Ej

where E is the included angle at the internal node and E M 1 it is factored out, so that Eq. (11) becomes M E j 1

M

j 1 Ej

j 1

¦ ³ q dE | ¦ q j 'E j

0

E1 . Since the radius R is a constant,

(12)

244


th ( q j q j 1 ) / 2 is taken as the average flux in the j zone within the included angle

Where q j 'E j

E j 1 E j . It is this additional Eq. (12) that is implemented in resolving the closure problem

at an internal node where more than two zones meet. As in previous flux GEM formulations, the complete solution information is made available for each element after solving the global matrix equation (6). That allows for the calculation of the solution at any point to involve only integrations on the element in which the point is located. The preclusion of other elements from the integration process greatly enhances the efficiency of the formulation. The direct calculation of the normal fluxes with an interpolation distribution that is of the same order as that of the primary variable enhances the formulation’s accuracy but gives rise to an escalation in the number of unknowns. The enhanced accuracy allows for the use of coarse grids (demonstrated in the next section) that compensates for the increased number of unknowns.

2

C K 1

3

1

3

q

P

1

S

3 1 3

K3 q4

q5 K4

K5

5

C

2

q

q4

4

K1

q

2

q

K3

2

q2

ns

3

K2

2

P

R

4

5

q5

4

K4

K5

q1

1

5

(a) (b) Figure 1: Establishing the compatibility equation at an internal node where more than 2 zones meet. 3. Simulated Examples The current flux GEM is applied to two examples. These two examples had been solved with a steady flux GEM formulation proposed by Lorinci et al. [6]. Here the first example is solved in a transient manner with the current formulation, while in both cases much fewer elements are used because of the high accuracy associated with the current formulation. 3.1 Example 1 In this example, discontinuities of hydraulic conductivity are of the order of magnitude of 104. The domain is rectangular [0,1] × [0,2] with four zones with hydraulic conductivity values: K 1 1 for the zone [0,0.5] × [0,1], K 2 10 2 for the zone [0.5,1] × [0,1], K 3 104 for the zone [0,0.5] × [1,2] and K 4 10 1 for the zone [0.5,1] × [1,2]. There is no forcing term, so f 0 . The example is simulated as a transient problem by considering the potential h to be everywhere zero initially, and thereafter suddenly raised to unit value at y 0 . The boundary conditions are: q (0, y, t ) q (1, y, t ) 0, 0 d y d 2 h( x,0, t ) 1, h( x,2, t ) 0, 0 d x d 1

(13)

A uniform time step 't 10 and the fully implicit scheme are adopted in the simulations. The domain is discretized into 8 × 16 linear rectangular elements. The simulation is carried out till steady state is achieved. This was achieved when the time is about 2000. The results at time of 400


245

is presented in Fig. 2a while the steady state solution is presented in Fig. 2b. The steady state result agrees with that of Lorinci et al. [6] which used 64 × 128 linear rectangular elements or 64 times more number of elements than employed in the current formulation. As expected in the results presented in Figs. 2a and 2b, the higher hydraulic conductivity at the lower half of the medium allows flow to quickly proceed into the medium so that steady state is more quickly attained than in the upper half with lower hydraulic conductivity. 2

K=10

2

-4

1.75

K=10 -4

1.75

K=10 -1

1.5

1.5

1.25

1.25

1

y

y

K=10 -1

0.75

1

0.75

-2

-2

K=10 0.5

K=10 0.5

0.25

0.25

K=1

K=1

0 0

0.25

0.5

0.75

1

x

0 0

0.25

0.5

0.75

1

x

(a) (b) Figure 2: Contour of potential in a 4-zoned medium with O(104) of hydraulic conductivity (a) at time 400, (b) steady state. 3.2 Example 2 In this example, two faults are in a rectangular domain [0,1] × [0,2]. Faults are common features in geological formations and they are shear planes when tectonic movements occur. Each fault is 0.75 long and 10-3 wide. The first fault is located at [0.25,1] × [0.4995,0.5005] and the second at [0,0.75] × [1.4995,1.5005]. The domain has a uniform hydraulic conductivity value of unity, while two cases are simulated with the faults having hydraulic conductivity values: K f 10 3 and Kf

10 4 . The boundary conditions are: q (0, y ) q (1, y ) 0, 0 d y d 2 h( x,0) 1, h( x,2) 0, 0 d x d 1

(14)

The steady state simulation of the problem is carried with 8 × 14 linear rectangular elements, and the results for the two cases presented in Figs. 3a and 3b. The steady state result for the first case agrees with that of Lorinci et al. [6] which used 16 × 34 linear rectangular elements. The constriction to the flow by the faults is observed by the steep gradient in the potential across the faults, and this more pronounced for the case with K f 10 4 . 4. Conclusion A flux GEM formulation has been applied to transient flow problem in heterogeneous media in which abrupt changes of several orders of magnitude of the hydraulic conductivity are encountered.

246


It calculates the potential and normal fluxes at every grid point using the same order of interpolating functions for both quantities. The closure anomaly at internal nodes where more than 2 zones of different hydraulic conductivity meet is resolved by an additional equation that is derived in this paper. It is derived in a general way such that it is independent of whether or not one uses regular or irregular grids. The accuracy of the formulation is evident by the coarse grid that is used to achieve accuracy comparable to those from other methods which use much finer grid. The coarseness of the discretization makes up for the large number of unknowns generated by the formulation due to the direct calculation of the normal fluxes at every grid point. 2

1.75

1.75

1.5

1.5

1.25

1.25

Faults

Faults

1

y

1

y

2

0.75

0.75

0.5

0.5

0.25

0.25

0

0 0

0.25

0.5

0.75

1

0

0.25

0.5

(a) Figure 3: Contour of potential for medium with 2 faults, (a) K f

0.75

1

x

x

(b) 10 3 , (b) K f

10 4 .

5. Acknowledgements Special thanks go to the National Research Foundation which provided the financial support for this research work. 6. References [1] Taigbenu A.E. The flux-correct Green element formulation for linear, nonlinear heat transport in [2] [3] [4] [5] [6] [7]

heterogeneous media, EABE, 32 (2008) 52-63. Taigbenu A.E., The flux-correct Green element method for linear and nonlinear potential flows, BeTeq VI, ed: A.P.Selvadurai, C.L. Tan & M.H. Aliabadi, EC Ltd UK (2005) 245-250. A.E. Taigbenu, Improvements in heat conduction calculations with flux-based Green element method, In 5th UK conference on BIM, ed. Ke Chen, University of Liverpool (2006) 190-199. R. A. Archer and R. N. Horne, Green element method and singularity programming for numerical well test analysis Engrg. Anal. with Boundary Elements, 26 (2002) 537-546. Liggett JA. & Liu PL-F. The Boundary Integral Equation Method for Porous Media Flow, George Allen & Unwin, UK (1983). P. Lorinczi, S.D. Harris and L. Elliot, Modelling of highly-heterogeneous media using a flux-vectorbased Green element method, EABE, 30 (2006) 818-833. Taigbenu, A.E. Implementing the flux Green element method on non-rectangular elements, In 6th UK conference on BIM, ed. J. Trevelyan, Durham University (2007) 203-212.


247

A procedure for 2-D elastostatic analysis of functionally graded materials by the Boundary Element-Free Method L. S. Miers1, J. C. F. Telles2 Programa de Engenharia Civil ± COPPE/UFRJ ± Cidade Universitária ± Centro de Tecnologia ± Bloco I, sala I 200 ± Ilha do Fundão, Rio de Janeiro / RJ ± Brasil 1

[email protected] [email protected]

2

KEYWORDS: functionally graded materials, meshless methods, boundary integral equations, nonhomogeneous media.

Abstract. A technique based on the coupling of standard boundary integral equations (BIE) and meshless interpolation schemes is proposed for 2-D elastostatic analysis applied to functionally graded materials (FGM). FGM are non-homogenous materials in which some properties (e.g. density, Young modulus, Poisson rate, etc.) vary as a function of spatial coordinates. Here, the numerical method adopted is the boundary element-free method (BEFM), which is a meshless technique based on BIE, such as the Local Boundary Integral Equation (LBIE) method and the Boundary Node Method (BNM), differing from them with respect to the integration domain and the approximation scheme. Introduction Conventional computational mesh-based methods, such as Finite Element Method (FEM) and Boundary Element Method (BEM), are not well suited for a certain number of problems involving remeshing processes, like large deformation analysis and crack propagation. To deal with this drawback, mesh-free counterparts of well-established numerical formulations have been developed. A series of so-called Meshless Local Petrov-Galerkin (MLPG) methods [1] has been developed in recent years for a great number of engineering applications. From the classical Boundary Integral Equation (BIE) formulation derives the Local Boundary Integral Equation (LBIE) method [2], which has been recently applied to potential problems with discontinuities [3], fracture mechanics problems [4] and non-homogeneous material analysis [5]. Other BIE-based meshless methods are the Boundary Node Method (BNM) [6] and the Boundary Element-Free Method (BEFM) [7]. Basically, they differ from LBIE in the chosen boundaries of integration, which in BNM and BEFM are the real boundary of the problem and in LBIE are circular (2-D) or spherical (3-D) sub-boundaries inside the global boundary, which characterizes LBIE as a domain method. In all three methods, the interpolation scheme used is based on least-squares approximations. In BNM and LBIE, the scheme used is the so-called Moving Least-Squares (MLS) [8] and in BEFM is used the Orthogonal Moving Least-Squares (OMLS), also called Improved Moving Least-Squares (IMLS) [7]. Both schemes are mesh-independent and the main difference between MLS and OMLS is the basis functions used in each one. In MLS the basis is composed by a set of complete linear independent monomial functions whereas in OMLS such functions are orthogonal polynomials. One of the recent sources of research for meshless applications is the analysis of Functionally Graded 0DWHULDOV )*0¶V >@ )*0¶V DUH QRQ-homogenous materials in which some properties (density, Young modulus, Poisson rate, etc.) vary as a function of spatial coordinates. This characteristic is very interesting for industry in general. Several FGM analyses have been proposed such as LBIE for heat conduction [10], elasticity [5] and viscoelasticity [11], among others. In this work a complete meshless technique based on BEFM for 2-D elastostatic analysis applied to )*0¶VLVSURSRVHG7KHPDWKHPDWLFDOEDVLVFRPSXWDWLRQDOLPSOHPHQWDWLRQDQGVRPHLOOXVWUDWLYHH[DPSOHV comparing results with other well-established techniques are presented. BEM formulation for non-homogeneous analysis For the solution of general elastostatic functionally graded material problems by the BE technique, an alternative boundary integral equation, not directly derived from the partial differential equation, can be obtained through weighted residual procedures or in the light of simple reciprocal statements as seen in [12]. Herein, only its final form is shown (body forces are omitted for simplicity),

248


cij (ȟ )u j (ȟ)

³

uij* (ȟ, x) p j (x)d*(x)

*

³

³

pij* (ȟ, x)u j (x)d*(x) H *jki (ȟ, x)V njk (x)d:(x)

*

(1)

:

where : represents the domain of the body, * its boundary and cij is the usual free coefficient found in elastic analysis. The following notation is used: x u ij* , p ij* and H *jki : displacement, traction and strain components at point x due to a unit concentrated ORDGDSSOLHGLQ³i´GLUHFWLRQDWSRLQW[ (homogeneous fundamental solution). x u j , p j and V njk : displacePHQWVWUDFWLRQVDQGILFWLWLRXV³FRUUHFWLQJ´VWUHVVHVWKDWFRPSHQVDWHIRUWKH variation of functional properties of the problem to be solved. It is worth mentioning that eq.(1) is meant to be valid for general 3D and 2D problems, provided subscripts are assumed to vary between 1-3 and 1-2 respectively. 6LQFHLQWKHFDVHRI)*0¶VWKHFRQVWLWXWLYHHTXDWLRQVDUHIXQFWLRQVRIWKHSRVLWLRQZLWKLQWKHERG\WKH computation of actual internal stresses is of great importance in the solution procedure. Therefore, the derivatives of eq.(1), written for [ : , can be combined to represent the internal stresses in the form

V ij (ȟ)

* ijk

³u *

* (ȟ, x) pk (x)d*(x) ³ pijk (ȟ, x)uk (x)d*(x) *

(2)

* n n ³ H ijkl (ȟ, x)V kl (x)d:(x) gij (V kl )

:

where the last two terms introduce the fictitious correcting stress influence. It should be mentioned that the derivatives of the domain integral of eq.(1) need careful evaluation and generate a Cauchy principal value integral (third integral on the right) together with a free term represented by the coefficient gij. Two-dimensional OMLS approximation scheme Just like in non-linear inelastic analysis, all boundary methods based on BIE (e.g., BEM [12], BNM [6], BEFM [7]), require that certain variable values must be determined within the domain of the problem for the complete solution (in certain applications, this computation can be carried out with the aid of reciprocity procedures). To deal with this, two kinds of interpolation are commonly used: domain interpolation, which is made based on the same dimension of the problem, and boundary interpolation, which is parameterized in one dimension less than the problem. The scheme adopted in this work to approximate boundary and domain variables is the so-called orthogonal moving least-squares (OMLS), also found in literature as improved moving least-squares (IMLS) [7]. This scheme is based on a standard moving least-squares method (MLS), differing only on the basis used. In MLS, the basis is composed by a complete set of linear independent monomial functions pT (xi ) 1, J (xi ), J (xi )2 , J (xi )m 1 or (3) j 1

>

p j (xi )

@

>J (xi )@

,

j 1, 2, ...., m 1

for boundary interpolation (m defines the order of the interpolation ± linear, quadratic, etc.; Jxi) is the 1-D parameterized co-ordinate of node i) and

p T (x i ) p T (x i )

>1,

>1,

xi ,

yi @

xi ,

yi , xi yi , xi2

for linear basis (m yi2

@

3)

for quadratic basis (m

(4)

6)

for domain interpolation, where xi is the real 2-D coordinates of node i,

yi @ (5) The basis used in OMLS is a complete set of weighted orthogonal polynomial functions that can be generated from the basis in eq.(3) by the Schmidt method as follows xi

>xi ,

p1 1

pi , pk p , k 1 pk , pk

i 1

pi

pi ¦ k

i

2, 3, ..., m

where ( pi, pj ) denotes the weighted inner product as follows [7]

(6)


249

n

p , p ¦ w p (x ) p (x ) i

j

k

i

k

j

k

(7)

k 1

If the set of polynomial functions is orthogonal then

0 if i z j ® ¯ Aij if i j

p , p i

j

(8)

The above interpolation leads to an approximation scheme that neither deals with fictitious quantities nor requires further post-processing of solutions, which are inconveniences of the MLS. This procedure is the same for boundary and domain interpolation. The OMLS approximation has the following form for displacements

ui (x)

ĭ ( x) u i

ui (x) I j (x)ui (x j ),

j 1, 2, ..., n

(9)

where ui is the vector containing the nodal values of displacement in i direction, n is the number of nodes that contribute to the interpolation at node i (i.e., the number of nodes at the domain of definition of node i) and ) is the shape function of OMLS. The exact same procedure is followed for traction interpolation. (10) ĭ(x) pT (x)A1 (x)B(x) where

A ( x) B ( x)

P T W ( x) P P T W ( x) T

P

ª p (x1 ) º « T » «p (x 2 )»; « ... » « T » «¬p (x n )»¼

W ( x)

0 ... 0 º ª w1 (x) « 0 » ( ) w x 2 « » « ... » ... « » 0 ( ) w x n ¬ ¼

(11)

wi(x) is the weight function, here chosen to be a Gaussian distribution function, as follows

wi (x)

e

§d · ¨¨ i ¸¸ © ci ¹

2k

e §r ¨¨ i © ci

§r · ¨¨ i ¸¸ © ci ¹ · ¸¸ ¹

2k

2k

for 0 d di d ri or

(12)

1 e wi (x) 0 for di t ri di = ||J(x) ± J(xi)||, ci is a constant that controls the shape of wi, ri is the size of the support of wi associated with xi (see Fig.1 for boundary interpolation and Fig.2 for domain interpolation) and k is a parameter here chosen as 1. There are many other functions suitable for use in MLS approximation, like cubic and quadric spline functions [1,13], but for many applications found in the literature, the best results were obtained with the Gaussian function. The domain of definition of a node i is composed by all nodes that contain it inside their supports.

Figure 1: Support of node i for boundary interpolation

250


Figure 2: Support and domain of definition of node i for domain interpolation Because of the basis orthogonality, matrix A(x) is diagonal, being naturally well conditioned and very easy to invert. In MLS, matrix A(x) does not have this property and sometimes its conditioning is found poor [7]. The normal strain along the boundary, H ** , can be calculated as follows

H ** (x)

w ĭ ( x) u* wJ (x)

(13)

where u * contains the value of the displacements in the direction of the boundary and the derivative term is

w ĭ ( x) wJ (x)

wp T (x) 1 wA 1 (x) w B ( x) A ( x ) B ( x) p T ( x ) B (x) p T (x) A 1 (x) wJ (x) wJ (x) wJ (x)

(14)

in which

wA 1 (x) wJ (x)

A 1 (x)

wA (x) 1 A ( x) wJ (x)

(15)

Spatial discretization A node cloud is spread over the boundary *and domain : of the problem. The boundary is divided in sub-boundaries *n, n «N (N is the total number of sub-boundaries (see Fig.3)), each connecting two neighbour boundary nodes. It is important to mention that the OMLS shape functions do not depend on them; their only purpose is to define the boundary geometry for numerical integration.

Figure 3: Global boundary * and sub-boundaries *n Since the interpolation of the domain variables over : uses a mesh-independent scheme, no internal cells are needed either to interpolate or to integrate over :. Also, stresses at boundary nodes are calculated employing the interpolated displacements and tractions over each boundary part as seen in [12], differing


251

only in the form of determination of the normal strains along the boundary, which is here carried out according to eq.(13). Substituting the displacements, tractions and fictitious stresses by their respective interpolated expressions in Eqs.(1) and (2) lead to n*

³ u (ȟ, x)¦I (x) * ij

cij (ȟ)u j (ȟ)

I

I 1

*

n*

*

p j (x I )d*(x) ³ pij* (ȟ, x)¦II (x) * u j (x I )d*(x) *

I 1

n:

(16)

* n ³ H jki (ȟ, x)¦IJ (x) :V jk (x J )d:(x) J 1

:

V ij (ȟ)

* ijk

³u *

n*

n*

* (ȟ, x)¦II (x) * pk (x I )d*(x) ³ pijk (ȟ, x)¦II (x) * uk (x I )d*(x) I 1

*

I 1

n:

(17)

* n n ³ H ijkl (ȟ, x)¦IJ (x) :V kl (x J )d:(x) gij (V kl )

:

J 1

where n* and n: indicate the number of nodes in the domain of definition of [ respectively, for boundary and domain interpolation and ( )|*and( )|: are the boundary and domain OMLS shape functions. The matrix forms of eqs.(16) and (17) are, respectively (18) Hu Gp Qın and (19) ı G' p H'u Q*ın where matrices H, +¶, Q, etc. are classical boundary element matrices. Note that matrix Q* also includes the contributions of the free coefficient gij. After the application of the displacement and traction boundary conditions, eqs.(18) and (19) can be written as (20) Ay f Qın and (21) ı A' y f' Q*ın Equation (20) can then be solved for the boundary unknowns included in vector y (22) Kın m where m represents the homogeneous material solution to the boundary problem. Substituting (22) in (21)

y

and rearranging,

ı

Sı n n

(23)

in which vector n represents the solution in terms of stresses for the homogeneous medium and

K S

A 1Q Q* A' K

m A 1f n f' A' m Non-homogeneous material modelling Consider that the Young modulus of a certain functionally graded material [9] varies as a function of spatial coordinates as (24) E (x) E EÖ (x) where E is a constant and EÖ (x) is the part that varies with the spatial coordinates. Hence, the real stresses can be written as (25) ı( E (x)) ı l ( E ) ın ( EÖ (x))

252


where Vl LV WKH VWDQGDUG RU ³UHIHUHQFH´ VWUHVV REWDLQHG Fonsidering only E and Vn is the correcting part calculated with EÖ (x) . If H contains the real strains, then

İ

D( E )ı l ( E )

(26)

where D(E ) is the elastic tensor that relates the real strains with the reference stresses. The relation between the real stresses and strains is as follows

ı( E (x)) C( E (x))İ

(27)

where C( E (x)) is the space varying operator calculated with E (x) . Substituting eq.(26) in (27) leads to

ı( E (x)) C( E (x))D( E )ı l ( E )

(28)

and considering Eq.(25), the following relation arises

ın ( EÖ (x))

I C(E(x))D( E ) ıl ( E )

(29) which is the relation between the correcting stresses and the reference or standard stresses involved in the solution process. The reference stresses can be computed if eq.(23) is modified as follows (30) ı l S I ın n and substituting eq.(29) in (30) leads to (31) ıl S I I C( E (x))D( E ) ıl ( E ) n Hence, rearranging (32) Bıl (E ) n

where matrix B is inherently well-conditioned, due to its very nature, as seen below

I S I I C( E(x))D( E )

(33) After solving the system of equations defined in eq.(32), the real stresses can be calculated using eq.(28) and the strains using eq.(26).

B

Examples Elastic strip. The first example is defined as the elastic strip depicted in Fig.4. It was previously analyzed in [11] and is subjected to a uniform unitary load on one edge and presents x-displacements restricted on another.

Figure 4: Example 1: Elastic catilevered strip in tension. In this example Q = 0.0 and the Young modulus varies as follows

E (x)

E eJx Here, E = 1.0 and J = 0.2 in the first analysis and J = 0.4 in the second. The node cloud used has 52

boundary nodes and 57 domain nodes. The comparison of the displacements ux of the nodes lying originally in x = 3.0 (the loading edge) obtained with this technique and the one presented graphically in [11] is shown in Table 1, in which the difference in percentage is computed with reference to the LBIE results. Table 1: Displacement ux at the loaded edge LBIE BEFM variation J values 0.2 2.251 2.255 0.17% 0.4 1.738 1.746 0.40%


253

Square plate. Consider the square plate under uniform unitary loading applied over the horizontal edges, as seen in Fig.5, with Q = 0.35 and Young modulus defined as follows ( E 1.23 u 10 5 ):

E (x) 1.23 u 10 5 1 0.5 x2 2 The node cloud used in this problem was composed by 44 boundary nodes and 81 domain nodes.

Figure 5: Example 2: Square plate in tension. The results obtained by BEFM technique are compared with the presented in [5]. Three results are analyzed: u1 and u2 displacements along the right vertical edge (Figs.6a and 6b) and V22 stress along the superior horizontal edge (Fig.6c).

Figure 6: (a) u1 displacements; (b) u2 displacements; (c) V22 stress Conclusions In the present work, a complete meshless technique based on BEFM for 2-D elastostatic analysis DSSOLHG WR )*0¶V LV SURSRVHG %HFDXVH RI WKHLU WKHUPDO DQG PHFKDQLFDO FKDUDFWHULVWLFV )*0¶V DUH YHU\ interesting for industry and the research on their applications is far from ending since new applications are continuously being developed for a number of engineering applications. As indicated by the results, the technique described is seen to be suitable for non-homogeneous media analyses. In addition, the development of new FGM applications with BEFM, as well as using other approaches like LBIE, for inelastic material behaviour, like plasticity, is currently under way and will be the object of future publications. References S.N.Atluri and S.Shen CMES - Computer Modelling in Engineering and Sciences, 3(1), 11-51 (2001). S.N.Atluri, J.Sladek, V.Sladek and T.Zhu Computational Mechanics, 25(2), 180-198 (2000). L.S.Miers and J.C.F.Telles Structructural Integrity and Durability, 1(3), 225-232, (2005). L.S.Miers and J.C.F.Telles CMES - Computer Modelling in Engineering and Sciences, 14(3), 161-169 (2006). [5] J.Sladek, V.Sladek and S.N.Atluri Computational Mechanics, 24(6), 456-462 (2000). [6] V.S.Kothnur, S.Mukherjee and Y.X.Mukherjee International Journal of Solids and Structures, 36(8), 1129-1147 (1998).

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

254


[7] K.M.Liew, Y.Cheng and S.Kitipornchai International Journal for Numerical Methods in Engineering, 65(8), 1310-1332 (2006). [8] P.Lancaster and K.Salkauskas Mathematics of Computation, 37(155), 141-158 (1981). [9] S.Suresh and A.Mortensen Fundamentals of Functionally Graded Materials (1998). [10] J.Sladek, V.Sladek and S.N.Atluri CMES - Computer Modelling in Engineering and Sciences, 6(3), 309-318 (2004). [11] J.Sladek, V.Sladek Ch.Zhang and M.Schanz Computational Mechanics, 37(3), 279-289 (2006). [12] J.C.F.Telles Boundary element method applied to inelastic problems: Lecture Notes in Engineering. Spriger-Verlag (1983). [13] S.N.Atluri and T.Zhu Computational Mechanics, 22(2), 117-127 (1998).


255

A comparative study of two time-domain BEM for 2D dynamic crack problems in anisotropic solids Michael Wünsche1,2, Chuanzeng Zhang2, Felipe García-Sánchez3 and Andres Sáez1 1

Departamento de Mecánica de Medios Continuos, Teoría de Estructuras e I. del Terreno, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain 2

3

Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany

Departamento de Ingeniería Civil, de Materiales y Fabricación, Universidad de Málaga, Plaza El Ejido s/n, 29013 Málaga, Spain

Keywords: Hypersingular time-domain BEM, anisotropic elastic solids, dynamic stress intensity factors, Galerkin-method, collocation method, convolution quadrature

Abstract. This paper presents two time-domain boundary element methods (TDBEM) for dynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic and linear elastic solids subjected to an impact loading. A combination of the classical displacement boundary integral equations (BIEs) and the hypersingular traction BIEs is applied. The spatial discretization is performed by a Galerkin-method in both cases. A collocation method is adopted for the temporal discretization in the first TDBEM, while the convolution quadrature of Lubich is implemented for the temporal discretization in the second TDBEM. An explicit time-stepping scheme is developed to compute the unknown boundary data and the crack-openingdisplacements numerically.

Introduction This paper presents a transient dynamic crack analysis for 2D, homogeneous, generally anisotropic and linear elastic solids. Stationary cracks in both infinite and finite solids under impact loading are considered. Two different time-domain BEM are developed and compared. A combination of the strongly singular displacement BIEs and the hypersingular traction BIEs is applied in both TDBEM. On the external boundary of the cracked solid the strongly singular displacement BIEs are used, while on the crack-faces the hypersingular traction BIEs are implemented. In the first TDBEM, time-domain elastodynamic fundamental solutions for anisotropic solids derived by Wang and Achenbach [4] via Radon transform are implemented, while the Laplace-transformed elastodynamic fundamental solutions for anisotropic solids given by Wang and Zhang [5] are used in the second TDBEM. To solve the time-domain BIEs numerically, an explicit time-stepping scheme is developed. The first TDBEM uses a collocation method for the temporal discretization. By using linear temporal shape-functions, time integrations in the system matrices can be carried out analytically. The second TDBEM uses the convolution quadrature of Lubich [2,3] for the temporal discretization. The spatial discretization is performed by the Galerkin-method in both TDBEM. Strongly singular and hypersingular boundary integrals are computed by special analytical and numerical techniques. To describe the local behavior of the crack-opening-displacements (CODs) at the crack-tips properly, square-root crack-tip shape-functions are applied for elements on the crack-faces near the cracktips, while linear shape-functions are chosen for all other boundary elements. The arising line-integrals in the elastodynamic fundamental solutions over a unit circle are computed numerically by standard Gaussian quadrature. Several numerical examples for computing transient elastodynamic stress intensity factors (SIFs) are presented and discussed to verify and compare the accuracy, the stability and the efficiency of the two different time-domain BEM.

Problem formulation and time-domain BIEs Let us consider a homogeneous, anisotropic and linear elastic solid with a crack of arbitrary shape. In the absence of body forces, the cracked solid satisfies the equations of motion

256


V iD , D ( x, t )

Uui (x, t ) ,

(1)

Hooke’s law

ViD (x, t ) CiDjE u j,E (x, t ) ,

(2)

u i ( x, t ) u i ( x, t ) 0 for t d 0 , t i (x, t ) 0 for t d 0 ,

(3) (4)

the initial conditions

and the boundary conditions t i ( x, t ) t i ( x, t ) , x *t , (5) (6) u i ( x, t ) u i (x, t ) , x *u . Here, ui, ıiD and ti=ıiDeD represent the displacement, the stress and the traction components, eD is the outward normal vector, ȡ is the mass density, CiDjE is the elasticity tensor, īc± denotes the upper and the lower crack-faces, īt and īu stand for the external boundaries where the tractions ti and the displacements ui are prescribed. A comma after a quantity represents spatial derivatives while a dot over a quantity denotes time differentiation. Greek indices take the values 1 and 2, while Latin indices take the values 1, 2 and 3. Unless otherwise stated, the conventional summation rule over repeated indices is implied. In the sense of the weighted residual, the time-domain displacement Galerkin-BIEs for a cracked solid can be written as

³ \(x) u (x, t ) d* ³ \(x) ³ >u j

x

*b

*b

G ij ( x, y , t ) t i ( y , t )

@

t ijG (x, y , t ) u i ( y, t ) d*y d*x

*b

(7)

\(x) t ijG (x, y , t ) 'u i (y , t ) d*y d*x ,

³

³

*b

*

c

where uijG(x,y,t) und tijG(x,y,t) are the elastodynamic displacement and traction fundamental solutions, ȥ(x) is the test function, ǻui are the crack-opening-displacements, Ƚb=Ƚu+Ƚt,, and an asterisk denotes the Riemann convolution which is defined by t

g ( x, t ) h ( x, t )

³ g(x, t W)h(x, W) dW .

(8)

0

The time-domain traction Galerkin-BIEs can be obtained by substituting eq (7) into Hooke’s law (2), taking the limit process xĺȽc± and considering the boundary conditions (5)-(6)

³ \(x) t (x, t) d* ³ \(x) ³ >v j

x

*

*

c

c

³ \ ( x) ³ w

* c

G ij ( x, y , t ) t i ( y , t )

@

w ijG (x, y , t ) u i (y , t ) d*yd*x

*b

G ij ( x, y , t ) 'u i ( y , t )

d*yd*x .

(9)

* c

where vijG(x,y,t) and wijG(x,y,t) are the traction and the higher-order traction fundamental solutions, which are defined by vijG (x, y , t ) CiDpEeD (y )u Gpj,E (x, y , t ) t Gji (x, y , t ) , (10) (11) w G ( x, y , t ) C e ( x ) C e ( y ) u G ( x , y , t ) . ij

iJpG J

jDkE D

pk , GE

The displacement BIEs (7) are strongly singular, while the traction BIEs (9) are hypersingular. The elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids derived by Wang and Achenbach [4] are implemented in the first TDBEM, while the Laplace-transformed elastodynamic fundamental solutions for anisotropic solids derived by Wang and Zhang [5] are used in the second TDBEM. It should be noted here that both elastodynamic fundamental solutions cannot be given in closedforms in contrast to that for homogeneous, isotropic and linear elastic solids. In 2D case, they can be represented by line-integrals over a unit circle. Both the time-domain and the Laplace-transformed fundamental solutions can be divided into singular static and regular dynamic parts. The singular static parts of the elastodynamic fundamental solutions can be reduced to an explicit expression and correspond to the elastostatic fundamental solutions. The regular dynamic parts can be given only as line-integrals over a unit circle.


257

Numerical solution of the time-domain BIEs To solve the strongly singular displacement BIEs (7) and the hypersingular traction BIEs (9), two different numerical solution methods are implemented. In both TDBEM, the Galekin-method is applied for the spatial discretization. At the crack-tips, special crack-tip elements are used to describe the local behavior of the CODs properly. This ensures an accurate and direct calculation of the dynamic stress intensity factors from the numerically computed CODs. Strongly singular and hypersingular boundary integrals are computed analytically and numerically by special techniques. The temporal discretization is performed by a collocation method in the first TDBEM. By using linear temporal shape-functions, time integrations can be performed analytically. In the second TDBEM, the convolution quadrature of Lubich [2,3] is implemented for the temporal discretization, which requires the Laplace-transformed instead of the time-domain fundamental solutions. This special feature distinguishes the second TDBEM from the first TDBEM and the classical time-domain BEM. The line-integrals over the unit circle arising in the elastodynamic fundamental solutions are computed numerically by using standard Gaussian quadrature formula. The essential features of the two TDBEM are summarized in Table 1. Variants

Spatial discretization

Temporal discretization

TDBEM 1

Galerkin-method

Collocation method

TDBEM 2

Galerkin-method

Convolution quadrature

Fundamental solutions Time-domain fundamental solutions Laplace-transformed fundamental solutions

Table 1: Two different TDBEM After temporal and spatial discretiozations and invoking the initial conditions (3) and (4), an explicit timestepping scheme can be obtained as K 1

A1 u K k

k

B1 t K

¦ (B

K k 1 k

t A K k 1 u k ) ,

(12)

k 1 K

where A and B are the system matrices, u is the vector containing the boundary displacements and the CODs, and tK is the traction vector for the external boundary and the crack-faces. By considering the boundary conditions (5) and (6), eq (12) can be rearranged as K 1 ª º x K (C1 ) 1 «D1 y K (B K k 1 t k A K k 1 u k )» , (13) k 1 ¬ ¼ K K in which x represents the vector with the unknown boundary data, while y denotes the vector with the prescribed boundary data. Eq. (13) is an explicit time-stepping scheme for computing the unknown boundary data including the CODs time-step by time-step.

¦

Numerical results In the explicit time-stepping scheme, the accuracy and the stability of the TDBEM are dependent on the used time-step. The following relation between the temporal and the spatial discretization is introduced to asses the stability of the present TDBEM c max 't C 22 / U , cT C66 / U . O , cL (14) le In eq (14), le represents the element-length, cmax ist the larger one of the wave velocities cL and cT, and ȡ is the mass density. For isotropic solids it is often reported in the literature that good results can be obtained by using Ȝ§1. To compare the two time-domain BEM, several numerical examples are investigated. For convenience, the following normalized dynamic SIFs are introduced (15) K*I ( t ) K I ( t ) / V0 Sa , K*II ( t ) K II ( t ) / V0 Sa , where ı0 is the loading amplitude. As first example, we consider a finite crack of length 2a in an infinite, homogeneous, anisotropic and linear elastic solid as depicted in Fig. 1. The crack is subjected to an impact tensile crack-face loading of the form

258


ı(t)=ı22H(t), where ı22 is the loading amplitude and H(t) is the Heaviside step function. A Graphite-epoxy composite with 65% graphite and 35% epoxy is considered, which has the following elastic constants 0 0 0 º ª155.43 3.72 3.72 « 16 . 34 4 . 96 0 0 0 »» « « 16.34 0 0 0 » Cij « (16) » GPa 3.37 0 0 » « « sym 7.48 0 » » « 7.48¼» ¬« and the mass density ȡ=1600kg/m3. The elasticity matrix (16) is rotated with the angle T=30° around x3-axis to obtain a general material anisotropy. The crack is divided into 20 elements. The effects of the time-step on the stability and the accuracy of the two different TDBEM for a fixed spatial discretization are investigated. Numerical calculations are carried out for the following normalized time-steps: cTǻt/a=0.04, 0.06, 0.08, 0.1, 0.16, 0.2, 0.3 and 0.4. According to eq (14) these values lead to Ȝ=0.4, 0.6, 0.8, 1, 1.6, 2, 3 and 4 respectively. Plane strain condition is assumed. x2 V 22 ( t )

a

a

x1

Fig. 1: A finite crack in an infinite anisotropic solid subjected to an impact tensile loading The normalized dynamic SIFs obtained by the first time-domain BEM (TDBEM 1) and the second one (TDBEM 2) are presented in Fig. 2 and Fig. 3 for different values of Ȝ.

Fig. 2: Normalized dynamic SIFs obtained by TDBEM 1 for different Ȝ-values Stable numerical results for the normalized mode-I and mode-II dynamic SIFs are obtained and they show a good agreement for all Ȝ-values. The maximum value of KI(t) provided by TDBEM 2 is slightly higher. The effects of the numerical damping for large time-steps can be clearly recognized by KII(t) in Fig. 2. A choice of the normalized time-steps cTǻt/a m are polyharmonic functions and become zero for r = 0.

Discretization of the boundary integral equation and representative length parameter Discretizing Eq.(13) by using constants elements gives NB N 1 l ∗ j l j u im (x , y ) d j (x ) qi (x j ) u m (y ) = 2 j i=1 j=1 K ∗ j l j j qim (x , y ) d j (x ) u i (x ) + α k u ∗N m (zk , yl ), − j

(17)

k=1

where N B denotes the number of boundary elements, x j andyl denote the centers of geometry of elements j and l, respectively. In Eq.(17), the unknowns are u i (x j ), qi (x j ), and α k , and their total number becomes N × N B + K . Although from Eq.(17), we obtain N × N B equations, we still need K equations in order to solve for N × N B + K unknowns. Therefore, we use additional K collocation points in the domain and specify the values for the source term u 2 (zk ), (k = 1, · · · , K ). It is achieved by using the component for m = 2 of Eq.(17) and substituting zk , (k = 1, · · · , K ) into yl as NB N ∗ u 2 (zl ) = u i2 (x j , zl ) d j (x j ) qi (x j ) −

j

i=1 j=1

j

∗ qi2 (x j , zl ) d j (x j )

u i (x ) + j

K

α k u ∗N 2 (zk , zl ).

(18)

k=1

The final linear algebraic equations can be written, by collecting all the unknown quantities on the left-hand side, as (19) [A] {x} = {b}, where [A] is the coefficient matrix, {x} the unknown vector, and {b} is the right-hand side vector calculated from known quantities. Iterative solvers are often used for solving equations like Eq.(19). Some of the rows of Eq.(19) are generated by applying Eq.(18) at K internal collocation points. Since u 2 is prescribed at these points in Eq.(18), the unknowns are the coefficients α k , (k = 1, 2, · · · , N ). We observe that u ∗N 2 (zk , zl ) becomes zero for zk = zl and the corresponding diagonal components of the matrix [A] corresponding to α k , (k = 1, 2, · · · , N ) also become zero. This is inappropriate for preconditioning of [A] to obtain fast convergence of iterative solution. Therefore, we modify the fundamental solution so that it becomes maximum for r = 0 and decreases


279

z 10

10

L O

y

10 x

Fig. 2

A cubic region.

monotonously in accordance with the increase of r by adding some of homogeneous solutions of Eq.(9). The modified fundamental solutions contain a parameter L, a representative length of the domain, and can be given as follows: ⎡ ⎤ 1 0 0 0 ⎢ ⎥ 4πr ⎢ ⎥ ⎢ ⎥ 1 L r ⎢ ⎥ + 0 0 ⎥ − ⎢ ⎢ ⎥ 8π 8π 4πr ⎥ (20) u∗ = ⎢ ⎢ ⎥ 1 r3 Lr 2 L3 r L ⎢ ⎥ − + − + 0 ⎥ ⎢ ⎢ ⎥ 96π 48π 96π 8π 8π 4πr ⎢ ⎥ ⎣ r3 1 ⎦ Lr 4 L 3r 2 L5 Lr 2 L3 r L r5 + − + − + − + − 2880π 960π 576π 96π 96π 48π 96π 8π 8π 4πr Numerical Examples Consider a cubic region having a edge of 10 as shown in Fig.2. All the sides are divided uniformly into 1200 triangular constant elements with 602 nodes. Also, 27 internal collocation points are placed uniformly. Following Poisson’s equation is calculated for the cubic region using the above mentioned methodD πz =0 10

(21)

0 at z = −5 100 at z = 5

(22)

at x, y = ±5

(23)

∇ 2 u 1 (x) + 10 cos The boundary conditions are given as follows: u 1 (x) = q1 (x) = 0

The number of simultaneous Poisson’s equations used in the present example is N = 3. The representative length L is taken as the length of the diagonal line of the cubic region. GMRES is used as the iterative solver with the convergence criterion ε = 10−5 and Point-Jacobi preconditioner. Fig. 3 shows the results for u 1 (z) at the internal collocation points. The results obtained by the present approach show good agreements with the exact solutions. Fig. 4 shows the change in the residual against the number of iterations. We observe that the results obtained by the method which uses the modified fundamental solutions with a representative length parameter can reduce the number of iterations by applying the preconditioner.

280


160 exact BEM

140 120

u1 (z )

100 80 60 40 20 0 -4

-2

0 z

2

4

Results for u 1 (z) at internal collocation points.

Fig. 3

0

10

not preconditioned preconditioned

-1

10

Residual

10-2 -3

10

-4

10

10-5 -6

10

0

10

20

30

40

50

60

Number of iterations

Fig. 4

Residuals versus number of iterations.

Concluding Remarks A boundary element method for Poisson’s equation has been presented. The source term is approximated with simultaneous coupled Poisson’s equations. The fundamental solutions of the coupled Poisson’s equations have been derived and the corresponding boundary integral equation has been presented. Some homogeneous solutions of the simultaneous Poisson’s equations are added to the fundamental solutions so that the convergence property of the iterative solutions is improved. The numerical test example has demonstrated the effectiveness of the present approach. References [1] P.W.Partridge, C.A.Brebbia and L.C.Wrobel: The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, (1992). [2] T.Matsumoto, A.Guzik, M.Tanaka: Int. J. Numer. Meth. Engng, Vol. 64 (2005), pp. 1432–1458. [3] A.J.Nowak, in: The Multiple Reciprocity Boundary Element Method edited by A.C.Neves, Computational Mechanics Publications, Southampton, (1994). [4] Y.Ochiai: Eng. Anal. Bound. Elem. Vol. 28-12 (2004), pp. 1445–1453. [5] Y.Ochiai: Int. J. Numer. Methods Eng. Vol. 63-12 (2005), pp. 1741–1756. [6] Y.Ochiai, V.Sladek and J.Sladek: Eng. Anal. Bound. Elem. Vol. 30-3 (2006), pp. 194–204 .


281

Scaled Boundary Finite-Element Analysis of Dynamic Stress Intensity Factors and T-stress Chongmin Song School of Civil and Environmental Engineering, University of New South Wales Sydney, NSW 2052, Australia. E-mail: [email protected] Keywords: Scaled boundary finite-element method; Stress intensity factor; T-stress; Stress singularity; Structural dynamics. Abstract: The dynamic stress intensity factors and T-stress at crack tips are determined by using the scaled boundary finite-element method. Only the boundary of a bounded domain is discretized with elements. The inertial effect at high frequencies is represented by expressing the dynamic stiffness as continued fractions. By introducing auxiliary variables, an equation of motion is formulated with a high-order static stiffness matrix and a high-order mass matrix. Standard methods in structural dynamics can be directly applied to perform modal, frequency-domain and time-domain analyses. The dynamic stress intensity factors and T-stress are determined directly from their definitions. This technique does not require a fundamental solution or an asymptotic expansion of the singular stress field. Numerical examples demonstrate the simplicity and high accuracy of the present technique. Introduction For the dynamic analysis of fracture problems, the inertial effect has to be considered in computing the stress fields and extracting the stress intensity factors and the T −stress. The mesh and time step should be sufficiently fine to represent the highest frequency of interest. Many studies on the dynamic stress intensity factors for cracks in homogeneous plates have been reported (see, e.g., [1–5]). Literature reviews can be found in [5, 6]. The T -stress at interface cracks is rarely studied [7, 8]. The scaled boundary finite-element method has emerged as an attractive alternative to model problems with singularities. In a single analysis, this method obtains [9]: (a) orders of singularity; (b) power-logarithmic singularities, when exist; (c) stress intensity factors; (d) T -stress and higher order terms in the asymptotic expansion; (e) angular distributions of stresses corresponding to individual terms in the asymptotic expansion. In addition, the stress intensity factors and T -stress are extracted directly based on their definitions without evaluating singular functions close to a singular point. Crack propagation is modeled in [10]. In this paper, a numerical procedure to determine the dynamic stress intensity factors and the dynamic T -stress is presented. Summary of the Scaled Boundary Finite-Element Method A so-called scaling centre O is chosen in a zone from which the total boundary must be visible (Fig. 1). Without losing generality, the origin of the Cartesian coordinates x, ˆ yˆ is chosen at the scaling centre. The boundary S is discretized with line elements. The geometry of an element is interpolated using the mapping functions [N(η )] in the local coordinate η and the nodal coordinates {x}, {y}. The domain V is described by scaling the boundary with a dimensionless radial coordinate ξ pointing ˆ y) ˆ inside away from the scaling centre O. ξ = 0 at O and ξ = 1 on the boundary is chosen. A point (x, the domain is expressed as x( ˆ ξ , η ) = ξ [N(η )]{x};

y( ˆ ξ , η ) = ξ [N(η )]{y}

(1)

282

Eds: R Abascal and M H Aliabadi η

y^

y^

θ x^

H @nxn û ( x )`nx1 >G @nxn ®

Assuming that the matrix [G ] can be inverted, equation (2) may be recast as:

(2)

290


wu ( x ) ½ ® ¾ ¯ wn ¿

1

>G @ > H @û ( x )`

(3)

Figure 1: Radiating body in unbounded acoustic domain

Figure 2: Definition of auxiliary measuring surfaces

Starting from the Boundary Integral Equation (1) it is possible to determine a vector containing the acoustic potential u i ( x d ) in m points (i=1,m) within the unbounded acoustic domain ( x d :inf ) :

û ( x d )`mu1

ªGˆ º ® wu ( x ) ¾½ ¬ ¼ mun ¯ wn ¿

nu1

ª¬ Hˆ º¼

mun

û ( x )`nu1

(4)

The vector containing the Neumann data on the boundary * R , ^wu ( x ) / wn` , given in (3), may be inserted into expression (4) leading to:

û ( x d )`mu1

^ª¬Gˆ º¼

1

>G @

mun

nu n

> H @nun ª¬ Hˆ º¼ mun `û ( x )`nu1 > R @mun û ( x )`nu1

(5)

with

^R`mun

^ª¬Gˆ º¼

1

mun

>G @

nun

> H @nun ª¬ Hˆ º¼ mun `

(6)

In equation (5) the radiating operator [R], defined in (6), relates the acoustic potential in n points at the boundary * R to the acoustic potential of m points within the domain :inf . The Inverse Problem.

The inverse problem consists of determining the acoustic potential {u ( x )} over the n points of the boundary * R from the known m points {u ( x d )} on the domain :inf . Formally this can be accomplished by inverting the radiating operator [R] in (5): 1

û ( x )`nu1 > R @

mun

û ( x d )`mu1

(7)


291

In practice this task requires the inversion of a rectangular and ill-conditioned matrix [2]. Two distinct schemes have been devised to improve or to regularize the inversion of the operator [R]. The idea is to decrease the condition number N of the matrix [R], which can defined as the ratio between the largest and the smallest singular values of [R], N ( R) V max / V min , [2, 3]. The Singular Value Decomposition – SVD. The generalized inverse, or pseudo-inverse, [ R] of a rectangular matrix [R] can be expressed as a function of its singular values as [2,3]:

> R@

>V @>6 @

ª¬U H º¼

(8)

The matrix > 6 @ is a diagonal matrix, the elements of which are the inverse of the singular values of [R]. It can be shown that the regularization consists in deleting from > 6 @ the singular values which are smaller than a given value. The success of the regularization procedure lays exactly on the proper choice of the smallest, non-negligible, singular value V S [3]. A strategy to determine this value will be given in the next session. The Tikhonov regularization procedure. The scheme proposed by Tikhonov [2, 4] requires the determination of the smallest singular value V S which allows the inversion of the matrix [RH R]. The matrix [RH] is the Hermitian of [R]. Introducing a unit diagonal matrix [I], the complete inversion equation for the Tikhonov scheme may be written as [4]:

û ( x )`

^ª¬ R

H

`

º¼ > R @ V S > I @

1

ª¬ R H º¼ û ( x d )`

(9)

Regularization Strategy.

The solution strategy proposed in this article is illustrated in figure 2. The idea is to recover the acoustic velocity potential û ( x )` at the boundary of the radiating body x * R from acoustic potential values measured, or numerically determined, over a closed surface that circumscribes the body û ( x d )` , x d * AUX 1 . To establish the smallest singular value V S of [R] to be considered in the SVD or in the Tikhonov regularization strategies, leading to an accurate inversion of [R], the acoustic potential is measured, or numerically determined, at a second closed surface * AUX 2 , which lays closer to the radiating object. This second auxiliary surface * AUX 2 is used to calibrate the inversion parameters, that is, the smallest singular value V S to be considered. Once this calibration is performed, the pseudo-inverse of [R] is used to determine the acoustic potential on the actual boundary * R . Numerical examples. A vibrating cylinder. Consider a vibrating cylinder with unit diameter a=1m, as shown in figure 3. The boundary * R is discretized in n=80 constant elements, upon which Neuman boundary conditions are applied, wu / wn cos T . The unbounded domain :inf is considered to be air with

sound velocity cs=345m/s. The acoustic potential û ( x d )` is numerically determined through equation (4) for m=120 equally spaced points at the auxiliary surface * AUX 1 with radius [1=2,0m and with m=80 points at the closer auxiliary surface * AUX 2 with radius [2=1.05m. In the second step, to calibrate the regularization processes, the values of the m=80 points on auxiliary surface

292


* AUX 2 are determined as an inverse problem by means of equations (7) and (9) from the known data (m=120) already known on * AUX 1 .

Table 1 shows the (largest) errors between the values of the acoustic potencial û ( x d )` , directly determined by the discretized integral equation (4) and the values recovered on * AUX 2 by inverting the radiating operator [R] using the SVD strategy. The comparison is performed for distinct values of the classical dimensionless wave number ka [1]. The accuracy of the recovered solution is determined for distinct limiting singular values V S . For each limiting singular value

V S and wave number ka, the system condition number N ( R) V max / V min is also furnished.

Figure 3: Vibrating cylinder

An analysis reveals that for low wave numbers even relative small values of V S furnishes a good solution of the inverse problem on the auxiliary surface. As expected lower values of the condition number N ( R) results in more accurate solutions. Table 2 furnishes the results for the same type of analysis but considering the Tikhonov regularization procedure. The Tikhonov strategy does not present the same behavior of the SVD solution. It can be clearly seen that there is an optimum limiting value for the singular value V S . For this example the best solution is obtained for VS=10-3. After this calibration procedure the acoustic potential may be recovered at the actual boundary * R of the vibrating cylinder using both inversion strategies. The acoustic potential recovered using the Tikhonov strategy with limiting value VS=10-3 and for the wave number ka=1.0, may be seen in figure 4. The potential on all the n=80 points of the cylinder boundary are shown. The largest error of the acoustic potential determined by this procedure is smaller than 0.2%. The SVD solution is

even more accurate for this case. A loudspeaker profile. The recovery of the acoustic potential on a 2D surface that resembles a loudspeaker profile, shown in figure 6, is discussed next. The loudspeaker profile (surface * R ) with dimensions H=2,4m, B=2,0m and L1=1,5m and immersed in air with cs=345m/s is divided into n=34 constant boundary elements. On this surface unit Neumann boundary conditions are imposed wu / wn 1 . An auxiliary surface * AUX 1 is established with a radius [1=2,0m. Upon this surface the acoustic potential is determined in m=102 points. Table 3 shows the largest errors for the potential recovered at the original n=34 boundary points on * R for 3 distinct limiting singular values. The dimensionless wave numbers are ka=10 and ka=20.


293

Table 1: Estimation of paremeter VS in SVD regularization ka

VS = 10-1 error %

N ()

VS = 10-2

VS = 10-3

N ()

error

error %

VS = 10-4

N ()

%

error

VS = 10-5

N ()

error

%

0.5

0.0004

8.35

0.0007

83.76

0.0005

1.0

0.0003

3.968

0.0007

9 51.23

2.0

0.28

3.655

0.28

63 66.67

4.0

2.27

7.265

2.27

57.99

796.1

VS = 10-6

N ()

error

%

N ()

%

0.035 7529.73

0.45

91740.5

8.62

109357

0.0028 774.06

0.04

4630.33

0.26

35.35

6.69

0 806265

0.28

3 799.31

0.34

8366.27

0.50

83579.2

3.98

740707

2.27

485.03

2.29

7462.42

2.41

40390.7

4.83

631115

8.0

16.16

7.890

16.16

75.72

16.16

4 688.90 16.15 7087.18 16.16 65221.3 16.91 480527

16.0

27.21

7.567

27.21

79.40

27.21

460.93 27.21 3320.53

ka

VS = 10-1

27.3

73651.2 28.18 74099.2

Table 2: Estimation of paremeter VS for Tikhonov regularization VS = 10-2

VS = 10-3

VS = 10-4

N ()

VS = 10-5

VS = 10-6

error%

N ()

error%

N ()

error%

N ()

error%

error%

N ()

error%

N ()

0.5

11.2

4.038

1.25

12.419

0.17

39.143

1.17

123.784 35.10

392.85

94.14 1289.93

1.0

11.7

2.581

1.32

7.593

0.18

23.826

1.27

75.30

2.0

12.1

1.742

1.36

4.621

0.19

14.3021

1.25

45.13

35.5

238.61

92.06 771.379

34.01

142.75

92.61 454.871

4.0

12.2

1.29

1.37

2.802

0.18

8.339

1.72

26.996

55.55

82.81

94.01 262.436

8.0

12.2

1.104

1.38

1.7868

0.24

4.7883

2.98

14.841

49.14

46.844

94.14 148.292

16.0

12,2

1.0314

1.38

1.280

0.27

2.7174

1.68

8.0525

54.98 25.2884 90.07 79.9558

294


Figure 5: Comparison between recovered potential (Tikhonov VS=10-3) and direct numerical calculation on the surface of the vibrating cylinder Solutions for SVD and Tikhonov strategies are reported. Both strategies furnish accurate results. Nevertheless, for this example the SVD strategy is more precise. A directional pattern of the recovered potential on the loud speaker surface can be seen in Fig 7.

Figure 6: loud speaker profile

Figure 7: Directional representation of the acoustic potential recovered boundary (ka=20)

Table 3: Accuracy of recovered potential on n=34 boundary points SVD.- error % TIKHONOV - erro% ka VS = 10-3 VS = 10-4 VS = 10-5 VS = 10-3 VS = 10-4 VS = 10-5 10.0 0.002 0.002 0.002 0.169 0.16 0.45 20.0 0.0005 0.0005 0.0005 0.208 0.021 0.021 Concluding remarks.

The BEM has been applied to recover acoustic potentials on the surface of a radiating body by inverting data, measured or numerically determined, at a given auxiliary closed surface. Two regularization strategies have been applied to solve the inversion problem. A second auxiliary surface has been proposed to calibrate the regularization strategies. Presented data show that the procedure leads to accurate results. Further investigations are being conducted to assess the role of discretization densities on the actual an auxiliary surfaces. Aknowledgements.

The research leading to this article has been supported by UFMS, Fapesp, CNPq, Capes and Fapex/Unicamp. This is gratefully acknowleged. References

[1] RD Ciskowski, CA Brebbia, Boundary Element Methods in Acoustics. Computational Mechanics Publications, Southampton-Boston, 1991. [2] PA Nelson, SH Yoon, Estimation of Acoustic Source Strength by Inverse Methods: Part I, Conditioning of the Inverse Problem. Journal of Sound and Vibration (2000), 233(4), 643-668. [3] GH Golub, CF van Loan, Matrix Computations, The Johns Hopkins University Press; third edition edition (October 15, 1996) [4] P Linz, Theoretical Numerical Analysis – An Introduction to Advanced Techniques. John Wiley, 1979.


295

Error controlled analysis of non-linear problems using the Boundary Element Method ¨ 1,a , Gernot Beer1,b Klaus Thoni 1

Institute for Structural Analysis, Lessingstraße 25, 8010 Graz, Austria a

[email protected],

b

[email protected]

Keywords: non-linear analysis, initial stress approach, internal cells, error controlled adaptivity, elasto-plasticity.

Abstract. This paper presents a new method to control the error of a non-linear boundary element analysis. For the evaluation of the domain integral, which arises from the initial stress formulation of non-linear problems, a mesh of either area or volume cells is used. The area and volume cells used in the formulation are of the type C 0 , and therefore, they cannot reproduce the expected initial stress field in the non-linear zone. This discontinuity between the cells is used to estimate the error and to refine the cell mesh. It is a general method which can be used to solve 2-dimensional as well as 3-dimensional problems. Moreover, it is not restricted to certain non-linear models. The applicability and the accuracy of this new method will be demonstrated on some elasto-plastic problems. Introduction The Boundary Element Method (BEM) is a useful tool for solving linear as well as nonlinear numerical problems. With this method, usually only the boundary of the problem has to be discretized. Thus, the dimension of the problem is reduced by one. This is a significant advantage regarding mesh generation, data storage, post-processing and the dimension of the system of equations. When dealing with non-linear problems, not only boundary integrals but also domain integrals arise. In the standard 2d or 3d approach, a mesh of area or volume cells is used respectively for the evaluation of the domain integrals (see Beer et al. [2], Venturini [12]). By using this method the main advantage of the BEM with respect to mesh generation is partially lost because cells have to be generated by the user, either in the whole domain or in parts of the domain that are expected to behave non-linearly. However, the size of the system of equations does not depend on the domain discretization, which means that no additional degrees of freedom will be introduced using internal cells. In addition, internal cells can be generated automatically (see Ribeiro et al. [10]). Basic equations For the derivation of the boundary integral equation for small strain non-linear problems, the following basic equations are relevant: • the Cauchy’s infinitesimal strain tensor, also called Green’s strain tensor 1 εij = (ui,j + uj,i ) 2

(1)

296


• the total strain, which is expressed as a sum of the elastic and non-linear components εij = εeij + ε0ij ,

(2)

• the Navier-Cauchy equation of equilibrium for the stress tensor σij,i + bj = 0

(3)

• the equilibrium condition on the boundary tj = σij ni

(4)

• and as far as the elastic strains are concerned, generalized Hooke’s law σij = Dijkl εekl .

(5)

By using eq. 2 and eq. 5 the non-linear part can be separated as follows σij = Dijkl (εkl − ε0kl ) = σije − σij0

(6)

where σije is a notational elastic stress (corresponding to the strain εkl ), and σij0 is termed the “initial” stress. However, the negative sign in this equation emphasizes the difference between this rather artificial decomposition of the stress (see Fig. 1). The non-linear part σij0 depends in general on the stress-strain relationship of the chosen non-linear model. Thus, the initial stress σij0 is related via the elastic constitutive relationship to the irrecoverable component of the strain ε0kl (e. g. Telles [11], Herding and Kuhn [7]).

σ

plasticity

σ

damage

σp

σ

non-linear elasticity

σd σe

∆σ σe

σ

σ1 σ2

σ Ed ε

E1 ε

E2 ε

Fig. 1: Schematic stress decomposition where σ p , σ d or ∆σ correspond to the initial stress σ 0

Non-Linear Boundary Integral Formulation For a domain Ω with boundary Γ, the standard integral representations are derived by applying Betti’s reciprocal principle or Green’s second identity. In particular, displacement, strain and stress integral representations are easily derived and may be found in Bonnet [3].


297

In the non-linear case, compared to the linear case, we have to deal with incremental quantities denoted by the superposed period (for example σ). ˙ The initial stress integral equation, which does not consider body forces, can be expressed in the incremental form as u˙ j (p) = Uij (p, Q)t˙j (Q)dΓ − Tij (p, Q)u˙ j (Q)dΓ Γ Γ (7) 0 (q)dΩ + Eijk (p, q)σ˙ jk Ω

where Q is a point on the boundary Γ and p and q are points in the domain Ω. u˙ j (Q) is 0 (q) the initial stress field. Uij (p, Q) and the displacement field, t˙j (Q) the traction field and σ˙ jk Tij (p, Q) are the Kelvin fundamental solutions for displacements and tractions at the field point Q in the j-th direction due to a unit load at the source point p in the i-th direction. Eijk is the strain kernel which corresponds to the displacement field Uij and is obtained from Eijk (p, q) = (Uij,k + Uik,j )/2 .

(8)

Eq. 7 is only applicable for internal points. For points on the boundary Γ, the limiting form is obtained by allowing the source point to approach the boundary (p → P ). This yields to the direct initial stress boundary integral equation Cij (P )u˙ j (P ) = Uij (P, Q)t˙j (Q)dΓ − Tij (P, Q)u˙ j (Q)dΓ Γ Γ (9) 0 (q)dΩ + Eijk (P, q)σ˙ jk Ω

where Cij (P ) is the so called jump term which is influenced by the geometry of the boundary. Apart from the additional domain integral, which considers the influence of the initial stresses, eq. 9 is the same as in linear elastic analysis. It should be mentioned that an alternative formulation is possible in which initial strains are employed rather than initial stresses. For more details see for example Cisilino and Aliabadi [4]. 0 In order to solve eq. 9 the initial stresses σ˙ jk (q) have to be determined. This requires an additional integral equation for stresses within the domain Ω. The strains can be computed by differentiating eq. 7. Furthermore, the following stress integral equation can be obtained by using generalized Hooke’s law σ˙ ij (p) = Dijk (p, Q)t˙k (Q)dΓ − Sijk (p, Q)u˙ k (Q)dΓ Γ Γ (10) 0 0 (q)dΩ + Fijkl σ˙ kl (p) + Wijkl (p, q)σ˙ kl Ω

where Dijk , Sijk and Wijkl are fundamental solutions and Fijkl are the free terms which arise from the singularity at p. This equation is valid for points located inside the domain Ω. In case the point is located at the boundary Γ, the limit of the integral when the source point p approaches the boundary can be taken. However, this procedure is computationally expensive due to the occurrence of hyper-singular integrands in the boundary integrals. To overcome this problem, an alternative and relatively simple method of evaluating boundary stresses from tractions and displacement tangential derivatives can be used (see Gao and Davies [5], Cisilino and Aliabadi [4] and Beer et al. [2]).

298


Error Controlled Domain Mesh Refinement In the past, several strategies for self-adaptive linear-elastic boundary element analysis were developed. An overview of such methods for example can be found in Kita and Kamiya [8] and Liapis [9]. However, this has not been the case for non-linear analysis. Only a few contributions have been made in this area until today. For example, Astrinidis et al. [1] used an adaptive scheme based on a total strain smoothing error criterion for elasto-plasticity. Gaspari and Aristodemo [6] introduced a refinement criterion based on a check of the elastic stress level for general material non-linearities. Nevertheless, this work presents a new method which automatically controls the domain discretization error. For evaluating the domain integrals in eq. 9 and 10, which consider the non-linear influence, either a mesh of area or volume cells is used. The internal cells used are of the type C 0 , and therefore, they cannot reproduce the expected initial stress field in the non-linear zone. The initial stress field is approximated for each cell by σij0 =

c

Nn σij0n ,

(11)

n=1

where N are the shape functions and c the number of nodes of the cell. The proposed selfcorrecting domain mesh algorithm uses the derivative of the approximated initial stress field by the cells as error indicator. The derivative is computed for all internal nodes by differentiating eq. 11 with respect to the Cartesian directions. This gives 0 σij,i =

c

Nn,i σij0n .

(12)

n=1

Since each of the internal nodes has at least 2 adjacent internal cells, the derivative gives different values (see Fig. 2). The difference between such values can be taken to estimate the local error. In the algorithm this error is bounded by a certain value. If the computed value is higher then the given value the adjacent cells will be refined.

initial stress distribution

error

cell 2 node of interest cell 1

Fig. 2: Schematic illustration of computing the local error in one direction

Non-Linear Solution Algorithm In continuum mechanics, non-linear problems are commonly solved by adopting a load incremental procedure and by iterating on the strain-stress constitutive equation. In this work


299

a simple explicit procedure based on the classical initial stress approach with constant stiffness (see e. g. Gao and Davies [5]) was chosen. The general unknowns are boundary displacements, tractions and initial stresses. The advantage of these techniques is that the system matrices are kept constant and only the right hand side changes. However, their drawback is the increase in the number of iterations. In the incremental procedure the load is divided into several increments. After computing the first trial for the initial stresses at each incremental step the quality of the domain mesh is checked and in the regions where the continuity of the initial stress field is not given the domain mesh automatically becomes finer. Numerical Results

p

To demonstrate the applicability and the accuracy of this new method an example which is well known in the literature is presented. A plate with a hole subject to constant tension is considered (see e. g. Astrinidis et al. [1]). Plane stress conditions are assumed. Due to the symmetry of the problem, only one half of the problem is analyzed. The geometry of the problem is shown in Fig. 3. To discretize the problem quadratic boundary elements and quadratic cells are used. The following material parameters are assumed (von Mises yield criterion): Elastic modulus Plastic modulus Yield stress Poisson’s ratio Loading

B = 20 cm

A———–A H = 36 cm r = 5 cm

E = 7 000 kg/mm2 H = 224 kg/mm2 fy = 24.3 kg/mm2 ν = 0.2 p = 0.91 · fy

p

Fig. 3: Geometry

The following figures show the automatic non-conforming mesh refinement at different load increments, from the initial mesh (a) to the final mesh (e):

(a)

(b)

(c)

(d)

(e)

Fig. 4: Cell mesh refinement (the dark cells indicate the ones which have to be refined) In Fig. 5 a comparison of the stress variation σyy along the minimum section A-A is made between some published results and the new method proposed in this paper. A good agreement is observed.

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30 Astrinidis et al. [1] new method 25

σyy [kg/mm2]

20

15

10

5

0 5

6

7

8

9

10

x [mm]

Fig. 5: σyy variation along minimum section A-A References [1] E. Astrinidis, R.T. Fenner, and G. Tsamasphyros. Elastoplastic analysis with adaptive boundary element method. Computational Mechanics, 33(3):186–193, 2 2004. [2] G. Beer, I. Smith, and C. Duenser. The Boundary Element Method with programming. Springer Wien New York, 2008. [3] M. Bonnet. Boundary Integral Equation Methods for Solids and Fluids. John Wiley & Sons, Ltd, 1999. [4] A.P. Cisilino and M.H. Aliabadi. A boundary element method for three-dimensional elastoplastic problems. Engineering Computations, 15:1011–1030, 1988. [5] X.W. Gao and T.G. Davies. Boundary Element Programming in Mechanics. Cambridge University Press, Cambridge, 2002. [6] D. Gaspari and M. Aristodemo. An algorithm for nonlinear BEM with adaptive refinement of the domain mesh. In Advances in Boundary Element Techniques VIII, pages 199–204, 2007. [7] U. Herding and G. Kuhn. A field boundary element formulation for damage mechanics. Engineering Analysis with Boundary Elements, 18(2):137–147, 1996. [8] E. Kita and N. Kamiya. Error estimation and adaptive mesh refinement in boundary element method, an overview. Engineering Analysis with Boundary Elements, 25(7):479– 495, 2001. [9] S. Liapis. A review of error estimation and adaptivity in the boundary element method. Engineering Analysis with Boundary Elements, 14(4):315–323, 1994. [10] T.S.A. Ribeiro, G. Beer, and C. Duenser. Efficient elastoplastic analysis with the boundary element method. Computational Mechanics, 41(5):715–732, 2008. [11] J.C.F. Telles. The Boundary Element Method Applied to Inelastic Problems. SpringerVerlag, Berlin, 1983. [12] W.S. Venturini. Boundary Element Method in Geomechanics. Springer-Verlag, Berlin, 1983.


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On the use of Boundary Element Methods for Inverse Problems of Damage Detection in Structures Ariosto B. Jorge1,2, Patricia S. Lopes1,3 and Maurício E. Lopes1,4 1

Federal University of Itajubá, Mechanical Engineering Institute, Itajubá, MG, Brazil. 2 Corresponding author. Email: [email protected] 3 Email: [email protected] 4 Email: [email protected]

Keywords: Inverse Problems; Damage detection; Boundary Element Methods; Elastostatics; Acoustics; Uncertainties; Optimization.

Abstract. In this work, the inverse problem of identifying damages in a plate structure is modeled using optimization techniques. The numerical modeling consists in two problems: a direct problem, in which the Boundary Element Method (BEM) is used to obtain the potential, stress, or acoustical distribution throughout the damaged structure; and an inverse problem, in which an optimization model is used to locate the damage in the structure, given measured information on the quantity of interest at some interior points (sensor locations). This paper discusses issues related to the use of BEM as the direct method, in order to improve damage identification and also the confidence in the damage location and size results. The discussion includes: i) the use of quantities obtained from derivatives of the original densities instead of the densities themselves, as the main variables in the objective functions and constraint equations, for faster convergence of the optimization procedure; ii) the ability of the direct method to properly capture the proximity between the “numerical” damage and the “real” damage; iii) the use of independent scalar quantities (for example, invariants of the stress tensor for the elasticity problem) at the interior points of interest (where measured and numerical information are compared in the optimization procedure), in order to avoid comparing derivatives obtained in different planes and directions at a particular interior point; iv) the influence of the BEM discretization (mesh refinement) in the numerical results. Introduction In this work, the inverse problem of identifying the presence, location and size of damages, such as cracks and holes, in a plate structure is modeled using optimization techniques. The numerical modeling consists in two main parts. The first part on the modeling is the direct problem, in which a model is required to obtain information on the distribution of the quantity of interest throughout the structure, given the boundary conditions and the presence of the damage. The models investigated in this work include the elastostatics and acoustics (potential) formulations. For the potential (acoustics) problem, the quantities of interest are the potential and its gradient at interior points. Also, for the elastostatics problem, the quantities of interest are the interior point displacements, strains, and stresses. The modeling of the structure is carried out using the Boundary Element Method (BEM) in all cases. The second part of the modeling consists in the inverse problem, in which a model is required for the procedure of locating the damage in the structure given some (partial) information on the quantity of interest at some particular locations (for example, where some sensors are placed). For this inverse problem, an optimization technique (genetic algorithm), will be used for all comparisons performed. This paper discusses four issues related to the use of BEM as the direct method, in order to increase damage identification and also to improve the confidence in the damage location and size results. Several runs were made for the various BEM models, for elastostatics and potential (acoustics). The first issue is the use of quantities obtained from derivatives of the original densities (such as strain or stress, for example) instead of the densities themselves (such as the displacement), as the main variables in the objective functions and constraint equations. The use of derivatives of the densities is expected to lead to faster convergence of the optimization procedure, as these derivative fields are expected to be more sensitive (local changes more evident) than the density fields, in the presence of damage. The second issue is the ability of the direct method to properly capture the proximity between the “numerical” damage (the damage included in the numerical model for a particular run of the BEM code) and the “real” damage (the damage to be detected in the real structure). In this work, the damage was simulated by a hole from which information on the density values was available at interior points corresponding to sensor locations. For this discussion, a comparison was done between the numerical results obtained with the “real” hole and results from several “numerical” holes approaching the “real” hole, to simulate the desired effect of proximity between holes. The third issue is the use of independent scalar fields, and not vector or tensor fields, as the variables of interest for the optimization procedure, in order to avoid comparing derivatives obtained in different planes and directions at a particular interior point. With this

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procedure, all interior point information to be used in the optimization procedure is directionindependent. For example, the invariants of the stress tensor (mean stress and octahedral stress, for the elasticity problem) at the interior points of interest (where measured and numerical information are compared in the optimization procedure) can be used, instead of the original stresses obtained with the BEM formulation. In this work, the mean stress is used, as the scalar quantity of interest evaluated at interior points. The fourth issue is the influence of the numerical errors due to the BEM discretization of a particular problem. Different meshes and different approaches for mesh refinement may have an influence in the numerical results. Thus, the discretization is expected to interfere with the optimization results, and consequently to have an impact in the quality of the damage identification information. Inverse Problems of Damage Detection in Structures The detection of damage in structures is an important engineering issue, with applications in flight safety and aircraft maintenance. The development of damage detection techniques can contribute to a better structural integrity analysis of a structure. The analysis of a damaged structure must involve the numerical treatment of data gathered from sensors spread throughout critical points in the structure, and the comparison of this data with numerical results used as reference (results from the same structure, undamaged or with known damage). To analyze a damage detection problem in a structure, first the modeling of the direct problems is required, to obtain the behavior of this structure in the presence of one or more pre-established damages, with assumed format and size, and at given positions (see references [1] and [2] for damage detection problems). In this work, two methods of analysis need to be given particular attention: 1) the study of stress and strain distributions in damaged structural elements, performed with a displacement-BEM model for elastostatics (see references [3] to [5] for elastostatics); and 2) the study of the distribution of sound waves (emitted from a pre-established source) in the damaged element, performed with a BEM model for acoustics (see references [6] to [9] for acoustics). The next step is to study the inverse problem, which consists of two parts: 1) monitoring the structural integrity, with sensors spread throughout the structure, to obtain some knowledge about the distribution of the quantity of interest (for example, stresses or strains, strain-gages, or the acoustic potential or pressure using microphones, accelerometers, or other sensors); and 2) computation of a functional obtained from adding differences (evaluated at all measurement points) between the values evaluated using the numerical model from the first part (direct problem), for an assumed damage, and the experimental values measured in the same points for the structure with the real damage. This functional is a function of the damage location, either numerical or measured from the real structure. This functional is expected to increase in value when the assumed numerical damage is far away from the real damage, and to reach its minimum value when both damages (numerical and real) coincide. Direct Problem: Boundary Element Methods The boundary element method is a numerical procedure well adapted for the modeling of a structure with a damage. In this method, the distribution of the quantities of interest in the domain is obtained from the information of the distribution of certain quantities in the boundary. Thus, the problem is described based on what happens in its boundaries, reducing the dimension of the problem and simplifying numerically the treatment. When modeling the damage detection problem by means of an analysis of the elastic or acoustic response of the structure under excitation, perturbations in the expected response imply in the presence of damage. Thus, the damage in the structure shall characterize its behavior, static or dynamic. Inverse Problem: Optimization Techniques The inverse problem might be modeled by means of optimization techniques. For the discussion, a simple direct boundary element problem for the distribution of a potential field in a domain is considered. The damage is simulated by the presence of small holes in the domain, and the goal is to obtain size and location of the damage. The direct method (BEM) provides one piece of information (the potential) for any desired point in the domain. Without the hole, the distribution of the potential is known a priori. If a small hole is included, the potential distribution is unknown and must be obtained numerically from the BEM solution. In this problem, an inverse method (optimization) is implemented. The results obtained for the inverse method by means of this technique are used to find the location and size of the hole. Increasing the problem complexity, the BEM for the elasticity problem can be used. In this case, boundary conditions for the displacement and traction shall be provided. Differently from the BEM for the potential, the BEM for elasticity (in a 2D problem) provides two pieces of information at a single interior point – one normal stress and one shear stress. But this information cannot be used directly in the optimization problem, as it depends on the system of coordinates being used, or on the normal direction of the cutting plane that passes through the point of interest. Therefore, a choice is made to adopt the


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stress invariants of the stress tensor at the point of interest– in 2D, the mean stress and the octahedral stress – as the vector field to be analyzed and used in the optimization problem. In order to increase the complexity of the damage detection problem using the BEM formulation, the goal could be to identify and locate the presence in the plate of one or more different types of damages, such as circular holes (number of holes, radius, and location of each hole), and cracks (number of cracks, orientation, size, and location of each crack). For all the different direct problems, the same optimization technique can be used to solve the inverse problem for damage detection. From the point of view of the inverse problem, the direct model is just a ‘black box’ to be supplied to give the numerical information needed to be used in the optimization procedure. Damage detection by means of optimization techniques. The “measured” or “real” data is obtained at sensor locations spread throughout the structure, for the assumed size and location of the real damage, simulated using BEM. The potential values at interior points simulate the information collected by the sensors at these points. In order to solve the inverse problem, an optimization algorithm, such as the Genetic Algorithm (GA), is used. The evaluation (fitness) function is formulated as a functional defined as a difference between measured (simulated) values of the local difference in the potential (between the undamaged plate and the plate with the damage) and the values of the same differences in potential calculated at the same points by the code (assuming several different locations and sizes for the ‘numerical’ damage). The general form for the functional to be minimized is given by Eq. (1).

Jj

1 n ¦ (measuredi calculated ji ) 2 2i1

(1)

where: n - Number of internal points i (“sensors” placed in the plate) where differences are evaluated; measuredi - Vector of simulated values for the differences obtained using BEM, for a given damage; calculatedji - Vector of differences in potential calculated by the code for each individual j. Figure 1(a) represents an undamaged thin plate with sensors indicating the points where the measurement of the quantities of interest (such as differences in potential or in stresses) is being performed. In order to solve the damage detection problem, an initial population is given to the GA. This initial population is formed by individuals constituting a possible solution for the problem. These individuals are chromosomes, which are themselves constituted by genes. Each gene in a chromosome represents one variable in the problem (such as position and size of a hole). As an example, Fig. 2(b) represents three possible configurations of chromosomes. While the location and size of the hole varies, the number and location of the sensors remains the same, for all chromosomes. The information on the quantity of interest is collected at these sensor locations for all cases.

(a) (b) Figure 1 – (a) Undamaged plate with four representative sensors; (b) Plate with a hole: three possible configurations for the chromosomes For a review of calculus-based optimization algorithms, see reference [10]. For heuristics based in the imitation of behaviors found in nature, such as genetic algorithms, see references [11] and [12]. Numerical Results and Discussion Several runs were made for the various BEM models, for elastostatics and acoustics (potential), in order to discuss several aspects related to the use of BEM as the direct method, and the influence of these aspects in the ability of the optimization algorithm to identify the damage and also in the reliability in the damage location and size results. For the various runs a plate problem of a square domain was considered, where the defect is simulated by the presence of a circular hole. For the elasticity problem, a BEM model was built for the plate with a hole with the boundary conditions illustrated in Fig. 2(a). Two discretizations were implemented for the external contour, a coarse mesh with 12 elements and a fine mesh with 48 elements. Fig. 2(b) shows the discretization for the case of 48 elements in the outer boundary and 12 elements in the hole, as well as the position of the nine sensors.

304


ı=0

(a) (b) Figure 2 – Elasticity plate model: (a) dimensions, loading, boundary conditions. Insert shows a stressfree hole; (b) boundary discretization (fine mesh) and sensor locations. Insert shows hole discretization. For the acoustics problem, a model was built for the plate with a hole with the dimensions and location of sound source as illustrated in Fig. 3(a). Fig. 3(b) shows the boundary conditions and also the discretization for the case of 24 elements in the outer boundary and 24 elements in the hole. y (cm) 3,0 2,0

du = 0

1,0

0,24 cm

u=0

0

x (cm)

-1,0

sound source

-2,0 -3,0

1,0

2,0

3,0

4,0

5,0

6,0

(a) (b) Figure 3 - acoustic model for the plate with a hole: (a) dimensions and position of the sound source; (b) loading, boundary conditions and plate discretization. The insert shows the discretization of the hole. The numerical results and discussion were obtained considering four issues. The first issue is the use of quantities obtained from derivatives of the original densities instead of the densities themselves, as the main variables in the objective functions and constraint equations. For the acoustic model of the plate, several interior point results were obtained, both for the potential field and for its derivative. These results were obtained as a function of time (where t = 0 represents the instant when the source has emitted a sound), for a plate with and without a hole. Fig. 4 presents comparisons through time for an illustrative interior point, where the potential results are shown in Fig. 4(a), and the results for the derivative of the potential are presented in Fig. 4(b). 0,30

with center hole without hole

Potential "u"

0,10

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0,00 0,30

0,35

0,40

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0,65

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0,15

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without hole with center hole

0,20 0,15 0,10 0,05 0,00 0,30 -0,05 -0,10

0,35

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-0,15

(a) (b) Figure 4 - results through time at an interior point, for the plate with and without hole (a) comparison of the distribution of the potential field; (b) comparison of the distribution of the derivative of the potential.


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One can see from the numerical results in Fig. 5 that the derivative of the density is more sensitive to the presence of the hole than the density itself. Thus, one can expect that the use of the derivative of the potential may lead to a faster convergence for the optimization algorithm used for the detection of the presence of damage (such as the hole in this example), when comparing to the use of the potential in the model of the direct problem in this algorithm. The second issue is the ability of the direct method to properly capture the proximity between the “numerical” hole (the hole included in the numerical model for a particular run of the BEM code) and the “real” hole (the hole to be detected in the real structure). A “real” hole was simulated in this work as a central hole, and the “measured” information on the sensor locations was obtained simply as numerical results for the interior points corresponding to these sensor locations. With this approach, a comparison was done between the numerical results obtained with the “real” hole and the numerical results obtained for several “numerical” holes approaching the center of the plate, to simulate the desired effect of proximity between holes. For this discussion, several runs were made using the acoustic model, with the “numerical” holes approaching the “real” hole, as illustrated in Fig. 5(a). Numerical results for the time distribution of the potential at interior points were obtained for the various cases. Fig. 5(b) illustrates the fact that, for a particular interior point, the differences in the potential (throughout a given time interval) decrease when the “numerical” hole approaches the “real” hole. Thus, the area between the “real” curve and the curve obtained from the “numerical” hole decreases in the vicinity the real hole. 7,0E-04

0,08

far away hole (1.0,-2.0) real hole (3.0,0.0) near hole (2.5,-0.5)

3,0

0

x (cm)

Potential "u"

1,0

Areas between curves of potencials

6,0E-04 5,0E-04

Ȉ(u1 – ui)^2

0,06

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4,0E-04

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0,00 0,87

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1

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Far away hole

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4

Near hole

(a) (b) (c) Figure 5 – influence of the proximity between “numerical” and “real” holes for the acoustic model: (a) setup of “numerical” holes approaching the “real” hole; (b) comparison for potential results at an interior point; (c) comparison of areas between curves for the time distribution of the potential at these points. In this work, a metric was established for a numerical measure of the approximate area between these curves of the potential u for the various cases, evaluated as shown in Eq. (2):

A

t

2

i t

³ f (t ) f (t ) dt | ¦ ¬ª f t f t ¼º 0

1

1

2

i

2

i

2

(2)

i 1

where: - f1 - potential at a given interior point for the “real” hole; - f2 - potential at a given interior point for the “numerical” hole being simulated. Fig. 5(c) shows the plot of this metric, used to approximate the áreas between curves, where teh integrals were approximated by summations, in the form of Eq. (2). One can note that the metric used for the areas between curves of potential at the interior point considered has decreased when the “numerical” hole has approached the “real” hole, thus showing that this metric has the ability to properly capture the proximity between the “numerical” hole and the “real” hole. For further developments and discussions, one can expect that a more adequate metric to indicate the proximity between the “numerical” hole and the “real” hole could be given by the summation of the areas between curves of potential for a greater set of interior points, where sensors would be placed in the real structure. Furthermore, following the above discussion of the first issue, one can expect also that the information of the areas between curves of derivatives of the potential would be more sensitive to the proximity between the “real” and “numerical” holes than the information on the differences of areas for the potential. Either by using the potential or the derivative of the potential, a functional for the differences among areas between curves, for the various interior points corresponding to sensor locations, could be built in a form similar to Eq. (1), by replacing the potential information by the areas between curves. One such functional J could be used in an optimization algorithm, in the form of a minimization problem ( min J Ai ), as the inverse problem to detect the presence of the hole in the struture.

¦

The optimization algorithm is currently being implemented for the acoustics problem, and is already implemented for the elastostatics problem. In what follows, the issues proposed were analyzed and discussed using numerical results for the inverse problem using genetic algortithm (GA) for the optimization, and using the elastostatics model as the direct problem.

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The third issue is the use of independent scalar fields as the variables of interest for the optimization procedure, to avoid comparing derivatives obtained in different planes and directions at a particular interior point. The elastostatics problem shown in Fig. 2 was solved for the mean stress at interior points of the damaged plate, and also for the stress components in the x and y directions. The plots in Fig. 6(a) show a comparison between the good damage identification results (in this case, location of the ycoordinate of the center of the hole) using this scalar field, and the poor results (both in terms of mean value and of uncertainty) for the identification when some elements of the original stress tensor (say, normal or shear stresses in the given directions x and y) where used as variables in the optimization procedure. Fig. 6(b) and (c) show the plots of the location and size of the holes obtained from 10 different runs of the genetic algorithm (GA) used in the optimization procedure. The genetic algorithm, due to its own randomness, generates a different optimal solution every time it is run, but the GA results obtained from the 10 runs using the mean stress present a tendency to be more concentrated near the “real” hole than the GA results using other variables. This feature is clearly ilustrated in Fig. 6(b) and (c), for locating the hole using the mean stress (ım) and the normal stress in x-direction (ıx), respectively.

(a) (b) (c) Figure 6 - comparison between results using a scalar quantity (ım) versus vector components (ıx, ıy, Ĳxy) as variables in the optimization procedure (a) location of y-coordinate of identified hole (mean values and uncertainty); (b) plot of holes located using ım; (c) plot of holes located using ıx.

15

0,2 12 elements 48 elements

10 5

0,15 COV

percentual error [%]

Also, one must note that the results obtained using the mean stress and/or the octahedral stress are direction-independent, while a simple rotation of axis could have led to completely different results for the identification procedure using the original components of the stress tensor as variables for the optimization problem. The fourth issue is the influence of the numerical errors due to the BEM discretization in the optimization results. A comparison is shown in Fig. 7 for the optimization results (using 10 runs of a GA algorithm) and for the quality of the damage identification for two different holes, for two meshes (a coarse mesh and a fine mesh) using the elastostatics formulation for the plate shown in Fig. 2(a). Fig. 7(a) shows illustrative results for the mean values of the error in the location (x and y coordinates) and size (radius r) of a center hole, while Fig. 7(b) shows illustrative results for the values of the coefficient of variation (COV) of the error in the location and size of a hole located at a different position (x=2.5, y=3.0).

12 elements 48 elements

0,1 0,05

0

0 x

y r hole location and size

x y hole location and size

r

(a) (b) Figure 7 - influence of BEM discretization in illustrative damage identification results: (a) mean values for the error in location and size of a center hole; (b) coefficient of variation (COV) of the error in the location and size of a hole located at a different position. The results presented in the plots in Fig. 7 indicate the influence of the mesh refinement, showing a trend in which the fine mesh has led to better damage identification results, both in mean values and in terms of


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uncertainty, as expected. These results are preliminary, as several different runs need yet to be made, for different damage types, sizes and locations, and for different degrees of freedom in the mesh refinement. Conclusions In this work, four issues were discussed concerning the use of the Boundary Element Method (BEM) as the direct problem in an inverse problem of identifying damages in a plate structure using optimization techniques. The BEM codes discussed include potential, elastostatics and acoustical distributions throughout the damaged structure. The focus in the discussion was the improvements in the damage identification and also in the confidence in the damage location and size results. The numerical results obtained led to some preliminary conclusions the issues discussed, as follows: i) the use of quantities obtained from derivatives of the original densities instead of the densities themselves, as the main variables in the objective functions and constraint equations, is expected to lead to faster convergence of the optimization procedure. Results were presented in this work for the acoustics case, but available preliminary results (not shown) for the elasticity case indicate the same trend; ii) the BEM codes used in the direct method have the ability properly capture the proximity between the “numerical” damage and the “real” damage. Results were presented in this work were for the acoustics case in time domain. One can expect results for the acoustics problem in frequency domain to present the same trend (ongoing research). Available preliminary results (not shown) for the elasticity case also indicate the same trend; iii) the use direction-independent scalar quantities (in this work, invariants of the stress tensor for the elasticity problem) at the interior points of interest (where measured and numerical information are compared in the optimization procedure), has led to consistently accurate results for the damage identification, when comparing to the results obtained using direction-dependent quantities directly obtained from the BEM output in the optimization algorithm; and iv) the BEM discretization (mesh refinement) has shown an influence in the numerical results, for the elastostatics case evaluated, as the damage identification results for the finer meshes tested were more accurate and presented less uncertainty than in the case of the coarse meshes tested.. Acknowledgements The authors acknowledge the support received from the Brazilian Agencies: CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico and FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais. References [1] Banks, HT; Inman, DJ; Leo, DJ & Wang, Y (1996): An experimentally validated damage detection theory in smart structures, Journal of Sound and Vibration, 191, 5, p. 859-880 [2] Burczynski, T & Beluch, W (2001): The identification of cracks using boundary elements and evolutionary algorithms, Engineering Analysis with Boundary Elements, 25, p. 313-322 [3] Brebbia,CA; Dominguez,J (1992): Boundary elements-an introductory course, 2nd Ed, McGraw-Hill [4] Paris F, Cañas J (1997): Boundary element method-fundamentals & applications, Oxford Univ. Press [5] Aliabadi, MH (2002): The boundary element method V.2:applications in solids and structures, Wiley [6] Mansur, JW (1983), A Time-Stepping Technique to Solve Wave Propagation Problems Using The Boundary Element Method, PhD Dissertation, University of Southampton, UK [7] Dominguez, J. (1993): Boundary Elements in Dynamics, Elsevier [8] Beskos, DE (1997): Boundary element methods in dynamic analysis: Part II (1986-1996), Appl. Mech. Rev., V. 50, 3, p. 149-197 [9] Wrobel,LC(2002): The boundary element method:applications in thermo-fluids&acoustics,V1,Wiley [10] RAO, SS (1996): Engineering Optimization - Theory and Practice, 3rd Ed., John Wiley [11] Goldberg, DE (1989): Genetic algorithms in search, optimization & machine learning, AddisonWesley [12] Bäck, T; Schwefel, H-P (1993): An overview of evolutionary algorithms for parameter optimization, Evolutionary Computation 1, 1, p. 1-23

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309

Algebraic preconditioning techniques for large-scale boundary integral equations in electromagnetism: a short survey Bruno Carpentieri1, a 1

Institute for Mathematics and Scientific Computing, Karl-Franzens University, Graz (Austria) a

[email protected]

Keywords: boundary integral equations, electromagnetism, preconditioning, radar cross section calculation.

Krylov

subspace

methods,

Abstract. The Fast Multipole Method has been introduced by Greengard and Rokhlin to reduce the complexity of large particle simulations. Its use has been successively extended to the analysis of wave scattering problems. For Maxwell and Helmholtz equations, it can reduce significantly the algorithmic and memory cost of one matrix-vector product with matrices arising from the Galerkin discretization, avoiding the storage and computation of all the entries. However, for most integral equations of practical interest the number of iterations remains unacceptably large and the use of robust preconditioning is urged. In this paper we survey some recent advances in the design of algebraic preconditioners for this problem class. Nowadays realistic simulations involving many million unknowns can be carried out in only a few hours of CPU time on a moderate number of processors. Introduction Recent developments on boundary element techniques have contributed to increase significantly the popularity of integral equation methods for the solution of Helmholtz and Maxwell equations on 2D and 3D problems as an efficient alternative to differential equation methods. Scattering applications address the physical issue of detecting the diffraction pattern of the electromagnetic (EM) radiation that is scattered by a large and complex body illuminated by an incident incoming radiation. The relevant interest for EM scattering problems is due to the fact that their accurate numerical solution is required in the simulation of many industrial processes, such as the prediction of the Radar Cross Section (RCS) of arbitrarily shaped 3D objects like aircrafts, the analysis of EM compatibility of electrical devices with their environment, the design of antennas, absorbing materials, and many others. Objects of interest in industrial applications generally have large dimension in terms of the wavelength, and the computation of their scattering cross section can be very demanding in terms of computer resources. Classical discretization schemes like the finite-element method (FEM) or the finite-difference method (FDM) can be used to discretize the continuous model and give rise to a sparse linear system of equations. The domain outside the object is truncated and an artificial boundary is introduced to simulate an infinite volume (see e.g. [1,2]). In the last thirty years a growing research attention has been devoted to integral equation methods, because they require a simple surface description of the target by means of triangular facets; this means that a 3D volume problem is reduced to solving a 2D problem on the surface simplifying the mesh generation significantly. This feature is especially advantageous for modeling moving objects and avoiding grid dispersion errors which typically occur in the discretization of large 3D domains by differential equation solvers (see e.g. [3]). The Method of Moments discretization of integral equations [4] gives rise to dense and complex-valued linear systems. The number of unknowns N of the pertinent system grows linearly with the size of the scatterer and quadratically with the frequency of the incoming radiation. In fact, for physical consistency it is necessary to use at least ten discretization points per wavelength [5] to recover the oscillating behavior of the Green's function as well as geometric or physical singularities of the scatterer. An electromagnetic scattering simulation of a

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realistic aircraft illuminated at one GHz of frequency may lead to linear systems with a few millions unknowns. In-core direct solvers, that have O(N2) memory complexity and O(N3) algorithmic complexity in a straightforward implementation, are not feasible even for solving medium-size problems. Indeed, the large dimension of these linear systems is often the main computational bottleneck limiting the viability of integral equations in real-life simulations. Iterative Krylov methods can solve the memory bottleneck of direct methods provided fast matrix-vector (M-V) product operations are employed. For boundary integral equations, efficient algorithms have been proposed to carry out fast and approximate M-V products using less than the straightforward O(N2) arithmetic operations and memory locations. The Fast Multipole Method (FMM) by Greengard and Rokhlin is an algorithm in this class [6,7]. Originally proposed in the context of particles simulations to evaluate rapidly the potential and force fields in systems involving a large number of interactions, it was successively applied to develop fast Helmholtz and Maxwell solvers expressed in an integral formulation in acoustic and electromagnetic scattering applications (see e.g. [8,9]). The algorithm computes interactions amongst degrees of freedom in the mesh at different levels of accuracy, depending on their physical distance. A two-level implementation of FMM reduces both the memory and the complexity of a matrix-vector product operation from O(N2) to O(N1.5), a three level to O(N4/3), and the Multilevel Fast Multipole Algorithm (MLFMA) to O(N log N). From a linear algebra point of view, with the help of the addition theorem the FMM can be represented as a decomposition of the coefficient matrix A as the sum of three terms A = Adiag + Anear + Afar,

(1)

where Adiag is the block diagonal part of A associated with interactions of basis functions belonging to the same box, Anear is the block near-diagonal part of A associated with interactions of basis functions belonging to one level of neighboring boxes (they are 9 in 2D and 26 in 3D), and Afar is the far-field part of A. In the M-V product operation, the contributions Adiag· x and Anear· x can be derived from MoM and are computed exactly, while the product Afar· x is computed approximately by MLFMA; note that MLFMA is applied to off-diagonal matrix elements only, which are two or three orders of magnitude less than diagonal elements for EM scattering problems. A matrix problem involving N unknowns can be solved using Krylov subspace solvers in Į · niter· O(Ax) flops where the constant Į depends on the implementation of the specific iterative method, niter is the number of required iterations to achieve a certain accuracy for the approximate solution and O(Ax) is the algorithmic complexity of each M-V product. The number of iterations niter can be unacceptably large for some integral operators. Analytical preconditioners are based on regularized techniques that may lead to integral formulations requiring less iterations, but they are more expensive and in general problem dependent. Another approach is to use algebraic preconditioners; a preconditioner M is a matrix that transforms the original linear system into an equivalent one which is more amenable to the iterative solution. Preconditioning is generally not required for solving smooth and compact integral operators [10,11] while it is essential for non-compact operators associated with singular integral equations. Also, properties of the target like geometry and material may affect the speed of convergence. Problems with cavities or open surfaces are likely to require many more iterations than closed objects of the same physical size; nonuniform meshes are known to produce ill-conditioned MoM matrix equations as well. In this short paper, we survey recent results with algebraic preconditioners for this problem class. Preconditioning boundary integral equations For surface integral equations, three formulations are generally considered. For open targets, the so-called Electric Field Integral Equation (EFIE) is mandatory to use; for closed targets, the Magnetic Field Integral Equation (MFIE) can be used. Both formulations suffer from


311

nonuniqueness of the solution. This problem can be solved by combining linearly EFIE and MFIE. The resulting formulation, known as Combined Field Integral Equation (CFIE), is the formulation of choice for closed targets. The standard Galerkin method can be used to discretize the integral equation and generate a dense linear system of equations. The coefficient matrix of the linear system is symmetric for EFIE, nonsymmetric for CFIE and MFIE. The unknowns of the linear system are associated with the vectorial flux across an edge in the mesh, and the right-hand side depends on the frequency and the direction of the illuminating wave. CFIE leads to a Fredholm equation of the second kind and the coefficient matrix of the discretized system is well conditioned with most of the eigenvalues grouped in the right half-plane [12]. On CFIE, unsymmetric Krylov solvers scale as O(N0.25). For EFIE, most of the eigenvalues are scattered in the left half-plane and some are grouped close to zero; near-singularity of the discretized operator is also revealed by pseudospectra analysis [13]. As a result, Krylov methods scale as O(N0.5) and preconditioning is crucial to accelerate the convergence. Most of the preconditioners for dense matrices follow a common design pattern: given a decomposition of the system of the form (S+B)x=b

(2)

where S is sparse, it retains the most relevant contributions to the singular integrals and is easy to invert, while B can be dense, we compute M from S using only local information. The transformed preconditioned system has the form (I + S-1 B) x = S-1b.

(3)

The motivation to consider decompositions of the form (2) can be settled in the framework of splitted operators [14]; of course, the choice of S is important for performance. The simplest approach is to compute S from A by means of sparsification strategies that use either algebraic, graph or mesh information. Algebraic strategies compute S by dropping all the entries lower than a prescribed threshold [15,16]; graph-based strategies exploit information extracted from the connectivity graph of the underlying physical mesh [16, 17] by performing a breadth-first search on the neighbors of each edge; mesh-based strategies use the spatial coordinates of the nodes in the mesh describing geometric neighborhoods amongst the edges [18]. Comparative experiments reported in [18] suggest that there is little to choose. All different approaches can provide good approximations to the dense coefficient matrix for very low sparsity ratios (up-to 2%). However, geometric information can take into account possible deformations of the geometry and are particularly suited to capture geometric singularities of non-smooth scatterers, like in the presence of breaks on the surface, cavities, disconnected parts. A splitted form of the discrete operator is directly available when fast integral solvers like the Fast Multipole Method are considered. We recall that fast methods partition the mesh of the object by recursive subdivision into disjoint aggregates or boxes of small size compared to the wavelength, and the coefficient matrix of the pertinent system can be written in the form (1). In this case, we may take S = Adiag + Anear as sparse local matrix that is determined using information from the underlying physical mesh. In the next sections we examine the behavior of some popular algebraic preconditioning methods for this problem class. Incomplete factorization. The Incomplete LU (ILU) preconditioner is one of the most robust and efficient algebraic method for preconditioning a linear system. It computes an approximate triangular factorization of S by performing an incomplete Gaussian elimination; sparsity may be imposed on the triangular factors extracting information from either the graph of the matrix or the magnitude of the entries of the approximate factors [19]. Early experiments with ILU on small systems arising from the discretization of boundary integral equations are found in [20]. The most

312


basic preconditioner in this class is ILU(0) that uses the sparsity pattern of S for the triangular factors; it can be considered the method of choice for preconditioning CFIE as it significantly improves block Jacobi and often delivers rates of convergence very similar to LU [13]. The setup of ILU(0) is completely automatic so that the iterative solver maintains the same algorithmic and memory cost of FMM. More sophisticate methods that allow some fill-in in the factors may result in a slight reduction of the number of iterations but for CFIE their use does not pay off in general as the construction and the application are more expensive. On the other hand, on indefinite formulations like EFIE, ILU(0) is not robust enough, and it is necessary to introduce some fill in the factors. The reason is that EFIE gives rise to linear systems whose coefficient matrix is indefinite and thus more difficult to solve. To control memory storage, the maximum number of entries per column in the approximate factors can be taken to be of the same order as the number of nonzeros in the near-field matrix; this value is easy to determine from the problem after the initial setup. A threshold parameter can be also introduced during the factorization to drop small entries that contribute little to the quality of the preconditioner. For sequential runs, the incomplete factorization is computationally attractive and competitive because the construction is quite cheap. The parallelization is not straightforward but can be carried out efficiently using domain decomposition techniques with moderate computational overhead. At our knowledge, the largest reported experiment so far with ILU preconditioners on the EFIE has O(105) unknowns [13]; the scalability of ILU for very large problems remains a relevant open research issue to explore. Due to indefiniteness, on the EFIE the triangular factors may be ill-conditioned and the triangular solves numerically unstable so that the results may be unpredictable (see Table 1). This numerical issue may be determined by eccessive dropping and/or occurrence of small pivots during the factorization. The true reason of failure can be detected using condition estimators. The conditioning should be checked before starting the iterations at the additional cost of one forward and backward substitution. If it reveals potential ill-conditioning of the factors, the reason of failure is mostly due to the presence of small pivots. In this case, the matrix needs to be preprocessed using either diagonal shifts [20], reordering schemes, or pivoting [13]. Pivoting seems the most robust stabilization approach. Table 1: Experiments with incomplete factorization preconditioner with no pivoting on a model problem (size n=2048). Density of sparsification of A = 2% IC (level) Density of L Cond (L) GMRES(30) IC(0) 2.0% 2·103 378 IC(1) 5.1% 1·103 79 IC(2) 9.1% 9·102 58 Density of sparsification of A = 4% IC (level) Density of L Cond (L) GMRES(30) IC(0) 4.0% 6·109 IC(1) 11.7% 2·105 IC(2) 19.0% 7·103 40

Approximate inverse methods. In the last ten years, a considerable amount of research work has been devoted to approximate inverse methods for solving boundary integral equations (see e.g. [18,21,22,23]). The idea is to compute an explicit sparse approximation M of A-1 or of its factors and use it as preconditioner at each step of an iterative solver. This approach is clearly appealing for massively parallel implementations as the preconditioning operation reduces to carry out one or more sparse matrix-vector products. In the general case it is not obvious how to determine a good sparsity pattern for M that captures most of the largest entries of A-1; indeed, it is known that the


313

inverse of an irreducible matrix is structurally dense. However, for dense matrices arising from the discretization of boundary integral equations selecting a suitable structure of M turns out not to be a critical issue if M is computed in unfactorized form. Owing to the rapid decay of the discrete Green's function, the entries of A-1 tends to decay very rapidly far from the diagonal and many of them are very small. The discrete Green’s function can be considered as a row or as a column of the exact inverse depicted on the physical computational grid. Additionally, the pattern of A-1 after dropping the small entries resembles very closely to the pattern of a sparsification of the coefficient matrix (see Figure 1).

Figure 1: Nonzero pattern of A (on the left) A-1 (on the right) after dropping the small entries.

This means that a very sparse matrix can effectively approximate the exact inverse. One natural option is to impose on M the same pattern of the near-field matrix S; this strategy generally provides very good results. Thus the preconditioner can be efficiently combined with FMM techniques resulting in a purely gray-box method for this problem class. Pattern selection strategies based on graph or mesh or algebraic information for approximate inverse methods are thoroughly discussed in [18,21,22]. Frobenius-norm minimization methods have been successfully used by several authors on this problem class; they compute the approximate inverse as the matrix M that minimizes I AM F , subject to certain sparsity constraints. The Frobenius-norm is generally chosen since it allows the decoupling of the constrained minimization problem into N independent linear least-squares problems, one for each column (resp. row) of M when preconditioning from the right (resp. left). The independence of the least-squares problems follows immediately from the identity I AM

2 F

n

¦e j 1

j

Am• j

2 2

where ej is the j-th canonical unit vector and m• j is the column vector representing the j-th column of M. Both the construction and the application of M are embarrassingly parallel. The box-wise decomposition of the domain carried out by FMM naturally leads to an a priori pattern selection strategy for M using geometric information, that is on the spatial distribution of its degrees of freedom. The nonzero structure of the column of the preconditioner associated with a given edge is defined by retaining all the edges within its leaf box and those in one level of neighbouring boxes. The approximate inverse turns out to have a sparse block structure; each block is dense and is

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associated with one leaf box. Indeed the least-squares problems corresponding to edges within the same box are identical because they are defined using the same nonzero structure and the same set of entries of A. It means that only one block QR factorization per leaf box is necessary and can be computed using BLAS-3 that enables to exploit data locality in the cache memory and reduce the setup cost [24]. It is evident that the construction of the preconditioner is straightforward to parallelize.

Multilevel methods The lack of scalability with respect to the frequency of the problem is an important limit of local preconditioners. Indeed, the number of nonzeros in the MoM part of the FMM matrix tends to decrease with the frequency. When the matrix S becomes very sparse, information related to the farfield is completely lost and some global mechanism has to be introduced to recover information on the numerical behavior of the discrete Green’s function. Multilevel mechanisms are designed to enhance the robustness and the scalability of the iterative solver, possibly preserving inherent parallelism. The multipole matrix is a suitable candidate to consider also in the design of global preconditioners. The two-level algorithm described in [24] perfoms a few steps of an inner Krylov method for the preconditioning operation (see Figure 2). The outer solver must be able to work with variable preconditioners; amongst various possibilities, we mention FGMRES [19]. From a numerical point of view, the efficiency of the algorithm relies on two main factors: the inner solver has to be preconditioned so that the residual in the inner iterations can be significantly reduced in a few steps, and the matrix-vector products within the inner and the outer solvers can be carried out with a different accuracy. The desirable feature of using different accuracies for the matrix-vector products is enabled by the use of FMM: a highly accurate FMM has to be used within the outer solver, as it governs the final accuracy of the computed solution. A less accurate FMM is used within the inner solver, as it is a preconditioner for the outer scheme. Inner-outer schemes can enable the solution of problems involving several millions unknowns on parallel computers [24,25]. Iterative solver: flexible Krylov method Do{ x x

M-V product: y = AFMM · x Preconditioning : z = M-1 y (GMRES, QMR, ...)

For i=1,2,... x M-V product : M § AFMM x Preconditioning : Approximate inverse } until convergence Figure 1: Inner-outer iterative schemes with different levels of accuracy for the M-V product operations.

Other approaches. Efficient multigrid methods having the same memory and arithmetic cost of the M-V product operation have been proposed for integral equations problems, like the hypersingular and the singlelayer potential integral operators arising from the Laplace equation (see e.g. [26]). Geometric multigrid can be tricky to implement as they need a hierarchy of grids; on the other hand, algebraic multigrid can offer the advantage to preserve good convergence properties using only single grid information. Combining multigrid with fast solvers makes it possible to maintain the cost of the preconditioning procedure of the same order of a M-V multiplication up to some constant factor


315

[27]. Finally, we mention that another active research area is currently addressing the development of efficient preconditioners based on wavelet transformations for this problem class (see e.g. [28]). Nowadays the main bottleneck is rapidly moving from the design and simulation to the pre- and postprocessing of the results, as the tools are not yet available to easily manipulate large meshes with millions of degrees of freedom. Table 2: Number of inner+outer iterations and CPU time for experiments on a realistic aircraft simulation on a Compaq machine (8 processors, 1.2 Gflops). GMRES() Iter Time 213084 973 7h 19m 591900 1461 16h 42m • 1160124 -• dof

FGMRES(30,60) Iter Time 30+1740 6h 10m 57+3300 1d 10h 51+2940 16h 41m •

Notation: - : memory or CPU time exceeded; • = run on 64 processors; d=day, h= hour, m=minutes.

References [1] J.-P. Berenger: A perfectly matched layer for the absorption of electromagnetic waves, J. Comp. Phys. Vol. 114 (1994), p. 185-200. [2] M. J. Grote: Nonreflecting Boundary Conditions for Electromagnetic Scattering, Int. J. Numer. Model. Vol. 13 (2000), p. 397-416. [3] W. R. Scott Jr: Errors due to spatial discretization and numerical precision in the finite-element method, IEEE Trans. Ant. Prop. Vol. 42(11) (1994), p. 1565-1569. [4] A.F. Peterson and S.L. Ray and R. Mittra: Computational Methods for Electromagnetics (IEEE Press, 1997). [5] A. Bendali: Approximation par elements finis de surface de problemes de diffraction des ondes electro-magnetiques, Ph.D. thesis Université Paris VI, 1984. [6] J. Carrier, L. Greengard and V. Rokhlin: A Fast Adaptive Multipole Algorithm for Particle Simulations, SIAM J. Scientific Computing Vol. 9(4) (1988), p. 669-686. [7] L. Greengard and V. Rokhlin: A Fast Algorithm for Particle Simulations, Journal of Computational Physics Vol. 73 (1987), p. 325-348. [8] V. Rokhlin: Rapid solution of integral equations of scattering theory in two dimensions, J. Comp. Phys. Vol. 86 (2) (1990), p. 414-439. [9] J.M. Song and W.C. Chew: Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering, Mico. Opt. Tech. Lett. Vol. 10(1) (1995). [10] S. Amini and K. Chen: Conjugate gradient method for second kind integral equations applications to the exterior acoustic problem, Engineering Analysis with Boundary Elements Vol. 6(2) (1993), p. 72-77. [11] P.M. Hemker and H.Schippers: Multigrid methods for the solution of Fredholm integral equations of the second kind, Mathematics of Computation Vol. 36 (1981), p. 215-232.

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[12] J.R. Mautz and R.F. Harrington: H-Field, E-Field, and Combined Field Solutions for Conducting Bodies of Revolution, Arch. Elektron. Übertragungstech Vol. 32(4) (1978), p. 159-164. [13] T. Malas and L. Gürel: Incomplete LU preconditioning with multilevel fast multipole algorithm for electromagnetic scattering, SIAM J. Scientific Computing Vol. 29(4) (2007). [14] K. Chen: An analysis of sparse approximate inverse preconditioners for boundary integral equations, SIAM J. Matrix Analysis and Applications Vol. 22(3) (2001), p. 1058-1078. [15] L. Yu. Kolotilina: Explicit preconditioning of systems of linear algebraic equations with dense matrices, J. Sov. Math. Vol. 43 (1988), p. 2566-2573 [16] S.A. Vavasis: Preconditioning for boundary integral equations, SIAM J. Matrix Analysis and Applications Vol. 13 (1992), p. 905-925. [17] K. Chen and P.J. Harris: Efficient preconditioners for iterative solution of the boundary element equations for the three-dimensional Helmholtz equation, Applied Numerical Mathematics Vol. 36(4) (2001), p. 475-489. [18] B. Carpentieri and I.S. Duff and L. Giraud: Sparse pattern selection strategies for robust Frobenius-norm minimization preconditioners in electromagnetism, Numerical Linear Algebra with Applications Vol. 7(7-8) (2000), p. 667-685 [19] Y. Saad: Iterative Methods for Sparse Linear Systems (PWS Publishing, New York, 1996) [20] B. Carpentieri, I.S. Duff, L. Giraud and M.~Magolu monga Made: Sparse symmetric preconditioners for dense linear systems in electromagnetism, Numerical Linear Algebra with Applications Vol. 11(8-9) (2004), p. 753-771. [21] G. Alléon and M. Benzi and L. Giraud: Sparse Approximate Inverse Preconditioning for Dense Linear Systems Arising in Computational Electromagnetics, Numerical Algorithms Vol. 16 (1997), p. 1-15. [22] K. Chen: On a class of preconditioning methods for dense linear systems from boundary elements, SIAM J. Scientific Computing Vol. 20(2) (1998), p. 684-698. [23] A.R. Samant and E. Michielssen and P. Saylor: Approximate inverse based preconditioners for 2D dense matrix problems, Technical Report CCEM-11-96, University of Illinois (1996) [24] B. Carpentieri and I.S. Duff and L. Giraud and G. Sylvand: Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations, SIAM J. Scientific Computing Vol. 27(3) (2005), p. 774-792. [25] T. Malas, O. Ergül and L. Gürel: Sequential and Parallel Preconditioners for Large-Scale Integral-Equation Problems. In Proceedings of the CEM’07 Conference, Izmir, August 2007. [26] U. Langer, D. Pusch and S. Reitzinger: Efficient Preconditioners for Boundary Element Matrices Based on grey-box Algebraic Multigrid Methods, International Journal for Numerical Methods in Engineering Vol. 58(13) (2003), p. 1937-1953. [27] U. Langer and D. Pusch: Data-sparse algebraic multigrid methods for large scale boundary element equations, Applied Numerical Mathematics Vol. 54 (3-4) (2005), p. 406-424. [28] S.C. Hawkins and Ke Chen: An Implicit Wavelet Sparse Approximate Inverse Preconditioner, SIAM J. Scientific Computing Vol. 27(2) (2005), p. 667-686.


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Solution of Nonlinear Reaction-Diffusion Equation by Using Dual Reciprocity Boundary Element Method with Finite Difference or Least Squares Method 1,a

G. Meral 1

1,b

M. Tezer-Sezgin

Department of Mathematics, Middle East Technical University, 06531,Ankara, Turkey. a

[email protected], b [email protected]

Keywords: Dual Reciprocity Boundary Element Method, Finite Difference Method, Least Squares Method, Nonlinear Reaction-Diffusion Equation.

Abstract: In this study, the system of time dependent nonlinear reaction-diffusion equations is solved numerically by using the dual reciprocity boundary element method (DRBEM). As the time integration method both the finite difference method (FDM) with a relaxation parameter and the least squares method (LSM) are made use of. The DRBEM is applied for spatial derivatives keeping the nonlinear term and the time derivative as nonhomogenity. The resulting time dependent system of ordinary differential equations (ODE) is solved by using the FDM with a relaxation parameter as well as the LSM for obtaining accurate and computationally efficient results. The computations are carried out for one nonlinear reaction-diffusion equation which has an exact solution and for one system of reaction-diffusion equations. The solution obtained with both methods agree well with the exact solution in the first example and with the other numerical results in the second example. The comparison of both time integration schemes shows that the FDM with a relaxation parameter gives better accuracy when the optimal value of the parameter is used. This makes the solution procedure time consuming and computationally expensive comparing to the LSM which is a direct application. 1. Introduction The nonlinear reaction diffusion equation as well as the system of nonlinear reaction-diffusion equations are very attractive in recent years, since they have practical applications in many fields of science and engineering. A number of combined methods for the time dependent partial differential equations is applied in the literature. For solving these problems classical methods discretize the spatial domain of the problem with one of the known methods such as boundary element method(BEM), finite element method(FEM), differential quadrature method(DQM) and finite difference method(FDM); then the resulting system of time dependent equations is solved by using the time integration schemes such as FDM, RKM(Runge-Kutta Method),FEM, LSM etc. In the linear case, for the one-dimensional convection-diffusion equation and for the two-dimensional diffusion equation, the FDM is used in both the space and the time directions in [1]and [2], respectively. Also, the heat and the heat conduction equations are solved by the DRBEM and with one and two step LSM in [3] and [4], respectively. They have also examined the three and four step LS schemes for the heat equation in [5] and have found that one and two step methods are more efficient. Recently, the nonlinear reaction-diffusion equation is solved by the combined application of DRBEM and FDM in [6] and with DQM and FDM as well as the one-dimensional Fisher’s equation in [7]. In the study of [8], the two-dimensional reaction-diffusion Brusellator system is solved by using DRBEM for space and FDM for time derivatives which includes a linerization procedure. In the present paper, the nonlinear reaction-diffusion equation as well as the system of nonlinear reaction-diffusion equations is solved by using the DRBEM for the spatial derivatives.Then for the time integration two different schemes are introduced, namely the FDM with a relaxation parameter

318


and the LSM. Then the effect on the convergence behaviour of the proposed FDM and the comparison with the LSM are presented. We also compare the computational efficiency (based on the computational time and accuracy) of both methods. 2. Definition of the Problem We consider the following system of nonlinear reaction-diffusion equations ∂uj = ν∇2 uj + pj (x, y) ∈ Ω, t > 0 j = 1 or j = 2 ∂t with the initial and mixed-type boundary conditions

(1)

uj (x, y, 0) = gj (x, y)

(2)

βj (x, y, t)uj + γj (x, y, t)qj = 0

(x, y) ∈ Γ, t > 0

(3)

∂uj , ∂n

n being the outward normal on the boundary. where Γ is the boundary of the domain Ω and qj = Here the nonlinearity pj depends on the unknowns, i.e. for a single equation the nonlinearity is p1 (u1 ) and for the system, the nonlinearities are p1 (u1 , u2 ) and p2 (u1 , u2 ) for the first and second equations in Eq. 1. 3. DRBEM Formulation Eq. 1 can be weighted by the fundamental solution u∗ = obtain ν(ci (uj )i +

Γ

(q ∗ uj − u∗ qj )dΓ) =

Ω

(

1 2π

ln 1r of Laplace operator in order to

∂uj − pj )u∗ dΩ ∂t

(4) ∗

after the application of the Divergence Theorem [9]. Here i denotes the source(fixed) point, q ∗ = ∂u ∂n and ci = Ω ∆(xi , yi , x, y)dΩ. The nonhomogenity, i.e. the time derivative and the nonlinear term, can be approximated using radial basis functions fj (x, y) resulting with a linear system of equations to be solved [F ] {α (t)} = {b}

(5)

where N and L are the number of boundary and selected interior nodes, respectively, and [F ] contains fj s as coloumns . The radial basis functions fj are related to other distance functions uˆj (x, y) through the relation ∇2 uˆj = fj in order to obtain the boundary integral only form after the application of the Divergence theorem [10].This leads us to the following matrix-vector formulation after the usage of the relationship (5) and substitution of the nonhomogenity vector b

∂u ˆ [F ]−1 − {p (u)} ν ([H] {uj } − [G] {qj }) = [H] Uˆ − [G] Q ∂t

(6)

where H and G denote the whole matrices of boundary elements with kernels q ∗ and u∗ , respectively. ˆ compromise the coordinate function column vectors uˆj and qˆj . The sizes of all the matrices Uˆ and Q in (6) are (N +L)×(N +L) and the vectors are of size (N +L)×1. Defining a new (N +L)×(N +L) matrix C as ˆ [F ]−1 . (7) [C] = − [H] Uˆ − [G] Q


Eq. 6 can be rearranged as ∂uj [C] + ν [H] {uj } − ν [G] {qj } = [C] {pj } . ∂t

319

(8)

4. Time Integration 4.1 Finite Difference Method The system of ordinary differential Eq.s 8 for the unknown uj can be written as

∂uj βj ¯ ¯ G + H {uj } (9) = {pj } − ∂t γj ¯ = ν [C]−1 [G], after the application of the boundary condition (3) with the matrices G −1 ¯ H = ν [C] [H]. In this subsection we will employ Euler scheme for the time derivative [10]

m+1 m βj ¯ ¯ m = uj +∆t pm − . (10) uj G + H uj j γj Since the method is explicit, the stability problems can be encountered and ∆t must be taken carefully. A relaxation procedure is employed with a parameter 0 ≤ µ ≤ 1 for the unknown uj in the form m+1 positioning the values of uj between the time levels m and m + 1. Then uj = (1 − µ)um j + µuj the Eq. 10 takes the form m+1 m

¯ + H ¯ ¯ ¯ G uj = I − ∆t (1 − µ) dm uj + ∆t pm I + ∆tµ dm+1 j G + H j j (11) βm j m where dm j = γ m evaluated at the time level t .

j

4.2 Least Squares Method When the boundary conditions are applied to the Eq. 8, then it can be rewritten as ∂uj [C] (12) + [Kj ] {uj } = [C] {pj } ∂t β where [Kj ] = ν [H] + γjj [G] . In a typical time element of length ∆t we approximate vectors uj m+1 and qj as uj ≈ Φ1 (t) um where φ1 (t) = 1 − ξ, φ2 (t) = ξ with ξ = t − tm are the j + Φ2 (t) uj linear interpolation functions.We construct the error functional Πj as in [4] tm rT rdt (13) Πj = tm+1

where r is the residual vector, which is obtained by substituting the approximate solution in Eq. 12. The LSM solution is obtained after minimizing the unknown uj from the system of equations as

= [B1 ] um + [B2 ] pm [A] um+1 j j j where [A] =

1 1 T 1 T T T [C] [Kj ] + [Kj ] [C] + [Kj ] [Kj ] [C] [C] + ∆t2 2∆t 3

(14)

320


µ 0.7 0.8 0.9

∆t=0.01 5.3 1.2(−3) 1.2(−3)

t=0.5 ∆t=0.1 1.7(−3) 2.0(−3) 2.3(−3)

∆t=0.5 1.2(−2) 1.3(−2) 1.4(−2)

∆t=0.01 7.07 1.1(-3) 1.1(-3)

t=2.0 ∆t=0.1 ∆t = 0.5 2.3(-3) 1.6(-2) 2.8(-3) 1.9(-2) 3.3(-3) 2.3(-2)

∆t=1.0 4.0(-2) 4.6(-2) 5.3(-2)

Table 1: Maximum absolute errors at small times with FDM for Problem 1 µ 0.7 0.8 0.9

∆t=0.01 7.3 4.5(−4) 4.5(−4)

t=8.0 ∆t=0.1 ∆t = 0.5 8.1(−4) 5.7(-3) 8.7(−4) 6.1(-3) 9.2(−4) 6.5(-3)

∆t=1.0 1.1(−2) 1.3(−2) 1.4(−2)

∆t=0.01 7.4 6.0(−7) 6.0(−7)

t=20.0 ∆t=0.1 ∆t = 0.5 3.3(−6) 1.7(-5) 3.6(−6) 1.8(-5) 3.8(−6) 1.8(-5)

∆t=1.0 3.2(−5) 3.5(−5) 3.8(−5)

Table 2: Maximum absolute errors for steady state with FDM for Problem 1

1 1 1 T T T T [C] [C] − ] [K ] [K [C] [K ] + [K ] [C] − j j j j ∆t2 2∆t 6

1 1 [C]T [C] + [Kj ]T [C] . [B2 ] = ∆t 2

[B1 ] =

The solution of the system of nonlinear reaction-diffusion equations is obtained from the solution of the algebraic Eq.s 11 or 14 . Note that in both time integration schemes the nonlinearity p (u) is approximated only at the time level tm in order to obtain a linear system of equations at the end. 5. Numerical Results Problem 1.We consider the nonlinear reaction-diffusion equation [2] with ν = 12 , p1 (u1 ) = (1 − u1 ) in the square region {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} for t ≥ 0 . The initial condition and the mixed type boundary condi1 . tions are taken appropriate with the exact solution u1 (x, y, t) = 1+ep(x+y−pt) To measure the quality of the approximate solutions with both methods we use the maximum absolute error which is defined by max |uexact − unum | where uexact , unum denote the exact and numerical solutions for the problem. The results in terms of these maximum absolute errors for different time increments are presented in Tables 1-2 at the point x = 12 for several values of relaxation parameter. Table 3 shows maximum absolute errors for the LSM solution at the same time levels with the same time increments, again at x = 12 . One can see from these tables that, the time increment ∆t = 0.1 is the right choice for both the methods but an optimal value of relaxation parameter is required in FDM. This is of course computationally expensive procedure. The LSM is preferred as a direct method although there is a one digit drop for small times and two digits drop for steady-state in the accuracy. u21

Problem 2. Next we solve the system of nonlinear reaction-diffusion equations with the nonlin1 in the same unit square earities p1 (u1 , u2 ) = 1 + u21 u2 − 23 u1 , p2 (u1 , u2 ) = 21 u1 − u21 u2 with ν = 500 ∆t 0.01 0.1 0.5 1.0

t=0.5 5.3(-1) 2.8(-2) 2.0(-2)

t=2.0 6.3(-1) 6.9(-2) 1.6(-1) 1.3(-1)

t=8.0 1.17 6.5(-2) 2.4(-1) 2.0(-1)

t=20.0 9.6(-1) 1.6(-4) 9.6(-4) 1.4(-1)

Table 3: Maximum absolute errors with LSM for Problem 1


321

in problem 1, subject to the initial and Neumann type boundary conditions

1 2 1 3 1 2 1 3 (x, y) ∈ Ω (u1 (x, y, 0) , u2 (x, y, 0)) = x − x, y − y 2 3 2 3

∂u1 ∂u2 , ∂n ∂n

= (0, 0)

(x, y) ∈ Γ, t > 0.

Figure 1: Solution of Problem 2

For this problem we cannot compare the convergence of both methods since we do not have any exact solution. But we see that both methods show the same expected behaviour with the reference solution in [8], i.e. u1 → 1 and u2 → 1/2 as time increases. We have found the optimal value of the relaxation parameter in FDM as 0.3 for the solution. Fig. 1 shows that the behaviour of the solution agrees with the behaviour of the solution in [8]. The solution is obtained with a considerable small number of boundary elements (N =8) in the use of both FDM and LSM. 6. Conclusion The system of nonlinear reaction-diffusion equations is solved by using the coupling of the method DRBEM in space with both the FDM and the LSM in time. The DRBEM results in a system of ODE’s in time and FDM(Euler) with a relaxation parameter gives quite good accuracy without the need of very small time increment. The optimal value of relaxation parameter is time consuming. In the LSM the accuracy drops one or two digits in small and large time values, respectively but it does not require very small time increment and relaxation parameter. The DRBEM requires very small number of boundary elements for obtaining a reasonable accuracy when it is coupled either with FDM or LSM time integration schemes .

References [1] M.M. Chawla, M.A. Al-Zanaidi, M.G. Al-Aslab, Computers and Mathematics with Applications, 39,71-84(2000). [2] M.M. Chawla, M.A. Al-Zanaidi, Computers and Mathematics with Applications, 42, 157168(2001). [3] K. M. Singh and M. S. Kalra, Engineering Analysis with Boundary Elements,18, 73-102(1993). [4] K. M. Singh and M.S. Kalra, Computer Methods in Applied Mechancis and Engineering, 190, 111-130(2000).

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[5] K. M. Singh and M.S. Kalra, Communications in Numerical Methods in Engineering, 12, 425431(1996). [6] G. Meral, Proceedings of the International Conference BEM/MRM 27, 133-140(2005). [7] G. Meral and M. Tezer-Sezgin, International Journal of Computer Mathematics, (in print). [8] W.T. Ang, Engineering Anlaysis with Boundary Elements, 27, 897-903(2003). [9] C.A. Brebbia and J. Dominguez, Boundary Elements an Introductory Course, Computational Mechanics Publications McGraw-Hill book company(1992). [10] P.W. Partridge, C.A. Brebbia and L.C. Wrobel, The dual reciprocity boundary element method, Computational Mechanics Publications Elsevier Applied Science(1992).


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A Galerkin Boundary Element Method with the Laplace transform for a heat conduction interface problem Roman Vodiˇcka Technical University of Koˇsice, Faculty of Civil Engineering, Vysokoˇskolska´ 4, 042 00 Koˇsice, Slovakia [email protected]

Keywords: boundary elements, domain decomposition, non-matching meshes, heat conduction, Laplace transform.

Abstract. The solution of a heat conduction problem with domain decomposition by a Laplace-transform method based on a boundary element technique is treated. A series of the complex valued boundary value problems for Helmholtz equation is solved in the frequency domain by the complex Symmetric Galerkin Boundary Element Method and subsequently transformed back to the time domain by the inverse Laplace transform. The algorithm of the domain decomposition solution includes generally curved interfaces and independent meshing of each side of an interface producing non-matching boundary element meshes. The examples present obtained numerical results and they are compared with analytical solutions. Introduction Time-dependent problems that are modeled by initial-boundary value problems (IBVP) can be treated by boundary integral equation (BIE) method. Such methods are widely and successfully being used also for numerical modeling of problems in heat conduction. A nice survey of BIE applications for time dependent problems is given in [2]. The present paper introduces an approach utilizing the Laplace transfrom. Such methods solve the problems in frequency domain, usually for complex frequencies. For each fixed frequency, the heat conduction problem reduces to a BIE for a boundary value problem (BVP) of the modified Helmholtz equation. The transformation back to the time domain includes special methods for inversion of the Laplace transform guided by the choice of the time dependence approximation. The split of the space domain into several parts (due to physical properties of materials or for a parallelization of the solving process etc.) usually includes domain decomposition techniques to be used [6, 10]. In the present approach, these methods due to interfaces are applied very naturally to the BIE system for the Helmholtz equation solved in the frequency domain as BIEs use unknowns directly on the boundary so that techniques derived for elliptic problems can be applied. The Boundary Element Method (BEM) as a numerical tool for solving BIE is also used in this context [4, 5, 8]. The numerical implementation of a curved interface is very important – it is naturally included in the discussed formulation together with the non-matching meshes along both sides of interfaces. As documented in the aforementioned papers, the curved boundaries present a strong ability of each problem with an interface approach. Various numerical implementations of the data transfer between two non-matching meshes via calculation of integrals over the discretized curved surfaces has been given in [1]. The present approach uses the implementation of data transfer using an auxiliary interface mesh referred to as common-refinement mesh. The heat conduction problem with an interface Let us consider a body defined by a domain, Ω ⊂ R2 in a fixed cartesian coordinate system xi (i = 1, 2), with a bounded Lipschitz boundary ∂Ω = Γ. Note, that Γ may include corners but not cracks and cusps. Let ΓS ⊂ Γ denote the smooth part of Γ, i.e. excluding corners, edges, points of curvature jumps, etc. Let n denote the outward unit normal vector defined on ΓS . Although the developed formulation is valid in 3D space as well, for the sake of simplicity we confine ourselves only to the analysis in 2D continuum.

324


The presence of interfaces cause the domain Ω to be split into several parts. For the sake of simplicity, let us consider a split into two non-overlapping parts ΩA and ΩB whose respective boundaries we denote ΓA and ΓB . There exists a common part of both boundaries ΓA and ΓB , let us denote this coupling boundary by Γc . The initial-boundary value problem with an interface (IBVPwI) of a heat conduction problem for a temperature distribution u(x, t) without volume heat sources can be stated as ∂uη (x, t) − aη2 ∆uη (x, t) = 0, x ∈ Ωη , t > 0, η = A, B, ∂t uη (x, 0) = uη0 (x), x ∈ Ωη , ∂uη (x, t) q η (x, t) = −k η uη (x, t) = g η (x, t), x ∈ Γηu , = hη (x, t), x ∈ Γηq , t ≥ 0 ∂n uA (x, t) − uB (x, t) = 0, q A (x, t) + q B (x, t) = 0, x ∈ Γc , t ≥ 0, where the material parameter aη2 =

kη c η ρη

(1a) (1b) (1c) (1d)

includes specific heat cη , density ρη and thermal conductivity k η . The

split of each boundary Γη due to the boundary and interface conditions can be written as: Γη = Γηu ∪ Γηq ∪ Γc (∅ = Γηu ∩ Γηt = Γηu ∩ Γc = Γηt ∩ Γc ). Therefore, functions g η (x, t) and hη (x, t) introduce given boundary conditions, while the function uη0 (x) defines the initial condition. The unilateral Laplace transform is applied to obtain the solution in the frequency domain, let the transform temperature solution be u(x, p) =

∞

0

u(x, t)e−pt dt,

p ≥ σ0 ,

(2)

with σ0 being the abscissa of convergence for the Laplace transform. Moreover, let the boundary conditions η being defined by functions with separated variables, i.e. g η (x, t) = gx (x)gtη (t), hη (x, t) = hηx(x)hηt (t). Then the IBVPwI (1) is transformed into a domain BVP with an interface (BVPwI) for the modified Helmholtz equation with the parameter p η

puη (x, p) − aη2 ∆uη (x, p) = uη0 (x),

u (x, p) =

η gx (x)gηt (p), B

x∈

Γηu ,

uA (x, p) − u (x, p) = 0,

η

q (x, p) =

x ∈ Ωη ,

hηx(x)hηt (p),

(3a) x ∈ Γηq ,

qA (x, p) + qB (x, p) = 0, x ∈ Γc .

(3b) (3c)

Naturally, the temperature solution u(x, t) is finally found by the inverse Laplace transform of u(x, p). The boundary integral equation system The problem (3) can be solved by a system of BIEs for a fixed parameter p. The fundamental solution of the governing differential equation and its necessary derivatives are known — they are given by the modified Bessel functions of the second kind Ki (z), (i = 0, 1, 2). All necessary functions and derivatives can by introduced by the following relations U η (x, y, p) = −(2πk η )−1 K0 (

√

p x − y ), a

Qη (x, y, p) = −k η

Qη∗ (x, y, p) = −k η

∂U η (x, y, p) , ∂ny

∂U η (x, y, p) , ∂nx Dη (x, y, p) = k η2

∂ 2 U η (x, y, p) . (4) ∂nx∂ny

The BIE system for BVPwI to be solved includes the boundary conditions (3b) and also the interface conditions (3c) in a weak form when solved by the Galerkin method. The formulation is based on a variation technique derived in [8]. In order to write the BIE system in a compact and transparent matrix form, we introduce an operator notation: η η ωrη Zrs ws = ω η (y)Z η (x, y, p)wη (x, p) dΓ(x) dΓ(y), (5a) Γηr Γηs η ωrη ZrΩ wη = ω η (y)Z η (x, y, p)wη (x) dΩ(x) dΓ(y), (5b) Γηr

Ωη


325

where ω stands for a weighting function, w stands for u or q, r and s stand for u, q or c, Z stands for U , Q, Q∗ or D. Moreover, the inner integral can be regular, singular or, in the case of (5a), also Hadamard finite-part integral. Hereinafter, indices u, t or c introduce a restriction of a function or of an operator to a part of boundary Γ with the same index. The matrix form of the resulting system then reads ⎞T⎛−U A uu ϕA u A∗ A ⎜ ϑq ⎟ ⎜ Qqu ⎜ A ⎟⎜ ⎜ A ⎜ϕc ⎟⎜ −Ucu ⎜ A ⎟⎜ ⎜ ϑc ⎟⎜ QA ∗ cu ⎜ B ⎟⎜ ⎜ϕu ⎟⎜ 0 ⎜ B ⎟⎜ ⎜ ϑq ⎟ ⎜ 0 ⎜ ⎟⎜ ⎝ϕB ⎠⎝ 0 c ϑB c 0 ⎛

⎞⎛ ⎞ A QA −Uuc QA 0 0 0 0 uq uc qA ⎟⎜ uA ⎟ A A∗ A −Dqq Q qc −Dqc 0 0 0 0 ⎟⎜uq ⎟ ⎟⎜ A ⎟ A A AB QA −Ucc − 21 Icc +QA 0 0 0 Icc ⎟⎜ qc ⎟ cq cc ∗ ⎟⎜ A ⎟ A A A −Dcq − 12 Icc +QA cc −Dcc 0 0 0 0 ⎟⎜uc ⎟ ⎟ B⎟ B B B B qu ⎟ 0 0 0 −Uuu Quq −Uuc Quc ⎟⎜ ⎟⎜ ∗ ∗ B⎟ B B ⎟⎜ 0 0 0 QB qu −Dqq QB qc −Dqc ⎜uq ⎟ ⎟ 1 B ⎝ qB ⎠ B B B⎠ 0 0 0 −Ucu QB −U I +Q c cq cc cc 2 cc ∗ BA B 1 B B∗ B uB c 0 Icc 0 QB cu −Dcq −Dcc 2 Icc +Q cc ⎛ ⎞ A A A A Uuq − 12 Iuu − QA 0 0 − akA 2 UuΩ 0 ⎛ A ⎞T uu A ϕu ⎜ 1 A ∗ ∗ ⎟⎛ ⎞ k A A Duu 0 0 QA 0 ⎟ A ⎟ ⎜ 2 Iqq − Q qq ⎜ ϑA aA 2 A uΩ ⎟ h ⎜ qA ⎟ ⎜ k A A A ⎟⎜ A ⎟ U −T 0 0 − U 0 ⎜ ϕc ⎟ ⎜ 2 A cq cu cΩ ⎟⎜ g ⎟ a ⎜ A⎟ ⎜ A ∗ ∗ ⎟⎜ B ⎟ ⎜ k A A A ⎜ ϑc ⎟ ⎜ −Q cq D 0 0 0 ⎟⎜h ⎟ 2 Q cΩ A cu ⎟ a =⎜ ⎜ ⎟ B ⎟ , (6) B ⎟ ⎜ϕ B 1 B k B B B ⎟ ⎜ 0 0 Uuq − 2 Iuu − Quu 0 − aB 2 UuΩ ⎟⎜ u⎟ ⎜ B ⎜g ⎟ ⎜ ⎜ ϑq ⎟ ⎜ B ⎠ ∗ ∗ ⎟⎝uA 1 k B B B B 0 ⎜ ⎟⎜ 0 0 I − Q D 0 Q 2 qq uΩ ⎟ uu 2 qq aB B ⎠⎜ ⎝ϕ B ⎟ uB 0 c k B B B ⎝ 0 0 Ucq −Qcu 0 − aB 2 UuΩ ⎠ ϑB c ∗ ∗ kB B 0 0 −QB cq Dcu 0 QB uΩ aB 2

η AB , I BA ) operators. Function ϕη where Irr is formal representation (5) of the identity (Irr ) and projection (Icc r cc and ϑηr should be chosen to form the basis of the function spaces, where the functions qηr and uηr , respectively, belong to. The found functions solve (3), while the solution of the original IBVPwI of heat conduction can be found by inverse Laplace transform. Some details of the numeric procedure applied will be noted in what follows.

Notes on the numerical solution Boundary element approximation. The numerical solution of BIE system (6) by a boundary element technique includes discretization of the boundary Γη of each subdomain Ωη by Weη boundary elements Γη k , k = 1, 2, . . . Weη . For making the expressions simpler, let us omit the superscript index η in this paragraph. The present discretization utilizes conforming isoparametric elements. Let us consider an n-th order polynomial parameterization of k-th boundary element Γk given by the relation x = N k (ξ), ξ ∈ 0, 1. Then the approximation of the functions u(x, p), q(x, p) can be written in the following form u(x, p) =

Wu

ϑm (x)um ,

m=1

q(x, p) =

Wq

ϕm (x)qm ,

(7)

m=1

where the nodal shape functions ϑm (x), resp. ϕm (x) are equal to element shape functions N kj (x) for some k, j such that x ∈ Γk , j ∈ {1, 2, . . . n + 1}. Moreover, the functions ϑm should be continuous, while the functions ϕm do not have to be. Nevertheless, it is clear that all the functions defined by the parameterization are smooth together with all their derivatives along an element. The relation between the parameterization of boundary N k (ξ) and the element shape functions N kj (x) is N kj (x) = N kj (N k (ξ)) = N j (ξ), where the polynomials N j (ξ) form the basis of the n-th order polynomials space. An example of the second order continuous elements and their parameterization is shown on Figure 1. It can be seen that the support of a function ϑm (x) can cover up to two neighbour elements: compare ϑm (x) and ϑm+1 (x). Calculation of integrals. The discretization of the BIE system (6) also includes calculation of the integrals (5) in the form 1 1 N li (y)Z(x, y, p)N kj (x)dΓ(x)dΓ(y) = N i (υ)Z(N k (ξ),N l (υ), p)N j (ξ) N k (ξ) N l (υ) dξdυ (8) Γl Γk

0 0

326


ϑm+1 (x) = N k3 (x)

ϑm+1 (x) = N (k+1)1 (x)

ϑm (x) = N k2 (x) Γk+1

ξk

=1 Ω Γk ξk

N kj(

=0

Γq

x) =

N j(ξ

1

N 2 (ξ)

)

N 1 (ξ) N 3 (ξ)

0

1 2

1

ξ

Figure 1: Second order elements and their shape functions for each particular integral kernel Z(x, y, p) and for each pair of boundary elements Γk and Γl . The idea of the calculation is to split the integral into a regular part and into a singular part if it occurs. The regular parts then can be evaluated e.g. by a standard Gauss-Legendre quadrature rule, while the singular parts are treated separately according to the type of the singularity. The integrals with Z = U and Z = Q can contain only logarithmic singularities, the other case Z = D may result even in an hypersingular integral, see also [7]. In order to demonstrate the way of calculation, let us discuss the most singular case in more detail. The √ p integral (8), with respect to (4) and to the notation r = |r|, r = x − y = N k (ξ) − N l (υ) and c = aη , reads 1 1 1 1 ∂ 2 K0 (c r) j = N i (υ) N (ξ) N k (ξ) N l (υ) dξdυ = N i (υ)N j (ξ) N k (ξ) N l (υ) × ∂nx∂ny 0 0 0 0

2 (r T nx)(r T ny ) 1 1 nTx ny K dξdυ+ + (c r) − ln r I (c r) − −K2 (c r) − ln r I2 (c r) + 2 2 1 1 c r r2 c cr r 1 1 (r T nx)(r T ny ) I1 (c r) T + nx ny dξdυ− N i (υ)N j (ξ) N k (ξ) N l (υ) ln r I2 (c r) r2 cr 0 0 1 1 T T 1 (r nx)(r ny ) 1 = N i (υ)N j (ξ) N k (ξ) N l (υ) 2 2 2 − nTx ny dξdυ = DR + DL − 2 DH , (9) c r r2 c 0 0 where Ii , i = 1, 2 are the modified Bessel functions of the first kind. The terms are reordered for the first integral DR to contain regular functions only, the second integral DL to contain a logarithmic singularity for r → 0 and the last term DH to be a hypersingular integral if r = 0 is a point of the integration domain. The logarithmic singularity should be treated in such a way that a suitable weighted quadrature formula could be used. Two cases should be distinguished for calculation of the integral DL : first, when the integrals in (8) are calculated with respect to the same elements, i.e. k = l and, second, when the elements are neighbouring in the mesh, having one common point. In the former case, the distance function r renders r = N k (ξ) − N k (υ) = N k (ξ)(ξ − υ) − 12 N k (ξ)(ξ − υ)2 + · · · = (ξ − υ)g(ξ, υ)

(10)

for a sufficiently smooth non-vanishing vector function g(ξ, υ) if ξ = υ. Therefore DL can be written as 1 1 1 1 1 1 DL = ln r G(ξ, υ)dξdυ = ln |g(ξ, υ)| G(ξ, υ)dξdυ + ln |ξ − υ| G(ξ, υ)dξdυ =DLR +DLL , (11) 0 0

0 0

0 0

with G(ξ, υ) being the non-singular and smooth rest of the integrand and with the split of the result into the regular part DLR and the singular part DLL . To be able to correctly apply a quadrature rule which requires the logarithmic function at the end point of the interval, following substitutions are required: ξ ∈ 0, 1 ∧ υ ∈ 0, ξ :

ξ = γ, υ = γ(1 − δ), J = γ,

γ ∈ 0, 1 ∧ δ ∈ 0, 1

(12a)

ξ ∈ 0, υ ∧ υ ∈ 0, 1 :

υ = γ, ξ = γ(1 − δ), J = γ,

γ ∈ 0, 1 ∧ δ ∈ 0, 1

(12b)


327

to obtain DLL =

1

1

0 0

˜ δ)γdγdδ = ln(γδ) G(γ,

1 0 0

1

˜ δ)dγdδ + γ ln γ G(γ,

0

1

ln δ

0

˜ δ)dγ dδ. γ G(γ,

1

(13)

In the latter case, the relation to be satisfied for boundary parameterization is e.g. N k (1) = N l (0). The integration domain is split into two triangles along the diagonal ξ = υ, with the following substitution to be applied to the triangle with the singularity: ξ ∈ 0, 1 ∧ υ ∈ 0, ξ :

ξ = γ(1 − δ) + 1, υ = γδ, J = γ,

γ ∈ 0, 1 ∧ δ ∈ 0, 1.

(14)

r = N k (γ(1 − δ) + 1) − N l (γδ) = N k (1) [γ(δ − 1)] − N l (0) (γδ) + · · · = γh(γ, δ),

(15)

At the vicinity of the singularity, the distance function r can be expressed as

with a function h(γ, δ) sufficiently smooth and non-vanishing for γ = 0, as the Lipschitz condition supposed for the boundary makes the expression N k (1) (δ − 1) − N l (0) δ not to vanish. The integral DL obeys the relation 1 υ 1 1 ˜ ˜ δ)γdγdδ = DL = ln r G(ξ, υ)dξdυ + ln(γ|h(γ, δ)|) G(γ, 0 0 0 0 1 1 1 1 ˜ ˜ ˜ δ)dγdδ + ˜ δ)dγdδ, (16) DLR + γ ln |h(γ, δ)| G(γ, γ ln γ G(γ, 0 0

0 0

where neither the first term DLR – the integral over the triangle without singularity, nor the second term contain any singular function and only the last integral includes a logarithmic function with zero argument, though smoothed by Jacobian γ. The last term in (9), DH , requires exceptional treatment as it includes hypersingular integral. This singularity has been obtained in a limit procedure, where a point x at the boundary Γ(x) has been approached by moving a point y from the interior of the domain Ω . Therefore, for the calculation of the integral, the boundary Γ(y) can be shifted by a small ε inwards to Ω resulting in an integral, which does not contain any singularity. to calculate the integral DH introducing a complex function z(ξ, υ) = N1k (ξ) − N1l (υ) + Itk is also useful l i N2 (ξ) − N2 (υ) because a simple calculation via integration by parts renders following result 1

ξ=1 υ=1 ∂z(ξ, υ) ∂z(ξ, υ) + dξdυ = − ln rN i (υ)N j (ξ) ξ=0 ∂ξ ∂υ υ=0 0 0 ξ=1 υ=1 1 1 1 1 N j (ξ) ln r N i (υ)dυ + N i (υ) ln r N j (ξ)dξ − ln r N j (ξ)N i (υ)dξdυ. (17)

DH =

1

N i (υ)N j (ξ)

0

1

z 2 (ξ, υ)

ξ=0

0

υ=0

0 0

The basic polynomial functions N j (ξ) and N i (υ) can be smoothly differentiated along an element, moreover they are (already parameterized) restrictions to particular boundary elements of continuous nodal shape functions ϑmj (x) and ϑmi (y), respectively. In the calculation, the integrals over the support of pertinent ϑfunctions, which vanish at the end points of their support, can be gathered together. Therefore, the three free terms also vanish and only the last integral remains. Finally, the aforementioned limit procedure should be performed. The integral, however, has remained only with logarithmic singularity, which can be treated as above. The inverse Laplace transform. The algorithm of numerical inverse of the Laplace transform due to Weeks has been applied, see [9]. It is based on the fact that the Laplace transform of a function expressed by a series with respect to appropriately scaled orthonormal Laguerre functions can be easily modified to calculate the approximation of its inverse Laplace transform by the quarter wave cosine fast Fourier transform. The numerical procedure includes several parameters which should be carefully chosen to obtain a quickly convergent and accurate results. The discussion of the choice of the parameters has been also presented in [3]. The parameters which are necessary for calculation are: abscissa of convergence of Laplace transform σ0 , abscissa of

328


evaluation of the inverse Laplace transform σ, scaling parameters of the Laguerre functions b and of the series coefficients evaluation r and also number N of functions used from the series, equal to the number of points in frequency domain for the inverse transform evaluation and the time interval determined by maximum required time instance tmax . An example

ΓB q

ΩB ΓB q

0.4

E ΓA q

S3 S2 Γc x 2 S1

ΓA u

F

0.2

An example has been chosen to document the numerical behaviour of the proposed numerical method. It includes a simple square divided into two subdomains, see Figure 2, with the interface defined by a cubic spline passing through the points E, Si and F , where the coordinates of Si are as follows: S1 [0.5; 0.4], S2 [1.0; 1.1], S3 [1.5; 1.6]. 2

ΓA q x1

ΩA

O Figure 2: A square domain with a curved interface with the pattern of element nods in the interface The solution of the problem has been chosen so that it is analytically expressed by the series formula ∞ ∞ (2m+1)πx2 (2m+1)πx2 2 2 −t nπx1 u(x, t) = x1 x2 1 − e + Tnm (t) sin cos 2 + T0m (t) sin , (18a) 4 4 m=1

T0m (t) =

32 3π(2m+1)

Tnm (t) =

·

n=1

λ0m −1−λ0m e−t +e−λ0m t λ0m (λ0m −1)

128(−1)n π 3 n2 (2m+1)

·

+

32 π 2 (2m+1)2

(−1)m −

2 π(2m+1)

−t +e−λ0m t −t −λ0m t 0m e , (18b) + 4 λ0m −1−λ × − 83 e λ−e λ0m (λ0m −1) 0m −1

λnm −1−λnm e−t +e−λnm t λnm (λnm −1)

λnm =

−

n2 π 2 4

1024 π 4 n2 (2m+1)2

+

(−1)m −

(2m+1)2 π 2 . 16

2 π(2m+1)

e−t −e−λnm t λnm −1 ,

(18c) (18d)

It means that the vanishing initial conditions for simplicity (no need to compute the volume integrals in (6) due to (5b)) are prescribed. The boundary conditions, which types are shown on Figure 2, read u ((x1 , 0), t) = 0, q ((2, x2 ), t) = −4x22 1 − e−t , (19a) q ((x1 , 2), t) = −4x21 1 − e−t . q ((0, x2 ), t) = 0, (19b) The discretization by boundary elements has been made in such a way that the lengths of all elements are approximately the same. Each side of the outer square contour is divided into 20 linear elements. The interface is meshed accordingly: the non-matching mesh taken contains 24 equally spaced elements along the face of the domain ΩA , the coarser mesh, and 27 elements with respect to the other domain, finer mesh. The parameters of the numerical inverse of the Laplace transform has been set to the following values: N = 16 points in the frequency domain, abscissa of the convergence σ0 set to zero, maximum time evaluation being unity, parameters r = 0.99, b = 16, and abscissa of evaluation σ being 8. The results obtained in the interface has been focused on: first, the solution evolution in time and its error along the interface is shown on Figure 3 for the temperature u and on Figure 4 for the flux q, second, the


0.02 0.016 0.012 0.008 0.004 0

uA

|uAn − uAa |

10 8 6 4 2 0

329

0.0 0.2 0.4 0.6 0.8 1.0 t

2.0 1.5 1.0 0.5 0.0

x1

x1

2.0 1.5 1.0 0.5 0.0

0.0 0.2 0.4 0.6 0.8 1.0 t

2.0 1.5 1.0 0.5 0.0

0.3 0.25 0.2 0.15 0.1 0.05 0

0.0 0.2 0.4 0.6 0.8 1.0 t

2.0 1.5 1.0 0.5 0.0

x1

|q An − q Aa |

7 6 5 4 3 2 1 0

x1

−q A

Figure 3: Distribution of the temperature u and of its error from the mesh of the domain ΩA along the interface and with respect to the time t

0.0 0.2 0.4 0.6 0.8 1.0 t

Figure 4: Distribution of the flux q and of its error from the mesh of the domain ΩA along the interface and with respect to the time t distribution of both functions (u and q) and their errors are evaluated at a specific time instance on Figure 5. All graphs use the x1 -coordinate of interface points as the abscissa for the values of pertinent functions at a time instance. Let us comment some observations. The time evolution graphs show nice relation between the error magnitudes and the high gradients of the solutions obtained along the coarser mesh of the interface, with respect to the domain ΩA . The errors are naturally worse for the fluxes q. The graphs contain the absolute errors, i.e. |un −ua |, and |q n −q a |, where the superscript ‘n’ denotes numerical solution obtained by the BIE system (6) and the inverse Laplace transform and the superscript ‘a’ denotes the analytical solution obtained by (18a) truncated within each sum to 20 terms, however, the maximum relative errors can be estimated from them: it is about 0.002 for the temperatures u and about 0.04 for the fluxes q. The same observation can be also done from Figure 5, which has been made for the maximum time value. Moreover, the pictures also include the results obtained along the finer mesh, so that both data can be compared mutually. As the solutions and their errors are plotted in the same graph, the mutual relation between the magnitude of the error and the descent steepness is even more obvious. The interface mesh pattern shown on Figure 2 can help to understand the oscillating character in the error distributions: the errors cannot coincide because the meshes have few common points. Nevertheless, the errors are not significant and in the current graphs the results of both functions u and q, plotted for the domain ΩB , nicely fit with the analytical solution.

330


Conclusion A test of a boundary element approach for solving interface heat transfer IBVP has been performed. The formulation utilizing the Laplace transform of the solution solves the problem in the frequency domain numerically by a series of BVPs for Helmholtz equation. The numerical method used here includes the complex symmetric Galerkin BEM. The idea of the approach has been taken from a method previously derived by the authors in [8] for problems of elasticity and it seems to work well with the presented example. The investigations and numerical tests planned for the proposed method will give it more rigorous explanation for its applicability in wider range of the initial-boundary value problems with interfaces.

0.008

6.0

0.004

4.0

0.200

|q An − q Aa | |q Bn − q Ba | qB exact

0.150 0.100

2.0

0.050

|q n − q a |

8.0

0.006

2.0 0.0 0.0

0.010

q

|uAn − uAa | 8.0 |uBn − uBa | uB 6.0 exact 4.0

|un − ua |

u

10.0

0.002 0.5

1.0 x1

1.5

0.000 2.0

0.0 0.0

0.000 0.5

1.0 x1

1.5

2.0

Figure 5: Results along the interface for the time t = 1

Acknowledgement The author gratefully acknowledge the Scientific Grant Agency VEGA for supporting this work under the Grant No. 1/4198/07. References [1] Boer, A. de, Zuijlen, A.H. van, Bijl, H.: Review of coupling methods for non-matching meshes. Comput. Methods Appl. Mech. Engrg, 196, pp. 1515–1525, 2007. [2] Costabel, M.: Time-dependent problems with boundary integral equation method. In: Encyclopedia of Computational Mechanics, John Wiley & Sons, Eds. Stein, de Borst, Hughes, vol. 1, chap. 25, 2004. [3] Garbow, B.S., Giunta, G., Lynnes, J.N., Murli, A.: Software for an implementation of Weeks’ method for the inverse Laplace transform problem. ACM T. Math. Software, 14, pp. 163–170, 1988. [4] Hsiao, G.C., Steinbach, O., Wendland, W.L.: Domain decomposition methods via boundary integral equations. J. Comp. Appl. Math., 125, pp. 521–537, 2000. [5] Langer, U., Steinbach, O.: Boundary element tearing and interconnecting method. Computing, 71, pp. 205 – 228, 2003. [6] Puso, M.A.: A 3D mortar method for solid mechanics. Int. J. Num. Meth. Engrg., 59, pp. 315–336, 2004. [7] Vodiˇcka, R.: On evaluation of integrals arising in SGBEM solution of modified Helmholtz equation. In: VIII. vedecká konferencia Stavebnej fakulty v Koˇsiciach, TU v Koˇsiciach, Stavebná fakulta, pp. 91–96, 2007. [8] Vodiˇcka, R., Mantiˇc, V., Par´ıs, F.: 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, pp. 173–203, 2007. [9] Weeks, W.T.: Numerical inversion of Laplace transform using Laguerre functions. Jornal of the Association for Computing Machinery, 13, pp. 419–426, 1966. [10] Wohlmuth, B.I.: Discretization Methods and Iterative Solvers Based on Domain Decomposition, Lecture Notes in Computational Science and Engineering, vol.17, Springer, Berlin, 2001.


331

Assembled Plate Structures by the Boundary Element Method D. D. Monnerat1, a, J. A. F. Santiago2, b and J. C. F. Telles2, c 1

Exactum Consultoria e Projetos Ltda, Rua Sete de Setembro, 43 20050-003 Rio de Janeiro, RJ, Brazil

2

Programa de Engenharia Civil - COPPE/UFRJ, Caixa Postal 68506 21941-972 Rio de Janeiro, RJ, Brazil

a


Keywords: Boundary element; Plates; Reissner’s plate theory.

Abstract. This paper deals with the analysis of assembled plate structures, subjected to arbitrary loadings, for which two dimensional plane stress elasticity and shear deformable plate bending theories are coupled. To this end, direct boundary element formulations, based upon Reissner’s plate theory and 2-D elasticity, are presented for elastostatic problems. The multi-region technique is employed to assemble the plates. Several plates sharing common interface boundaries are accommodated, including inclined ones. After a standard coordinate transformation, each region can be combined taking into account displacement compatibility and equilibrium equations in order to obtain the final equation system. 1. Introduction Structures composed of assembled plane elements with close or open cross sections have been employed in several branch of engineering, such as civil, mechanic, naval, aeronautics, etc.; mainly due to the advantage of attaining high flexural rigidity with low self weight. Papers by Palermo [1] and Palermo et al. [2] discuss plate assembles with close and open cross sections using Kirchhoff’s plate theory and two-dimensional elasticity. Dirgantara and Aliabadi [3] and Baiz and Aliabadi [4] also analyzed assembled plate structures under arbitrary loadings using Reissner’s plate theory and two regions sharing common interfaces. In the present work, the direct boundary element formulation for the multi-region technique, based upon Reissner´s plate and 2-D elasticity theories are presented for elastostatic problems, considering isotropic materials, small deformations and small displacements. Several plates sharing common interface boundaries are accommodated, including inclined ones, generalizing previous analyses [5]. Several numerical examples are presented and results are compared with exact analytical and finite element solutions to demonstrate the accuracy of the proposed formulation. 2. Boundary Integral Equations The equations will be presented here in indicial notation. Here roman indices vary from 1 to 3 and Greek indices vary from 1 to 2. The integral equations adopted to represent displacement components can be written as i) Reissner’s plate bending: C ij [ w j [ ³ pij* [ , x w j x d* x *

§

Q

·

³ wij [ , x p j x d*x ³ ¨¨© wi3 [ , x 1 Q O2 wiD ,D [ , x ¸¸¹qx d:x *

*

:

*

*

(1)

332


ii) Two dimensional plane stress elasticity: * [ , x u E x d*x CDE [ u E [ ³ tDE *

³ uDE [ , x t E x d*x , *

(2)

*

where the boundary integrals on the left-hand side are interpreted in sense of Cauchy’s principal value. In Equations 1 and. 2, wD are the rotations about the xD axes, uD are the displacements on the plane x1 x 2 and w3 is the displacement in the x 3 direction. The terms p j are the bending moments (j = 1, 2) and shear force (j = 3), which are given as pD

M DE n E and p3

the other hand tD represent in plane tractions, given by tD

O

QD nD , respectively. On

N DE n E , q is the distributed load and

10 / h , h being the thickness. The coefficients Cij [ depend on the boundary geometry at the

source point [. The kernels wij* [ , x and pij* [ , x represent the fundamental solution for plate * * [ , x and tDE [ , x represent the fundamental solution for plane elasticity. bending, whereas uDE

The complete expressions for them can be found in references [6-8]. Equations 1 and 2 represent five integral equations per functional node of a structure under bending and extensional effects in the local coordinate system. Three degrees of freedom come from Reisser’s plate theory and two degrees of freedom from plane stress elasticity.. The domain integral of the equation 1 can be transformed in a boundary integral applying the divergence theorem. Here q is considered constant (uniformly distributed load), hence Equation 1 can be written as Cij [ w j [ ³ pij* [ , x w j x d* x *

§ * · Q * * ³ wij [ , x p j x d*x q ³ ¨¨©X i,D [ , x 1 Q O2 wiD [ , x ¸¸¹nD x d*x , * *

where X i* is the particular solution of the equation X i*,DD

(3)

wi*3 . The expression for X i*,D can be found

in reference [8]. 3. Assemble of the Equations System

Equations 2 and 3 represent the basic expressions for the solution of spatial assembled plate problems using boundary element method (BEM). In general terms, the boundaries and interfaces of a structure are discretized in elements, for which displacements and tractions are interpolated by means of functional node values. The integral equations for plane stress elasticity and shear deformable plate bending are applied to all functional nodes, in a corresponding local coordinate system, for every region, generating a linear equation system of the following matrix form: ªH S « ¬ 0

0 º uS ½ » ® ¾ H P ¼ 5 x 5 ¯w P ¿ 5 x1

ªG S « ¬ 0

0 ½ 0 º ts ½ ® P¾ ® ¾ , P» G ¼ 5 x 5 ¯p ¿ 5 x1 ¯b ¿ 5 x1

(4)


333

where u S û1 u 2 `T , w P ^w1 w2 w3 `T , t S ^t1 t 2 `T and p ^p1 p 2 p3 `T are the displacements (u and w) and tractions (t and p) for plane stress elasticity and plate bending, respectively, b ^0 0 q`T is the domain load vector and G S , H S , G P and H P are the element influence matrices for plane stress elasticity (S) and plate bending (P), respectively. To solve the problem the equation system of each region must be referred to the same global coordinate system. Therefore, the local equation system (referred to the individual plate) is transformed to the global one by employing the coordinate transformation matrix M as u1 ½ °u ° ° 2° °° w3 °° ® ¾ ° w1 ° ° w2 ° ° ° ¯° 0 ¿°

u1 ½ °u ° ° 2° u °°u °° M ® 3¾, °I1 ° °I 2 ° u ° ° ¯°I3 ¿°

Mu and MT u

t1 ½ °t ° ° 2° °° p3 °° ® ¾ ° p1 ° ° p2 ° ° ° ¯° 0 ¿°

t1 ½ °t ° ° 2° p °° t °° M ® 3 ¾, ° m1 ° °m 2 ° p ° ° ¯°m3 ¿°

Mp (5)

MT p

where u û1 u 2 u 3 I1 I 2 I3 ` and p ^t1 t 2 t 3 m1 m2 m3 ` represent the displacements and tractions vectors, respectively, referred to the global coordinate system. One can observe in equations 5 that a new rotation I 3 (or bending moment m3 ) about the x3-axis is added to the previous u (or p) vector. Therefore a new equation per functional node is needed, namely a restrain equation, given by T

\ 31 m1 \ 32 m 2 \ 33 m3

0,

T

(6)

in which \ ij is the cosine of the angle between xi local and xj global axes. After a standard coordinate transformation, the sub-regions can be combined taking into account displacement compatibility and equilibrium of tractions along the interface boundaries, in order to obtain the final global equation system. Notice that now six degree of freedom per functional node are considered. These equations (compatibility and equilibrium equations) can be written as follows:

i) Displacement compatibility u11

u12

u1i

u 12

u 22

u 2i

u 31

u 32

u 3i

I11

I12

I1i

I 21

I 22

I 2i

I31

I32

I3i

ii) Equilibrium of tractions

(7)

334


t11 t12 t1i

0

t 2i

0

t 31 t 32 t 3i

0

t 12

t 22

(8)

m11 m12 m1i

0

m12 m22 m2i

0

m3i

0,

m31

m32

where the index i is the sub-region (individual plate) sharing common interface boundaries. After the boundary conditions are enforced, the global equation system can be solved in order to produce the global unknowns (displacements and tractions) on external boundaries and interfaces of the structure.

4. Numerical Examples Several examples of 3-D assembled plate structures under flexural and extensional loads, simultaneously, are analyzed. The results are compared with beam theory and finite element solutions to demonstrate the accuracy of the proposed formulation.

4.1. Cantilever I Beam In this example a cantilever beam with an I cross section (see figure 1) is studied. Dimensions and properties are: L1 = L2 = 40 cm, L3 = 100 cm, t1 = 0.5 cm, t2 = 1 cm, E = 21000 kN/cm2 and Ȟ = 0.3.

Figure 1: Geometrical dimensions of the beam The beam is subjected to a linear distributed load q, varying from -100kN/cm to 100kN/cm, on the web of the opposite end of the vertical support, as indicated in figure 2.


335

Figure 2: Cantilever beam subjected to linear distributed load To analyze this example, 42 elements and 100 functional nodes have been employed over the external boundaries and interfaces of the beam. The obtained results are compared with beam theory and the finite element method. For beam theory the solution is given by ux

M 2 x , 2 EI

(9)

where M is the applied bending moment and I is the moment of inertia with respect to the neutral axis. The obtained results for the vertical displacements of the beam along the neutral axis, are presented in figure 3. 0

10

20

30

40

50

60

70

80

90

100

0,000 -0,025

w (cm)

-0,050 -0,075 -0,100 -0,125 -0,150 -0,175 -0,200 x (cm) Beam Theory

BEM

FEM

Figure 3: Comparison of vertical displacements As seen in figure 3, the beam theory and both methods (BEM and FEM) are in close agreement, confirming the validity of the proposed method.

4.2. L-shaped Plate Structure In this second example three rectangular plates with different sizes and the same thickness were assembled in order to form the L-shaped plate structure, showed in figure 4

336


The geometric constants are: L1 = L2 = 100 cm, L3 = L4 = 50 cm, t = 5 cm, ș = 120º. The modulus of elasticity and Poisson´s ratio are taken to be 7000 kN/cm2 and 0.33, respectively.

Figure 4: Plate assembly geometry The L-shaped plate structure is loaded by the uniformly distributed load qy = qz =0.05 KN/cm, in the y and z directions, along the tip edge of the horizontal plate, as depicted in figure 5.

Figure 5: Cantilever plate subjected to the uniformly distributed loading in y and z directions The problem is modelled with three sub-regions, each having 16 elements and 42 functional nodes along the interfaces and external boundaries. The results for vertical and horizontal displacements along the cross section at x = 50 cm are shown in figure 6. These results are here compared with the finite element method (FEM). In order to improve the comparison, figure 6 presents the deformed shape in expanded scale with a factor of 10. As can be seen, the BEM result is in excellent agreement with the FEM.


337

130 120 110 100 90

w (cm)

80 70 60 50 40 30 20 10 0 0

10

20

30

40

50

60

70

80

90 100 110

v (cm) Undeform ed shape

BEM (deform ed)

FEM (deform ed)

Figure 6: Deformed structure

5. Conclusion The presented analysis of structures composed of 3-D associations of plane panels subjected to co-occurring bending and extension loads has been considered quite satisfactory, with accurate results in comparison to existing alternative procedures. In addition, the discussed BEM implementation has been found to lead to acceptable solutions even in case of rather coarse discretization alternatives, employing a reduced number of elements. It can, therefore, be recommended for such panel assembled structures existing in current engineering practice.

References [1] L. Palermo Jr., M. Rachid and W.S. Venturini: Analysis of Thin Walled Structures using the Boundary Element Method, Engineering Analysis with Boundary Elements. Vol. 9 (1992), pp. 359-363. [2] L. Palermo Jr., Analysis of Thin Walled Structures as Assembled Plates by Boundary Element Method (in Portuguese). Thesis of Doctor of Science (D.Sc.), Escola de Engenharia de São Carlos / USP, São Carlos, SP, Brazil, (1989). [3] T. Dirgantara and M.H. Aliabadi, Boundary Element Analysis of Assembled Plate Structures. Commun. Numer. Meth. Engng. Vol.17 (2001), pp. 749-760. [4] P.M. Baiz and M.H. Aliabadi, Local Buckling of Thin Walled Structures by BEM, Advances in Boundary Element Techniques, pp. 39-44, (2007).

338


[5] D.D. Monneratt, Analysis of Assembled Plate Structures using the Boundary Element Method (in Portuguese), Dissertation of Master of Science (M.Sc), COPPE / UFRJ, Rio de Janeiro, RJ, Brazil, (2008). [6] C.A. Brebbia, J.C.F Telles and L.C. Wrobel: Boundary Element Techniques: Theory and Applications in Engineering, Springer-Verlag, Berlin, (1984). [7] L.C. Wrobel and M.H. Aliabadi, The Boundary Element Method, Wiley, Chichester, (2002). [8] F. Vander Weeën, "Application of the Boundary Integral Equation Method to Reissner's Plate Model", International Journal for Numerical Methods in Engineering. Vol. 18 (1982), pp. 110., (1982).


339

Development of a time-domain fast multipole BEM based on the operational quadrature method in 2-D elastodynamics Takahiro SAITOH1,a, Sohichi HIROSE 2,b and Takuo FUKUI3,c 1 University of Fukui, 3-9-1, Bunkyo, Fukui-shi, Fukui, Japan Tokyo Institute of Technology, 2-12-1,O-okayama, Meguro-ku, Tokyo, Japan 3 University of Fukui, 3-9-1, Bunkyo, Fukui-shi, Fukui, Japan a [email protected], b [email protected], c [email protected] 2

Keywords: Operational Quadrature Method (OQM), Fast Multipole Method (FMM), Time-domain, Elastodynamics. Abstract. This paper presents a new time-domain Fast Multipole Boundary Element Method in 2-D elastodynamics. In general, the use of direct time-domain BEM sometimes causes the instability of time-stepping solutions and needs much computational time and memory. To overcome these difficulties, in this paper, the Operational Quadrature Method (OQM) developed by Lubich is applied to establish the stability behavior of the time-stepping scheme. Moreover, the Fast Multipole Method (FMM) is adapted to improve the computational efficiency for large size problems. The proposed method is tested for large-size elastic wave scattering by many cavities.

Introduction Since the Boundary Element Method (BEM) is known as a suitable numerical approach for wave analysis, time-domain transient problems have been solved by many researchers using BEM by Mansur and Brebbia[1], and Hirose[2]. In general, transient problems can usually be solved for unknown timedependent quantities by a direct time-domain BEM with a time-stepping scheme. However, the use of direct time-domain BEM sometimes causes two problems. The one is the instability problem of timestepping procedure and the other one is computational efficiency problem for a large size problem. Recently, to overcome the former problem, the Operational Quadrature Method (OQM), proposed by Lubich[3], has been used for the BEM formulation for some engineering problems such as 2-D scalar wave problem[4], poroelastic problem[5] and 2-D anisotropic problem[6]. In the formulation of BEM based on OQM (OQBEM), the convolution integral is numerically approximated by a quadrature formula whose weights are determined by the Laplace transformed fundamental solution and a linear multistep method. The computational complexity becomes O(LM 2 N) for the problem with M elements, N time steps, and L expansion terms. On the other hand, the latter problem still remain because it is difficult to solve a large scale problem with the large number of M by using OQBEM. In this paper, we propose a new time-domain fast multipole BEM based on OQM in 2-D elastodynamics. The Fast Multipole Method (FMM), proposed by Greengard and Rokhlin[7], is applied to the OQBEM to resolve the computational efficiency problem. After the description of basic concept and formulation of proposed method in 2-D elastodynamics, numerical examples for elastic wave scattering are demonstrated by using the proposed method. The computational efficiency of the proposed method is also confirmed.

340


Figure 1 A elastic wave scattering model. Operational Quadrature Method In this section, the operational quadrature method (OQM) is briefly described. The Operational Quadrature Method (OQM), first proposed by Lubich, approximates the convolution f ∗ g(t) by a discrete convolution using the Laplace transform of the time dependent function f(t − τ ). In general, the convolution integral is defined as follows: f ∗ g(t) =

t 0

f(t − τ )g(τ )dτ , t ≥ 0

(1)

where ∗ denotes the convolution. The convolution integral defined by Eq. 1 is approximated by OQM as follows: f ∗ g(nt)

ωn−j (t)g(jt)

(2)

j

where time t was divided into N equal steps t. Moreover, ωj (t) denotes the quadrature weights which are determined by the coefficients of the following power series with complex variable z, namely δ(ζ) )= ωn (t)z n . t n=0 ∞

F(

(3)

In Eq. 3, F is the Laplace transform of the time dependent function f. The power series defined in Eq. 3 can be calculated by Cauchy’s integral formula. Considering a polar coordinate transformation, the Cauchy’s integral is approximated by a trapezoidal rule with L equal steps 2π/L as follows: 1 ωn (t) = 2πi

F |ζ|=ρ

δ(ζ) t

ζ

−n−1

L−1 −2πinl δ(ζl) ρ−n dζ F e L . L t

(4)

l=0

where δ(ζ) is the quotient of the generating polynomials of a linear multistep method and ζl is given by ζl = ρe2πil/L . In addition, ρ is the radius of a circle in the domain of analyticity of F .


341

Time-Domain BEM Formulation in 2-D Elastodynamics ¯ in an exterior elastic media D as shown in We consider the 2-D elastic wave scattering by a scatterer D ¯ scattered waves are Fig. 1. When the incident wave uin hits the boundary surface S of a scatterer D, ¯ Assuming the zero initial conditions, i.e., ui (x, t = generated by the interaction with the scatterer D. 0) = 0 and ∂ui(x, t = 0)/∂t = 0, the governing equations and boundary conditions are written as follows:

µui,jj (x, t) + (λ + µ)uj,ij (x, t) = ρ ui = uî on S1 ,

ti = tî on S2,

∂ 2 ui (x, t) in D ∂t2 S2 = S \ S1

(5) (6)

where ui and ti show the displacement and traction respectively, ρ is the density of elastic media D, and λ and µ indicate Lamé constants. In Eq. 6, uî and tî are given boundary values. The time-domain boundary integral equation in 2-D elastodynamics can be expressed by Cij (x)ui (x, t) = uin i (x, t) +

S

Uij (x, y, t) ∗ tj (y, t)dSy −

S

Tij (x, y, t) ∗ uj (y, t)dSy .

(7)

In Eq. 7, Uij (x, y, t) and Tij (x, y, t) denote the time-domain fundamental solution and its double layer kernel for 2-D elastodynamics and Cij is the free term[8]. Normally, Eq. 7 is discretized by using the appropriate interpolation functions for the unknown values and solved by a time-stepping algorithm. However, there are mainly two disadvantages of the conventional time-domain BEM. The first one is an instability encountered in the time-stepping procedure. The other is the difficulty in solving large scale problems.

Time-Domain FMBEM Based on OQM in 2-D Elastodynamics To overcome the disadvantages of the conventional time-domain BEM, the Operational Quadrature Method (OQM) and the Fast Multipole Method (FMM) are introduced. BEM Formulation Based on OQM In solving the system of the boundary integral equation (7) numerically, the boundary surface S is discritized into M elements due to a piecewise constant approximation of the unknown displacement ui and traction ti . Taking the limit of x ∈ D → x ∈ S and applying Eq. 2 and Eq. 4 in OQM to the convolution integrals in Eq. 7 yields the following discritized boundary integral equations for time increment t and n steps as follows: n M n−k 1 ui (x, n∆t) = uin Aij (x, y α)tαj (k∆t) − Bijn−k (x, y α )uαj (k∆t) (x, n∆t) + i 2 α=1 k=1

m where Am i and Bi are the influence functions which are defined by

(8)

342


L−1 2πiml ρ−m Uîj (x, y, sl)e− L dSy L S l=0 L−1 −m ρ 2πiml Tîj (x, y, sl )e− L dSy . Bijm (x, y) = L S

Am ij (x, y) =

(9) (10)

l=0

In Eq. 9 and Eq. 10, sl is given by sl = δ(ζl)/(t). The parameter ρ has to be ρ < 1 and is taken as √ ρL = where shows the assumed error in the computation of Eq. 9 and Eq. 10. Uîj (x, y, s) and ˆ Tij (x, y, s) are Laplace domain fundamental solutions in 2-D elastodynamics as follows: 1 îj (x, y, s) = 1 U K0 (sT r)δij − 2 [K0 (sT r) − K0 (sL r)],ij 2πµ sT

2 2 îk,k (x, y, t) + ρc2 Uîj,k (x, y, t) + Uîk,j (x, y, t) nk (y) Tîj (x, y, s) = nj (y)ρ(c − 2c )U L

T

T

(11) (12)

where cL and cT are the wave velocity of longitudinal and transversal waves respectively, r is given by r = |x − y|, Kn is the modified Bessel function of the second kind in Eq. 11 and ni (y) is the component of a outward unit normal vector with respect to y. Note that sL and sT are defined by sL = δ(z)/(cLt) and sT = δ(z)/(cT t) due to the simple expression. To determine δ(ζl), we use the backward differential formula (BDF) of order two as follows: δ(ζl ) = (1 − ζl ) +

(1 − ζl2 ) . 2

(13)

Note that Eq. 9 and Eq. 10 are identical to the discrete Fourier transform. Therefore, the calculations of Eq. 9 and Eq. 10 can be evaluated by means of the FFT algorithm. After arranging Eq. 8 according to the boundary conditions, we can obtain M 0 1 ui (x, nt) + Bij (x, y α)uαj (nt) − A0ij (x, y α)tαj (nt) 2 α=1

uin =nt) + i (x,

n−1 M

α α n−k α α An−k ij (x, y )tj (kt) − Bij (x, y )uj (kt) .

(14)

α=1 k=1

For the n-th time step, all the quantities on the right-hand side are known. Therefore, the unknown values uαi and tαi can be obtained by solving the above equation. Unfortunately, we cannot solve a large scale problem with the large number of M by the timedomain BEM based on OQM because the required computational complexity and memory become O(LM 2 N) and O(M 2 L) in Eq. 14, respectively. Therefore, the time-domain BEM based on OQM is accelerated by the Fast Multipole Method (FMM) in this research.

Time-Domain Fast Multipole BEM Formulation Based on OQM The FMM proposed by Greengard and Rokhlin is a technique to reduce the computational time and memory for a large scale problem. In recent years, Fast Multipole BEM, which is the coupling method


343

of BEM and FMM, has been developed to improve the computational efficiency for various large scale problems in many engineering fields, e.g., the 2-D scalar wave problem[9][10], and the 3-D sound and environmental vibration problems[11]. Since FMBEM algorithm has been described in detail in other published papers (for example, see the paper of Nishimura[12]), we will summarize only the essential formulas here. Now, the fundamental solution in Eq. 11 is transformed into the following equation; îj (x, y, s) = 1 ΦU + e3ij ΨU U ,i ,j 2 µsT

(15)

where ΦU and ΨU are displacement potentials with respect to P and S-waves, which are defined by 1 K0 (sL |x − y|),k 2π 1 ΨU = e3kl K0 (sT |x − y|),l . 2π ΦU =

(16) (17)

To apply FMM, we consider a point o near the source point y. Locations of field point x and source point y are expressed as (r, θ) and (ρ, φ), respectively in polar coordinate system originated at the point o. Using Graf’s addition theorem, we obtain the multipole expansions of the displacement potentials as follows:

ΦU =

∞ 1 M U Kn (sL r)einθ 2π n=−∞ n

(18)

ΨU =

∞ 1 U N Kn (sT r)einθ 2π n=−∞ n

(19)

where the coefficients MnU and NnU are called the multipole moments, which are given by ∂ In (sL ρ)e−inφ ∂yk ∂ [In(sT ρ)e−inφ ]. NnU = −e3kl ∂yl

MnU = −

(20) (21)

In Eq. 20 and Eq. 21, In shows the modified Bessel function of the first kind. Multipole moments MnT and NnT for Tîj (x, y) is similarly obtained. Once the multipole moments are obtained, we can quickly evaluate the matrix-vector products of the discritized integral equation (14) using the fast multipole algorithm[7]. The translation formulas (M2M, M2L and L2L) are also derived from Graf’s addition theorem of the fundamental solutions defined in Eq. 11 and Eq. 12. The modified Bessel function In (z) , in practice, tends exponentially to infinity for large argument z. This fact sometimes causes the instability of the translation formulas when the cell size is large. To resolve the problem, we introduced the scaling of the multipole and local expansion coefficients.

Numerical Examples Time-domain BEM based on OQM (OQBEM) is applied to analyze the transient behaviors of a cavity with radius a as shown in Fig. 2. The boundary of the cavity is supposed to be traction free. The

344

Eds: R Abascal and M H Aliabadi 㪈㪉

B

cavity

x2

㪏 0

A

Pao and Mow-A OQBEM-A Pao and Mow-B OQBEM-B Pao and Mow-C OQBEM-C

㪈㪇

a

u / 1 u

C

㪍

㪋

x1

incident wave

㪉

㪇

㪇

㪉

㪋

㪍

㪏

㪈㪇

c t/a

L

Figure 3 Analytic and numerical solutions u1 /u0 at A, B and C. The solutions obtained by OQBEM are indicated by symbols and the analytical numerical results of Pao and Mow are shown by solid lines.

Computing time ( sec )

Figure 2 A scattering model.

10

9

10

8

10

7

10

6

10

5

3a

OQBEM FM - OQBEM

x2

cavity

2a

10

4

10

3

10

2

10

x1 incident wave

1

10

1

10

2

10

3

10

4

10

5

The number of Elements Figure 4 The comparision of CPU time between OQBEM and the FM-OQBEM.

Figure 5 A multiple scattering model.

number of elements is 64 and time increment is cL t/a = 0.0625. The parameters N and L are given by N = L = 128. In addition, ρ is assumed to be ρ = 0.95609320 ( = 10−10 ). The displacement components of the incident wave are given by uin i (x, t) = u0 δi1 [(cL t − x1 − a)/a]H(cLt − x1 − a).

(22)

Fig. 3 shows the displacement u1 /u0 as a function of time at A, B and C on the boundary of the cavity. This problem has been analytically solved in the frequency domain by Pao and Mow[13]. The transient solution can be obtained by superposing the results in the frequency domain by means of the fast Fourier transform. The results by OQBEM are in good agreement with the analytical-numerical results of Pao and Mow. Fig. 4 shows the CPU time needed in order to solve scattering problems of the incident waves by the cavity using time-domain BEM based on OQM (OQBEM) or fast multipole BEM based on OQM (FM-OQBEM). In this analysis, the number of elements is adjusted by changing the size of the element. We cannot solve the case that the number of elements is 512 or more with OQBEM


345

because of the restriction of the memory. We can see that FM-OQBEM is faster than OQBEM when the number of elements is several thousands or more as shown in Fig. 4. Finally, we consider the scattering problem of an incident wave with wave length a/2 by 8 × 8 cavities with the radius a of the individual cavities and the cavity spacing 3a between two adjacent cavities along the x1 and x2 axis as shown in Fig. 5. The components of the incident wave are given by uin i (x, t) = u0 (1 − cos πΘ) ,

0 ≤Θ = t−

x1 + 11.5a ≤ 2π. cL

(23)

The parameters are taken as N = L = 256, ρ = 0.95609320 ( = 10−10 ) and cL t/a = 0.125. The number of DOF in each time step is 8192. This problem cannot be solved by OQBEM. Therefore, the fast multipole method is applied to accelerate the matrix vector products of discritized boundary integral equation and to save the memory. Also, OpenMP with 8 threads is used to parallelize this analysis. Fig. 6 (a)-(d) show the time variations of the wave fields u1/u0 around cavities. We can see that scattered waves are generated by the interaction of the incident wave and cavities. Thus, time-domain fast multipole BEM based on OQM is very effective in both aspects of the computational time and required memory for a large scale problem.

Conclusions In this paper, the time-domain fast multipole BEM formulation based on OQM was developed for 2-D elastodynamics. The convolution integrals were discritized by the operational quadrature method and the fundamental solutions in Laplace domain were used for the calculations of influence functions. The fast multipole method was applied to accelerate the calculations of matrix-vector products for the retarded potential and to reduce the memory requirement. As numerical examples, scattering problems of incident waves by cavities were demonstrated and the computational efficiency of the proposed method was confirmed. In near future, we will develop the time-domain fast multipole BEM based on OQM in 3-D elastodynamics.

Acknowledgement This work is supported by the Japan Society of the Promotion of Science.

References [1] W. J. Mansur and C. A. Brebbia Transient elastodynamics using a time-stepping technique, In; Boundary Elements, C. A. Brebbia, T. Futagami and M. Tanaka (Eds), 677-698 (1983). [2] S. Hirose Boundary Integral equation method for transient analysis of 3-D cavities and inclusions, Engineering analysis with Boundary Elements, vol.8, No.3, 146-153 (1991). [3] C. Lubich Convolution quadrature and discretized operational calculus I , Numer. Math.,52, 129145 (1988). [4] A. I. Abreu, J. A. M. Carrer and W. J. Mansur Scalar wave propagation in 2D: a BEM formulation based on the operational quadrature method, Engineering analysis with Boundary Elements, 27, 101-105 (2003).

346


x2/a

x2/a

x1/a (a)

x2/a

x1/a (b)

x2/a

x1/a (C )

x1/a (d)

Figure 6 Time variations of displacements u1 /u0 around cavities. (a) cL t/a = 6.25 (b) cL t/a = 12.5 (c) cL t/a = 19.5 (d) cL t/a = 25.0. [5] M. Schanz and V. Struckmeier Wave propagation in a simplified modelled poroelastic continuum: Fundamental solutions and a time domain boundary element formulation, Numer. Math.,64, 18161839 (2005). [6] Ch. Zhang Transient elastodynamic antiplane crack analysis of anisotropic solids, Int. J. Solids and Structures, vol. 37, 6107-6130 (2006). [7] L. Greengard and V. Rokhlin A fast algorithm for particle simulations, Journal of Computational Physics, 73, 325-348 (1987). [8] S. Kobayashi Wave Analysis and Boundary Element Methods, Kyoto University Press (in Japanese), (2000). [9] T. Fukui and J. Katsumoto Fast multipole algorithm for two dimensional Helmholtz equation and its application to boundary element method. Proc of the 14th Japan National Symposium on Boundary Element Methods (in Japanese), 81-86 (1997). [10] T. Saitoh, S. Hirose, T. Fukui and T. ISHIDA, Development of a time-domain fast multipole BEM based on the operational quadrature method in 2-D wave propagation problem, Advances in Boundary Element Techniques VIII, 355-360 (2006). [11] T. Saitoh A study on effective 3-D numerical analysis of environmental vibration and noise induced by a moving train, doctral thesis in Tokyo Institute of Technology, (2006). [12] N. Nishimura, Fast multipole accelerated boundary integral equation methods, Appl. Mech. Rev., 55, 299-324 (2002). [13] Y.-H. Pao and C. C. Mow, Diffraction of Elastic Waves and Dynamics Stress Concentrations, Crane and Russak, New York, (1973).


347

Characteristic matrix in the bending plate analysis by SBEM Panzeca T.1,a, Cucco F. 2,b, Salerno M. 1,c 1

Diseg, Viale delle Scienze, 90128 Palermo, Italy 2

Via E. Tricomi 8, 90127 Palermo, Italy

a


Keywords: bending plate, Symmetric Boundary Element Method, Kirchhoff shear force.

Abstract. This paper deals with the thin bending plate analysis by using the symmetric approach of Boundary Element Method (SBEM). A formulation is used in which the plate boundary is discretized into boundary elements and is subjected to appropriate distributions of shear forces and couples, as well as of vertical displacement and rotations. These distributions are the causes and are modelled through appropriate shape functions, whereas the generalized effects are obtained, according to the Galerkin approach, as weighting of the displacements and the rotations, as well as of the shear forces and moments. In the equations system the algebraic operator is a symmetric matrix whose coefficients are defined as double integrals with high order singularities, all computed in closed form. Introduction The object of this paper is to consider some computational aspects regarding the thin bending plate analysis with the SBEM (Bonnet et al. [1]). Respect to the collocation BEM, in which nonsymmetric boundary integral formulations for bending plates have been studied by various authors (Beskos [2], Aliabadi [3]), the SBEM approach shows few contributions about the plate analysis (Tottenham [4], Frangi and Bonnet [5], Perez-Gavilan and Aliabadi [6]). The computation of the solving equation coefficients presents considerable difficulties for the presence of high order singularities in the kernels of the integrals and it is carry out either by means of a derivative transfer technique, employing the integration by parts to reduce the singularity of the fundamental solutions (Frangi and Bonnet [5]), or by means of an approach based on a limit process (Frangi and Guiggiani [7] in the collocation context). In the present paper a general computational methodology that makes easier the generation and the check of the coefficients calculus is shown. In this methodology, already applied to the in-plane loaded plate in Panzeca et al. [8], the bending plate is studied without considering the actual constraint and boundary load conditions. For the plate an appropriate algebraic operator, so-called characteristic matrix, which connects the kinematical and mechanical quantities along the boundary, is introduced. This matrix is singular and its coefficients are valued by imposing in a sequential way some distributions of causes on the boundary elements, and by computing the effects in the same boundary through a weighting process of the response. This approach is particularly useful when the matrix coefficients are determined; indeed the rigid body technique allows to verify the effectiveness of the coefficients computation. In the first section the peculiarities of the characteristic matrix are shown and an appropriate rearrangement and employment of this matrix is explained in order to obtain the algebraic operators of the mixed value elastic problems. In the second section a technique to compute a coefficient of the characteristic matrix is illustrated. The kernels of the integrals are defined as distributions in the Schwartz sense, specifically the distribution definition is employed as the limit of a succession of functions. This approach makes it

348


possible to naturally cancel out the singularities of higher order in the cause integration, whereas the lower order singularities are smoothed by the effect shape functions and eliminated by the outer integration. 1. Symmetric characteristic operators Let us consider the bending problem for a linearly elastic plate of domain : and boundary * , distinguished into constrained *1 and free * 2 . The plate is subjected to the following external actions (Fig. 1a): - body forces p normally applied to the middle surface in the domain : ; T - displacements and rotations u u n Mn 0 imposed on the constrained boundary *1 ; T - forces and couples f fn cn 0 given on the free boundary * 2 . The elastic response to the known external actions may be obtained in terms of boundary quantities, defined along an element characterized by the outward normal n : T - shear forces and couples f f n c n csn on *1 ; T - displacement and rotations u u n Mn Msn on * 2 ; To study the plate a general strategy is used, based on the introduction of a matrix, called characteristic. The plate is embedded in an unlimited domain : f (Fig. 1b) having the same Young’s modulus E , Poisson’s ratio Ȟ and the same thickness h of the plate, and as a consequence its boundary may be considered as boundary * of : or * of the complementary domain : f \ : . jn

G1

fn

jn p

p

W

8

fn

un

un

G2

G+

G cn

cn

a)

b)

Fig. 1: a) A polygonal plate, b) the plate embedded in : f . To derive the characteristic matrix no distinction is made between the constraint or free boundaries. It involves that the entire boundary is subjected to a distribution of layered mechanical actions f and to a distribution of double layered kinematical discontinuities 'u u . It is known that the response in terms of kinematical u and mechanical t quantities at every point of the on the boundary * is given by the Somigliana Identities (SI). By imposing the boundary conditions u + 0 and t + 0 on * , which replace the classical Diriclet and Neumann conditions u u and t f on * , these SI may be written in compact form in the following way: u+

t+

1 u [f ] u PV [u] u uˆ [p] 0 2 1 t PV [f ] t t [-u] tˆ [p] 0 2

(1.a)

(1.b) where the following positions are valid:

u [f ]

³G *

uu

f d*, u PV [u]

v³ G *

ut

(u) d*, uˆ [p]

³G

:

uu

p d:

(2a-c)


v³ G

t PV [f ]

tu

f d*,

³G

t [u]

*

tt

(u) d* , tˆ [p]

*

³G

349

tu

p d:

(2d-f)

:

being G hk ( h, k = u, t) the fundamental solutions matrices defined in Panzeca et al. [9]. Let us operate the discretization of the boundary and introduce appropriate shape functions :t and :u to model the layered mechanical and the double layered kinematical quantities: f

:t F ,

u

:u U

(3a,b)

where F and U are the vectors collecting nodal quantities. Linear shape functions are assumed for :t , whereas quadratic ones in the bending rotation and hermitian ones in the displacement and torsional rotation discontinuity are assumed for :u . Let us perform the weighing process in accordance with the Galerkin approach in the eqs. (1a,b), so obtaining the following boundary integral equations: W+ P+

1 ˆ A uu F A ut ( U ) Cut ( U ) W 2 1 A tu F A tt ( U) Ctu F Pˆ 0 2

(4a)

0

(4b)

where the following positions are set: W+

³: u T t

ˆ d* , W

³ : [³ G T t

*

A ut

³ : [v³ G T t

*+

P+

*+

ut

:

T + u

d* + , Pˆ

³ : [³ G T u

*+

tu

uu

:t' d* ]d* ,

*

³ : [³ G

tt

: d*]d* ' u

(5a-l) T u

*+ T u

*+

T t

³ : [ v³ G

p d:]d* + , A tu

:

1 [ :uT :t d* + ], A tt 2 *³+

³ : [³ G

1 [ :tT :u d* + ], 2 *³+

+

*

³: t

p d:]d* + , A uu

*

: d* ]d* , Cut ' u

*+

Ctu

uu

tu

:t' d*]d* + ,

*

+

*

In compact form one has: B X Lˆ

(6)

0

with

B

( A + C), A

ª A uu «A ¬ tu

A ut º , C A tt »¼

1 2 Cut º ª 0 , X « 1 2 C 0 »¼ tu ¬

ªF º ˆ «U » , L ¬ ¼

ˆ º ªW « » «¬ Pˆ »¼

(7a-e)

The matrix A is symmetric, whereas the matrix C , which includes the free terms, is emisymmetric. The matrix B is unsymmetric and singular: the singularity depends on the circumstance that the plate may be subjected to a rigid motion. In the present paper all the coefficients of the matrix B have been computed in closed form. The matrix B is used to solve the mixed value elastic problems. Indeed it allows to generate both the pseudostiffness matrix and the load vectors due to the mechanical actions applied on the free

350


boundary * 2 and to the kinematical quantities imposed at the constrained boundary *1 , as it has been made by Panzeca et al. [8] in the in-plane loaded plate. In order to get this aim, a rearrangement of the rows and columns of the characteristic matrix is performed: indeed the vectors F and U , W and P are redefined in the following way T

T

T

T

F = ª¬F1T F2T º¼ , U = ª¬ U1T U T2 º¼ , W = > W1 T W2 T @ , P = > P1 T P2 T @

(8a-d)

Let us to introduce the Dirichlet and Neumann generalized conditions on the * boundaries: W1

0 on *1 ,

P2

0 on * 2

(9a-b)

The rearrangement introduced in the eqs. (9a-b) allows to derive the following equations: ª W2 º ªB u 2u 2 «W » « B « 1 » = « u1u 2 « P2 » « B t 2u 2 « » « ¬« P1 ¼» ¬« B t1u 2

B u 2u1 B u 2t 2 B u1u1 B u1t 2 B t 2u1 B t 2t 2 B t1u1 B t1t 2

B u 2t1 º B u1t1 »» B t 2t1 » » B t1t1 ¼»

ª F2 º ª W2p º « » « » « F1 » + « W1p » p « U 2 » « P2p » « » « » P ¬« U1 ¼» ¬« 1p ¼»

ª0º «0» « » «0» « » ¬« 0 »¼

(10)

In compact form: K X + Lˆ = 0

(11)

where the following positions are made: B º ªB ª F º K = « u1u1 u1t 2 » ; X= « 1 » ; Lˆ = L t F2 + L u (-U1 ) + L p p ; B B ¬ t 2u1 t 2t 2 ¼ ¬ U 2 ¼ ª W1p º ªB º ªB º L t « u1u 2 » ; L u « u1t1 » ; L p « » ; ¬ B t 2u 2 ¼ ¬ B t 2t1 ¼ ¬ P2p ¼

(12a-f)

In eq. (11) K is the pseudostiffness matrix, symmetric and non singular, whereas Lˆ is the generalized load vector. 2. Coefficients analytical computation in the Kirchhoff model. In the Kirchhoff model the following Love-Kirchhoff hypotheses have been introduced: - kinematical assumption: in the plate boundary, having normal n and tangent s , the torsional rotation is the tangential derivative of the vertical displacement; - mechanical assumption: the distributed torsional moment along the plate boundary may be replaced by an appropriate distribution of transversal shear forces, leading to the so-called Kirchhoff shear, which is the sum of the shear force and the tangential derivative of the torsional moment. To get a symmetric formulation, a kinematical quantity, associated to the Kirchhoff shear force and defined the Kirchhoff displacement discontinuity, has to be introduced. The shape functions employed for the modelling process are assumed to be linear for the forces and couples, quadratic for the rotation discontinuities, whereas they are assumed to be hermitian for the Kirchhoff displacements discontinuities. The shape functions employed for the weighting process


351

are assumed to be linear for the generalized displacements and rotations, quadratic for the generalized moments, whereas they are assumed to be hermitian for the generalized Kirchhoff shear forces. In the evaluation of the matrix coefficients the following steps are used: - to impose a unitary value at the node according to the local node system (Fig. 2b), - to model the cause along the boundary elements through appropriate shape functions according to the local side system (Fig. 2e), - to employ the Somigliana Identities, - to perform the weighting process of the boundary quantities by means of appropriate shape functions according to the Galerkin strategy. In the case of a rectangular plate, with sides parallel to the Cartesian axes, let us suppose to determine the generalized Kirchhoff shear force T K associated with node 1 as the effect of a Kirchhoff displacement discontinuity 'U K 1 imposed at the same node (Fig. 2) between the two frontiers * and *+ . In the following schemes the use of the fundamental solutions is shown, with the objective to take both the Kirchhoff kinematical and mechanical assumptions into account: kinematical assumption Msn

wu ws '

'u

'u 'Mn G tt

t ª tt11 m n « tt 21 msn «¬ tt 31

'Msn

tt12 tt 22 tt 32

tt13 º int egration by parts tt 23 » o G tt » tt 33 ¼

mechanical assumption t K

G tt

'u ª§ wtt13 · t « ¨ tt11 ¸ ws ' ¹ «© wtt · § m n «¨ tt 21 23 ¸ «© ws ' ¹ « w § msn « ¨ tt 31 tt 33 ·¸ «¬ © ws ' ¹

'Mn º tt12 » » tt 22 » o G tt » » tt 32 » »¼

t 'u

'Mn

ª§ wtt13 · º t « ¨ tt11 ¸ tt12 » s ' w © ¹ « » wtt · § m n «¨ tt 21 23 ¸ tt 22 » «© » ws ' ¹ « » msn « §¨ tt 31 wtt 33 ·¸ tt 32 » «¬ © »¼ ws ' ¹

wmsn ws 'Mn

ª§ wtt º · t K «¨ tt11 13 » ¸ s ' w tt w § · 32 «¨ » ¸ ¨ tt12 ¸ wtt 33 · ¸ © ws ¹ » «¨ w § tt «¨ ws ¨ 31 ws ' ¹¸ ¸ » ¹ «© © » wtt 23 · « § » tt 22 m n « ¨ tt 21 ¸ »¼ ws ' ¹ ¬ ©

K ª tt11 « tt K ¬ 21

K º tt12 tt 22 ¼»

As a consequence, the displacement discontinuity imposed at node 1 is transferred along the boundary sides next to the node as the sum of two distributions: one concerning the vertical displacement and the other one concerning the tangential derivative of the torsional rotation. The generalized Kirchhoff shear force T K associated with the node 1 can be obtained as the sum of the weighted shear force along the boundary sides a and b. On every side the weighted shear force is obtained as the sum of two contributions and specifically the vertical force and the tangential derivative of the torsional moment. At a point of each boundary side the Kirchhoff fundamental K solution tt11 , that has to be modelled by the cause shape functions and has to be weighted by the effect shape function, may be expressed as the sum of four terms:

352


K tt11

tt11

wtt13 w § wtt · tt 31 33 ¸ ws ' ws ¨© ws ' ¹

(13)

K where tt11 is the Kirchhoff traction at point x of normal n caused by: - a displacement discontinuity -u 1 applied at a point x ' with normal n' ( tt11 wtt 31 ws ). - a tangential derivative of the torsional rotation -Msn 1 applied at a point x ' with tangent s ' ( wtt13 ws ' w ws wtt 33 ws ' );

NODE QUANTITIES

s n

b G

a

x

x

s

n

Mx , Fx

My , Fy

x

G+ y

1

T

a)

K

=?

y

x

s’ n’

G’

b

+

G’

z

b)

c)

T, U

z

x’ y’

x’

a

n’

s’

x’

1 D U K =-U=1 z’

BOUNDARY SIDES DISTRIBUTIONS Distributions of displacement Distributions of shear forces and discontinuities and of tangential derivative of tangential derivative of the torsional moment of the torsional rotation

a

d)

b 1

Cn , Fn

G

a Csn , Fsn

f)

b 1

n

s

e)

F, W

Fig. 2: a) Weighted Kirchhoff equivalent shear force, b) node local system, c) generalized Kirchhoff displacement discontinuity, d) distributions of the effects, e) side local system, f) distributions of the causes. The causes are imposed according to the node local system and their boundary effects are evaluated according to the side local system. The generalized shear force T K associated with node 1 as the effect of a Kirchhoff displacement discontinuity 'U K 1 imposed at the same node derives from the double integration of the product K between the fundamental solution tt11 and the shape functions < 'u (x ') and < u (x) . The fundamental solution is defined in the boundary elements a and b, where it is specified as the K K K K , tt11 following forms tt11 , tt11 , tt11 , in which the double indices represent the sides where aa ab bb ba the effect and cause distributions are located. Consequently one has: Inner integrals: - The Kirchhoff shear force at a point of the side a, caused by the kinematical distributions associated to the Kirchhoff displacement discontinuity in the sides a and b:


-

1

³ tt 0

K 11 aa

1

K < ua ([ ') d[ ' ³ tt11 < ub ([ ') d[ ' ab 0

F

t aK

353

· E h 3 § 24 3(3 Q) ([) ¸ ¨ 24S(1 Q) © [ 1 [ ¹

(14)

The Kirchhoff shear force at a point of the side b, caused by the kinematical distributions associated to the Kirchhoff displacement discontinuity in the sides a and b: 1

³ tt 0

K 11 bb

1

K < ub ([ ') d[ ' ³ tt11 < ua ([ ') d[ ' ba 0

E h 3 § 24 3(5 Q) ¨ 24S(1 Q) © [ 1 [

F

t Kb

· ([ ) ¸ ¹

(15)

F

F

where ([) and ([) are functions containing logarithmic singularity only. K In eqs. (14) and (15), the fundamental solution tt11 shows a singularity of order 1 r 4 in the ii K shows the same singularity only for interval (0,1), instead the fundamental solution tt11 ij [ [' 0 . Outer integral: - Let us compute the primitives T aK , T bK of the Kirchhoff shear forces t aK , t Kb weighted through hermitian shape functions: T aK

³
0. Ra, P r and K are the Rayleigh number, Prandtl number, and material parameter respectively. ψ and w are the stream function and the vorticity with ∂v u = ∂ψ , u = − ∂ψ and w = ( ∂x − ∂u ). The initial and the boundary conditions are taken as ∂y ∂x ∂y (1 +

w=T =N =0

when t = 0

x=0:

0 ≤ y ≤ 1,

u = v = 0,

T = 0.5,

N =0

x=1:

0 ≤ y ≤ 1,

u = v = 0,

T = −0.5,

N =0

y = 0, 1 :

0 ≤ x ≤ 1,

u = v = 0,

∂T /∂y = 0,

(2)

N =0.

The vorticity boundary conditions are derived from the Taylor series expansion of the stream function equation. For K = 0, stream function, vorticity transport and energy equations describe the classical problem of natural convection of a Newtonian fluid in a differentially heated square cavity, first considered by Vahl Davis [8]. The heat transfer coefficient in terms of the local Nusselt number, N u, and the average Nusselt number, N uav at the vertical walls are defined by 1 ∂T |x=0,1 , N uav = − N udy . Nu = − (3) ∂x 0 Application of DRBEM The equations in (1) are weighted through the domain Ω as in [7], by the fundamental solution 1 1 ln u∗ = 2π r


365

of Laplace equation in two dimensions in which r is the distance between the source and the fixed points. Applying Green’s second identity, we have the following integral equations for each source point i: ci ψi + (ψq ∗ ψ − ψ ∗ ψq )dΓ = (−w)ψ ∗ dΩ Γ

Ω

(1 + K)ci wi + (1 + K)

Γ

∗

∗

(wq w − w wq )dΓ

=

( ∂w Ω ∂t

+ u ∂w + v ∂w + K∇2 N − ∂x ∂y

Ra ∂T )w∗ dΩ P r ∂x

1 ∂T ∂T ∗ ∂T (Tq ∗ T − T ∗ Tq )dΓ +u +v )T dΩ = ( P r ∂t ∂x ∂y Γ Ω (1 + K2 )ci Ni + (1 + K2 ) Γ (Nq ∗ N − N ∗ Nq )dΓ = Ω ( ∂N + u ∂N + v ∂N + 2KN − Kw)N ∗ dΩ ∂t ∂x ∂y (4) 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. Expanding the nonhomogeneties in each equation in terms of the radial basis functions fj ’s 1 ci Ti + Pr

−w

=

N +L

αj fj (x, y)

j=1

∂w ∂w Ra ∂T ∂w +u +v + K∇2 N − ∂t ∂x ∂y P r ∂x

=

N +L

α ¯ j (t)fj (x, y)

j=1

∂T ∂T ∂T +u +v ∂t ∂x ∂y

=

∂N ∂N ∂N +u +v + 2KN − Kw ∂t ∂x ∂y

=

N +L

(5) α ˜ j (t)fj (x, y)

j=1 N +L

α ˘ j (t)fj (x, y)

j=1

¯j , α ˘ j and α ˜ j are undetermined coefficients. The numbers of boundary and selected where αj , α internal nodes are denoted by N and L , respectively. The radial basis (coordinate) functions fj are linked to the particular solutions of each equation with the Laplace operator. Substituting these expansions in Eq. (4) and the application of Green’s second identitiy to the right hand sides will result in matrix vector equations for each unknown ψ, w, T and N . Hψ − Gψq = (H ψˆ − Gψˆq )α (1 + K)(Hw − Gwq )

= (H wˆ − Gwˆq )¯ α

1 (HT − GTq ) = (H Tˆ − GTˆq )˜ α Pr K ˆ − GNˆq )˘ (HN − GNq ) = (H N α 1+ 2 where G and H are (N + L) × (N + L) matrices defined by 1 ∂ 1 1 1 Hij = ci δij + Gij = ln( ) dΓj . ln( ) dΓj , 2π Γj ∂n r 2π Γj r

(6)

366


ˆ w, ˆ are constructed by taking the corresponding particular solutions The matrices ψ, ˆ Tˆ and N as columns. Evaluation of the right hand sides of each equation in (5) at all boundary and interior (N +L) points gives Hψ − Gψq

= (H ψˆ − Gψˆq )F −1 {−w}

∂w ∂w ∂w Ra ∂T +u +v + K∇2 N − ∂t ∂x ∂y P r ∂x ∂T 1 ∂T ∂T = (H Tˆ − GTˆq )F −1 (HT − GTq ) +u +v Pr ∂t ∂x ∂y K ˆ − GNˆq )F −1 ∂N + u ∂N + v ∂N + 2KN − Kw 1+ (HN − GNq ) = (H N 2 ∂t ∂x ∂y

(1 + K)(Hw − Gwq )

= (H wˆ − Gwˆq )F −1

(7) where F is the (N + L) × (N + L) matrix containing coordinate functions fj ’s as columns. Derivatives of w, T and N are approximated by the DRBEM idea ∂w ∂F −1 = F w, ∂x ∂x

∂F −1 ∂w = F w ∂y ∂y

∂F −1 ∂T = F T, ∂x ∂x

∂F −1 ∂T = F T ∂y ∂y

∂N ∂F −1 = F N, ∂x ∂x

∂F −1 ∂N = F N ∂y ∂y

∂ 2 F −1 ∂2N = F N, 2 ∂x ∂x2

(8)

∂2N ∂ 2 F −1 = F N. 2 ∂y ∂y 2

Substituting convection terms back into Eq. (7), and finally rearranging, we end up with the following system of ordinary differential equations for w, T and N respectively ˜ + Gw ˜ q − SF w˙ − Hw

= 0

˜t T + G˜t Tq T˙ − H

= 0

N˙ − Hñ N + Gñ Nq − SF 1 = 0 and a linear system of equations for ψ Hψ − Gψq = −Sw

(9)


367

˜ G, ˜ Hñ , G ˜ n, H ˜t and G ˜ t are with S = (H ψˆ − Gψˆq )F −1 and the matrices SF , H, Ra ∂T − K∇N, SF 1 = Kw P r ∂x ˜ = S −1 (1 + K) H − u ∂F F −1 + v ∂F F −1 H ∂x ∂y SF =

∂F K ∂F −1 F −1 + v F − 2K Hñ = S −1 (1 + ) H − u 2 ∂x ∂y

(10)

˜t = S −1 1 H − u ∂F F −1 + v ∂F F −1 H Pr ∂x ∂y ˜ = S −1 (1 + K) G , G

K Gñ = S −1 (1 + ) G, 2

1 G˜t = S −1 G. Pr

For the derivatives of w, T and N in Eq. (9) implicit central differences are used assuming the previous two time level solutions are known. Results and Discussion A numerical model was developed to validate the accuracy for the solutions of 2D unsteady natural convection flow of micropolar fluids in a square cavity given in Eq. (1) and (2). The no-slip boundary conditions of the velocities are assumed. The horizontal walls are adiabatic, while the vertical walls are isothermally heated. Solutions are obtained by using N = 120 boundary elements and L = 400 interior nodes. Computations are carried out for Ra = 103 , 104 and 105 with the time increments ∆t = 0.5, 0.01 and 0.003 respectively. The material parameter K is taken as 0, 0.5, 1 and 2. An increase in Rayleigh number results in intensified circulation inside the cavity, and thinner thermal boundary layers for all the variables, stream function, vorticity and isotherms near the heated and cooled walls. For Ra = 103 the vortex at the center was in circular pattern. With the increase in Rayleigh number the vortex changes its shape to elliptical form. Since the viscous forces are dominating when Ra = 103 , there is not enough convective motion of the fluid within the cavity. The isotherms are almost vertical in this case. As the Rayleigh number increases, the isotherms undergo an inversion at the central region of the cavity. These behaviors can be seen from Fig. 1 and Fig. 2. The effect of varying Ra on the average Nusselt number at the heated wall is shown in Table 1 for some values of K and a fixed value of Prandtl number, P r = 0.71. It shows that for a fixed value of Ra, an increase in K reduces the heat transfer. In addition, the Newtonian fluid (K = 0) is found to have higher average heat transfer rates than a micropolar fluid (K = 0). This is because an increase in the vortex viscosity would result in an increase in the total viscosity of the fluid flow, thus decreasing the heat transfer. The results are in good agreement with the results given in [6].

368


Figure 1: Streamlines, vorticity contours, isotherms for Ra = 103 , K = 0.5, K = 1 and K = 2


Figure 2: Streamlines, vorticity contours, isotherms for Ra = 105 , K = 0.5, K = 1 and K = 2

369

370


K 0 0.5 1 2

Ra = 103 Present 1.118 1.057 1.031 1.012

Ra = 103 [6] 1.118 1.057 1.034 1.016

Ra = 104 Present 2.217 1.925 1.735 1.529

Ra = 104 [6] 2.234 1.947 1.771 1.545

Ra = 105 Present 4.476 4.023 3.703 3.302

Ra = 105 [6] 4.486 4.033 3.729 3.314

Table 1: The effect of K on the average Nusselt number N uav for different values of Ra

Conclusion The unsteady natural convective heat transfer of micropolar fluids in a differentially heated square cavity is computationally studied using the DRBEM. Time derivative is discretized using implicit central difference scheme. The results are obtained for all variables, stream function, vorticity and temperature also for a Newtonian fluid for comparison. Simulations are performed to investigate the effects of the Rayleigh number, Ra, and the material parameter, K, on the momentum and heat transfer. As the Rayleigh number increases boundary layer formation starts and the average Nusselt number increases. However, an increase in the material parameter reduces the average Nusselt number. References [1 ] AC. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966); 1-18. [2 ] D.C Lo , D.L Young and C.C Tsai, High resolution od 2D natural convection in a cavity by the DQ method, JCAM, 203 (2007); 219-236. [3 ] S. Roy and T. Basak, Finite element analysis of natural convection flows in a square cavity with non-uniformly heated walls, Int. J. Engrg. Sci., 43 (2005); 668-680. [4 ] T.H Hsu and C.K. Chen, Natural convection of micropolar fluids in a rectangular enclosure, Int. J. Engrg. Sci., 34(4) (1996); 407-415. [5 ] T.H. Hsu, P.T. Hsu and S.Y. Tsai, Natural convection flow of micropolar fluids in an enclosure with heat sources, Int. J. Heat Mass Transfer, 40, No.17 (1997); 4239-4249. [6 ] O. Aydin and I. Pop, Natural convection in a differentially heated enclosure filled with a micropolar fluid, Int. J. of Thermal Sciences, 46 (2007); 963-969. [7 ] 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). [8 ] G. Vahl Davis, Natural convection in a square cavity: A benchmark solution, Int. J. Numer. Meth. Fluids, 3 (1983); 249-264.


371

Two-dimensional Thermo-Poro- mechanic fundamental solution for unsaturated soils Pooneh Maghoul a, Behrouz Gatmiri c,b, Denis Duhamel a a

b

Université Paris-Est, Institut Navier, LAMI, Ecole des Ponts, Paris, France Email: [email protected]

Université Paris-Est, Institut Navier, CERMES, Ecole des Ponts, Paris, France Email: [email protected] c

Department of Civil Engineering, University of Tehran, Tehran, Iran

Keywords: Boundary element method; fundamental solution; time domain; unsaturated soil; porous media; thermo-poro-elastic behaviour

Abstract. In this article, after a brief discussion on the unsaturated soils’ governing differential equations including the equilibrium, air, moisture and heat transfer equations, the closed form transient thermal fundamental solution of the governing differential equations for an unsaturated two-dimensional deformable porous medium with linear elastic behaviour for a symmetric polar domain have been introduced. The derived fundamental solution has been verified mathematically by comparison with the previously introduced corresponding fundamental solution. Introduction There are numerous media encountered in engineering practice whose behaviour is not consistent with the principles and concepts of classical saturated soil mechanics. An unsaturated porous medium can be represented as a three-phase (gas, liquid, and solid) or three-component (water, dry air, and solid) system. The liquid phase is considered to be pure water containing dissolved air and the gas phase is assumed to be a binary mixture of water vapour and “dry” air [1, 2]. The air in an unsaturated soil may be in an occluded form when the degree of saturation is relatively high. At a lower degree of saturation, the gas phase is continuous. Commonly it is the presence of more than two phases that results in a medium that is difficult to deal with in engineering applications. The phenomenon of coupled heat and moisture transfer in a deformable partly saturated porous medium is important in relation to several problems, including underground storage of hot and cold materials, disposal of high-level nuclear waste and so on. Thermallyinduced moisture movements can lead to changes in both the thermal and isothermal properties of the soil which can subsequently affect the functioning of the soil for its intended purpose. In order to model unsaturated soil behaviour, firstly the governing partial differential equations should be derived and solved. Because of the complicated forms of governing partial differential equations the different numerical methods are presented for solving them. Among them, the boundary element method as the most efficient is going to be employed for more complicated and coupled ones regarding the behaviour and consequently the governing differential equations. As in this method, during formulation boundary integral equations, the applied mathematics concept of the Green functions has been employed. This type of fundamental solutions for the governing partial differential equations should be first derived. Indeed, attempting to solve numerically the boundary value problems for unsaturated soils using boundary element method leads one to search for the associated Green functions [3]. The comprehensive state-of-the-art review by [4, 5, and 6] provides clearly presented information on the fundamental solution applied in the saturated soil. For unsaturated soils, the first Green functions for the nonlinear governing differential equations for static and quasi-static poroelastic media for both two and three-dimensional problems have been derived by [6, 7]. The thermo-poro-elastic Green functions for the nonlinear governing differential equations for static and for both two and three-dimensional problems have been derived by [8]. The present research is an attempt to derive these Green functions for two-dimensional deformable quasi-static unsaturated soil. Following some reasonable and necessary simplifications, the fundamental solutions will be introduced in both frequency and time domains.

372


Governing Equations For an unsaturated material influenced by heat effects, the governing partial differential equations considered are of four main groups: equilibrium equations, moisture transfer equations, air transfer equations and heat diffusion equations [9]. Solid Skeleton. The equilibrium equation and the constitutive law for the soil’s solid skeleton including the effects of suction and temperature [10]: V ij G ij Pa Pa ,i bi 0 (1) ,j

d V ij G ij Pa

Where F s

1 s

D. D and FT

we

ET

Dd H V F sG ij d Pa Pw FT G ij d T

1 e wT

1 T

1 s

(2) 1 T

E s m and D

D. D . With D

ET m in which E s

we 1 e w Pa Pw

and

and m ª¬1 1 0 º¼ .

The linear elastic matrix D is written as: Dijkl OG ij G kl P G ik G jl G il G jk

(3)

Considering the strain-deformation relations: 1 ui , j u j ,i 2

H ij

(4)

One can conclude: O P ui ,ij P u j ,ii F s 1 Pa , j F s Pw, j FT T, j b j

0

(5)

Continuity and transfer equations for moisture. This part will be divided into vapour transfer and liquid transfer formulation as follows:

U

Liquid phase transfer. According to [11] the unsaturated flow equation can be written as qw / U w K w . \ z

(6)

The capillary potential \ varies with moisture content and temperature. The capillary potential in a reference temperature in terms of suction will take the following form: (7) \ r (T ) Pg Pw / J w The variation of capillary potential according to the temperature is considered by the introduction of surface tension V (T ) . \ (T , T ) \ r (T ).V (T ) / V r (8) Where V (T ) is surface tension of the water and V r and \ r (T ) are respectively, the surface tension of the water and the capillary potential in a reference temperature. Substituting from eqs (7, 8) and after manipulating some mathematical operations a new suction-based formulation of water movement equation can be found. U qw / U w DTwT DPw Pg Pw Dwz (9) Where DTw is thermal liquid diffusivity K w

\ r (T ) dV (T ) V (T ) , DPw is isothermal liquid diffusivity K w and dT V r .J w Vr

Dw is gravitational diffusivity, K w .

Vapour transfer. The equation of vapour diffusion in porous media according to [12] theory is given as D.v.n.Uvap (10)

qvap

In order to find Uvap as a function of temperature and moisture content, local thermodynamic equilibrium should be assumed. Under this assumption the following thermodynamic relationships can be introduced:


Uvap

373

U 0 .h \ .g

h

exp(

R .T

(11) (12)

)

Where U0 is the density of saturated water vapour and h is is the relative humidity. In this equation U0 is a function of T only and h is a function of T only. Then: dU dh U vap h 0 T U0 T dT dT Substituting Eq (13) in Eq (10) and considering the hypotheses presented in [13]: qvap / U w DTv T DPv Pg Pw

(13)

(14)

Where D ( T ) a ° .v.n. T ® Uw ° 0, ¯

DTv

§ d U0 ¨ h. © dT

· ¸, T % n , ¹ T n

and DPv

g V (T ) D , T%n ° .v.n. U vap RT V r .J w ® Uw °0, T n ¯

Total moisture transfer. The total moisture movement in unsaturated soil due to temperature gradient and its resulting moisture content gradient is equal to the sum of the flows which take place in both phases, vapour and liquid. Thus q / U w V U DT T DP Pg Pw Dwz (15) Where DT is the thermal water diffusivity, and is equal to DTvap DTw , and D P is isothermal water diffusivity and is equal to DPvap DPw . Moisture mass conservation. The conservation law for moisture mass is written: w U w .n.S r U vap .n.(1 S r ) div( U w . V U ) wt

^

`

0

(16)

It seems reasonable to dispense with the variations of K a and K w due to the variations of S r and consequently of Pg Pw for simplicity, since deriving the considered Green functions will become too difficult, at least with usual methods, due to the nonlinearity of or existence of non-constant coefficients in the governing differential equations. In this manner, the effects of S r have been considered in air and water coefficients of permeability by assuming K a and K w as a multi-linear function of

P

g

Pw for each finite domain [6].

However, Eq (16) may be written as:

U

w

Sr U vap .(1 S r )

wwnt n. U

w

U vap

wwSt

r

U w DT 2T U w DP 2 Pg U w DP 2 Pw

(17)

Considering that, the porosity equals the volumetric strain: HX

n

H kk

(18)

uk , k

And the definition of the degree of saturation in the linear form:

^

`

1 bs ( Pg Pw ) 1 d s (T T0 ) ,

Sr

(19)

The governing equation for the moisture becomes:

U

w

Sr U vap (1 Sr )

wuk , k wt

n. U w Uvap . g1

w( Pg Pw ) wt

n. U w U vap . g 2

wT wt

U w DT 2T U w DP 2 Pg U w DP 2 Pw

(20) Where g1

wSr / w Pg Pw and g 2

wS r / wT .

Continuity and transfer equations for air. Considering the generalized Darcy’s law, the air flow equation can be given as: Vg

qg / U g

K g . Pg / J g z

Considering that Pg is a function of temperature, this equation can be written as

(21)

374

Vg


K g wPg . .T K g . Pg / J g z J g wT

(22)

Using the thermodynamic state equations for gases, the 1 wPg

Pg Patm

J g wT

T 273 .J g

1 wPg . can be replaced by

J g wT

E pg

(23)

This yields Vg

§ § Pg K g .E pg .T K g . ¨ ¨ ¨ ¨ Jg © ©

· · ¸ z ¸ ¸ ¸ ¹ ¹

(24)

With Kg

U g .g d . ªe.(1 Sr ) ¼º Pg ¬

c.

(25)

c and d are constants. With the same approach presented for the moisture equations, the mass conservation of air can be written as: wU (26) div U v 0 wt

U a . 1 H 1 S r

wuk , k wt

U a .n H 1 g1

w ( Pa Pw ) wt

U a .n H 1 g 2

wT wt

(27)

§ · §k · 2 2 2 U a . ¨ H .DPw Pw ¨ a H .DPw ¸ Pa k a .E Pg H .DTw T ¸ 0 © Ja ¹ © ¹

Heat flow. Total flow of latent and sensitive heat in an unsaturated porous medium is given based on Philip and De Vries’ theory as: q

O .T [Cmw . U w .U Cmv . U w .Vv Cmg .U g .Vg ] T T0 U w .h fg .Vv U v .h fg .Vg

(28)

Energy Conservation Equation. The differential equation for heat flow is a description, in mathematical terms, of the law of conservation of energy. The energy conservation equation in a porous medium can be expressed by (29) wM wt

div( q )

0

In which q is heat flux and M is the volumetric bulk heat content of medium which can be defined by M cT . T T0 n T .Uv .h fg

(30)

Where cT is the volumetric heat capacity of unsaturated mixture and can be written as: cT

(1 n ). U s .Cms T . U w .Cmw ( n T ). Uv Cmv ( n T ).U g Cmg

(31)

Combining the above equations yield the general differential equation of energy conservation as: F1

wuk , k

F 2 g1

wt

wPg wt

F 2 g1

wPw wT ^F 2 .g 2 F3 ` wt wt

F 4 2T F 5 2 Pg F 6 2 Pw

Where F1 Cms . U s Cmw .S r .U w Cmv .(1 S r ) Uv Cmg .(1 Sr ).U g T T0 Uv .h fg .(1 S r )

F3

C C

F4

Om Cmw .U w . DTw Cmv .U w . DTv Cmg . K g .E Pg .U g T T0 U w .h fg .DTv U v .h fg .K g .E Pg

F2

mw

ms

.n. U w Cmv .n. Uv Cmg .n. U g

T T U .h 0

v

fg

.n

. U s .(1 n ) Cmw .Sr . U w Cmv .(1 Sr ) Uv Cmg .n.(1 S r ). U g

(32)


F5

C

F6

Cmw .U w .DPw Cmv .U w .DPv T T0 U w .h fg .DPv

mw

. U w . DPw Cmv . U w . DPv Cmg . K g . U g

T T U 0

w

375

.h fg . DPv U v .h fg .

Kg

Jg

Set of governing equations. The governing partial differential equations based on the linearization assumptions considered may be summarized and simplified as: & & & & E1 .(w tU )( x, t ) E 2 .U ( x, t ) - ( x, t ) 0, x S, t 0 (33) &

Where - ( x, t )

^b1

of E1 are: E1 i j

T

0 0 0` ,

b2

E1 kl E1 44

& U ( x, t )

û

1

u2

Pw

Pg

T

T

`

&

, x ( x, y ) , 2 D

E1 3i a6 .wi E1 33 a7 E1 34 a15 E1 45 a16 E1 43 a14 E1 5i a20 .w i E1 53 And the components of E 2 are: E 2 i j a1 .w i .w j G ij a2 . ' E 2 i 3 E 2 l k 0 E 2 33 a10 ' E 2 34 a11 ' E 2 35 a12 ' E 2 43 E 2 53 a24 ' E 2 54 a25 ' E 2 55 a26 ' 0

0

a8 a21 a3 .w i a17 '

E1 35 E1 54 E 2 i 4 E 2 44

and a9 a22 a4 .w i a18 '

the

E1 4 i E1 55 E 2 i 5 E 2 45

components a13 .w i a23 a5 .w i a19 '

Where i, j , k 1, 2, l 3, 5, and ' is Laplace operator. Laplace transform domain fundamental solution The objective of this section is to derive the fundamental solution associated with equation (33) which is the response of the medium to unit point excitation (continuous unit line excitation in 2D). The general solution procedure developed by Kupradze [14] is used in this study for the derivation of the fundamental solution [5]. For a continuous unit line force in the i th direction suddenly applied at the origin, i.e. & & & - ( x, t ) G ( x ).H (t ) where H (t ) is the Heaviside step function, the Laplace transform of which is 1/ p G ( x ) . Then, one can rewrite equation (33) in the following form: & & 1 & p.E1 . D ( x, p ) E 2 . D ( x, p ) I G ( x ) p ( x, p ) I Ĭ (wx, p)D

1 G ( x) p

0,

& x R2 ,

(34)

0,

(35)

Where I denotes the unit matrix of order 5, D = ª¬ D ij º¼ is the transformed fundamental solution matrix and 5u5 Ĭ (wx, p ) p.E1 ( x) E 2 ( x) is the differential operator matrix with the components as follows:

4ij (wx, p )

a1 .w i w j G ij a2 . 2

4i 3 (wx, p )

a3 w i

4i 4 (wx, p)

a4 w i

4i 5 (wx, p )

a5 w i

43 j (wx, p)

p.a6 w j

4 4 j (wx, p)

p.a13 w j

45 j (wx, p)

p.a20 w j

433 (wx, p )

p.a7 a10 . 2

434 (wx, p)

p.a8 a11 . 2

435 (wx, p )

p.a9 a12 . 2

443 (wx, p )

p.a14 a17 . 2

444 (wx, p )

p.a15 a18 . 2

4 45 (wx, p )

2

453 (wx, p )

p.a21 a24 . 2

4 55 (wx , p )

p.a23 a26 . 2

454 (wx, p ) wD

p.a16 a19 .

p.a22 a25 .

w / wxD , D

2

(36)

2

1, 2. and is the Laplacian operator.

The first stage is to determine Ĭ (wx, p) , adjoint differential operators of Ĭ (wx, p ) , which is defined by 4ik (wx, p ) 4 kj (wx, p ) G ij det Ĭ(wx, p) (37) In which the determinant of Ĭ (wx, p ) is given by det Ĭ (wx, p )

D1 . p 3 . 4 D2 . p 2 . 6 D3 . p.8 D4 .10

(38)

376


Where D1 , D2 , D3 and D4 are constants including above aij coefficients. Built from the cofactors of Ĭ (wx, p ) , the element of differential operator Ĭ (wx, p ) can be expressed as:

D ij* ( x, p ) G ij B1 . p 3 . 2 B2 . p 2 . 4 B3 . p . 6 B4 .8 B5 . p 3 .w i .w j B6 . p 2 . 2 .w i .w j B7 . p. 4 .w i .w j B8 . 6 .w i .w j D i*3 ( x, p ) B9 . p 2 . 2 .w i B10 . p. 4 .w i B11 . 6 .w i D i*5 ( x, p ) B15 . p 2 . 2 .w i B16 . p. 4 .w i B17 . 6 .w i D * ( x, p ) B . p 2 . 4 B . p . 6 B .8

D i*4 ( x, p ) B12 . p 2 2 .w i B13 . p. 4 .w i B14 . 6 .w i D 3*i ( x , p ) B18 . p 3 . 2 .w i B19 . p 2 . 4 .w i B20 . p. 6 .w i D * ( x, p ) B . p 2 . 4 B . p . 6 B .8

* D 35 ( x, p ) * D ( x, p )

D 4*i ( x, p ) B30 . p 3 . 2 .w i B31 . p 2 . 4 .w i B32 . p. 6 .w i * D 44 ( x, p ) B36 . p 2 . 4 B37 . p. 6 B38 .8 * D ( x, p ) B . p 3 . 2 .w B . p 2 . 4 .w B . p. 6 .w

33

21

22

23

34

B27 . p 2 . 4 B28 . p. 6 B29 .8

B33 . p 2 . 4 B34 . p . 6 B35 .8 43 * D45 ( x, p ) B39 . p 2 . 4 B40 . p . 6 B41 .8 * D 53 ( x, p ) B45 . p 2 . 4 B46 . p . 6 B47 .8 * D ( x, p ) B . p 2 . 4 B . p . 6 B .8 55

51

52

24

5i

25

i

42

* D 54 ( x, p )

26

43

i

44

i

B48 . p 2 . 4 B49 . p. 6 B50 .8

(39)

53

For the second stage, we assume that M r , p is a scalar solution to the equation det Ĭ (wx, p ) M r , p

1 G x p

0

(40)

Which gives Ĭ(wx, p )Ĭ* (wx, p)

1 IG ( x ) p

0

(41)

Consequently, we get Ĭ* wx, p M D

(42)

Equation (42) enables us to determine the twenty five functions D ij by applying the differential operator Ĭ* wx, p to the single unknown function M r , p . Now, equations (38) and (40) may be combined to yield § 6 D3 · D D . p. 4 2 . p 2 . 2 1 . p 3 ¸ ª¬ p . D4 . 4 M º¼ G x 0 (43) ¨ D4 D4 D4 © ¹ This relation, with the introduction of ) as ) p.D4 . 4 M , leads to the following differential equation:

2

O12 2 O22 2 O32 ) G x

0

(44)

After manipulating some mathematical operations, one may obtain the M r , p function as follows: M

§ · K 0 O1 .r K 0 O2 .r K 0 O3 .r 1 ¨ ¸ 2 2 2 2 2 4 2 2 2 4 ¸ 2S . p.D4 ¨ O22 O12 O32 O12 O14 O O O O O O O O O O 1 2 3 2 2 1 3 2 3 3 ¹ ©

In which: O1 m1 p , O2

m2

p , O3

m3

p

(45)

(46)

And the mi coefficients in eq (46) are: §1 · §1 · D D 3 .i 3 .i . h t ¸ , m3 ¨ h t 3 .h t ¸ ¨ h t 3 ¨2 ¸ ¨2 ¸ 3 2 3 2 D D 4 4 © ¹ © ¹ 1 1 1 , m5 , m6 m2 m1 m3 m1 m1 m2 m3 m2 m1 m3 m2 m3

§ D3 ¨ht 3D4 ©

m1 m4

· ¸ , m2 ¹

(47)

Where § D2 § D D D · D · D ( 3 )2 ¸ / 9, r ¨ 9 3 . 2 27 1 2( 3 )3 ¸ / 54 (48) ¨3 D4 ¹ D4 D4 ¹ © D4 © D4 D4 Finally, by applying the differential operator Ĭ* wx, p to M r , p and by definition of the Ȇi interim functions, h

3

r q3 r 2 , t

3

r q3 r 2 , q

we get the 2D fundamental solution of (33) as: Ȇ1

C1 .:11 C2 .:12 C3 .:13 C4 .:14

Ȇ2

C5 .:11 C6 .:12 C7 .:13 C8 .:14

(49)


Ȇ3

377

C5 .: 21 C6 .: 22 C7 .: 23 C8 .: 24

D i j ( x, p ) G ij .Ȇ1

r . xi . x j r

3

. Ȇ2

2. x .x i

r 2G ij

j

r

3

.Ȇ

xi . C9 .: 22 C10 .: 23 C11 .: 24 r x D i 5 ( x, p ) i . C15 .: 22 C16 .: 23 C17 .: 24 r x D4i ( x, p ) i . C30 .: 27 C31 .: 25 C32 .: 26 r D i 3 ( x , p )

3

xi . C12 .: 22 C13 .: 23 C14 .: 24 r x D 3i ( x , p ) i . C18 .: 27 C19 .: 25 C20 .: 26 r x D 5i ( x, p ) i . C42 .: 27 C43 .: 25 C44 .: 26 r D 34 ( x, p ) C24 .:12 C25 .:13 C26 .:14 D ( x, p ) C .: C .: C .: D i 4 ( x, p )

43

33

12

34

13

35

D 33 ( x, p ) C21 .:12 C22 .:13 C23 .:14

(50)

D 35 ( x, p ) C27 .:12 C28 .:13 C29 .:14 D 44 ( x, p ) C36 .:12 C37 .:13 C38 .:14 D ( x, p ) C .: C .: C .:

14

D 45 ( x, p ) C39 .:12 C40 .:13 C41 .:14 D 54 ( x, p ) C48 .:12 C49 .:13 C50 .:14

53

45

D 55 ( x, p )

12

46

13

47

14

C51 .:12 C52 .:13 C53 .:14

In which the Cij coefficients are constants and the : kl interim functions are: :11

ª º K 0 O1 .r K 0 O2 .r K 0 O3 .r » 2 2 p2 . « 2 2 2 2 2 2 2 2 2 2 2 2 2 « O1 O2 O3 O2 O2 O1 O3 O2 O3 O3 ¼» ¬ O2 O1 O3 O1 O1

:12

ª K O .r K O .r K O .r 2 02 2 2 2 02 3 2 p . « 2 02 1 2 2 « O2 O1 O3 O12 O O O O O O3 O2 O32 1 2 3 2 1 ¬

:13

ª O 2 . K O .r O 2 . K O .r O 2 . K O .r 1 0 1 « 2 3 2 0 32 2 2 2 0 22 2 2 2 2 2 « O2 O1 O3 O1 O1 O2 O3 O2 O1 O3 O2 O32 ¬

:14

O . K 0 O2 .r O . K 0 O3 .r 1 ª O . K 0 O1 .r 2 2 .« 2 2 2 2 p « O22 O12 O32 O12 O O O O O O22 O32 1 2 3 2 1 O3 ¬

: 21

ª º K1 O1 .r K1 O2 .r K1 O3 .r » 2 2 p2 . « 2 2 2 2 3 2 2 2 3 « O2 O12 O32 O12 O13 O O O O O O O O O O 1 2 3 2 2 1 3 2 3 3 » ¬ ¼

: 22

ª º K1 O1 .r K1 O2 .r K1 O3 .r » p. « 2 2 2 2 2 2 2 2 2 « O2 O12 O32 O12 O1 O O O O O O O O O O 1 2 3 2 2 1 3 2 3 3 » ¬ ¼

º » » ¼

º » » ¼ º »

4 3

4 2

4 1

»¼

ª O1 . K1 O1 .r O .K O .r O . K O .r º « » 2 2 2 1 22 2 3 2 1 32 2 2 2 2 2 « O1 O2 O3 O2 O1 O3 O2 O32 »¼ ¬ O2 O1 O3 O1 O 3 . K O .r O 3 . K O .r O 3 . K O .r 1 ª 2 2 2 1 22 2 3 2 1 32 . « 2 1 2 1 12 2 2 p « O2 O1 O3 O1 O1 O2 O3 O2 O1 O3 O2 O32 ¬

: 25

§ O . K O .r O . K O .r O . K O .r 2 2 2 1 22 2 3 2 1 32 p . ¨ 2 1 2 1 12 2 ¨ O2 O1 O3 O12 O O O O O O3 O2 O32 1 2 3 2 1 ©

· ¸ ¸ ¹

: 26

§ O . K1 O1 .r O .K1 O2 .r O .K1 O3 .r ¨ 2 2 2 2 2 2 ¨ O22 O12 O32 O12 O O O O O O22 O32 1 2 3 2 1 O3 ©

: 27

§ · K1 O1 .r K1 O2 .r K1 O3 .r ¸ p2 . ¨ 2 2 2 2 2 2 2 2 2 ¨ O2 O12 O32 O12 O1 ¸ O O O O O O O O O O 1 2 3 2 2 1 3 2 3 3 ¹ ©

: 23

: 24

3 3

3 2

3 1

º » » ¼

· ¸

¸¹

(51)

378


Transient fundamental solution To obtain the time domain fundamental solution, one needs to evaluate the analytical inversion of D ij . First, it is necessary to find out the inverse transform \ kl > r , t @ of the functions containing the modified Bessel functions : kl > r , t @ . By the use of the following formulas [15]: ȁ0 ª¬ m j .r , t º¼

L

1

ȁ1 ª m j .r , t º ¬ ¼

L

1

ȁ2 ¬ª m j .r , t ¼º

L

1

ªK « 0 « ¬

m j .r . p º » » p ¼

ªK « 0 « ¬

m j .r . p º » » p. p ¼

ªK « 1 « ¬

m j .r . p º » » p ¼

2 1 § m j .r ī ¨ 0, ¨ 2 © 4t

§ m j .r 2 exp ¨ ¨ 4.t ©

· ¸ ¸ ¹

· m j .r 2 § m j .r 2 . ī ¨ 0, ¸ ¸ ¨ 4 4t ¹ ©

§ m j .r 2 exp ¨ ¨ 4.t m j .r © 1

· ¸ ¸ ¹

(52)

· ¸ ¸ ¹

Expressions of the interim functions are obtained: § m4

\ 11 > r , t @ L 1 ^:11 ` ¨

© m1

\ 12 > r , t @ L 1 ^:12 ` \ 13 > r , t @ L

1

\ 14 > r , t @ L

1

^:13 ` ^:14 `

· m m / 0 ª m1 .r , t º 5 / 0 ª m2 .r , t º 6 / 0 ª m3 .r , t º ¸ ¬ ¼ m ¬ ¼ m ¬ ¼ 2 3 ¹

m / ª¬ m .r , t º¼ m / ª¬ m .r , t º¼ m / ª¬ m .r , t º¼ m .m ./ ª¬ m .r , t º¼ m .m ./ ª¬ m .r , t º¼ m .m ./ ª¬ m .r , t º¼ m .m ./ ª¬ m .r , t º¼ m .m ./ ª¬ m .r , t º¼ m .m ./ ª¬ m .r , t º¼ 4

0

1

4

1

4

2 1

5

0

1

0

0

2

5

1

5

2

2 2

6

0

0

2

2

0

3

6

6

3

2 3

0

3

0

3

\ 21 > r , t @ L

1

^:21`

§ m · m5 m6 4 ¨ / ª m .r , t º / ª m .r , t º / ª m .r , t º ¸ ¼ m m 1¬ 2 ¼ m m 1¬ 3 ¼¸ ¨m m 1¬ 1 2 2 3 3 © 1 1 ¹

\ 22 > r , t @ L

1

^:22 `

§ m · m m ¨ 4 / ª m .r , t º 5 /1 ª m2 .r , t º 6 /1 ª m3 .r , t º ¸ ¼ ¬ ¼ ¬ ¼¸ ¨ m 1¬ 1 m m 1 2 3 ¹ ©

\ 23 > r , t @ L

1

^: 23 `

m .

\ 24 > r , t @ L

1

\ 25 > r , t @ L \ 26 > r , t @ L

1

1

\ 27 > r , t @ L

^:24 `

m1 ./1 ª¬ m1 .r , t º¼ m5 . m2 ./1 ª¬ m2 .r , t º¼ m6 . m3 ./1 ª¬ m3 .r , t º¼

m4 . m1 ./ 2 ª¬ m1 .r , t º¼ m5 . m2 ./ 2 ª¬ m2 .r , t º¼ m6 . m3 ./ 2 ª¬ m3 .r , t º¼

^:27 `

r

xi . C12 .\ 22 C13 .\ 23 C14 .\ 24 r x D 3i ( x, p ) i . C18 .\ 27 C19 .\ 25 C20 .\ 26 r x D 5i ( x, p ) i . C42 .\ 27 C43 .\ 25 C44 .\ 26 r D 34 ( x, p ) C24 .\ 12 C25 .\ 13 C26 .\ 14 D ( x, p ) C .\ C .\ C .\

33

12

§ m · m m ¨ 4 / 2 ª m1 .r , t º 5 / 2 ª m2 .r , t º 6 / 2 ª m3 .r , t º ¸ ¬ ¼ ¬ ¼ ¬ ¼ ¸ ¨ m m2 m3 1 ¹ ©

r

43

m4 .m1 m1 ./ 2 ª¬ m1 .r , t º¼ m5 .m2 m2 ./ 2 ª¬ m2 .r , t º¼ m6 .m3 m3 ./ 2 ª¬ m3 .r , t º¼

Finally, the fundamental solutions are obtained: r . xi . x j 2.xi . x j r 2G ij .b D i j ( x, p ) G ij .b1 .b 2 3 3 3 D i 4 ( x, p )

m4 .m1 m1 ./1 ª¬ m1 .r , t º¼ m5 .m2 m2 ./1 ª¬ m2 .r , t º¼ m6 .m3 m3 ./1 ª¬ m3 .r , t º¼

^: 25 `

^:26 `

1

4

(53)

34

13

35

14

D 45 ( x, p ) C39 .\ 12 C40 .\ 13 C41 .\ 14

xi . C9 .\ 22 C10 .\ 23 C11 .\ 24 r x D i 5 ( x, p ) i . C15 .\ 22 C16 .\ 23 C17 .\ 24 r x D 4i ( x, p ) i . C30 .\ 27 C31 .\ 25 C32 .\ 26 r D i 3 ( x, p )

D 33 ( x, p )

C21 .\ 12 C22 .\ 13 C23 .\ 14

D 35 ( x, p ) C27 .\ 12 C28 .\ 13 C29 .\ 14 D 44 ( x, p ) C36 .\ 12 C37 .\ 13 C38 .\ 14 D 53 ( x, p ) C45 .\ 12 C46 .\ 13 C47 .\ 14


D 54 ( x, p )

379

D 55 ( x, p )

C48 .\ 12 C49 .\ 13 C50 .\ 14

C51 .\ 12 C52 .\ 13 C53 .\ 14

(54)

Where b1

B1 .\ 11 B2 .\ 12 B3 .\ 13 B4 .\ 14

b2 b3

B5 .\ 11 B6 .\ 12 B7 .\ 13 B8 .\ 14 B5 .\ 21 B6 .\ 22 B7 .\ 23 B8 .\ 24

(55)

For instance, the derived Green functions are shown through Figs. 1 to 2:

0.001 0.0005 z

0.00001

0 -0.0005

5´ 10

-6

0 y

-0.00001 -6 -5 ´ 10 x

0

0 -0.5 z -1 -1.5 -2

1000 800

0.002 002

-5 ´ 10 - 6

0.004

5´ 10 - 6 0.00001 -0.00001

600 400 t 0.006 r 0.008

200 0.01

Fig 2: Green function D34

Fig 1: Green function D11 Solid skeleton displacment in direction one due to a unit point load in direction one.

Variations of water pressure due to a unit increment in air pressure.

Verification Firstly, the new transient fundamental solution is compared to the steady THHM fundamental solution [8] by substituting the coefficients of terms in which the time variation is present, by zero:

G F

D i 3 ( x, p )

F

. 6 .w i M

D i 4 ( x, p )

F

. 6 .w i M

D i j ( x, p )

0,

i

D 34 ( x, p )

F

D 43 ( x, p )

F

D 45 ( x, p )

F

.8 M

D 54 ( x, p )

F

.8 M

Where M

1 r 8 25 12 Ln r and Fij2 are the constant coefficients. 3, 538, 944.S . p . D4

2 11

ij

2 13

2 14

2 22

2 33

2 42

.8 F122 . 6 .w i .w j M

F132 §1 · xi Ln r ¸ 4S . p .D4 ¨© 2 ¹

F142 §1 · Ln r ¸ xi 4S . p .D4 ¨© 2 ¹

F152 §1 · Ln r ¸ xi 4S . p. D4 ¨© 2 ¹

D i 5 ( x, p )

F

. 6 .w i M

D 33 ( x, p )

F

.8 M

F212 Ln r 2S . p. D4

F222 Ln r 2S . p. D4

D 35 ( x, p )

F

.8 M

F232 Ln r 2S . p. D4

F312 Ln r 2S . p. D4

D 44 ( x, p )

F

.8 M

F322 Ln r 2S . p. D4

D 53 ( x, p )

F

.8 M

F412 Ln r 2S . p. D4

3, 5, j 1, 2,

.8 M

F332 Ln r 2S . p. D4

F422 Ln r 2S . p. D4

.8 M

2 31

ª § xi x j · º F122 · 1 2 2 2 § «G ij ¨ Ln r 2 F11 F12 ¸ F12 ¨ 2 ¸ » , i , j 1, 2, 4S . p . D4 «¬ © 2 ¹ © r ¹ »¼

D ij ( x, p )

D 55 ( x, p )

2 15

2 21

2 23

2 32

2 41

F

2 43

.8 M

(56)

F432 Ln r 2S . p . D4

Secondly, if the coefficients representing the thermal behaviour of the phenomenon and Henry’s coefficient approach to zero the steady fundamental solution (eq. 56) will approach the corresponding isothermal solutions [7]:

380


J w .F s . xi . 1 2 ln r

D i j ( x, p )

ª¬(O P ) 2(O 3P ).ln r º¼ .r 2 .G ij 2(O P ). xi . x j 8S .P .(O 2 P ).r 2

D i 3 ( x, p )

D i 4 ( x, p )

J g .( F s 1). xi . 1 2 ln r 8. p .S .(O 2P ). K g

D 3i

D 33 ( x, p )

J w .ln r 2S . p . K w

D 44 ( x, p )

J g .ln r 2S . p .K g

D 34

0

D i 5

0,

D 43

D 4i

D 5 j

8S . p.(O 2P ).K w

(57)

0

i , j 1, 4,

Also, it is evident that while F s approaches zero, the fundamental solution in eq (57) approaches elastostatic fundamental solution [16, 17]: ª(O P ) 2(O 3P ).ln r ¼º .r 2 .G ij 2(O P ). xi . x j J g . xi D i j ( x, p ) ¬ . 1 2 ln r D i 4 ( x, p ) 8.S .(O 2 P ).K g .U g 8S .P .(O 2 P ).r 2 D 33 ( x, p ) D i 5

D 5 j

J w .ln r 2S . K w . U w 0,

i , j 1, 4,

D 44 ( x, p )

J g .ln r 2S . p . K g . U g

D i 3 ( x, p )

0

D 3i

D 4i

0

D 34

D 43

0

(58)

References [1] D.W Pollock Water resources research, 22 (5), 765-775 (1986). [2] S.Olivella, J.Carrera, A.Gens, E.E.Alonso Transport Porous Media, 15, 271–293(1994). [3] E.Jabbari, B.Gatmiri, 7th International Conference on Boundary Element Techniques (B.Gatmiri, A.Sellier, M.H.Aliabadi), EC:, Paris, 247-248 (2006). [4] B.Gatmiri, M.Kamalian International Journal of Geomechanics 2(4), 381–398 (2002). [5] B.Gatmiri, K.V.Nguyen Communications in Numerical Methods in Engineering 21 (3), 119–132 (2005). [6] B.Gatmiri, E.JabbariInternational Journal of Solids and Structures42, 5971–5990 (2005). [7] B.Gatmiri, E.Jabbari 5th International Conference on Boundary Element Techniques (M.H.Aliabadi, V.M.A.Leitão), EC:, Lisbon, 217-221 (2004). [8] E.Jabbari, B.Gatmiri International Journal of Computer Modelling in Engineering and Sciences 18(1), 31-43 (2007). [9] B.Gatmiri, P.Delage, M.Cerrolaza Advances in Engineering Software 29(1), 29-43 (1998). [10] B.Gatmiri, P.Delage, 1th International Conference on Unsaturated Soils (E.E.Alonso, P.Delage), EC:, Paris, 1049-1056 (1995). [11] L.A.Richards J. Physics 1, 318-333 (1931). [12] J.R.Philip, D.A.de Vries Trans. Am. Geophys 38, 222-232 (1957). [13] J.Ewen, H.R.Thomas Géotechnique, 39(3), 455-470 (1989). [14] V.D.Kupradze et al. Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland, Netherlands (1979). [15] M.Abramowitz, I.A.Stegun Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C. (1965). [16] P.K.Banerjee The Boundary Element Methods in Engineering, McGraw-Hill Book Company, England (1994). [17] G.Beer Programming the Boundary Element Method, John Wiley and Sons, England (2001).


381

A Three-Step MDBEM for Nonhomogeneous Elastic Solids X.W. Gao1, a, J. Hong1,b and Ch. Zhang2,c 1

Department of Engineering Mechanics, Southeast University, Nanjing 210096, PR China 2

Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany a

[email protected], b [email protected], c [email protected]

Keywords: Boundary element method; Nonhomogeneous elastic solids; Multi-domain technique; Functionally graded materials (FGMs).

Abstract. In this paper, a three-step boundary element method (BEM) is presented for solving boundary value problems in two-dimensional (2D) and three-dimensional (3D) nonhomogeneous and linear elastic solids by using the multi-domain boundary element method (MDBEM). Fundamental solutions for homogeneous and linear elastic solids are adopted in the MDBEM. Boundary-domain integral equations expressed in terms of normalized displacements and tractions are formulated for each sub-domain. The first step is the elimination of internal variables, and the second one is the elimination of boundary unknowns used only by individual sub-domains, and the last step is the establishment of a system of linear algebraic equations according to the continuity/discontinuity conditions of the displacements and the tractions at common nodes on the interfaces. Discontinuous elements are utilized to model the traction discontinuity at corner nodes. Numerical examples are presented to demonstrate the accuracy and the efficiency of the present three-step MDBEM. Introduction The material properties of continuously nonhomogeneous solids such as functionally graded materials (FGMs) are dependent on spatial positions [1]. Although the boundary element method (BEM) has been successfully developed and applied to homogeneous and linear elastic solids since many years, its extension and applications to continuously nonhomogeneous and linear elastic solids are not straight-forward. The main reason is the fact that the required fundamental solutions or Green's functions for general nonhomogeneous and linear elastic solids are either mathematically too complicated or not available, which makes an easy and efficient numerical implementation difficult. One remedy for this difficulty is the use of fundamental solutions for homogeneous and linear elastic solids in the BEM formulation, which involves domain-integrals in the boundary-domain integral equations. Although the domain integrals can be converted to global boundary integrals [2] or local ones [3], the resulting system of algebraic equations involves unknown quantities consisting of both boundary unknowns and internal displacements. The numerical solution of such a system with mixed boundary and internal variables is in general very time consuming or even unfeasible for large-scale problems using the single-domain BEM. An efficient numerical way to solve boundary value problems in nonhomogeneous and linear elastic solids is the multi-domain boundary element method (MDBEM) [4,5], where the considered domain is divided into several sub-domains. There exist two different efficient solution techniques in the MDBEM. One is the variable condensation technique and the other is the iterative technique. The first technique results in a small system of equations by eliminating some variables in an intermediate step, while the latter solves the large system of algebraic equations by using a fast iterative scheme such as the Krylov’s iteration method [6]. In this paper, a three-step BEM is presented for solving boundary value problems in two-dimensional (2D) and three-dimensional (3D) nonhomogeneous and linear elastic solids by using the MDBEM [5]. Fundamental solutions for homogeneous and linear elastic solids are adopted in the MDBEM. Boundary-domain integral equations with normalized displacements and tractions are formulated for sub-domains. The first step of the method is the elimination of internal variables for each sub-domain. The second step is the elimination of boundary unknowns defined over nodes used only by the sub-domain itself. And the third step is the establishment of a system of linear

382


algebraic equations according to the continuity/discontinuity conditions of the displacements and the tractions at common nodes on the interfaces. Discontinuous elements are implemented to properly model the traction discontinuity at corner-nodes. The present three-step MDBEM has two important features, namely, only interfacial displacements are unknowns in the final system of algebraic equations, and the coefficient matrix is blocked and sparse. Consequently, large-scale 2D and 3D boundary value problems in nonhomogeneous and linear elastic solids can be dealt with efficiently. Integral equations for functionally graded materials (FGMs) In isotropic, continuously nonhomogeneous, and linear elastic solids, such as functionally graded materials, the shear modulus P is a function of spatial coordinates, while, for most cases, the Poisson’s ratio Q can be regarded as a constant. Under this assumption, integral equations for boundary and internal nodes can be derived by using Gauss’ divergence theorem as follows [5,7] cui (x p )

³U *

ij

(x, x p )t j (x)d * ³ Tij (x, x p )u j (x)d * ³ Vij (x, x p )u j ( x) d : , *

:

(1)

where c=1 for internal points and c=1/2 for smooth boundary points, U ij and Tij are the Kelvin displacement and traction fundamental solutions [8] for an isotropic, homogeneous and linear elastic solid with P 1 , and

Vij

1 ^P,k r,k [(1 2Q )G ij E r,i r, j ] (1 2Q )(P,i r, j P, j r,i )` , 4SD (1 Q )r D

(2)

where E=2 for 2D and E=3 for 3D problems and D=E-1. In Eqs. (1) and (2), u j (x) and P are the

normalized displacements and shear modulus defined by ui (x)

P (x) ui (x) ,

P (x) log P (x) .

(3)

From Eq. (1), it can be seen that no displacement gradients are involved in the boundary-domain integral equations. This is attributed to the use of the normalized quantities defined in Eq. (3). Comparison of Eq. (1) to the conventional boundary integral equations for isotropic, homogeneous and linear elastic solids [8] shows that there is a domain integral appearing in the integral equations. This domain integral is converted into an equivalent boundary integral using the radial integral method (RIM) [2]. Since the unknown variables u j are included in the domain integral, some internal points are required to be placed inside the domain : to improve the computational accuracy in the approximation of u j in terms of the radial basis functions (RBFs) in the application of RIM [2,7]. Consequently, the resulting system of algebraic equations includes both boundary unknowns and internal normalized displacements as the system unknowns. To solve such a system using a single domain technique, the computational time and the required memory storage would be huge for complicated 3D problems. Therefore, a robust MDBEM solution technique is desired for solving such large-scale problems. Three-step MDBEM for isotropic, nonhomogeneous and linear elastic solids As shown in Fig. 1, the domain of concern is divided into a number of sub-domains. The nodes used for each sub-domain are classified into three types: “self nodes”, “internal nodes”, and “common nodes”. To efficiently exploit the MDBEM technique, the order of the three types of nodes is arranged in such a way that the self nodes are numbered first, followed by the common nodes and finally the internal nodes. To model the traction discontinuity at a corner or an edge, discontinuous element [5] is used and more than one nodes are defined at an internal corner at which different sub-domains meet


383

and at a boundary corner at which at least one of the components is specified with the displacement boundary condition (see Fig. 1).

tj

u u

tj

:i

u u

:k

t

t

:j

self node internal node common node Fig. 1. Definition of three types of nodes After using the node arrangement strategy described above and applying RIM [2] to transform the domain integral into a boundary integral, the boundary-domain integral equations (1) can be converted into a system of algebraic equations for each sub-domain, which can be written in the matrix form as H bs us᧧H bc uc H bi ui

Gbs ts᧧Gbc tc

(4)

H is us᧧H ic uc H ii ui

Gis ts᧧Gic tc

(5)

for boundary nodes, and

for internal nodes. In Eqs. (4) and (5), the subscript b denotes quantities for boundary nodes consisting of self nodes and common nodes, and the subscripts s, i and c represent quantities for self, internal and common nodes, respectively. Also, us , uc , ui , t s and tc are displacement and traction vectors corresponding to the three types of nodes. It is noted that for piecewise homogeneous solids, the matrix H ii is an identity matrix and H bi is a zero matrix. After invoking all specified displacement and traction boundary conditions in Eqs. (4) and (5), the following equation set can be obtained for each sub-domain Abs xs᧧H bcuc H bi ui

yb᧧Gbctc ,

(6)

Ais xs᧧H ic uc H ii ui

yi᧧Gic tc ,

(7)

where xs is the vector consisting of unknown displacements and unknown tractions over the self nodes, and yb and yi are the known vectors formed by multiplying all given boundary displacements and tractions with their corresponding matrix elements. To solve Eqs. (6) and (7) for the unknown vectors xs , uc , tc and ui , a three-step solution technique is applied, which is described in the following.

384


Step 1: Eliminating internal unknowns for each sub-domain

The matrix H ii in Eq. (7) is a square matrix and well-posed, so eliminating the internal displacements ui from Eqs. (6) and (7) it follows for each sub-domain Abs xs᧧H bcuc

yb᧧Gbc tc ,

(8)

where

Abs

Abs H bi ( H ii ) 1 Ais ,

H bc

H bc H bi ( H ii ) 1 H ic ,

Gbc

Gbc H bi ( H ii ) 1 Gic ,

yb

yb H bi ( H ii ) 1 yi .

(9)

It is noted again that for piecewise homogeneous solids, the matrix H ii is an identity matrix and H bi is a zero matrix, and therefore it is unnecessary to form Eq. (8) since the matrices to be formed in Eq. (9) reduce to the original matrices. Step 2: Eliminating boundary unknowns for each sub-domain

Since for each sub-domain the boundary nodes are composed of the self nodes and the common nodes, Eq. (8) can be divided into two sets of equations for self nodes and common nodes, i.e., Ass xs᧧H sc uc

ys᧧Gsc tc ,

(10)

Acs xs᧧H ccuc

yc᧧Gcc tc .

(11)

All matrices in Eqs. (10) and (11) are sub-matrices of the corresponding matrices in Eq. (8). Now, elimination of xs from Eqs. (10) and (11) yields Hˆ cc uc

yˆ c᧧Gˆ cc tc ,

(12)

where Hˆ cc H cc Acs ( Ass ) 1 H sc , Gˆ cc Gcc Acs ( Ass ) 1 Gsc , yˆ c

(13)

1

yc Acs ( Ass ) ys .

Step 3: Assembling the system of equations from all sub-domain’s contributions

Equations (10) and (11) can be applied to every sub-domain. For the n-th sub-domain, the traction vector tc for the common nodes can be expressed as tc( n )

(Gˆ cc( n ) ) 1 ( Hˆ cc( n )uc( n ) yˆ c( n ) ) .

(14)

Assembling all sub-domain’s contributions for the global common nodes and applying the traction equilibrium condition

¦t

(n) c

0 results in the final system of algebraic equations as

n

K ccU c where

Yc ,

(15)


K cc

¦ (Gˆ

385

) Hˆ cc( n )Q ( n ) ,

( n ) 1 cc

(16)

n

Yc

¦ (Gˆ

( n ) 1 cc

) yˆ c( n ) ,

(17)

n

where Q ( n ) is the location matrix consisting of 0 and 1, which relates the local displacement vector uc( n ) to the global one U c . Solving Eq. (15) for U c , we can obtain the displacements at all common interface nodes, and then substituting it back into Eq. (14) we can compute the tractions at each domain’s common nodes. Using these results, the boundary unknowns at self nodes of each sub-domain can be calculated by applying Eq. (10). Equation (15) shows that the number of degrees of freedom of the system is only the number of degrees of freedom of the common interface nodes, which is much smaller than those of all boundary and internal nodes. It is noted that for piecewise homogeneous solids, the present three-step solution technique reduces to a two-step solution technique consisting of the last two steps as described above.

Numerical example The numerical example considered here is a multi-planar tubular DX-Joint as depicted in Fig. 2. This example has been analyzed in reference [8]. The DX-Joint consists of a large diameter tube (chord) intersected orthogonally by two smaller diameter tubes (braces). The outer radii of the chord and braces are 228.6mm and 28.58mm, respectively, while the inner radii are one-half of these values. For simplicity, only a short section of the tubes is analyzed and the eight-fold symmetry is exploited (Fig. 3). The half-lengths of the tubes are given by Lx 900mm and Ly Lz 660mm . A constant

Poisson’s ratio Q = 0.3 is used and a tensile load F=0.234GPa is applied to each end of the DX-Joint.

F

F

F

F F F Fig. 2. Multi-planar tubular DX-Joint subjected to axial loads

386


The computational domain is divided into three sub-domains as shown in Fig. 3. The BEM mesh consists of 1660 linear quadrilateral boundary elements with 1606 boundary nodes (including 56 interface elements and 60 interface nodes) and 329 internal nodes.

z

:3

:2

:1

y

x Fig. 3. BEM mesh of the DX-Joint Case 1: Homogeneous DX-Joint

For the purpose of the validation of the present MDBEM, numerical calculations are first carried out for the homogeneous case by setting the Young’s modulus E = 200GPa for all the three sub-domains. Figure 4 shows the distribution of the displacement ux along the middle line of the outer surface of the chord. For comparison, the results using the single-domain code BEMECH listed in [8] are also given. It can be seen that the two sets of numerical results are in very good agreement.

Displacement (mm)

1.4

BEMECH

1.2

Present

1 0.8 0.6 0.4 0.2 0 0

150

300

450

600

750

900

x (mm)

Fig. 4. Displacement for homogenous DX-Joint with E = 200GPa


387

Case 2: Functionally graded DX-Joint

The second computation is performed by assuming E1

ELx D x , E 2

E Ly e E y and E3

E Lz e J z for

the sub-domains :1 , : 2 and :3 , respectively. The parameters D, E and J are determined by

D where E=200GPa, ELx

ELx E , Lx

10 E , ELy

E

log ( ELy / E ) , Ly

5E and ELz

J

log ( ELz / E ) , Lz

10 E . Figure 5 plots the distribution of the

displacement ux along the middle line of the outer surface of :1 , and Fig. 6 shows the displacements uy and uz along the middle lines of the outer surfaces of : 2 and :3 , respectively. Figure 7 illustrates

the axial stresses V yy and V zz along the middle lines of the domains : 2 and :3 , respectively. From Figs. 6 and 7 it can be seen that the axial displacements along domains 2 and 3 are quite different, since the variation of the Young’s modulus is different for the two domains. On the other hand, Fig. 7 shows the same pattern for the axial stresses of the two domains 2 and 3. This indicates that the distribution of the axial stresses mainly depends on the applied loads in the considered case. 0.32

Displacement (mm)

ux (over surface of domain 1) 0.24

0.16

0.08

0 0

150

300

450

600

750

900

x (mm)

Fig. 5. Variation of ux over the outer surface of :1

Dispacements (mm)

0.5 0.4

uy (over surface of domain 2)

0.4

uz (over surface of domain 3)

0.3 0.3 0.2 0.2 0.1 0.1 0.0 200

300

400

500

600

700

y or z over surface 2 or 3 (mm)

Fig. 6. Variations of uy and uz over the outer surfaces of : 2 and :3

388


0.68 Sigma-yy (over domain 2)

Stresses

0.60

Sigma-zz (over domain 3)

0.52 0.44 0.36 0.28 0.20 200 255 310 365 420 475 530 585 640 y or z over surface 2 or 3 (mm)

Fig. 7. Variations of axial stresses along middle lines of : 2 and :3 Summary

A three-step MDBEM is presented in this paper for the numerical solution of boundary value problems in 2D and 3D isotropic, continuously nonhomogeneous and linear elastic solids. Fundamental solutions for isotropic, homogeneous and linear elastic solids are implemented in the present MDBEM. Boundary-domain integral equations expressed in terms of normalized displacements and tractions are formulated for each sub-domain. The domain-integrals are transformed to boundary integrals by using the radial integration method (RIM). Through a two-step elimination procedure, a system of linear algebraic equations for the displacements at common nodes is established. Discontinuous elements are adopted to model the traction discontinuity at corner nodes. The number of unknowns in the present three-step MDBEM is much smaller than that of the classical single-domain BEM. Numerical examples show that the present three-step MDBEM is efficient and suitable for solving large-scale problems in isotropic, continuously nonhomogenous and linear elastic solids. Acknowledgement

Support by the German Research Foundation (DFG) under the project number ZH 15/10-1 is gratefully acknowledged. References

[1] [2] [3] [4] [5] [6] [7] [8]

X.W. Gao, Ch. Zhang, J. Sladek and V. Sladek: Compos. Sci. Tech. Vol. 68 (2008), p. 1209. X.W. Gao: Eng. Anal. Bound. Elem. Vol. 26 (2002), p. 905. J. Sladek, V. Sladek and Ch. Zhang: Building Research Journal, Vol. 53 (2005), p. 71. S. Ahmad and P.K. Banerjee: Int. J. Numer. Meth. Engng., Vol. 26 (1988), p. 891. X.W. Gao, L. Guo, and Ch. Zhang: Eng. Anal. Bound. Elem. Vol. 31 (2007), p. 965. K. Davey, S. Bounds, I. Rosindale and M.T.A. Rasgado: Comp. & Struct. Vol. 80 (2002), p. 643. X.W. Gao, Ch. Zhang and L. Guo: Eng. Anal. Bound. Elem. Vol. 31 (2007), p. 974. X.W. Gao and T.G. Davies: Boundary Element Programming in Mechanics. Cambridge University Press, 2002.


389

DBEM for Fracture Analysis of Stiened Curved Panels (Plates and Shallow Shells Assemblies) P.M. Baiz and M.H. Aliabadi Department of Aeronautical Engineering, Imperial College London South Kensington campus, London SW7 2AZ Keywords: Fracture Mechanics, Shear Deformable, Plate and Shallow Shell Assemblies.

Abstract. This paper presents applications where the DBEM formulation presented by Dirgantara and Aliabadi [3] is combined with the multi region BEM presented recently by Baiz and Aliabadi [2], for the analysis of cracked shear deformable plates and shallow shell assemblies. Stress intensity factors are obtained using the CTOD technique. Several examples are solved to demonstrate the capabilities of the proposed technique. Comparing DBEM with FEM models, it was clear that good accuracy and e!ciency can be achieved with the present multi region DBEM approach. Introduction. Cracks are present in most structural members either as a result of the manufacturing process or due to a localized damage during service life. These cracks may grow by fatigue, corrosion or creep, decreasing the strength and leading to the failure of the structure. The dual boundary element method (DBEM) is based on the use of dierent equations on each crack surface (displacement and traction integral equations). During the past years, the dual boundary element method has emerged as a robust numerical method for fracture mechanics problems [1]. Applications of the dual boundary element method to fracture mechanics of shear deformable plates have been reported independently by Rashed, Aliabadi and Brebbia [6] and Ahmadi-Brooghani and Wearing [8] while DBEM for shear deformable shallow shells have been derived by Dirgantara and Aliabadi [3]. Multi-region BEM (Plate and Shallow Shells). Lets consider M assembled cylindrical shallow shells or plates joined at Jq as shown in Figure 1a. The global coordinate system is given by p p -x1 -x2 -x3 , and the local coordinate systems for each region by -xp 1 -x2 -x3 (m = 1; M ). The plates or shallows shells have an uniform thickness h, Young’s modulus E, Poisson’s ratio º. As shown in Figure 1c, w represent rotations of the middle surface, w3 denotes the out-of-plane displacement, and u represent in-plane displacements. And generalized tractions are denoted as: p due to the stress couples, p3 due to shear stress resultant and t due to membrane stress resultants. Shallow shells are defined using a curvilinear coordinate system. This means that contrary to flat plates which have a fix normal (local coordinate system) through the whole plate, the local coordinate system of a shallow shell changes with the curvature (see Figure 1b). In the simple case of two shallow shells or plates with the same axis orientation at the junction line (see Figure 1c), the continuity and equilibrium equations along the joint can be written as follows:

= up+1 ; up wlp = wlp+1 ;

P X

tp =0

p=1 P X

(1)

pp m =0

p=1

Because two or more angled plates or shallow shells joined together are considered, an approach similar to that proposed in [10] is developed. To simplify this approach, the local coordinate systems of each region is assumed to be defined such that the xp 2 directions are all aligned with the global direction x2 , following the implementation for plate assemblies presented by Wen et. al. [9] or Di

390


:m

x3m

Jn

Shallow Shell Base Plane

x1m

D c

Jn

w3

m

x2

D c

x3m

u1

x1m

x3

R

x3

x2

Dc

x2

*m

x1

x1

a)

m

x2 m

x2

x2

x3m

m+1

x1m

x2

x3m

m+1

x3

x1m

m+1

x3

D

b)

m+1

x2

m+1

x1

m+1

x1

x3 wm2

x1

Displacements

um2

w3m

m+1

w2

w1m

um+1 2

u1m

c)

m

p1

Tractions

wm+1 3 wm+1 1 um+1 1 pm+1 1

m

p3

t2m t1m

p2m

pm+1 3

t2m+1

m+1

p2 t1m+1

Figure 1: a) Plate and Shallow Shell Assembly, b) Local Coordinate System of Shallow Shell, c) Simple Assemblies. Pisa [4]. Based on the above simplification, w3 and u1 displacements for any given shallow shell at a junction line (Jq ) can be presented as shown in Figure 1b. Therefore, compatibility equations for each pair of adjacent shallow shells (e.g. m = 1 and m = 2) could be written as follows: 1 1f 1 1 1f 1 1f 1 2 2f 2 2f 2 2 2f 2 2f 2 u11 (n1f 11 n11 + n31 n13 ) + w3 (n13 n11 + n33 n13 ) = u1 (n11 n11 + n31 n13 ) + w3 (n13 n11 + n33 n13 ) 1 1f 1 1 1f 1 1f 1 2 2f 2 2f 2 2 2f 2 2f 2 u11 (n1f 11 n31 + n31 n33 ) + w3 (n13 n31 + n33 n33 ) = u1 (n11 n31 + n31 n33 ) + w3 (n13 n31 + n33 n33 )

u12 = u22

w11 = w12 w21 = 0

w22 = 0

(2)

where np ln are the components of the rotation matrix of the shallow shell base plane m from local to global coordinates [10], and npf ln are the components of the rotation matrix from the curvilinear coordinate system to the shallow shell base plane m. The components of npf ln are given by:

®p f

= cos(®p npf 11 f );

npf 12 = 0;

p npf 13 = cos(90 + ®f )

npf 21 npf 31

npf 22 = 1; pf n32 = 0;

npf 23 = 0 pf n33 = cos(®p f )

= 0; = cos(90

¡ ®p f );

(3)

is measured with respect to the shallow shell base plane, as shown in Figure 1b. where Equations in (2) result in a system of 5M ¡4 compatibility conditions, and have to be supplemented


391

with 4 equilibrium conditions as follows: P X

pf p pf p p pf p pf p [tp 1 (n11 n11 + n31 n13 ) + p3 (n13 n11 + n33 n13 )] = 0

p=1 P X

pf p pf p p pf p pf p [tp 1 (n11 n31 + n31 n33 ) + p3 (n13 n31 + n33 n33 )] = 0

p=1 P X

tp = 0 2

p=1 P X

pp = 0 1

(4)

p=1

to produce the required 5M equations. This approach relies on the assumption that the plate or shallow shell flexural rigidity in its own plane is so large that it is possible to ignore its associated deformation, in another words, there is no drilling rotation. Boundary Integral Formulation. The dual boundary element method is based on the use of two independent equations, the displacement and traction boundary integral equations, at each pair of coincident source points on the surfaces that define a crack. The displacement integral equations for collocation points on one crack surface (x+ 2 K+ ), can be written as follows [3]:

Z Z 1 1 zm (x+ ) + zm (x3 ) + ¡ SlmW (x+ > x)zm (x) gK (x) = ZlmW (x+ > x)sm (x) gK (x) 2 2 K K · ¸ Z ¢ ¡ 1 ¡ 2 x> (X) + x> (X) + x!>! (X) gl (X) ¡ Zl3W x+ > X n E 2 1¡ l Z ¢ ¡ ¡ Zl3W x+ > X n E ((1 ¡ )n + n!! ) z3 (X) gl (X) l Z + Zl3W (x+ > X)t3 (X)gl(X)

(5)

l

Z 1 1 W (x+ > x)x (x)gK(x) x (x+ ) + x (x3 ) + ¡ W 2 2 K

and,

Z + K

W X (x+ > x)E [n (1 ¡ ) + n!! ] z3 (x)q (x)gK(x)

Z

W X (x+ > X)E [n (1 ¡ ) + n!! ] z3> (X)gl(X) Z Z W W (x+ > x)w (x)gK(x) + X (x+ > X)t (X)gl(X) = X

¡

l

K

(6)

l

In order to avoid an ill-conditioned system, the traction integral equations are used for collocations on the other crack surface (x3 2 K3 ) [3]:

Z Z 1 1 W W (x3 > x)z (x)gK(x) + q (x3 ) ¡ S3 (x3 > x)z3 (x)gK(x) s (x3 ) ¡ s (x+ ) + q (x3 ) = S 2 2 K K Z Z W W = q (x3 ) ¡ Z (x3 > x)s (x)gK(x) + q (x3 ) Z3 (x3 > x)s3 (x)gK(x) K K µ ¶ Z 1¡ 2 W x># (X) + x#> (X) + x!>! (X) # Z3 (x3 > X)gl(X) ¡q (x3 ) n# E 2 1¡ l

392

Eds: R Abascal and M H Aliabadi Z ¡q (x3 )

W n# E ((1 ¡ )n# + # n!! ) z3 (X) Z3 (x3 > X)gl(X)

l

Z +q (x3 ) l

W Z3 (x3 > X)t3 gl(X)

(7)

Z Z 1 1 W W s3 (x3 ) ¡ s3 (x+ ) + q (x3 ) ¡ S3 (x3 > x)z (x)gK(x) + q (x3 ) = S33 (x3 > x)z3 (x)gK(x) 2 2 K K Z Z W W (x3 > x)s (x)gK(x) + q (x3 ) ¡ Z33 (x3 > x)s3 (x)gK(x) = q (x3 ) Z3 K

K

µ ¶ 1¡ 2 W ¡q (x3 ) n# E x># (X) + x#> (X) + x!>! (X) # Z33 (x3 > X)gl(X) 2 1¡ l Z W ¡q (x3 ) n# E ((1 ¡ )n# + # n!! ) z3 (X) Z33 (x3 > X)gl(X) Z

l

Z +q (x3 ) l

and

W Z33 (x3 > X)t3 gl(X)

(8)

Z 1 1 W (x3 > x)x (x)gK(x) w (x3 ) ¡ w (x+ ) + q (x3 ) = W 2 2 K Z W (x3 > x)E [n# (1 ¡ ) + # n!! ] z3 (x)q# (x)gK(x) +q (x3 ) ¡ X K

Z ¡q (x ) 3

l

W X (x3 > X)E [n# (1 ¡ ) + # n!! ] z3> (X)gl(X)

Z Z W W (x3 > x)w (x)gK(x) + q (x3 ) X (x3 > X)t gl(X) = q (x3 ) ¡ X K

l

1 + q (x3 )E [(1 ¡ )n# + # n!! ] z3 (x3 ) 2

(9)

Equations (5-6) and (7-9) represent displacement and traction integral equations on the crack surfaces, respectively; and together with the displacement integral equations (see equations 3.1 and 3.2 in [3]) for collocation on the rest of the boundary ¡h , form the dual boundary integral formulation in shallow shell problems. It is worth notice that as the source points x+ and x3 are coincident, extra free terms appear in equations (5-9) for collocation on both crack surfaces. Solution Strategy. The implementation of the dual boundary element formulation requires that boundary ¡ to be discretized. In the case of shallow shell regions several uniformly distributed domain points are required for the application of the dual reciprocity method (DRM). In the case of the boundary: continuous, semi discontinuous and discontinuous quadratic isoparametric boundary elements are used to describe the geometry of each region (plate or shallow shell). The detailed modelling strategy is similar to the one described by Portela Aliabadi and Rooke [5] and can be summarized as follows [3]: ² The crack boundaries are discretized with discontinuous quadratic elements (each node of one crack surface is coincident with another node on the opposite crack surface). ² Continuous quadratic elements are applied along the remaining boundary of the structure, except at the intersection between a crack and an edge or at corners, where semi-discontinuous elements are required in order to avoid a common node at intersections. ² The traction integral equations are used for collocation on one crack surface (x3 2 K3 ).


393

2.00E-04 ABAQUS

x1/h1

DBEM

u2 /h1

0.00E+00 1.00E+02

1.50E+02

2.00E+02

-2.00E-04

-4.00E-04

E=71.016 GPa h1=0.0025 m v=0.33 h2=0.005 m q=0.006 MPa h =0.0075 m 3 b=0.75 m

h1

-6.00E-04

x3 x1

h3 0.075 m

0.035 m

x2

h2 0.040 m

Figure 2: Crack Opening Displacement for Curved Stiened Panel. ² The displacement integral equations are used for collocation on the opposite crack surface (x+ 2 K+ ). ² In the non-crack boundaries (x0 2 Kh ) the common displacement integral equations are employed. ² For shallow shell, several uniformly distributed DRM points are used in the domain. This simple strategy is very robust, making the DBEM an eective tool for the modeling of general edge or embedded crack problems. Numerical Example. A curved stiened panel with a centre crack as shown in Figure 2 is analyzed. The material properties and dimensions are also presented in Figure 2. The curved panel is simply supported along all the boundary and is subjected to an uniform internal pressure q. In order to validate the DBEM formulation in shallow shell and plate assemblies, results are compared with FEM solutions. The FEM half model has a total of 5922 elements and 18133 nodes. The DBEM model contains 7 shallow shells and 6 flat plates, with a total of 248 elements and 214 DRM points (14 elements per crack side). Figure 2 presents the in-plane displacement u2 along the symmetry line of the central cracked shallow shell (crack opening displacement). From Figure 2 is evident the good agreement between both numerical solutions (DBEM and FEM). Conclusion. In this work, applications of the DBEM for the fracture mechanics analysis of shear deformable plate and shallow shells assemblies was presented. A multi-region technique was used to model plate and shallow shell assembled structures subjected to arbitrary loading. Additional equations were obtained by imposing compatibility and equilibrium equations along the interface boundaries. The DBEM shallow shell formulation was developed by coupling boundary element formulations of shear deformable plate bending and two dimensional plane stress elasticity; as a result, domain integrals appear in the formulation and are treated with the Dual Reciprocity Technique.

394


Traction integral equations were applied on one crack surface and the usual displacement integral equations on the other crack surface and non-crack boundaries. Special crack tip elements are used to model accurately the displacement field. These displacements were used for the evaluation of SIF using the CTOD technique. Comparing DBEM with FEM models, it was clear that good accuracy and e!ciency can be achieved with the present multi region DBEM approach.

References [1] Aliabadi, M.H., The Boundary Element Method, vol II: application to solids and structures, Chichester, Wiley (2001). [2] P.M. Baiz, M.H. Aliabadi, Local Buckling of Thin-Walled Structures (Plate and Shallow Shell Assemblies) by the Boundary Element Method. Submitted to International Journal of Solids and Structures. [3] Dirgantara, T., Aliabadi, M.H., Dual boundary element formulation for fracture mechanic analysis of shear deformable shells, International Journal of Solids and Structures, 28, 7769-7800 (2001). [4] Di Pisa, C., Boundary Element Analysis of Multi-layered Panels and Structures, PhD Thesis, Department of Engineering, Queen Mary University of London (2005). [5] Portela, A., Aliabadi, M.H. and Rooke, D.P., The dual boundary element method: eective implementation for crack problems, International Journal for Numerical Methods in Engineering, 33, 1269-1287 (1992). [6] Rashed, Y. F., Aliabadi, M. H. and Brebbia, C. A., Hyper-singular boundary element formulation for Reissner plates, International Journal of Solids and Structures, 35, 2229-2249 (1998). [7] Reissner, E., On a Variational Theorem in Elasticity, Journal of Mathematics and Physics, 29, 90-95 (1950). [8] Wearing, J.L., Ahmadi-Brooghani, S.Y., Fracture analysis of plate bending problems using boundary element method, in Plate Bending Analysis with Boundary Elements, Advanced in Boundary Element Series, M. H. Aliabadi (Ed.), Computational Mechanics Publications, Southampton (1998). [9] Wen, P.H., Aliabadi, M.H., Young, A., Crack growth analysis for multi-layered airframe structures by boundary element method, Engineering Fracture Mechanics, 71, 619-631 (2004). [10] Zienkiewicz, O.C., Taylor, R., The Finite Element Method, Vol 2: Solid Mechanics, B-H, Oxford, (2000).


395

An Incremental Technique to Evaluate the Stress Intensity Factors by the Element-Free Method P.H. Wen1 and M.H. Aliabadi2 1 2

Department of Engineering, Queen Mary, University of London, London, UK, E1 4NS Department of Aeronautics, Imperial College, London, UK, SW7 2BY

Abstract In this paper an incremental technique was developed to evaluate stress intensity factors accurately by the use of the element-free Galerkin method based on the variation of potential energy. The stiffness matrix is evaluated with a domain integral by the use of radial basis function interpolation without elements in the domain. The Laplace transformation technique and the Durbin inversion method are used to obtain the time domain physical values. The applications of the proposed incremental technique to two-dimensional fracture mechanics have been presented. Comparisons have been made with benchmark analytical solutions and boundary element method. Key words: Element-free Galerkin method, stress intensity factor, Laplace transformation method, Fracture mechanics, moving least square interpolation.

1. Introduction Although the FEM and BEM have been very successfully established and applied in engineering as numerical tools, the development of new advanced methods nowadays is still attractive in computational mechanics. Meshless approximations have received much interest since Nayroles et al [1] proposed the diffuse element method. Later, Belyschko et al [2] and Liu et al [3] proposed element-free Galerkin method and reproducing kernel particle methods, respectively. One key feature of these methods is that they do not require a structured grid and are hence meshless. Recently, Atluri and his colleagues presented a family of Meshless methods, based on the Local weak Petrov-Galerkin formulation (MLPGs) for arbitrary partial differential equations [4] with moving least-square (MLS) approximation. MLPG is reported to provide a rational basis for constructing meshless methods with a greater degree of flexibility. However, Galerkin-base meshless methods, except MLGP presented by Atluri[5] still include several awkward implementation features such as numerical integrations in the local domain. A comprehensive review of meshless methods (MLPG) can be found in the book [6] by Atluri. A variety of local interpolation schemes that interpolate the randomly scattered points is currently available. The moving least square and radial basis function interpolations are two popular approximation techniques recently. With comparisons of these two techniques, the moving least-square approximation is generally considered to be one of the best schemes with a reasonable accuracy, particularly for static elasticity demonstrated by Wen et al [7]. In this paper, the mesh free Garlerkin method is presented with the radial basis function interpolation and an incremental technique has been developed to calculate the stress intensity factors with high accuracy. In addition, the enriched radial basis function and high density of node distribution

396


near the crack front are not needed in this approach. The accuracy of proposed method has been demonstrated through benchmark examples.

2. Variation of potential energy and MLS Based on the variation of potential energy, the element free Galerkin method is developed on the basis of finite element method by the use of radial base function interpolation in this paper to evaluate static and dynamic stress intensity factor with an incremental technique. For a linear two dimensional elasticity, the equilibrium equations can be written as V ij , j f i Uui (1) where V ij denotes the stress tensor, f i the body force, U is the mass density, ui

w 2 u i / wt 2 the

acceleration. Considering the variation of the total potential energy, with respect to each nodal displacement, and the relations u ĭuˆ , İ Buˆ and ı Dİ yields a linear algebraic equation system in a matrix form as >K @2 N u2 N uˆ 2 N >M @2 N u2 N uˆ f 2 N (2) where N is the total number of node in the domain ȍ. The stiffness and mass matrices are: K ³ B T (x, y )D(y )B(x, y )d:(y ), M U ³ ĭ T (x, y )ĭ(x, y )d:(y ) (3) :

f

:

x i i 1,2,...N , and nodal force vector is defined by

in which x

³ĭ

T

:

where ĭ(y, x)

(x, y )b(y )d:(y ) ³ ĭ T (x, y )t (y )d*(y )

(4)

*V

Î (y, x ),I 1

1

2

ûˆ , uˆ ,..., uˆ `

(y, x 2 ),..., I n ( y ) (y , x n ( y ) )` and uˆ i

1 i

2 i

n(y ) T i

are vector of

^

`

x1( k ) , x 2( k ) , shape functions and nodal values of displacement. The collocation points x k k 1,2,..., n(y ), I k are the shape functions and n(y) the total number of nodes in the local domain named as supported domain as shown in Figure 1. For a two dimensional plane stress case, we can rearrange the above equation in a matrix form as T u(y ) û1 , u 2 ` ĭ(y, x)uˆ ; uˆ ûˆ11 , uˆ 12 , uˆ12 , uˆ 22 ..., uˆ1n ( y ) , uˆ 2n ( y ) ` (5) For convenience of analysis, the tilde (^) is removed in the following discussion. Applying the Laplace transform to the equation (2) yields >K @ s 2 >M @ u~ ~f (6) where s is the Laplace parameter. We assume that the displacements u(y) at the point y can be approximated in terms of the nodal values in a local domain (see Figure 1) as T

n(y )

u i (y )

¦I

k

(y , x k )uˆ ik

ĭ (y , x)uˆ i

(7)

k 1

where ĭ (y , x)

Î (y, x ),I 1

1

2

(y , x 2 ),..., I n ( y ) (y , x n ( y ) )`, uˆ i

^

`

ûˆ , uˆ 1 i

2 i

`

T

,..., uˆ in ( y ) , i 1,2 and

uˆ i (x) is the nodal values at point x k x1( k ) , x2( k ) , k 1,2,..., n(y ) . For the two dimensional plane stress case, we can rearrange the above relation as follows


397

node xi

ȍ

sub-domain ȍy

ȍ

*

field point y

Figure 1. Sub-domain ȍy for MLS/RBF interpolation of the field point y and support domains.

û1 , u2 `T

u(y )

0 º ªI1 0 I2 0 ... In ( y ) « 0 I 0 I ... 0 I », 1 2 n( y ) ¼ ¬

ªĭ 0 º » « ¬ 0 ĭ¼

ĭ( y , x) uˆ

ĭ(y, x)uˆ ,

ûˆ , uˆ , uˆ , uˆ ..., uˆ 1 1

1 2

2 1

n(y ) 1

2 2

(8)

, uˆ2n ( y ) ` . T

3. Radial bases function The distribution of function u in the sub-domain : y over a number of randomly distributed notes ^x i `, i 1,2,..., n(y ) can be interpolated, at the point y, by n(y )

¦ R (y, x )a (y , x) ^R (y , x), R

u (y )

i

i

R T (y , x)a(y )

i

i 1

where R T

1

2

(9)

(y , x),..., Rn ( y ) (y , x)` is the set of radial basis functions centred at

the point y, â ` are the unknown coefficients to be determined. The radial basis function has been selected to be the following multi-quadrics n( y ) k k 1

Rk ( y , x )

c2 y xk

2

(10)

with a free parameter c and in this paper, we select c=h (h is specified length in each example). From the interpolation strategy in Eq. (9) for RBF, a linear system for the unknowns coefficients a is obtained by (11) R 0a u It is apparent that the interpolation of field variable is satisfied exactly at each node. As the RBFs are positive definite, the matrix R 0 is assured to be invertible. Therefore, we can obtain the vector of unknowns from Eq. (11) a R 01 (x)u(x) (12) So that the approximation u(y) can be represented, at domain point y, as

398


n(y )

- 4 - u (y )

R T (y , x)R 01 (x)u(x)

ĭ (y , x)u

¦I u k

k

k 1

(13) where the nodal shape function are defined by ĭ (y, x) R T (y , x)R 01 (x) (14) It is worth noticing that the shape function depends uniquely on the distribution of scattered nodes within the support domain and has the Kronecker Delta property. As the inverse matrix of coefficient R 01 (x) is a function only of distributed node xi in the support domain, it is much simpler to evaluate the partial derivatives of shape function. From Eq. (13), we have n(y )

u .k ( y )

ĭ ,k (y , x)u

¦I

i ,k

ui

(15)

i 1

4. Incremental technique To derive the integral for stress intensity factor for dynamic problem, one static reference problem has to be determined first. Let *t and *u be the traction and displacement boundaries respectively. If there is an increment of crack surface 'a at crack tip, an increment of displacement 'u k in the domain and on the traction boundary and 't k on the displacement boundary will occur. The stress intensity factor in the static case can be written as 1/ 2

§ P · § 't · § 'u k · ¨ (16) ¸t k d* ³ ¨ k ¸u k d* ¸ ¨ ³ ¨ (1 Q ) * © 'a ¹ ¸ 'a ¹ *u © t © ¹ The numerical results of static case can be used directly to the dynamic problem. The relationship between the stress intensity factors for the reference problem and the real dynamic problem can be written as ª 'u k ~ º 't 'u P ~ KI tk d* ³ k u~k d* Us 2 ³ k u~k d: » (17) « a a ' ' (1 Q ) K I0 ¬«*³t 'a *u : ¼» In order to evaluate the stress intensity factor in the time domain, the Durbin’s inverse method is employed in this paper K ~§ 2e K t ª 1 ~ 2kS · § 2kS t ·½º (18) f (t ) i ¸ exp¨ ¸ ¾» « f (K ) ¦ Re® f ¨K T ¬ 2 T ¹ © T ¹ ¿¼ k 0 ¯ © ~ where f ( s k ) is the transformed variables in the Laplace transform domain when the parameter K I0

s k K 2kS i / T . The selection of parameters K and T affects the accuracy slightly. In the computations, we have chosen K 5 / t0 and T / t0 20 in the following examples, where E (1 Q ) / U (1 Q )(1 2Q ) . Two numerical t 0 h / c1 , h is the height of cracked sheet and c1 examples are given to demonstrate the accuracy and efficiency of the proposed technique.


V0

399

1.50

1.40

1.325 [8]

KI/ı0¥ʌa

1.30

h

y2 a

crack tip

1.20

Increment technique 1.10

handbook

n

y1 1.00

b

0

1

2

3

4

5

(a) (b) Figure 2. Square plate with a central crack (h=b) under tension V 0 : (a) a quarter of the plate; (b) normalized stress intensity factor, where ǻa/a=10-n. 3.0

3.5

This method

2.5

3.0

This paper BEM

2.5

BEM

KI/ı0¥ʌa

KI/ı0¥ʌa

2.0

1.5

1.0

2.0

1.5

1.0

0.5 0.5

c1t/h

c1t/h

0.0 0

2

4

-0.5

6

8

10

12

14

16

0.0 0

2

4

6

8

10

12

14

16

-0.5

(a) (b) Figure 3. Square plate with a central crack subjected to dynamic tension V 0 H (t ) : (a) h=b; (b) h=2b.

5. Examples 5.1 A central crack in rectangular sheet under uniform static load V 0 A square plate of width 2b and height 2h containing a centred crack of 2a subjected to a uniform shear load V 0 on the top and the bottom is analysed. Due to the symmetry, a quarter of plate is considered as shown in Figure 2 (a). Here Poisson’s ratio Ȟ=0.3. A set of 11×11 ( N total 121 ) uniformly distributed nodes is used and the integration is performed by dividing the square into 10×10 cells with 4×4 Gauss points. The support domain is selected as a circle of radius d y centered at field point y, which is determined such that the minimum number of nodes in the sub domain n(y ) t N 0 , here the number N 0 is selected to be 10 for all following examples. Figure 2(b) shows the convergence of the normalized stress intensity factor

400


K I0 / V 0 Sa against the length of the increment of crack surface 'a . Excellent agreement with

Reference [8] can be achieved when 'a / a 10 3 . 5.2 A Single central crack in rectangular plate under tension Consider a rectangular plate of width 2b and length 2h with a centrally located crack of length 2a. It is loaded dynamically in the direction perpendicular to the crack by a uniform tension V 0 H (t ) on the top and the bottom. Due to the symmetry, a quarter of plate is considered as shown in Figure 2(a). Poisson’s ratio Ȟ=0.3 and Young’s modulus is unit. Two geometries of rectangular plate are considered in this example, i.e. h=b and h=2b. To demonstrate the accuracy of mesh free method, the results given by Wen [9] using the indirect boundary element method (fictitious load method) are plotted for comparison. Normalize dynamic stress intensity factors K I / V 0 Sa by these two techniques are shown in the Figures 3(a) and 3(b). Apparently before the arrival time of dilatation wave traveling from the top of plate, the stress intensity factor should remain to be zero. The agreement between the solutions is considered to be good.

6. References [1] B. Nayroles, G. Touzot & P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics, 10, 307-318, 1992. [2] T. Belytschko, Y.Y. Lu & L. Gu, Element-free Galerkin method, Int. J. Numerical Methods in Engineering, 37, 229-256, 1994. [3] W.K. Liu, S. Jun & Y. Zhang, Reproducing kernel particle methods, Int. J. Numerical Methods in Engineering, 20, 1081-1106, 1995. [4] S.N. Atluri & T. Zhu, A new meshless local Peyrov-Galerkin (MLPG) approach to nonlinear problems in computational modelling and simulation, Comput Model Simul Engng, 3, 187-196, 1998. [5] S.N. Atluri & T. Zhu, The meshlesss local Peyrov-Galerkin (MLPG) approach for sovling problems in elasto-statics, Comput Mech, 25, 169-179, 1999. [6] S.N. Atluri, The Meshless Method (MLPG) for Domain and BIE Discretizations, Forsyth, GA, USA, Tech Science Press, 2004. [7] P.H. Wen and M.H. Aliabadi, An Improved Meshless Collocation Method for Elastostatic and Elastodynamic Problems, Communications in Numerical Methods in Engineering, 2007 (to appear). [8] D.P. Rooke and D.J. Cartwright, A Compendium of Stress Intensity Factors, HMSO, London, 1976. [9] P.H. Wen, Dynamic Fracture Mechanics: Displacement Discontinuity Method, Computational Mechanics Publications, Southampton, 1996.


401

Stress analysis of composite laminated plates by the boundary element method F. L. Torsani, A. R. Gouveia, E. L. Albuquerque, and P. Sollero Faculty of Mechanical Engineering, State University of Campinas 13083-970, Campinas, Brazil, [torsani,adriana,ederlima,sollero]@fem.unicamp.br

Keywords: Plates, boundary element method, laminated composites, and stress analysis. Abstract. This paper presents a boundary element analysis of stresses in laminate composite plates following Kirchhoff hypothesis. Stress integral equations are derived from the transversal displacement integral equation. All derivatives of anisotropic thin plate fundamental solutions are computed analytically. Stresses are computed in each lamina at internal points of the plate. A numerical example is presented in order to assess the proposed method. Results are compared with solutions found in literature, showing good agreement. Introduction The material anisotropy presents two different hands. On one hand, it turns the material analysis extremely hard due the large number of variables necessary to represent its mechanical properties. On the other hand, the use of anisotropic materials allows the designer to control their mechanical properties along each direction, increasing the material strength without increasing the weight. With the demand by optimization of natural resources and the large offer of computational resources, the designer target are changing from simple analysis to optimized performance. In this way, the use of high performance composite materials is an interesting option because they allow the control of their mechanical properties either by the choice of their components, matrix and reinforcement, or by the component order inside the material. Together with the demand by high performance composite materials, it has also increased the demand by reliable and accurate numerical procedures for this materials analysis. The complexity of the anisotropic material analysis is evident in literature. It can be noted that the number of references in which the boundary element method is applied for anisotropic materials is significantly smaller than those treating isotropic materials. However, in the last ten years, important advances on boundary element techniques applied to anisotropic materials were published in the literature. For example, plane elasticity problems were analyzed by [1, 2], [3], and [4, 5, 6, 7], out of plane elasticity problems by [8], tri-dimensional problems by [9, 10, 11], and Kirchhoff plates by [12]. Boundary element formulations have been applied to plate bending anisotropic problems considering Kirchhoff as well as shear deformable plate theories. [13] presented a boundary element analysis of plate bending problems using fundamental solutions proposed by [14] based on Kirchhoff plate bending assumptions. [15] proposed a formulation in which the singularities were avoided by placing source points outside the domain. [16] presented an analytical treatment for singular and hypersingular integrals of the formulation proposed by [13]. Shear deformable plates have been analyzed using the boundary element method by [17, 18] with the fundamental solution proposed by [19]. In this work the calculation of internal point stresses of anisotropic plates using the boundary elements method is presented. Stress integral equations are derived from the transversal displacement integral equation. Stresses are computed in every lamina at internal points of the plate. Results are compared with solutions found in literature, showing good agreement.

402


Boundary integral equation As shown by [12], the boundary integral equation for transversal displacements w in a boundary point of an anisotropic plate can be written as:

1 w(Q) + 2

Γ

Ω

Γ

Vn∗ (Q, P )w(P ) − m∗n (Q, P )

Vn (P )w∗ (Q, P ) − mn (P )

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

Nc ∂w∗ (Q, P ) dΓ(P ) + Rci (P )wc∗i (Q, P ) + ∂n i=1

g(P )w∗ (Q, P )dΩ.

(1)

∂ where ∂n is the derivative in the direction of the outward vector n that is normal to the boundary Γ; mn and Vn are, respectively, the normal bending moment and the Kirchhoff equivalent shear force on the boundary Γ; Rc is the thin-plate reaction of corners; wc is the transverse displacement of corners; P is the field point; Q is the source point; and an asterisk denotes a fundamental solution. Fundamental solutions required at equation (1) are given by [12].

Strain and displacement in laminate composite plates Laminates are fabricated such that they act as an integral structural element. To assure this condition, the bond between two laminae in a laminate should be infinitesimally thin and not shear deformable to avoid the laminae slip over each other, and to allow displacement continuity along the bond [20]. Thus, we could consider that strains are continuous along its thickness. However, as each laminae is compounded by different materials, stresses present discontinuities along laminate interfaces. In Kirchhoff plates, strains are given by: ∂2w , ∂x2 2 ∂ w = −z 2 , ∂x ∂2w . = −2z ∂x∂y

εx = −z εy γxy

(2)

So, in order to obtain strains, second order derivatives of integral equations (1) need to be computed. For internal points, these derivatives are given by: ∂ 2 w(Q) = ∂x2

2 ∗ ∂ Vn

∂x2

Γ

Γ

Vn (P )

(Q, P )w(P ) −

Nc 2 ∗ ∂ Rc i ∂ 2 m∗n ∂w(P ) (Q, P ) (Q, P )wci (P ) − dΓ(P ) + 2 ∂x ∂n ∂x2 i=1

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

Ω

g(P )

∂ 2 w∗ (Q, P )dΩ, ∂x2

(3)


∂ 2 w(Q) = ∂y 2

2 ∗ ∂ Vn

∂y 2

Γ

Γ

403

(Q, P )w(P ) −

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

Vn (P )

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

Ω

g(P )

∂ 2 w∗ (Q, P )dΩ, ∂y 2

(4)

and ∂ 2 w(Q) = ∂x∂y

2 ∗ ∂ Vn Γ

Γ

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

Vn (P )

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

Ω

g(P )

∂ 2 w∗ (Q, P )dΩ. ∂x∂y

(5)

As presented by [20], stresses at each laminae can be evaluated from strain given by equation (2) as following: ⎧ ⎫ ⎪ ⎪ ⎨ σx ⎬

⎡

⎤⎧

⎫

⎪ Q11 Q12 Q16 ⎪ ⎨ x ⎬ ⎥ Q22 Q26 ⎦ y , ⎪ ⎪ Q26 Q66 ⎩ γxy ⎭

⎢ σy = ⎣ Q12 ⎪ ⎩ τ ⎪ ⎭ Q16 xy

(6)

where matrix Q is given by:

Q = [T ]−1 [Q] [T ] .

(7)

The transformation matrix [T ] is given by: ⎡ ⎢ [T ] = ⎣

⎤

cos2 θ sin2 θ 2 sin θ cos θ ⎥ sin2 θ cos2 θ −2 sin θ cos θ ⎦ , 2 2 − sin θ cos θ sin θ cos θ cos θ − sin θ

(8)

where θ is the angle between the fibre orientation and the direction of axis x. The stiffness matrix [Q] is given, in terms of engineering constants, by: ⎡

EL 1−ν ν

LT T L ⎢ νLT ET [Q] = ⎣ 1−ν LT νT L

0

νLT ET 1−νLT νT L ET 1−νLT νT L

0

⎤

0 ⎥ 0 ⎦, GLT

(9)

where EL is the elastic modulus in the parallel to the fibre direction, ET is elastic modulus in the transversal to the fibre direction, GLT is the shear modulus in the plane of the laminae, and νLT is the principal Poisson ratio in the plane of the laminae. Numerical results To validate the procedures implemented, a nine-layered, symmetrical angle-ply laminate with stacking sequence [+θ/ − θ/ + θ/ − θ/ + θ/ − θ/ + θ/ − θ/ + θ] with 0 ≤ θ ≤ 45o was chosen. The plate is square with edge length a = 1 m. All edges are simply-supported and all layers have the same thickness. The total thickness is equal to h = 0.01 m and material properties are: EL = 207 GPa, ET = 5.2

404


GPa, GLT = 3.1 GPa, and νLT = 0.25. Figures 1 and 2 show the effect of the variation of θ on the displacement and bending stress resultants, respectively, at the centre of the plate. They are compared with finite element results obtained by [21]. It is worth to say that the finite element formulation considers the effect of the shear deformation. As it can be seen, in both cases the agreement between the boundary element thin plate and the finite element shear deformable plate is very good. 4.5

1000 × (wET h3 /qa4 )

w, this work w, Reference [21] 4

3.5

3

2.5

2 0

5

10

15

20

25

30

35

40

45

θ (degrees)

Figure 1: Effect of the orientation θ in the transversal displacement response at the centre of the plate.

100 × (Mx /qa2 ), 100 × (My /qa2 )

14 Mx, this work My, this work Mx, Reference [21] My, Reference [21]

12 10 8 6 4 2 0 0

5

10

15

20

25

30

35

40

45

θ (degrees)

Figure 2: Effect of the orientation θ in the moment response at the centre of the plate. Figure 3 shows the stress distribution (σx ) along the thickness of the plate. It can be seen that stress are discontinuous at the interface and vary linearly along each laminae.


405

2.5

(10 × σx × h2 )/q

2

1.5

1

0.5

0 0

0.1

0.2

z/h

0.3

0.4

0.5

Figure 3: Stress distribution (σx ) along the thickness for θ = 45o at the centre point of the plate.

Conclusions This paper presented a boundary integral formulation for the computation of stress in internal points of anisotropic thin plates. An integral equation for the second displacement derivative is developed and all derivatives of the fundamental solution are computed analytically. The obtained results are in good agreement when compared with finite element thick plate results. Acknowledgment The authors would like to thank the State of S˜ ao Paulo Research Foundation (FAPESP) for financial support for this work (grant number: 03/09498-0).

References [1] P. Sollero and M. H. Aliabadi, Fracture mechanics analysis of anisotropic plates by the boundary element method. International Journal of Fracture, 64: 269-284, 1993. [2] P. Sollero and M. H. Aliabadi, Anisotropic analysis of composite laminates using the dual boundary elements methods. Composite Structures, 31:229-234, 1995. [3] A. Deb, Boundary elements analysis of anisotropic bodies under thermo mechanical body force loadings. Computers and Structures, 58:715-726, 1996. [4] E. L. Albuquerque, P. Sollero and M. H. Aliabadi, The boundary element method applied to time dependent problems in anisotropic materials. International Journal of Solids and Structures, 39:1405-1422, 2002. [5] E. L. Albuquerque, P. Sollero and P. Fedelinski, Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems. Computers and Structures, 81:1703-1713, 2003. [6] E. L. Albuquerque, P. Sollero and P. Fedelinski, Free vibration analysis of anisotropic material structures using the boundary element method. Engineering Analysis with Boundary Elements, 27:977-985, 2003.

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[7] E. L. Albuquerque, P. Sollero and M. H. Aliabadi, Dual boundary element method for anisotropic dynamic fracture mechanics. International Journal for Numerical Method in Engineering, 59:11871205, 2004. [8] Ch. Zhang, Transient elastodynamics antiplane crack analysis of anisotropic solids. International Journal of Solids and Structures, 37:6107-6130, 2000. [9] M. Kogl and L. Gaul, A boundary element method for transient piezoelectric analysis. Engineering Analysis with Boundary Elements, 24:591-598, 2000. [10] M. Kogl and L. Gaul, A 3-d boundary element method for dynamic analysis of anisotropic elastic solids. CMES-Computer Modeling in Engineering and Science, 1:27-43, 2000. [11] M. Kogl and L. Gaul, Free vibration analysis of anisotropic solids with the boundary element method. Engineering Analysis with Boundary Elements, 27:107-114, 2003. [12] E. L. Albuquerque, P. Sollero, W. S. Venturini and M. H. Aliabadi, Boundary element analysis of anisotropic kirchhoff plates. International Journal of Solids and Structures, 43:4029-4046, 2006. [13] G. Shi and G. Bezine, A general boundary integral formulation for the anisotropic plate bending problems. Journal of Composite Materials, 22:694-716, 1988. [14] B. C. Wu and N. J. Altiero, A new numerical method for the analysis of anisotropic thin plate bending problems. Computer Methods in Applied Mechanics and Engineering, 25:343-353, 1981. [15] C. Rajamohan and J. Raamachandran, Bending of anisotropic plates by charge simulation method. Advances in Engineering Software. 30:369-373, 1999. [16] W. P. Paiva, P. Sollero and E. L. Albuquerque, Treatment of hypersingularities in boundary element anisotropic plate bending problems. Latin American Journal of Solids and Structures, 1:49-73, 2003. [17] J. Wang and K. Schweizerhof, Study on free vibration of moderately thick orthotropic laminated shallow shells by boundary-domain elements. Applied Mathematical Modelling, 20:579-584, 1996. [18] J. Wang and K. Schweizerhof, Free vibration of laminated anisotropic shallow shells including transverse shear deformation by the boundary-domain element method. Computers and Structures, 62:151-156, 1997. [19] J. Wang and K. Schweizerhof, The fundamental solution of moderately thick laminated anisotropic shallow shells. International Journal of Engineering and Science, 33:995-1004, 1995. [20] B. D. Agarwal and L. J. Broutman, Analysis and performance of fiber composites. 2nd Edition, John Wiley & Sons Inc, New York, 1990. [21] H. V. Lakshminarayana and S. S. Murthy, 1984. A shear-flexible triangular finite element model for laminated composite plates. International Journal for Numerical Methods in Engineering, 20: 591–623, 1984.


407

Boundary Element Formulation for Dynamic Analysis of Cracked Sheets Repaired with Anisotropic Patches

M. Mauler , P. Sollero , E. L. Albuquerque 1

2

3

Faculty of Mechanical Engineering, State University of Campinas 13083-970, Campinas, Brazil, [martimmn 1 ,sollero2 ,ederlima3 ]@fem.unicamp.br Keywords:

Abstract.

Boundary Element Method, Anisotropy, Bonded Repair, Dynamic Fracture Mechanics.

The aim of this paper is to present the boundary formulation and a solution procedure to

perform the dynamic analysis of cracked sheets repaired with an anisotropic patch. The numerical method that is used to perform the modeling of the crack is the dual boundary elements method (DBEM). The interaction eect between the sheet and the anisotropic repair is modeled using the dual reciprocity boundary elements method (DRBEM). The inertial eects are also modeled using the DRBEM. A transient solution procedure is presented.

Introduction The presence of cracks in mechanical or structural components under dynamic loads decreases its mechanical and fatigue resistance due to the high stress concentration at the crack tip.

Fracture

mechanics problems are a major concern in the aeronautic industry, since there is the need of projects with high reliability, high resistance, and low costs. composed by metallic sheets and stiners.

Aeronautical structures are usually

Cracked sheets in aeronautical structures are usually

repaired by bonding, bolting, or screwing a metallic repair over the crack region. The use of bolted or screwed repairs creates holes in the structure, which are major stress concentrators and where cracks are likely to initiate at. The bonded repairs have being used successfully and are regarded by the industry as an ecient solution for this kind of problem. In this paper, the boundary element method (BEM) is applied for the analysis of the dynamic response of cracked sheets repaired with an anisotropic patch. The elastostatic response of this system has been previously presented by Useche, Sollero and Albuquerque [1].

Due to strong geometrical singularities, it is not suitable to obtain

good results for displacements and tractions in the outskirts of the crack using the conventional BEM formulation.

Therefore, another technique must be used in order to obtain good results

for tractions and displacements at this region.

The dual boundary element method (DBEM) is a

boundary modelling technique aiming fracture mechanics problems, allowing to discretize the crack in a single region. This technique has been successfully described and implemented by Dirgantara [2], and regarded as an ecient technique to simulate fracture mechanics problems. The consideration of interaction and inertial eects of the components introduces domain integrals in the boundary equilibrium equations, which must be transformed into boundary integrals.

The dual reciprocity

method (DRBEM) has been successfully used to overcome this problem for isotropic and anisotropic sheets, as shown by Kögl and Gaul [3] and Albuquerque, Sollero and Aliabadi [4]. In order to have a broader range of results, a transient solution procedure is adopted, allowing to obtain results for dierent load cicles. Loeer and Mansur [6].

This procedure has been described and implemented by Houbolt [5] and However, a transient solution is usually more time consuming than a

solution on the frequency domain.

408


Boundary Element Formulation

Figure 1: Cracked plate adhesively bonded with anisotropic patch Considering that the sheet shown in Fig.1 is under a dynamic load, the integral equation for the sheet (S) in a source point (x ) is given by: cSij x uSj x + 1 hS

ˆ

ˆ

TijS x , x uSj x dΓ =

ΓS

UijS

x , x bSj x dΩ +

ΩR

ˆ

ˆ

UijS x , x tSj x dΓ+

ΓS

S UijS x , x ρ¨ uj x dΩ .

(1)

ΩS

Similarly, the integral equation for the repair (R) is obtained substituting the (S) indexes in eq.(1) by (R). The coecient cij (x ) depends on the position of the source point (x ) in relation to the boundary which is being integrated, tj (x ) and uj (x ) are the tractions and displacements of the system, Tij (x , x) and Uij (x , x) are the fundamental solutions for tractions and displacements, h is the thickness of the component and ρ is the mass density of the component material. The rst three terms from eq.(1) refer to the classical elastostatics formulation, the fourth term refers to the assembling of the plate and the patch, and the last term refers to the eect of body forces due to the masses of the sheet and the patch under dynamic load. The shear reactions in the adhesive bj (x ) will be calculated by the dierence between the displacements of the sheet and the patch[1]: bj (x) =

SA S u x − uR , j x hA j

(2)

where SA is the shear module of the adhesive material and hA is the thickness of the adhesive layer. It is important to remember that the indexes (S) and (R) in eq.(1), eq.(2), and further equations do not imply on any summation, and are used only to address which component is being considered.

Dual Boundary Element Method (DBEM) The DBEM technique consists on applying a displacement equation in one of the sides of the crack, and a traction equation on the remaining side of the crack. Since the only cracked component in this analysis is the isotropic sheet, this procedure is not applied to the anisotropic patch. The displacement equation is given by eq.(1). The traction equation, obtained by the dierentiation of eq.(1) [2], is given by:


409

ˆ ˆ S S 1 S S S Sijk Dijk x , x tk x dΓ+ x , x uk x dΓ = n(i) x tj x + n(i) x 2 ΓS ΓS ˆ ˆ S S 1 S S , x b Dijk uk x dΩ , x x dΩ + n x x , x ρ¨ n D x (i) (i) ijk k hS ΩR

(3)

ΩS

of derivatives of Tij (x , x) and Uij (x , x). Sijk (x , x) and D (x , x) are linear combinations −2 , and D (x , x) exhibits strong singularWhen x → x, Sijk (x , x) exhibits hypersigularity O r −1 , where r (x , x) is the distance between the source node and the integration point and ity O r ni (x ) is a unitary vector, normal to the source point boundary. The eq.(3) is known as hypersinwhere

gular equation for plane elasticity, and, alongside with eq.(1), constitutes the basis of the DBEM technique.

Dual Reciprocity Boundary Element Method (DRBEM) The eq.(1) contains both domain and boundary integrals. The DRBEM allows approximating a domain integral by a sum of boundary integral functions. Applying the DRBEM in this elastodynamic problem consists on approximate the inertial and interaction eects of eq.(1) by: D d αkd fjk bj x = x, x

and

D

is the number of nodes shared by the patch and the sheet and

nodes of the component which is being considerated.The coecients coecients and

d (x , x) fjk

are given by:

(4)

e=1

d=1

where

E e ρ¨ uj x = βke qjk x, x ,

e (x , and qjk

x)

αkd

E

is the total number of

and

βke

are interpolation

are interpolation functions, which, for the isotropic sheet,

d xd , x = (1 − r) δjk fjk

and

e qjk (xe , x) = (1 − r) δjk .

(5)

For the anisotropic patch[3, 4], the interpolation functions are given by:

d fjk xd , x = Cjilm [c r (r,m r,i δlk + δim δlk )]

e and qjk (xe , x) = Cjilm [c r (r,m r,i δlk + δim δlk )] ,

(6) where

c

Cjilm

is the elastic constant tensor from the equilibrium equation of the patch. The constant

is chosen randomly when particular solutions for displacements

adopted[3]:

where

nm

u ˆkj = c r3 δkj

and

u ˆkj (x )

tˆkj = σkjm nm ,

is a unitary vector, normal to the source point boundary and

σkjm

u ˆkj

d fjk

(x )

are (7)

σkjm

3r2 = Ckmrs c (r,s δjr + r,r δjs ) . 2

In order to obtain useful solutions for solutions for

and tractions tˆkj

is given by: (8)

e , it is important to remember that the particular and qjk

must respect the equilibrium equations given by[3]: d = Cjilm u ˆdlk,im fjk

and

e qjk = Cjilm u êlk,im .

(9)

410


Coupling the eq.(4) with the eq.(1), the equilibrium equation for the sheet is given by: ˆ

cSij x uSj x + D 1 d ⎢ S d d S d αk ⎣cij x u ˆkj x + S h d=1

UijS

⎢ S e eS e ˆkj (x ) + ⎣cij (x ) u

e=1

ˆ

UijS x , x tSj x dΓ+

⎤ ˆ ⎥ d dS d dS d S ˆkj x dΓ⎦ + x , x tkj x dΓ − Tij x , x u d

ΓR

⎡ βke

ˆ

ˆ

ΓS

ΓS

⎡

E

TijS x , x uSj x dΓ =

ΓR

UijS

e

(x ,

x) tekjS

ˆ

e

(x ) dΓ −

ΓS

(10)

⎤

TijS

e

(x ,

x) u êkjS

⎥ (x ) dΓ⎦ e

.

ΓS

Similarly, the equilibrium equation for the patch is obtained substituting the (S) indexes in eq.(10) by (R).

Discretized Boundary Element Formulation

In order to obtain the elastodynamic response of the system, the boundary is divided into boundary elements. The elements taken in consideration are quadratic continuous elements to model the patch and quadratic discontinuous elements to model the sheet. For matters of convenience, four vectors with dimension (2 × E) are created: u = ϕu(i) ; uˆ = ϕû(i) ; t = ϕt(i) ; ˆt = ϕtˆ(i) , (11) where ϕ is the vector of quadratic shape functions, u (2 × E) and t (2 × E) are the vectors of nodal displacements and tractions of the system, and uˆ (2 × E) and ˆt (2 × E) are the vectors of particularly solutions for nodal displacements and tractions of the system. Coupling the eq.(11) with eq.(10), and calling ˆ

Γj

U ϕdΓ =

G

ˆ

and Γj

T ϕdΓ =

H

(12)

,

it is possible to rewrite the equilibrium equation for the sheet as:

HSljuSj = GSljtSj + h1S

D d=1

HSljudj S − GSljtdj S

αSd +

E

HSljuej S − GSljtej S βSe

e=1

.

(13)

In a similar way, the equilibrium equation for the patch is obtained substituting the (S) indexes in eq.(13) by (R). H and G are similar to H and G, but obtained by integration on the repair boundary. The D vectors αd (2 × 1) and the E vectors βe (2 × 1) can be assembled in two vectors α (2D × 1) and β (2E × 1). Therefore, the eq.(4) can be rewritten as:

b = Fα

and

p = Qβ

,

(14)

where b contains the shear reactions of the adhesive for the nodes shared by the patch and the sheet and p contains the body forces of the component under consideration for each one of its nodes. The matrix F contains the values of the function fjd (x , x) for the nodes shared by the patch and the sheet. Similarly, the matrix Q contains the values of the function qje (x , x) for the nodes of the component which is being considerated. The matrix form for the equilibrium equations of the components are obtained coupling the eq.(14) and with eq.(13), and can be written as:

HSΓuSΓ − GSΓtSΓ = ASΓαSΓ−BSΓpSΓ

HRΓuRΓ − GRΓtRΓ = ARΓαRΓ−BRΓpRΓ


uRΩ − HRΩ uRΩ = ARΩ αRΩ − BRΩ pRΩ

uSΩ − HSΩ uSΩ = ASΩ αSΩ − BSΩ pSΩ where

A and B are given by:

ˆ D − GT ˆ D F−1 A = HU ˆ T

and

ˆ U

411

(15)

ˆ E − GT ˆ E Q−1 B = HU

and

,

are matrixes of traction and displacement fundamental solutions.

(16)

.

Making use of the

DRBEM and the relation given by eq.(2) it is possible to rewrite the term for the interaction sheetrepair as:

uSΩ − uR =

hA S S Fα SA

uR − uSΩ =

and

hA R R F α . SA

(17)

Finally, coupling the equations for the sheet and the repair using the DRBEM, the equation system, which rules the problem, is given by:

(H − A)S (H − A)R

AS AR

uS uR

=

GS tS + BS pS BR pR

(18)

Transient Solution Procedure To solve the linear system given by eq.(18), a transient solution procedure is used. This procedure was proposed by Houbold[5], which is the most indicated to solve problems of time integration alongside with the DRBEM [6]. Considering that the inertial eects of the components are due an acceleration eld given by:

p = ρü = Qβ the eq.(18) can be rewritten for an instant of time

(H − A)S (H − A)R

AS AR

uSτ +τ uRτ +τ

τ + τ

τ

as:

GS tSτ +τ + BS ρS u¨ Sτ +τ BR ρR u¨ Rτ +τ

=

In order to proceed with the time integration, the

(19)

,

period is divided in

N

.

time steps , where:

τ = N τ . Assuming that the solution for the eq.(20) is known at

(20)

(21)

τ = 0, τ, 2τ..., the acceleration at τ + τ

is approximated by the expression[5]:

u¨ τ +τ

=

1 2uτ +τ − 5uτ + 4uτ −τ − uτ −2τ . τ 2

(22)

Inserting eq.(22) into eq.(20), the following system of equation is obtained:

⎡ ⎣

(H − A)S − ρS τ2 2 BS (H − A)R

AS AR − ρR τ2 2 BR

⎤ ⎦

uSτ +τ uRτ +τ

⎧ ⎫ S S 1 S S S ⎨ G S tS ⎬ τ +τ + B ρ τ 2 −5uτ + 4uτ −τ − uτ −2τ . R R 1 R R R ⎩ ⎭ B ρ τ 2 −5uτ + 4uτ −τ − uτ −2τ

=

(23)

412


Conclusions In this paper, a formulation for dynamic analysis of cracked sheet repaired with adhesively bonded patches was presented, and a procedure to solve the problem was proposed as well. The procedure described in this paper is more time consuming than a procedure on a frequency domain. However, allows a broader range of solutions with application of dierent load cicles. This procedure also allows a further implementation of dynamic stress intensity factor determination, which is an important variable in the design and project of aeronautical structures.

Acknowledgments The authors would like to thanks FAPESP (The State of São Paulo Research Foudation) and CNPq (National Research Council) for the nancial support of this work.

References [1] J. Useche, P. Sollero and E. L. Albuquerque, Boundary element analysis of cracked sheets repaired with adhesively bonded anisotropic patches , BeTeq: International Conf. On Boundary Element Techniques, Paris, (2006). [2] T. Dirgantara, Boundary element analysis of cracks in shear deformable plates and shells , Topics in Engineering, V.43, Southampton, WTI Press. 2002. [3] M. Kögl and L. Gaul, A boundary element method for transient piezoelectric analysis. Engn. Anal. With Boundary Elements, 24:591-598 (2000) [4] E. L. Albuquerque, P. Sollero and M. H. Aliabadi The boundary element method applied to time dependent problems in anisotropic materials , Int. Journal of Solids and Structures, 39; 1405-1422, 2002. [5] J. C. Houbolt, A recurrence matrix solution for the dynamic response of elastic aircraft , Journal of Aeronautical and Science, 17:540-550 (1950) [6] C. Loeer and W. J. Mansur, Analysis of time integration schemes for boundary element applications to transient wave propagation problems. In Brebbia and W. S. Venturini, editors, Boundary Element Techniques: Applications in stress analysis and heat transfer, 105-122, Computational Mechanics Publications, Southampton. (1987)


413

Homogenization of nonlinear composites using Hashin-Shtrikman principles and BEM P.Prochazka1,a and Z. Sharif Khodaei2,b 1 Society of Science, Research and Advisory, Czech Association of Civil Engineers Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Prague, Czech Republic a [email protected] b [email protected]

2

Keywords: Homogenization of composites, nonlinear problem, Hashin-Strikman principle, boundary elements, eigenparameters, Lippmann-Swinger equation

Abstract. Boundary element methods suffer from one disadvantage, which is description of inhomogeneous and plastic structures. It appears that one useful trick can bridge this issue. This is generated by generalized Hashin-Strikman variational principles, in which the eigenparameters are involved. Such an approach leads to separation of phases in the composites. In very many cases one phase behaves linearly (it is mostly the fiber) and the other (matrix in prevailing cases) nonlinearly. Using an equivalent integral formulation of the H-S principles yields exclusion of fiber influence in the problem. This is an impact of identification of material properties of the fiber with that of comparative medium. New unknown strains or stresses in the matrix occur instead of linear mechanical properties defined in the fibers. When considering nonlinear material behavior of the matrix, these strains or stresses play a role of new quantities appearing in iteration steps needed for identification of plastic behavior. 1. Introduction The approach for calculating plastic deformations (or alternatively relaxation stresses) selected in this paper starts with the idea of Hashin-Strikman variational principles, [1], which lead to variational bounds of linear composites. Extending the principles by introducing eigenparameters into the formulation, [2], new free parameters occur in the postulation of the problem and they will serve plastic deformation or relaxation stresses. They were successfully used in optimization of prestress in laminated composites in [3], for example. Survey of access to the homogenization techniques of periodic composites using numerical procedures can be found in [4]. In this paper basic relationships are derived and the approach is briefly described. An example is presented in the end of this paper. The plastic behavior is described in [5] for the Mises hypothesis.

2. Basic considerations Before we tackle the formulation of the problem of finding overall material properties of a composite nonlinear structure using boundary element method a useful approach derived in a similar way as Hashin-Shtrikman variational principles will be mentioned. For this reason the H-S principles will be briefly mentioned in the sequel. Only primary principle is considered for application to lower bounds (lower estimate of the overall properties – deformation method) while the dual principle can be applied to force method of finding upper estimate of the effective material properties.

414


Let a domain ȍ R3 with its boundary ī

īu ī p , īu ī p

0 , be given, describing a shape

of the body under consideration. Moreover, let us assume that along the boundary īu of the body

1,2,3 , be prescribed and along the boundary ī p tractions be given. Influence

displacements ui , i

of volume weight forces is neglected. The approach is split into two steps: 1st step: displacements u 0 { {ui0 } , tractions p 0 { { pi0 } , strains İ 0 { {İij0 } and stresses ı 0 { {ı ij0 } ( i, j 1,2,3 ) are calculated for homogenized, isotropic comparative medium and they are considered to be known in the next. Statical equations for the stress field and geometric boundary conditions are valid:

wı ij0 wx j

0 ,

ui0

u i on īu , pi0

p i on ī p

(1)

and also homogeneous and isotropic Hooke’s law holds valid with material stiffness matrix L0 : ı ij0

L0ijkl İkl0 in ȍ

(2)

2nd step: quantities for inhomogeneous anisotropic medium are to be determined, i.e. displacements u , tractions p , strains İ and stresses ı are unknown. In this step geometrically same body is assumed with prescribed boundary displacements u and given tractions p from the first step. Hooke’s law is now valid for material stiffness matrix of the whole body (no more isotropic homogeneous) involving also eigenstress field Ȝ : ı ij

Lijkl İkl Ȝij in ȍ ,

u i on ī { īu , pi

ui

p i on ī { ī p

(3)

Eij İij ( uc( x )), İijc ! 0

(4)

Periodic unit cell is considered, for which it holds:

ui ( x )

Eij x j uic( x ), İij ( x )

Eij İijc ( x )

where Eij are components of the overall (macroscopic) strain tensor, uci are components of fluctuating displacement vector. In periodic structure it holds for the unit cell: uc is same at each boundary point in

the direction of invariance of the layer, while tractions are there antisymmetric. Similarly to the classical H-S principle symmetric polarization tensor Ĳ is introduced, defined as:

ıij

L0ijkl İklc Ĳij

(5)

Inasmuch as ı is statically admissible, it also holds:

w ( L0ijkl İklc Ĳij ) wx j Next, subtracting (5) and (3), considering (4), provides:

Ĳij [ Lijkl ]İkl L0ijkl Ekl Ȝij

0

in

0

ȍ,

(6)

>L@

L L0

(7)


415

3. Lippmann-Swinger equation for plasticity

In this section integral formulation to the H-S variational problem is created. Multiplying (7) successively by test functions ĳi , i 1,2,3 , integrating the result over the domain ȍ , applying two times Green’s theorem and setting for ĳi the kernel displacement components yield: uic( ȟ )

³ uik ( x, ȟ ) pkc ( x )dī ( x ) ³ pik ( x, ȟ )ukc ( x )dī ( x ) ³ İikl ( x, ȟ ) Ĳkl ( x )dȍ( x ), *

*

ī

*

ī

ȟ ȍ

(8)

ȍ

where * ( x, ȟ ) İikl

wuik* ( x, ȟ ) , pik* ( x, ȟ ) wx j

L0ijlm

wulk* ( x, ȟ ) nj wxm

* ( x, ȟ )n j L0ijlm İlmk

Here and in what follows geometrically and materially symmetric unit cell is considered, so that the boundary displacements uc vanish.

Positioning ȟ on the boundary ī and taking into account the boundary conditions (4), the integral equation equivalent to (6) is then:

³ uik ( x, ȟ ) pkc dī ( x ) ³ İikl ( x, ȟ ) Ĳ kl ( x )dȍ ( x ), *

0

*

ī

ȟī

(9)

ȟ ȍ

(10)

ȍ

Differentiating (8) with respect to ȟ j provides the expression İijc ( ȟ )

³ hijk ( x, ȟ ) pkc dī ( x ) ³ ȥijkl ( x, ȟ ) Ĳ kl ( x )dȍ( x ), *

*

ī

ȍ

and the volume integral is taken in Hadamard’s sense. Equations (9) and (10) can be expressed in a comprehensive form: there is an operator G (Ĳ ) which relates the fluctuating strain and polarization tensor as: İ c( x ) G ( Ĳ ( x )) (11) Let us substitute for Ĳkl ( x ) from (7) into the latter equation and split the integration over fiber and matrix, which provides: İijc ( ȟ )

³ hijk ( x, ȟ ) pkc dī ( x ) ³ ȥijkl ( x, ȟ ){[ Lijkl ]( İkl (ȟ ) µij (ȟ )) Lijkl Ekl }dȍ ( x ) *

*

ī

ȍ

0

m

* ³ ȥijkl ( x, ȟ ){[ Lijkl ]İkl ( ȟ ) L0ijkl Ekl }dȍ ( x ), ȟ ȍ

(12)

ȍf

f , the latter equation as there is no eigenstrain (plastic strain) inside of a fiber. If we set L0ijkl { Lijkl simplifies as:

416


³ hijk ( x, ȟ ) pkc dī ( x ) ³ ȥijkl ( x, ȟ ){[ LĮȕkl LĮȕkl ]( İĮȕc (ȟ ) µkl (ȟ )) Lijkl Ekl }dȍ( x ) *

İijc ( ȟ )

m

*

f

f

ȍm

ī

f * Lijkl Ekl ³ ȥijkl ( x, ȟ )dȍ ( x ),

(13) ȟ ȍ

ȍ

Next, equation (13) can be written in increment form (note that no differentiation is carried out, it is not permitted here for the singular integrals involved in such a process do not admit it):

ǻİijc ( ȟ )

³ hijk ( x, ȟ )ǻpkc dī ( x ) ³ ȥijkl ( x, ȟ ){[ Lijkl Lijkl ](ǻİklc (ȟ ) ǻµij (ȟ ))}dȍ( x ), *

m

*

ī

ȍ

f

ȟ ȍ m (14)

m

and the points of observer ȟ belong now only to the matrix. For completeness let us write integral equations which are equivalent to (9): 0

³ uik ( x, ȟ )ǻpkc dī ( x ) ³ İikl ( x, ȟ )[ LĮȕkl Lijkl ](ǻİklc (ȟ ) ǻµkl (ȟ ))dȍ( x ), *

*

ī

ȍ

m

f

ȟ ȍm

(15)

m

Equations (9) and (13) establish a simultaneous system for computation of elastic state in inhomogeneous material composed form fiber and matrix due to unit strain impulses. The latter two equations (14) and (15) create a simultaneous system for improvement of boundary tractions and fluctuating strains due to plasticity. In conclusion, the strain components at any point of the domain are dependent on boundary tractions and strains and eigenstrains in matrix only, and the boundary tractions are dependent on the strains and eigenstrains in matrix. The latter equations lead us to an approach, which is described in more details in the following section. 4. Calculation of plasticity effect First, let us consider geometry of a composite unit cell. The solution of such a cell in periodic structure is concisely described in [4], for example. Applying appropriate unit displacements to the boundary of the cell, its elastic material properties are identified with that of fiber, null superscript quantities are straightforwardly obtained, and even on much more complicated geometry than ours. The values of displacements, strains and stresses are also speculated at starting levels.

Figure 1:

Unit cell used in the study

Dividing the external boundary of the unit cell into subsurfaces in 3D or subintervals in 2D as

¦ pij (ȟ )ț ( I j ), j 1

M

M

N

pi ( ȟ )

İij ( ȟ )

¦ İijk (ȟ )ț (Qk ),

k 1

µij ( ȟ )

¦ µijk (ȟ )ț (Qk )

k 1

(16)


417

where ț are the characteristic functions being equal to one for ȟ inside the subregion I i or ȍi and zero otherwise, I i is a boundary subregion, ȍi is a subregion in the domain, M , N are respectively numbers of domain subregions and boundary subregions. If we put pij ( ȟ )

İijk ( ȟ )

İij (ȟ k )

İijk and µijk ( ȟ )

µij ( ȟ k )

pi ( ȟ j )

pij , ȟ j is centered in I j ,

µijk , ȟ k is identical with the center of gravity of the subregion

ȍk , all quantities are uniformly distributed inside their subregions (elements). Obviously, I i is a boundary element while ȍk is an internal cell. Applying these approximations, putting first µij 0 overall, from (12) and (13) strains, tractions and stresses follow. Then stresses are obtained from (5). Testing them for plastic rules, distribution of ǻµij is specified and (14) with (15) delivers the increments of strains using only the domain of the matrix. This is the greatest advantage of the above described approach, which consists in concentration of the problem of improvement of boundary tractions and fluctuating strain to the domain of matrix only, while fiber is no longer involved into computation.

5. Example

A plane square unit cell is considered with fiber volume ratio equal to 0.6 according to Fig. 2, so that only first quarter is assessed. The following elastic material properties of phases are assumed: Young's modulus of fiber Ef = 210 GPa, Poisson's ratio Ȟ f = 0.16; on the matrix Em = 17 GPa, and Ȟ m = 0.3. The Mises plasticity is considered with plastic coefficient k 0 20 MPa . Displacements in the planar principal directions are drawn in Fig. 2 for unit displacement excitation in x-direction.

Figure 2: Distribution of elastic and plastic displacements in the first quarter of the unit cell

418


6. Conclusions

In this paper useful treatment of plastic behavior on a unit cell is described when using boundary element method. This splendid numerical method suffers from one important problem: inhomogeneous material properties (so is plasticity, for example) are hardly involved in the computation. It appears that for particular problems there is a way on how to bridge this problem. Extended Hashin-Shtrikman variational principles are used for equivalent formulation, which covers polarization tensor. A special choice of elastic material properties (identification of comparative medium with fiber) leads us to elimination of fibers from the nonlinear computations and basic simplification of iterative procedure. From the pictures describing distribution of displacements in the first quarter of the cell at elastic and plastic states it is seen that principal redistribution of extreme displacements is attained. This result is obtained in very short computational time, as only two iterations are necessary to get significant values of the displacements. The continuation of iterations is no more necessary, is insignificant. Moreover, in each iteration step only multiplication of vectors and matrices is required; there is no solution of equations. Only in the elastic state standard linear algebraic equations are solved. Acknowledgments: The financial support of this work provided for the first author by grants No. GACR 103/08/1197 and CZE MSM 6840770001 of the Grant Agency of the Czech Republic are gratefully acknowledged. References

[1] Z. Hashin, S. Shtrikman: On some variational principles in anisotropic and nonhomgeneous elasticity, J Mech Phys Solids 10(3), (1962), p. 35-42. [2] P.P. Procházka, J. Šejnoha: Extended Hashin-Shtrikman variational principles, Applications of Mathematics 49 (4), (2004), p. 357-372. [3] G.J. Dvorak, P.P. Procházka: Thick-walled composite cylinders with optimal fiber prestress, Composites Part B 27B, (1996), p. 643-649. [4] P.M. Suquet: Homogenization techniques for composite media, Lecture Notes in Physics 272, Berlin, Springer (1985), p. 194-278. [5] M. Duvant, J.-P. Lions: Variational inequalities in mechanics, Dunod, Paris (1978).


419

Transient Dynamic Analysis of Interface Cracks in 2-D Anisotropic Elastic Solids by a Time-Domain BEM Stefanie Beyer1, a, Chuanzeng Zhang2, b, Sohichi Hirose3, c, Jan Sladek4, d and Vladimir Sladek4, e 1 2 3

Energy Sector, Siemens AG, D-02826 Görlitz, Germany

Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany

Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo 152-8552, Japan 4

Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia a

[email protected], b [email protected], c [email protected], d

[email protected], e [email protected]

Keywords: Interface cracks; Layered anisotropic elastic solids; Dynamic stress intensity factors; Time-domain BEM; Multi-domain technique.

Abstract. In this paper, transient elastodynamic analysis of an interface crack in two-dimensional (2-D), layered, piecewise homogenous, anisotropic and linear elastic solids subjected to an impact loading is presented. A time-domain boundary element method (BEM) is developed for this purpose. Displacement boundary integral equations (BIEs) in conjunction with a multi-domain technique are applied in the present time-domain BEM. Collocation method is used for both the spatial and the temporal discretizations. Numerical examples for computing the complex dynamic stress intensity factor are presented and discussed. Introduction Interface cracks and interface debonding are the most distinct failure mechanisms in composite materials because of the mismatch in the material properties. To characterize the crack-tip stress and deformation fields, stress intensity factors (SIFs) and energy release rates are often used in linear elastic fracture mechanics. Because of the mathematical complexity of the interface crack problems, only very few investigations on interface cracks in generally anisotropic and linear elastic solids under impact loading conditions can be found in literature. Although the time-domain boundary element method (BEM) can be utilized in principle for this purpose, its numerical implementation and applications to dynamic analysis of interface cracks in homogeneous, generally anisotropic and linear elastic solids have been reported in literature only very recently. The main reason for this deficiency is due to the fact that the required elastodynamic fundamental solutions in the time-domain BEM do not have explicit closed-form analytical expressions and they have a very complex mathematical structure. This paper presents a transient dynamic analysis of interface cracks in two-dimensional (2-D), layered, anisotropic and linear elastic solids. A time-domain BEM is developed for this purpose. A multi-domain BEM is applied, which divides the inhomogeneous layered anisotropic solid into homogeneous and anisotropic layers. Time-domain elastodynamic fundamental solutions for homogeneous, anisotropic and linear elastic solids and displacement boundary integral equations are used for each layer. For both temporal and spatial discretizations of the boundary integral equations, collocation methods are adopted. By using the continuity/discontinuity conditions of the displacements and the stresses on the interfaces and the crack-faces and initial conditions, an

420


explicit time-stepping scheme is obtained for computing the discrete unknown boundary data including the crack-opening-displacements (CODs). An efficient technique for computing the complex dynamic SIFs from the numerically calculated CODs is presented and discussed. To demonstrate the accuracy and the efficiency of the present time-domain BEM, numerical examples are presented and discussed. Initial boundary value problem and time-domain BIEs We consider a layered, anisotropic and linear elastic solid with an interface crack as shown in Fig. 1. In the absence of body forces, the layered solid satisfies the equations of motion V ij , j U ui , (1) Hooke’s law V ij Eijkl uk ,l , (2) the initial conditions ui ( x , t ) ui ( x , t ) 0 for t d 0 , (3) the boundary conditions (2) fi ( x , t ) 0 , x * c * (1) (4) c *c , * (5) f ( x, t ) f ( x, t ) , x * , i

i

f

ui ( x, t ) u ( x, t ) , x *u ,

(6)

* i

an the continuity/discontinuity conditions on the interface *int and the crack-faces * (1,2) c ui(1) ( x, t ) ui(2) ( x, t ) , V ij(1) ( x , t ) V ij(2) ( x , t ) , x *int , (2) (2) 'ui ( x, t ) ui(1) ( x * (1) x *c . c , t ) ui ( x * c , t ) ,

(7) (8)

In Eqs. (1)-(8), ui , V ij and fi

V ij n j represent the displacement, the stress and the traction components, n j is the outward normal vector, U is the mass density, Eijkl is the elasticity tensor, *c

(2) * (1) denotes the crack-faces, * ex c *c

* f *u stands for the external boundary with * f

and *u being the boundary parts with prescribed tractions fi * and displacements ui* , * int is the interface, and 'ui is the crack-opening-displacements (CODs), respectively. A comma after a quantity represents spatial derivatives while a dot over a quantity denotes time differentiation. Greek indices take the values 1 and 2, while Latin indices take the values 1, 2 and 3. Unless otherwise stated, the conventional summation rule over repeated indices is implied.

Material 1

Domain : 1

* ex

* (1) c

*int * (2) c

* int

Material 2

Domain : 2 Fig. 1: An interface crack in a layered anisotropic and linear elastic solid


421

To each sub-domain of the cracked solid, the following time-domain displacement BIEs are applied cij u j ( x , t )

³ u

G ij

f j - tijG u j ds y ,

, x * ex *int * (1,2) c

(9)

*

where cij is the free-term depending on the smoothness of the boundary, uijG ( x , y; t ,W ) and tijG ( x , y; t ,W ) are the elastodynamic displacement and traction fundamental solutions for

homogeneous, anisotropic and linear elastic solids, x and y represent the observation and the source points, and denotes Riemann convolution t

³ f (t W ) g (W )dW .

f g

(10)

0

For smooth boundaries cij 0.5G ij , where G ij is the Kronecker-delta function. The elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids derived by Wang and Achenbach [1] are implemented in the present time-domain BEM. Note here that the elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids cannot be given in closed forms in contrast to homogeneous, isotropic and linear elastic solids. In 2-D case, they can be represented by line-integrals over a unit circle. It should be also remarked here that the displacement BIEs (9) have a strong singularity in the sense of Cauchy-principal value integrals. Numerical solution procedure

To solve the strongly singular displacement BIEs (7), a numerical solution procedure is developed. Collocation method is applied for both the temporal and the spatial discretizations. The displacements and the tractions are approximated by ui ( y,W )

M

P

¦¦ P

p

m (u )

( y ) K(pu ) (W ) u i m ,

(11)

m 1 p 1

f i ( y ,W )

M

P

¦¦ P

m( f )

( y ) K(pf ) (W ) f

p

i m

,

(12)

m 1 p 1

where Pm () ( y ) is the spatial shape functions, K(p) (W ) is the temporal shape function, (ui )mp and ( f i ) mp are discrete values at the m-th collocation point and p-th time-step. Also, M is the total element number and the time is divided into P equal time-steps, i.e., W P 't . In this analysis, constant spatial and linear temporal shape functions are adopted for simplicity, i.e., 1, y * m , ¯0, y * m

P m u ( y ) P m f ( y ) ® K(pu ) (W ) K(pf ) (W )

1 ^ >W ( p 1)'t @ H >W ( p 1)'t @ 2 >W p't @ H >W p't @ 't >W ( p 1)'t @ H >W ( p 1)'t @ ` ,

(13)

(14)

where H [] is the Heaviside step function. In Fig. 2, the used spatial and temporal shape functions are depicted.

422


Pm ( ) ( y )

m 1

K(p) (W )

m

l1

m

2

3

y

l3

l2

W ( p 1)'t

p't

( p 1)'t

Fig. 2: Spatial and temporal shape functions By substituting Eqs. (11) and (12) into the time-domain BIEs (9), applying the discretized BIEs to each sub-domain and invoking the initial conditions (3), a system of linear algebraic equations can be obtained as A1 u N

N -1

B1 f N ¦ B N -n 1 f n A N -n 1 un ,

(15)

n 1

where An and B n are the system matrices, u N is the vector containing the boundary displacements, and f N is the traction vector for the external boundary and the crack-faces. By considering the boundary conditions (4)-(6) and the continuity conditions (7), equation (12) can be rearranged as xN

1 -1

C

N -1 ª º « D1 y N ¦ B N -n 1 f n A N -n 1 un » , n 1 ¬ ¼

(16)

in which x N represents the vector with unknown boundary data, while y N denotes the vector with known boundary data. The explicit time-stepping scheme (16) is applied for computing the unknown boundary data time-step by time-step. By using constant spatial and linear temporal shape-functions (13) and (14), time and spatial integrations can be performed analytically. Strongly singular integrals are computed analytically by a special regularization technique. Only the line-integrals over the unit circle in the elastodynamic fundamental solutions have to be computed numerically by using standard Gaussian quadrature formula. More details on the numerical implementation of the time-domain BEM can be found in the recent work of Beyer et al. [2] and Beyer [3]. Computation of the complex dynamic stress intensity factor

For an interface crack, the displacement and the stress fields near the crack-tip can be characterized by a complex stress intensity factor which is defined as K

K1 i K 2 ,

(17)

where K1 and K 2 are the real and the imaginary parts of the complex stress intensity factor. The amplitude and the phase angle of the complex dynamic stress intensity factor can be obtained by using the following equations


K (t )

(18)

K12 (t ) K 2 2 (t )

K (t )

1 4H 2 4 cosh SH

423

2S

>t2 'u1 (G , t )

2

2

t2 'u 2 (G , t ) @ > d1 'u 2 (G , t ) d 2 'u1 (G , t ) @

G

d1 t2 t1 d 2 tan M (t )

K 2 (t ) K1 (t )

d 2 d1 > 'u2 (G , t ) / 'u1 (G , t ) @ t1 > 'u2 (G , t ) / 'u1 (G , t ) @ t2

, (19)

.

In Eqs. (18) and (19), G is a small distance to the crack-tip, H is the bi-material constant, and the constants d , dD , tD ( D 1, 2 ) can be found in [4]. Since the stress field near the tip of an interface crack shows a very complicated oscillating singularity, no special crack-tip elements are implemented in the present time-domain BEM for simplicity. For this reason, the local behavior of the crack-opening-displacements (CODs) cannot be described properly by using the present time-domain BEM. To minimize the numerical error in the computation of the complex dynamic stress intensity factor from the CODs, a least-squares technique based on the minimization of the quadratic deviations of the displacements on the crackfaces is applied. Numerical examples

As first numerical example, we consider an interface crack of length 2a in a rectangular plate consisting of two homogeneous, anisotropic and linear elastic materials as depicted in Fig. 3. The plate is subjected to an impact tensile loading of the form V V 0 H (t ) , where V 0 is the loading amplitude and H (t ) is the Heaviside step function.The geometry of the cracked plate is defined by 2w = 20,0mm, 2h = 40,0mm and 2a = 4,8mm. The outer boundary of the plate is discretized by 80 constant elements with 20 elements for each side, and the crack is discretized by 15 constant elements. A time-step 't 0.3P s is applied. Plane strain condition is assumed in the numerical calculations. Two different Graphite-epoxy composites are considered in the first example, which have the following material constants

Cij(1)

0 0 º ª122.77 3.88 18.88 0 « » 16.34 4.79 0 0 0 « » « 28.3 0 0 0 » « » GPa , 3.91 0 0 » « « 21.83 0 » sym « » 6.94 ¼» ¬«

U (1)ҏ = 1600kg/m3

Cij(2)

0 0 0 º ª 65.41 4.29 26.17 « 16.34 4.39 0 0 0 »» « « 54.99 0 0 0 » « » GPa , 5.27 0 0 » « « 33.96 0 » sym « » 5.58¼» ¬«

U (2)ҏ = 1600kg/m3

Numerical results for the normalized amplitude of the complex dynamic stress intensity factor K (t ) / K Ist ( K Ist V 0 S a ) are presented in Fig. 4. A comparison with the numerical results of

424


Wünsche [5] shows a very good agreement. Wünsche used a time-domain BEM based on a collocation method for the temporal discretization and a Galerkin-method for the spatial discretization. Figure 4 shows that the normalized amplitude of the complex dynamic stress intensity factor is zero before the wave arrival time at the crack-tip t h / cL(1) 6.25P s , where cL(1) is the velocity of the quasi-longitudinal wave of the domain : . After the wave arrival, K (t ) / K st I

1

increases rapidly and it reaches its maximum value K (t ) / K Ist 3.82 at t 9.9 P s . Then it decreases to a local minimum and thereafter it increases again. A second peak is obtained at t 18.3P s .

V

4.0 3.5 3.0

:1 2a

2h

~ K (t) / KIst

V(t) V0

:2

2.5 2.0 1.5

t

2w

1.0 WÜNSCHE [5] TDBEM

0.5 0.0

V

0

4

8

12

16

20

t [Ps]

Fig. 4: Normalized amplitude of the complex dynamic stress intensity factor

4.0

4.0

3.5

3.5

3.0

3.0

2.5

2.5

~ K (t) / KIst

~ K (t) / KIst

Fig. 3: An inner interface crack

2.0 a = 3 mm a = 4 mm a = 4 mm a = 6 mm a = 7 mm a = 8 mm

1.5 1.0 0.5

2.0 a = 9 mm a = 10 mm a = 11 mm a = 12 mm a = 13 mm a = 14 mm

1.5 1.0 0.5 0.0

0.0 0

5

10

15

t [Ps]

20

25

0

5

10

15

20

25

30

t [Ps]

Fig. 5: Effects of the crack-length on the normalized amplitude of the complex dynamic stress intensity factor

35


425

By using the same temporal and spatial discretizations, the effects of the crack-length on the normalized amplitude of the complex dynamic stress intensity factor are investigated. The corresponding numerical results are presented in Fig. 5. Figure 5 reveals that the crack-length has significant influences on K (t ) / K Ist . Both the maximum value and the corresponding time instant are dependent on the crack-length. For small crack-length the first peak of K (t ) / K st is also its I

maximum value, while for large crack-length the second peak becomes its maximum. In the second example, we consider an edge interface crack of length a in a rectangular plate consisting of two different anisotropic and linear elastic materials as depicted in Fig. 6. The geometry of the cracked plate is defined by 2w = 20,0mm, 2h = 40,0mm and a = 4,8mm. The plate is subjected to an impact tensile loading of the form V V 0 H (t ) . The outer boundary of the plate is discretized by 80 constant elements with 20 elements for each side, and the crack is discretized by 15 constant elements. A time-step 't 0.4 P s is chosen. Plane strain condition is assumed in the numerical calculations. The same material combination as in the first example is investigated. Figure 7 shows the normalized amplitude of the complex dynamic stress intensity factor versus the time. Here again, a comparison of the present numerical results with that of Wünsche [5] shows again a good agreement. The K (t ) / K Ist -curve shows a more smooth increase with time after the h / cL(1) 6.25P s than in the first example for a central interface crack. The maximum value of K (t ) / K Ist is attained at about t 13.6 P s and a second peak is observed at about t 19.2 P s .

wave arrival time t

V

4.5 4.0 3.5

:1

a

3.0

~ K (t) / KIst

2h

V(t) V0

:2

t 2w

2.5 2.0 1.5 1.0

WÜNSCHE [5] TDBEM

0.5 0.0

V

0

4

8

12

16

20

t [Ps]

Fig. 6: An inner interface crack

Fig. 7: Normalized amplitude of the complex dynamic stress intensity factor

Finally, the effects of the length of the edge interface crack on the normalized amplitude of the complex dynamic stress intensity factor are investigated by using the same temporal and spatial discretizations. Figure 8 presents the corresponding numerical results for K (t ) / K Ist . Similar to the first example for a central interface crack, the crack-length may affect the behaviour of the K (t ) / K Ist -curve considerably. The numerical results given in Fig. 8 confirm again that both the maximum value and the associated time depend strongly on the crack-length. From Fig. 8 it can be concluded that in comparison to the K (t ) / K Ist -curve for a central interface crack as shown in Fig.

426


5, the variations of the K (t ) / K Ist -curve with time are even more complex in the case of an edge interface crack. 6.0

6.0 a = 3,0 mm a = 4,0 mm a = 5,0 mm

5.5 5.0 4.5

5.5 5.0 4.5 4.0

~ K (t)/ KIst

~ K (t) / KIst

4.0 3.5 3.0 2.5

3.5 3.0 2.5

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

a = 6,0 mm a = 7,0 mm a = 8,0 mm

0.0

0

5

10 15 20 25 30 35 40 45 50

t [Ps]

0

5

10 15 20 25 30 35 40 45 50

t [Ps]

Fig. 8: Effects of the crack-length on the normalized amplitude of the complex dynamic stress intensity factor Summary

This paper presents a time-domain BEM for transient elastodynamic analysis of an interface crack in 2-D, layered, piecewise homogeneous, anisotropic and linear elastic solids. Time-domain displacement BIEs in conjunction with a multi-domain technique are applied in the present timedomain BEM. For both the temporal and the spatial discretizations, a collocation method is adopted. Constant spatial shape functions and linear temporal shape functions are used for simplicity, which allow us to perform the temporal and the spatial integrations analytically. Only the line-integrals appearing in the elastodynamic fundamental solutions have to be computed numerically. An explicit time-stepping scheme is obtained for computing the unknown boundary data numerically. An efficient least-squares technique is applied for accurately compute the complex dynamic stress intensity factor from the CODs. Numerical results for the complex dynamic stress intensity factor are presented and compared with available reference solutions. Acknowledgement

This work is supported by the German Research Foundation (DFG) under the project numbers ZH 15/5-1 and ZH 15/5-2, which is gratefully acknowledged. References

[1] C.-Y. Wang and J.D. Achenbach: Geophys. J. Int. Vol. 118 (1994), p. 384. [2] S. Beyer, Ch. Zhang, S. Hirose, J. Sladek and V. Sladek: Structural Durability & Health Monitoring Vol. 3 (2007), p. 177. [3] S. Beyer: PhD Thesis (in German), TU Bergakademie Freiberg, Germany, 2008. [4] S.B. Cho, K.R. Lee, Y.S. Choy and R. Yuuki: Engineering Fracture Mechanics Vol. 43 (1992), p. 603. [5] M. Wünsche: PhD Thesis (in German), TU Bergakademie Freiberg, Germany, 2008.


427

Modelling of topographic irregularities for seismic site response F. J. Cara1, B. Benito2, I. Del Rey1, E. Alarcón1. 1 ETSI Industriales Universidad Politécnica de Madrid José Gutierrez Abascal, 2 28006, Madrid, Spain [email protected] 2

ETSI Topografía, Geodesia y Cartografía Universidad Politécnica de Madrid Carretera de Valencia, km 7 28031, Madrid, Spain

Keywords: BEM, wave propagation, layered media, irregular interfaces, wave scattering, site effects

Abstract. Seismic evaluation methodology is applied to an existing viaduct in the south of Spain, near Granada, which is a medium seismicity region. The influence of both geology and topography in the spatial variability of ground motion are studied as well as seismic hazard analysis and ground motion characterization. Artificial hazard-consistent ground motion records are synthesised applying seismic hazard analysis and site effects are estimated through a diffraction study. Direct BEM is used to calculate the valley displacement response to vertically propagating SV waves and transfer functions are generated allowing the transformation of free field motion to motion at each support. A closed formulae is used to estimate these transfer function. Finally, the results obtained are compared. Introduction Performance Based Engineering is the reassurement of the classical line which tries to use the most advanced and comprehensive procedures to give confidence to the designer and quantify structural damages in terms that both owner and society can understand the involved risks. Among others, this new paradigm includes hazard analysis and site effects for important bridges. The hazard analysis will produce two main results. The first is the characterization of the hazard itself. The second one is the identification of most-likeable scenarios, allowing selection or generation of records compatible with them. Seismic hazard analysis is beyond the scope of this paper, although a brief summary is presented below. Then site effects will be investigated, assessing the need of considering multiple support excitations. Direct Boundary Element Method (DBEM) will be used to compute transfer functions allowing the transformation of free field motion to motion at the foundation of each pier. The influence of topographic details in earthquake accelerations has been recognized for a long time. As the analytical solution is limited to very simple geometric types, all research on realistic cases is based on numerical techniques that can be the so-called Indirect Boundary Element method (IBEM) [1], a pure Direct Boundary Element Method (DBEM) [2,3], or a mixture of Finite and Boundary elements [4]. The boundary element method is especially well suited for the analysis of the seismic response of valleys of complicated topography and stratigraphy. Infinite regions are naturally represented, and the radiation of waves towards infinity is automatically included in the model which is based on an integral representation valid for internal and external regions. The main focus of this work is related to the evaluation of the different accelerations at the pier foundations of bridges, and generating semi analytically recommendations that can be applied to the general response spectra.

428


Description of the viaduct and the valley The bridge is part of the Spanish Highway network, which means that the Spanish code considers it as an important infrastructure with a design life of 100 years. The bridge has two-decks, supported by u-shaped piers in their tops. The total length is 305 m distributed in six spans: 45,50 m + 4x53,50 m + 45,50 m. The pier heights are (from left to right) 27, 64 m, 74, 79 m and 50 m. The valley geometry can be seen in Figure 1. The left hillside has a slope of approximately 30º and the right one is about 40º. The central part is almost horizontal around the river bed.

Figure 1. Valley geometry.

The geology is shown in Figure 2. There is an erosive contact between the Pliocene and the Tertiary rocks that produces a shallow layer of conglomerates on a dome of schists. Inside this one a dolomitic inclusion has been detected near pier number 4. Both abutments are founded on conglomerates.

Figure 2. Valley geology. S waves velocity properties, the density ȡ and the shear modulus G of the different materials are shown in Table 1.

Conglomerates Schists Dolomites Table 1. Material properties.

Cs (m/s) 1000 1800 2400

ȡ ( kg/m3 ) 2200 2400 2700

G ( GPa) 2,20 7,76 15,55


429

Seismic hazard analysis Seismic hazard assessment at the bridge site is computed following the standard zonified probabilistic approach, where earthquake occurrence is modelled as a poissonian process and earthquake recurrence is characterized by a doubly-truncated Gutenberg-Richter relation [5]. There is a relatively poor knowledge on potential fault sources in the study area. However, geomorphologic and neotectonic analyses provide rough estimates on present-day activity and maximum possible magnitudes of some faults. According to these studies, it is possible to relate a magnitude Mw 5.5-6.5, epicentral distance Repi 30-40km event with the Ventas de Zafarraya and Pinos-Puente principal faults. That scenario was used to performance simulations of hazard-consistent acceleration-time histories. Site modelling Structural analysis of the bridge showed that one of the limit conditions was related to the longitudinal bridge behaviour. So it was decided to conduct a bidimensional study in the plane containing the bridge vertical and longitudinal directions, and to study for the valley response to the incidence of vertically propagating SV waves. The objective is to obtain, in the frequency domain, the transfer functions between the displacements at every pier foundations and a reference point in a fictitious outcrop of schists far from the site that will be supposed to be subjected to the seismic hazard defined in the previous point. The use of a DBEM is especially appropriate for this kind of analysis. In this case 4 subregions have been discretized. The boundary element mesh has been interrupted 2000 m from the centre of the valley where the behaviour is similar to a monodimensional column of stratified soil. The total number of elements is 640, with parabolic interpolation of displacements and tractions. At the surface interface, the element size is equal to 15 meters and, at internal interfaces, the element size has been chosen taking into account the maximum shear wave velocity of the two materials separated by the interface.

Figure 3. Time domain displacement of the valley to incident vertical SV waves. The incident time signal is a Ricker wavelet. The stations are located along the free surface of the half-space, at a horizontal dimensionless horizontal coordinate x=a.

430


Synthetic seismograms were computed using the FFT algorithm for a Ricker wavelet. Time series were obtained from the transfer functions estimated at receivers placed along the free surface. Figure 3 shows a sample of the synthetic seismograms, computed at the surface of the model defined in Figure 2, for the vertical incidence of SV waves (Ricker wavelet’s central frequency is fp=1.0cs/2a = 2.90 Hz). The response spectra of the horizontal component are plotted in Figure 4b at every support of the bridge. The response spectra have been calculated from a simulated acceleration-time history and are presented in terms of pseudo-acceleration as a function of time. Figure 4b shows the comparison between the spectra generated at every support combining the free field motion and the transfer functions from Figure 4a and the spectra computed directly from the free field acceleration time history. In all cases, the differences are noticeable, and at some periods are as high as the 50%. These results show that it is mandatory to conduct a diffraction study.

(a)

(b)

Figure 4. (a) Horizontal displacement transfer functions obtained for BEM model. (b) Elastic response spectra generated at every support taking into account BEM transfer functions and free field response spectrum. Closed form solution Bridge designers not always have the time or the specific techniques to develop complex diffraction problems. In addition, the uncertainties involved in the quantification of seismic action suggest the possibility of using the closed form solution associated to the transfer function of a layered media to estimate the relative displacement between piers. In addition, main interest of designers is centred around response spectra method, so that this section tries to compare the relative differences that can be found when the approach is applied to this complicated layer media. Consider a soil deposit consisting of 2 horizontal layers resting on a semi-infinite media. Assuming that an incident harmonic SV vertical wave is propagating in the semi-infinite media, satisfaction of the requirements of equilibrium and compatibility at each interface give rise to a system of simultaneous equations, which allows the amplitudes for the reflected and refracted waves to be expressed in terms of the amplitude of the incident SV wave, so that, one can define a transfer function H f ( H 1 , U1 ,cs1 ,[1 ,H 2 , U 2 ,cs2 ,[ 2 , U3 ,cs3 ,[ 3 ,Z ) relating the horizontal displacement amplitude at the free boundary to the incident waves (where H is layer height, ȡ is material density, cs is SV-wave propagation velocity and ȟ is the material damping ratio, for superficial layer (1), intermediate layer (2) and semi-infinite media (3); Ȧ is the angular frequency of the incoming waves). Applying Haskell propagator methodology [6]:


H

2

431

2

·

1 P1 eik1H1 1 P1 e ik1 H1 1 P2 eik2 H 2 E1 1 P2 eik2 H 2 P1

G1k1 , G2 k2

P2

G2 k2 G3 k3

(2) ik1 H1

ik1 H 1

E1

1 P1 e 1 P1 e 1 P1 eik1H1 1 P1 e ik1H1 kn

(1)

(3)

Z

(4)

1

csn 1 i2[ n 2

Figure 5 shows the transfer functions obtained with the closed form solution, where ȡ, cs are the same from Table 1, ȟ is equal to 0.05 in all cases, H2 is equal to 200 m and using for H1 the depth of the stratum under each pier.

(a)

(b)

Figure 5. (a) Horizontal displacement transfer functions obtained for closed formulae. (b) Elastic response spectra generated at every support taking into account BEM transfer functions, closed formulae transfer functions and free field response spectrum. Figure 5b shows the comparisons between the spectra generated through those transfer function at every support and those from Figure 4b. For all practical purposes, these response spectra are similar to those obtained with BEM, and therefore, for practical design the approach seems valid. Summary Principal seismic codes define the seismic actions by means of elastic response spectrum. Therefore, bridge designers use elastic response spectrum in bridge projects and, at least, for the general proportion of structural members, they are mainly interested for simle procedures. This paper presents how BEM can be used to obtain acceleration-time histories at every bridge pier foundations taking into account site effects, valleys of irregular geometry and complicated strata with irregular interfaces. Elastic response spectra obtained with these transfer functions have been compared with free field elastic response spectrum. The results show that free file spectrum introduces significant errors and justified the consideration of site effects. Finally, it has been shown that close form solutions produce acceptable results from the engineering practical view point, especially if, as usual, before been applied those spectra are smoothed.

432


Acknowledgement This work has been produced as a part of the research founded by Spanish Ministero de Fomento, within the National Plan of Scientific Research, Development and Technological Innovation 2004-2007, with number of project 80007/A04. The authors gratefully acknowledge the kindness of Prof. R. Gallego who provided the boundary element code LiPoSo [5]. References [1] Sánchez-Sesma FJ, Campillo M. Diffraction of P, SV, and Rayleigh waves by topographic features: a boundary integral formulation. Bull Seismol Soc Am 1991; 81: 2234–53. [2]

Dominguez, J. Boundary elements in dynamics. ElSevier & CMP 1993.

[3] Alvarez-Rubio S., Benito J.J., Sanchez-Sesma F.J. and Alarcon E. The use of direct boundary element method for gaining insight into complex seismic site response. Computers & Structures 83; 821-835. 2005. [4] Faccioli E, Paolucci R, Vanini M. TRISEE 3D site effects and soilfoundation interaction in earthquake and vibration risk evaluation. In: European Commission, Directorate General XII for Science, Research and Development. 1994–1998 Environments and Climate Programme-Climate and Natural Hazards Unit [5] Gaspar-Escribano J.M., Benito, B. Ground-Motion Characterization of Low-to-Moderate Seismicity Zones and Implications for Seismic Design: Lessons from Recent Mw 4.8 Damaging Earthquakes in Southeast Spain. Bulletin of the Seismological Society of America, Apr 2007; 97: 531 - 544. [6]

Kramer, S.L. Geotechnical Earthquake Engeneering. Prentice Hall. New Jersey. 1996.

[5] Gallego R. BEM program LiPoSo. Internal report. ETSI Caminos, Canales y Puertos. Universidad de Granada.


433

A fast 3D BEM for anisotropic elasticity based on hierarchical matrices I. Benedetti1,a, A. Milazzo1,b, M.H. Aliabadi2,c 1

Dipartimento di Tecnologie ed Infrastrutture Aeronautiche, Viale delle Scienze, Edificio 8, 90128 Palermo - Italy 2

Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW72AZ, London, UK a


Keywords: Fast BEM solvers, hierarchical matrices, anisotropic elasticity. Abstract. In this paper a fast solver for three-dimensional anisotropic elasticity BEM problems is developed. The technique is based on the use of hierarchical matrices for the representation of the collocation matrix and uses a preconditioned GMRES for the solution of the algebraic system of equations. The preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. The application of hierarchical matrices to the BEM solution of anisotropic elasticity problems has been numerically demonstrated highlighting both accuracy and efficiency leading to almost linear computational complexity.

Introduction. The use of composite materials in many engineering applications enables improved design for structures, equipment and devices. The performance of such inherently anisotropic materials must be carefully evaluated to meet increasing requirements in critical engineering applications. Much effort has been devoted to experimental studies for composite materials characterization. On the other hand numerical modeling and analysis have recently gathered significant momentum. Computational methods such as the finite difference method (FDM), the finite element method (FEM) and the boundary element method (BEM) have been widely exploited to carry out numerical analyses of structural problems involving both isotropic and anisotropic materials. The boundary element method is particularly appealing for many structural applications, although its extensive industrial usage, especially when large scale computations are involved, is hindered by some limitations, mainly related to the features of the solution matrix. Such matrix is generally fully populated, thus resulting in increased memory storage requirements as well as increased solution time with respect to other numerical methods for problems of the same order. Moreover, the analysis of three-dimensional anisotropic elastic solids in the framework of the BEM requires some additional considerations. The lack of anisotropic Green’s functions for the construction of the boundary integral representation [1] results in the use of either the integral expression of the fundamental solutions [2-5] or explicit expressions with complex calculations [6-14]. Due to the form of the 3D fundamental solutions, BEM techniques for anisotropic elasticity applications resulted in slower computations with respect to the isotropic case, for which analytical closed form fundamental solutions are known. Many investigations have been carried out to overcome such limitations. In particular, fast multipole methods (FMMs) have been developed to solve efficiently boundary element formulations for elasticity problems [15]. Although FMMs are very effective, they require the knowledge of the kernel expansion in advance in order to carry out the integration and this is particularly complex for anisotropic elasticity problems, for which analytic closed form expressions of the kernels are not available. In the present paper the Fast BEM based on hierarchical matrices and their algebra proposed in reference [17] for three-dimensional isotropic elasticity is extended to anisotropic applications. The main step is the construction of the approximation of suitable blocks of the boundary element matrix based on the computation of only few entries of the original blocks. This approximation, in conjunction with the use of Krylov subspace iterative solvers, leads to relevant numerical advantages, namely reduced memory storage requirements and reduced computational time for the solution. The effectiveness of the technique for the analysis of anisotropic solids is numerical demonstrated in the reported applications.

434


The boundary element model The boundary integral equation governing the behavior of a three-dimensional anisotropic body with boundary *, in absence of applied body forces, is given by

cij x0 u j x0 ³ Tij x0 , x u j x d * *

³U x , x t x d * ij

0

j

(1)

*

where u j and t j are the boundary displacement and tractions. U ij and Tij are the anisotropic fundamental solution kernels, whose expression is given in references [10] where different approaches for their computation are discussed. After standard BEM discretization [1] eq. (1) leads to a linear system of the form Hu

Gt

(2)

where u and t are the vectors collecting the components of the nodal values of displacement and boundary tractions, respectively. The solution of system (2), after forcing the boundary conditions in terms of prescribed nodal values, provides the values of the unknown displacement and tractions on the body boundary. The system of algebraic equations presents a coefficient matrix which is fully populated and neither symmetric nor definite. This results in increased memory requirements as well as increased assembly and solution time with respect to other numerical methods for problems of numerical comparable size. Moreover, in the case of anisotropic elasticity, all the matrix entries need to be computed integrating kernels with complex expression, due to the lack of closed form Green’s functions for three-dimensional anisotropy. The use of hierarchical matrices for the approximation and solution of BEM systems of equations arising in isotropic elasticity applications has been proposed in [16,17]. Such technique seems very appealing for anisotropic BEM problems where the computation and integration of the integral equation kernels is very involved. In the following the basic principles and the practical steps needed to generate the BEM matrix approximation using hierarchical matrices are summarized. The reader is referred to the references [16,17] for more details. The construction of the fast BEM solver is based on a hierarchical representation of the collocation matrix. Such representation is built by representing the matrix as a collection of sub blocks, some of which admit a special approximated and compressed format. Such blocks, referred to as low rank blocks, can be approximated by computing only some of the entries of the original blocks (18) through adaptive algorithms known as Adaptive Cross Approximation (ACA) (19,20). Low rank blocks represent the numerical interaction, through asymptotic smooth kernels, between sets of collocation points and clusters of integration elements which are sufficiently far apart from each other. The distance between clusters of elements enters a certain admissibility condition, based on some selected geometrical criterion, for the existence of a low rank approximant. The blocks that do not satisfy such condition are called full rank blocks and they need to be computed and stored entirely. The low rank representation of the collocation matrix allows to reduce memory storage requirements as well as to speed up operations involving the matrix. The process leading to the subdivision of the matrix into low and full rank blocks is based on geometrical considerations on the boundary mesh of the analyzed body, as schematically illustrated in fig. (1). Each block in the collocation matrix is related to two sets of boundary elements, the one containing the collocation points corresponding to the matrix row indices and the one grouping the elements over which the integration is carried out, that contains the nodes corresponding to the matrix columns. If these two sets of boundary elements are separated, then the block will be represented and stored in low rank format and it is called admissible, while it will be entirely generated and stored in full rank format otherwise. The admissibility of a candidate block is based on the inequality

min diam : xo , diam : x d K dist (: xo , : x )

(3)

where K ! 0 is a parameter influencing the number of admissible blocks on one hand and the convergence speed of the adaptive approximation of low rank blocks on the other hand [21]. The matrix block-wise subdivision and classification is based on a previous hierarchical partition of the matrix index set aimed at grouping subsets of indices corresponding to contiguous nodes and elements on the basis of some computationally efficient geometrical criterion. The partition is stored in a binary tree of index subsets, or cluster tree, that constitutes the basis for the following construction of the hierarchical block subdivision that will be stored in a quaternary block tree. A possible algorithm, leading to geometrically balanced trees, is given in reference [22] and considers for simplicity boxes framing the considered clusters.


435

Figure 1. Schematic of the boundary subdivision. As the admissible blocks have been located, their approximation is generated through ACA algorithms which allow to reach adaptively the a priori selected accuracy H c . Additionally, to optimize memory storage requirements and reduce the overall computational complexity, the low rank blocks are recompressed without accuracy penalties, taking advantage of the reduced Singular Value Decomposition (SVD) [23]. Moreover, since the initial matrix partition is generally not optimal, once the blocks have been generated and recompressed, the entire structure of the hierarchical block tree can be modified through a coarsening procedure, which reduces the storage requirements and speeds up the solution maintaining the preset accuracy [24]. As an almost optimal representation is obtained, the solution of the system can be tackled either directly, through hierarchical matrix inversion [25], or indirectly, through iterative methods [26]. In both cases, the efficiency of the solution relies on the use of a special arithmetic, i.e. a set of algorithms that implement the operations on matrices represented in hierarchical format, such as addition, matrix-vector multiplication, matrix-matrix multiplication, inversion and hierarchical LU decomposition. A collection of algorithms that implement many of such operations is given in [21] while the hierarchical LU decomposition is discussed in [26]. The use of iterative methods takes full advantages of the hierarchical representation exploiting the efficiency of the low-rank matrix-vector multiplication. The convergence of iterative solvers can be improved, or sometimes obtained from a non convergent scheme, by using suitable preconditioners. In the present approach an LU preconditioner matrix is built in hierarchical format starting from a coarse approximation with accuracy H p of the collocation matrix [26]. An iterative GMRES algorithm is finally used in conjunction with such precoditioner for solving the system.

Numerical experiments and discussion The hierarchical computational scheme described in [17] has been modified to perform analyses on anisotropic bodies. In particular the anisotropic fundamental solutions are computed by using the technique proposed by Wilson and Cruse in reference [5]. In this technique the dependence of the fundamental solution kernels on the source point to field point direction is obtained by interpolation from a database containing the so called modulation functions for the fundamental solution displacement and displacement derivatives. The database is actually constituted by tables containing the modulation functions for different source point to field point directions, described in terms of two angles -1 and -2 in spherical coordinates. All the computations have been performed using an Intel® CoreTM Duo processor T9300 (2.5 GHz) with 2 GB of RAM. In order to compare the obtained results with those obtained for isotropic bodies, the same configuration proposed in reference [17] with anisotropic material properties has been analyzed. The analyzed mechanical element loaded anti-symmetrically at the holes and clamped at the center cylinders is shown in fig. 2 with the features of the considered meshes.

436


Nr of Nodes 412 1150 3094

Mesh A Mesh B Mesh C

Nr of Elements 138 384 1032

Figure 2. Geometry and mesh data. A first set of analyses has been carried out on the mesh C to investigate the effect of the preconditioner accuracy H p on the convergence of the hierarchical GMRES iterative solution. For this purpose, the accuracy of the collocation matrix has been set to H c 10 5 , the admissibility parameter has been chosen as K

2, 8

the minimal block size has been set to nmin 36 [17] and the GMRES relative accuracy has been set to 10 . The results obtained are shown in Table 1 where the preconditioner percentage of storage, the solution times and the solution speed up ratios with respect to standard anisotropic BEM are given. A second set of analyses has been performed to study the influence of the admissibility parameter K on the solution. The results obtained are presented in Table 2 in terms of percentage storage before and after coarsening, speed up ratios and accuracy with respect to the standard BEM solution. The storage memory requested by the fast hierarchical BEM is independent from the admissibility parameters. The same is not true for the solution time and consequently for the speed up of the solution which are affected by the chosen value of K . Therefore, as expected, there is an optimum value of K which set the best block partition of the matrix and minimize the time requested for the solution. For a detailed discussion abut the influence of K see reference [17]. Table 3 reports memory requirements before and after coarsening, assembly time and speed up ratio, solution time and speed up ratio and the accuracy of the final solution at different values of the collocation matrix requested accuracy with the other parameter set at the values shown in the table. The memory requirements, assembly times and solution times decrease when the preset accuracy decreases, as the average rank of the approximation is reduced. Table 1. Effect of the preconditioner matrix accuracy (mesh C, H c 10 5 , K

2 , nmin

36 )

Hp

Preconditioner Storage %

Preconditioner Time (s)

Precond. LU time

GMRES time

Total Solution time (s)

Solution Speed up

Total Speed up

106 105 104 103 102 101

33.60 33.60

616.91 444.13

616.23 443.45

1.90 1.62

750.88 578.58

0.61 0.47

25.10 17.26 10.49 4.51

336.83 186.2 90.3 39.67

278.62 128.04 57.57 19.34

2.18 2.96 5.46 369.37

476.28 326.6 230.6 542.69

0.37 0.25 0.19 0.43

0.63 0.51 0.42 0.33 0.28 0.47

Table 2. Storage, times and speed up ratios for different K (mesh C, H c 10 5 , H p 10 2 , nmin K

0.5 2 2 4 6

36 )

Storage % before coars. 73.37 48.27

Storage % after coars. 33.55 33.60

Assembly time (s) 183.68 191.92

Solution time (s) 313.86 230.65

Assembly Speed up 0.67 0.70

Solution Speed up 0.23 0.19

Total Speed up 0.31 0.28

L2 norm 2.1 10 4 4.3 10 4

44.23 36.68 36.14

33.67 33.31 32.88

193.15 215.09 223.37

210.50 163.49 173.75

0.71 0.78 0.81

0.16 0.13 0.13

0.26 0.25 0.26

8.5 10 4 4.8 10 4 7.0 10 4

Table 3. Effect of hierarchical matrix accuracy (mesh C, K 4 , H p 10 2 , nmin

36 )

Hc

Storage % before coars.

Storage % after coars.

Assembly time (s)

Solution time (s)

Assembly Speed up

Solution Speed up

Total Speed up

10 2

26.54

10.45

142.74

104.21

0.52

0.08

0.16

L2 norm 5.7 10 1


10 3 10 4 10 5 10 6

29.27 32.67 36.68 45.52

17.22 24.94 33.31 45.04

158.63 178.48 215.09 955.80

437

136.86 156.47 163.49 451.60

0.58 0.65 0.78 3.47

0.10 0.13 0.13 0.36

1.3 10 1 9.6 10 3 4.8 10 4 3.4 10 5

0.18 0.23 0.25 0.93

Table 4. Effect of fundamental solution description accuracy (mesh C, H c 10 5 , K 4 , H p 10 2 , nmin

36 )

Storage % before coars.

Storage % after coars.

Assembl y time (s)

Solution time (s)

Assembl y Speed up

Solution Speed up

Total Speed up

L2 Norm

5.0q

42.93

41.16

526.27

254.00

1.96

0.21

1.1 10 3

'- 1.0q

37.93

34.81

368.75

201.77

1.36

0.16

0.52 0.37

'-

36.68

33.31

215.09

163.49

0.78

0.13

0.25

4.8 10 4

'-

0.5q

4.6 10 4

Regarding the collocation matrix hierarchical approximation, an interesting issue which need to be investigated is the effect of the degree of accuracy in the interpolation of the modulation functions of the fundamental solutions. In particular, with reference to the approach employed in the present paper, this effect has been studied by performing analyses with three different accuracy of the tables describing the modulation functions. Each tables is characterized by a different number of entries for the source point to field point direction, obtained by setting the angular separation trough the different directions to a fixed value '-1 '-2 '- . The results are shown in Table 4, where it is evidenced that the degree of accuracy in the description of the kernels affects the performance of the ACA block approximation. High accuracy in the kernel description determines a lower average rank in the ACA approximation of admissible blocks, with better resulting performances of the method. Finally, the performances of the method are highlighted in Fig. 3, where the memory usage and the solution time obtained by analyzing three different meshes are shown. The variation of memory usage and solution time with respect to the degrees of freedom clearly shows that, also for anisotropic bodies, the presented approach requires almost linear computational complexity. Its efficiency improves with the problem dimension and it appears very appealing for large scale systems. 1600 Standard BEM Fast Hierarchical BEM

0.6

Solution Time [sec]

Memory Usage [GB]

0.8

0.4

0.2

0

Standard BEM Fast Hierarchical BEM

1200

800

400

0 0

2000

4000

6000

DoF

8000

10000

0

2000

4000

6000

8000

10000

DoF

Figure 3. Memory usage and solution time with respect to problem DoF.

Conclusions A Fast solver for 3D anisotropic elasticity BEM problems based on hierarchical matrices has been developed. The performed tests demonstrated the applicability of the technique to the analysis of anisotropic bodies. The performances of the method, namely relevant memory storage and solution time savings, previously demonstrated for isotropic BEM systems, make the technique appealing also for anisotropic problems. Almost linear computational complexity at increasing degrees of freedom has been evidenced. Such features make the technique very appealing for large scale applications.

438


References [1] M.H. Aliabadi, The Boundary Element Method: Applications in Solids and Structures, vol. 2. John Wiley & Sons Ltd (2002). [2] D.M. Barnett, Phys. Stat. Sol. (b), 49, 741-748 (1972). [3] L.J. Gray, A. Griffith,L. Johnson, P.A. Wawrzynek, Electron. J. Bound. Elem , 1,. 68-94 (2003). [4] S.M. Vogel, F.J. Rizzo, J. Elast., 3, 203-216 (1973). [5] R.B. Wilson, T.A. Cruse, Int. J. Numer. Meth. Eng., 12, 1383-1397 (1978). [6] T. Chen, F.Z. Lin, Comput. Mech. ,15,. 485-496 (1995). [7] T.C.T. Ting, V.G. Lee, Q. J. Mech. Appl. Math., 50, 407-426 (1997). [8] G. Nakamura, K. Tanuma, Q. J. Mech. Appl. Math., 50, 179-194, (1997). [9] E. Pan, F.G. Yuan, Int. J. Numer. Meth. Eng., 48, 211-237 (2000). [10] N.A. Schclar, Anisotropic analysis using boundary elements. Comp. Mech. Publ. (1994). [11] M.A. Sales, L.J. Gray, Comput. Struct., 69, 247-254 (1998). [12] F. Tonon, E. Pan, B. Amadei, Comput. Struct. ,79, 469-482 (2001). [13] V.G. Lee, Mech. Res. Commun., 30, 241-249 (2003). [14] C.Y. Wang, M. Denda, Int. J. Solids Structures., 44 7073-7091 (2007). [15] V. Popov, H. Power, Eng. An. Bound. Elem., 25, 7–18 (2001). [16] M. Bebendorf, R. Grzhibovkiskis, Math. Meth. Appl. Sciences, 29, 1721-1747 (2006). [17] I. Benedetti, M.H. Aliabadi, G.Davì, Int. J. Solids Structures, 45, 2355-2376 (2008). [18] E. E. Tyrtyshnikov, Calcolo, 33, 47-57, (1996). [19] M. Bebendorf, Numerische Mathematik, 86, 565-589, (2000). [20] M. Bebendorf, S. Rjasanow, Computing, 70, 1-24, (2003). [21] S. Börm, L. Grasedyck and W. Hackbusch, Eng. An. Bound. Elem., 27, 405–422, (2003). [22] K. Giebermann, Computing, 67, 183-207, (2001). [23] M. Bebendorf, Effiziente numerische Lösung von Randintegralgleichungen unter Verwendung von

Niedrigrang-Matrizen, Ph.D. Thesis, Universität Saarbrücken, 2000. dissertation.de, Verlag im Internet, ISBN 3-89825-183-7, (2001) [24] L. Grasedyck, Computing, 74, 205-223, (2005). [25] L. Grasedyck, W. Hackbush, Computing, 70, 295-334, (2003). [26] M. Bebendorf, Computing, 74, 225-247, (2005).


439

A BEM approach in nonlinear acoustics V. Mallardo1 and M. H. Aliabadi2 1

2

Department of Architecture, University of Ferrara, Italy, [email protected] currently at the Imperial College London as research associate

Department of Aeronautics, Imperial College London, UK, [email protected]

Keywords: Nonlinear acoustics, integral equations, dual reciprocity. Abstract. The present paper deals with a novel application of the Boundary Element Method (BEM) to two-dimensional (2D) nonlinear acoustics. The acoustic waves are supposed to be of finite-amplitude and the analysis is performed in the frequency domain. By applying the perturbation technique, the governing differential equations are transformed into a system of two Helmholtz equations, one homogeneous and the other one inhomogeneous. The Dual Reciprocity Boundary Element Method (DRBEM) is used to transfer the domain integral to the boundary. The procedure is validated by comparison with an ad-hoc analytical solution and tested for different basis functions. Introduction The interaction of an acoustic signal with matter is said to be linear if the response of the material and the strength of the ouput signal vary linearly with the strength of the input signal. For instance, in one dimension (1D) the equation of motion reduces to the linear wave equation only if the acoustic Mach number M is negligible in comparison with unity. Furthermore, any wave will become distorted, no matter how small M is, if it can propagate a sufficient distance (see [1]). For high input signal strengths, or for materials with some special properties, some nonlinear effects may occur. Some of these effects are increasingly used for nondestructive characterization of materials and damage detection in industrial products. A review of the nonlinear acoustic applications for material characterization can be found in [2]. The nonlinear phenomena which are linked to finiteamplitude waves, i.e. when the acoustic Mach number is not negligible in comparison with the unity, are involved in the mechanisms which determine a great number of practical applications. For instance in solids, plastic and metal welding, machining and cutting, material forming. In fluids, particle filtration, defoaming, drying. The BEM is a numerical approach for solving field problems based on the boundary integral equation (BIE) formulations. The BEM has been used to solve exterior and interior linear acoustic problems for many years (see for instance [3]) because of its boundary only discretisation and automatically satisfaction of the radiation condition at infinity. A review on the applications in elastostatics, thermoelasticity, elastoplasticity, contact and fracture mechanics can be found in [4]. The interest of extending such a tool to realistic modeling of the nonlinear acoustic field generated by finite-amplitude acoustic waves is evident. This is the main purpose of the present work which is limited to the 2D analysis. The governing differential equations are threaten by a perturbative approach, then transformed into homogeneous/inhomogeneous integral equations and finally numerically solved by coupling the conventional BEM with the DRBEM. Some numerical examples are presented in order to demonstrate the efficiency of the proposed procedure. The governing equations The equations which govern the general motion of an unviscous fluid are mass conservation, momentum conservation and thermodynamic state (see for instance [1]): ρ0 − ρ =∇·u ρ 1

(1a)

440


∂2u + ∇P = 0 ∂t2 γ ρ −Q P =A ρ0 ρ0

(1b) (1c)

where ρ and ρ0 are the actual and the ”no-perturbation” mass density, respectively, u is the fluid displacement vector, P is the thermodynamic pressure, γ = cp /cv is the ratio of the specific heats at constant pressure (cp ) and constant volume (cv ) and Q is a constant to be determined from experimental data. Dissipation mechanisms are neglected. By Taylor expanding the density of the fluid ρ up to the second order term of the acoustic disturbance p = P − P0 , the following governing wave equation can be obtained: P,tt β (2) ∇2 P − 2 = − 4 (p2 ),tt c0 ρ0 c0 where comma derivative notation is adopted, c0 is the sound speed in linear acoustics and β = (1+γ)/2 is referred to as the coefficient of nonlinearity. For instance, the water at 20◦ C has β = 3.5. By adopting the following approximation: p = p¯l + p¯c

(3)

where bar indicates the dependance on the time variable, p¯l represents the first order approximation of the acoustic disturbance and p¯c furnishes its second order correction, and by assuming time-harmonic waves, i.e. p¯l = pl eiωt and p¯c = pc e2iωt , the differential Eq.(2) furnishes the following nonlinear system of differential equations (see [6] for details): ∇2 pl + kl2 pl = 0 2

∇ pc +

kc2 pc

(4a)

4ω 2 β 2 = p ρ0 c40 l

(4b)

where kl = ω/c0 and kc = 2kl . Third or higher order terms are neglected. This means that finite but of moderate amplitude waves are considered. In conclusion, the total acoustic pressure is obtained by solving the above system of differential equations in terms of pl and pc , i.e.: p(x, t) = pl eiωt + pc e2iωt

(5)

The system of Eqs.(4) can be solved analytically only in very simple cases. For complex geometries numerical techniques must be involved. The numerical implementation Finite element methods (FEM) for time-harmonic acoustics governed by the Helmholtz equation have been an active research area for nearly 40 years. The BEM has demonstrated to be efficient especially for scattering problems: in no other field is the BEM used so intensively by industry. Furthermore, recent advances such as the fast multipole and the panel clustering methods have tremendously improved its efficiency both for pulsating and for internal problems. The nonlinear acoustic boundary integral equations (NABIEs) can be derived by applying the weighted residual technique together with the Green’s identities (see [3]) to the Eqs.(4) to give: (6a) c(ξ)pl (ξ) + − q ∗ (ξ, x)pl (x)dΓ(x) − p∗ (ξ, x)ql (x)dΓ(x) = 0 Γ Γ c(ξ)pc (ξ) + − q ∗ (ξ, x)pc (x)dΓ(x) − p∗ (ξ, x)qc (x)dΓ(x) = p∗ (ξ, x)b(pl (x))dΩ(x) (6b) Γ

Ω

Γ

2


where b(pl (x)) =

441

4ω 2 β 2 p (x) = cβ p2l (x) ρ0 c40 l

(7)

The integral on the left hand side is to be interpreted in the sense of Cauchy principal value and the free term c(ξ) is equal to 0.5 if the tangent line at ξ is continuous. The symbols ξ and x denote the source and the field points, respectively, Γ is the boundary of the domain Ω under analysis. The fundamental solutions p∗ , q ∗ are given in any BEM book (see for instance [5]). In 2D they are expressed in terms of the modified zero and first order Bessel functions of the second kind. In the conventional BEM the boundary is divided into NE elements (quadratic in the present paper) and the Eq.(6a) is collocated in each boundary node to furnish a discrete system of equations in terms of the acoustic either pressure or flux in the boundary nodes. The final system is solved by any numerical technique after applying the boundary conditions. Such a procedure cannot be applied directly to Eq.(6b) if a boundary-only formulation is required. The domain integral can be transformed into the sum of boundary integrals by the DRBEM (see [7]). The keypoint is the approximation of the right hand side of Eq.(6b), i.e. b = b(pl (x)), by a finite series of basis functions for which a particular solution is available. It can be written as: b(pl (x)) ≈

N +L

fj αj =

j=1

N +L

f (x, η j )αj

(8)

j=1

where fj is function of the distance between the field point x and the dual collocation point η j . The αj coefficients are unknown and they are determined by collocating Eq.(8) at N + L (N on the boundary and L in the domain) arbitrary points. Most applications concern elastostatics and elastodynamics as well as the Poisson equation. So far no applications have been proposed concerning the Helmholtz equation even if much effort has been made to determine various particular solutions [8-9]. In the present paper various approximating functions fj are compared, i.e. the well trodden 1 +r along with the 1 +r2 , the thin plate spline (TPS) r2 Logr and the augmented thin plate spline (ATPS) r2 Logr + αN +L+1 + αN +L+2 x1 + αN +L+3 x2 . The corresponding particular solutions can be found in [6], [8-9]. On the basis of the above considerations, the NABIEs governing the propagation of finite but moderate amplitude acoustic waves can be written: (9a) c(ξ)pl (ξ) + − q ∗ (ξ, x)pl (x)dΓ(x) − p∗ (ξ, x)ql (x)dΓ(x) = 0 ξ ∈ ∂Ω Γ Γ ◦ (9b) c(ξ)pl (ξ) + − q ∗ (ξ, x)pl (x)dΓ(x) − p∗ (ξ, x)ql (x)dΓ(x) = 0 ξ ∈Ω Γ Γ c(ξ)pc (ξ) + − q ∗ (ξ, x)pc (x)dΓ(x) − p∗ (ξ, x)qc (x)dΓ(x) = p∗ (ξ, x)b(pl (x))dΩ(x) ξ ∈ ∂Ω (9c) Γ

Γ

Ω

The Eq.(9b) is included in the system of equations when some (let’s say L) internal points are considered to better evaluate αj in the application of the Dual Reciprocity (DR) approach to Eq.(9c). In the numerical procedure the modified Bessel functions needs to be computed at the integration points. Their accurate computation is very important. Two different series expansions, as suggested in [5], are adopted, i.e. one for small arguments and the other for large arguments. The numerical scheme is a standard collocation one. The boundary is discretised into quadratic, isoparametric elements. The discrete system of equations is solved by the LU decomposition. Numerical results In order to demonstrate the efficiency of the proposed procedure, some numerical examples are acquainted. They all refer to the wave propagation inside the cylinder of radius R = 1 depicted in Fig. 1(a) and with the following parameters: ρ0 = 100, c0 = 100, β = 3.5, all in compatible units.

3

442


First of all the numerical solution is compared to the analytical one. In such a simple geometry, in fact, an analytical solution can be obtained in terms of a power series. The comparison is performed for kl R = 1 and kl R = 10. p0 = 1000 is imposed on the whole boundary. D C B

θ

A

(a)

(b)

(c)

Figure 1: (a) Geometry. Adopted internal points: (b) kl R = 1, (c) kl R = 10. Fig. 2 convey the behavior of pl , see Fig. 2(a), and pc , see Fig. 2(b), in the case of kl R = 1. The agreement is excellent. Twelve quadratic boundary elements and 25 internal points, as shown in Fig. 1(b), are sufficient to obtain a relative error of less than 0.5%, but it must be underlined that 9 internal points would furnish an error of less than 2%. 0

1400

Analytical solution BEM solution


-2 -4

1300

Pressure pc

Pressure pl

-6 1200

-8 -10 -12

1100

-14 -16

1000

0

0.2

0.4

0.6

Distance from the center

0.8

-18

1

0

(a)

0.2

0.4

0.6


0.8

1

(b)

Figure 2: Comparison between analytical and numerical results (a) pl and (b) pc . kl R = 1 Fig. 3 draw the behavior of the pressure for higher frequency, i.e. in the case kl R = 10. A finer boundary discretisation and more internal points are necessary due to the lower value of the wavelength. In fact, 144 boundary elements and 841 internal points, as depicted in Fig. 1(c), are necessary to converge to the analytical solution. The very good agreement can be noticed in this case too. In order to acquaint the efficiency of the DR approach, the procedure is tested with reference to two special expressions of the source term for which an analytical solution is easily obtained. For both k = 1. The first equation is: (10) ∇2 p(x) + p(x) = x1 with solution: pA1 = sinx1 + sinx2 + x1

(11)

∇2 p(x) + p(x) = 4x21 + 4x22 + 12x1 x2 + 3x31 x2 + 2x21 x22 − x1 x32

(12)

The second differential equation is:

4


with solution:

443

pA2 (x) = 3x31 x2 + 2x21 x22 − x1 x32

(13)

Seventy-two boundary elements are used in order to cancel any error source due to the boundary discretisation. The value of the flux corresponding to both the analytical solutions at the boundary nodes illustrated in Fig. 1(a) is reported in Table 1 and in Table 2. 2000

80

1000

60

-1000 -2000

20 0

-3000


-4000 -5000


40

Pressure pc

Pressure pl

0

0

0.2

0.4

0.6


0.8

-20 -40

1

0

0.2

(a)

0.4

0.6


0.8

1

(b)

Figure 3: Comparison between analytical and numerical results (a) pl and (b) pc . kl R = 10 The last column of each table furnishes the highest value, among the four points A,..,D, of the relative error for each basis function. Concerning the solution A1, the error results to be less than 2% for all the basis functions, but it is reduced to 0.2% for the one which best fits, i.e. AT P S. The error is slightly higher with reference to the analytical solution A2. The last column does not consider the error at the node A where the exact value is zero. Analytic 1+r 1 + r2 TPS AT P S

A 1.540 1.560 1.542 1.512 1.541

B 1.836 1.855 1.838 1.810 1.838

C 1.782 1.798 1.784 1.763 1.784

D 1.295 1.304 1.297 1.285 1.297

err(%) 1.3 0.2 1.9 0.2

Table 1: Flux at some boundary points for different basis function. First analytical solution. 0 internal points

Analytic 1+r 1 + r2 TPS AT P S

A 0. 4.9e-2 857. -1.99e-2 1.82e-2

B 4.41 4.55 1450. 4.45 4.49

C 4. 4.14 2090. 4.03 4.07

D 0.414 0.524 1370. 0.417 0.456

err(%) 21 – 0.8 9.2

Table 2: Flux at some boundary points for different basis function. Second analytical solution. 25 internal points It must be underlined that 1 + r2 fails in furnishing an acceptable solution for the analytical solution A2. The reason is well reported in literature and it is related to the ill-conditioned feature of 5

444


the matrix involved in Eq.(8). Furthermore, the convergence criterion of the radial basis function r2n in 2D is not supported by a mathematical proof and, hence, its use is not recommended. Conclusions A numerical method for studying the nonlinear 2D propagation of high-intensity acoustic waves has been presented. A perturbation theory up to the second-order approximation has been first applied. The nonlinearity provokes a domain integral which has been transformed into boundary integrals by the DRBEM. The efficiency of the procedure has been verified in a case for which an analytical solution can be obtained. Different basis functions have been tested for some special cases and the results have been compared and discussed. Finite-amplitude waves are directly involved in the mechanisms that determine a great number of pratical applications in industrial processing. Numerical models are clearly advantageous over other approaches as giving solutions for a large number of different cases and for any irregular geometry or special boundary conditions. Thus, the method above presented can be an excellent tool to determine the acoustic field in the industrial processing system and to assist the experimental tests. The main advantage over the FE approach stands in the possibility not to discretise the domain under analysis, but to limit the discretisation to the boundary only. Furthermore, it furnishes very accurate results in terms of both pressures and fluxes. References [1] R. T. Beyer. Nonlinear acoustics. In Physical Acoustics Edited by W.P. Mason, Academic, New York, 1965. [2] Y. Zheng, R. G. Maev, I. Y. Solodov. Nonlinear acoustic applications for material characterization: A review. Can. J. Phys., 77:927–967, 1999. [3] L. C. Wrobel. The Boundary Element Method Volume 1: Applications in Thermo-Fluids and Acoustics. Wiley, Chichester, West Sussex, 2002. [4] M. H. Aliabadi. The Boundary Element Method Volume 2: Applications in Solids and Structures. Wiley, Chichester, West Sussex, 2002. [5] J. Dominguez. Boundary Elements in Dynamics. Computational Mechanics Publications, Southampton, Boston, 1993. [6] V. Mallardo, M. H. Aliabadi. The Dual Reciprocity Boundary Element Method (DRBEM) in nonlinear acoustic wave propagation. Computer and Experimental Simulations in Engineering and Science CESES, in print, 2008. [7] P. W. Partridge, C. A. Brebbia, L. C. Wrobel. The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton, London & New York, 1992. [8] S. Zhu. Particular solutions associated with the Helmholtz operators used in DRBEM. Bound. Elem. Abstracts, 4(6):231–233, 1993. [9] A. H. D. Cheng. Particular solutions of Laplacian, Helmholtz-type, and polyharmonic operators involving higher order radial basis functions. Engineering Analysis with Boundary Element, 24:531–538, 2000.

6


445

TIME CONVOLUTED DYNAMIC KERNELS FOR 3D SATURATED POROELASTIC MEDIA WITH INCOMPRESSIBLE CONSTITUENTS M. Jiryaei Sharahi1, M. Kamalian2 and M.K. Jafari3 1

International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran, [email protected]

2


3


Keywords: boundary element, dynamic poroelasticity, time convoluted dynamic kernels Abstract. This paper presents the explicit and simple analytical time domain convoluted kernels that appear in the discretized governing BIE of the three-dimensional well known u-p formulation of saturated porous media with incompressibile fluid and solid particles. At first, the corresponding boundary integral equations are obtained for the governing differential equations which are established in terms of solid displacements and fluid pressure. Subsequently, the analytical time domain convoluted kernels that appear in the BIE are derived. Finally, a set of numerical results are presented which demonstrate the accuracies and some salient features of the proposed solutions.

INTRODUCTION The dynamic analysis of saturated porous media, is of interest in various fields, such as geophysics, acoustics, soil dynamics and many earthquake engineering problems. From a macroscopical point of view, saturated soil is a two-phase medium constituted of solid skeleton and fluid. Dynamic behaviours of each phase as well as that of the whole mixture are governed by the basic principles of continuum mechanics. In phenomena with medium speeds, such as earthquake problems, it is reasonable to neglect the fluid inertial effects, and to reduce the complete dynamic governing differential equations to the simple commonly called u-p formulation [1,2,3]. The governing differential equations could be further simplified by neglecting the compressibility of the solid particles and fluid, which could be reasonably assumed incompressible compared to the soil skeleton [4,5]. The BEM is one of the most efficient numerical mehods for solving wave propagation problems in elastic media, because of its efficiency in dealing with semi-infinite or infinite domain problems that has long been recognized. Predeleanu [6], Manolis & Beskos [7] and Wiebe & Antes [8] were among the firsts who developed boundary integral equations and fundamental solutions governing the dynamics of poroelastic media, in terms of solid skeleton displacement and fluid displacements components. Later, Cheng et al. [9], Dominguez [10], Chen & Dargush [11] and recently Schanz [12] developed another forms of boundary integral equations and fundamental solutions of dynamic poroelasticity in terms of less independent variables. But their algorithms were based on transformed domain fundamental solutions. Obviously, time domain BEM for modeling the transient behaviour of media is preferred than the transformed domain BEM, because formulating the numerical procedure entirely in time domain and combining it with the FEM, provides the basis for solving nonlinear wave propagation problems. Proper displacement and traction fundamental solutions are one of the key ingredients required for solving wave propagation problems in saturated porous media by the BEM.Considering the independent practical variables of solid skeleton displacement and fluid pressure, Kaynia [13] was the first who presented

446


approximate transient 3D displacement fundamental solutions for the special case of short-time. Chen [14, 15]proposed another approximate transient 2D and 3D displacement solutions for the special case of short time as well as the general case, which were too complicated to be applied in BE algorithms. Gatmiri & Kamalian [16] showed that Chen's approximation could not be used in the simplified case of u-p formulation. They derived another approximate transient 2D displacement fundamental solutions for the u-p formulation which were still too complicated to be used in BE algorithms. Later Gatmiri & Nguyen [5] proposed much less complicated transient 2D fundamental solutions for the u-p formulation of saturated porous media consisted of incompressible constituents. Recently Kamalian et al[17,18] derived the transient displacement and traction fundamental solutions for the simplified u-p formulation of 3D poroelastic media with incompressible constituents. Presentation of the analytical time domain convoluted dynamic kernels that appear in the discretized BIE for the u-p formulation of 3D saturated porous media with incompressible constituents, constitutes the main essence of this paper. Some numerical results are plotted to show the accuracies and some salient features of the proposed solutions.

GOVERNING EQUATIONS The governing equations of dynamic poroelasticity were first derived by Biot [19] using the concept of variational formulation. These equations were later recast by using other theories such as the theory of mixtures (Prevost [2]) and the principles of continuum mechanics (Zienkiewicz and Shiomi [1], Gatmiri [3], etc.). Following the procedure outlined by Zienkiewicz and Shiomi [1], one can write the equations describing, respectively, the conservation of total momentum, the constitutive equation of the solid skeleton, the flow conservation for the fluid phase and the generalized Darcy’s law as follows: Equilibrium equation:

V ij , j f i

ȡui ȡ f Zi

(1)

Constitutive relation:

V ij

Ou i ,i P (u i , j u j ,i ) DpG ij

(2)

Flow conservation for fluid phase:

w i ,i J

Du i ,i

p Q

(3)

Generalized Darcy’s law:

p ,i

1 i w i U f ui mw k

(4)

where

m

Uf n,

U

(1 n) U s nU f ,

1Q

n K f (D n) K s , D

1 K Ks

ui represents the displacement of the solid skeleton, p denotes the excessive fluid pore pressure and wi represents the average displacements of the fluid relative to the solid. ıij represents the total stress, the elastic constants Ȝ and µ denote the drained Lame constants and ț=k/Ș is the permeability coefficient, with Ș and k


447

denoting the fluid dynamic viscosity and the intrinsic permeability of the solid skeleton, respectively. ȡs is the solid density, ȡf denotes the fluid density, ȡ represents the density of solid-fluid mixture, m denotes the mass parameter and n is the porosity. In addition, Į and Q are material parameters which describe the relative compressibility of the constituents. Ks and Kf denote the bulk modulus of the solid grains and the fluid while K represents the bulk modulus of the solid skeleton. Finally, fi and Ȗ denote the body force and the rate of fluid injection into the media, respectively. Omiting all terms of fluid acceleration in equation (1) as well as all dynamic terms in equation (4) and eliminating wi from equations (3) and (4), the well known governing u-p formulation of a poroelastic media with incompressible solid particles and fluid, in which all coefficients 1/Ks, 1/Kf as well as 1/Q tend towards zero, could be easily obtained in the Laplace transform domain as follows:

~ Pu~i , jj (O P )u~ j , ji Us 2 u~i D~p,i f i k~ p ,ii Du~i ,i J~

(5)

0

0

(6)

the tilde denotes the Laplace transform and s demonstrates the Laplace transform parameter. In equations (5) and (6), the contributions due to initial conditions are neglected.

BOUNDARY INTEGRAL EQUATIONS The governing boundary integral equations will be deduced starting from the equilibrium equation and using the well known weighted residual method. Weighting equation (5) by the displacement type function u'i, integrating over the body ȍ, using integrations by parts twice and finally grouping the corresponding terms together, one finds the following expression:

~~ ~ ~ ~~ ~ ~ ~~ ~ ~ ³ t u c t cu d* ³ f u c f cu d: ³ D pH c p cH d: i

i

i

*

i

i

i

i

i

:

ii

ii

0

(7)

:

Also weighting equation (6) by the pressure type function p', integrating over the body ȍ, using integrations by parts twice and finally grouping the corresponding terms together, one finds the following expression:

³ k ~p c ~p *

,n

~ p~ p ,cn d* ³ ~J ~ p c - ~J c ~ p d: ³ s D ~Hiic ~ p - ~Hii ~ p c d: :

0

(8)

:

Eliminating the common term from equations (8) and (9) and returning to the time domain, one obtains the governing boundary integral equation as follows:

³ t *

i

p - ~Jc~ p c d: 0

uic tic ui d* ³ p qc - pc q d* ³ f i uic fic ui d: ³ ~Jc ~ *

:

(9)

:

where

q

k

wp wn

ti and q denote, respectively, the traction vector on and the flux normal to the boundary (ī).

(10)

448


By Assuming zero body forces (fi and Ȗ) and assigning proper unit Heaviside point forces and supplementary impulse scalar sources to f'i and Ȗƍ as

G ( x1 )G ( x 2 )G ( x3 ) H (t )

(11)

J ( x, t ) G ( x1 )G ( x 2 )G ( x3 )G (t )

(12)

f ( x, t )

we can obtain the Somigliana type integral equations:

cij ( x0 ).ui ( x0 , t ) c( x0 ). p ( x 0 , t )

³ (t * G *

³ (t *

i

i

ij

Fij * ui )d* ³ (G4 j * q p * F4 j ).d*

(13)

*

F * u )d* (G * q p * F ).d* *G i4 i4 i 44 ³ 44

(14)

*

x is the source point; x0 is field point; cĮȕ(x0) is a matrix of constants, depend only upon the local geometry of the boundary at x0; Į, ȕ=1, 2, 3, 4. GĮȕ and FĮȕ are the time domain displacement and traction fundamental solutions are derived by Kamalian et al.[16, 17] as

Gij

Aij

Bij C ij § Aij · § r · e6 t ¨¨ e1 r , t e2 r , t 2 e3 r , t ¸¸ H ¨¨ t ¸¸ U Uv d Uv d © 2EU ¹ © vd ¹ C ij Bij § Aij · § r ¨¨ e5 r , t 2 Dij ¸¸ H ¨¨ t e4 r , t Uv s Uv s © 2U ¹ © vs

(15)

G4 i

r 2Er,i § · § r ¨¨ ,i 2 e7 r , t g 2 r , t ¸¸ H ¨¨ t rv v 4 SD r 4 SD d d © ¹ ©

r · ¸¸ ,i 2 e8 t 4 SD r ¹

(16)

Gi 4

r,i 2 Er,i § · § r ¨¨ e2 r , t ¸¸ H ¨¨ t e r , t 2 1 rv 4 v SD r SD 4 d d © ¹ ©

r,i · ¸¸ e9c t 4 SD r2 ¹

(17)

º · ¸¸e7 r , t exp 2Et » ¹ ¼

(18)

Fij

(OGkj ,k DG 4 j )ni P (Gij ,m Gmj ,i )nm

(19)

Fi 4

(OG k 4,k DG 44 )ni P (G i 4,m G m 4,i )nm

(20)

F4 j

k (G4 j ,m )nm

(21)

F44

k (G 44,m )n m

(22)

G44

where

· ¸¸ ¹

r 1 ª § «H ¨ t 4Skr ¬ ¨© v d


449

D2 2 Uk

E

(23)

O 2P U

vd

(24)

P U

vs

i, j 1,2,3 , r 2

(25)

xi xi , r,i

xi r

(26)

Aij

1 (3r,i r, j G ij ) 4Sr 3

(27)

Bij

1 (3r,i r, j G ij ) 4Sr 2

(28)

1 (r,i r, j ) 4Sr

(29)

C ij

Dij

1 G ij 4SrP

(30)

Functions ei(r,t) and gi(r,t) are given in Appendix.

TIME DOMAIN CONVOLUTED DYNAMIC KERNELS Implementation of boundary integral equations needs approximation in temporal variations of the field variables. For temporal integration the time axis is divided into N equal steps and the field variables are assumed to remain constant during a time step, so that they can be taken out of the convolution integral, thus the time integration involves only the kernels and is expressed by

³t *

i

* G ij d*

N

³ ¦t *

n i

GijN 1 n d*

(31)

n 1

n't

GijN 1 n

³ G t W dW ij

(32)

n 1 't

then by T

N 't W :

GijN 1 n by similar way

>G T @ ij

N n 1 't N n 't

(33)

450


G4Nj1 n

>G

T @ NN nn '1t 't

(34)

j4

GiN4 1 n

>G

4i

T @ NN nn '1t 't

(35)

G44N 1 n

>G

44

T @ NN nn '1t 't

(36)

FijN 1 n

>F T @

Fi 4N 1 n

>F

ij

i4

N n 1 't

(37)

N n 't

T @ NN nn '1t 't

(38)

F4Nj 1 n

> kG

j 4,m

T nm @ NN nn '1t 't

(39)

F44N 1 n

> kG

44 , m

T nm @ NN nn '1t 't

(40)

with the temporal discretization described above, equations (13) and (14) transforms into:

c ij u iN ½ ® N ¾ ¯ c p ¿

N

° § ªGijN 1 n

¦ ®°³ ¨¨ «G n 1

¯ī © ¬

N 1 n i4

G 4Nj1 n º ª t in º ª FijN 1 n »« »« G 44N 1 n ¼ ¬q n ¼ ¬ Fi 4N 1 n

½ F4Nj 1 n º ª u in º · ¸ dī °¾ N 1 n » « n » ¸ F44 °¿ ¼ ¬ p ¼¹

(41)

BEHAVIOUR OF TRANSIENT KENRNELS AT LARGE TIME STEP One of the important properties of the transient kernels is that at a very large time step the convoluted kernels should reduce to the corresponding steady state kernels. At the first time step N=1 so that T 'T :

> @

lim 't of Gij1

lim t of Gij t

ª§ 1 xi x j «¨¨ 3 ¬«© 8S r

§ 1 ¨¨ © 2 P 2PQ

· ·º ¸¸ ¸¸» ¹ ¹¼»

ª xi x j º 1 3 4Q G ij » « 16rSP 1 Q ¬ r 2 ¼ ss

Gij

(42)

Similarly

> @

0

(43)

> @

0

(44)

lim 't of Gi14 lim 't of Fi14

> @

1 lim 't of G 44

1 4Srk

ss

G44

(45)


> @

lim 't of F441

ss

F44

xj D 8Sk O 2 P r

> @

ª G im r,i r, m º Dn m « » 8S (O 2P ) ¬ r ¼

lim 't of F41j ss

4Sr 2

lim 't of G41 j

> @

where

n m r, m

451

(46)

(47)

(48)

GDE and ss FDE are steady state fundamental solutions. as can be seen, all the convoluted transient

kernels reduce to corresponding elastostatic kernels at a very large time step.

NUMERICAL RESULTS A set of numerical results are presented in this section to demonstrate the accuracy and some salient features of the proposed transient convoluted kernels. A saturated soft soil with incompressible solid grains and pore water was considered in which the material properties were defined in the metric system as follows: Ȝ=12.5 MPa, µ=8.33 MPa, ȡ =2120 kg/m3, Į=1, ț=1×10-7 m4/Ns. The point force (or fluid source) is applied at the coordinate (0,0,0) at time t=0 and the receiver is located at coordinate (0.01,0.02,0.03). Figures 1-4 depict the 1 1 1 1 , G14 , G 41 , and G 44 components respectively. As can be presented analytical closed form kernels of G11 seen, dynamic kernels gradually move toward static kernels at a very large time step. It is also interesting to note the arrival times of the pressure (vp=), diffusive (vd=117.3 m/s) and shear (vs=62.7 m/s) waves, which could be detected by sudden changes appearing in the dynamic kernels. The notable initial values of 1 1 the pressure components G14 and G 44 differ from zero, because the pressure wave with its wave propagation velocity of infinity arrives immediately and affects the media's response.

CONCLUSION Boundary integral equation, temporal discretization of BIE and analytical closed form expressions for transient dynamic kernels are presented for the well known u-p formulation of 3D saturated porous media with incompressible constituents, in terms of the practical variables of solid skeleton displacement and fluid pressure. A set of numerical results are presented that demonstrate some salient features of the dynamic kernels. The derived kernels could be simply implemented in time domain BEM for modeling the transient behaviour of saturated porous media and provides the basis to develop more effective numerical hybrid BE/FE methods for solving 3D nonlinear wave propagation problems in the near future.

452

Eds: R Abascal and M H Aliabadi 0.2

KERNEL G11 (µm)

0.15 0.1 0.05 0 -0.05

Dynamic Static

-0.1 0

0.5

1

1.5

2

2.5

3

3.5

Time (ms) 1 Figure 1. dynamic and static kernels of G11 at (0.01,0.02,0.03)

0.004

KERNEL G41(pa)

0.0035 0.003 0.0025 0.002 0.0015 0.001 Dynamic Static

0.0005 0 0

0.5

1

1.5

2

2.5

3

3.5

Time (ms) 1 at (0.01,0.02,0.03) Figure 2. dynamic and static kernels of G 41

12 Dynamic Static

KERNEL G14(m)

10 8 6 4 2 0 0

0.5

1

1.5

2 Time (ms)

2.5

3

3.5

1 Figure 3. dynamic and static kernels of G14 at (0.01,0.02,0.03)


453

KERNEL G44(Mpa)

24.9 19.9 14.9 9.9 4.9

Dynamic Static

-0.1 0

0.5

1

1.5

2

2.5

3

3.5

Time (ms) 1 Figure 4. dynamic and static kernels of G 44 at (0.01,0.02,0.03)

APPENDIX: FUNCTIONS e i r , t AND g i r , t

e1 r , t e2 r , t e3 r , t

e4 r , t

1 2 UE 1

³

t

r vd

Uv d

³

1 Uv d2

³

t

r vd t

r vd

t

Urv d2

ª 1 exp 2E t W º «t W 2E »g1 r ,W dW 2E ¬ ¼ g 2 r ,W dW g1 r ,W dW

g1 r , t

E g 2 r , t Uv d3

ª 2 t º 1 1 exp 2Et » «t E 2E 2 ¬ ¼

e5 r , t

1 2U

e6 r , t

³ >1 exp 2E t W @g r,W dW

e7 r , t g1 r , t g 2 r , t

t

r vd

³

t

r vd

1

exp 2 E t W g1 r ,W dW

ª Er v § r 2 d I §¨ E t 2 r vd ·¸ G ¨¨ t exp E t « 2 1© 2 ¹ vd « t r vd © ¬ 2 exp E t ª I 0 §¨ E t 2 r vd ·¸º «¬ © ¹»¼

·º ¸¸» ¹»¼

454


REFERENCES 1. Zienkiewicz OC and Shiomi T. Dynamic behavior of saturated porous media, the generalized Biot formulation and it’s numerical solution. Int. J. Numer. Anal. Methods Geomech 1984; 8: 71-96. 2. Prevost JH. Dynamics of porous media. Geotechnical Modeling And Applications. S M Sayed, ed. Gulf Publishing Company 1987; 76-146. 3. Gatmiri B. A simplified finite element analysis of wave-induced effective stresses and pore pressures in permeable sea beds. Geotechnique 1989; 40: 15-30. 4. Gatmiri B and Nguyen KV. Time 2D fundamental solutions for saturated porous media with incompressible fluid. Commun. Numer. Meth. Engng. 2004; 21(3): 119-132. 5. Schanz M., Pryl D., “Dynamic fundamental solutions for compressible and incompressible modeled poroelastic continua”, International Journal of Solids and Structures, 41: 4047-4073, 2004. 6. Predeleanu M., “Development of Boundary Element Method to Dynamic Problems for Porous Media”, Appl. Math. Modelling, 8: 378-382, 1984. 7. Manolis G.D. & Beskos D.E., “Integral Formulation and Fundamental Solutions of Dynamic Poroelasticity and Thermoelasticity”, Acta Mechanica, 76: 89-104, 1989. 8. Wiebe T.H. and Antes H., “A Time Domain Integral Formulation of Dynamic Poroelasticity”, Acta Mechanica 90: 125-137, 1991. 9. Cheng A.H.D. and Badmus T., “Integral Equation for Dynamic Poroelasticity in Frequency Domain with BEM Solution”, Journal of Engineering Mechanics-ASCE, 117(5): 1136-1157, 1991. 10. Dominguez J., “Boundary Element Aproach for Dynamic Poroelasticity Problems”, J. of Numer. Meth. Engng , 35: 307-324, 1992. 11. Dargush G F and Banerjee P K. A time domain boundary element method for poroelasticity. Int. J. Numer. Methods In Eng. 1989; 28: 2434-2449. 12. Schanz, M., “poroelastodynamic Boundary Element Formulation”,Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach, Springer-Verlag publication, 77-98, 2001. 13. Kaynia A.M., “Transient greens functions of fluid-saturated porous media”, Computers & Structures, 44: 19-27, 1992. 14. Chen J. Time Domain Fundamental Solution To Biot’s Equations Of Dynamic Poroelasticity. PartI: Two-Dimensional Solution. Int. J. of Solids & Structures 1994; 31(10): 1447-1490. 15. Chen J. Time Domain Fundamental Solution To Biot’s Equations Of Dynamic Poroelasticity. PartII: Three-Dimensional Solution. Int. J. of Solids & Structures 1994; 31(2): 169-202. 16. Gatmiri B., Kamalian M., “On the Fundamental Solution of Dynamic Poroelastic Boundary Integral Equations in Time Domain”, ASCE; The International Journal of Geomechanics, 2(4): 381-398, 2002. 17. Kamalian M., Gatmiri B. and Jiryaee Sharahi M., “Time domain 3D fundamental solutions for saturated porelastic media with incompressible constituents”, Proc. of the 7th International Conference on Boundary Element Techniques (BeTeq2006-Paris), 2006. 18. Kamalian M. and Jiryaee Sharahi M., “Traction transient fundamental solutions for 3D saturated porelastic media with incompressible constituents”, Proc. of the 4th International Conference on Earthquake Geotechnical Engineering (4ICEGE-Thessaloniki), 2007. 19. Biot M.A., “Theory of propagation of elastic waves in a fluid-saturated porous solid: I. Low-frequency range, II. higher frequency range”, J. Acoust. Soc. Am., 28: 168-191, 1956.


455

Stability analysis of composite plates by the boundary element method E. L. Albuquerque1, P. M. Baiz2, and M. H. Aliabadi3 1

Faculty of Mechanical Engineering, State University of Campinas Campinas, Brazil, [email protected] Currently at Imperial College London as an academic visitor. 2


3


Keywords: Stability of structures, linear buckling, composite plates, radial integration method. Abstract. This paper presents a boundary element formulation for the stability analysis of symmetric laminate composite plates where only the boundary is discretized. Body forces are written as a sum of approximation functions multiplied by coefficients. Domain integrals which arise in the formulation are transformed into boundary integrals by the radial integration method. Plate buckling equations are written as a standard eigenvalue problem. The accuracy of the proposed formulation is assessed by comparison with results from literature. Buckling coefficients and buckling modes are obtained using this formulation. Introduction Demand by an accurate stability analysis of anisotropic materials has increase with the increasing use of composite materials in engineering projects. In general, composites panels are very light structures that present high stiffness and strength. However, due to their slenderness, buckling is one of the main concern during their design. The boundary element method (BEM) has provided a powerful solution to the field of plate buckling. Syngellakis and Elzein [1] presented a boundary element solution of the plate buckling based on Kirchhoff theory under any combination of loadings and support conditions. Nerantzaki and Katsidelakis [2] developed a boundary element method for buckling analysis of plates with variable thickness. Elastic buckling analysis of plates using boundary elements can also be found in [3]. Buckling analysis of shear deformable isotropic plates was presented in [4]. To the best of author’s knowledge, the only work that presents a boundary element formulation applied to non-isotropic plates is due to [5] who presented an orthotropic formulation with a domain discretization. In this paper, a boundary element formulation for the stability analysis of general anisotropic plates with no domain discretization is presented. Classical plate bending and plane elasticity formulations are used and the domain integrals due to body forces are transformed into boundary integrals using the radial integration method. Numerical results are presented to assess the accuracy of the method. Buckling coefficients computed using the proposed formulation are in good agreement with results available in literature.

Boundary integral equations In the absence of body forces, the governing equation of the anisotropic thin plate buckling is given by:

Nij,j = 0,

(1)

456


D11 u3,1111 + 4D16 u3,1112 + 2(D12 + D66 )u3,1122 + 4D26 u3,1222 + D22 u3,2222 = Nij u3,ij ,

(2)

where i, j, k = 1, 2; uk is the displacement in directions x1 and x2 , u3 stands for the displacement in the normal direction of the plate surface; Nij are the in-plane stress components, D11 , D22 , D66 , D12 , D16 , and D26 are the anisotropic thin plate stiffness constants. The boundary integral equation for in-plane displacements, obtained by applying reciprocity and Green theorems at equation (1), is given by [6]:

cij uj (Q) +

Γ

t∗ik (Q, P )uk (P )dΓ(P ) =

Γ

u∗ik (Q, P )tk (P )dΓ(P )

(3)

where ti = Nij nj is the traction in the boundary of the plate in the plane x1 − x2 , and nj is the normal at the boundary point; P is the field point; Q is the source point; and asterisks denote fundamental solutions. The anisotropic plane elasticity fundamental solutions can be found, for example, in [7]. The constant cij is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. The in-plane stress resultants at a point Q ∈ Ω are written as:

cik Nkj (Q) +

Γ

∗ Sikj (Q, P )uk (P )dΓ(P ) =

Γ

∗ Dijk (Q, P )tk (P )dΓ(P )

(4)

where Dikj and Sikj are linear combinations of the plane-elasticity fundamental solutions. The integral equation for the plate buckling formulation, obtained by applying reciprocity and Green theorems at equation (2), is given by:

Ku3 (Q) +

Γ

Nc

=

i=1

Rci (P )u∗3ci (Q, P ) +

+λ

Vn∗ (Q, P )w(P ) − m∗n (Q, P )

Ω

Nij u∗3,ij dΩ +

Γ

Vn (P )u∗3 (Q, P ) − mn (P )

Γ

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

∂u∗3 (Q, P ) dΓ(P ) ∂n

ti u∗3 u3,i − ti u3 u∗3,i dΓ ,

(5)

where ∂() ∂n is the derivative in the direction of the outward vector n that is normal to the boundary Γ; mn and Vn are, respectively, the normal bending moment and the Kirchhoff equivalent shear force on the boundary Γ; Rc is the thin-plate reaction of corners; u∗3ci is the transverse displacement of corners; λ is the critical load factor; the constant K is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. As in the previous equation, an asterisk denotes a fundamental solution. Fundamental solutions for anisotropic thin plates can be found, for example, in [8]. A second integral equation is necessary in order to obtain the thin plate buckling boundary element formulation. This equation is given by:

K

∂u3 (Q) + ∂m =

Nc i=1

Γ

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

Rci (P )

+λ

∂u∗3ci (Q, P ) + ∂m

Γ

Vn (P )

∂ 2 u∗3 ∂u∗3 (Q, P ) − mn (P ) (Q, P ) dΓ(P ) ∂m ∂n∂m

∂u∗3,ij ∂u∗3,i ∂u3,i u3 Nij ti u∗3 dΩ + − ti u3 ∂m ∂m ∂m Ω Γ

dΓ ,

(6)


457

∂() is the derivative in the direction of the outward vector m that is normal to the boundary where ∂m Γ at the source point Q. As can be seen in equations (5) and (6), domain integrals arise in the formulation owing to the contribution of in-plane stresses to the out of plane direction. In order to transform these integrals into boundary integrals, consider that a body force b is approximated over the domain Ω as a sum of M products between approximation functions fm and unknown coefficients γm , that is: M

b(P ) ∼ =

m=1

γm fm .

(7)

The approximation function used in this work is: fm = 1 + R,

(8)

Equation (7) can be written in a matrix form, considering all boundary and domain source points, as: b = Fγ

(9)

γ = F−1 b

(10)

Thus, γ can be computed as:

Body forces of integral equations (5) and (6) depend on displacements. So, using equation (10) and following the procedure presented by Albuquerque et al. [9], domain integrals that come from these body forces can be transformed into boundary integrals.

Matrix Equations After the discretization of equations (5) and (6) into boundary elements and collocation of the source points in all boundary nodes, a linear system is generated. It is worth notice that the only loads considered in the linear buckling equations are that related to the in-plane stress Nij and tractions ti that are multiplied by the critical load factor λ. This means that all the known values of u3 , ∂u3 /∂n, Mn , Vn , wci , Rci (boundary conditions) are set to zero. Dividing the boundary into Γ1 and Γ2 (Figure 1), this linear system can be written as: 3 Γ1: u3 = ∂u ∂n = 0

Ω

Γ2: Vn = Mn=0

Figure 1: Domain with constrained and free degrees of freedom.

H11 H12 H21 H22

w1 w2

−

G11 G12 G21 G22

V1 V2

=λ

M11 M12 M21 M22

w1 w2

,

(11)

where Γ1 stands for stands for the part of the boundary where displacements and rotations are zero and Γ2 stands for the part of the boundary where bending moment and tractions are zero. Indices 1 and 2 stand for boundaries Γ1 and Γ2 , respectively. Matrices H, G, and M are influence matrices of the boundary element method due to integral terms of equations (5) and (6).

458


As w1 = 0 and V2 = 0, equation (11) can be written as: H12 w2 − G11 V1 = λM12 w2 , H22 w2 − G21 V1 = λM22 w2

(12)

ˆ 2, ˆ 2 = λMw Hw

(13)

ˆ = H22 − G21 G−1 H12 , H 11 ˆ = M22 − G21 G−1 M12 . M 11

(14)

or

ˆ and M ˆ are given by: where H

The matrix equation (13) can be rewritten as an eigen vector problem 1 w2 , λ

(15)

ˆ ˆ −1 M. A=H

(16)

Aw2 = where

Provided that A is non-symmetric, eigenvalues and eigenvectors of equation (15) can be found using standard numerical procedures for non symmetric matrices.

Numerical results The numerical results are presented in terms of the dimensionless parameter Kcr which is given by: Kcr =

Ncr a2 D22

(17)

where Ncr is the critical load (Ncr = λ× the applied load) and a is the edge length of the square plate. Consider a square graphite/epoxy plate under different boundary conditions. The thickness of the plate is h = 0.01 m. The material properties are: elastic moduli E1 = 181 GPa and E2 = 10.3 GPa, Poisson ratio ν12 = 0.28, and shear modulus G12 = 7.17 GPa. The mesh used has 28 quadratic discontinuous boundary elements of the same length (7 per edge) and 49 (7 × 7) uniformly distributed internal points. The plate is under uniformly uniaxial compression and the critical load parameter Kcr is computed considering all edges simply-supported (SSSS), all edges clamped (CCCC), and two edges clamped and two edges simply supported (CSCS). In the last case, the two edges where the load is applied are simply supported and the two remaining edges are clamped. The results are shown in Table 1 together with results obtained by [5] using a boundary element formulation with domain discretization and the analytical solution presented by [10]. As it can be seen, there is a good agreement between the results obtained in this work and those presented in literature. Critical buckling modes for each case are shown in figures 2, 3, and 4.

Conclusions This paper presented a boundary element formulation for the stability analysis of symmetric laminated composite plates where domain integrals are transformed into boundary integrals by the radial integration method. As the radial integration method doesn’t demand particular solutions, it is easier to implement than the dual reciprocity boundary element method. Results obtained with the proposed formulation are in good agreement with results presented in literature.


Figure 2: Critical buckling mode of cases 1, 3 and 5.

Figure 3: Critical buckling mode of case 2.

Figure 4: Critical buckling mode of cases 4 and 6.

459

460


Table 1: Critical load parameter Kcr for a graphite/epoxy plate with different boundary conditions. Case 1 2 3 4 5 6

Boundary conditions SSSS SSSS CCCC CCCC CSCS CSCS

Loadings N1 = 0 N2 = 0 N1 = 0 N2 = 0 N1 = 0 N2 = 0

This work 130.82 71.53 493.70 168.27 161.47 146.47

Reference [5] – 71.36 481.21 168.16 163.24 143.89

Reference [10] 129.78 69.46 – – 162.03 141.33

Acknowledgment The first author would like to thank the CNPq (The National Council for Scientific and Technological Development, Brazil), AFOSR (Air Force Office of Scientific Research, USA), and FAPESP (the State of S˜ ao Paulo Research Foundation, Brazil) for financial support for this work.

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] J. Purbolaksono and M. H. Aliabadi. Buckling analysis of shear deformable plates by boundary element method. International Journal for Numerical Methods in Engineering, 62:537–563, 2005. [5] G. Shi. Flexural vibration and buckling analysis of orthotropic plates by the boundary element method. J. of Solids and Structures, 26:1351–1370, 1990. [6] M. H. Aliabadi. Boundary element method, the application in solids and structures. John Wiley and Sons Ltd, New York, 2002. [7] P. Sollero and M. H. Aliabadi. Fracture mechanics analysis of anisotropic plates by the boundary element method. Int. J. of Fracture, 64:269–284, 1993. [8] 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. [9] E. L. Albuquerque, P. Sollero, and W. P. Paiva. The radial integration method applied to dynamic problems of anisotropic plates. Communications in Numerical Methods in Engineering, 23:805– 818, 2007. [10] S. G. Lekhnitskii. Anisotropic plates. Gordon and Breach, New York, 1968.


461

BEM model of mode I crack propagation along a weak interface applied to the interlaminar fracture toughness test of composites L. Távara, V. Mantiþ, E. Graciani, J. Cañas, F. París Grupo de Elasticidad y Resistencia de Materiales, Escuela Técnica Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, España [email protected], [email protected], [email protected], [email protected], [email protected] Keywords: composites, interlaminar fracture, weak interface, linear elastic-brittle law, BEM

Abstract. A numerical study of damage propagation in composite laminates is presented. Interlaminar fracture toughness (GIc) test of two unidirectional carbon fiber laminates bonded by an adhesive layer is studied. Displacement control is used in the numerical test simulation to ensure stable crack propagation. The adhesive layer is modelled in the 2D Boundary Element Method (BEM) code developed as a weak interface by means of a continuous distribution of springs governed by a linear elastic-brittle law. In this law, the normal stresses across the interface are proportional to the relative normal displacements (opening) up to a certain maximum stress value. It is shown that this approach provides a good representation of the actual adhesive behaviour. An important feature of the BEM approach developed is that the parameters governing the springs are independent of the boundary element mesh, i.e. distances between springs and element types used. This fact allows us to perform an easy mesh refinement if required. It is shown that the local properties of the numerical solution obtained near the crack tip agree with the predictions obtained with the weak interface theory. The present model permits the study of both crack propagation and crack initiation. An excellent agreement is observed between the load – displacement diagrams obtained in the BEM analysis and in the laboratory tests. The computational procedure developed can be used to estimate the maximum allowed load of a structure including similar adhesive bonded joints of laminates. Introduction Traditionally, the methods that simulated crack propagation were based on Linear Elastic Fracture Mechanics (LEFM) assuming the presence of a crack, which made difficult the study of damage and/or crack initiation occurring in the first step of fracture process. Recently, other models have been intensively developed as cohesive crack model which assumes hypotheses different to those adopted in LEFM avoiding the presence of a stress singularity at the crack tip. These models are suitable to study both crack initiation and crack propagation, and also to estimate the fracture energy and the maximum allowable load of a structure. In many practical situations, the behavior of adhesive joints can be described modeling the thin adhesive layer as a continuum spring distribution [1] with an appropriate stiffness parameter. This interface model is usually called weak interface or elastic interface [2,3]. In the present work linear elastic – brittle constitutive law of these springs is adopted in order to allow an easy modeling of crack propagation along a weak interface. In the present work the above described weak interface behavior has been implemented in a 2D BEM code [4,5], whose original version allowed modeling of plane elastic problems, including several linear elastic anisotropic solids with strong interfaces or contact zones between them. The new feature incorporated to this code is the incorporation of the possibility of defining weak interfaces between the elastic solids where required. Another feature of the code is that the equilibrium and compatibility conditions, along contact zones and strong or weak interfaces, are imposed using a weak formulation allowing an easy use of non-conforming discretizations [4,5,6]. The understanding of the adhesive layer behavior is very important in the quality evaluation of this kind of joints, and particularly in determining the parameters that characterize its resistance to fracture and failure. These parameters can then be used in design and quality control of the productive process. The quality of an adhesive joint between composite laminates is usually evaluated by an interlaminar fracture test, where an estimation of the critical interlaminar fracture energy (GIc) is obtained. An extensive experimental study and a numerical study by Finite Element Method of this test and of different adhesives were recently carried out by the present authors and their co-workers [7,8].

462


Weak interface According to Lenci and co-workers [2,3], a weak interface is considered as a model of a thin linear elastic adhesive layer between two surfaces. In the present wok, adhesive damage and/or rupture are modeled as a free separation of both surfaces. Thus, the springs that simulate an adhesive layer are governed by the following linear elastic-brittle law, shown also in Fig. 1:

V

kG

if

G Gc ,

and

V

0

if

G ! Gc

(1)

where ı is the normal stress in a spring, į is the relative opening of the extremes of the spring (separation between surfaces), k is a stiffness parameter, ıc and įc are, respectively, the critical normal stress and the critical relative normal displacement leading to the spring rupture. Fracture mechanics is indirectly involved through the area under the linear law line in Fig. 1 given by GIc value, G Ic 12 V cG c . ı ıc

GIc

įc

İ

Figure 1. Linear elastic-brittle law of a spring. According to the weak interface theory [2,3], interface tractions are bounded at the tip of a crack situated along a weak interface, whereas these tractions are singular (unbounded) at the tip of an interface crack situated along a perfect interface (called also strong interface, where no relative displacements of bonded surfaces are allowed). Thus, during crack growth along a weak interface these tractions are kept bounded. It appears that local normal tractions in the zone close to the interface crack tip follow the law [2]:

V # V 0 V 1[ >ln([ ) 1@

(2)

where V and V are constants and x = aȟ, where x is the distance from the crack tip to a point (in the bonded part of the interface) where these tractions are evaluated and a is a characteristic length, usually the crack length or semilength. Mode I weak interface implementation in the 2D BEM code Incremental formulation. The numerical solution of the non-linear problem formulated is based on a gradual application, by means of a load factor Ȝ, 0 Ȝ 1, of the loads and displacements imposed. The solution procedure is given by a series of lineal stages, “load steps”. At the beginning of each load step an actual adhesively bonded zone is defined, which defines the actual linear system of equations. By solving this system the corresponding elastic solution is obtained. This solution fulfills all the conditions of the weak interface formulation up to a certain maximum value of the load factor Ȝ associated to this load step. A further increment of the load factor leads to rupture of some springs. Thus, the solution of the problem will be divided into a number M (a priori unknown) of load steps where the values of the problem variables vary linearly:

I ( x , O ) O' mI ( x )

(3)

with Ȝm-1 Ȝ Ȝm, m = 1,..., M, and Ȝ0=0, and where I ( x ,O ) is the value of any problem variable at a point x after Ȝ fraction of load is applied, ' mI ( x ) being the variable value obtained in the solution of the linear system corresponding to the m-th load step.


463

464


This procedure can be repeated as many times as necessary to reach the equilibrium after the whole load is applied. Nevertheless, changing the conditions node by node can make the crack propagation to be very smooth (especially for fine meshes), in opposite to the experimental evidence for some industrial adhesives that show crack growing by small jumps. That is why for a specific case of adhesive, like that simulated in the present work, the end of a load step can be defined by a situation where a fixed number of consecutive nodes (number 20 is chosen in the present study) do not fulfill the condition (7). Laboratory test Test description. The test used in the aeronautical industry to evaluate the interlaminar fracture toughness in composite-composite joints is performed following AITM 1.0005 [9] and/or I+D-E 290 [10] standards. The specimen used is the Double Cantilever Beam (DCB) shown in Fig. 3(a). The DCB specimen is formed by two laminates joined by a thin adhesive layer. The laminates are processed according to EN 2565 standard, and the specimens are cut after the panel has been cured. The specimen is fixed to the grips of the universal testing machine through small tabs bonded to laminates as shown in Fig. 3(b). During crack propagation the load (P) and the displacement (d) of the wedge grips are continuously registered. w

t

P

0º

L = 250 ± 5 mm L1 = 25 ± 1 mm w = 25.0 ± 0.2 mm t = 3.0 ± 0.2 mm

d

L l1

Figure 3. (a) Scheme of the DCB specimen, (b) Test configuration. Adhesive type. In a study of experimental results obtained from GIc tests for different kinds of adhesive [7], it was observed that the adhesives FM 300K0.5 and EA 9695 K.05 present falls in the experimental load – displacement curve. This behavior was explained by the presence of a polyester support in these adhesives. Evaluation of the adhesive model parameters. As the parameters of the adhesive model adopted here are a priori unknown, they are adjusted by fitting the experimental and numerical load-displacement curves. Experimental results provide estimations of GIc values, the crack length for some load values and the load – displacement curves. With these data, and using equation (9), a first estimation of the critical displacement (įc) and the slope k is obtained. After a comparison between the numerical and experimental results these values can be adjusted better.

G Ic

kG c2 2

(9)

Other trial and error methods to obtain an estimation of Kadh value are presented in [7], the estimated values of Kadh obtained therein and of k obtained here are in a good agreement according to (4). Numerical Results

In the present numerical study, a 2D model has been solved using the BEM code described above, where the plane strain and linear elastic behavior hypotheses have been assumed. The laminate considered is a 8552/AS4 carbon fiber – epoxy composite (0º plies), with the following orthotropic properties: Ex=135GPa, Ey=10GPa, Ez=10GPa, Gxy=5GPa, Gxz=5GPa, Qxy=0.3, Qyz=0.4 and Qxz=0.3. The adhesive used is EA 9695


465

K.05, an epoxy adhesive with a polyester support. The estimated properties of the adhesive spring model are: k=1514GPa/m and Vc =1514MPa. A maximum displacement of 25 mm was progressively applied in the direction normal to the specimen boundary at a 15 mm distance from the specimen extreme where the initial crack is situated. The normal stresses along the bonded zone obtained in the last load step, are shown in Fig. 4(a). The initial longitude of the adhesive layer 225 mm is discretized by 468 or 936 springs placed between the nodes of the conforming boundary element meshes on A and B sides of the weak interface. 1.8 1.6

468 nodes 936 nodes

Normal stress (MPa)

1.4 1.2

ı/ıc 1

Y a

0.8

1

X

P

0.8

0.6

0.6 0.4

0.4

0.2 0

0.2

-0.2 -0.4 0

0.025 0.05 Distance to the crack tip (m)

0.075

0.01

0.02

0.03

0.04

0.05

0.06

0.07

ȟ

Figure 4. (a) Normal stresses near the crack tip, (b) Fitting of the normalized local stress solution by an analytic expression (x = aȟ) (10). It is noteworthy that the local stress solution near the crack tip agrees very well with the predictions of the weak interface theory (2). In Fig. 4(b), the normalized stresses ı/ıc represented as a function of ȟ (the initial adhesive layer being modeled by 468 springs) are compared with the curve of expression (10), obtained from (2) by applying the least square method.

V 1.00658 .89711[ ln[ . Vc

(10)

Comparison between the experimental and numerical load - displacement diagrams

As can be observed in Fig. 5, the numerical results obtained provide a good approximation of the experimental results. Therefore, the use of the weak interface formulation seems to be a promising approach to model composite adhesive joints. 350

numerical experimental

300

Load P (N)

250 200 150 100 50 0 0.000

0.010

0.020 0.030 Displacement d (m)

0.040

0.050

Figure 5. Comparison between the experimental and numerical load - displacement diagrams, and a detail of the polyester support of the adhesive used.

466


Conclusions

As shown by the numerical results presented, the weak interface formulation modeled by a spring distribution, correctly describes the behavior of adhesive joints in the aeronautical industry. An analytic expression for the local solution of normal tractions at the crack tip, deduced in the weak interface theory, has been successfully compared with the present numerical solution. Noteworthy is the bounded character of stresses along the weak interface, the maximum value of stresses being achieved at the crack tip. The spring constitutive law introduced and included in the incremental algorithm of the BEM code has the advantage of being independent of the number of springs used in the interface. It has been proved that the real behavior of an adhesive layer with a polyester support that joins two unidirectional laminates can be approximated very well by means of BEM and a distribution of springs which follow a linear elastic-brittle constitutive law, by adjusting the parameters of the discrete model (k, Vc, and the number springs that breaks in a load step). This fact will allow predicting the real behavior of structures that include similar adhesive joints by the model developed here. From laboratory test and fractographic analysis it has been concluded that the jumps appearing in the experimental load – displacement curve are caused by the polyester support of the adhesive resin. The results obtained in this work can also be considered as a starting point for a study of adhesives including different kinds of adhesive support. Acknowledgements

The authors acknowledge the support of the Junta de Andalucía (Projects of Excellence TEP-1207 and TEP 02045) and also the support by the Spanish Ministry of Education and Science through Projects TRA200506764 and TRA2006-08077. References

[1] Erdogan F., Fracture mechanics of interfaces, In: Damage and Failure of Interfaces, Rossmanith ed., Balkema, Rotterdam, (1997). [2] Lenci S., Analysis of a crack at a weak interface, International Journal of Fracture, 108: 275-290, (2001). [3] Geymonat G., Krasucki F., Lenci S., Mathematical analysis of a bonded joint with a soft thin adhesive, Mathematics and Mechanics of Solids, 4: 201-225, (1999). [4] Graciani E., Formulación e implementación del método de los elementos de contorno para problemas axisimétricos con contacto. Aplicación a la caracterización de la interfase fibra matriz en materiales compuestos, Tesis Doctoral, ETSI – Universidad de Sevilla, (2006). [5] Graciani, E., Mantic, V., París, F. y Blázquez, A., Weak formulation of axi-symmetric frictionless contact problems with boundary elements. Application to interface cracks, Computer and Structures, 83: 836-855, (2005). [6] Blázquez A., París F., Mantiþ V., BEM solution of two-dimensional contact problems by weak application of contact conditions with nonconforming discretizations, International Journal of Solids and Structures, 35: 3259-3278, (1998). [7] Jiménez M. E., Modelización del ensayo de tenacidad a la fractura interlaminar en materiales compuestos, Proyecto de Fin de Carrera, ETSI – Universidad de Sevilla, (2006). [8]Jiménez M. E., Cañas J., Mantiþ V., Ortiz J. E., Estudio numérico y experimental del ensayo de tenacidad a fractura interlaminar de uniones adhesivas composite-composite, MATCOMP ‘07, Valladolid, (2007). [9] AITM 1-0005, Determination of interlaminar fracture toughness energy. Mode I. Issue 2, AIRBUS, (1994). [10] I+D-E-290, Ensayo de tenacidad a la fractura interlaminar sobre estratificados de fibra de carbono, CASA, (1988).