Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

Abstract The solution of three-dimensional elastostatic problems using the Symmetric Galerkin Boundary Element Method (SGBEM) gives rise to fully populated (albeit symmetric) matrix equations, entailing high solution times for large models. This paper is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for elastic solid with cracks. Arbitrary geometries and boundary conditions may be considered. Numerical results on test problems involving a cube, single or multiple cracks in an unbounded medium, and a cracked cylindrical solid are presented. BEM models involving up to 10 6 BEM unknowns are considered, and the desirable predicted trends of the elastostatic FM-SGBEM, such as a O ( N ) complexity per iteration, are verified.

[1]  Tetsuya Sakuma,et al.  A TECHNIQUE FOR PLANE-SYMMETRIC SOUND FIELD ANALYSIS IN THE FAST MULTIPOLE BOUNDARY ELEMENT METHOD , 2005 .

[2]  Weng Cho Chew,et al.  Fast multipole method as an efficient solver for 2D elastic wave surface integral equations , 1997 .

[3]  J. Nédélec,et al.  Integral equations with non integrable kernels , 1982 .

[4]  Marc Bonnet,et al.  Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity , 1995 .

[5]  Weng Cho Chew,et al.  Fast integral equation solvers in computational electromagnetics of complex structures , 2003 .

[6]  Toru Takahashi,et al.  A fast BIEM for three-dimensional elastodynamics in time domain☆ , 2003 .

[7]  I. N. Sneddon,et al.  Crack Problems in the Classical Theory of Elasticity , 1969 .

[8]  Lothar Gaul,et al.  A comparison of FE–BE coupling schemes for large‐scale problems with fluid–structure interaction , 2009 .

[9]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.

[10]  Kenichi Yoshida,et al.  Applications of Fast Multipole Method to Boundary Integral Equation Method , 2001 .

[11]  Yi-Shao Lai,et al.  Fast boundary element method for three-dimensional solids containing many cracks , 2003 .

[12]  Marc Bonnet,et al.  Fast multipole method applied to elastostatic BEM-FEM coupling , 2005 .

[13]  Stefan Kurz,et al.  The adaptive cross-approximation technique for the 3D boundary-element method , 2002 .

[14]  Stéphanie Chaillat,et al.  A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain , 2008 .

[15]  M. Aliabadi,et al.  Dual boundary element method for three-dimensional fracture mechanics analysis , 1992 .

[16]  J. H. Kane,et al.  Symmetric Galerkin fracture analysis , 1995 .

[17]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[18]  M. Aliabadi.  Boundary Element Formulations in Fracture Mechanics , 1997 .

[19]  N. Nishimura.  Fast multipole accelerated boundary integral equation methods , 2002 .

[20]  Huy Duong Bui,et al.  An integral equations method for solving the problem of a plane crack of arbitrary shape , 1977 .

[21]  Gernot Beer,et al.  The Boundary Element Method with Programming: For Engineers and Scientists , 2008 .

[22]  D. Zorin,et al.  A kernel-independent adaptive fast multipole algorithm in two and three dimensions , 2004 .

[23]  R. Barsoum.  On the use of isoparametric finite elements in linear fracture mechanics , 1976 .

[24]  Naoshi Nishimura,et al.  A fast multipole boundary integral equation method for crack problems in 3D , 1999 .

[25]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[26]  Attilio Frangi,et al.  3D fracture analysis by the symmetric Galerkin BEM , 2002 .

[27]  M. Bonnet.  Stability of crack fronts under Griffith criterion: a computational approach using integral equations and domain derivatives of potential energy , 1999 .

[28]  Stefano Miccoli,et al.  A galerkin symmetric boundary‐element method in elasticity: Formulation and implementation , 1992 .

[29]  Kenichi Yoshida,et al.  Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D , 2001 .

[30]  J. E. Gómez,et al.  A multipole direct and indirect BEM for 2D cavity flow at low Reynolds number , 1997 .

[31]  T. A. Cruse,et al.  Numerical solutions in three dimensional elastostatics , 1969 .

[32]  Frank J. Rizzo,et al.  An integral equation approach to boundary value problems of classical elastostatics , 1967 .

[33]  M. Bonnet.  Boundary Integral Equation Methods for Solids and Fluids , 1999 .

[34]  Yijun Liu,et al.  Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems , 2009 .

[35]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .

[36]  Guillaume Sylvand.  La méthode multipôle rapide en électromagnétisme. Performances, parallélisation, applications , 2002 .

[37]  Leonard J. Gray,et al.  Symmetric Galerkin boundary integral formulation for interface and multi-zone problems , 1997 .

[38]  G. Maier,et al.  Symmetric Galerkin Boundary Element Methods , 1998 .

[39]  S. Li,et al.  Symmetric weak-form integral equation method for three-dimensional fracture analysis , 1998 .

[40]  T. Cruse.  Boundary Element Analysis in Computational Fracture Mechanics , 1988 .

[41]  V. Rokhlin.  Rapid solution of integral equations of classical potential theory , 1985 .

[42]  Attilio Frangi,et al.  Multipole BEM for the evaluation of damping forces on MEMS , 2005 .

[43]  Johannes Tausch.  Sparse BEM for potential theory and Stokes flow using variable order wavelets , 2003 .

[44]  Olaf Steinbach,et al.  The fast multipole method for the symmetric boundary integral formulation , 2006 .

[45]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .