Fast solution of BEM systems for elasticity problems using wavelet transforms

This paper describes a fast approach for solving elasticity problems using boundary element method. The key idea is based on compressing matrices, to sparsify them, using fast wavelet transforms. Compactly supported orthogonal wavelets have been used to compress matrices arising from applying BEM to practical engineering problems, which can be added as a black box to existing BEM codes. Dense and fully populated matrices arising from BEM have been changed to semi-banded sparse matrices by simultanesously applying wavelet and permutation matrices. Numerical results including that of a precise study on thresholding parameter, of solution accuracy for displacements and stresses and of saving in computer time and memory are presented. The results show that the proposed method is efficient for complex problems.

[1]  Kazuhisa Abe,et al.  A practical determination strategy of optimal threshold parameter for matrix compression in wavelet BEM , 2003 .

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

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

[4]  K. Chen,et al.  DISCRETE WAVELET TRANSFORMS ACCELERATED SPARSE PRECONDITIONERS FOR DENSE BOUNDARY ELEMENT SYSTEMS , 1999 .

[5]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[6]  Carlos Magluta,et al.  On the block wavelet transform applied to the boundary element method , 2004 .

[7]  Christian Lage,et al.  Wavelet Galerkin algorithms for boundary integral equations , 1997 .

[8]  Stephen A. Vavasis,et al.  Fast Wavelet Transforms for Matrices Arising from Boundary Element Methods , 1994 .

[9]  Webe João Mansur,et al.  Solution of BEM systems of equations via iterative techniques , 1992 .

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

[11]  Matjaž Hriberšek,et al.  The wavelet transform for BEM computational fluid dynamics , 2004 .

[12]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[13]  M. Aliabadi,et al.  The Boundary Element Method , 2002 .

[14]  Carlos Magluta,et al.  Fast solution of problems with multiple load cases by using wavelet‐compressed boundary element matrices , 2003 .

[15]  Wolfgang Dahmen,et al.  Multiscale methods for the solution of the Helmholtz and Laplace equations , 1997 .

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

[17]  Kevin Amaratunga.  A wavelet-based approach for compressing kernel data in large-scale simulations of 3D integral problems , 2000, Comput. Sci. Eng..

[18]  Tomás F. Pena,et al.  Parallel iterative solvers involving fast wavelet transforms for the solution of BEM systems , 2002 .

[19]  Fritz Keinert,et al.  Wavelets and Multiwavelets , 2003 .

[20]  John R. Williams,et al.  Introduction to wavelets in engineering , 1994 .

[21]  J.C.F. Telles,et al.  Iterative solution of bem equations by GMRES algorithm , 1992 .