January 25 - 27, 2007

Fast multipole method formulation of the boundary element method for 3D elastodynamics in frequency-domain

Stéphanie Chaillat (Ecole Polytechnique)

In different areas where the BEM is used (electromagnetism, acoustics,...), considerable speedup of solution time and decrease of memory requirements have been achieved through the development, over the last decade, of the Fast Multipole Method (FMM). The goal of the FMM is to speed up the matrix-vector product computation. This is achieved by (i) using a multipole expansion of the relevant Green's tensor, which (unlike in the standard BEM) allows to reuse element integrals for all collocation points, and (ii) defining a (recursive, multi-level) partition of the region of space enclosing the domain boundary of interest into cubic cells, allowing to optimally cluster influence computations according to the ratio between cluster size and distances between two such clusters. Moreover, the governing matrix is never explicitly formed. The FMM-accelerated BEM therefore achieves substantial savings in both CPU time and memory: O(N logN) instead of O(N2) for the classical BEM.

In this work, the FMM is extended to the 3-D frequency-domain elastodynamics and applied to the computation of sit effects in 3-D. This communication is organized as follows. First, the main features of the elastodynamic FMM-BEM formulation are presented. Then, numerical efficiency and accuracy, after the determination of the optimal parameters of the method, are assessed on the basis of numerical results obtained for problems having known solutions (performed on a single PC computer for problem sizes of up to N = O(106)). In particular, numerical results are in agreement with the expected theoretical complexity of the FMM-accelerated elastodynamic BEM. Finally, the present FMM-BEM is demonstrated on seismology-oriented examples.

This is joint work with M. Bonnet and J.F. Semblat.

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