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Next: Boundary Integral Equation Up: Application of a diagonal Previous: Abstract

Introduction

The Fast Multipole Method (FMM), originally proposed by Rokhlin[1] as a fast solver of boundary integral equations in two dimensional Laplace's equation, found applications in various fields of science and technology in these fifteen years. To the best of our knowledge, however, applications of FMM to boundary integral equations in three dimensional elastodynamics are found only in Fujiwara[2] and Yoshida et al.[3] Fujiwara uses diagonal forms (high frequency FMM. See Rokhlin[4]) in his FMM formulation which requires 6 moments. On the other hand, Yoshida et al. use the original FMM with the Wigner-3j symbols (low frequency FMM) and a 4 moment FMM formulation. The diagonal form FMM is based on a plane wave representation of the fundamental solution, while the original (low frequency) FMM uses a series expansion of the fundamental solution. In high frequency problems the complexity of the Fast Multipole Boundary Integral Equation Method (FM-BIEM) for Helmholtz' equation with diagonal forms is known to scale as $ O(N\log N)$ in problems with $ N$ unknowns, while that for the original FMM is $ O(N^2)$, since their translation formulae including the Wigner-3j symbols are not diagonal. Therefore, the FM-BIEM with diagonal forms is considered to be more efficient than that with Wigner-3j symbols in high frequency problems. In low frequency problems, however, the complexity of the FMM with the Wigner-3j symbols reduces to $ O(N)$, while diagonal forms are known to lead to numerical instabilities in such cases (See Dembart and Yip[5] for the Helmholtz case). Hence, these formulations are complementary and both of them are considered to be necessary in the development of fast methods for elastodynamics. The purpose of this paper is to investigate the improvement of the performance of the high frequency FMM brought about by the 4 moment formulation in the analysis of three dimensional scattering of elastic waves by cracks. We assume that the wave field is time-harmonic and the time factor is $ e^{-i\omega
t}$.


next up previous
Next: Boundary Integral Equation Up: Application of a diagonal Previous: Abstract
2001-12-14