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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
in
problems with unknowns, while that for the original FMM is
, 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 ,
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
.

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