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Introduction

In spite of the advantage of reduction of dimensionality BIEM has been applied to relatively small problems so far, because the resulting matrix is dense. Indeed, this property leads to a serious exhaustion of the memory of a computer, since the memory requirement of BIEM is O(N2), where N is the number of unknowns. When one attempts to solve matrix equations with direct methods such as Crout's method, Gaussian elimination, etc. the required computational cost increases to O(N3). But the appearance of FMM changed the circumstances drastically. FMM reduces the computational cost to $O(N^{1+\alpha}(\log N)^{\beta})$ and the memory requirements to O(N), where $\alpha$ and $\beta$ are nonnegative numbers. With the help of FMM, BIEM can be applied to large scale problems.

FMM was initially investigated by Rokhlin[1] as a fast solver for integral equations for the two-dimensional Laplace equation, and then was applied to multibody problems with Coulombic potential by Greengard[2]. Since then FMM has been developed as a fast solution method for large scale problems. An application of FMM to BIEM has been investigated by several authors: e.g., by Nishimura et al.[3] for crack problems for the three-dimensional Laplace equation, by Fu et al.[4], Fukui et al.[5] and Takahashi et al.[6] for ordinary problems for three-dimensional elastostatics, by Yoshida et al.[7,8] for crack problems in three-dimensional elastostatics and by Fujiwara[9] and by Yoshida et al.[10] for three-dimensional elastodynamics.

In FMM the computational cost for the M2L translation dominates the performance especially in three-dimensional problems or problems dealing with the Helmholtz equation. In view of this Rokhlin introduced the diagonal form[11,12] so as to reduce the computational cost for the M2L translation. Recently the number of researches using diagonal forms is increasing. The use of the diagonal form has been investigated by several authors: Koc and Chew[13], Epton and Dembart[14], for example. However, the diagonal form proposed by Rokhlin is known to have numerical instabilities in dealing with the Laplace equation[15] or low frequency problems for the Helmholtz equation[16]. In order to overcome these problems Hrycak and Rokhlin[17] proposed a new FMM for the two-dimensional Laplace equation, Greengard and Rokhlin[18] and Cheng et al.[19] for the three-dimensional Laplace equation, and Greengard et al.[20] for the three-dimensional Helmholtz equation. Nishimura et al.[21] applied the new FMM to crack problems for the two-dimensional Laplace equation. An application of the new FMM to three-dimensional elastostatics is mentioned in Fu et al.[22] but they present only an integral representation for the fundamental solution of anisotropic elastostatics without FMM formulation or numerical examples. In this paper we discuss an application of the new FMM to three dimensional elastostatic crack problems. The results show that the new FMM is more efficient than the original FMM.


next up previous
Next: Formulation for integral equation Up: Application of New Fast Previous: Application of New Fast
Ken-ichi Yoshida
2001-03-26