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Background

In contrast with domain methods such as Finite Element Method (FEM) or Finite Differential Method (FDM), Boundary Integral Equation Method (BIEM), also called Boundary Element Method (BEM), discretises only the boundary of the domain. Because of this reduction of the dimensionality BIEM was expected to be advantageous in large-scale problems. However, the application of this method has so far been limited to relatively small problems. This was because coefficient matrices in BIEM are full, due to which both the operation count and the memory requirements for the matrix equation buildup are of the order of O(N2), where N is the number of unknowns. The operation count even increases to O(N3) as one attempts to solve the matrix equations with conventional direct solvers such as Crout's method and Gaussian elimination. In particular, the full matrix property leads to a serious exhaustion of the memory of a computer and is an obstacle for applications of BIEM to large-scale problems. On the other hand, coefficient matrices in domain methods are banded and both computational complexity for matrix buildup and memory requirements are O(N). This is why applications of domain methods to large-scale problems are relatively easy. However, the appearance of Fast Multipole Method (FMM), also referred to as Fast Multipole Algorithm (FMA), drastically changed the circumstances around BIEM. The use of FMM in conjunction with iterative solvers such as Conjugate Gradient (CG) method and Generalised Minimal RESidual (GMRES) method has been shown to reduce the memory requirements to O(N) and the operation count to $O(N(\log
N)^{\alpha})$, where $\alpha$ is a small non-negative number. Thus FMM enables us to apply BIEM to large-scale problems. Thanks to FMM, BIEM is no longer what it was.

FMM was initially introduced by Rokhlin[69] as a fast solution method for integral equations for two-dimensional Laplace's equation and then developed by Greengard[33,35] as a fast evaluation method for the pairwise force calculation in multibody problems with Coulombic potential. FMM has been applied to problems in various fields such as BIEM and Molecular Dynamics (MD).

The author has been studying this topic in order to apply Fast Multipole Boundary Integral Equation Method (FM-BIEM) to practical problems in fracture mechanics and earthquake engineering. In this thesis the author discusses applications of FM-BIEM to various fundamental boundary value problems in three dimensions and show its efficiencies so as to make the first step toward practical applications.


next up previous contents
Next: Organization of thesis Up: Introduction Previous: Introduction
Ken-ichi Yoshida
2001-07-28