Application of Fast Multi-pole Boundary Element Method to 2-D Acoustic Scattering Problem

摘要

The Fast Multi-pole Method (FMM) is a very effective approach to accelerate the numerical solutions of the boundary element method (BEM) for the problems with large scale computation. This paper discusses an application of the FMM to two-dimensional boundary integral equation method for acoustic scattering problem. We seek the solution of Helmholtz equation Δu+k^2u=0 in the form of a combined single- and double-layer potential. The boundary integral equation is discretized with Nystr?m method. It is obvious that the kernel of integral operator is unsymmetrical. If the resulting linear system is solved by the conjugate gradient method of unsymmetrical linear system, both the products of matrix A with vector x and A^T with x should be repeatedly evaluated. In this paper, we construct the hierarchical cell structures of FMM with two different methods, and the multi-pole expansion, local expansion and translations of the coefficients are given for the second integral operator A and its conjugate operator A^T. The boundary integral equation is solved by FMM. The numerical results show that FMM is more efficient than direct computation approach.