Integral Equation Method via Domain Decomposition and Collocation for Scattering Problems

Abstract

In this paper an exterior domain decomposition (DD) method based on the boundary element (BE) formulation for the solutions of two or three-dimensional time-harmonic scattering problems in acoustic media is described. It is known that the requirement of large memory and intensive computation has been one of the major obstacles for solving large scale high-frequency acoustic systems using the traditional nonlocal BE formulations due to the fully populated resultant matrix generated from the BE discretization. The essence of this study is to decouple, through DD of the problem-defined exterior region, the original problem into arbitrary subproblems with data sharing only at the interfaces. By decomposing the exterior infinite domain into appropriate number of infinite subdomains, this method not only ensures the validity of the formulation for all frequencies but also leads to a diagonalized, blockwise-banded system of discretized equations, for which the solution requires only O(N) multiplications, where N is the number of unknowns on the scatterer surface. The size of an individual submatrix that is associated with a subdomain may be determined by the user, and may be selected such that the restriction due to the memory limitation of a given computer may be accommodated. In addition, the method may suit for parallel processing since the data associated with each subdomain (impedance matrices, load vectors, etc.) may be generated in parallel, and the communication needed will be only for the interface values. Most significantly, unlike the existing boundary integral-based formulations valid for all frequencies, our method avoids the use of both the hypersingular operators and the double integrals, therefore reducing the computational effort. Numerical experiments have been conducted for rigid cylindrical scatterers subjected to a plane incident wave. The results have demonstrated the accuracy of the method for wave numbers ranging from 0 to 30, both directly on the scatterer and in the far-field, and have confirmed that the procedure is valid for critical frequencies.