Invited Paper A New BEM Formulation Of Acoustic Scattering Problems

Abstract

Kussmaul's (1969) formulation of acoustic scattering problems involves the superposition of a double-layer distribution and a simple-layer distribution on the boundary.This ensured the existence of a unique solution for all wave numbers of the incoming wave. However,the boundary integral equation involves a highly singular kernel which proves difficult to treat numerically. An alternative approach is to place the double-layer distribution on an interior boundary similar (similarly situated) to the given boundary.This avoids the singularity problems and enables excellent numerical results to be achieved.Some numerical results are presented. INTRODUCTION Problems of hard acoustic scattering may be formulated in terms of boundary integral equations.Direct formulations are based upon the Helmholtz formula;indirect formulations utilise layer potentials.This is a Neumann boundary-value problem which has a unique solution subject to regular behaviour at infinity.However the relevant classical integral equations,whether direct or indirect,fail at wave numbers corresponding with the eigenfrequencies of the interior Dirichlet problem.In this paper we show that a relatively simple, nonsingular,supplement to the classical indirect integral equation ensures a a unique solution at all wave numbers. An immediate consequence is that meaningful numerical solutions can be achieved,for a wide range of wavenumbers,by computationally effective boundary-element methods (BEMs). Some results for scattering by a hard sphere are provided.For completeness we mention some recent papers presenting alternative resolutions of the difficulty. Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 282 Boundary Elements Rego Silva et al [1] Numerical Implementation of a Hypersingular BEM formulation for Acoustic Radiation Problems,following the approach of Burton and Miller [2].Here the hypersingular integral is evaluated numerically by an algorithm of Guiggiani et al [3] building essentially upon Hadamard's finite-part integration. Liu and Rizzo [4] A weakly-singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems,again following the approach of Burton and Miller. However here the hypersingular integral is weakened by a regularisation technique,which enables it to be evaluated numerically by straightforward methods. Wu et al [8] Vectorization and parallelization of the acoustic boundary element code BEMAP on the IBM ES/3090 VF,exploits the supercomputing facilities to improve the CHIEF method originated by Schenck [15]. Kirkup and Henwood [5] Computational Solution of Acoustic Radiation Problems by Kussmaul's Boundary Element Method,adopting Kussmaul's representation [6] of the wave function as the superposition of a simple-layer and a double-layer potential generated from sources on the boundary.This formulation also involves a hypersingular integral,which is evaluated numerically by the method of Terai [7],an early exploitation of Hadamard's finite part integration. Kussmaul's approach may be modified by locating the double-layer sources on an interior boundary,similar and similarly situated to the given boundary.This eliminates the hypersingular integral and at the same time provides a unique solution at all wave numbers. Brief details are given in the following section. KUSSMAUL FORMULATION Any solution, ,of the Helmholtz equation existing on the infinite domain B exterior to a closed (simply connected) surface dB (fig. 1) of order 0(r~') as /•co can generally be represented by a simple-layer potential dq ; q € dB , p € B+dB (1) dB where o(q) is the source density at q and dq is the area element at q ; also Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 283 /-',-'» ; r=|f-g| (2) where k is the wave number associated with , defined by