A high frequency boundary element method for scattering by a class of nonconvex obstacles

Abstract

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

“…For a smooth boundary, as is considered herein, (22) and (23) can be shown to be non-singular, so can be computed using standard quadrature techniques. Equation (24) includes a stronger singularity, but in practice the matrix will be evaluated using the following statement [41,42], which only contains a weak singularity: (28) This leaves only weak singularities in (21) and (28); these were regularised using the Sato transform [58] of order 5, as has been shown to be optimal for weakly-singular oscillatory kernels [59].…”

“…a circular cylinder (right), and algorithms that compute edge diffraction by truncating the infinite wedge canonical problem (left) also exist [21,22]. Some HNA-BEM algorithms can be considered to follow a similar principal in that they use different oscillatory functions on different boundary sections [13,14,23].…”

“…The spatial windowing that they enable is an essential ingredient of both POU-BEM [12], where it means a set of oscillatory functions that don't exactly match the incident wave or its scattering may still approximate it, and HNA-BEM, where differing oscillatory functions will be defined on sections of the boundary that are 'illuminated' differently [13,14,23].…”

“…Whether this is possible and how best to do it is currently unknown; a first step might be to join up the two problems studied herein to make the torpedo problem in [40]. Following that, consideration of how to model wedge diffraction in this framework, as depicted in Figure 1, would be a high priority, since it would enable the modelling of polygons, as has been successfully done with HNA-BEM [13,23]. The algorithm would ideally also be taken into three-dimensions.…”

A High Frequency $hp$ Boundary Element Method for Scattering by Convex Polygons

“…Further, much of the analysis, in particular our regularity and best approximation results should be extendable to that case, and our hp algorithms are also potentially adaptable to curvilinear polygons (see [16,29] for h-version results in these directions). More challenging is any extension to nonconvex polygons: see [14,13]. With possible extensions to non-convex scatterers in mind, some of the results in the current paper, in §2 and §4 in particular, are stated and proved in more generality than is required for the convex case.…”