Implicit boundary integral methods for the Helmholtz equation in exterior domains

Chieh Chen,Richard Tsai

doi:10.1186/s40687-017-0108-y

Abstract

We propose a new algorithm for solving Helmholtz equations in exterior domains with implicitly represented boundaries. The algorithm not only combines the advantages of implicit surface representation and the boundary integral method, but also provides a new way to compute a class of the so-called hypersingular integrals. The keys to the proposed algorithm are the derivation of the volume integrals which are equivalent to any given integrals on smooth closed hypersurfaces, and the ability to approximate the natural limit of the singular integrals via seamless extrapolation. We present numerical results for both two- and three-dimensional scattering problems at near resonant frequencies as well as with non-convex scattering surfaces.

Highlights

Let Γ be a closed and compact C2 hypersurface that separates Rm, m = 2, 3, into a connected and bounded open region Ω and its complement
We consider the solution of the following Neumann boundary value problem for the Helmholtz equation:
We focus on a boundary integral formulation that is derived from the

Summary

Background

The novelty of the proposed algorithm involves (i) the assumption that Γ := ∂Ω is defined by the zero level set of a signed distance function or the closest point mapping to it and (ii) a new way of computing surface integrals of the so-called hypersingular kernels. The computation of the signed distance functions and the closest point mappings are considered standard routines in the level set methods [17] and can be done to high-order accuracy in many different ways, e.g., [1,5,21,24], where by extending the interface coordinates as constants along interface normals, PΓ can be computed to fourth-order in the grid spacing. We follow the approach of [11] which replaces the values of KΓ (x, y) by a function K (PΓ (x), r0), where r0 corresponds to a small regularization parameter

Regularization of the double-layer potentials

Extrapolative integrals

Scattering in three dimensions by a “Bean” shape

Summary