Regularized combined field integral equations for acoustic transmission problems

TL;DR

This paper introduces Generalized Combined Source Integral Equations (GCSIE), a new class of well-conditioned boundary integral equations for acoustic scattering problems involving homogeneous penetrable scatterers, improving solution stability.

Contribution

The paper develops GCSIE, which use regularizing operators approximating admittance operators, resulting in second-kind integral equations for smooth interfaces in 2D and 3D.

Findings

GCSIE are well-conditioned for smooth interfaces.

They are formulated as second-kind integral equations.

Applicable to 2D and 3D acoustic scattering problems.

Abstract

We present a new class of well conditioned integral equations for the solution of two and three dimensional scattering problems by homogeneous penetrable scatterers. Our novel boundary integral equations result from suitable representations of the fields inside and outside the scatterer as combinations of single and double layer potentials acting on suitably defined regularizing operators. The regularizing operators are constructed to be suitable approximations of the admittance operators that map the transmission boundary conditions to the exterior and respectively interior Cauchy data on the interface between the media. The latter operators can be expressed in terms of Dirichlet-to-Neumann operators. We refer to these regularized boundary integral equations as Generalized Combined Source Integral Equations (GCSIE). The ensuing GCSIE are shown to be integral equations of the second kind in the case when the interface of material discontinuity is a smooth curve in two dimensions and a smooth surface in three dimensions.