Method of virtual sources using on-surface radiation conditions for the Helmholtz equation

TL;DR

This paper introduces a novel virtual source method for the Helmholtz equation that improves boundary integral formulations by displacing Green's function singularities, resulting in well-conditioned, accurate solutions for wave propagation problems.

Contribution

The paper presents a new virtual source approach that displaces Green's function singularities, enabling boundary discretization with continuous kernels and improved system conditioning.

Findings

Well-conditioned systems with stable spectra

Accurate solutions across various wavelengths and mesh refinements

Effective implementation in 2D and 3D settings

Abstract

We develop a novel method of virtual sources to formulate boundary integral equations for exterior wave propagation problems. However, by contrast to classical boundary integral formulations, we displace the singularity of the Green's function by a small distance $h>0$. As a result, the discretization can be performed on the actual physical boundary with continuous kernels so that any naive quadrature scheme can be used to approximate integral operators. Using on-surface radiation conditions, we combine single- and double-layer potential representations of the solution to arrive at a well-conditioned system upon discretization. The virtual displacement parameter $h$ controls the conditioning of the discrete system. We provide mathematical guidance to choose $h$, in terms of the wavelength and mesh refinements, in order to strike a balance between accuracy and stability. Proof-of-concept implementations are presented, including piecewise linear and isogeometric element formulations in two- and three-dimensional settings. We observe exceptionally well-behaved spectra, and solve the corresponding systems using matrix-free GMRES iterations. The results are compared to analytical solutions for canonical problems. We conclude that the proposed method leads to accurate solutions and good stability for a wide range of wavelengths and mesh refinements.