A single layer representation of the scattered field for multiple scattering problems

TL;DR

This paper introduces a novel integral representation for the scattered field in multiple scattering problems, enabling efficient solutions without spherical harmonic expansions and extending the Fast Multipole Method to more complex geometries.

Contribution

It generalizes the scattered field representation to arbitrary smooth surfaces, facilitating improved computational methods for multiple scattering problems.

Findings

Integral representation supports any smooth surface enclosing scatterers.

Extension of Fast Multipole Method to non-ball subsets.

Enables efficient multiple scattering computations.

Abstract

The scattering of scalar waves by a set of scatterers is considered. It is proven that the scattered field can be represented as an integral supported by any smooth surface enclosing the scatterers. This is a generalization of the series expansion over spherical harmonics and spherical Bessel functions for spherical geometries. More precisely, given a set of scatterers, the field scattered by any subset can be expressed as an integral over any smooth surface enclosing the given subset alone. It is then possible to solve the multiple scattering problem by using this integral representation instead of an expansion over spherical harmonics. This result is used to develop an extension of the Fast Multipole Method in order to deal with subsets that are not enclosed within non-intersecting balls.