On the construction of scattering matrices for irregular or elongated enclosures using Green's representation formula

TL;DR

This paper introduces a flexible Green's function-based method for constructing scattering matrices for irregular or elongated enclosures, expanding the applicability of multiple scattering techniques beyond traditional geometries.

Contribution

It develops a new approach using Green's representation formula that applies to a broader class of geometries, including irregular and elongated enclosures, in both 2D and 3D.

Findings

Effective in multi-particle scattering scenarios

Demonstrated flexibility with layered media

Compatible with commercial PDE software

Abstract

Multiple scattering methods are widely used to reduce the computational complexity of acoustic or electromagnetic scattering problems when waves propagate through media containing many identical inclusions. Historically, this numerical technique has been limited to situations in which the inclusions (particles) can be covered by nonoverlapping disks in two dimensions or spheres in three dimensions. This allows for the use of separation of variables in cylindrical or spherical coordinates to represent the solution to the governing partial differential equation. Here, we provide a more flexible approach, applicable to a much larger class of geometries. We use a Green's representation formula and the associated layer potentials to construct incoming and outgoing solutions on rectangular enclosures. The performance and flexibility of the resulting scattering operator formulation in two-dimensions is demonstrated via several numerical examples for multi-particle scattering in free space as well as in layered media. The mathematical formalism extends directly to the three dimensional case as well, and can easily be coupled with several commercial numerical PDE software packages.