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

TL;DR

This work presents a Green's representation–based framework to build scattering matrices on rectangular proxy surfaces for irregular or elongated enclosures governed by the 2D Helmholtz equation $\Delta u + k^2 u = 0$. By combining layer potentials, Green's identities, and a careful discretization of both obstacles and proxy surfaces, the authors construct a scattering operator $\mathrm{A}^{\Gamma}_{P}$ that maps incoming data on a proxy surface to the corresponding scattered data, enabling efficient multi-particle scattering computations. The method extends naturally to layered media via a layered Green's function $g^{LM}$ and Sommerfeld-type corrections, yielding a unified approach for free-space and layered configurations. Numerical results demonstrate spectral convergence with respect to proxy-point resolution, scalable fast algorithms (FMM/NUFFT) for large ensembles of scatterers, and practical accuracy in complex geometries, illustrating potential for design optimization and integration with commercial PDE solvers.

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.

Figures (1)

• Figure 1: In multiple scattering methods, each inclusion with boundary $\Gamma$ is enclosed in a proxy surface $P$. For any incoming field, sampled on $P$, the scattering matrix yields a representation for the outgoing field induced on $\Gamma$ (but only in the exterior of $P$). These methods work well when inclusions are spatially separated. When $P$ is a disk, as in (a), even fairly close interactions can be accurately computed, requiring only that the enclosing disks are disjoint. For high aspect ratio inclusions, as in (b), they fail since the enclosing disks overlap even when there is a significant gap between them. Here, we extend the approach by constructing scattering matrices on more closely contoured enclosures such as a rectangle, as depicted in (c).

Theorems & Definitions (2)

• Remark 1
• Remark 2