An efficient boundary integral equation solution technique for solving aperiodic scattering problems from two-dimensional, periodic boundaries

TL;DR

This paper introduces an efficient boundary integral equation method for two-dimensional aperiodic scattering problems on periodic boundaries, leveraging Floquet--Bloch transform and low rank algebra for significant speed improvements.

Contribution

It develops a novel, fast boundary integral technique that avoids quasiperiodic Green's functions, enabling efficient solutions for aperiodic scattering on periodic structures.

Findings

Method is 20-30 times faster than Green's function approach

Achieves high accuracy with fewer discretization points

Utilizes precomputation and low rank algebra for efficiency

Abstract

This manuscript presents an efficient boundary integral equation technique for solving two-dimensional Helmholtz problems defined in the half-plane bounded by an infinite, periodic curve with Neumann boundary conditions and an aperiodic point source. The technique is designed for boundaries where one period does not require a large number of discretization points to achieve high accuracy. The Floquet--Bloch transform turns the problem into evaluating a contour integral where the integrand is the solution of quasiperiodic boundary value problems. To approximate the integral, one must solve a collection of these problems. This manuscript uses a variant of the periodizing scheme by Cho and Barnett which alleviates the need for evaluating the quasiperiodic Green's function and is amenable to a large amount of precomputation that can be reused for all of the necessary solves. The solution technique is accelerated by the use of low rank linear algebra. The numerical results illustrate that the presented method is 20-30 faster than the technique utilizing the quasiperiodic Green's function for a stair-like geometry.