A boundary integral equation method for wave scattering in periodic structures via the Floquet-Bloch transform

TL;DR

This paper introduces a boundary integral equation method utilizing the Floquet-Bloch transform for high-accuracy simulation of acoustic wave scattering in locally perturbed periodic structures, effectively handling non-quasi-periodic wavefields.

Contribution

It develops a novel boundary integral equation approach with efficient algorithms and spectral discretization for wave scattering in perturbed periodic media, improving accuracy and computational efficiency.

Findings

Method achieves high accuracy in numerical experiments.

Computational complexity is comparable to quasi-periodic problems.

Efficient algorithms for Green's functions and wavefield computation are demonstrated.

Abstract

This paper is concerned with the problem of an acoustic wave scattering in a locally perturbed periodic structure. As the total wavefield is non-quasi-periodic, effective truncation techniques are pursued for high-accuracy numerical solvers. We adopt the Green's function for the background periodic structure to construct a boundary integral equation (BIE) on an artificial curve enclosing the perturbation. It serves as a transparent boundary condition (TBC) to truncate the unbounded domain. We develop efficient algorithms to compute such background Green's functions based on the Floquet-Bloch transform and its inverse. Spectrally accurate quadrature rules are developed to discretize the BIE-based TBC. Effective algorithms based on leap and pullback procedures are further developed to compute the total wavefield everywhere in the structure. A number of numerical experiments are carried out to illustrate the efficiency and accuracy of the new solver. They exhibit that our method for the non-quasi-periodic problem has a time complexity that is even comparable to that of a single quasi-periodic problem.