Numerical Methods for Quasi-Periodic Incident Fields Scattered by Locally Perturbed Periodic Surfaces

TL;DR

This paper develops and extends Bloch transform-based finite element methods to efficiently handle non-decaying incident fields, such as plane waves, in wave scattering problems involving locally perturbed periodic surfaces.

Contribution

It introduces a standard and high order numerical method for non-decaying incident fields, expanding the applicability of Bloch transform techniques in wave scattering simulations.

Findings

Numerical experiments validate the effectiveness of the methods.

The methods are extendable to generalized cases with less efficient standard algorithms.

High order methods improve accuracy for faster decaying incident fields.

Abstract

Waves scattering from unbounded structures are always complicated problems for numerical simulations. For the case that the non-periodic incident field scattered by (locally perturbed) periodic surfaces, with the help of the Bloch transform, the problem could be solved by some finite element methods, if the incident fields decay at certain rate at the infinity. For faster decaying incident fields, a high order numerical method is also available. However, in these cases, the plain waves, which belong to a very important family of incident fields but do not decay at the infinity, are not included. In this paper, we aim to develop the Bloch transform based standard finite method for this certain case, and then establish the high order method afterwards. Numerical experiments have been carried out for both the standard and high order numerical methods. Based on the algorithms for incident plain waves, we could also extend the numerical methods to more generalized cases when only the not so efficient standard method is available.