Kuanrong Shen | Searcharxiv

arXiv2025

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

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.

math.NA↗

arXiv2023

Does PML exponentially absorb outgoing waves scattering from a periodic surface?

The PML method is well-known for its exponential convergence rate and easy implementation for scattering problems with unbounded domains. For rough-surface scattering problems, authors in [5] proved that the PML method converges at most algebraically in the physical domain. However, the authors also asked a question whether exponential convergence still holds for compact subsets. In [25], one of our authors proved the exponential convergence for periodic surfaces via the Floquet-Bloch transform when the wavenumber is positive and not a half integer; when the wavenumber is a positive half integer, a nearly fourth-order convergence rate was shown in [26]. The extension of this method to locally perturbed cases is not straightforward, since the domain is no longer periodic thus the Floquet-Bloch transform doesn't work, especially when the domain topology is changed. Moreover, the exact decay rate when the wavenumber is a half integer remains unclear. The purpose of this paper is to address these two significant issues. For the first topic, the main idea is to reduce the problem by the DtN map on an artificial curve, then the convergence rate of the PML is obtained from the investigation of the DtN map. It shows exactly the same convergence rate as in the unperturbed case. Second, to illustrate the convergence rate when the wavenumber is a half integer, we design a specific periodic structure for which the PML converges at the fourth-order, showing that the algebraic convergence rate is sharp. We adopt a previously developed high-accuracy PML-BIE solver to exhibit this unexpected phenomenon.

math.NA↗