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.

💡 Research Summary

This paper presents a novel and highly accurate numerical solver for acoustic wave scattering in two-dimensional periodic structures that contain a local perturbation. The central challenge addressed is that the total wavefield loses quasi-periodicity due to the perturbation, making it difficult to truncate the unbounded domain effectively for numerical simulation.

The authors’ core strategy is to leverage the Floquet-Bloch (FB) transform to convert the original non-quasi-periodic problem into an integral over a continuum of quasi-periodic sub-problems. The key component is the background Green’s function G(x; x_s) for the unperturbed periodic structure, which satisfies the Helmholtz equation with a point source and Dirichlet boundary conditions on the original obstacles. Instead of computing G directly in the infinite domain, the authors apply the FB transform to represent it as an integral of quasi-periodic Green’s functions G_qp(α, x; x_s) over the FB parameter α ∈