A Spectral Element Method with Transparent Boundary Condition for Periodic Layered Media Scattering

TitleA Spectral Element Method with Transparent Boundary Condition for Periodic Layered Media Scattering
Publication TypeJournal Article
Year of Publication2013
AuthorsHe, Y, Min, M, Nicholls, D
JournalJournal of Computational Physics
Other NumbersANL/MCS-P5018-0913

We present a transparent boundary operator for a high-order spectral element approach for solving exterior scattering problems governed by a scalar Helmholtz equation. In particular, we consider incident waves at arbitrary angles impinging on scattering surfaces with periodic gratings, where the scattering solutions are represented in a quasi periodic form. We rewrite our governing equation into a formula that eliminates the quasi-periodicity and solve the reformulated scalar Helmholtz equation with periodic, Dirichlet, and transparent boundary conditions. We construct a spectral element Dirichlet-to-Neumann boundary operator for the transparent boundary condition that ensures nonreflecting outgoing waves on the artificial boundaries in the truncated computational domain. We present an explicit formula that accurately computes the Fourier data involved in the boundary operator on the spectral element discretization space. Our solutions are represented by the tensor product basis of the one-dimensional Legendre Lagrange interpolation polynomials based on the Gauss-Lobatto-Legendre grids. We study scattered field solutions in single-and double-layer media with smooth and nonsmooth scattering surfaces. Geometric structures of the scattering surfaces include rectangular, triangular, and sawtooth grooves that are accurately represented by the body-fitted quadrilateral elements. We use a GMRES iteration technique to solve the resulting linear system. We validate our results provided with spectral convergence in comparison with exact solutions and the results by the transformed field expansion method, including the energy defect measure.