Spectrally-accurate numerical method for acoustic scattering from doubly-periodic 3D multilayered media

Abstract

A periodizing scheme and the method of fundamental solutions are used to solve acoustic wave scattering from doubly-periodic three-dimensional multilayered media. A scattered wave in a unit cell is represented by the sum of the near and distant contribution. The near contribution uses the free-space Green's function and its eight immediate neighbors. The contribution from the distant sources is expressed using proxy source points over a sphere surrounding the unit cell and its neighbors. The Rayleigh-Bloch radiation condition is applied to the top and bottom layers. Extra unknowns produced by the periodizing scheme in the linear system are eliminated using a Schur complement. The proposed numerical method avoids using singular quadratures and the quasi-periodic Green's function or complicated lattice sum techniques. Therefore, the proposed scheme is robust at all scattering parameters including Wood anomalies. The algorithm is also applicable to electromagnetic problems by using the dyadic Green's function. Numerical examples with 10-digit accuracy are provided. Finally, reflection and transmission spectra are computed over a wide range of incident angles for device characterization.

On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

“…together with the transmission conditions (14) and the conditions of radiation at infinity. (The latter concept, together with the related reciprocal lattice Λ * and the associated Rayleigh expansion, are described in what follows.)…”

“…Wave-scattering by periodic media, including RW anomalous configurations, at which the quasiperiodic Green function ceases to exist, has continued to attract significant attention in the fields of optics [17,22,33,34,35,36,39,45,50] and computational electromagnetism [3,8,4,9,10,31,14,26,42,39,18]. Classical boundary integral equations methods [43,49,52] have relied on the quasi-periodic Green function (denoted throughout this work as G q κ ), which is defined in terms of a slowly converging infinite series (equation (27)).…”

Taylor expansion based fast multipole method for 3-D Helmholtz equations in layered media

“…This is different from formulations based on free space Green's functions, which will need additional unknowns on the infinite material layer interfaces (cf. [6,7,8]). For the solution of the resulting linear system from the discretized IEs, iterative solvers such as GMRES are usually used, which require the product of a full matrix, from the discretization of the integral operator, and a solution vector.…”