Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

Abstract

This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction-used to properly impose the Rayleigh radiation condition-yield a robust second-kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh-Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix-vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.

A Complex-Scaled Boundary Integral Equation for Time-Harmonic Water Waves

“…Although the WGF proves effective at solving Helmholtz problems in the context of layered and periodic media as well as waveguides [8,9,7,39], its direct application to the water wave problem under consideration is not feasible. While it may be tempting to attribute the failure of the WGF method to the logarithmic growth of the free space Green's function, an argument can be made that by combining an easy-to-evaluate harmonic function with the free-space Green's function, a decaying Green's function can be formed (in a manner akin to the construction used in [12]).…”