An Accurate Solution to Periodic Grating Structure Scattering Using Nystrom Method with An Over-determined Testing Scheme

Abstract

Characterizations of artificial periodic structure scattering is of great importance in various engineering designs. However, accurate theoretical modeling remains an issue especially at large incidence angles. In this paper, an integral equation (IE) formulation is developed to solve the sub-wavelength periodic grating scattering problem. To achieve high accuracy, the Nystrom method is adopted to discretize the IE with the Gaussian-Legendre quadrature rule and the local correction scheme. Meanwhile, an over-determined testing scheme of the IE is proposed to overcome the potential internal resonance issue associated with the surface integral equation. This over-determined testing scheme turns out crucial to the validity of the IE methods when applied to the grating scattering problem, not limited to the internal resonance frequencies. In computing the impedance matrix elements, the periodic Green's functions are evaluated to high accuracy and efficiency by combing a recently developed imaginary wavenumber extraction technique and an integral transformation approach. The effectiveness of the proposed method is demonstrated by modeling the extraordinary acoustic transmission (EAT) phenomenon associated with the near grazing incidence of a plane wave upon a periodic array of closely spaced circular cylinders, for which case the commercial software Comsol yields non-physical transmittance that exceeds unity. The proposed method can be applied to arbitrary periodic gratings and is of great significance to the design and simulation of such structures.