A numerical method for elastic wave scattering in multi-layered periodic media based on the scattering matrix and BEM

Abstract

This paper presents a numerical method for analyzing elastic wave scattering by multi-layered periodic structures in two dimensions. The proposed method is based on the boundary element method (BEM) and scattering matrix representing a relationship between incoming and outgoing waves in the vicinity of a periodic structure. First, we define the scattering matrix using a plane-wave expansion of the solution of the elastic problem. Then, we show the integral formulae that convert the solution of a system of boundary integral equations into elements of the scattering matrix. The proposed scattering matrix method with the BEM enables us to reduce the scattering problem in a multi-layered structure to a purely algebraic one in terms of the scattering matrix of each layer, resulting into less requirement in computational resources than in the methods which are based on the meshing of an entire boundary and solution of the corresponding boundary integral equations. Such a procedure called layer-doubling method is implemented in a numerically stable manner based on the eigendecomposition of a relevant matrix. Moreover, some scattering properties and phononic band structures of semi-infinite phononic crystals are yielded through an asymptotic analysis on the scattering matrix. Some numerical examples are presented to demonstrate that the proposed method can solve accurately scattering problems and also can calculate some related eigenmodes including the Bloch modes and guided waves within stacked structures.

Keywords

• layer-doubling method
• scattering matrix
• elastic wave
• boundary element method
• phononic crystal
• elastic metamaterial