Discrete Green's functions for time-harmonic wave problems on unbounded domains with periodic variation of material properties

Abstract

In this paper we present a numerical method suitable for solution of steady state wave problems in which the material properties vary periodically throughout the domain. Discrete Green's functions, in finite element sense, are selected as the representatives of the problems. The Green's functions are evaluated on unbounded domains and this involves satisfaction of radiation conditions in the solutions. The formulation, given in this paper, is the extension of the one recently proposed by the authors for domains with homogenous materials. Here, the principles of Floquet theory for solution of partial differential equations, with periodic coefficients, are used. First, the fundamental exponential-like wave bases are obtained through the dispersion relations and then the radiation conditions are satisfied by selecting the wave bases. For selecting the wave bases, a quadrant of the main unbounded domain together with a set of appropriate boundary conditions is considered. For satisfaction of the boundary conditions a discrete transformation technique, proposed by the authors, is used. Application of the method is shown on a sample problem and the results are compared with those obtained form exact solution of a rather similar problem with homogenized material. The comparison shows the validity of the solution method. © 2006 Civil-Comp Press.