Discrete Green's functions for unbounded using finite element method

Abstract

In this report dynamic solution of unbounded domains using pure finite element method is presented. The problems of concern are those with governing differential equations of constant coefficient. When the grid is of repeatable pattern the solution of an unbounded domain is reducible to a solution over a smaller domain with a grid consisting of few numbers of repeatable patterns. It is shown that by use of the proportionality property, having roots in governing equations, both conditions required for a unique solution, i.e. the decay and radiation of energy, are met through a spectral formulation. As a key point, a consistent transformation approach is proposed in order to employ such a spectral formulation for the solution. The transformation technique is analogous to those conventionally used for solution of partial differential equations but of course in a matrix form. To demonstrate the applicability of method, Green’s functions for two dimensional scalar and elastic wave equations are obtained numerically in frequency domain. The method is also capable of giving Green’s functions for dynamic solution of domains with repeatable material properties. Comprehensive discussions are given for accuracy and convergence of the solution.