An accurate solution of elastodynamic problems by numerical local Green's functions

Abstract

Green's function based methodologies for elastodynamics in both time and frequency domains, which can be either numerical or analytical, appear in many branches of physics and engineering. Thus, the development of exact expressions for Green's functions is of great importance. Unfortunately, such expressions are known only for relatively few kinds of geometry, medium and boundary conditions. In this way, due to the difficulty in finding exact Green's functions, specially in the time domain, the present paper presents a solution of the transient elastodynamic equations by a time-stepping technique based on the Explicit Green's Approach method written in terms of the Green's and Step response functions, both being computed numerically by the finite element method. The major feature is the computation of these functions separately by the central difference time integration scheme and locally owing to the principle of causality. More precisely, Green's functions are computed only at t = Δt adopting two time substeps while Step response functions are computed directly without substeps. The proposed time-stepping method shows to be quite accurate with distinct numerical properties not presented in the standard central difference scheme as addressed in the numerical example.