A Duhamel integral based approach to one-dimensional wave propagation analysis in layered media

Abstract

In this paper, a new method for analysing one-dimensional wave propagation in a layered medium is presented. It is based on Duhamel integrals in combination with the convolution quadrature method (CQM) [9, 10]. The CQM is a technique which approximates convolution integrals, in this case the Duhamel integrals, by a quadrature rule whose weights are determined by Laplace transformed fundamental solutions and a multi-step method. Duhamel integrals are used to ensure equilibrium between the layers. The methodology is closely related to structural engineering and should be more familiar to engineers in practice than the usual boundary element method. In order to investigate the accuracy and the stability of the proposed algorithm, two benchmark problems are studied. The method is presented for one-dimensional problems, namely rods, but it can be readily extended to two- or three-dimensional dynamic interaction problems, e.g., dynamic soil-structure interaction. The results are very stable with respect to time step size and they are in very good agreement with analytical solutions.