The spectral element method for elastic wave equations - Application to 2-D and 3-D seismic problems

A spectral element method for the approximate solution of linear elastodyna mic equations, set in a weak form, is shown to provide an efficient tool fo r simulating elastic wave propagation in realistic geological structures in two- and three-dimensional geometries. The computational domain is discret ized into quadrangles, or hexahedra, defined with respect to a reference un it domain by an invertible local mapping. Inside each reference element, th e numerical integration is based on the tenser-product of a Gauss-Lobatto-L egendre 1-D quadrature and the solution is expanded onto a discrete polynom ial basis using Lagrange interpolants. As a result, the mass matrix is alwa ys diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are intro duced in variational form to simulate unbounded physical domains. The time- discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multicorrector format. Lo ng term energy conservation and stability properties are illustrated as wel l as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two-dimensional problems obtained by other methods. The p otentiality of the method is then illustrated by studying a simple three-di mensional model. Very accurate modelling of Rayleigh wave propagation and s urface diffraction is obtained at a low computational cost. The method is s hown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three-dimensional s urface topographies. Copyright (C) 1999 John Wiley & Sons, Ltd.