A Time-Domain Formulation of Elastic Waves in Heterogeneous Unbounded Domains

Abstract

A time-domain formulation and its numerical implementation of three-dimensional elastic waves in unbounded domains are discussed. To simulate elastic waves in unbounded domains, it is necessary to truncate the originally infinite extent and remove artificial reflections at truncated boundaries. In this work, perfectly matched layer (PML) is used to surround the truncated regular domain and to absorb outgoing waves beyond the truncated boundaries. To derive governing equations for elastic waves in the PML-truncated domain, complex coordinate stretching is applied to equilibrium, constitutive, and compatibility equations in the frequency domain. Then the stretched equations were converted to the time domain by applying the inverse Fourier transform to yield a system of mixed displacement-stress equations. Introducing a mixed finite element method leads to mixed semi-discrete equations of motion which can be integrated in time for displacements and stresses. Numerical experiments show that reflections from truncated boundaries are significantly reduced by using the developed PML method. For the reduction of computational cost in higher dimensions, an MPI-based parallel computation has been performed on assembly and time integration routines of the finite element wave simulation, which shows good scalability.