Non-convolutional second-order complex-frequency-shifted perfectly matched layers for transient elastic wave propagation

Abstract

The numerical simulation of wave propagation in heterogeneous unbounded media using domain discretization techniques requires truncation of the physical domain: at the truncation boundary, Perfectly Matched Layers (PMLs) – buffers wherein wave attenuation is imposed – are often used to mimic outgoing wave motion and prevent waves from re-entering the interior computational domain. The PML’s wave-dissipative properties derive from a coordinate mapping concept, where the physical coordinate is mapped onto a frequency-dependent complex coordinate in the PML via a complex stretching function. The choice of the stretching function controls not only the spectral character and the absorptive properties of the PML, but also, more critically, the long-time stability when the computational domain-PML ensemble is used for transient wave simulations.The standard PML stretching function can lead to error growth, particularly when propagating waves impinge at grazing incidence on the truncation boundary. By contrast, a modification to the standard stretching function that has led to the Complex-Frequency-Shifted CFS-PML has been shown to alleviate the temporal instability. However, whereas PML formulations using the standard stretching function can lead to second-order in time semi-discrete forms, affording multifaceted benefits, all CFS-PML formulations to date require the evaluation of convolutions. In this paper, we discuss a new CFS-PML formulation that avoids the evaluation of convolutions, while preserving the second-order temporal character of elastic waves. It is shown that, upon spatial discretization, the CFS-PML can be completely described by a triad of stiffness, damping, and mass matrices, which can be readily incorporated into existing finite element codes originally designed for interior problems, to endow them with wave simulation capabilities on unbounded domains. Numerical experiments in the time-domain demonstrate the efficacy of the proposed approach; we also report long-time stability for problems involving waveguides and grazing wave incidence.