Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media

We are concerned with elastic waves arising in plane-strain problems in an elastic semi-infinite arbitrarily heterogeneous medium. Specifically, we discuss the development of a new mixed displacement–stress formulation for forward elastic wave simulations in perfectly-matched-layer (PML)-truncated heterogeneous media.To date, most PML formulations split the displacement and stress fields, resulting in non-physical components for each field. In this work, we favor unsplit schemes, primarily for the relative ease by which the resulting forms can be incorporated into existing codes, the ease by which the resulting semi-discrete forms can be integrated in time, and the ease by which they can be used in adjoint formulations arising in inverse problems, contrary to most past and current developments. We start by following classical lines, and apply complex-coordinate-stretching to the governing equations in the frequency domain, while retaining both displacements and stress quantities as unknowns. With the aid of auxiliary variables the resulting mixed form is rendered second-order in time, thereby allowing the use of standard time integration schemes. We report on numerical simulations demonstrating the stability and efficacy of the approach.