A Mixed Perfectly-Matched-Layer for Transient Wave Simulations in Axisymmetric Elastic Media

We are concerned with elastic wave simulations arising in elastic, semi-infinite, heterogeneous, three-dimensional media with a vertical axis of symmetry through the coordinate origin. Specifically, we discuss the development of a new mixed displacement-stress formulation in PML-truncated axisymmetric media for forward elastic wave simulations. Typically, a perfectly-matched-layer (PML) is used to surround a truncated finite computational domain in order to attenuate outwardly propagating waves without reflections for all non-zero angles-of-incidence and frequencies. To date, standard formulations use split fields, where the displacement components are split into normal and parallel to the PML interface components. In this work, we favor unsplit schemes, primarily for the computational savings they afford when compared against split-field methods. We use complex-coordinate stretching in the frequency-domain, but retain both unsplit displacements and stresses as unknowns prior to inverting the stretched forms back into the time-domain. We use a non-classical mixed finite element approach, and an extended Newmark-beta scheme to integrate in time the resulting semi-discrete forms, which in addition to the standard terms, include a jerk or jolt term. We report on numerical simulations demonstrating the stability and efficacy of the approach.