Time-domain PML formulation for modeling viscoelastic waves with Rayleigh-type damping in an unbounded domain: Theory and application in ABAQUS

The Perfectly Matched Layer (PML) method is a powerful approach to absorb outgoing waves from a finite computational domain in various media, such as elastic, poroelastic, anisotropic, and viscoelastic, by means of layers of artificial material placed at the finite domain boundaries. However, its use in most finite-element method (FEM) codes, which take into account viscous damping by employing only the Rayleigh and Lindsday damping formulation, is inadequate. This paper introduces a viscoelastic Perfectly Matched Layer (PML) formulation with Rayleigh-type damping for finite-element time-domain analyses. The PML formulations in the frequency- and time-domain are derived using a two-parameter complex coordinate stretching function. The displacement field is the only unknown variable in the formulation, so that the approach can be readily implemented in general finite-element codes. The weak form formulation in the time-domain is implemented in ABAQUS/Standard by merging it with a user-defined element (UEL) subroutine in Fortran90. The UEL is developed for 2D plane-strain (PE4ML) and plane-stress (PS4ML) four-node, linear, isoparametric, quadrilateral elements. The Hilber-Hughes-Taylor implicit time integration scheme is used to evaluate the unknown displacements, velocities, and accelerations. The validity and efficiency of the viscoelastic PML formulation and the UEL subroutine are examined with two numerical examples, namely a single-layer and a multi-layered soil profile with different damping ratios and properties. The results also highlight the poor performance of elastic PML layers, when they are placed at the boundaries of a viscoelastic interior domain, because, as the damping ratio increases, the reflection at the interface of the interior domain and the elastic PML domain increases. Long-time stability of the model is also examined, and no instabilities are observed.