Review and Recent Developments on the Perfectly Matched Layer (PML) Method for the Numerical Modeling and Simulation of Elastic Wave Propagation in Unbounded Domains

TL;DR

This review comprehensively discusses the development and implementation of PML methods for elastic wave simulation in unbounded domains, introducing a new mixed displacement-strain PML formulation compatible with standard finite element codes.

Contribution

The paper introduces a novel mixed displacement-strain PML formulation for elastodynamics that is second-order in time and easily integrable into existing finite element frameworks.

Findings

The proposed PML achieves high accuracy in wave absorption.

The hybrid implementation enhances computational efficiency.

Numerical results demonstrate stability and effectiveness in 2D problems.

Abstract

This review article revisits and outlines the perfectly matched layer (PML) method and its various formulations developed over the past 25 years for the numerical modeling and simulation of wave propagation in unbounded media. Based on the concept of complex coordinate stretching, an efficient mixed displacement-strain unsplit-field PML formulation for second-order (displacementbased) linear elastodynamic equations is then proposed for simulating the propagation and absorption of elastic waves in unbounded (infinite or semi-infinite) domains. Both time-harmonic (frequency-domain) and time-dependent (timedomain) PML formulations are derived for two-and three-dimensional linear elastodynamic problems. Through the introduction of only a few additional variables governed by low-order auxiliary differential equations, the resulting mixed timedomain PML formulation is second-order in time, thereby allowing the use of standard time integration schemes commonly employed in computational structural dynamics and thus facilitating the incorporation into existing displacementbased finite element codes. For computational efficiency, the proposed timedomain PML formulation is implemented using a hybrid approach that couples a mixed (displacement-strain) formulation for the PML region with a classical (displacement-based) formulation for the physical domain of interest, using a standard Galerkin finite element method (FEM) for spatial discretization and a Newmark time scheme coupled with a finite difference (Crank-Nicolson) time scheme for time sampling. Numerical experiments show the performances of the PML method in terms of accuracy, efficiency and stability for two-dimensional linear elastodynamic problems in single-and multi-layer isotropic homogeneous elastic media.