[Paper Review] Perfectly matched layer for second-order time-domain elastic wave equation: formulation and stability

Hisham Assi, R.S.C. Cobbold|arXiv (Cornell University)|Dec 13, 2013

Electromagnetic Simulation and Numerical Methods32 references3 citations

TL;DR

This paper presents a compact second-order time-domain formulation for perfectly matched layers (PML) in elastic wave simulations of 2D anisotropic solids, using complex coordinate stretching with a two-parameter stretch function. It achieves superior stability for unstable anisotropic media by increasing the scaling parameter, reducing computational cost through only six equations (two second-order PDEs and four auxiliary equations), and enabling stable long-term simulations with minimal numerical reflections.

ABSTRACT

A time domain system of equations is proposed to model elastic wave propagation in an unbounded two-dimensional anisotropic solid using perfectly matched layer (PML). Starting from a system of first-order frequency domain stress-velocity equations and using complex coordinate stretching approach with a two-parameter stretch function, a second-order formulation is obtained. The final system, which consists of just two second order equations along with four auxiliary equations, is smaller than existing formulations, thereby simplifying the problem and reducing the computational cost. The discrete stability of the solutions for a given mesh size is examined with the help of a plane-wave analysis of the corresponding continuous problem. It is shown that increasing the scaling parameter of the stretch function leads to significant stability improvements for certain anisotropic media that have known issues. Numerical computations for different isotropic and anisotropic media are used to illustrate the results.

Motivation & Objective

• To address the challenge of spurious reflections in unbounded elastic wave simulations using a stable PML approach.
• To reduce computational cost and simplify implementation by minimizing the number of equations in time-domain PML formulations.
• To improve discrete stability in PML for anisotropic media that are known to be unstable in classical PML formulations.
• To provide a simple, effective method to enhance stability through scaling parameter adjustment without complex multi-parameter tuning.

Proposed method

• Derives a second-order time-domain PDE system from first-order frequency-domain stress-velocity equations using complex coordinate stretching with a two-parameter stretch function.
• Introduces four auxiliary equations to maintain the second-order structure and enable time-domain implementation.
• Applies plane-wave analysis to assess discrete stability and identify unstable modes in the PML region.
• Uses a variable scaling parameter (α̃j) that increases from 1 to higher values (e.g., 20, 90) across the PML to suppress unstable wave modes.
• Employs finite element method (FEM) with COMSOL Multiphysics for numerical validation and simulation of wave propagation in isotropic and anisotropic media.
• Applies polynomial transitions in PML coefficients to ensure smooth variation and reduce numerical reflections at the physical domain interface.

Experimental results

Research questions

• RQ1 Can a second-order time-domain PML formulation be derived for elastic wave propagation in 2D anisotropic solids that reduces the number of equations compared to existing methods?
• RQ2 How does the choice of stretch function parameters affect the discrete stability of the PML in anisotropic media?
• RQ3 Can increasing the scaling parameter α̃j stabilize otherwise unstable anisotropic media in the PML without requiring complex multi-parameter formulations?
• RQ4 To what extent does the proposed formulation outperform classical PML in terms of long-term stability and computational efficiency?
• RQ5 What is the impact of mesh size and wave incidence angle on the stability of the PML in second-order formulations?

Key findings

• The proposed formulation uses only six equations (two second-order PDEs and four auxiliary equations), making it the smallest reported time-domain PML formulation for elastic waves.
• For material IV, the PML remains stable even after 20 ms of simulation, demonstrating long-term stability with the optimized scaling parameters.
• Increasing the scaling parameter α̃j from 1 to 20 and 90 significantly suppresses instabilities in material III, particularly in the x₁ direction, though residual instability persists in the x₂ direction.
• For material V, increasing α̃j to 10 and 1 improved stability across all directions, eliminating instabilities observed at lower values.
• Plane-wave analysis confirmed that higher α̃j values shift unstable roots out of the discretely resolved spatial frequency range, explaining the stability improvement.
• Energy evolution plots (Figure 11) show a clear reduction in reflected energy in the physical domain when higher α̃j values are used, confirming improved absorption and stability.