Time domain modeling of elastic waves using a stress-based unsplit-field perfectly matched layer with enhanced numerical stability

Abstract

A stress-based unsplit-field perfectly matched layer (PML) method with enhanced numerical stability is proposed to simulate elastic waves in heterogeneous unbounded domains. The proposed PML method attenuates the waves by forcing the satisfaction of mixed-field equations constructed using a complex coordinate stretching in the frequency domain. Unlike other unsplit-field PMLs, which introduce coordinate stretching before combining the constitutive and compatibility equations, the proposed method first combines the two equations and then stretches the resulting equation. The stretched equation is converted back to the time domain using the inverse Fourier transform to yield transient elastic wave equations in the PML-truncated domain. Semi-discrete equations of motion for displacement and stresses were obtained using standard Lagrange-family finite elements. In a series of numerical examples involving semi-infinite elastic media, the accuracy of the wave solutions calculated using the proposed PML method was evaluated using a displacement L2-norm, a normalized L2-error, and an energy norm. The examples further validated the wave absorption performance of the PML and evaluated the long-term stability of solutions utilizing the Ricker pulse excitation with various central frequencies. The numerical experiments indicate that the solutions obtained using the proposed PML are significantly more stable than those obtained using the existing strain-based unsplit-field PMLs. The proposed method is particularly useful for applications requiring explicit time integration of semi-discrete equations of motion because the mass matrix can be diagonalized. The PML method and its associated solution strategies can be applied to various engineering problems, such as soil-structure interaction analysis, monitoring of structures, and geophysical probing.