A multiaxial perfectly matched layer (M-PML) for the long-time simulation of elastic wave propagation in the second-order equations

Abstract

Abstract In order to conquer the spurious reflections from the truncated edges and maintain the stability in the long-time simulation of elastic wave propagation, several perfectly matched layer (PML) methods have been proposed in the first-order (e.g., velocity–stress equations) and the second-order (e.g., energy equation with displacement unknown only) formulations. The multiaxial perfectly matched layer (M-PML) holds the excellent stability for the long-time simulation of wave propagation, even though it is not perfectly matched in the discretized M-PML equation system. This absorbing boundary approach can offer an alternative way to solve the problem of the late-time instability, especially for anisotropic media, which is also suffered by the convolutional perfectly matched layer (C-PML) that is supposed to be competent to handle most stable problems. The M-PML termination implementation in the first-order formulations is well proposed. The common drawback of the implementation of the first-order M-PML formulations is that it necessitates fundamental reconstruction of the existing codes of the second-order spectral element method (SEM) or finite element method (FEM). Therefore, we propose a nonconvolutional second-order M-PML absorbing boundary condition approach for the wave propagation simulation in elastic media that has not yet been developed before. Two-dimensional numerical simulation validations demonstrate that the proposed second-order M-PML has good performances: 1) superior efficiency and stability of absorbing the spurious elastic wavefields, both the surface waves and body waves, reflected on the boundaries; 2) superior stability in the long-time simulation even in the isotropic medium with a high Poisson's ratio; 3) superior efficiency and stability in the long-time simulation for anisotropic media. This method hence makes the SEM and FEM in the second-order wave equation formulation more efficient and stable for the long-time simulation.