An efficient finite element time-domain formulation for the elastic second-order wave equation: A non-split complex frequency shifted convolutional PML

The perfectly matched layer (PML) technique has demonstrated very high efficiency as absorbing boundary condition for the elastic wave equation recast as a first-order system in velocity and stress in attenuating non-grazing bulk and surface waves. This paper develops a novel convolutional PML formulation based on the second-order wave equation with displacements as the only unknowns to annihilate spurious reflections from near-grazing waves. The derived variational form allows for the use of e. g. finite element and the spectral element methods as spatial discretization schemes. A recursive convolution update scheme of second-order accuracy is employed such that highly stable, effective time integration with the Newmark-beta (implicit and explicit with mass lumping) method is achieved. The implementation requires minor modifications of existing displacement-based finite element software, and the stability and efficiency of the proposed formulation is verified by relevant two-dimensional benchmarks that accommodate bulk and surface waves. Copyright (C) 2011 John Wiley & Sons, Ltd.