Accurate absorbing boundary conditions for anisotropic elastic media. Part 2: Untilted non-elliptic anisotropy

Abstract

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general elastic media, we investigate the accuracy aspects of local ABCs designed for untilted non-elliptic anisotropy in the frequency domain (time-harmonic analysis). While simple space–time transformations are available to treat the wavemodes with opposing phase and group velocities present in elliptic anisotropic media, no such transformations are known to exist for the case of non-elliptic anisotropy. In this paper, we use the concept of layer groupings along with an unconventional stretching of the finite element mesh to guarantee the accuracy of local ABCs designed to treat all propagating wavemodes, even those with opposing phase and group velocities. The local ABC used here is the perfectly matched discrete layer (PMDL) which is a simple variant of perfectly matched layers (PMLs) that is also equivalent to rational approximation-based local ABCs (rational ABCs); it inherits the straightforward approximation properties of rational ABCs along with the versatility of PML. The approximation properties of PMDL quantified through its reflection matrix allow us to (a) show that it is impossible to design an accurate PMDL with wavenumber-independent parameters, (b) theoretically demonstrate the ability of wavenumber-dependent parameters to ensure accuracy, and finally (c) design a practical though unconventional stretching of the finite element PMDL mesh that facilitates the implementation of wavenumber-dependent parameters. The validity of this work is demonstrated through a series of numerical experiments.