Optimized first-order absorbing boundary conditions for anisotropic elastodynamics

Abstract

We propose new forms of low-order absorbing boundary conditions (ABC) for time-dependent elastic waves in isotropic and anisotropic media. The configuration considered is that of a two-dimensional elastic waveguide. The ABC shares with the widely-known Lysmer–Kuhlemeyer (LK) boundary condition the ease of implementation, especially in a finite-element (FE) setting, while offering gains in accuracy over the LK-like condition (or the LK condition in the isotropic case). The new conditions are proved to be energy-stable, even in the case when inverse modes are present, for which the normal components of the phase and group velocities have an opposite direction. The developed ABCs are particularly convenient to use in a Finite Element setting as they preserve the symmetry and positivity of the formulation and are simple to implement. The ABC features several parameters, which are optimized for accuracy. The optimization criterion is based on the notion of the energy-rate reflection coefficient. We test the performance of the scheme via a numerical example. While the gain in accuracy is found to be moderate (around 20% relative to the LK condition), the main result of this investigation lies in the development of ABCs for anisotropic elastodynamics that are proved to be stable from the outset, and are amenable for accuracy optimization. We plan to design in a future study analogous stable and optimized high-order ABCs using the methodology developed here.