On exact truncation of backward waves in elastrodynamics

TL;DR

This paper develops an exact truncation method for elastic wave scattering in anisotropic media, addressing backward waves, by deriving a Green's tensor, classifying wave behavior, and proposing a new transparent boundary condition validated through experiments.

Contribution

It introduces a novel Green's tensor derivation, a classification of wave propagation, and an exact boundary condition for elastic waves with backward waves, advancing computational elastrodynamics.

Findings

Successfully classifies propagation behavior of Green's tensor.

Proposes an exact transparent boundary condition applicable to backward waves.

Validates the method's accuracy and efficiency through numerical experiments.

Abstract

For elastic wave scattering problems in unbounded anisotropic media, the existence of backward waves makes classic truncation techniques fail completely. This paper is concerned with an exact truncation technique for terminating backward elastic waves. We derive a closed form of elastrodynamic Green's tensor based on the method of Fourier transform and design two fundamental principles to ensure its physical correctness. We present a rigorous theory to completely classify the propagation behavior of Green's tensor, thus proving a conjecture posed by B\'ecache, Fauqueux and Joly (J. Comp. Phys., 188, 2003) regarding a necessary and suffcient condition of the non-existence of backward waves. Using Green's tensor, we propose a new radiation condition to characterize anistropic scattered waves at infinity. This leads to an exact transparent boundary condition (TBC) to truncate the unbounded domain, regardless the existence of backward waves or not. We develop a fast algorithm to evaluate Green's tensor and a high-accuracy scheme to discretize the TBC. A number of experiments are carried out to validate the correctness and efficiency of the new TBC.