A contrast-source integral-equation approach for three-dimensional modeling of elastic wave problems

Abstract

In this paper, a recently proposed formulation of an integral equation for solving three-dimensional elastic wave scattering problems is numerically implemented. The approach is formulated in terms of the stress tensor and particle velocity vector, where the symmetric tensors of rank two are decomposed into their omnidirectional and deviatoric constituents. Subsequently, this integral equation is used to obtain a contrast-source type integral equation. For solving these integral equations we employ a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme, which is based on quadrature formulas that provide (second-order) accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via Fast Fourier Transforms. As linear solvers we employ the Conjugate Gradient for Normal Residual (CGNR) scheme, which is always monotonically convergent, but has a slow convergent rate, and the Bi-Conjugate Gradient Stabilized (BiCGSTAB) scheme, which is more efficient, but it is less stable. The convergence rates of iterative schemes are further improved through the use of a simple diagonal preconditioner. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed approaches.