A CG-FFT approach to the solution of a stress-velocity formulation of three-dimensional elastic scattering problems

Abstract

In this paper we introduce a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme for the numerical solution of the integral equation formulating three-dimensional elastic scattering problems. The formulation is in terms of the stress tensor and particle velocities as the unknown field variables. In contrast with the formulation based on particle displacements, this approach leads to integral representations that do not involve derivatives of the unknown fields, thus resulting in simplified and more stable numerics. The numerical procedure is based on suitable 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 FFTs. The scheme is further improved through the introduction of (approximation-based) pre-conditioners that are shown to accelerate the convergence of the CG iterations. Numerical results are presented that demonstrate the accuracy and efficiency of the proposed methodology.