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A CG-FFT approach to the solution of a stress-velocity formulation of three-dimensional elastic scattering problems. (English) Zbl 1154.74051

Summary: We introduce a conjugate gradient fast Fourier transform (CG-FFT) scheme for numerical solution of the integral equation governing three-dimensional elastic scattering problems. The formulation is in terms of stress tensor and particle velocities as unknown field variables. In contrast with the formulation based on particle displacements, this approach leads to integral representations that do not involve derivatives of 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 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.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74J20 Wave scattering in solid mechanics
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
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