GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Generalized Gaussian quadrature rules for systems of arbitrary functions
SIAM Journal on Numerical Analysis
Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators
SIAM Journal on Scientific Computing
The fast multipole method: numerical implementation
Journal of Computational Physics
Nonlinear Optimization, Quadrature, and Interpolation
SIAM Journal on Optimization
Accelerating Fast Multipole Methods for the Helmholtz Equation at Low Frequencies
IEEE Computational Science & Engineering
Efficient fast multipole method for low-frequency scattering
Journal of Computational Physics
A wideband fast multipole method for the Helmholtz equation in three dimensions
Journal of Computational Physics
A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures
SIAM Journal on Scientific Computing
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In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Green@?s tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Green@?s tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil-structure interaction, with applications to seismology or civil engineering. However, unlike the full-space Green@?s tensor, the elastodynamic half-space Green@?s tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about N=6x10^5 degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved.