Fast multipole method applied to 3-D frequency domain elastodynamics

Abstract

This article is concerned with the formulation and implementation of a fast multipoleaccelerated BEM for 3-D elastodynamics in the frequency domain, based on the so-called diagonal form for the expansion of the elastodynamic fundamental solution, a multi-level strategy. As usual with the FM-BEM, the linear system of BEM equations is solved by GMRES, and the matrix is never explicitly formed. The truncation parameter in the multipole expansion is adjusted to the level, a feature known from recent published studies for the Maxwell equations. A preconditioning strategy based on the concept of sparse approximate inverse (SPAI) is presented and implemented. The proposed formulation is assessed on numerical examples involving O(10 5 ) BEM unknowns, which show in particular that, as expected, the proposed FM-BEM is much faster than the traditional BEM, and that the GMRES iteration count is significantly reduced when the SPAI preconditioner is used.

A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

“…On the basis of the decomposed expression, a variety of FMMs for elastodynamics have been developed for the last decade in terms of the following problems: 3D crack problems in lowfrequency regime (using the original FMM) [24,25] and high-frequency regime (using the diagonal form FMM) [26], 3D (ordinary) problems [35][36][37] for high-frequency regime, 2D problems for lowfrequency regime [27] and 2D/3D periodic problems for low-frequency regime [28][29][30][31]. We note that the decomposition is applicable to the time-domain analysis too [32][33][34].…”

The development of the pFFT accelerated BEM for 3-D acoustic scattering problems based on the Burton and Miller's integral formulation

“…To overcome these difficulties, several fast BEMs were developed in the past decades. The BEM accelerated by the fast multipole expansion method (FMM) [2][3][4][5][6] had been developed and applied to nearly all the fields that the conventional BEM can be employed. The BEM accelerated by the precorrected fast Fourier transform method (pFFT) [7] was widely applied as well [8][9][10][11][12].…”

Error analysis and novel near-field preconditioning techniques for Taylor series multipole-BEM

“…Therefore, preconditioning is essential to the effective solution of the linear systems using the iterative methods [4][5][6]. In order to improve the convergence rates, many preconditioning techniques have been proposed in the published literatures [6][7][8][9][10][11][12][13]. The popular and remarkable preconditioning techniques are sparse approximate inverse (SPAI) preconditioning [8][9][10][11].…”