Calderon's preconditioning for periodic fast multipole method for elastodynamics in 3D

Isakari, Hiroshi; Niino, Kazuki; Yoshikawa, Hitoshi; Nishimura, Naoshi

doi:10.1002/nme.3332

Cited by 21 publications

(14 citation statements)

References 14 publications

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“…However, the final result is correct. We also note that a similar accumulation of eigenvalues at two points can be found in the PMCHWT formulation in elastodynamics where the square of the system operator is given in the form similar to (15) [16].…”

Section: Formulationsupporting

confidence: 63%

Calderón preconditioning approaches for PMCHWT formulations for Maxwell's equations

Niino

Nishimura

2012

Int J Numerical Modelling

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Preconditioning methods based on Calderón's formulae for the Poggio-Miller-Chang-Harrington-Wu-Tsai formulations for Maxwell's equations in 3D are discussed. Five different types of formulations are proposed. The first three use different basis functions for surface electric and magnetic currents. The first type is a preconditioning just by appropriately ordering the coefficient matrix using the Gramian matrix as the preconditioner. Other two types utilise preconditioners constructed using matrices needed in the main fast multipole method algorithms. The fourth and fifth types are similar to the second and third types, but they use the same basis functions for both surface electric and magnetic currents. We make several numerical experiments with proposed preconditioners to confirm the efficiency of these proposed methods.AT À1 y ¼ b followed by x ¼ T À1 y:This approach is called method 1. This approach is based on the observation that (15) holds. 564 K. NIINO AND N. NISHIMURA

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Section: Formulationsupporting

confidence: 63%

Calderón preconditioning approaches for PMCHWT formulations for Maxwell's equations

Niino

Nishimura

2012

Int J Numerical Modelling

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“…Improved versions of this approach were more recently proposed in [19,82], with numerical examples involving BE models of size up to N = O(10 6 ). Moreover, a 2D elastodynamic version of the wideband approach has been recently proposed in [77], while periodic 3D elastodynamic problems are addressed in [46] using a low-frequency separable decomposition of periodic Green's tensors.…”

mentioning

confidence: 99%

“…In particular, preconditioners based on the Calderón identities verified by the integral operators, have produced good results, for both the standard BEM and the FM-BEM. Calderón analytic preconditioners have for example been proposed for the Laplace equation [74], the Helmholtz equation [23], non-periodic problems for 2D Helmholtz [4] and periodic FMMs for 3D elastodynamics [46]. Other strategies have been proposed in electromagnetism to reformulate the integral operator, at the continuous level, to obtain well-conditioned formulations.…”

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confidence: 99%

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

Chaillat

Bonnet

2013

Wave Motion

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To cite this version:Stéphanie Chaillat, Marc Bonnet. Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics. Wave Motion, Elsevier, 2013Elsevier, , 50, pp.1090Elsevier, -1104 Abstract This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green's tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green's tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process.

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“…With the reciprocal theorem between p andp in Ω \ Ω ε , and Eqs. (13), (21), (22), (25) and (26), we can evaluate δJ as follows:…”

Section: 2mentioning

confidence: 99%

A level-set-based topology optimisation for acoustic–elastic coupled problems with a fast BEM–FEM solver

Isakari

Kondo

Takahashi

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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This paper presents a structural optimisation method in three-dimensional acoustic-elastic coupled problems. The proposed optimisation method finds an optimal allocation of elastic materials which reduces the sound level on some fixed observation points. In the process of the optimisation, configuration of the elastic materials is expressed with a level set function, and the distribution of the level set function is iteratively updated with the help of the topological derivative. The topological derivative is associated with state and adjoint variables which are the solutions of the acoustic-elastic coupled problems. In this paper, the acoustic-elastic coupled problems are solved by a BEM-FEM coupled solver, in which the fast multipole method (FMM) and a multi-frontal solver for sparse matrices are efficiently combined. Along with the detailed formulations for the topological derivative and the BEM-FEM coupled solver, we present some numerical examples of optimal designs of elastic sound scatterer to manipulate sound waves, from which we confirm the effectiveness of the present method.

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Calderon's preconditioning for periodic fast multipole method for elastodynamics in 3D

Cited by 21 publications

References 14 publications

Calderón preconditioning approaches for PMCHWT formulations for Maxwell's equations

Calderón preconditioning approaches for PMCHWT formulations for Maxwell's equations

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

A level-set-based topology optimisation for acoustic–elastic coupled problems with a fast BEM–FEM solver

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