Application of the multi-level time-harmonic fast multipole BEM to 3-D visco-elastodynamics

Grasso, Eva; Chaillat, Stéphanie; Bonnet, Marc; Semblat, Jean‐François

doi:10.1016/j.enganabound.2011.11.015

Cited by 26 publications

(25 citation statements)

References 50 publications

(86 reference statements)

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“…The applicability of the single-domain and multidomain elastodynamic FM-BEM formulations to weakly dissipative visco-elastic media characterized (for fixed frequency) by complex-valued wavenumbers k ⋆ α = k α (1 + iζ α ) (α = P,S), where k ⋆ α are the elastic pressure and shear wavenumbers and ζ α ≪ 1 are the damping ratios, has been examined in [39]. In particular, the validity of selection rules such as (13) in the complex-wavenumber case, an issue only sparingly addressed in the available literature, was examined.…”

Section: Multi-domain Elastodynamic Formulationmentioning

confidence: 99%

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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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Section: Multi-domain Elastodynamic Formulationmentioning

confidence: 99%

“…An initial series of investigations on the elastodynamic FMM [19,20,39], whose main features and findings are summarized in this section, has concentrated on the kernel-dependent approach based on the full-space Green's tensor (3).…”

Section: Formulations Based On the Full-space Green's Tensormentioning

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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“…It may (consequently) lead to huge numerical costs in 3D elastodynamics, • the spectral element method has been increasingly considered to analyse 2D/3D wave propagation in linear media with a good accuracy due to its spectral convergence properties [38][39][40] but the spurious wave reflections still have to be removed [30][31][32][33][34][35][36][37], • the boundary element method allows a very good description of the radiation conditions but is preferably dedicated to weak heterogeneities and linear constitutive models 3 [11,17,24,33,41,46]. It is thus difficult to deal with strong heterogeneities in 3D except if considering the original developments recently proposed to reduce the computational cost of the method [43][44]47], • the Aki-Larner method takes advantage of the frequency-wavenumber decomposition but is limited to simple geometries [45,46], • the scaled boundary finite element method is a kind of solution-less boundary element method [48], • other methods such as series expansions of wave functions [49].…”

Section: Modeling Seismic Wave Propagation By the Bem 21 Numerical Mmentioning

confidence: 99%

“…This equation is written in the framework of linear elasticity but, since the analysis is performed in the frequency domain, damped mechanical properties may be considered through the complex modulus of the medium [33,47] (it is discussed in the following).…”

Section: Elastodynamicsmentioning

confidence: 99%

Site effects in an alpine valley with strong velocity gradient: Interest and limitations of the ‘classical’ BEM

Delépine

Semblat

2012

Soil Dynamics and Earthquake Engineering

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-Seismic waves may be strongly amplified in deep alluvial basins due to the velocity contrast (or velocity gradient) between the various layers as well as the basin edge effects. In this work, the seismic ground motion in a deep alpine valley (Grenoble basin, French Alps) is investigated through various 'classical' Boundary Element models. This deep valley has a peculiar geometry ("Y"-shaped) and involves a strong velocity gradient between surface geological structures. In the framework of a numerical benchmark [21-23], a representative cross section of the valley has been proposed to investigate 2D site effects through various numerical methods. The 'classical' Boundary Element Method is considered herein to model the strong velocity gradient with a 2D piecewise homogeneous medium. For a large incidence angle, the transfer functions estimated from plane SH waves are close to the one computed with shallow SH point sources. The fundamental frequency is estimated at 0.33 Hz (SH wave) and the agreement with previous experimental results (spectral ratios) is good. Comparisons between 1D and 2D amplification are then performed: the values of the fundamental frequency and the corresponding amplitude are larger in 2D. Converting frequency domain results into the time domain, we underline surface waves generated at the valley edges and the directivity effect for the amplified wave-field. In the time domain for plane SV-wave, we also computed the ground motion for a strong seismic event (M=6): time duration and peak ground velocity are found to be 3 times larger than for the input signal. Such 2D models involving basin effects are then capable to recover the high amplification level measured in the field. Nevertheless, to deal with complex 3D basins as also proposed in the "ESG" benchmark [21][22][23], the capabilities of the classical Boundary Element Method are limited. As shown recently [43,47,52], such improvements as the fast multipole formulation (FM-BEM) may be a promising alternative for future 3D simulations.

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“…In such cases, both the potentially large number of additional DOFs required for the free surface and the selection of a suitable truncation radius are serious issues. An acceptable size of the meshed part of the free surface was empirically estimated in [6,18] as about 3-5 times the radius of the surface irregularity of interest. Assuming a uniform mesh density (with element sizes typically set to a fixed fraction, say 1/10, of the shear wavelength), this results in BE model sizes of 10-15 times that of the meshed irregularity.…”

Section: Introductionmentioning

confidence: 99%

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

Chaillat

Bonnet

2014

Journal of Computational Physics

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Abstract. 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 fullspace 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 = 6×10 5 degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved. IntroductionThe main advantage of the boundary element method (BEM) is that only the domain boundaries (and possibly interfaces) are discretized, leading to a reduction of the number of degrees of freedom (DOFs) relative to domain-discretization methods. Moreover, elastodyamic field equations and radiation conditions are exactly satisfied by the formulation [26], which avoids cumulative effects of grid dispersion and makes the BEM well suited for modeling wave propagation in unbounded domains. However, the standard BEM [3] leads to fully-populated matrices, which results in high computational complexity (O(N 2 ) per iteration using an iterative solver such as GMRES, where N denotes the number of boundary degrees of freedom (DOFs) of the BE model) and severe problem size limitations induced by the O(N 2 ) size of the influence matrix.The advent of accelerated BE methodologies has dramatically improved the capabilities of BEMs for many areas of application, in a large part owing to the rapid development of the Fast Multipole Method (FMM) over the last 15-20 years [29]. Such approaches have resulted in considerable solution speedup, memory reduction and model size increase. The FMM inherently relies on iterative solvers (usually GMRES), so as to avoid actual computation and storage of the fully-populated influence matrix, and is known to require O(N log N )...

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Application of the multi-level time-harmonic fast multipole BEM to 3-D visco-elastodynamics

Cited by 26 publications

References 50 publications

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

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

Site effects in an alpine valley with strong velocity gradient: Interest and limitations of the ‘classical’ BEM

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

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