A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain

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

doi:10.1016/j.cma.2008.04.024

Cited by 103 publications

(153 citation statements)

References 51 publications

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“…The evaluation of a single-layer potential S[t](x) of the form (7) for a given density t, which is typically one of the main computational tasks involved in the iterative solution of integral equations such as (6), is now addressed. As indicated earlier, attention is focused on the computation of the complementary potential S C [t](x) introduced in (10), as the other contributions S ∞ [t](x) and S ∞ [t](x) can be evaluated using known FMM procedures.…”

Section: Fast Multipole Methodsmentioning

confidence: 99%

“…Applying iterative linear solvers, such as GMRES [34], to the integral equation (6) essentially entails the evaluation of single-layer and double-layer elastic potentials of the form…”

Section: Standard Boundary Integral Equationmentioning

confidence: 99%

“…A FMM treatment of integral equation (6) thus may exploit the "standard" FMM associated with the diagonal form-based decomposition of the full-space Green's tensor [6], for the evaluation of con-…”

Section: Standard Boundary Integral Equationmentioning

confidence: 99%

“…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%

“…[5] and [16] for the computation of seismic waves. The methodology of [16] has later been improved in [6] for homogeneous semi-infinite elastic propagation domains, through incorporation of recent advances of FMM implementations for Maxwell equations [10], allowing to run BEM models of size up to N = O(10 6 ) on a single-processor PC. Other investigations address e.g.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Fast Multipole Methodsmentioning

confidence: 99%

“…Applying iterative linear solvers, such as GMRES [34], to the integral equation (6) essentially entails the evaluation of single-layer and double-layer elastic potentials of the form…”

Section: Standard Boundary Integral Equationmentioning

confidence: 99%

Section: Standard Boundary Integral Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Chaillat

Bonnet

2014

Journal of Computational Physics

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Boundary Integral Equation Methods for Elastic and Plastic Problems

Bonnet

2017

Encyclopedia of Computational Mechanics Second Edition

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This chapter deals with formulations based on boundary integral equations (BIEs) for elastic and plastic problems. After a brief review of the basic integral identities of solid mechanics and issues associated with the singular character of the fundamental solutions, the collocation and symmetric Galerkin BIE formulations and the associated boundary element methods (BEMs) are presented in their conventional form where the complete matrix equation is set up using numerical integration and stored. This approach being inadequate for large problem sizes, fast solution techniques are then reviewed, with emphasis on the fast multipole method. Next, BEMs in both collocation and symmetric Galerkin form are described for fracture mechanics and small‐strain elastoplasticity. The treatment of the hypersingular integrals arising in the integral representation of tractions on the crack surface or of strains at potentially plastic interior points is discussed, along with other issues. Shape sensitivity analysis techniques are also presented, based on either the direct differentiation of the primary elastic BIE or an adjoint solution. Finally, the symmetric Galerkin BIE is used to define a symmetric formulation for BEM–FEM coupling.

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Data‐driven execution of fast multipole methods

Ltaief

Yokota

2013

Concurrency and Computation

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SUMMARYFast multipole methods (FMMs) have scriptO (N) complexity, are compute bound, and require very little synchronization, which makes them a favorable algorithm on next‐generation supercomputers. Their most common application is to accelerate N‐body problems, but they can also be used to solve boundary integral equations. When the particle distribution is irregular and the tree structure is adaptive, load balancing becomes a non‐trivial question. A common strategy for load balancing FMMs is to use the work load from the previous step as weights to statically repartition the next step. The authors discuss in the paper another approach based on data‐driven execution to efficiently tackle this challenging load balancing problem. The core idea consists of breaking the most time‐consuming stages of the FMMs into smaller tasks. The algorithm can then be represented as a directed acyclic graph where nodes represent tasks and edges represent dependencies among them. The execution of the algorithm is performed by asynchronously scheduling the tasks using the queueing and runtime for kernels runtime environment, in a way such that data dependencies are not violated for numerical correctness purposes. This asynchronous scheduling results in an out‐of‐order execution. The performance results of the data‐driven FMM execution outperform the previous strategy and show linear speedup on a quad‐socket quad‐core Intel Xeon system.Copyright © 2013 John Wiley & Sons, Ltd.

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A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain

Cited by 103 publications

References 51 publications

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

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

Boundary Integral Equation Methods for Elastic and Plastic Problems

Data‐driven execution of fast multipole methods

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