Theory and implementation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">H</mml:mi></mml:math>-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

Chaillat, Stéphanie; Desiderio, Luca; Ciarlet, Patrick

doi:10.1016/j.jcp.2017.09.013

Cited by 35 publications

(36 citation statements)

References 33 publications

(83 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is known that the H-matrix representations in their current form are not optimized for very high nondimensional wavenumbers [19]. Table 3, which presents speedups over unpreconditioned FM-BEM, indicates however that this possibly is not the case when employing a coarse approximation only as a preconditioner; indeed, one can appreciate the significant speedups in all three cases, including for increasing wave numbers.…”

Section: Implementation Details and Validations Of The Preconditionermentioning

confidence: 96%

“…The cost of this part of the matrix-vector product is then reduced. The interested reader is referred to [19] for full details concering the H-matrix construction for oscillatory Helmholtz kernels.…”

Section: Principles Of the H-matrix Representation Of The Bem System mentioning

confidence: 99%

“…The addition of an H-matrix construction for the nested preconditioner results in a slightly longer precomputation time. Although the complexity of constructing the H-matrix representation is generally higher than the FMM complexity, it can be shown [7,19] that the growth in complexity is bounded as O(N 3/2 log N ). This property is also verified in the figure where we have additionally plotted the setup time t Hmat,ε setup .…”

Section: Implementation Details and Validations Of The Preconditionermentioning

confidence: 99%

“…The efficiency of hierarchical matrices relies on the possibility to approximate, under certain conditions, the underlying kernel function by low-rank matrices. The approach has been shown to be very efficient for asymptotically smooth kernels (e.g., the Laplace kernel) and relatively efficient in a large frequency range for oscillatory kernels such as Helmholtz or elastodynamic kernels [25,19,39]. These two approaches have advantages and drawbacks in the context of the acceleration of BEMs.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

Amlani

Chaillat

Loseille

2019

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

This paper presents an efficient algebraic preconditioner to speed up the convergence of Fast Multipole accelerated Boundary Element Methods (FM-BEMs) in the context of time-harmonic 3D wave propagation problems and in particular the case of highly non-uniform discretizations. Such configurations are produced by a recently-developed anisotropic mesh adaptation procedure that is independent of partial differential equation and integral equation. The new preconditioning methodology exploits a complement between fast BEMs by using two nested GMRES algorithms and rapid matrix-vector calculations. The fast inner iterations are evaluated by a coarse hierarchical matrix (H-matrix) representation of the BEM system. These inner iterations produce a preconditioner for FM-BEM solvers. It drastically reduces the number of outer GMRES iterations. Numerical experiments demonstrate significant speedups over non-preconditioned solvers for complex geometries and meshes specifically adapted to capture anisotropic features of a solution, including discontinuities arising from corners and edges.

show abstract

Section: Implementation Details and Validations Of The Preconditionermentioning

confidence: 96%

Section: Principles Of the H-matrix Representation Of The Bem System mentioning

confidence: 99%

Section: Implementation Details and Validations Of The Preconditionermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

Amlani

Chaillat

Loseille

2019

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…This has been proposed and discussed by Rjasanow and Weggler [37] for electromagnetic problems and later used in Ref. [13] for elastodynamics.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Cross Approximation for BEM in Elasticity

Haider

Schanz

2019

J. Theor. Comp. Acout.

View full text Add to dashboard Cite

Boundary element methods (BEM) have obtained a mature state in the last years so that industrial applications are possible. However, to treat real-world problems, the so-called fast methods are necessary to reduce the original quadratic complexity to an almost linear order. Essentially, two methods are popular, the so-called fast multipole method, which uses a kernel expansion, and the algebraic approach based on [Formula: see text]-matrices with the adaptive cross approximation (ACA) to compress the matrix blocks. The latter is frequently used for scalar-valued problems, but for vector-valued problems, a modification of the pivot strategy is required. It has been suggested to search for the largest singular value out of all minimal singular values of the fundamental solution blocks. This strategy has been proposed by Rjasanow and Weggler and is studied here for elastostatics and elastodynamics. It is shown with numerical experiments that this strategy is mostly robust and results in an almost linear complexity.

show abstract

A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

Mavaleix-Marchessoux

Bonnet

Chaillat

et al. 2020

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

Three-dimensional (3D) rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretization (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretization) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the -transform and the convolution quadrature method, we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency-domain BEMs. Then, taking advantage of a well-designed high-frequency approximation, we drastically reduce the number of frequency-domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regard to the time discretization and O(N log N) for the spacial discretization, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid-structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity and the scattering of an abrupt wave by simple and realistic geometries.

show abstract

Theory and implementation of H-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

Cited by 35 publications

References 33 publications

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

Adaptive Cross Approximation for BEM in Elasticity

A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

Contact Info

Product

Resources

About