scholarly journals Computationally Efficient Boundary Element Methods for High-Frequency Helmholtz Problems in Unbounded Domains

2017 ◽  
pp. 215-243 ◽  
Author(s):  
Timo Betcke ◽  
Elwin van ’t Wout ◽  
Pierre Gélat
Keyword(s):  
Boundary Element ◽  
High Frequency ◽  
Unbounded Domains ◽  
Boundary Element Methods ◽  
Computationally Efficient

Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Steffen Börm
Keyword(s):  
Helmholtz Equation ◽  
Boundary Element ◽  
High Frequency ◽  
Boundary Element Methods ◽  
Directional Interpolation

Abstract Boundary element methods for the high-frequency Helmholtz equation require efficient compression techniques for the resulting matrices. Directional interpolation converges exponentially and is very robust and fast, but high accuracies lead to very large storage requirements. This problem can be solved by combining interpolation with algebraic recompression techniques that significantly reduce the storage requirements while keeping the accuracy and robustness and only moderately increasing the runtime.

scholarly journals Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

2017 ◽  
Vol 341 ◽  
pp. 429-446 ◽  
Author(s):  
Stéphanie Chaillat ◽  
Marion Darbas ◽  
Frédérique Le Louër
Keyword(s):  
Boundary Element ◽  
High Frequency ◽  
Boundary Element Methods ◽  
Scattering Problems

scholarly journals Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

2020 ◽  
Vol 1 (4) ◽  
Author(s):  
A. Gibbs ◽  
D. P. Hewett ◽  
D. Huybrechs ◽  
E. Parolin
Keyword(s):  
Least Squares ◽  
Boundary Element ◽  
High Frequency ◽  
Singular Integrals ◽  
Weighted Least Squares ◽  
Computational Cost ◽  
Boundary Element Methods ◽  
Two Dimensional ◽  
Numerical Instabilities ◽  
Value Decomposition

Abstract We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.

Weighted Finite Difference and Boundary Element Methods Applied to Groundwater Pollution Problems

1991 ◽  
Vol 23 (1-3) ◽  
pp. 517-524
Author(s):  
M. Kanoh ◽  
T. Kuroki ◽  
K. Fujino ◽  
T. Ueda
Keyword(s):  
Boundary Element Method ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Element ◽  
Difference Method ◽  
Groundwater Pollution ◽  
Small Region ◽  
Boundary Element Methods ◽  
Computational Time ◽  
Element Method

The purpose of the paper is to apply two methods to groundwater pollution in porous media. The methods are the weighted finite difference method and the boundary element method, which were proposed or developed by Kanoh et al. (1986,1988) for advective diffusion problems. Numerical modeling of groundwater pollution is also investigated in this paper. By subdividing the domain into subdomains, the nonlinearity is localized to a small region. Computational time for groundwater pollution problems can be saved by the boundary element method; accurate numerical results can be obtained by the weighted finite difference method. The computational solutions to the problem of seawater intrusion into coastal aquifers are compared with experimental results.

scholarly journals Analytical preconditioners for Neumann elastodynamic boundary element methods

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Stéphanie Chaillat ◽  
Marion Darbas ◽  
Frédérique Le Louër
Keyword(s):  
Boundary Element ◽  
Boundary Element Methods
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