Polar coordinate transformation approach for treatment of singular integrals in boundary element methods

1988 ◽  
Vol 9 (10) ◽  
pp. 959-967 ◽  
Author(s):  
Liu Yong-hui ◽  
Lu Xin-sen
Keyword(s):  
Boundary Element ◽  
Coordinate Transformation ◽  
Singular Integrals ◽  
Boundary Element Methods ◽  
Transformation Approach ◽  
Polar Coordinate

Modelling ground vibration from railways using wavenumber finite- and boundary-element methods

2005 ◽  
Vol 461 (2059) ◽  
pp. 2043-2070 ◽  
Author(s):  
X Sheng ◽  
C.J.C Jones ◽  
D.J Thompson
Keyword(s):  
Boundary Element ◽  
Singular Integrals ◽  
Moving Load ◽  
Three Dimensional ◽  
Ground Surface ◽  
Ground Vibration ◽  
Element Model ◽  
Boundary Element Methods ◽  
Element Discretization ◽  
Environmental Vibration

A mathematical model is presented for ground vibration induced by trains, which uses wavenumber finite- and boundary-element methods. The track, tunnel and ground are assumed homogeneous and infinitely long in the track direction ( x -direction). The models are formulated in terms of the wavenumber in the x -direction and discretization in the yz -plane. The effect of load motion in the x -direction is included. Compared with a conventional, three-dimensional finite- or boundary-element model, this is computationally faster and requires far less memory, even though calculations must be performed for a series of discrete wavenumbers. Thus it becomes practicable to carry out investigative study of train-induced ground vibration. The boundary-element implementation uses a variable transformation to solve the well-known problem of strongly singular integrals in the formulation. A ‘boundary truncation element’ greatly improves accuracy where the infinite surface of the ground is truncated in the boundary-element discretization. Predictions of vibration response on the ground surface due to a unit force applied at the track are performed for two railway tunnels. The results show a substantial difference in the environmental vibration that could be expected from the alternative designs. The effect of a moving load is demonstrated in a surface vibration example in which vibration propagates from an embankment into layered ground.

Symmetric Galerkin Boundary Element Methods

1998 ◽  
Vol 51 (11) ◽  
pp. 669-704 ◽  
Author(s):  
Marc Bonnet ◽  
Giulio Maier ◽  
Castrenze Polizzotto
Keyword(s):  
Boundary Element ◽  
Quadratic Forms ◽  
Singular Integrals ◽  
Single Layer ◽  
Initial Boundary ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Weight Functions ◽  
Generalized Variables ◽  
Initial Boundary Value

This review article concerns a methodology for solving numerically, for engineering purposes, boundary and initial-boundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form. The discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager’s sense. As main consequences of the above provisions, symmetry is exhibited by matrices with a key role in the algebraized versions; some quadratic forms have a clear energy meaning; variational properties characterize the solutions and other results, invalid in traditional boundary element methods enrich the theory underlying the computational applications. The present survey outlines recent theoretical and computational developments of the title methodology with particular reference to linear elasticity, elastoplasticity, fracture mechanics, time-dependent problems, variational approaches, singular integrals, approximation issues, sensitivity analysis, coupling of boundary and finite elements, and computer implementations. Areas and aspects which at present require further research are identified, and comparative assessments are attempted with respect to traditional boundary integral-elements. This article includes 176 references.

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
Sign in / Sign up

Export Citation Format

Share Document