NUMERICAL QUADRATURES FOR NEAR-SINGULAR AND NEAR-HYPERSINGULAR INTEGRALS IN BOUNDARY ELEMENT METHODS

2009 ◽  
Vol 109A (1) ◽  
pp. 49-60
Author(s):  
Michael Carley
Keyword(s):  
Boundary Element ◽  
Boundary Element Methods ◽  
Hypersingular Integrals ◽  
Numerical Quadratures

Review of Dual Boundary Element Methods With Emphasis on Hypersingular Integrals and Divergent Series

1999 ◽  
Vol 52 (1) ◽  
pp. 17-33 ◽  
Author(s):  
J. T. Chen ◽  
H.-K. Hong
Keyword(s):  
Integral Representation ◽  
Boundary Element ◽  
Series Representation ◽  
Review Article ◽  
Boundary Element Methods ◽  
Divergent Series ◽  
Current Status ◽  
Hypersingular Integrals ◽  
Dual Representations ◽  
Regularization Techniques

This article provides a perspective on the current status of the formulations of dual boundary element methods with emphasis on the regularizations of hypersingular integrals and divergent series. A simple example is given to show the dual integral representation and the dual series representation for a discontinuous function and its derivative and thereby to illustrate the regularization problems encountered in dual boundary element methods. Hypersingularity and the theory of divergent series are put under the framework of the dual representations, their relation and regularization techniques being examined. Applications of the dual boundary element methods using hypersingularity and divergent series are explored. This review article contains 249 references.

Regularization Techniques Applied to Boundary Element Methods

1994 ◽  
Vol 47 (10) ◽  
pp. 457-499 ◽  
Author(s):  
Masataka Tanaka ◽  
Vladimir Sladek ◽  
Jan Sladek
Keyword(s):  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Scalar Potential ◽  
Fundamental Solutions ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Element Formulation ◽  
Hypersingular Integrals ◽  
Regularization Techniques

This review article deals with the regularization of the boundary element formulations for solution of boundary value problems of continuum mechanics. These formulations may be singular owing to the use of two-point singular fundamental solutions. When the physical interpretation is irrelevant for this topic of computational mechanics, we consider various mechanical problems simultaneously within particular sections selected according to the main topic. In spite of such a structure of the paper, applications of the regularization techniques to many mechanical problems are described. There are distinguished two main groups of regularization techniques according to their application to singular formulations either before or after the discretization. Further subclassification of each group is made with respect to basic principles employed in individual regularization techniques. This paper summarizes the substances of the regularization procedures which are illustrated on the boundary element formulation for a scalar potential field. We discuss the regularizations of both the strongly singular and hypersingular integrals, occurring in the boundary integral equations, as well as those of nearly singular and nearly hypersingular integrals arising when the source point is near the integration element (as compared to its size) but not on this element. The possible dimensional inconsistency (or scale dependence of results) of the regularization after discretization is pointed out. Finally, we discuss the numerical approximations in various boundary element formulations, as well as the implementations of solutions of some problems for which derivative boundary integral equations are required.

scholarly journals Divergent Integrals in Elastostatics: General Considerations

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
V. V. Zozulya
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Hypersingular Integrals ◽  
Integral Equation Methods ◽  
Weakly Singular ◽  
Easy Calculation ◽  
2D And 3D

This article considers weakly singular, singular, and hypersingular integrals, which arise when the boundary integral equation methods are used to solve problems in elastostatics. The main equations related to formulation of the boundary integral equation and the boundary element methods in 2D and 3D elastostatics are discussed in details. For their regularization, an approach based on the theory of distribution and the application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular, and hypersingular integrals, have been constructed.

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.

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