THE EVALUATIONS OF CAUCHY PRINCIPAL VALUE INTEGRALS AND WEAKLY SINGULAR INTEGRALS IN BEM AND THEIR APPLICATIONS

1996 ◽  
Vol 39 (6) ◽  
pp. 1017-1028 ◽  
Author(s):  
J. ZHU ◽  
A. H. SHAH ◽  
S. K. DATTA
Keyword(s):  
Singular Integrals ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Weakly Singular ◽  
Principal Value

scholarly journals Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Chunxiao Yu ◽  
Lingling Wei
Keyword(s):  
Singular Integrals ◽  
Riemann Integral ◽  
Piecewise Constant ◽  
Cauchy Principal ◽  
Numerical Examples ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value ◽  
Error Expansion ◽  
Superconvergence Results

The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.

A General Algorithm for Multidimensional Cauchy Principal Value Integrals in the Boundary Element Method

1990 ◽  
Vol 57 (4) ◽  
pp. 906-915 ◽  
Author(s):  
M. Guiggiani ◽  
A. Gigante
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Singular Integrals ◽  
Applied Mechanics ◽  
Full Generality ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value ◽  
Element Method

This paper presents a new general method for the direct evaluation of Cauchy principal value integrals in several dimensions, which is an issue of major concern in any boundary element method analysis in applied mechanics. It is shown that the original Cauchy principal value integral can be transformed into an element-by-element sum of regular integrals, each one expressed in terms of intrinsic (local) coordinates. The actual computation can be performed by standard quadrature formulae and can be easily included in any existing computer code. The numerical results demonstrate the accuracy and efficiency of the method, along with its insensitivity to the mesh pattern. This new method has full generality and, therefore, can be applied in any field of applied mechanics. Moreover, there are no restrictions on the numerical implementation, as the singular integrals may be defined on surface elements or internal cells of any order and type.

scholarly journals An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

2011 ◽  
Vol 2011 ◽  
pp. 1-21
Author(s):  
Samir A. Ashour ◽  
Hany M. Ahmed
Keyword(s):  
Numerical Evaluation ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Finite Part ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value

Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals , and extend Clenshow's algorithm to evaluate these integrals in a stable way.

scholarly journals Rational Transformations for Evaluating Singular Integrals by the Gauss Quadrature Rule

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 677
Author(s):  
Beong In Yun
Keyword(s):  
Numerical Evaluation ◽  
Singular Integrals ◽  
Quadrature Rule ◽  
Transformation Method ◽  
Gauss Quadrature ◽  
Cauchy Principal ◽  
Weakly Singular ◽  
Principal Value ◽  
Transformation Methods ◽  
Gauss Quadrature Rule

In this work we introduce new rational transformations which are available for numerical evaluation of weakly singular integrals and Cauchy principal value integrals. The proposed rational transformations include parameters playing an important role in accelerating the accuracy of the Gauss quadrature rule used for the singular integrals. Results of some selected numerical examples show the efficiency of the proposed transformation method compared with some existing transformation methods.

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