Hypersingular Integrals in Integral Equations and Inequalities: Fundamental Review Study

2019 ◽  
pp. 687-717 ◽  
Author(s):  
Suzan J. Obaiys ◽  
Rabha W. Ibrahim ◽  
Ahmad F. Ahmad
Keyword(s):  
Integral Equations ◽  
Hypersingular Integrals ◽  
Review Study

scholarly journals Regularization of the Divergent Integrals. II. Application in Fracture Mechanics.

2007 ◽  
Vol 4 (2) ◽  
Author(s):  
Vladimir Zozulya
Keyword(s):  
Fracture Mechanics ◽  
Integral Equations ◽  
Convex Polygon ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Hypersingular Integrals ◽  
Divergent Integral ◽  
Contour Integrals ◽  
Weakly Singular ◽  
Arbitrary Convex

In this article the methodology for divergent integral regularization developed in [8] is applied for regularization of the weakly singular and hypersingular integrals, which arise when the boundary integral equations (BIE) methods are used to solve problems in fracture mechanics. The approach is based on the theory of distribution and the application of the Green theorem. The weakly singular and hypersingular integrals over arbitrary convex polygon have been transformed to the regular contour integrals that can be easily calculated analytically or numerically.

scholarly journals Optimal with respect to accuracy methods for evaluating hypersingular integrals

2021 ◽  
Vol 23 (4) ◽  
pp. 360-378
Author(s):  
Ilya V. Boykov ◽  
Alla I. Boykova
Keyword(s):  
Integral Equations ◽  
Electromagnetic Fields ◽  
Quadrature Formulas ◽  
Power Functions ◽  
Order Of Accuracy ◽  
Hypersingular Integrals ◽  
Hypersingular Integral ◽  
Real Positive Number ◽  
Physical Fields ◽  
Derivatives Of

In this paper we constructed optimal with respect to order quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M), Ω¯ur,γ(Ω,M), Ω=[−1,1]l, l=1,2,…,M=Const, and γ is a real positive number. The functions that belong to classes Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M) have bounded derivatives up to the rth order in domain Ω and derivatives up to the sth order (s=r+⌈γ⌉) in domain Ω∖Γ, Γ=∂Ω. Moduli of derivatives of the vth order (r<v≤s) are power functions of d(x,Γ)−1(1+|lnd(x,Γ)|), where d(x,Γ) is a distance between point x and Γ. The interest in these classes of functions is due to the fact that solutions of singular and hypersingular integral equations are their members. Moreover various physical fields, in particular gravitational and electromagnetic fields belong to these classes as well. We give definitions of optimal with respect to accuracy methods for solving hypersingular integrals. We constructed optimal with respect to order of accuracy quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M).

scholarly journals A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations

1992 ◽  
Vol 59 (3) ◽  
pp. 604-614 ◽  
Author(s):  
M. Guiggiani ◽  
G. Krishnasamy ◽  
T. J. Rudolphi ◽  
F. J. Rizzo
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Collocation Point ◽  
Three Dimensional ◽  
Careful Analysis ◽  
Boundary Integral ◽  
Hypersingular Integrals ◽  
Closed Surfaces ◽  
First Time ◽  
General Method

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

3D FRACTURE MECHANICS FOR A MODE I CRACK IN PIEZOELECTRIC MEDIA

2004 ◽  
Vol 01 (03) ◽  
pp. 445-456 ◽  
Author(s):  
MENG-CHENG CHEN
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Electric Displacement ◽  
Basic Density ◽  
Mode I ◽  
Singular Stress ◽  
Hypersingular Integrals ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
Electric Displacement Intensity Factor

This paper deals with mode I fracture problems for a planar crack in an infinite piezoelectric solid subjected to electric and tension loading. The finite-part integral concept is used to prove rigidly hypersingular integral equations for the crack by using three-dimensional linear piezoelectricity theory. Investigations on the singularities and the singular stress and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional hypersingular integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by the definitions similar to those of elasticity. Then, a numerical method for the solution of the hypersingular integral equations is developed, in which the displacement and electric potential differences across the crack surfaces are approximated with a product of basic density functions and polynomials. Numerical solutions of several typical planar cracks are obtained with high accuracy.

Complex hypersingular integrals and integral equations in plane elasticity

1994 ◽  
Vol 105 (1-4) ◽  
pp. 189-205 ◽  
Author(s):  
A. M. Linkov ◽  
S. G. Mogilevskaya
Keyword(s):  
Integral Equations ◽  
Plane Elasticity ◽  
Hypersingular Integrals

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.

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