Weakly singular, singular and hypersingular integrals in 3‐D elasticity and fracture mechanics

2015

Adv Comput Math

This article considers methods of weakly singular and hypersingular integral regularization based on the theory of distributions. For regularization of divergent integrals, the Gauss-Ostrogradskii theorem and the second Green's theorem in the sense of the theory of distribution have been used. Equations that allow easy calculation of weakly singular, singular, and hypersingular integrals in oneand two-dimensional cases for any sufficiently smooth function have been obtained. These equations are compared with classical methods of regularization. The results of numerical calculation using classical approaches and those based of the theory of generalized functions, along with a comparison for different functions, are presented in tables and graphs of the values of divergent integrals versus the position of the colocation point.

mentioning

confidence: 98%

mentioning

confidence: 99%

Regularization of divergent integrals: A comparison of the classical and generalized-functions approaches

2015

Adv Comput Math

“…One-dimensional (1-D) and multi-dimensional divergent integrals can also be calculated using the same method. For example, two-dimensional (2-D) hypersingular integrals that arise from the BIE solution of the 3-D static and dynamic problems of fracture mechanics have been considered in [9,10].…”

Section: Introductionmentioning

confidence: 99%

Regularization of the Divergent Integrals. I. General Consideration.

2007

EJBE

This article considers weakly singular, singular and hypersingular integrals, which arise when the boundary integral equation (BIE) methods are used to solve problems in science and engineering. For their regularization, an approach based on the theory of distribution and 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. Such approach may be easily generalized and applied to the calculation of multidimensional integrals with singularities of various types.

“…We applied the theory of distribution approach for the first time in [8], which was further developed in [10][11][12] and applied for fracture dynamics problems in our numerous publications [1][2][3]. In our earlier publications the regularization technique was based on Green theorem and for each divergent integral specific method had to be developed.…”

Section: Introductionmentioning

confidence: 99%

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

2007

EJBE

In this article the methodology for divergent integral regularization developed in [9] is applied for the regularization of the weakly singular and hypersingular integrals, which arise when the boundary integral equation (BIE) methods are used to solve fracture mechanics problems. 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 are transformed into the regular contour integrals that can be easily calculated analytically or numerically.