Abstract
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.
Highlights
A huge amount of publications is devoted to the boundary integral equation methods BIM and its application in science and engineering 1–5
One of the main problems arising in the numerical solution of the BIE by the boundary element method BEM is a calculation of the divergent integrals
The weakly singular integrals are considered as improper integrals; the singular integrals are considered in the sense of Cauchy principal value PV ; the hypersingular integrals had been considered by Hadamard as finite part integrals FP
Summary
A huge amount of publications is devoted to the boundary integral equation methods BIM and its application in science and engineering 1–5. Different methods have been developed for calculation of the divergent integrals. The correct mathematical interpretation of the divergent integrals with different singularities has been done in terms of the theory of distributions generalized functions. In our previous publications 9–19 , approach based on the theory of distributions has been developed for regularization of the divergent integrals with different singularities. The above mentioned approach for regularization of the hypersingular integrals has been used for the first time in 11 It was further developed for static and dynamic problems of fracture mechanics in 15, 18, 19 , respectively, and in 9, 10, 13. The approach for the divergent integral regularization based on the theory of distribution and Green’s theorem is further developed. The equations that enable easy calculation of the weakly singular, singular, and hypersingular integrals for different shape functions are presented here
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