Nearly singular approximations of CPV integrals with end- and corner-singularities for the numerical solution of hypersingular boundary integral equations.

*(English)*Zbl 1181.74147Summary: A local numerical approach to cope with the singular and hypersingular boundary integral equations (BIEs) in non-regularized forms is presented in the paper for 2D elastostatics. The approach is based on the fact that the singular boundary integrals can be represented approximately by the mean values of two nearly singular boundary integrals and on the techniques of distance transformations developed primarily in previous work of the authors. The nearly singular approximations in the present work, including the normal and the tangential distance transformations, are designed for the numerical evaluation of boundary integrals with end-singularities at junctures between two elements, especially at corner points where sufficient continuity requirements are met. The approach is very general, which makes it possible to solve the hypersingular BIE numerically in a non-regularized form by using conforming \(C^0\) quadratic boundary elements and standard Gaussian quadratures without paying special attention to the corner treatment.

With the proposed approach, an infinite tension plate with an elliptical hole and a pressurized thick cylinder were analyzed by using both the formulations of conventional displacement and traction boundary element methods, showing encouragingly the efficiency and the reliability of the proposed approach. The behaviors of boundary integrals with end- and corner-singular kernels were observed and compared by the additional numerical tests. It is considered that the weakly singularities remain but should have been cancelled with each other if used in pairs by the corresponding terms in the neighboring elements, where the corner information is included naturally in the approximations.

With the proposed approach, an infinite tension plate with an elliptical hole and a pressurized thick cylinder were analyzed by using both the formulations of conventional displacement and traction boundary element methods, showing encouragingly the efficiency and the reliability of the proposed approach. The behaviors of boundary integrals with end- and corner-singular kernels were observed and compared by the additional numerical tests. It is considered that the weakly singularities remain but should have been cancelled with each other if used in pairs by the corresponding terms in the neighboring elements, where the corner information is included naturally in the approximations.

##### MSC:

74S15 | Boundary element methods applied to problems in solid mechanics |

74K20 | Plates |

65N38 | Boundary element methods for boundary value problems involving PDEs |

##### Keywords:

Boundary element method; Hypersingular boundary integral equation; End-singularity; Corner-singularity; Nearly singular approximation; Distance transformation; Numerical evaluation
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\textit{H. Ma} and \textit{N. Kamiya}, Eng. Anal. Bound. Elem. 27, No. 6, 625--637 (2003; Zbl 1181.74147)

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