An improved singular curved boundary integral evaluation method and its application in dual BEM analysis of two- and three-dimensional crack problems

Abstract

In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and J-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.