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.

本文对一种有效的直接方法进行了改进，该方法用于对高阶奇异弯曲边界积分进行数值评估。然后，使用这种改进的奇异积分评估方法来求解双边界元方法（Dual BEM）中涉及的强和超奇异积分，该方法结合了位移和牵引边界积分方程的使用来解决单域公式中的裂纹问题。基于参数平面展开法和径向积分法，进行了奇异积分评估，提出了一种处理奇异径向积分的新方法，在该方法中起着至关重要的作用。在等参坐标系中，将奇异弯曲边界积分映射到固有坐标中的奇异方平面积分，然后采用径向积分法（RIM）将奇异的正方形平面积分变换为内在正方形平面轮廓上的规则线积分，并将其转换为从源点到内在正方形平面轮廓上的路径上的奇异径向积分。利用奇异性隔离技术将奇异的径向积分分为两部分，正常的径向积分通常可以使用高斯正交求值，奇异的径向可以通过将被积函数的非奇异部分扩展为幂来进行分析求值系列。与传统的局部插值方法处理奇异的径向积分相比，新提出的方法具有更严格的数学推导，并且可以获得更稳定，更精确的结果。基于成功执行奇异边界积分的直接评估，Dual BEM成功地用于解决二维和三维弹性裂纹问题，包括具有连续或不连续元素的直线和曲线裂纹路径。两种不同的方法，几何外推法和J积分法用于评估应力强度因子。给出了几个数值例子来验证所提出方法的有效性。