This paper focuses on near singularity in weakly singular integrals of three-dimensional boundary element method. In traditional methods, only one directional transformation is introduced to remove the weak singularity. However, near singularity is still existed. Thus in our paper, serval methods for removal of weak singularity are introduced firstly. Then, angular and sigmoidal transformations are introduced to remove the near singularity in the circumferential direction combined with polar transformation. Generalized Duffy transformation is introduced to remove weak singularity in u direction, which also separates the near singularity in v direction. Based on this method, sinh transformation is constructed with the integrand of v. Finally, several numerical examples considering large top opening angle of the singular integration element and large ratio of the two edge length of the angle (the max edge length/ the min edge length) are proposed. Results demonstrate that generalized Duffy transformation coupled with sinh transformation in v direction is a best choose for weakly singular integrals.

本文重点研究三维边界元方法的弱奇异积分中的近奇异性。在传统方法中，仅引入一种方向变换来消除弱奇异性。但是，仍然存在接近奇异的情况。因此在本文中，首先介绍了去除弱奇异点的服务方法。然后，引入角和S形变换，以消除周向与极性变换相结合的奇异性。引入了通用Duffy变换以消除u方向上的弱奇点，这也分离了v方向上的近奇点。在此方法的基础上，用v的整数构造正弦变换。最后，提出了几个数值示例，这些示例考虑了奇异积分元件的最大顶部张开角度和该角度的两个边缘长度（最大边缘长度/最小边缘长度）的比率较大。结果表明，广义Duffy变换与v方向上的sinh变换是弱奇异积分的最佳选择。