The nearly singular integral, arising in simulating thin coatings or close-boundary physical quantities, are not adequately dealt with in the isogeometric boundary element method (IGABEM), especially in 3D problems. In this paper, we propose a semi-analytical approach for the nearly singular integrals of 3D potential problems. We first expand all the kernel items by Taylor series up to second order accuracy. In order to employ the semi-analytical formulae when integrating in parametric space, coordinate $(\xi ,\eta )$ is further transformed to polar coordinate $(\rho ,\theta )$. We then use the subtraction technique to separate the integrals to near-singular parts and regular parts. For the near-singular parts, a semi-analytical treatment is performed where the integrations with respect to $\rho$ are expressed by analytical formulae recursively, while the ones related to $\theta$ are computed by Gaussian quadrature. The remaining regular integrals are treated numerically by the $sinh$ transformation method. By adding them together, we could efficiently handle the nearly singular integrals and therefore obtain accurate close-boundary potentials and flux densities in 3D potential problems. The accuracy of the presented method for nearly singular integrals to a curved element with different orders of singularities, namely the nearly weakly, strongly and highly singular integrals, are first tested. We then further consider potential problems of three typical 3D structures. All the presented results are compared with the recently proposed improved $sinh$ transformation method and analytical solutions. The above numerical examples fully show the efficiency and competitiveness of the presented semi-analytical schemes.

在模拟薄涂层或近边界物理量时产生的近奇异积分在等几何边界元法 (IGABEM) 中没有得到充分处理，尤其是在 3D 问题中。在本文中，我们针对 3D 潜在问题的近奇异积分提出了一种半解析方法。我们首先通过泰勒级数将所有内核项扩展到二阶精度。为了在参数空间中积分时使用半解析公式，坐标$(\xi ,\eta )$进一步转换为极坐标$(\rho ,\theta )$. 然后我们使用减法技术将积分分离为近奇异部分和规则部分。对于接近奇异的部分，进行半解析处理，其中积分相对于$\rho$由解析公式递归表示，而与$\theta$由高斯求积计算。剩余的正则积分由$辛$转化法。通过将它们加在一起，我们可以有效地处理近乎奇异的积分，从而在 3D 潜在问题中获得准确的近边界势和通量密度。首先测试了所提出的方法对具有不同奇点阶数的弯曲单元的近似奇异积分的准确性，即近似弱、强和高度奇异积分。然后我们进一步考虑三个典型 3D 结构的潜在问题。所有呈现的结果都与最近提出的改进方法进行了比较$辛$变换方法和解析解。上述数值例子充分展示了所提出的半解析方案的效率和竞争力。