For boundary element analysis of the orthotropic potential problems in thin structures, the higher order elements are expected to discretize the boundary. However, the use of the higher order elements leads to more complex forms of the integrands in boundary integral equations. The resulting nearly singular integrals on the higher order elements are difficult to be evaluated when the source point is very close to the integral element. In this paper, a semi-analytic algorithm is presented to evaluate the nearly singular integrals on the quadratic elements in two dimensional (2-D) orthotropic potential problems. By constructing the approximate singular integral kernels, the nearly singular integrals through subtraction technique are transformed into the sum of regular parts and singular parts. Then, the former are calculated by the conventional Gaussian quadrature and the latter are calculated by the analytical integral formulas. Numerical examples demonstrate that the present semi-analytic algorithm is efficient and accurate to calculate the nearly singular integrals on the quadratic elements. Especially, the BEM with the present semi-analytic algorithm is successfully applied to analyzing 2-D orthotropic potential problems in very thin structures.

对于薄结构中正交异性势问题的边界元分析，期望高阶元离散化边界。但是，使用高阶元素会导致边界积分方程中被积分数的形式更加复杂。当源点非常接近积分元素时，很难评估在高阶元素上产生的几乎奇异的积分。在本文中，提出了一种半解析算法来评估二维（2-D）正交各向异性势问题中二次元上的近奇异积分。通过构造近似奇异积分核，通过减法技术将近似奇异积分转换为正则部分和奇异部分之和。然后，前者由常规高斯求积法计算，后者由解析积分公式计算。数值算例表明，该半解析算法对于计算二次元上的近似奇异积分是有效且准确的。特别是，采用本半解析算法的边界元法成功地用于分析非常薄结构中的二维正交各向异性势问题。