A new fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D elastostatic problems by using linear and three-node quadratic elements. The use of higher-order elements in BEM analysis results in more complex forms of the integrands, in which the direct Gaussian quadrature is difficult to calculate the singular and nearly singular integrals. Herein, the complex notation is first introduced to simplify all integral formulations (including the near-field integrals) in FM-BEM for 2-D elasticity. In direct evaluation of the near-field integrals, the nearly singular integrals on linear elements are calculated by the analytic scheme, and those on quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the present method possesses higher accuracy than the FM-BEM with constant elements. The computed efficiency of FM-BEM with higher order elements for analyzing large scale problems is still O(N), where N is the number of linear system of equations. In particular, the proposed FM-BEM is available for solving thin structures.

提出了一种新的快速多极边界元方法 (FM-BEM)，用于使用线性和三节点二次元分析二维弹性静力学问题。在边界元分析中使用高阶元素会导致被积函数的形式更加复杂，其中直接高斯求积很难计算奇异和近似奇异的积分。在此，首先引入复数符号以简化 FM-BEM 中二维弹性的所有积分公式（包括近场积分）。在近场积分的直接计算中，线性单元上的近奇异积分采用解析式计算，二次单元上的近场积分采用稳健的半解析式计算。数值例子表明，本方法比具有恒定元素的FM-BEM具有更高的精度。O ( N )，其中N是线性方程组的数量。特别是，提出的 FM-BEM 可用于求解薄结构。