二维正交各向异性位势问题的高阶单元快速多极边界元法

研制了一种适用于二维正交各向异性位势问题的高阶单元(线性单元和二次单元)快速多极边界元法. 在快速多极边界元法中, 源点对于远场区域的积分采用快速多极展开式计算, 而对于近场区域的积分则直接进行计算. 高阶单元的使用使得近场积分, 尤其是奇异积分和几乎奇异积分的计算更加复杂. 通过引入复数表达对其进行简化, 若边界采用线性单元插值, 近场积分可直接解析计算; 若采用二次单元插值, 则给出一个半解析算法计算近场积分. 高阶单元奇异积分和几乎奇异积分计算难题的解决, 使得高阶单元快速多极边界元法不仅能够计算一般结构, 也能被应用于超薄体结构, 拓宽了高阶单元快速多极边界元法的适用范围. 数值算例表明, 若计算精度一定, 高阶单元快速多极边界元法较常值单元快速多极边界元法使用的单元数量显著减少, 且高阶单元快速多极边界元法计算时间与自由度数量成线性关系, 其计算效率仍处于$O(N)$量级, 因此高阶单元快速多极边界元法可更加高效求解大规模问题. 

ANALYSIS OF 2-D ORTHOTROPIC POTENTIAL PROBLEMS USING FAST MULTIPOLE BOUNDARY ELEMENT METHOD WITH HIGHER ORDER ELEMENTS

A new fast multipole boundary element method is proposed for analyzing 2-D orthotropic potential problems by using linear and three-node quadratic elements. In the fast multipole boundary element method, fast multipole expansions are used for the integrals on elements that are far away from the source point, and the direct evaluations are used for the integrals on elements that are close to the source point. The use of linear and three-node quadratic elements results in more complicated computations for near-field integrals, especially singular integrals and nearly singular integrals. In this paper, the complex notation is introduced to simplify the near-field integrals. If the boundary is discretized by linear elements, the near-field integrals are calculated by the analytic formulas, if the three-node quadratic element is used, a semi analytical algorithm is given to calculate the near-field integrals. Accurate evaluations of the singular integrals and nearly singular integrals on linear and three-node quadratic elements ensure that the present fast multipole boundary element method can be applied to the ultra-thin structure, which broadens the application of the fast multipole boundary element method with linear and quadratic elements. Numerical examples show that the number of elements required by the fast multipole boundary element method with linear and quadratic elements is significantly less than that with constant elements. In addition, the required CPU time is increased linearly with the increase of the number of degrees of freedom $(N)$, which demonstrates the computational efficiency is still in the complexity of $O (N)$. Therefore, the present method exhibits higher accuracy and efficiency for solving large-scale problems.