薄体结构温度场的高阶边界元分析

分析了二维问题边界元法3节点二次单元的几何特征，区分和定义了源点相对高阶单元的Ⅰ型和Ⅱ型接近度.针对二维位势问题高阶边界元中奇异积分核，构造出具有相同Ⅱ型几乎奇异性的近似核函数，在几乎奇异积分单元上分离出积分核中主导的奇异函数部分.原积分核扣除其近似核函数后消除几乎奇异性，成为正则积分核函数，并采用常规Gauss数值方法计算该正则积分；对奇异核函数的积分推导出解析公式，从而建立了一种新的边界元法高阶单元几乎奇异积分半解析算法.应用该算法计算了二维薄体结构温度场算例，计算结果表明高阶单元半解析算法能充分发挥边界元法优势，显著提高计算精度.

High-Order Boundary Element Analysis of Temperature Fields in Thin-Walled Structures

The geometric features of 3-node elements in the 2D BEM were analyzed, and the relative distance (namely the approach degree) from a source point to a high-order element was defined. Based on the geometric features, the approximate kernel functions were constructed with the same II-type singularity as the nearly singular kernel functions. For the nearly singular integrals, the dominant singular parts were separated from the original kernel functions through subtraction. After subtraction of the approximate kernel functions, the original kernel functions were rid of near singularity and turned into the sum of two integrals, of which one was a regular integral to be evaluated accurately with the conventional Gaussian quadrature, the other was a singular integral to be calculated with a series of analytical formulae derived herein. Then a new semi-analytical algorithm was established to compute the nearly singular integrals for the high-order elements effectively. In verification, the new method was applied to calculate several temperature field examples of thin-body structures for 2D potential boundary element analysis. The results indicate that the presented high-order-element semi-analytic algorithm takes full advantage of the BEM and has highly improved calculation accuracy and efficiency.