三维问题边界元法中几乎奇异积分的正则化算法

当源点靠近边界单元时,边界积分方程通常存在几乎奇异积分的计算难题.基于三角形单元,将源点到单元的距离与单元特征长度比值定义为接近度,用于度量边界单元中积分奇异性的程度.将单元上的面积分在局部的极坐标系ρθ下表示,利用一些初等函数的积分公式,获得对变量ρ作单层积分的解析表达式.几乎强奇异和超奇异面积分被转化为沿单元围道上一系列线积分,而Gauss数值积分能够有效计算这些线积分.应用该算法分析三维弹性薄壁结构获得了成功.

A REGULARIZATION ALGORITHM FOR THE NEARLY SINGULAR INTEGRALS IN 3-D BEM

The nearly singular integrals occur in the boundary element analysis for the thin-wall structures and calculating the quantities at interior points close to the boundary. In this paper, the triangular boundary element with three nodes is considered in the three-dimensional boundary element analysis, where the relative distance from a source point to the element is introduced to measure the singularities of the integrals. The smaller the relative distance is, the more difficult the integrals are to be evaluated. The surface integrals on the triangular element are expressed in a local polar coordinate system p.Then the integrals are analytically integrated with respect to the polar variable p by means of the integral formulae of some elementary functions. Thus the nearly singular surface integrals are transformed into a series of line integrals along the contour of the element. The resulting line integrals are computed efficiently by the Gauss numerical quadrature instead of the original singular surface integrals. Moreover, the regularization algorithm can also be applied to higher order elements by subdividing the elements into several triangular sub-elements. Here, the algorithm is employed successfully to analyze the thin-walled structures of the three-dimensional elasticity problems.