边界元中计算任意高阶奇异线积分的直接法

A DIRECT METHOD FOR EVALUATING LINE INTEGRALS WITH ARBITRARY HIGH ORDER OF SINGULARITIES

提出了一种精确计算任意高阶奇异曲线积分的直接计算法.首先将曲线单元上的各种几何量用投影线上的几何量来表示,然后通过幂级数展开和解析的方法显式地消除了积分的奇异性.还导出了计算等参坐标对局部直角坐标偏导数的表达式.由于这种方法涉及到的是总体尺度间的坐标变换,操作起来直观明了,可以处理二维问题边界元分析中出现的任意高阶奇异边界积分.最后用具体算例验证该方法的正确性.

This paper presents a new direct method for evaluating arbitrary singular boundary integrals appearing in 2D boundary element analysis. Firstly, geometry quantifies on a curved line element are expressed using those projected on a tangential line. Then, singularities involved in the integrals are analytically removed by expressing the non-singular part of the integration kernel as power series. A set of formulations for computing the first and second derivatives of intrinsic coordinates with respect to local orthogonal coordinates are also presented in the paper for the first time. Since the coordinate transformation is at the real spatial scale, the operation is straightforward and convenient, and can be applied to treat arbitrary high order of singular integrals. Finally, some examples are given to verify the correctness and stability of the presented method.