边界元近奇异积分计算的迭代sinh⁃sigmoidal组合式变换法

An Iterated Sinh⁃Sigmoidal Combined Transformation Method for Calculating Nearly Singular Integrals of Boundary Elements

精确有效地消除积分的近奇异性是三维边界元法在工程应用中的首要问题.当源点与三角形积分单元间的距离无限趋近于零时,会出现近奇异积分问题,积分单元的形状和投影点的位置都是影响近奇异积分计算精度的重要因素.现有的非线性变换法大多只关注径向上积分的近奇异性,而忽略了角度方向和积分单元形状的影响,在投影点接近三角形积分单元边界的情况下,无法获得令人满意的计算精度,并且对子三角形积分单元的形状非常敏感.因此提出了一种改进的基于自适应分块技术和不同坐标变换的迭代sinh⁃sigmoidal组合式变换法,分别消除径向和角度方向积分的近奇异性,在确保计算精度的同时,大大减小了计算规模.数值算例验证了该方法的有效性.

Accurate and effective elimination of the near singularity of integrals is the primary problem in engi⁃neering application of 3D BEM.Nearly singular integrals will appear when the distance between source points and the triangle integral element approaches zero indefinitely.The shape of the integral element and the position of the projection points are both important factors influencing the computation accuracy of nearly singular inte⁃grals.Most of the existing nonlinear transformation methods only focus on the near singularity of integrals in the radial direction,ignoring the angular direction and the shape of triangular integral elements,so the calculation accuracy is always unsatisfactory and very sensitive to the shape of the integral elements of sub⁃triangles in the case of projection points near the boundary.An improved iterated sinh⁃sigmoidal combined transformation method based on the adaptive partitioning technology and different coordinate transformations was proposed to eliminate the near singularity of integrals in radial and angular directions respectively,which can greatly reduce the calculation scale while ensuring the calculation accuracy.Numerical examples were presented to verify the proposed method.