三维变系数热传导问题边界元分析中几乎奇异积分计算

Evaluation of nearly singular integrals in boundary element analysis of 3D heat conduction problem with variable coefficients

在边界积分的数值计算过程中,当源点离积分单元很近时,边界积分就会具有几乎奇异性,此时不能直接用高斯数值积分公式计算几乎奇异积分。本文以三维非均质热传导问题为例,介绍了一种计算几乎奇异边界积分的新方法。首先,采用Newton-Raphson迭代算法确定积分单元上离源点最近的点;然后,将积分单元上任意一点的坐标在最近点处展开成泰勒级数,并计算源点到积分单元任意点的距离;最后,将距离函数代入几乎奇异边界积分中,并运用指数变换方法导出积分单元上几乎奇异积分的计算公式。文中给出了两个非均质热传导问题的算例来验证所述方法的正确性、有效性和稳定性。

When the source point is very close to the integrated element in the numerical evaluation of boundary integrals, nearly singularity will appear in the boundary integrals, which results in that the integral can＇t be calculated directly by using the Gaussian quadrature formulas. A new method for evaluating the nearly singular boundary integral is presented in the paper based on 3D non-homogeneous heat conduction problems. In the proposed method,the Newtonaphson iteration algorithm is adopted to determine the point on the boundary element which is closest to the source point; and then the distance from the source point to any point on the element is calculated by expanding the coordinates at the point as Taylor series of the closet point; finally, the integration formula for evaluation of the nearly singular boundary integral is derived by substituting the distance function into the nearly singular boundary inte- gral and using the exponential transform method. Two numerical examples for 3D non-homogeneous heat conduction problems are given to verify the correctness, effectiveness and stability of the presented method.