三维变系数热传导问题边界元分析中几乎奇异积分计算

摘要

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 in-tegral can’t be calculated directly by using the Gaussian quadrature formulas.A new method for evalua-ting the nearly singular boundary integral is presented in the paper based on 3D non-homogeneous heat conduction problems.In the proposed method,the Newton-Raphson iteration algorithm is adopted to de-termine 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.