基于球面细分法的高效高精度近奇异积分计算

精确高效地计算近奇异积分，对边界元法的成功实施至关重要，也是边界元法在实际工程计算中面临的主要障碍之一。论文提出了一种基于球面细分技术的近奇异积分计算方法，可以精确计算任意基本解类型、任意单元形状和任意源点位置的近奇异积分。该方法首先通过计算源点到单元的最近最远距离，来确定球面细分的初始半径和终止半径；然后通过一系列半径呈指数级增长的球面来分割积分单元，得到一系列三角形和四边形子单元；最后把细分后得到的子单元变成弧形状，即三角形和四边形子单元分别变成扇形和环形子单元。由于球面细分是直接在三维笛卡尔坐标系下进行的，所以它适用于任何类型的单元。此外，由于基本解主要是源点到场点距离的函数，因此在同等精度下，近奇异积分在子单元的环向上所需要的高斯积分点数将大大减少。在径向方向上，由于球半径系列呈指数级变化，各个子块可以做到等精度高斯积分。数值算例表明，与传统近奇异积分计算方法相比，论文提出的方法更加稳定，精度更高。

Evaluating nearly singular integrals accurately and efficiently based on sphere subdivision method

The boundary element method (BEM) has been widely used for solving engineering and scientific problems. Compared with the finite element method (FEM), the BEM is more attractive for its dimension reduction feature and higher accuracy. Accurate and efficient evaluation of nearly singular integrals is of crucial importance for successful implementation of the BEM. The nearly singular integrals in BEM have been studied for a long time, and many methods have been proposed. However, none of them can evaluate these integrals accurately and efficiently. In this paper, a method based on sphere subdivision technique is proposed for evaluating nearly singular integrals. With the method, the nearly singular integrals can be evaluated accurately and efficiently for cases of arbitrary type fundamental solution, arbitrary shape of element and arbitrary location of source point. In the proposed method, the minimum and maximum distances between the source point and the integration element are firstly computed, which determine the beginning and ending of sphere radius. Then triangular and quadrilateral sub-elements can be obtained by subdividing the integration element through a sequence of spheres with exponential increasing radius. Finally the obtained sub-elements are turned into arc-shape ones, i.e. the triangular and quadrilateral sub-elements are changed to flabellate and annular sub-elements, respectively. The sphere subdivision is performed in 3D Cartesian coordinate system, thus the proposed method is suitable for any elements. In addition, fundamental solution is a function of the distance between the source point and the field point, so in the same level of accuracy, the number of Gaussian point can greatly decrease in circular direction for evaluating nearly singular integrals on sub-elements. Because of exponential growth of sphere radius, the integration can be of the same level of accuracy in the radial direction. The numerical examples have demonstrated that the proposed method has much better stability and accuracy than conventional methods.