任意核函数单元形状源点位置高效高精度奇异和近奇异积分

精确高效地计算奇异和近奇异积分，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。关于奇异和近奇异积分的研究已有几十年的历史，但是迄今仍没有一个统一的，对任意基本解类型、任意单元形状、任意源点位置都适用的，高效高精度的计算方法。而且，现有的近奇异积分方法都不能正确处理时域基本解在小时间步长时出现的近奇异积分，以及弹性波基本解的非连续问题。本项目提出一个球面细分方案：通过一系列半径呈指数级增长的球面分割单元，得到的单元子块拥有适宜高斯积分的最佳形状与尺寸，可以一次性彻底地解决边界元法中的所有奇异和近奇异积分难题。研究内容包括：（1）三维面单元球面细分法。（2）三维体单元球面细分法。（3）针对时域基本解小步长和弹性波非连续基本解的球面细分法。（4）针对细分子单元块的高斯积分准则。（5）基于球面细分的面积分和体积分的并行计算方法。

Accurate and efficient evaluation of singular integrals and nearly singular integrals is of crucial importance for successful implementation of the boundary element method (BEM), which is also one of the main obstacles for the BEM applied to engineering problems. The boundary integrals in BEM have been studied for a long time, and many methods have been proposed. However, none of them can evaluate the singular integrals and nearly singular integrals accurately and efficiently for cases of arbitrary type fundamental solution, arbitrary shape of element and arbitrary location of the source point. In this project a sphere subdivision method is proposed, which subdivides an element into a number of patches through a sequence of spheres with increasing radius and the obtained patches are good in shape and size for Gauss Quadrature, and hence the difficulties associating with integrals in the BEM can be completely overcome. The research contents are as following: (1) Automatic sphere subdivision method for surface elements; (2) Automatic sphere subdivision method for volume elements; (3) Sphere subdivision method for time-domain fundamental solutions; (4) The Gaussian Quadrature criterion for integrals on patches. (5) Parallel implementation of the numerical integration in BEM based on the Sphere Subdivision Method.