三维位势问题新的规则化边界元法

本文致力于三维位势问题的间接变量规则化边界元法研究,提出了新的规则化边界元法的理论和方法.构造了与法向量关联的两个线性无关的特别切向量,建立与问题基本解有关的量的法向、切向梯度的特性定理,提出转化域积分方程为边界积分方程的极限定理,在此基础上,导出间接变量规则化边界积分方程.与广泛实践的直接边界元法比,本文具有优点：（1）降低了密度函数的连续性要求;（2）更适合求解薄体结构问题.因为所给方程中不含超奇异与几乎超奇异积分,积分的规则化算法更加有效;（3）可计算任何边界位势梯度.数值实施时,C0连续单元描述几何曲面,不连续插值逼近边界量.针对问题的特殊的边界曲面,提出一种精确几何单元.数值算例表明,本文算法稳定、效率高,所得数值结果与精确解相当地吻合.

A new regularized BEM for 3D potential problems

This presentation is mainly devoted to the research on the regularization of indirect boundary integral equations （IBIEs） for three-dimensional problems and establishes the new theory and method of the regularized BEM. The two special tangential vectors, which are linearly independent and associated with the normal vectors, are constructed, and then a characteristics theorem for the contour integrations of the normal and tangential gradients of some quantities, related with the fundamental solutions for 3D potential problems, is presented. A limit theorem for the transformation from domain integral equations into boundary integral equations （BIEs） is also proposed. Based on this, together with a novel decomposition technique to the fundamental solution, the regularized BIEs with indirect unknowns, which don＇t involve the direct calculation of CPV and HFP integrals, are derived for 3D potential problems. Compared with the widely practiced direct regularized BEMs, the presented method has many advantages. First, the continuity requirement for density function in the direct formulation can be reduced here. Second, it is more suitable for solving the structures of thin bodies, considering the solution process for boundary or field quantities doesn＇t involve the HFP integrals and nearly HFP integrals so the regularization algorithm to the considered singular or nearly singular integrals is more effective. Third, the proposed regularized BIEs can calculate the any potential gradients on the boundary, but not limited to the normal fluxes, and also independent of the potential BIEs. A systematic approach for implementing numerical solutions is proposed by adopting the CO continuous elements to depict the boundary surface and the discontinuous interpolation to approximate the boundary quantities. Especially, for the boundary value problems with elliptic surfaces or piecewise plane surfaces boundary, the exact elements are developed to model their boundaries with almost no error. The validity of the proposed scheme is demonstrated by several benchmark examples. Excellent agreement between the numerical results and exact solutions is obtained even with using small amounts of element.