三维位势问题的梯度边界积分方程的新解法

三维位势问题的边界元分析中,关于坐标变量的边界位势梯度的计算是一个困难的问题. 已有一些方法着手解决这个问题,然而,这些方法需要复杂的理论推导和大量的数值计算. 本文提出求解一般边界位势梯度边界积分方程的辅助边值问题法. 该方法构造了与原边界值问题具有相同解域的辅助边值问题,该辅助边值问题具有已知解,因此通过求解此辅助边值问题,可获得梯度边界积分方程对应的系统矩阵,然后将此系统矩阵应用于求解原边值问题,求解过程非常简单,只需求解一个线性系统即可获得原边值问题的解. 值得注意的是,在求解原边值问题时,不再需要重新计算系统矩阵,因此辅助边值问题法的效率并不很差. 辅助边值问题法避免了强奇异积分的计算,具有数学理论简单、程序设计容易、计算精度高等优点,为坐标变量梯度边界积分方程的求解提供了一个新的途径. 3个标准的数值算例验证了方法的有效性. 

A NEW METHOD FOR SOLVING THE GRADIENT BOUNDARY INTEGRAL EQUATION FOR THREE DIMENSIONAL POTENTIAL PROBLEMS 1)

In the boundary element analysis of three-dimensional potential problems, it is a very difficult task to calculate the boundary potential gradients with respect to the space coordinates instead of normal one. Several techniques have been proposed to address this problem so far. They, however, usually need complex and lengthy theoretical deduction as well as a large number of numerical manipulation. In this study, a new method, named auxiliary boundary value problem method (ABVPM), is presented for solving the gradient boundary integral equation (GBIE) for three dimensional potential problems. An ABVPM with the same solution domain as the original boundary value problem is constructed, which is an over-determined boundary value problem with known solution. Consequently, the system matrix of the GBIE, which is the most important problem for boundary analysis, will be obtained by solving this ABVPM. It can be used to solve original boundary value problem. The solution procedure is very simple, because only a linear system need to be solved to obtain the solution of the original boundary value problem. It is worth noting that when solving the original boundary value problem, it is not necessary to recalculate the system matrix, so the efficiency of the auxiliary boundary value method is not very poor. The proposed ABVPM circumvents the troublesome issue of computing the strongly singular integrals, with some advantages, such as simple mathematical deduction, easy programming and high accuracy. More importantly, the ABVPM provides a new idea and way for solving the GBIE. Three benchmark examples are tested to verify the effectiveness of the proposed scheme.