三维位势快速多极虚边界元最小二乘法

Fast Multipole Virtual Boundary Element Least Square Method for Solving Three dimensional Potential Problems

摘要:

将快速多极展开法（FMM）和广义极小残值法(GMRES)结合于三维位势问题的虚边界元最小二乘法，使求解方程的计算量和储存量与所求问题的计算自由度数成线性比例；欲达到数值模拟大规模自由度问题的目的.基于位势问题虚边界元最小二乘法的数值求解格式，将对角化和指数展开系数的概念引入到常规的快速多极展开法中，将三维位势问题的基本解推导为更适合于快速多极算法的展开格式，并用广义极小残值法求解方程组，旨在达到进一步提高效率且仍保证较高计算精度的目的.数值算例说明了该方法的可行性，及计算效率和计算精度.

Abstract:

The fast multipole method (FMM) and generalized minimal residual (GMRES) algorithm are applied to virtual boundary element least square method to solve three dimensional potential problems, so that the amount of the computational elapsed time and the memory volume of the storage problems with the calculation of demand are linearly proportional to the number of degrees of freedom of the problem to be solved. Then the numerical simulation large scale degrees of freedom question might be achieved by the method. Based on the numerical form of virtual boundary element least square method for potential problems, the fundamental solutions of three dimensional potential problems are derived as the numerical scheme to be more suitable for FMM, through the introduction of concepts of diagonalization and exponential expansion moments, in order to further improve the efficiency of the problem with almost the same high precison. The GMRES algorithm is adopted to find the solution of matrix equation. The numerical examples relating to simulation of large scale problems achieved by the method verify the feasibility, efficiency and calculation precision of the method.