自适应快速多极正则化无网格法求解大规模三维位势问题

Adaptive Fast Multipole Regularized Meshless Method for Large-Scale Three Dimensional Potential Problems

正则化无网格法(regularized meshless method,RMM)是一种新的边界型无网格数值离散方法.该方法克服了近年来引起广泛关注的基本解方法(method of fundamental solutions,MFS)的虚假边界缺陷,继承了其无网格、无数值积分、易实施等优点.另一方面,RMM方法同MFS方法的插值方程都涉及非对称稠密系数矩阵,运用常规代数方程的迭代法求解时都要求O(N2)量级的乘法计算量和存储量.随着问题自由度的增加,该方法的计算量增加极快,效率较低,一般难以计算大规模问题.为了克服这个缺点,利用对角形式的快速多级算法(fast multipole method,FMM)来加速RMM方法,发展了快速多级正则化无网格法(fast multipole regularized mesheless method,FM-RMM).该方法无需数值积分并且具有O(N)量级的计算量和存储量,可有效地求解大规模工程问题.数值算例表明,FM-RMM算法可成功在内存为4GB的Core(TM)Ⅱ台式机上求解高达百万级自由度的三维位势问题.

The regularized meshless method （RMM） is a new meshless boundary collocation method. This method overcame the perplexing fictitious boundary associated in the method of fundamental solutions （MFS）, while inherited all its merits being truly meshless, integrationfree, and easy-to-program. Like the MFS, the RMM also produced dense and nonsymmetric coefficient interpolation matrix, which required O（N^2） multiplication operations and memory requirements in an iterative solution procedure. Since the calculation operations would dramatically increase with the number of DOF, the RMM was computationally too expensive to solve large-scale problems. In order to overcome this bottleneck, this study combined the RMM with the popular diagonal form fast multipole method （FMM） to develop the fast multipole regularized meshiess method （FM-RMM）. The proposed scheme was integration-free and meshfree and significantly reduced the operations and the memory requirements to the order of 0 （N） and consequently could much efficiently solve large scale problems. Numerical examples illustrate that the present FM-RMM method can successfully solve three-dimensional potential problems with up to 1 million nodes on a Core（TM） II desktop with 4 GB memory.