随机有限元分析的区域分解二级预处理子并行求解算法  

A domain decomposition two-level preconditioner parallel algorithm for stochastic finite element analysis

作　　者：付朝江 王珂 王中华 Fu Chaojiang;Wang Ke;Wang Zhonghua(College of Civil Engineering,Fujian University of Technology,350118,Fuzhou,China;Fujian-Taiwan Cooperative Institute of Civil Engineering Technology in Universities of Fujian Province,350118,Fuzhou,China)

机构地区：[1]福建工程学院土木工程学院,福州350118 [2]闽台合作土木工程技术福建省高校工程研究中心,福州350118

出　　处：《应用力学学报》2020年第5期2183-2189,I0024,I0025,共9页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金面上项目(51378124)。

摘　　要：针对谱随机有限元分析,采取区域分解来设计一个适用于大规模方程系统的线性求解器。为迭代求解大规模线性方程组,采用二级预处理子,通过对一级诺伊曼-诺伊曼(Neumann-Neumann)预处理子增加一个粗网格,构造二级预处理子;算法实现涉及求解每个子区域的局部问题和在所有子区域中传播信息的粗问题;在分布式消息传递接口(MPI)环境下实现并行计算,将本文提出的二级Neumann-Neumann预处理子算法与传统的一级预处理子算法的并行性能进行比较。计算结果表明,本文提出的二级Neumann-Neumann预处理子算法(PCG-NNC)比传统的一级预处理子算法(PCG-NN)的计算时间明显减少,效率提高,算例方案2中8处理器时PCG-NNC算法时间为777.39s,效率为72.6%,而PCG-NN算法时间为1169.00s,效率为63.4%。PCG-NNC显示出更好的性能,优于传统的一级预处理子算法,可有效地进行随机有限元分析。A two-level domain decomposition method is used to devise a linear solver for the large-scale system in the spectral stochastic finite element method(SSFEM). A two-level preconditioner is presented in order to iteratively solve the large-scale linear system. The two-level preconditioner is constructed by adding a coarse grid to the one-level Neumann-Neumann preconditioner. The implementation of the algorithm involves solving a local problem on each subdomain and a coarse problem that propagates information globally among the subdomains. A distributed implementation of the parallel algorithm is carried out using MPI. The parallel performances of the Neumann–Neumann two-level preconditioner are contrasted with the one-level preconditioner. Numerical studiesindicate that the computing time of proposed Neumann–Neumann two-level preconditioner algorithm(PCG-NNC) is significantly shorter than that of the one-level preconditioner algorithms(PCG-NN), and the efficiency of PCG-NNC is improved. For example, in example scheme 2 the time of PCG-NNC is 777.39 s, the efficiency is 72.6%, while the time of PCG-NN is 1169.00 s, and the efficiency is 63.4%. PCG-NNC is superior in performance to PCG-NN. The proposed approaches provide an efficient treatment of stochastic finite element analysis.

关 键 词：区域分解 预处理子 随机有限元法 SCHUR补 并行计算 

分 类 号：O246[理学—计算数学] O302[理学—数学]