多重节点法并行求解大规模稀疏线性方程组  

Hierarchical node method for solving large-scale sparse linear equations in parallel

作　　者：王永庆 刘彬 Yongqing Wang;Bin Liu(Applied Mechanics Laboratory,Department of Engineering Mechanics,Tsinghua University,Beijing,100084,China)

机构地区：[1]Applied Mechanics Laboratory,Department of Engineering Mechanics,Tsinghua University,Beijing,100084,China

出　　处：《Acta Mechanica Sinica》2024年第4期159-171,共13页力学学报（英文版）

基　　金：the National Natural Science Foundation of China(Grant Nos.11720101002,11921002,and 11890674);Science Challenge Project(Grant No.TZ2018001).

摘　　要：本文以弹性力学为例,提出了多重节点法用于求解大规模稀疏线性方程组.多重节点法利用叠加性将复杂问题分解为简单的子问题,并利用局部性减少计算量.粗节点刚度阵和子区域刚度阵在迭代时不变,使得多重节点法适合求解多工况问题.粗节点和细节点之间的映射关系通过求解局部有限元模型得到而非插值法,使得多重节点法能够快速求解复杂拓扑问题.为了加快收敛速度,提出了最优粗节点刚度阵,并基于正则化最小二乘法给出了其计算方法.大规模数值实验表明,多重节点法具有很好的可扩展性以及近似线性加速比.与主流算法如多重网格法相比,多重节点法在求解多工况问题时收敛速度更快.In this paper,the hierarchical node method(HNM)is proposed for solving large-scale sparse linear equations in the context of elastic mechanics.HNM decomposes complex problems into simple subproblems using superposition and reduces computational costs using locality.In HNM,the coarse node stiffness matrix and subdomain stiffness matrix remain unchanged during the iteration process,which makes HNM suitable for solving multiple right-hand side problems.The mapping relationship between coarse nodes and fine nodes is obtained by solving local finite element problems rather than using the interpolation method,which enables HNM to solve complex topology problems rapidly.To accelerate the convergence rate of HNM,an optimal coarse node stiffness matrix is proposed,and a calculation method is provided based on the regularized least squares method.Large-scale numerical experiments show that HNM exhibits excellent scalability and a near-linear speedup.Compared to mainstream algorithms such as multigrid,HNM achieves faster convergence speeds when solving multiple right-hand side problems.

关 键 词：正则化最小二乘法 节点法 细节点 局部性 多重网格法 叠加性 并行求解 插值法 

分 类 号：O175[理学—数学]