基于分层高阶四边形亚参元分析的PCG法研究  

Study on PCG method based on hierarchical higher-order quadrilateral subparametric finite element analysis

作　　者：肖映雄[1] 谢凌洁 XIAO Ying-xiong;XIE Ling-jie(Civil Engineering and Mechanics College,Xiangtan University,Xiangtan 411105,Hunan,China)

机构地区：[1]湘潭大学土木工程与力学学院,湖南湘潭411105

出　　处：《云南大学学报（自然科学版）》2022年第5期877-887,共11页Journal of Yunnan University(Natural Sciences Edition)

基　　金：国家自然科学基金(11601462);湖南省教育厅科研项目(19A502)。

摘　　要：在二维有限元分析中,为了提高计算精度,往往需要采用高阶单元.这类单元具有某些特殊的优点,例如,能解决弹性问题的闭锁现象.但与线性元相比,高阶单元节点数多,单元分析复杂,相应离散化矩阵又具有病态性,利用通常的方法求解有限元方程时其效率将大大降低.针对分层高阶四边形亚参元分析中的离散化线性代数系统,通过利用其系数矩阵的分层结构特性以及对应分块矩阵的代数性质,设计了一种简单、有效的预条件子.该预条件子的计算主要化归为Q元离散化系统的求解,通过调用已有的代数多重网格(GAMG)法高效求解Q元系统,获得了内迭代计算效率得到显著提升的预条件共轭梯度(PCG)法.数值试验结果验证了PCG法的有效性,为分层高阶有限元分析提供了一种高效求解器.In order to improve the calculation accuracy of the finite element analysis in two dimensions,the higher-order elements are often used in that they are superior and necessary under certain conditions over the lowerorder ones,for example,they can overcome the Poisson’s ratio locking.However,the higher-order elements have many nodes,the element analysis is complex and the coefficient matrix of the resulting system of equations is ill conditioned.Thus,the efficiency of the commonly used solvers will be rapidly reduced.A simple and efficient preconditioner is then presented for the system of equations arising from the hierarchical higher-order quadrilateral subparametric finite element discretizations by combining the hierarchical structure of the coefficient matrix and the properties of the resulting block diagonal matrices.The basic idea of this method was to essentially turn the hierarchical higher-order discrete systems into mainly solving the corresponding Q 4 element discrete systems.In this way,we could obtain preconditioned conjugate gradient(PCG)method whose efficiency of inner iteration had been greatly improved by using the existing efficient GAMG methods.The results of some numerical experiments have verified the efficiency of the corresponding PCG method and this will provide a fast solver for the hierarchical higher-order finite element analysis.

关 键 词：分层高阶单元 病态矩阵 代数多重网格法 预条件子 数值模拟 

分 类 号：O343.3[理学—固体力学] TB115[理学—力学]