求解三维Wilson元离散化线性系统的PCG方法  

Preconditioned Conjugate Gradient Methods for the 3D Wilson Nonconforming FEM Discretizations

作　　者：肖映雄[1] 李真有 

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：《应用数学和力学》2016年第8期820-831,共12页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(10972191);湖南省自然科学基金(14JJ2063);湖南省教育厅资助科研项目(15A183)~~

摘　　要：非协调元方法是克服三维弹性问题体积闭锁的一种有效方法,它具有自由度少、精度高等优点,但要提高其有限元分析的整体效率还必须为相应的离散化系统设计快速求解算法.考虑了Wilson元离散化系统的快速求解.当Poisson(泊松)比ν→0.5时,该离散系统为一高度病态的正定方程组,预处理共轭梯度(PCG)法是求解这类方程组最为有效的方法之一.另外,在实际应用中,由于结构的特殊性,网格剖分时常常会产生具有大长宽比的各向异性网格,这也将大大影响PCG法的收敛性.该文设计了一种基于"距离矩阵"的代数多重网格(DAMG)法的PCG法,并应用于近不可压缩问题Wilson元离散系统的求解.这种基于"距离矩阵"的代数多重网格法,能更有效地求解各向异性网格问题,再结合有效的磨光算子,相应的PCG法对求解近不可压缩问题具有很好的鲁棒性(robustness)和高效性.The nonconforming finite element method（FEM） is an efficient method to overcome the volume locking trouble in 3D elasticity problems.This method has the advantages of a few degrees of freedom and high accuracy.In order to improve the overall efficiency of the FEM analysis,it is necessary to design some faster solvers for the corresponding system of discretization equations.The faster solvers for the Wilson nonconforming FEM discretizations were considered.When Poisson＇s ratio v was close to 0.5,the resulting system of equations was symmetric positive definite and highly ill-conditioned,and the preconditioned conjugate gradient（PCG） method was one of the most efficient methods for solving such FEM equations.Moreover,in practical applications,anisotropic meshes are often obtained due to the specificity of the structure considered,which will greatly decrease the convergence rate of the PCG method.A type of PCG method based on the DAMG was presented and then applied to the solution of the Wilson FEM discretizations.This DAMG was an algebraic multi grid（AMG） method based on the distance matrix and can be used to solve the system of equations discretized on anisotropic meshes.The numerical results show that,in combination with the effective smoothing operators,the proposed PCG method has high efficiency and robustness for nearly incompressible problems.

关 键 词：体积闭锁 WILSON非协调元 病态矩阵 代数多重网格法 预条件子 

分 类 号：O343.3[理学—固体力学] TB115[理学—力学]