求解三维Wilson元离散化线性系统的PCG方法

发布时间：2018-05-08 18:45

本文选题：体积闭锁 + Wilson非协调元　；参考：《应用数学和力学》2016年08期

【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.
[Abstract]:Non-conforming element method is an effective method to overcome the volume latchup of three-dimensional elastic problem. It has the advantages of less degree of freedom and higher accuracy. But in order to improve the overall efficiency of finite element analysis, it is necessary to design a fast solution algorithm for the corresponding discretization system. The fast solution of Wilson element discretization system is considered. When Poisson ratio is 0.5, the discrete system is a highly ill-conditioned positive definite system. The preconditioned conjugate gradient PCG method is one of the most effective methods for solving these equations. In addition, due to the particularity of the structure, anisotropic meshes with large aspect ratio are often generated in mesh generation, which will greatly affect the convergence of the PCG method. In this paper, an algebraic multigrid PCG method based on "distance matrix" is designed and applied to the solution of Wilson element discrete systems for nearly incompressible problems. This algebraic multigrid method based on "distance matrix" can solve anisotropic mesh problem more effectively. Combined with effective polishing operator, the corresponding PCG method has good robustness and high efficiency for solving nearly incompressible problems.
【作者单位】：湘潭大学土木工程与力学学院;
【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)~~
【分类号】：O343.2;O241.82

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