弹性动力学问题的质量集中杂交应力有限元法  

Hybrid Stress Finite Element Method with Mass Lumping forElastodynamic Problems

作　　者：王建威 郭元辉 WANG Jianwei;GUO Yuanhui(College of Mathematics,Sichuan University,Chengdu Sichuan 610065,China;Education and Information Technology Center,China West Normal University,Nanchong Sichuan 637009,China)

机构地区：[1]四川大学数学学院,成都610065 [2]西华师范大学教育信息技术中心,四川南充637009

出　　处：《西华师范大学学报（自然科学版）》2020年第4期377-384,共8页Journal of China West Normal University(Natural Sciences)

基　　金：西华师范大学英才科研基金项目(17YC180)。

摘　　要：在利用有限元方法求解偏微分方程的过程中,很多时候都涉及到质量矩阵的求逆,例如对于非稳态问题,每一个时间步都需要求质量矩阵的逆——这将导致较大的计算成本。“质量集中”是一种通过特殊数值积分将质量矩阵对角化以提高计算效率的技术。Yu和Xie在2015年提出一种求解线弹性动力学问题的全离散杂交应力四边形有限元法,位移采用连续的等参双线性插值,应力采用分片独立的5-参数模式,时间离散采用二阶中心差分格式。本文研究该方法的质量集中格式,利用以位移插值节点为求积节点的Gauss-Lobatto数值积分来实现质量矩阵对角化。特别地,在一定网格条件下给出了该数值积分在任意四边形网格上的截断误差估计。数值算例验证了质量集中格式的性能。In the finite element solution of partial differential equation,the inverse computation of mass matrix is often involved.For example,the inverse of mass matrix is computed at each time step for the unsteady state problem,resulting in a large computational cost.“Mass lumping”is a technique that improves computation efficiency by diagonalizing the mass matrix through special numerical integration.A fully discrete hybrid stress quadrilateral finite element method for linear elastodynamic problems was proposed by Yu&Xie in 2015.The method employs continuous parametric bilinear interpolation for displacement approximation,piecewise-independent 5-parameter mode for stress approximation,and second order central difference scheme for temporal discretization.This paper is devoted to the construction of mass lumping scheme for the fully discrete hybrid stress method.Gauss-Lobatto quadrature formula,with nodes of displacement interpolation being the quadrature nodes,is applied to diagonalize the mass matrix.In particular,the truncation error estimation of Gauss-Lobatto quadrature formula on quadrilateral meshes is derived under certain mesh condition.Numerical examples are provided to verify the performance of the proposed mass lumping scheme.

关 键 词：线弹性动力学问题 全离散格式 杂交应力有限元 Gauss-Lobatto数值积分 质量集中方法 

分 类 号：O241.82[理学—计算数学]