摘要
线弹性问题源自机械制造及航空等工程应用。线弹性问题中的微分方程较复杂,其解析解目前还是未知的。经典有限元法常被用于数值求解线弹性问题,但难以克服“闭锁”现象。为此研究者提出了弱有限元法,方法引入了弱微分算子(弱梯度、弱散度等),并且用间断多项式函数离散问题中的微分方程。在弱有限元法中,近似函数的连续性通常由边界函数及稳定子共同保证,即通过加入稳定子来保证数值格式的稳定性,这会导致数值格式结构复杂、计算量大且逼近速度慢。为了克服这些问题,研究者提出了无稳定子的弱有限元法,通过提高弱梯度算子的逼近多项式次数来保证格式的稳定性。本文提出了一种无稳定子的弱有限元法,引入了弱梯度和弱散度算子,并且分别用分片线性和分片二次多项式逼近单元内部的位移和边界位移。本文引入了辅助变量,给出了与问题等价的混合格式,并证明格式关于Lamé常数一致收敛,从而能够克服“闭锁”现象。相比含稳定子的弱有限元法,本文的格式结构简单,逼近速度更快。数值算例验证了理论结果。
Linear elastic problem originates from engineering applications such as machinery and aviation.Due to the complexity of differential equation in the linear elastic problem,the analytic solutions of the problem are still unknown.Although the classical finite element methods can be used to numerically solve the linear elastic problem,these methods cannot overcome the"locking"phenomenon.To address this problem,weak Galerkin methods are proposed,which introduces the weak differential operators(weak gradient,weak divergence,etc.)and uses the discontinuous polynomial functions to discrete the differential equation in the problem.In weak Galerkin method,the continuity of the approximation functions usually needs to be guaranteed by the boundary function and stabilizers.That is to say,the stability of the numerical scheme is guaranteed by adding the stabilizers,which makes the scheme become very complex and brings the problems of large amount of calculation and slow approximation speed.To keep the stability,the stabilizer-free weak Galerkin methods are proposed,in which the approximation polynomial degree of the weak gradient operator is increased.In this paper,a stabilizer-free weak Galerkin method is proposed for the linear elastic problem.Firstly,the weak gradient and weak divergence operators are introduced,and the piecewise linear and piecewise quadratic polynomials are used to approximate the internal displacement and boundary displacement of the element,respectively.Then,by introducing the auxiliary variables,a mixed scheme equivalent to the problem is given,and the uniform convergence of the scheme with respect to the Laméconstant is proven,that is,the method can overcome the"locking"phenomenon.In comparison with the weak Galerkin method with stabilizers,the proposed method is very simple and the approximation speed is faster.Numerical examples verify the theoretical results.
作者
许岳
陈豫眉
谢小平
XU Yue;CHEN Yumei;XIE Xiaoping(School of Mathematics,Sichuan University,Chengdu 610065,China;School of Mathematics and Information,China West Normal University,Nanchong 637009,China)
出处
《四川大学学报(自然科学版)》
北大核心
2026年第1期83-90,共8页
Journal of Sichuan University(Natural Science Edition)
基金
国家自然科学基金(12171340)。
关键词
线弹性问题
弱有限元方法
稳定子
linear elastic problem
weak Galerkin method
stabilizer
