非线性弹性问题的增强混合有限元方法

An augmented mixed finite element method for nonlinear elasticity problems

摘要:

为了研究非线性弹性问题的数值解，本文介绍了一种具有强对称应力张量的全增强混合有限元方法.该方法除了包括通常线弹性问题中的应力张量和位移外还将应变张量作为辅助未知量.通过引入Galerkin最小二乘项，本文得到了两层鞍点算子方程,并以此作为问题的弱方程.为了得到离散增强方程的适定性，本文以分片常量多项式来逼近应变张量,以分片线性多项式逼近应力张量和位移，得到了最优阶误差估计.数值算例验证了方法的有效性.

Abstract:

This paper introduces and analyses a full augmented mixed finite element method for the nonlinear elasticity problems with strongly imposed symmetry stress tensor. The mixed method includes the strain tensor as an auxiliary variable, which combines with the usual stress-displacement approach adopted in linear elasticity. By introducing different stability terms, we obtain an argmented mixed finite element variation formulation and a full argmented mixed finite element variation formulation. In order to obtain the well-posedness of the two schemes, we adopt different finite element spaces to approximate the variables and derive the optimal error estimate. Finally, a numerical example is presented to confirm the theoretical analysis.