线弹性问题的自适应混合有限元方法

Based on a posteriori error estimates and mesh adaptivity, the adaptive method is widely used to solve the finite element discrete problems and have been shown to be more competitive; mixed finite element method is a powerful tool for solving the linear elasticity problems, since it can approximate simultaneously the displacement and the stress. This project mainly concerns the adaptive symmetric mixed finite element method for linear elasticity problems. Firstly, we will study the a posteriori error estimates of the symmetric mixed finite elements for three-dimensional linear elasticity problems. The Helmholtz decomposition of the stress tensor will be given by investigating the three dimensional elasticity complex. Residual-based a posteriori error estimator will be proposed and its reliability will be proven by the Helmholtz decomposition. The efficiency of the estimator will be proven constructively by introducing some finite element of the fourth order problems, which is connected to the symmetric tensor finite element by the Airy function, as the auxiliary space. Secondly, we will study the residual based a posterior error estimates of the symmetric nonconforming mixed methods for linear planar elasticity problems. The stress error is decomposed into the conforming part and the nonconforming part by the Helmholtz decomposition. Reliability of the estimator is obtained by estimating the two parts separately. Finally, using the residual based a posteriori error estimators proposed above, we will design optimal adaptive symmetric mixed finite element algorithms to solve the linear elasticity problems, Further investigation on some basic ingredients, such as quasi-orthogonality, discrete Helmholtz decomposition, discrete upper bound and so on, will be done to prove the convergence and optimality of the adaptive algorithm..

以后验误差估计和自适应网格改进技术为核心的自适应方法已被广泛用于有限元离散问题的数值求解中，并表现出色；可同时逼近位移与应力的混合有限元方法是数值求解线弹性问题的强有力工具。本项目主要研究线弹性问题的自适应对称型混合有限元方法。我们首先研究三维线弹性问题对称型协调有限元方法的后验误差估计。利用三维弹性序列给出应力的Helmholtz分解，据此构造残量型的后验误差估计子并证明其可靠性；利用对称型混合元和四阶问题有限元之间的关系，构造性地证明估计子的有效性。其次研究线弹性问题对称型非协调混合元方法的残量型后验误差估计。应用Helmholtz分解把应力误差分解为协调误差和非协调误差两部分，然后分别估计得到误差估计子的可靠性。最后利用所构造的后验误差估计子设计求解线弹性问题的对称型混合元自适应算法，研究拟正交性、离散Helmholtz分解、离散上界等重要性质，证明算法的收敛性和最优性。

混合有限元方法可同时求解位移和应力，是数值求解线弹性问题的强有力工具。相对于标准有限元方法，混合有限元方法由于在计算中涉及到更多的未知量而使计算规模增大，因此如何构造混合元离散问题可靠且有效的后验误差估计子，优化网格加密策略，实现问题的高效自适应计算具有重要的应用价值。. 本项目主要研究了线弹性问题的对称型混合有限元方法及其离散问题的后验误差估计。首先我们构造了求解线弹性问题的一族对称型非协调混合有限元，这族元的应力和位移有限元空间具有很好的匹配性，在形式上关于空间维数具有一致性，可以推广到任意维问题。我们证明了混合元离散问题解的存在唯一性并给出了最优的先验误差估计。对二维、三维问题进行了数值实验，从数值上验证了所构造混合元的最优收敛性和超收敛性,并且从理论上证明了这族元的超收敛性。其次我们研究了二维和三维线弹性问题对称型协调混合元方法的后验误差估计。利用应力误差的Helmholtz正交分解，构造了自适应求解离散问题的残量型后验误差估计子，证明了估计子的可靠性和有效性。通过对不同边值问题的自适应数值计算，验证了所构造后验误差估计子的可靠性和有效性，数值计算表明我们所构造的自适应算法具有最优的收敛性。最后我们研究了对称型非协调混合元离散问题的后验误差估计。本项目现已发表SCI检索论文2篇。. 需要特别指出的是我们最近几年所得到的关于线弹性问题对称型混合有限元方法的研究成果，得到了工程界研究人员的关注，被用于求解一些工程问题并取得了比较好的计算效果，接下来我们将深入研究对称型混合元方法在实际工程计算中的应用。

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