平面强化有限单元法的h型网格自适应

自适应有限元中的单元细划会不可避免地引入非规则节点,针对非规则节点的多点约束处理方法存在自由度多、数值稳定性差等缺点,根据强化有限单元法（FEM++）中数学网格和物理网格分离的特点,构造一种能够统一处理任意阶非规则节点的位移模式关联法则——物理模式重构法,并导出相应的有限元列式.修正了常规有限元中的超收敛应力恢复法和ZZ（Zienkiwicz-Zhu）后验误差估计理论,提出强化有限单元法的后验误差估计方法及相应的h型网格自适应策略.物理模式重构法在单元层次上直接将非规则节点的位移用具有独立自由度的数学节点位移线性表达,既保证了单元间的位移连续性,又不需要约束处理的多余自由度,保证了数值稳定性.数值算例表明：强化有限单元法的后验误差估计方法及自适应策略能给出满足精度要求的网格.

h-adaptive enhanced finite element method for plane problems

The multipoint constraint method, which is currently applied to deal with the hanging node introduced by the h-refinement in standard finite element method, requires more degrees of freedom (DOFs) and degrades the numerical stability, therefore, a convenient method to treat the hanging node was presented in the framework of the enhanced finite element method (FEM++). According to separated mathematical and physical meshes, a new correlation rule for FEM++, namely reconstruction of physical displacement mode, was proposed to handle the hanging node with any irregularity index in a uniform way, and the corresponding finite element formulas were then derived. The superconvergent patch recovery and ZZ (Zienkiwicz-Zhu) error estimate method were improved, and the a posteriori error estimate method and corresponding h-refinement strategy suitable for FEM++ were then put forward. The DOFs of the hanging node can be represented by the independent DOFs of the mathematical nodes using reconstruction of physical displacement mode, so no extra DOF is needed, and the displacement continuity between the elements and the numerical stability can both be guaranteed. Numerical examples demonstrate that, meshes that satisfying an aim error can be obtained by the a posteriori error estimate method and the h-refinement strategy proposed here.