Reissner-Mindlin板杂交有限元方法的快速求解

As a parameter dependent problem, the numerical solution of Reissner-Mindlin (R-M) plate model has been always an important research topic in computational mathematics and engineering. Currently, the theory is already quite mature on how to construct the “robust” finite element method independent of plate thickness, while there has been relatively few works on solver for the linear system arising from R-M plate problem. Corresponding to a class of “robust” low-order quadrilateral hybrid finite element method, this project aims at fast solvers for the saddle point system. This class of hybrid method includes variables of displacements, shear force and bending moments, where we use continuous piecewise isoparametric bilinear interpolation for the approximation of displacements, while the shear force and bending moments are piecewise independent. Since the shear force and bending moments can be eliminated at the element level, only unknowns of nodal displacements remain in the total stiffness matrix equation (i.e. Schur Complement system). The focus of our research is the solvers for the Schur Complement system. We hope some algorithm such as preconditioning and mutigrid method independent of plate thickness and mesh size can be designed.

作为参数相关问题，Reissner-Mindlin (R-M)板模型的数值解法一直是计算数学界和工程界的重要研究课题。目前关于如何构造与板厚一致稳定的有限元方法的理论已经比较成熟，但是对于有限元离散后的线性方程组的快速求解算法的研究，尚处于初级阶段。本项目主要是针对一类“鲁棒的”的低阶四边形杂交元方法，研究其相应的鞍点系统的快速解法。这类杂交元方法以位移、剪切应力和弯矩为变量，其中位移采用连续的分片等参双线性插值，而剪切应力和弯矩则为分片独立的；由于剪切应力和弯矩可以通过静力凝聚作用在单元水平上消去(对应的系统即为Schur Complement系统)，最终的计算规模与双线性位移元相当。我们研究的重点是Schur Complement系统的快速求解算法，旨在构造与板厚参数以及网格尺寸一致无关的预处理及多重网格算法。

本项目原计划是系统研究Reissner-Mindlin (R-M)板模型的一类数值上“鲁棒的”低阶四边形杂交元方法对应的Schur Complement系统的快速求解算法。为了理论的完整性，我们首先研究了该类低阶四边形杂交元方法关于板厚的一致稳定性，而后考虑Schur Complement系统的快速求解算法。我们得到具体结果如下：.1. 讨论了一类以位移、剪切应力和弯矩为变量的低阶杂交元方法的Locking-free收敛性。我们基于混合有限元理论，建立了其与板厚一致无关的误差估计。.2. 针对上述杂交有限元方法对应的Schur Complement系统，我们提出了一种两重网格求解方式并进行了数值模拟，结果显示：对于给定的板厚，该求解算法的收敛性与网格参数一致无关。