Reissner-Mindlin板问题带约束非协调旋转Q_1有限元方法

本文利用带约束非协调旋转Q_1元逼近Reissner-Mindlin板问题中旋度的两个分量.并分别选择Wilson元、双线性元和带约束非协调旋转Q_1元逼近挠度,相应地选取不连续的矢量值分片线性函数空间、最低阶旋转Raviart-Thomas元空间和矢量值分片常数函数空间为离散的剪应力空间,在矩形网格上构造了三个板元.通过证明一个离散的Korn不等式,并借助MITC4元的解构造了旋度、挠度和剪应力一个具有某种特殊且关键的可交换性的插值.再利用Helmholtz分解分析相容性误差.我们证明了这三个矩形元在能量范数意义下与板厚无关的一致最优收敛性.数值算例验证了我们的理论结果.

CONSTRAINED NONCONFORMING ROTATED Q1 ELEMENT METHODS OF THE REISSNER-MINDLIN PLATE PROBLEM

Three rectangular elements are proposed for solving numerically the Reissner-Mindlin plate problem. In all of them, the constrained nonconforming rotated Q₁ element is used to approximate both components of the rotation variable while the Wilson element, the bilinear element, and the constrained nonconforming rotated Q₁ element are chosen to discretize the transverse deflection, respectively. Accordingly, the space of piecewise vector-valued polynomials of degree≤1, the space of the lowest order rotated Raviart-Thomas element, and the space of piecewise vector-valued constants, are taken as the approximate space of the shear force. Thanks to the approximate solution by the MITC4 plate element, an interpolation is constructed for the rotation, the deflection and the shear force, respectively. The feature of these three interpolations is that they admit a special and crucial commuting property. In order to analyze the consistency errors, the Helmholtz decomposition of the shear force is evoked. Finally, with the help of the discrete Korn inequality established, a uniformly optimal convergence rate with respect to the plate thickness t is proved for these plate elements. Numerical examples are provided to demonstrate the theoretical results.