In this work we develop new finite element discretisations of the shear-deformable Reissner--Mindlin plate problem based on the Hellinger-Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence $\boldsymbol{M} \in \mathcal{HZ} \subset H^{\mathrm{sym}}(\mathrm{Div})$. The latter results in highly accurate approximations of the bending moments $\boldsymbol{M}$ and in the rotation field being in the discontinuous Lebesgue space $\boldsymbol{\phi} \in [L]^2$, such that the Kirchhoff-Love constraint can be satisfied for $t \to 0$. In order to preserve optimal convergence rates across all variables for the case $t \to 0$, we present an extension of the formulation using Raviart-Thomas elements for the shear stress $\mathbf{q} \in \mathcal{RT} \subset H(\mathrm{div})$. We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting. This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries. The robustness of the formulations and the construction of the Hu-Zhang element are tested for shear-locking, curved geometries and an L-shaped domain with a singularity in the bending moments $\boldsymbol{M}$. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods.

翻译：本文基于对称应力的Hellinger-Reissner原理，发展了剪切变形Reissner-Mindlin板问题的新型有限元离散格式。具体而言，我们采用协调的Hu-Zhang单元在具有平方可积散度对称平方可积场空间$\boldsymbol{M} \in \mathcal{HZ} \subset H^{\mathrm{sym}}(\mathrm{Div})$内离散弯矩。这使得弯矩$\boldsymbol{M} $的逼近具有高精度，同时旋转场属于非连续Lebesgue空间$\boldsymbol{\phi} \in [L]^2$，从而可在$t \to 0$时满足Kirchhoff-Love约束。为在$t \to 0$情形下保持所有变量的最优收敛速率，我们提出了基于Raviart-Thomas单元剪切应力$\mathbf{q} \in \mathcal{RT} \subset H(\mathrm{div})$的扩展格式。我们证明了连续问题的存在唯一性，并借助精确复形继承离散问题的适定性。本文通过多面体模板方法及Legendre多项式在参考单元上高效构造了Hu-Zhang基函数，使其适用于hp-FEM。参考单元上的基函数通过新型多面体变换映射到物理单元，该变换同样适用于弯曲几何。通过剪切闭锁、弯曲几何及弯矩$\boldsymbol{M}$具有奇异性的L形域算例，验证了格式的鲁棒性及Hu-Zhang单元的构造有效性。此外，我们将新型格式与原始方法、MITC方法及近期提出的TDNNS方法进行了性能比较。