In this paper, we propose a robust low-order stabilization-free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress-hybrid principle. We refer to this approach as the Stress-Hybrid Virtual Element Method (SH-VEM). In this method, the Hellinger$-$Reissner variational principle is adopted, wherein both the equilibrium equations and the strain-displacement relations are variationally enforced. We consider small-strain deformations of linear elastic solids in the compressible and near-incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are $\textit{virtual}$. The stress field, similar to the stress-hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5-parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain-displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH-VEM is free of volumetric and shear locking, and it converges optimally in the $L^2$ norm and energy seminorm of the displacement field, and in the $L^2$ norm of the hydrostatic stress.

翻译：本文提出一种基于应力混合原理的鲁棒低阶无稳定化虚拟单元方法，用于四边形网格上的线弹性问题。我们将该方法称为应力混合虚拟单元方法（SH-VEM）。该方法采用Hellinger-Reissner变分原理，通过变分形式同时满足平衡方程和应变-位移关系。我们考虑可压缩与近不可压缩状态下线弹性固体的小变形问题，计算域采用四边形网格（包含凸四边形与凹四边形）。单元内部，位移场通过一组规范形函数的线性组合逼近，这些形函数是$\textit{虚拟}$的。应力场则采用对称张量多项式的线性组合表示，这与Pian和Sumihara的应力混合有限元方法类似。每个单元使用5参数展开的应力场，并在畸变四边形上应用应力变换方程。在应变-位移关系的变分表述中，通过散度定理将应力系数表示为节点位移的函数，从而得到仅以节点位移为未知量的表达式。针对线弹性领域的多个基准问题给出数值结果。我们证明SH-VEM方法完全消除了体积闭锁和剪切闭锁现象，并在位移场的$L^2$范数与能量半范数、以及流体静应力的$L^2$范数下均达到最优收敛阶。