In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. We adopt the Hellinger--Reissner variational principle to construct a weak equilibrium condition and a stress based projection operator. On applying the divergence theorem to the weak strain-displacement relations, the stress projection operator is expressed in terms of the nodal displacements, which leads to a displacement-based formulation. This stress-hybrid approach assumes a globally continuous displacement field while the stress field is discontinuous across each element. The stress field is initially represented by divergence-free tensor polynomials based on Airy stress functions. However, for flexibility in choosing basis functions, we also present a formulation that uses a penalty term to enforce the element equilibrium conditions. This method is referred to as the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numerical results are presented for PSH-VEM and SH-VEM, and we compare their convergence to the composite triangle FEM and B-bar VEM on benchmark problems in linear elasticity. The SH-VEM converges optimally in the $L^2$ norm of the displacement, energy seminorm, and the $L^2$ norm of hydrostatic stress. Furthermore, the results reveal that PSH-VEM converges in most cases at a faster rate than the expected optimal rate, but it requires the selection of a suitably chosen penalty parameter.

翻译：本文提出了一种基于六节点三角形网格的一阶应力混合虚拟单元法（SH-VEM），用于求解线性平面弹性问题。我们采用Hellinger-Reissner变分原理构建弱平衡条件及基于应力的投影算子。通过对弱应变-位移关系应用散度定理，应力投影算子可用节点位移表达，进而得到基于位移的公式。该应力混合方法假设位移场全局连续，而应力场在单元间不连续。应力场初始采用基于Airy应力函数的无散度张量多项式表示。为增强基函数选择的灵活性，我们还提出了一种通过罚项强制单元平衡条件的公式，称为罚应力混合虚拟单元法（PSH-VEM）。针对线性弹性基准问题，我们给出了PSH-VEM和SH-VEM的数值结果，并与复合三角形有限元法及B-bar VEM进行收敛性对比。SH-VEM在位移的$L^2$范数、能量半范数及静水压力$L^2$范数上均达到最优收敛阶。此外，结果表明PSH-VEM在多数情况下收敛速度超过预期最优阶，但需选取合适的罚参数。