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. In each element, 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, but we also present a formulation that uses a penalty term to enforce the element equilibrium conditions, 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 norm of the displacement, energy seminorm, and the 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.

中文翻译：

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六节点三角网格可压缩和近不可压缩线弹性的应力混合虚拟元法

在本文中，我们提出了一种用于线性平面弹性的六节点三角形网格的一阶应力混合虚拟单元法（SH-VEM）。我们采用 Hellinger-Reissner 变分原理来构造弱平衡条件和基于应力的投影算子。在每个单元中，应力投影算子都以节点位移表示，从而得出基于位移的公式。这种应力混合方法假设全局连续的位移场，而应力场在每个单元上是不连续的。应力场最初由基于艾里应力函数的无散张量多项式表示，但我们还提出了一种使用惩罚项来强制单元平衡条件的公式，称为惩罚应力混合虚拟单元法（PSH- VEM）。给出了 PSH-VEM 和 SH-VEM 的数值结果，并且我们在线性弹性基准问题上将它们的收敛性与复合三角形 FEM 和 B 杆 VEM 进行了比较。 SH-VEM 在位移范数、能量半范数和静水应力范数上最佳收敛。此外，结果表明，PSH-VEM 在大多数情况下以比预期最佳速率更快的速率收敛，但需要选择适当的惩罚参数。