In this paper, a hybrid variational framework for the Virtual Element Method (VEM) is proposed and a family of polygonal elements for plane elasticity is developed. Under specific assumptions, it is proved that the minimization of Total Potential Energy and the projection operation typical of enhanced VEM can be deduced from the stationary condition of the Hellinger–Reissner mixed functional. Since the designed elements can be regarded as either enhanced VEM or hybrid finite elements, they are named as Hybrid Virtual Element Method (HVEM). The primary variables are the displacements along the element boundary and the stress field within the element domain. The assumed stress field is expressed on a polynomial basis that satisfies the divergence-free condition. In the HVEM formulation, stabilization-free elements can be obtained using two concepts, namely and . In particular, the cases show the best solution in recovering both displacement and stress fields. Several numerical applications are developed, assessing the stability for a single distorted element. The proposed family of HVEM proves to be accurate, also if coarse meshes are used. Additionally, the effectiveness of the proposed HVEM is demonstrated for typical structural elements, testing the convergence rate and comparing the results with analytic or other numerical solutions.

中文翻译：

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二维弹性问题的混合虚拟单元公式

本文提出了虚拟单元法（VEM）的混合变分框架，并开发了一系列用于平面弹性的多边形单元。在特定的假设下，证明了总势能的最小化和增强VEM典型的投影运算可以从Hellinger-Reissner混合泛函的平稳条件推导出来。由于所设计的单元既可以看作是增强的VEM，也可以看作是混合有限元，因此被称为混合虚拟单元法（HVEM）。主要变量是沿单元边界的位移和单元域内的应力场。假设的应力场以满足无散条件的多项式形式表示。在 HVEM 公式中，可以使用两个概念获得无稳定化单元，即 和 。特别是，这些案例展示了恢复位移和应力场的最佳解决方案。开发了几种数值应用程序，评估单个扭曲单元的稳定性。即使使用粗网格，所提出的 HVEM 系列也被证明是准确的。此外，针对典型的结构单元证明了所提出的 HVEM 的有效性，测试了收敛速度并将结果与​​解析解或其他数值解进行了比较。