A hybrid virtual element formulation for 2D elasticity problems

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 hyper-stability and iso-stability. In particular, the iso-stable 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.