An enhanced VEM formulation for plane elasticity

In this paper, an enhanced Virtual Element Method (VEM) formulation is proposed for plane elasticity. It is based on the improvement of the strain representation within the element, without altering the degree of the displacement interpolating functions on the element boundary. The idea is to fully exploit polygonal elements with a high number of sides, a peculiar VEM feature, characterized by many displacement degrees of freedom on the element boundary, even if a low interpolation order is assumed over each side. The proposed approach is framed within a generalization of the classic VEM formulation, obtained by introducing an energy norm in the projection operator definition. Although such generalization may mainly appear to have a formal value, it allows to effectively point out the mechanical meaning of the quantities involved in the projection operator definition and to drive the selection of the enhanced representations. Various enhancements are proposed and tested through several numerical examples. Numerical results successfully show the capability of the enhanced VEM formulation to (i) considerably increase accuracy (with respect to standard VEM) while keeping the optimal convergence rate, (ii) bypass the need of stabilization terms in many practical cases, (iii) obtain natural serendipity elements in many practical cases, and (vi) effectively treat also nearly incompressible materials.