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The Virtual Element Method (VEM) for the elasticity problem is considered in the framework of the Hu-Washizu variational formulation. In particular, a couple of low-order schemes presented in [1], are studied for quadrilateral meshes. The methods under consideration avoid the need of the stabilization term typical of the VEM, due to the introduction of a suitable projection on higher-order polynomials. The schemes are proved to be stable and optimally convergent in a compressible regime, including the case where highly distorted (even non-convex) meshes are employed.

A recent mixed formulation of the Virtual Element Method in 2D elastostatics, based on the Hu-Washizu variational principle, is here extended to 2D elastodynamics. The independent modeling of the strain field, allowed by the mixed formulation, is exploited to derive first order quadrilateral Virtual Elements (VEs) not requiring a stabilization (namely, self-stabilized VEs), in contrast to the standard VEs, where an artificial stabilization is always required for first order quads. Lumped mass matrices are derived using a novel approach, based on an integration scheme that makes use of nodal values only, preserving the correct mass in the case of rigid-body modes. In the case of implicit time integration, it is shown how the combination of a self-stabilized stiffness matrix with a self-stabilized lumped mass matrix can produce excellent performances both in the compressible and quasi-incompressible regimes with almost negligible sensitivity to element distortion. Finally, in the case of explicit dynamics, the performances of the different types of derived VEs are analyzed in terms of their critical time-step size.

An original, variational formulation of the Virtual Element Method (VEM) is proposed, based on a Hu–Washizu mixed variational statement for 2D linear elastostatics. The proposed variational framework appears to be ideal for the formulation of VEs, whereby compatibility is enforced in a weak sense and the strain model can be prescribed a priori, independently of the unknown displacement model. It is shown how the ensuing freedom in the definition of the strain model can be conveniently exploited for the formulation of self-stabilized and possibly locking-free low order VEs. The superior performances of the VEs formulated within this framework has been verified by application to several numerical tests.

In this paper, we extend the hybridization procedure proposed in [D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, ESAIM Math. Model. Numer. Anal. 19 (1985) 7-32] to the Virtual Element Method for linear elasticity problems based on the Hellinger-Reissner principle. To illustrate such a technique, we focus on a specific 2D scheme, but other methods and 3D problems can be considered as well. We also show how to design a better approximation of the displacement field using a straightforward post-processing procedure. The numerical experiments confirm the theory for both two and three-dimensional problems.

The objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al., CMAME 2016] we are able to show an "almost uniform"error bound (in the sense that the unique term that depends in an unfavourable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robust in the convection dominated cases. We finally present some numerical result.

We present a Virtual Element Method for the 3D linear elasticity problems, based on Hellinger–Reissner variational principle. In the small strain theory, we propose a low-order scheme with a-priori symmetric stresses and continuous tractions across element interfaces. A convergence and stability analysis is developed and we confirm the theoretical predictions via some numerical tests.

The present paper is the second part of a twofold work, whose first part is reported in Artioli et al. (Comput Mech, 2017. doi:10.1007/s00466-017-1404-5), concerning a newly developed Virtual element method (VEM) for 2D continuum problems. The first part of the work proposed a study for linear elastic problem. The aim of this part is to explore the features of the VEM formulation when material nonlinearity is considered, showing that the accuracy and easiness of implementation discovered in the analysis inherent to the first part of the work are still retained. Three different nonlinear constitutive laws are considered in the VEM formulation. In particular, the generalized viscoelastic model, the classical Mises plasticity with isotropic/kinematic hardening and a shape memory alloy constitutive law are implemented. The versatility with respect to all the considered nonlinear material constitutive laws is demonstrated through several numerical examples, also remarking that the proposed 2D VEM formulation can be straightforwardly implemented as in a standard nonlinear structural finite element method framework.

The present work deals with the formulation of a virtual element method for two dimensional structural problems. The contribution is split in two parts: in part I, the elastic problem is discussed, while in part II (Artioli et al. in Comput Mech, 2017) the method is extended to material nonlinearity, considering different inelastic responses of the material. In particular, in part I a standardized procedure for the construction of all the terms required for the implementation of the method in a computer code is explained. The procedure is initially illustrated for the simplest case of quadrilateral virtual elements with linear approximation of displacement variables on the boundary of the element. Then, the case of polygonal elements with quadratic and, even, higher order interpolation is considered. The construction of the method is detailed, deriving the approximation of the consistent term, the required stabilization term and the loading term for all the considered virtual elements. A wide numerical investigation is performed to assess the performances of the developed virtual elements, considering different number of edges describing the elements and different order of approximations of the unknown field. Numerical results are also compared with the one recovered using the classical finite element method.