A Hu–Washizu variational approach to self-stabilized virtual elements: 2D linear elastostatics

Abstract

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.

A first-order stabilization-free Virtual Element Method

“…Recalling that ξ is constant, from (33) we infer ξ = 0, a contradiction since ξ ̸ = 0. Furthermore, decomposition (31) follows from a dimensional count, while the L 2 -orthogonality is simply (33).…”

“…This paper follows similar lines of the above-mentioned stabilisation-free attempts [12,13,30,31], but for the Laplacian problem written in the usual H(div)−L 2 mixed formulation. In particular, we consider a VEM version of the lowest order Raviart-Thomas Finite Element Method, see [4].…”

“…Virtual Element schemes for which no stabilisation form is required have been recently presented, in different 2D frameworks, in [12,13,30,31]. These approaches share the idea to employ a projection onto a polynomial space of higher degree than the one usually taken in standard VEM.…”

Stress‐hybrid virtual element method on quadrilateral meshes for compressible and nearly‐incompressible linear elasticity

“…More recently, in the spirit of assumed-strain methods, 10,11 projections onto higher order strains have been pursued in the VEM to devise stabilization-free schemes. [12][13][14][15][16] On a quadrilateral, the stabilization-free virtual element method (SF-VEM) 14 with projection onto an affine strain field (nine parameters) suffers from volumetric locking in the near-incompressible limit. In this article, as a point of departure, we appeal to the Hellinger-Reissner variational principle and assumed stress (referred to as hybrid stress or stress hybrid) techniques 17,18 to devise a virtual element formulation that is robust for compressible and nearly-incompressible linear elasticity over quadrilateral meshes.…”

“…Böhm et al 9 has provided a study of different virtual element methods for incompressible problems and compared the results to classical finite element techniques. More recently, in the spirit of assumed‐strain methods, 10,11 projections onto higher order strains have been pursued in the VEM to devise stabilization‐free schemes 12‐16 . On a quadrilateral, the stabilization‐free virtual element method (SF‐VEM) 14 with projection onto an affine strain field (nine parameters) suffers from volumetric locking in the near‐incompressible limit.…”