A Virtual Element Method for elastic and inelastic problems on polytope meshes

Abstract

We present a Virtual Element Method (VEM) for possibly nonlinear elastic and inelastic problems, mainly focusing on a small deformation regime. The numerical scheme is based on a low-order approximation of the displacement field, as well as a suitable treatment of the displacement gradient. The proposed method allows for general polygonal and polyhedral meshes, it is efficient in terms of number of applications of the constitutive law, and it can make use of any standard black-box constitutive law algorithm. Some theoretical results have been developed for the elastic case. Several numerical results within the 2D setting are presented, and a brief discussion on the extension to large deformation problems is included.

Ill‐conditioning in the virtual element method: Stabilizations and bases

“…Among the other references, we recall the following: the p and hp version of the method [11][12][13][14][15][16], parabolic problems [17], Cahn-Hilliard, Stokes, Navier-Stokes and Helmoltz equations [18][19][20][21][22], linear and nonlinear elasticity problems [23][24][25], general elliptic problems [26], PDEs on surfaces [27], Domain Decomposition [28], application to discrete fracture networks [29], serendipity VEM [30], VEM on surfaces [27]. The implementation of the method is described in [31], whereas the basic principles of the 3D version of the method are the topic of [32,33].…”

Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

“…In this case, we have multiplied the stabilizing term by the material/geometric parameter λt −2 to ensure (12). A proof of (11)-(12) for the above (standard) choices could be derived following the arguments in [13].…”

“…Indeed, by avoiding the explicit construction of the local basis functions, the VEM can easily handle general polygons/polyhedrons without complex integrations on the element (see [9] for details on the coding aspects of the method). The Virtual Element Method has been applied successfully in a large range of problems, see for instance [1,2,7,8,9,12,15,16,17,20,23,25,28,35,39,40,41,47,48].…”

Serendipity face and edge VEM spaces

“…More recently, they underwent rapid developments, with extension to various problems (see [10], [9], [12], [2], [13], [14], [30], [32]). …”