Curvilinear Virtual Elements for 2D solid mechanics applications

In the present contribution we develop a sharper error analysis for the Virtual Element Method, applied to a model elliptic problem, that separates the element boundary and element interior contributions to the error. As a consequence we are able to propose a variant of the scheme that allows to take advantage of polygons with many edges (such as those composing Voronoi meshes or generated by agglomeration procedures) in order to yield a more accurate discrete solution. The theoretical results are supported by numerical experiments.

Sharper error estimates for Virtual Elements and a bubble-enriched version.

Parallel block preconditioners for three-dimensional virtual element discretizations of saddle-point problems

An enhanced VEM formulation for plane elasticity

Among Numerical Methods for PDEs, the Virtual Element Methods were introduced recently in order to allow the use of decompositions of the computational domain in polytopes (polygons or polyhedra) of very general shape. The present paper investigates the possible interest in their use (together or in alternative to Finite Element Methods) also for traditional decompositions (in triangles, tetrahedra, quadrilateral or hexahedra). In particular their use looks promising in problems related to high-order PDEs (requiring Cp finite dimensional spaces with p ≥ 1), as well as problems where incompressibility conditions are needed (e.g. Stokes), or problems (like mixed formulation of elasticity problems) where several useful features (symmetry of the stress tensor, possibility to hybridize, inf-sup stability condition, etc.) are requested at the same time.

Finite Elements and Virtual Elements on Classical Meshes

A three-dimensional Hellinger–Reissner virtual element method for linear elasticity problems

We extend the hybridization procedure proposed in [Arnold, Brezzi, 1985, ESAIM: M2AN] to the Virtual Element Method for linear elasticity problems based on the Hellinger-Reissner principle. To illustrate such a technique, we focus on the 2D case, 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.

Hybridization of the Virtual Element Method for linear elasticity problems

We present a family of Virtual Element Methods for three-dimensional linear elasticity problems based on the Hellinger-Reissner variational principle. A convergence and stability analysis is developed. Moreover, using the hybridization technique and exploiting the information derived from this procedure, we show how to compute a better approximation for the displacement field. The numerical experiments confirm the theoretical predictions.

A family of three-dimensional Virtual Elements for Hellinger-Reissner elasticity problems

Abstract This note aims at illustrating the application of the Virtual Element Method to elasticity problems in mixed form, following the Hellinger-Reissner variational principle. In order to highlight the potential and the flexibility of our approach, we focus on a three-dimensional low-order Virtual Element scheme, but similar considerations apply to two-dimensional and higher-order methods.

Michele Visinoni

Papers

A three-dimensional Hellinger–Reissner virtual element method for linear elasticity problems

Hybridization of the Virtual Element Method for linear elasticity problems

A family of three-dimensional Virtual Elements for Hellinger-Reissner elasticity problems

Virtual Element Methods for three-dimensional Hellinger-Reissner elastostatic problems