The Virtual Element Method (VEM) is a recent methodology for the approximation of problems described by partial differential equations. VEM can be considered as a generalization of Finite Elements.

The main asset of VEM is the possibility of using mesh elements with arbitrary geometrical forms, thus allowing very complex meshes within a Galerkin type approximation. Such feature opens up a significant flexibility for the discretization and also the design of algorithms, even for low order ansatz spaces.
The VEM permits the use of polygonal elements for problems in 2D and polyhedral elements in 3D. Meshes can consist of elements with different shapes from convex to non-convex forms and different number of nodes. Despite this variety of different element geometries, the mesh provides a continuous discretization.
The aim of the course is to present theoretical fundamentals, the current state-of-the art, and future directions of “computational approaches for the analysis of the mechanics of materials” using VEM. So far, the method has been successfully applied in many areas of solid mechanics, such as elastic and inelastic numerical simulation of solids for small and finite strains, crystal plasticity, coupled problems, fracture mechanics, homogenization procedures, plate problems, contact problems. These areas will be covered throughout the course, so that the broad applicability of virtual element schemes will be evident to the participants. In addition, areas outside solid mechanics will be addressed, such as Navier-Stokes flow, magnetostatics and phase transition.

Naturally all methods have advantages and disadvantages; these will be discussed in depth to provide a clear view about the applicability of VEM to engineering problems. The course will also emphasize the mathematical formulation of the different ansatz spaces and investigate the relations with the Finite Element Method.

In the course we will explore the advantageous features of the virtual element method from the engineering and mathematical view. This includes the discussion of the use of VEM for non-matching meshes in contact mechanics, the arbitrary shape of the elements in homogenization of polycrystal materials, the exact imposition of the incompressibility constraint in fluid and solid mechanics, the treatment of cracks, the use of curved elements for exact geometry representation and other applications in engineering.

One of the keys to the understanding of a new technique is to illustrate it in the context of the necessary software. This will be done in workshop sessions based on MATLAB and the automatic tools AceGen and AceFem.

The language of Mechanics will be employed together with Mathematics as the transport vehicle, for providing lectures in a consistent, broad, and equilibrated format. The course will appeal to doctoral students and postdoctoral researchers from academia and industry with an interest in the response of materials and structures, and a background in civil or mechanical engineering or in material sciences.