Virtual Element Method (VEM)

The Virtual Element Method (VEM) is an innovative numerical technique designed for polygons in 2D and polyhedra in 3D. Avoiding the explicit definition of the shape functions and introducing an innovative construction of the stiffness matrixes, the method has important advantages with respect to standard Finite Element Method. Its mathematical foundations have been well established and it has been applied on numerous applications.

We propose an original, variational formulation of the Virtual Element Method (VEM), 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.

As it is well-known, in most cases the VE stiffness matrix requires a stabilization to avoid the development of zero-energy hourglass modes. Even though the stabilization has proved to be effective in guaranteeing the correct convergence order, its substantially empirical nature, being based on artificial stiffness coefficients, is viewed as a limitation of the method. For this reason, there is a significant interest in investigating the possibility to formulate self-stabilized VEs, i.e., not requiring any artificial stabilization. In view of the weak enforcement of compatibility, in the proposed Hu-Washizu formulation the strain model is independent of the displacement one. It is then rather natural to exploit this additional freedom for the formulation of self-stabilized VEs.

The superior performances of the VEs formulated within this framework has been verified by application to several numerical tests.

Figure:  Convergence test with analytical solution: comparison of standard and self-stabilized VEM for different quadrilateral meshes.

Lamperti A., Cremonesi M., Perego U., Russo A., Lovadina C. (2023) A Hu-Washizu variational approach to self-stabilized Virtual Elements: 2D linear elastostatics, Computational Mechanics (to appear)