A finite element procedure that works on polygonal meshes

The virtual element method (VEM) is a relatively new discretization technique that can be seen as a generalization of the finite element method. In VEM there is no need to know explicitly the finite element space in each element—thus the virtual name. The method allows for arbitrary order approximations on polygonal meshes, and the polygons need to even be convex! In this project we looked at the method from an engineering perspective, comparing it with standard FEM. We also looked at creating hybrid VEM and FEM meshes. We showed the method converges optimally and can provide tremendous flexibility in analyzing complex problems. In the figure below, the method is applied to the Flamant problem, a half-space loaded with a point force.

Flamant problem
VEM applied to the Flamant problem. The solution shows the use of nested VEM and FEM meshes.

Related publications

  • M. Mengolini, M. F. Benedetto, and A. M. Aragón. "An engineering perspective to the virtual element method and its interplay with the standard finite element method." Computer Methods in Applied Mechanics and Engineering 350 (2019), pp. 995–1023