A Hu-Washizu variational approach to self-stabilized quadrilateral Virtual Elements: 2D linear elastodynamics

Abstract

A recent mixed formulation of the Virtual Element Method in 2D elastostatics, based on the Hu-Washizu variational principle, is here extended to 2D elastodynamics. The independent modeling of the strain field, allowed by the mixed formulation, is exploited to derive first order quadrilateral Virtual Elements (VEs) not requiring a stabilization (namely, self-stabilized VEs), in contrast to the standard VEs, where an artificial stabilization is always required for first order quads. Lumped mass matrices are derived using a novel approach, based on an integration scheme that makes use of nodal values only, preserving the correct mass in the case of rigid-body modes. In the case of implicit time integration, it is shown how the combination of a self-stabilized stiffness matrix with a self-stabilized lumped mass matrix can produce excellent performances both in the compressible and quasi-incompressible regimes with almost negligible sensitivity to element distortion. Finally, in the case of explicit dynamics, the performances of the different types of derived VEs are analyzed in terms of their critical time-step size.