Development of a stabilised Petrov-Galerkin formulation for low-order tetrahedral meshes in Lagrangian fast solid dynamics

A stabilised low-order finite element methodology is presented for the numerical simulation of a mixed conservation law formulation in fast solid dynamics. The mixed formulation, where the unknowns are linear momentum, deformation gradient and total energy, can be cast in the form of a system of first order hyperbolic equations. The difficulty associated with locking effects commonly encountered in traditional displacement formulations is addressed by treating the deformation gradient as one of the primary variables. Such formulation is first discretised in space by using a stabilised Petrov-Galerkin (PG) methodology, a generalisation of the Variational Multi-Scale (VMS) approach. The semi-discretised system of equations are then evolved in time by employing a Total Variation Diminishing Runge-Kutta (TVD-RK) time integrator. The resulting formulation achieves optimal convergence with equal orders in velocity (or displacement) and stresses. A series of numerical examples are used to assess the performance of the proposed algorithm. The new formulation is proven to be very efficient in nearly incompressible and bending-dominated scenarios.