Development of a stabilised Petrov–Galerkin formulation for conservation laws in Lagrangian fast solid dynamics

• Novel PG computational methodology for fast transient dynamics.
• PG stabilisation derived from a variational point of view.
• Same order of convergence for stresses and displacements.
• Good behaviour in nearly incompressible and bending dominated scenarios.
• Good balance between accuracy and speed of computation.

A stabilised second 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 standard pure displacement formulations is addressed by treating the deformation gradient as one of the primary variables. The formulation is first discretised in space by using a stabilised Petrov–Galerkin (PG) methodology derived through the use of variational (work-conjugate) principles. The semi-discretised system of equations is then evolved in time by employing a Total Variation Diminishing Runge–Kutta (TVD-RK) time integrator. The formulation achieves optimal convergence (e.g. second order with linear interpolation) with equal orders in velocity (or displacement) and stresses, in contrast with the displacement-based approach. This paper defines a set of appropriate stabilising parameters suitable for this particular formulation, where the results obtained avoid the appearance of non-physical spurious (zero-energy) modes in the solution over a long term response. We also show that the proposed PG formulation is very similar, and under certain conditions identical, to the well known Two-step Taylor Galerkin (2TG). A series of numerical examples are presented in order to assess the performance of the proposed algorithm. The new formulation is proven to be very efficient in nearly incompressible and bending dominated scenarios.