Implicit finite incompressible elastodynamics with linear finite elements: A stabilized method in rate form

We propose a stabilization method for linear tetrahedral finite elements, suitable for the implicit time integration of the equations of nearly and fully incompressible nonlinear elastodynamics. In particular, we derive and discuss a generalized framework for stabilization and implicit time integration that can comprehensively be applied to the class of all isotropic hyperelastic models. In this sense the presented development can be considered an important extension and complement to the stabilization approach proposed by the authors in previous work, which was instead focused on explicit time integration and simple neo-Hookean models for nearly-incompressible elasticity. With the goal of computational efficiency, we also present a two-step block Gauss-Seidel strategy for the time update of displacements, velocities and pressures. Specifically, a mixed system of equations for the velocity and pressure is updated implicitly in a first stage, and the displacements are updated explicitly in a second stage. The proposed mixed formulation is then embedded in Newton-type strategies for the nonlinear solution of the equations of motion. Various implicit time integration strategies are considered, and, particularly, we focus on high-frequency dissipation time integrators, which are preferable in transient mechanics applications. An extensive set of numerical computations with linear tetrahedral elements is presented to demonstrate the performance of the proposed approach.