A variationally consistent fractional time-step integration method for incompressible and nearly incompressible Lagrangian dynamics

In this paper we present a fractional time-step method for Lagrangian formulations of solid dynamics problems. The method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly. Energy fluctuations are found to be minimal and stay bounded for long durations. In order to handle incompressibility, a mixed formulation in which the pressure appears explicitly is adopted. The velocity update over a time step is split into deviatoric and volumetric components. The deviatoric component is advanced using explicit time marching, whereas the pressure correction for each time step is computed implicitly by solving a Poisson-like equation. Once the pressure is known, the volumetric component of the velocity update is calculated. In contrast with standard explicit schemes, where the time-step size is determined by the speed of the pressure waves, the allowable time step for the proposed scheme is found to depend only on the shear wave speed. This leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems