Deformation extrapolation and initial predictors in large-deformation finite element analysis

Abstract

In implicit Lagrangian finite element formulations, it is generally necessary to extrapolate to time t ^k+1the motion in the converged time step t ^k−1→t ^k, in order to provide an initial predictor for the solution at time t ^k+1. This extrapolation is typically carried out by assuming a constant material velocity field in [t ^k−1, t ^k+1]. However, this practice can lead to a poor initial estimate of the equilibrium solution at t ^k+1, particularly when moderate or large rotation increments are involved. In some cases, the constant-material-velocity-field extrapolation is so inaccurate that convergence of the subsequent equilibrium iteration is precluded. The proposed technique involves extrapolation of the stretch and rotation fields rather than the displacement field. The extrapolated stretch and rotation are used to recompose a local deformation gradient which is, in general, incompatible. However, a compatible deformation field is recovered by minimization of a suitably-defined quadratic functional. This constrained minimization problem determines the displacement field that is closest, in a certain sense, to the extrapolated stretch and rotation. In this manner, an initial predictor is determined that is much more accurate, in many situations, than that given by the simple constant-nodal-velocity assumption. Three numerical examples are presented that illustrate the effectiveness of the proposed extrapolation method.