Is there an order-barrier $p\leq2$ for time integration in computational elasto-plasticity?

Abstract

This paper is devoted to the question, whether there is an order barrier $p\leq2$ for time integration in computational elasto-plasticity. In the analysis we use an implicit Runge-Kutta (RK) method of order $p=3$ for integrating the evolution equations of plastic flow within a nonlinear finite element framework. We show that two novel algorithmic conditions are necessary to overcome the order barrier, (i) total strains must have the same order in time as the time integrator itself, (ii) accurate initial data must be calculated via detecting the elastic-plastic switching point (SP) in the predictor step. Condition (i) is for a \emph{consistent} coupling of the global boundary value problem (BVP) with the local initial value problems (IVP) via displacements/strains. Condition (ii) generates consistent initial data of the IVPs. The third condition, which is not algorithmic but physical in nature, is that (iii) the total strain path in time must be smooth such that condition (i) can be fulfilled at all. This requirement is met by materials showing a sufficiently smooth elastic-plastic transition in the stress-strain curve. We propose effective means to fulfil conditions (i) and (ii). We show in finite element simulations that, if condition (iii) is additionally met, the present method yields the full, theoretical convergence order 3 thus overcoming the barrier $p\leq 2$ for the first time. The observed speed-up for a 3rd order RK method is considerable compared with Backward Euler.