Downwind and upwind approximations for primal and dual problems of elasto-plasticity with Prandtl–Reuss type material laws

Abstract

Modern adaptive finite element (FE) algorithms for solution of initial-boundary-value-problems (IBVP) employ goal-oriented error measures in order to assess the quality of computational results for the physical event under investigation. However, traditional time-stepping algorithms for solution of the corresponding dual problem run backwards-in-time, which due to additional storage requirements might become a serious drawback when an extensive number of time steps for the FEM simulation arises. In this paper, we take advantage of an end-boundary-value-problem (EBVP) associated to IBVP, with corresponding dual-problem running forwards in time. In order to obtain a unified framework for numerical approximation of primal and dual weak forms for both, IBVP and EBVP, respectively, we apply the concept of downwind and upwind approximations not only to the trial functions but also to the test functions. This results into eight different integration schemes. On this basis, as a main result of this contribution, a time-stepping algorithm is obtained, which runs forwards-in-time for the dual problem and therefore avoids the additional storage requirements of the traditional backwards-in-time stepping procedures. The presented algorithm is numerically tested and validated for a CT-specimen for elastic and elasto-plastic behavior, where the constitutive equations are written in Prandtl–Reuss type format.