Approximation of the dual problem for error estimation in inelastic problems

Abstract

In numerical simulations with the finite element method the dependency on the mesh -and for time-dependent problems on the time discretization -arises. Adaptive refinements in space (and time) based on goal-oriented error estimation [1] become more and more popular for finite element analyses to balance computational effort and accuracy of the solution. The introduction of a goal quantity of interest defines a dual problem which has to be solved to estimate the error with respect to it. Often such procedures are based on a space-time Galerkin framework for instationary problems [2]. Discretization results in systems of equations in which the unknowns are nodal values. Contrary, in current finite element implementations for pathdependent problems some quantities storing information about the path-dependence are located at the integration points of the finite elements [3], e.g. plastic strains etc. In this contribution we propose an approach -similar to [4] for sensitivity analysis -for the approximation of the dual problem which mainly maintains the structure of current finite element implementations for path-dependent problems. Here, the dual problem is introduced after discretization. A numerical example illustrates the approach.

Dual‐based adaptive FEM for inelastic problems with standard FE implementations

“…For some restrictions, we show that it is possible to obtain equivalent formulations. As a consequence, we call our proposed method an approximation of the exact problem with the space‐time Galerkin method, and the results are promising, see also our previous work for preliminary results. The main innovations of this paper can be summarized as follows: Goal‐oriented error estimation that drives adaptive refinement in space and time for inelastic problems in solid mechanics.…”