Materials such as composites are heterogeneous at the micro-scale, where several constituents with different material properties can be distinguished like elastic inclusions and the elasto-plastic matrix with isotropic hardening. One has to deal with these heterogeneities on the micro-scale and then perform a scale transition to obtain the overall behavior on the macro-scale, which is often referred to as homogenization. The present contribution deals with the combination of numerically inexpensive mean-field and numerically expensive full-field homogenization methods in elasto-plasticity coupled to adaptive finite element method (FEM) which takes into account error generation and error transport at each time step on the macro-scale. The proposed adaptive procedure is driven by a goal-oriented a posteriori error estimator based on duality techniques. The main difficulty of duality techniques in the literature is that the backwards-in-time al gorithm has a high demand on memory capacity since additional memory is required to store the primary solutions computed over all time steps. To this end, several down wind and upwind approximations are introduced for an elasto-plastic primal problem by means of jump terms . Therefore, from a computational point of view, the forwards-in time duality problem is very attractive. A numerical example illustrates the effectiveness of the proposed adaptive approach based on forwards-in-time method in comparison to backwards-in-time method.
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