Goal‐oriented adaptivity based on a model hierarchy of mean‐field and full‐field homogenization methods in elasto‐plasticity

Abstract

AbstractMaterials such as composites are heterogeneous at the micro scale, where several constituents with different material properties can be distinguished like elastic inclusions and an elasto‐plastic matrix. 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 an efficient combination of numerically low cost mean‐field and numerically high cost full‐field homogenization methods in elasto‐plasticity. To this end, model adaptivity is a promising methodology as it has been well‐established for linear elastic composite models [1]. Quite similarly to the adaptive FEM, it starts with an affordable homogenization method, and then through a loop wise error control, a local switch to accurate homogenization methods is performed to enhance the accuracy (referred to as model refinement). Like hierarchical FE structures for adaptive FEM, hierarchical model structures are established for model adaptivity. Additionally, the model adaptivity is coupled to the well established adaptive finite element method (FEM), such that both macro model and macro discretization errors are controlled. The proposed adaptive procedure is driven by a goal‐oriented a posteriori error estimator based on duality techniques. Finally, two numerical examples illustrate the effectiveness of the proposed adaptive approach.