Goal-oriented adaptivity based on a model hierarchy of mean-field and full-field homogenization methods in linear elasticity

Homogenization methods are drawing increasing attention for simulation of heterogeneous materials like composites. For balancing the accuracy and the numerical efficiency of such strategies, we deal with both model and discretization errors of the finite element method (FEM) on a macroscale. Within a framework of goal-oriented adaptivity, we consider linear elastic heterogeneous materials, for which first-order homogenization schemes apply. A novel model hierarchy is proposed based on mean-field and full-field homogenization methods. For the former, we consider several well-established schemes like Mori-Tanaka or self-consistent as basic models, and for the latter, as superior models, unit cell problems are solved via the FEM under an a priori chosen boundary condition. For a further stage of the model hierarchy, we consider hierarchical unit cells within the frame of the FEM toward an adaptive selection of the unit cell size. By means of several numerical examples, we illustrate the effectiveness of the proposed adaptive approach.