Model adaptivity on effective elastic properties coupled with adaptive FEM

The research field of model adaptivity is well established, aiming at adaptive selection of mathematical models from a well defined class of models (model hierarchy) to achieve a preset level of accuracy. The present work addresses its application to a class of linear elastic composite problems. We will show that the classical bounding theories can provide a model hierarchy in a natural and theoretically consistent manner, without combination of different methods using a priori knowledge. To arrive at computable higher order bounds, the classical singular approximation is made. As a new finding, this may, under certain circumstances, give rise to an overlap effect. To overcome this, a correction is proposed. 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. For efficient computation of the dual solution, a patch-based recovery technique is proposed and compared to existing methods. For illustration, numerical examples are presented.