Hierarchical modeling of heterogeneous bodies

In this paper, a methodology is introduced for the development of adaptive methods for hierarchical modeling of elastic heterogeneous bodies. The approach is based on the idea of computing an estimate of the modeling error introduced by replacing the actual fine-scale material tensor with that of a homogenized material, and to adaptively refine the material description until a prespecified error tolerance is met. This process generates a family of coarse-scale solutions in which the solution corresponding to the fine-scale model of the body, which embodies the exact microstructure, loading and boundary conditions, represents the highest level of sophistication in a family of continuum models. The adaptive strategy developed can lead to a new non-uniform description of material properties which reflects the loading and boundary conditions. A post-processing technique is also introduced which endows the coarse-scale solutions with fine-scale information, through a local solution process. Convergence of the adaptive algorithm is proven and modeling error estimates as a function of scale of material description are presented. Preliminary results of several numerical experiments are given to confirm estimates and to illustrate the promise of the approach in practical applications.