Estimation of Local Modeling Error and Goal-Oriented Adaptive Modeling of Heterogeneous Materials: I. Error Estimates and Adaptive Algorithms

A theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented. These quantities may, for example, represent averaged stresses on the surface of inclusions or mollifications of pointwise stresses or displacements or, in general, local features of the “fine-scale” solution characterized by continuous linear functionals. These estimators are used to construct goal-oriented adaptive procedures in which models of the microstructure are adapted to deliver local features to a preset level of accuracy. Algorithms for implementing these procedures are discussed and preliminary numerical results are given. The analysis is restricted to linear, static, heterogeneous, elastic materials.