Adaptive hierarchical modeling of heterogeneous structures

Predictability of the response of structural components to the action of external forces hinges on the selection of an appropriate mathematical and computational model of the governing physics. Invariably, this also involves decisions on what spatial and temporal scales are expected to be important in influencing the quality of the prediction. The process of model selection, particularly multiscale modeling, is not well defined and is often imprecise, heuristic, and the source of the most error in predicting physical behavior. This work presents a systematic technique for model selection and analysis of a class of multiscale problems encountered in the study of heterogeneous materials. The process, referred to as hierarchical modeling, consists of precisely characterizing a set of mathematical models of events of the smallest scale expected to influence the events of interest, and of developing rigorous a posteriori estimates of modeling error in the results obtained for one scale compared to models of finer scale. These estimated errors are then used in an adaptive process that automatically selects models and inherent spatial scales that produce simulations meeting preset error tolerances. The microstructures can be randomly distributed or deterministic or both, depending on the structure of models in the hierarchical set. The adaptive process can lead to models with non-uniform structure that depends upon boundary and initial data, loads and source terms, geometry, and other data. Several implementations of this process with applications to composite materials are described. (C) 1999 Elsevier Science B.V. All rights reserved.