Adaptive hierarchical modeling of heterogeneous structures

Predictability of the response of structural components to the action of ex ternal forces hinges on the selection of an appropriate mathematical and co mputational 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 selectio n, particularly multiscale modeling, is not well defined and is often impre cise, heuristic, and the source of the most error in predicting physical be havior. This work presents a systematic technique for model selection and a nalysis of a class of multiscale problems encountered in the study of heter ogeneous materials. The process, referred to as hierarchical modeling, cons ists of precisely characterizing a set of mathematical models of events of the smallest scale expected to influence the events of interest, and of dev eloping rigorous a posteriori estimates of modeling error in the results ob tained for one scale compared to models of finer scale. These estimated err ors are then used in an adaptive process that automatically selects models and inherent spatial scales that produce simulations meeting preset error t olerances. 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 depend s upon boundary and initial data, loads and source terms, geometry, and oth er data. Several implementations of this process with applications to compo site materials are described. (C) 1999 Elsevier Science B.V. All rights res erved.