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.