Hierarchical modeling of heterogeneous solids

The modeling of microscale effects required to describe physical phenomena such as the deformation of highly heterogeneous materials makes the use of standard simulation techniques prohibitively expensive. Most homogenization techniques that have been proposed to circumvent this problem lose small-s cale information and as a result tend to produce acceptable results only fo r narrow classes of problems. The concept of hierarchical modeling has been advanced as an approach to ov ercome the difficulties of multiscale modeling. Hierarchical modeling can b e described as the methodology underlying the adaptive selection of mathema tical models from a well-defined class of models so as to deliver results o f a preset level of accuracy. Thus, it provides a framework for the automat ic and adaptive selection of the most essential scales involved in a simula tion. In the present paper, we review the Homogenized Dirichlet Projection Method (HDPM) [J.T. Oden and T.I. Zohdi, Comput. Methods Appl. Mech. Engrg. 148 ( 1997) 367-391; T.I. Zohdi, J.T. Oden and G.J. Rodin, Comput. Methods Appl. Mech. Engrg. 138 (1996) 273-298] and present several extensions of its unde rlying theory. We present global energy-norm and L-2 estimates of the model ing error resulting from homogenization. In addition, new theorems and meth ods for estimating error in local quantities of interest,such as mollificat ions of local stresses are presented. These a posteriori estimates form the basis of the HDPM. Finally, we extend the HDPM to models of local failure and damage of two-phase composite materials. The results of several numeric al experiments and applications are given. (C) 1999 Elsevier Science S.A. A ll rights reserved.