Hierarchical modeling of heterogeneous solids
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The modeling of microscale erfects required to describe physical phenomena such as the deformation of highly heterogeneous materials makes the use of standard simulation techniques prohibitively expensive. Most homogenizatioll techniques that have been proposed to circumvent this problem lose small-scale information and as a result tend to produce acceptable results only for narrow classes of problems. The concept of hierarchicaill/odt'/illg has been advanced as an approach to overcome the difljcultics of lIlultiscale modeling. Hierarchical modeling can be described as the methodology underlying the adaptive selection of mathematical models from a well-detined class of models so as to deliver results of a prescl level of accuracy. Thus, il provides a framework for the autol1latic .lIld adaptive selection of the most essential scales involved in a simulation. In the present paper. we review the Homogenized Dirichlet Projection Method (I-IDPM) lJ.T. Odcn and T.I. Zohdi. Compu!. Methods Appl. Mech. Engrg. 148 (1997) 367-391: T.I. Zohdi. J.T. Oden and GJ. Rodin. Compu!. ~1ethods Appl. Mech. Engrg. 138 (1996) 273-2981 and present several extcnsions of its underlying theory. Wc present global energy-norm and L 2 estimates of the modeling error resulting from homogcnization. In addition. new theorems and methods for estimating error in local quantities of interest,such as mollifications of local stresses arc prcsented. These a posteriori estimates form the basis of thc HDPM. Finally, we extend the HDPM to models of local failure and damagc of two-phase composite materials. The results of scvcral llulllcrical cxperiments and applications are given. © 1999 Elsevier Science S.A. All rights reserved.