Hierarchical modeling of heterogeneous solids

Abstract

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-scale information and as a result tend to produce acceptable results only for narrow classes of problems. The concept of hierarchical modeling has been advanced as an approach to overcome the difficulties of multiscale modeling. Hierarchical modeling can be described as the methodology underlying the adaptive selection of mathematical models from a well-defined class of models so as to deliver results of a preset level of accuracy. Thus, it provides a framework for the automatic and adaptive selection of the most essential scales involved in a simulation. 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 underlying theory. We present global energy-norm and L 2 estimates of the modeling error resulting from homogenization. In addition, new theorems and methods for estimating error in local quantities of interest, such as mollifications 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 numerical experiments and applications are given.