Adaptive Modeling of Composite Structures: Modeling Error Estimation

The accurate simulation of the behavior of composite materials depends OIl many factors such as the character, size, topology, and mechanical properties of the microstructure, the definition of the domain of interest and the loads as well as the specification of accuracy desired and the goal of the simulation. There is, therefore, a need to develop a systematic technique to adaptively select the most appropriate scale that governs specific features of the response that are of interest. The concept of hierarchical modeling provides such a framework. In this approach, the accuracy of a given mathematical model, compared to a model of finer scale, is evaluated with the use of a posteriori estimates of the "modeling error" and these form the basis of an adaptive procedure. This investigation focuses on the analysis of the equilibrium of linearly elastic heterogeneous bodies characterized by highly oscillatory elastic coefficients. The control of modeling error in such systems was studied in earlier works [37, 32] using the HDPM : the Homogenized Dirichlet Projection Method. There, estimates of the error in the homogenized solutions in energy norms were used as a basis for an adaptive process of model selection. Following a brief review of the HDPM, we introduce an alternate method of model adaptation : the Hierarchical Adaptive Method for Model Enhancement and Refinement (HAMMER). In this method, we seek to generate a sequence of elasticity tensors such that the solutions they produce converge to the solution produced by the fine-scale microstructure. We extend the theory of a posteriori estimation of modeling error to "quantities of interest", by which we mean local features of the response such as the average of stresses on material interfaces, boundary displacements, or mollified point-values of the displacement field, or any feature of the solution that can be characterized as a continuous linear functional on the space of functions to which the solution belongs. We establish computable upper and lower bounds on errors in such quantities of interest. This theory represents a significant departure from

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