Estimation of Local Energy Norms of Modeling Error in Multi-Scale Modeling of Linearly Elastic Heterogeneous Solids

Abstract

This thesis presents the results of an investigation into the development of an estimator of modeling error in terms of local strain energy norms in multi-scale mod- eling of linear heterogeneous solids. Analysis of heterogeneous solids in engineering or computationally prohibitive problems presents challenges in that the media often possess a micro-structure too complex to numerically analyze for practical purposes. The macroscopic or 'global' behavior a of heterogeneous solid is often known to be predictable using an 'effective' or homogenized surrogate model that is computable. However, the ability to predict critical fine-scale features of the response is lost. Multi- scale modeling techniques have been introduced as a means of including fine-scale information in user specified regions of interest and the degree to which information is added is generally determined by a tolerance in terms of an error estimate. A means of assessing the error in modeling of heterogeneous solids is desired in order to determine validity of such surrogate models, both the homogenized and multi-scale models. Previous work in estimation of modeling error generally quantifies the error by using a residual-based methodology. This requires the solution of dual problems iv governing the quantity/feature of the model that is of interest to the analyst. The features of interest generally concern fine or micro-scale features of the response, since they play a crucial role in the initiation as well as the evolution of micro-scale failure mechanisms. Eventually, however they could lead to structural failure, i.e. on the macro-scale. This thesis adds to the modeling error estimation field by introducing an estimate of the modeling error in terms of a nonlinear quantity of interest, the local strain energy norm. The estimate is provided in terms of the integral of the error in the strain energy over a local domain of interest. Two estimates are explored, one being an equivalent quantifier of the local strain energy, and one a lower bound. Numerical verifications are provided, both including two-phase linearly elastic composites under loading.