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.