Local Estimation of Modeling Error in Multi-Scale Modeling of Heterogeneous Elastic Solids
University of Kansas
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This thesis presents the results of an investigation toward the development of a new methodology of local estimation of modeling error in the analysis of linear elastostatic problems of heterogeneous solids. Due to the increase in the use of multiphase composites in structural engineering applications in the past three or four decades, the numerical analysis of their mechanical response has accordingly gained importance. However, it is well known that the microscopic heterogeneity of the material data generally leads to computational problems that are beyond practical means. Several methods have been developed that essentially seek to replace the complex models with surrogate descriptions of the material behavior that lead to computational feasibility. In this work, a relatively recent approach, called Goal-Oriented Adaptive Modeling (GOAM), is considered, which was introduced by Oden, et al. The distinguishing feature in its concept is that it seeks to construct surrogate, multi-scale material models that are capable of providing accurate predictions of the microscopic features of the material response that are of practical interest to the analyst. To do so, a homogenized surrogate model is first established, using existing classical homogenization techniques, and its response is obtained. A goal-oriented assessment is then made of the quality of the solution from the surrogate model by computing estimates of the modeling error in the fine-scale response features that are of interest to the analyst. Depending on the estimated error, the model is accordingly enhanced in an iterative process by including some fine-scale heterogeneous data in a small portion of the domain near the domain of interest. A crucial aspect of this process is the error estimate. For the method to be of practical use, one must be able to verify the accuracy of the predictions of any of the surrogate descriptions. Only then can one accept or reject any of the surrogate descriptions, with or without some fine scale features. In other words: the estimates are necessary to validate the models. Current error estimates rely on the solution of a dual problem, related to the quantity of interest of the response. The dual problem in these approaches is defined globally, requiring the heterogeneous material data for the entire domain. Thus, the computation of the dual solution is just as prohibitively expensive as the solution of the exact problem. To overcome this, surrogate models have been proposed to compute the dual problem as well, further introducing an inaccuracy into the error estimate. In this thesis, these problems are resolved by introducing a new, local error estimator that can be computed exactly, therefore eliminating the need for expensive global computations and enhancement iterations in the error estimation process. The estimates are only applicable to quantities of interest involving stresses and gradients of displacement and quantities that can be expressed as linear functionals of the material response. A two-dimensional, heterogeneous, linearly elastic beam under bending and tensile loads is used as a model problem to present numerical studies.
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