Hierarchical models of thin elastic structures: Overview and recent advances in error estimation and adaptivity

In the numerical approximation of hierarchical models of thin bodies, there are two separate contributions to the total error: (a) modeling error is the error due to dimensional reduction, and (b) discretization error is the error due to the numerical approximation. Traditionally, estimation of these errors has been performed in terms of abstract quantities like the global energy norm, which does not give a clear representation of the error in the local features of the solution. Recently, a new class of finite element error estimation techniques has emerged wherein errors in a simulation are measured not in norms but in quantities of interest to the analyst. Following a brief overview of the literature, we focus on extending the theory of a posteriori estimation of the modeling error to local quantities of interest that have physical meaning. Independent estimates are obtained for the modeling and discretization errors in quantities of interest. These local estimates are used to develop goal-oriented adaptive modeling strategies in which the model is automatically adapted by controlling the two errors independently and effectively to obtain local quantities of interest to within a preset level of accuracy. Numerical examples are presented to demonstrate the accuracy of the error estimates and the adaptive strategy.