Modeling Error and Adaptivity in Nonlinear Continuum Mechanics

Abstract

In this report, computable global bounds on errors due to the use of various mathematical models of physical phenomena are derived. The procedure involves identifying a so-called fine model among a class of models of certain events and then using that model as a datum with respect to which coarser models can be compared. The error inherent in a coarse model, compared to the fine datum, can be bounded by residual functionals unambiguously defined by solutions of the coarse model. Whenever there exist hierarchical classes of models in which levels of sophistication of various coarse models can be defined, an adaptive modeling strategy can be implemented to control modeling error. In the present work, the class of models is within those embodied in nonlinear continuum mechanics.

Structural Simulations Using Multi-Resolution Material Models

“…Some results from using ELSE with the coarse mesh of 867 elements are given in Ref. [13]. However, some tabular data and general observations considering all of the generated results will be presented later on.…”

“…The mean diameter of this sample was 880 pm, which is sufficient for CT imaging. According to the literature [13], Young's modulus of glass does not vary significantly from one type of glass to another. Therefore, for glass beads, we adopted E=73 GPa (E-glass).…”

“…It is straightforward to show that the following global bound on modeling error holds (see [13,19]):…”

“…An interpretation of the error measure on the left-hand side of (3.8) as a global smoothing of total error is given in [13].…”