In this talk, we discuss a goal-oriented methodology for the accurate simulation of certain multiscale phenomena. In virtually every simulation, there are two sources of error: (a) the use of a simplified mathematical model that does not fully capture the fine-scale physics of the phenomenon results in modeling error, and (b) the use of a numerical method, such as the finite element method, introduces discretization or approximation error. To minimize the total error in the simulation, it is necessary to estimate and control both contributions. While the estimation and control of discretization error is relatively well-explored [1, 2], there remain several unanswered questions regarding the estimation and control of modeling error.

In [3, 4], Oden and Vemaganti put forward an adaptive mechanism for choosing the most appropriate mathematical model based on the goals of the simulation. This goaloriented approach relies on local a posteriori estimates of modeling error. We describe some recent work on the local estimation and control of modeling error and discuss applications to plate- and shell-shaped structures.

References

[1] Ainsworth, M. and Oden, J. T. (2000). A Posteriori Error Estimation in Finite Element Analysis. New York: John Wiley & Sons.

[2] Babuˇska, I. and Strouboulis, T. (2001). The Finite Element Method and its Reliability. New York: Oxford University Press.

[3] Oden, J. T. and Vemaganti, K. (2000). Estimation of Local Modeling Error and GoalOriented Adaptive Modeling of Heterogeneous Materials; Part I: Error Estimates and Adaptive Algorithms. J. Comp. Phys. 164 (1), 22–47.

[4] Vemaganti, K. and Oden, J. T. (2001). Estimation of Local Modeling Error and GoalOriented Adaptive Modeling of Heterogeneous Materials; Part I I: A Computational Environment for Adaptive Modeling of Heterogeneous Elastic Solids. Comp. Meth. Applied Mech. Engrg. 190, 6089–6124.