Multi Level Monte Carlo Methods for Uncertainty Quantification and Robust Design Optimization in Aerodynamics

The vast majority of problems that arise in aircraft production and operation require decisions to be made in the presence of uncertainty. An effective and accurate quantification and control of the level of uncertainty introduced in the design phase and during the manufacturing and operation of aircraft vehicles is imperative in order to design robust and risk tolerant systems. Indeed, the geometrical and operational parameters, that characterize aerodynamic systems, are naturally affected by aleatory uncertainties due to the intrinsic variability of the manufacturing processes and the surrounding environment. Reducing the geometrical uncertainties due to manufacturing tolerances can be prohibitively expensive while reducing the operational uncertainties due to atmospheric variability is simply impossible. The quantification of those two type of uncertainties should be available in reasonable time in order to be effective and practical in an industrial environment. The objective of this thesis is to develop efficient and accurate approaches for the study of aerodynamic systems affected by geometric and operating uncertainties. In order to treat this class of problems we first adapt the Multi Level Monte Carlo probabilistic approach to tackle aerodynamic problems modeled by Computational Fluid Dynamics simulations. Subsequently, we propose and discuss different strategies and extensions of the original technique to compute statistical moments, distributions and risk measures of random quantities of interest. We show on several numerical examples, relevant in compressible inviscid and viscous aerodynamics, the effectiveness and accuracy of the proposed approach. We also consider the problem of optimization under uncertainties. In this case we leverage the flexibility of our Multi Level Monte Carlo approach in computing different robust and reliable objective functions and probabilistic constraints. By combining our approach with single and multi objective evolutionary strategies, we show how to optimize the shape of transonic airfoils in order to obtain designs whose performances are as insensitive as possible to uncertain conditions.