A Fast and Robust Adaptive Methodology for Airfoil Design Under Uncertainties based on Game Theory and Self-Organising-Map Theory

Abstract

Robust Design Optimization is the most appropriate approach to face problems characterized by uncertainties in the operating conditions, that represent a crucial point of aeronautical research activities. The Robust Design methodology illustrated in this paper is based on the multi-objective approach: applying the statistical definition of stability, the method finds, at the same time, optimised solutions for performances and stability.Game Theory is an innovative and efficient numerical methodology that can be applied to solve this kind of multi-objective optimization problems. A Competitive Game Strategy is applied in this paper by linking a mono-objective algorithm, like Downhill Simplex, with a statistical analysis methodology, based on t-Student or on the correlation matrix, that allow to find the optimal variables decomposition between the players (objectives) in the course of the optimization.An alternative to this statistical procedure is given by the innovative Self-Organising-Maps (SOM) theory, used to find correlations between input or output variables and based on non-linear ordered regression for topology data mapping.The test case used to compare the different methodologies, after a preliminary test on mathematical functions, is the optimization of a symmetric airfoil in transonic and Eulerian flow field with uncertainties in the free stream Mach Number; once the most efficient algorithm is chosen, it is applied to the most demanding optimization of a RAE2822 airfoil in transonic and viscous flow field with uncertainties in the free stream Mach Number and in the angle of attack. In these optimization cases, an adaptive Response Surface Methodology, called DACE, has been used in order to reduce the number of computations required. I. Introduction: Competitive Game Strategies applied to Robust Design Optimization A. Robust DesignMultidisciplinary Design Optimization is achieving more and more agreement in the aerospace community: many optimization methodologies have been developed to produce solutions that are interesting for the practical industrial cases. In fact, most of the industrial processes are permeated by uncertainties: the numerical design is generally different, from a geometric point of view, from the manufactured product because of the dimensional tolerances, and, more frequently, the working point is not fixed, but is characterized by some fluctuations in the operating variables.

Multifidelity Physics-Based Method for Robust Optimization Applied to a Hovering Rotor Airfoil

“…In particular, its influence on the airfoil angle of attack is the most significant, whereas minor effects are noticed on the Mach and Reynolds numbers. The level of uncertainty considered for each variable is consistent with those typically employed in the literature of uncertainty-based analysis in fluid dynamics (e.g., [42][43][44]). Their combined effect results in a large variation of the angle of attack once it is propagated through the BEMT equations.…”

Analysis of geometric uncertainties in CFD problems solved by RBF-FD meshless method

“…The existing methods for uncertainty quantification can roughly be divided into two main categories: nonintrusive, or statistical, and intrusive, or non-statistical. Examples of non-intrusive approaches are Monte Carlo [30,31], Stochastic Collocation [32,33,34], Chaos Collocation [6,35,36], metamodel-based methods [37,14].…”

Cumulative global metamodels with uncertainty — a tool for aerospace integration

“…This was done with the goal of fabricating and wind-tunnel testing several different canopies, in order to provide a smooth path from the existing, reference canopy to the (very different) optimised canopy. While optimisation methods have been developed (34) that can incorporate robustness from the onset, the current straightforward approach did yield a final design deemed to be robust, i.e., which was relatively insensitive to small perturbations in The results of these initial calculations were used to form the first generation of response surfaces (one for each aerodynamic condition). A global search of the objective function over the response surfaces would then 'suggest' the next set of CFD runs.…”