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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.

The main purpose of this study is the development of an innovative methodology for Heat Exchangers (HE) design to replace the conventional design procedures. The new procedure is based on the definition of a software package managed by modeFRONTIER, a multi-objective optimization software produced by ESTECO, able to create HE virtual models by targeting several objectives, like HE performance, optimal use of material, HE minimal weight and size and optimal manufacturability.

This paper describes an application of Robust Design methodology in the transonic airfoil design. It has been observed that, minimizing the drag at a single design point (Mach number and angle of attack fixed), it is possible to find solutions characterized by poor offdesign performances (over-optimizing problem). For this reasons, the stability of the performances inside the range of operative conditions is an important objective in the design. Once the operative conditions are defined (range of Mach number and angle of attack), a Multi Objective approach is needed; in particular, two are the objectives to be optimized: the mean performances inside the range of operative conditions (optimise mean value of the aerodynamic coefficients) and the stability of the solution (minimize variance of the coefficients).In this Multi Objective optimization problem, we have applied a competitive Game Strategy, based on Nash equilibrium, combined with a particular mono-objective algorithm, the Simplex. The players are in charge of different objectives, corresponding to the two objectives, that have to be optimized by the Simplex algorithm. Since the variables space is split between the two players, each player influences the choices of the other one in the course of the optimisation, until an equilibrium point, corresponding to the best compromise between the objectives, is found.About the optimization test case, the range of operative conditions is Mach=0.73±0.05 and angle of attack 2°±0.5, and the original RAE2822 airfoil is parameterized. To reduce the high number of CFD analysis based on Navier-Stokes equations, a statistic extrapolation method, based on an adaptation of DACE, is used to define the required response surfaces.According to our results, the methodology seems to be a promising approach which offers a new possibility to the designer, in particular when a good compromise of performance and stability is required, with cheap computational resources.

In this article we describe a new method for the aerodynamic optimisation and inverse design problem resolution. This method is based on the coupling of a classical optimiser with a neural-network. A NavierStokes flow solver is used for an accurate computation of the objective function. At first the neural-network, which has been trained by an initial small database, is used to obtain, by the interpolation of the design sensitivities, a new design point, which is then computed by the Navier-Stokes solver in order to update the neural-network training database for further iterative step. Since the neural-network provides the optimiser with the derivatives, the objective function has to be evaluated only once at every step. By this method, the computational effort is significantly reduced with respect to the classical optimisation methods based on the design sensitivities, that are computed directly by the flow solver. The method proposed has been positively tested on the inverse design of a three-dimensional axial compressor blade, and a summary of the results is provided.