A Competitive Game Approach for Multi Objective Robust Design Optimization

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. Nomenclature E( fi( x , u )) = mean value of function fi of the variables x in the range of fluctuating control parameters u ? ( fi( x , u )) = standard deviation of function fi in the range of fluctuating control parameters u ???????????????????????? mean of the stochastic process in DACE extrapolation ?(x) = extrapolation error in the point x defined by a Gaussian distribution of Normal type (0,? ) RMSE = Root Mean Squared Error of the extrapolation errors