Concurrent Aerodynamic Optimization of Rotor Blades Using a Nash Game Method

A multi-objective strategy adapted to the aerodynamic concurrent optimization of helicopter rotor blades is developed. The present strategy is based on Nash Games from game theory, where the objective functions are minimized by virtual players involved in a non-cooperative concurrent game. A method is presented to split the design vector into two sub-spaces, defined to be the strategies of the players in charge of the minimization of the primary and the secondary objective functions respectively. This split of territory allows the optimization of the secondary function while causing the least possible degradation of the first one. This methodology is applied to the model rotor ER- ATO, seeking to maximize the Figure of Merit in hover while minimizing the required rotor power in forward flight. An initial constrained optimization in hover is conducted using a previously devel- oped adjoint-based technique using the 3D Navier-Stokes solver elsA along with the gradient-based CONMIN algorithm. The chord, twist and sweep distributions of the baseline blade are parametrized using Be zier and cubic splines for a total of 16 design variables. The obtained optimized rotor is then used as a starting point to launch constrained and unconstrained Nash games. The comprehensive rotor code HOST is used to evaluate forward flight performance and a surrogate model is built to obtain the hover performance at low computational cost. Twist and sweep distribution laws are op- timized independently at first and then a final joint optimization involving twist, sweep and chord is performed. The results demonstrate the potential of this technique to obtain helicopter rotor designs realizing interesting trade-offs between strongly antagonistic objectives.

[1]  Ron Kohavi,et al.  A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection , 1995, IJCAI.

[2]  N. A. Diamantidis,et al.  Unsupervised stratification of cross-validation for accuracy estimation , 2000, Artif. Intell..

[3]  Manfred,et al.  High-Fidelity Optimization Framework for Helicopter Rotors , 2010 .

[4]  J. Périaux,et al.  Multicriterion Aerodynamic Shape Design Optimization and Inverse Problems Using Control Theory and Nash Games , 2007 .

[5]  J. Nash,et al.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[6]  Joanne L. Walsh,et al.  Optimization Methods Applied to the Aerodynamic Design of Helicopter Rotor Blades , 1987 .

[7]  M. D. McKay,et al.  A comparison of three methods for selecting values of input variables in the analysis of output from a computer code , 2000 .

[8]  Gianluca Iaccarino,et al.  Helicopter Rotor Design Using a Time-Spectral and Adjoint-Based Method , 2008 .

[9]  W. von Grünhagen,et al.  HOST, a General Helicopter Simulation Tool for Germany and France , 2000 .

[10]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[11]  A. Le Pape,et al.  Numerical optimization of helicopter rotor aerodynamic performance in hover , 2005 .

[12]  A. Le Pape,et al.  Aerodynamic Shape Optimization of Hovering Rotors Using a Discrete Adjoint of the Reynolds-Averaged Navier–Stokes Equations , 2011 .

[13]  Li Liu,et al.  Helicopter vibration reduction throughout the entire flight envelope using surrogate-based optimization , 2007 .

[14]  Michael S. Eldred,et al.  DAKOTA : a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis. Version 5.0, user's reference manual. , 2010 .

[15]  George N. Barakos,et al.  Optimising Aspects of Rotor Blades in Forward Flight , 2011 .

[16]  Bryan Glaz,et al.  Surrogate based optimization of helicopter rotor blades for vibration reduction in forward flight , 2008 .

[17]  Gene J Bingham.  The Aerodynamic Influences of Rotor Blade Taper, Twist, Airfoils and Solidity on Hover and Forward Flight Performance. , 1982 .

[18]  Garret N. Vanderplaats,et al.  CONMIN: A FORTRAN program for constrained function minimization: User's manual , 1973 .

[19]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[20]  F. Menter.  Two-equation eddy-viscosity turbulence models for engineering applications , 1994 .

[21]  Laurent Cambier,et al.  Status of the elsA CFD Software for Flow Simulation and Multidisciplinary Applications , 2008 .

[22]  Jean-Antoine Désidéri.  Cooperation and competition in multidisciplinary optimization , 2012, Comput. Optim. Appl..

[23]  G. D. van Albada,et al.  A comparative study of computational methods in cosmic gas dynamics , 1982 .

[24]  Paola Cinnella,et al.  Aerodynamic rotor blade optimization at Eurocopter - a new way of industrial rotor blade design , 2013 .

[25]  Johan C. Kok,et al.  Resolving the Dependence on Freestream Values for the k- Turbulence Model , 2000 .

[26]  Farrokh Mistree,et al.  Kriging Models for Global Approximation in Simulation-Based Multidisciplinary Design Optimization , 2001 .