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Self-organizing Maps to Enhance Local Performance of Multi Objective Optimization

This talk focuses on using Self-Organizing Maps to improve the local performance of multi-objective optimization, particularly in the context of uncertain operating variables. It discusses the formulation of robust design and the application of game theory to solve these optimization problems.

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Self-organizing Maps to Enhance Local Performance of Multi Objective Optimization

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  1. Self-organizing Maps to Enhance Local Performance of Multi Objective Optimization Valentino Pediroda, Danilo Di Stefano Dipartimento di Ingegneria Meccanica Università di Trieste Trieste, ITALY Esteco Srl Trieste, Italy

  2. Basic formulation of Robust Design • 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. • More frequently, the working point is not fixed, but is characterized by some fluctuations in the operating variables. • In this talk we focus on the uncertainties in the operating variables; in the airfoil design case: • angle of attack • Mach Number

  3. Example in aeronautics Uncertainties on Mach number causes over-optimization Hicks R. M. and Vanderplaats G. N.,Application of numerical optimization to the design of supercritical airfoils without drag-creep, SAE Paper No. 770440, Business Aircraft Meeting, Wichita, 1977.

  4. Non-dominated solution in case of single point optimization algorithm Non-dominated solution with Robust Design (more stable) Basic formulation of Robust Design • What happens when we optimise a function in which the input design parameters are defined by themeanvalue (Xm) and thedeviation(d) ?

  5. Basic formulation of Robust Design • So when there is the presence of fluctuations a Multi Objective Approach is needed • Maximise the mean value of the function (performance) • Minimise the variance of the function (stability)

  6. probability distribution of uncertainties Basic formulation of Robust Design Mathematic formulation of the objective functions withRobust Design Theorybecomes:

  7. Game Theory GAMETHEORY multi objective optimization problems

  8. Game Theory • COOPERATIVE GAMES • PARETO • NON-COOPERATIVE GAMES • NASH • SEQUENTIAL GAMES • STACKELBERG

  9. PARETO GAME cooperative symmetric Optimization of XY player1 player2 Player1 Player2 … … X Y Optimization of OBJECTIVE 1 Optimization of OBJECTIVE 2 … …

  10. NASH GAME non-cooperative symmetric Optimization of XY player1 player2 Player1 Player2 … … X Y Optimization of OBJECTIVE 1 Y costant given by player 2 Optimization of OBJECTIVE 2 X costant given by player 1 X Y … …

  11. STACKELBERG GAME hierarchic competitive Optimization of XY player1 player2 LEADER FOLLOWER Player1 … Player2 Optimization of OBJECTIVE 2 X costant given by leader X Optimization step of OBJECTIVE 1 Y costant given by follower Y X Optimization of OBJECTIVE 2 X costant given by leader … Y

  12. High average performance but not stable Standard Deviation Stable solution but low average performance Solution in the middle Function (Average Value) Multi Objective Robust Design What do we need? We need the best compromises

  13. Application in robust design airfoil optimization • It is possible to illustrate the concept of Robust Design considering a 2D airfoil shape optimization problem in transonic field. It has been observed (Hicks and Vanderplaats, 1977) that minimizing drag at a single design point causes reduction of performances (D) at nearby off-design points. original D optimised at M=0.77 Thus, it is necassary to optimise drag with (two) input parameters given by mean values and deviation MACH=0.730.05 =2o0.5 Uniform density function

  14. Pressure field

  15. Application in robust design airfoil optimization • To understand the different optimisation techniques in relation to the Robust Design problem, we choose a simple case: • Symmetric airfoil (baseline NACA0012) • 0° incidence • MIN E(Cd), MIN (Cd) • Navier-Stokes code (MUFLO from EADS) • Parameterisation of the airfoil by means 9 design variables (Bezièr weighting points).

  16. Pressure field: mean and variance

  17. Pressure profile: mean and variance

  18. Multi Objective Robust Design Optimization • In the optimization process the achievable configurations have been determined by modifying an baseline configuration, the supercritical airfoil RAE 2822 designed by the Royal Aircraft Establishment • Parameterisation of the airfoil by means 18 design variables (Bezièr weighting points). • Navier-Stokes solver (MUFLO) • turbulence model: Johnson Coakley equations

  19. Multi Objective Robust Design Optimization • The research dominion in the Multi Objective Robust Design Optimization will be M=0.73 ± 0.05 and (angle of attack) aoa=2° ± 0.5°; • Four objectives functions Seven Constraints • MOGA is exploited to find the solutions;

  20. Multi Objective Robust Design Optimization

  21. Multi Objective Robust Design Optimization Classical Pareto Frontier Rappresentation

  22. Multi Objective Robust Design Optimization Lift and drag surfaces comparison

  23. Multi Objective Robust Design Optimization Excellent results especially for drag coefficient: performance (mean value) and stability (variance)

  24. Visualization in Multi-D (SOM) Clustering of data Self-Organizing Maps The Self-Organizing Map (SOM) is an unsupervised neural network algorithm that projects high-dimensional data onto a two-dimensional map With a n-dimensional space, the SOM makes an association between the data and n regular grids (one for every dimension)

  25. Self Organizing Maps • Iris example (classical example in data mining), 4 parameters: • Petal lenght • Petal width • Sepal lenght • Sepal with • 3 different classes of iris: • Setosa, Virginica or Versicolor SepalL SepalW PetalL PetalW Tipo 4.6 3.6 1.0 0.2 Setosa 5.1 3.3 1.7 0.5 Setosa 4.8 3.4 1.9 0.2 Setosa 5.0 3.0 1.6 0.2 Setosa 5.0 3.4 1.6 0.4 Setosa 6.5 2.8 4.6 1.5 Versicolor 5.7 2.8 4.5 1.3 Versicolor 6.3 3.3 4.7 1.6 Versicolor 4.9 2.4 3.3 1.0 Versicolor 6.6 2.9 4.6 1.3 Versicolor 7.6 3.0 6.6 2.1 Virginica 4.9 2.5 4.5 1.7 Virginica 7.3 2.9 6.3 1.8 Virginica 6.7 2.5 5.8 1.8 Virginica 7.2 3.6 6.1 2.5 Virginica 6.5 3.2 5.1 2.0 Virginica

  26. Self Organizing Maps Clustering, local correlations, No linear dependencies,

  27. Multi Objective Robust Design Optimization Classical Pareto Frontier Rappresentation

  28. Multi Objective Robust Design Optimization

  29. Multi Objective Robust Design Optimization Not only objectives, but design variables too!

  30. Multi Objective Robust Design Optimization V13 With the SOM it is possible the visualization between variables and performances

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