Selforganizing Maps to Enhance Local Performance of Multi Objective Optimization

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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

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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

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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.
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Basic formulation of Robust Design

• What happens when we optimise a function in which
  the input design parameters are defined by the
  mean value (Xm) and the deviation (d) ?

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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)

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Basic formulation of Robust Design
Mathematic formulation of the objective functions
with Robust Design Theory becomes
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Game Theory

• GAME THEORY

multi objective optimization problems
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Game Theory
COMPETITION
INDIVIDUALS
GAME
PLAYERS
DESIGN PROCESS
DESIGN TEAMS
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Game Theory

• COOPERATIVE GAMES
• 
  PARETO
• NON-COOPERATIVE GAMES
• 
  NASH
• SEQUENTIAL GAMES
• 
  STACKELBERG

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PARETO GAME
cooperative symmetric
Optimization of XY player1 player2
Player1
Player2


X
Y
Optimization of OBJECTIVE 1
Optimization of OBJECTIVE 2


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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


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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
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Multi Objective Robust Design
What do we need? We need the best compromises
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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 M0.77
Thus, it is necassary to optimise drag with (two)
input parameters given by mean values and
deviation MACH0.73?0.05 ?2o ?0.5
Uniform density function
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Pressure field
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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).

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Pressure field mean and variance
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Pressure profile mean and variance
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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

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Multi Objective Robust Design Optimization
The research dominion in the Multi Objective
Robust Design Optimization will be M0.73 0.05
and (angle of attack) aoa2 0.5 Four
objectives functions Seven Constraints MOGA
is exploited to find the solutions
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Multi Objective Robust Design Optimization
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Multi Objective Robust Design Optimization
Classical Pareto Frontier Rappresentation
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Multi Objective Robust Design Optimization
Lift and drag surfaces comparison
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Multi Objective Robust Design Optimization
Excellent results especially for drag
coefficient performance (mean value) and
stability (variance)
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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)
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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
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Self Organizing Maps
Clustering, local correlations, No linear
dependencies,
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Multi Objective Robust Design Optimization
Classical Pareto Frontier Rappresentation
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Multi Objective Robust Design Optimization
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Multi Objective Robust Design Optimization
Not only objectives, but design variables too!
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Multi Objective Robust Design Optimization
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With the SOM it is possible the visualization
between variables and performances