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204 Laurent Dumas • if ˜ J(x ng i ) 0 is called the attenuation parameter. The scalar coefficients (ψi)1≤i≤nc are obtained by solving the least square problem of size N × nc: minimize err(x)= N� (J(Ti) − ˜ J(Ti)) 2 nc � + λ i=1 where λ>0 is called the regularization parameter. In order to attenuate or even remove the dependence of this model to its attached parameters, a secondary global optimization procedure (a classical GA) has been over-added in order to determine for each x, thebestvalues (with respect to err(x)) of the parameters nc, r ∈ [0.01, 10], λ ∈ [0, 10] and Φ ∈{Φ1,Φ2,Φ3,Φ4}. Asthisnewstepintroduces a second level of global optimization, it is only reserved to cases where the time evaluation of x ↦→ J(x) is many orders of magnitude higher than the time evaluation of x ↦→ ˜J(x), as in the car drag reduction problem. j=1 ψ 2 j
7 CFD-based Optimization for Automotive Aerodynamics 205 Fig. 7.7 The Rastrigin function Rast2 7.3.4 Validation on Analytic Test Functions Before applying them on real world applicative problems, all the previous global optimization algorithms have been tested and compared on various analytic test functions and among them the well-known Rastrigin function with n parameters: Rastn(x)= n� � 2 xi − cos(2πxi) � + n (7.10) i=1 defined on O =[−5, 5] n , for which there exists many local minima and only a global minimum located at xm =(0, ...,0) and equal to 0 (see Fig. 7.7). 7.3.4.1 AHMs vs. Evolutionary Algorithms A rather exhaustive comparison has been made between the classical evolutionary algorithm ES and the AHM introduced in Sect. 7.3.2. The statistical results are summarized in Table 7.3 for the Rastrigin function with 6 parameters. In this table, the success rate represents the rate of runs which were able to locate the correct attraction basin of the global minimum after a given number of evaluations of the cost function (respectively 500, 1000 and 2000). Note that any gradient evaluation counts for n evaluations of the cost function as it is the case in a finite-difference approximation.
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Optimization and Computational Flui
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Editors: Prof. Dr.-Ing. Dominique T
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vi Preface Our first research proje
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viii Contents 2.4.1 GoverningEquati
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x Contents 6.2.3 Parameterization..
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xii Contents 9.4.1 Multi-objectiveO
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xiv List of Contributors Gábor JAN
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Part I Generalities and methods
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4 Dominique Thévenin 12 10 8 6 4 2
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6 Dominique Thévenin Objective 10
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8 Dominique Thévenin Objective 10
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10 Dominique Thévenin Parameter 2
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12 Dominique Thévenin three-dimens
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14 Dominique Thévenin Let us come
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16 Dominique Thévenin References 1
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18 Gábor Janiga 2.1 Introduction D
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20 Gábor Janiga y Tinlet = 293 K 0
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22 Gábor Janiga y [mm] 25 20 15 10
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24 Gábor Janiga Table 2.1 Number o
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26 Gábor Janiga A dominates the in
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28 Gábor Janiga be shown later, th
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30 Gábor Janiga INPUT FILE Gambit
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34 Gábor Janiga ∆T ∆P Position
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36 Gábor Janiga ∆P [mPa] 2.2 2.0
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38 Gábor Janiga Table 2.3 Speed-up
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40 Gábor Janiga For all computatio
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42 Gábor Janiga y [mm] 4 2 0 -1 0
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44 Gábor Janiga Air mass flow-rate
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46 Gábor Janiga Temperature variat
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50 Gábor Janiga The modified k-ω
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52 Gábor Janiga Error for Re∗ =
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54 Gábor Janiga Error for Re∗ 20
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56 Gábor Janiga References 1. Ali,
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58 Gábor Janiga 43. Janiga, G., Th
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Chapter 3 Mathematical Aspects of C
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3 Mathematical Aspects of CFD-based
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Chapter 4 Adjoint Methods for Shape
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4 Adjoint Methods for Shape Optimiz
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Part II Specific Applications of CF
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112 Nicolas R. Gauger Nomenclature
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114 Nicolas R. Gauger Fig. 5.1 Cost
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116 Nicolas R. Gauger and the four
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118 Nicolas R. Gauger CL := 1 � C
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120 Nicolas R. Gauger the cost func
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122 Nicolas R. Gauger Table 5.1 An
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124 Nicolas R. Gauger The main adva
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126 Nicolas R. Gauger Spalart-Allma
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128 Nicolas R. Gauger (a) (b) Fig.
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134 Nicolas R. Gauger Fig. 5.10 Plo
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136 Nicolas R. Gauger drag, lift, s
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138 Nicolas R. Gauger solution of t
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140 Nicolas R. Gauger the presence
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142 Nicolas R. Gauger Fig. 5.16 Con
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144 Nicolas R. Gauger optimization
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Chapter 6 Numerical Optimization fo
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6 Numerical Optimization for Advanc
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6 Numerical Optimization for Advanc
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8 Multi-objective Optimization in C
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Chapter 9 CFD-based Optimization fo
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9 CFD-based optimization for paperm
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292 Index heat exchanger, 222, 224,
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