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250 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto Table 8.3 Values of the design variables for the selected linear piecewise channels ID i ID ii ID iii ID iv ID v L 2.00 1.64 1.54 1.58 1.47 h 0.400 0.400 0.383 0.400 0.400 ϕin 60.00 ◦ 59.04 ◦ 60 ◦ 59.04 ◦ 60.00 ◦ ϕout 60.00 ◦ 56.04 ◦ 50.16 ◦ 51.24 ◦ 49.32 ◦ transl 0.066 0.102 0.202 0.236 0.298 is highlighted. Due to the simplicity of the parametrization, the dominant set can probably be taken as the limit performance of this kind of geometry. In fact, further optimization process has not given appreciable improvements on design objectives. From the analysis of the shape of the channels along the Pareto front, sketched for convenience in the same figure, no sensible fluctuation is evident for variables like ϕin and h, whichremaincloseto60 ◦ and 0.4, respectively. What makes the difference is the translation of the upper profile, responsible for the development of the separation bubble induced by the corrugation. In Table 8.3, the values of the design variables required for the definition of the channels, marked in Fig. 8.12 for illustrative purpose, are presented. 8.8.2 NURBS Optimization As already stated, the increased number of degrees of freedom causes a more expensive optimization task. In contrast with linear piecewise optimization, the Full Factorial algorithm has not been used as first exploration of the design space. Since there are 11 degrees of freedom, the number of individuals to be computed would have risen to 3 11 , that means 177, 147, with a three-level Full Factorial. The Sobol algorithm [14] has been chosen to define an initial population of 50 individuals. The optimization algorithm chosen was again MOGA-II. The optimization has started allowing great freedom to the design variables, and no constraint has been imposed. In Table 8.4, the summary of design variable ranges and the number of steps (basis), which notches them to discrete form, are presented. The choice of the variables and their range might dramatically affect the convergence rate to a good solution. In our case, the generation of input strings leading to incoherent geometries has to be avoided as much as possible. One strategy is to scale the x-direction Cartesian variables to the length of the channel L. Therefore, all parameters are proportional to each other. Figure 8.13(a) shows the Pareto front after this first optimization stage. Five channels, representative of different combination of the two objectives, are highlighted. From Fig. 8.13(a), it is clear that all the individuals selected have in common the presence of closing bends
8 Multi-objective Optimization in Convective Heat Transfer 251 Fig. 8.13 Pareto front comparison between different optimization processes for the 2D NURBS channels: (a) firstMOGA;(b) first MOGA with constraint; see also Fig. 8.14 in wall profile. The same happens for almost all the other channels. Taking into account the industrial feasibility, though only from a methodological point of view, these channels would be very difficult to realize, for example, in a forming process. Therefore, a sort of fabricability check has been implemented, and its aim was to discard those geometries whose wall profile
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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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192 Laurent Dumas 7.1 Introducing A
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194 Laurent Dumas C Fig. 7.2 Wake f
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196 Laurent Dumas Table 7.1 Example
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198 Laurent Dumas Fig. 7.5 Numerica
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