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248 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto Optimization TOOL Scheduler FALSE Fig. 8.11 Optimization work-flow Define initial set of designs(population) by DOE Define Input/Design variables SOLVER Bbuild geometry generate mesh ssolve forced Cconvection problem compute f and Nu Output objectives Stop condition TRUE Post-processing there is an ample variety of numerical techniques to perform optimization tasks. EAs are the most robust ones, and have been used in this optimization process. In particular, MOGA-II algorithm, has been employed. The optimization process, sketched in Fig. 8.11, follows these tasks: • Automatically the optimization software generates a set of numbers, i.e., the geometrical design variables, representing the shape of a channel (called individual). • These variables are written in an input file, which is sent to the CFD solver. This, in turn, computes the flow and thermal fields, and from these, it evaluates the friction factor and the Nusselt number, which represent the objective functions. • The numerical values of the objective functions are sent back to the optimizer, which generates another set of geometrical parameters.
8 Multi-objective Optimization in Convective Heat Transfer 249 f/f 0 10 8 6 4 2 Iii Pareto front Ii 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Nu/Nu 0 Fig. 8.12 Pareto front for the 2D linear piecewise optimization process The results are expressed in terms of the ratios Nu/Nu0 and f/f0, where Nu0 and f0 are the Nusselt number and the friction factor for a parallel plate channel, taken as reference geometry. 8.8 Results and Discussion 8.8.1 Linear Piecewise Optimization The first analysis has been conducted on the linear piecewise geometry type. The variables are small in number and clearly related to the shape of the channel: the depth of the asperity, the forward side angle, the backward side angle, the length of the channel and the translation of the upper wall. The Full Factorial algorithm [14] with three levels has been used (DOE) to sample the design space. This method gives the best homogeneous distribution of the samples. The number of Full Factorial samples is 243 and the parameters ranges are given in Table 8.1. In a GA optimization, the size of the population within the design space affects the convergence ratio. After having performed the numerical simulation on this set, its Pareto front has been chosen as the initial population of multi-objective optimization with GA. This preliminary Pareto front is a good selection in order to limit the number of starting channels. The optimization has been performed with MOGA-II along 30 generations of 20 individuals each: so the total number of designs evaluated is 600. The results are sketched in Fig. 8.12 where the Pareto front Iv Iiii Iiv
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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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