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244 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto 8.7.2.1 Aggregating Functions The most straightforward approach in handling multiple objectives is the use of an arithmetical combination of all the objectives. Thus, the resulting single function can be studied with any of the single-objective algorithms either evolutionary or classical. Aggregating approaches are the oldest mathematical programming methods found in literature [45]. Applied to EA, the aggregate function approach does not require any change to the basic search mechanism. Therefore, it is efficient, simple, and of easy implementation. It can be successfully used on simple multi-objective optimizations that present continuous convex Pareto fronts. An example of this approach is a linear sum of weights of the form: � m� � min wifi(X) i=1 where the weight wi represent the relative importance of the m-th objective function. The weighting coefficients are usually assumed to sum at 1: m� wi =1. i=1 Aggregate function may be linear as in the previous example or nonlinear. Both types of function have been used with evolutionary algorithms but, generally speaking, aggregating methods have some limitations in generating complex Pareto fronts, though nonlinear aggregating function do not necessarily present such limitations [8]. Aggregating functions are widely used in Multi-Criteria Decision Analysis (MCDA) or Multi-Criteria Decision Making (MCDM) [5]. MCDA is a discipline aimed at supporting decision makers who are faced with making numerous and conflicting evaluations. MCDM is frequently used by EA users for a posteriori analysis of optimization results, but the discipline can be applied in a much more sophisticated way for a priori analysis. MCDM is briefly introduced in Sect. 8.7.3. 8.7.2.2 Population-based Approaches In these techniques the population of an EA is used to diversify the search, but the concept of Pareto dominance is not directly incorporated into the selection process. The first approach of this kind is the Vector Evaluation Genetic Algorithm (VEGA) introduced by Schaffer [48]. At each generation this algorithm performs the selection operation based on the objective switching rule, i.e., selection is done for each objective separately, filling equal portions of mating pool (the new generation) [12]. Afterwards, the mating pool is shuf-
8 Multi-objective Optimization in Convective Heat Transfer 245 fled, and crossover and mutation are performed as in basic GAs. Solutions proposed by VEGA are locally non-dominated, but not necessarily globally non-dominated. This comes from the selection operator which looks for optimal individuals for a single objective at a time. This problem is known as speciation. Groups of individuals with good performances within each objective function are created, but non-dominated intermediate solutions are not preserved. 8.7.2.3 Pareto-based Approaches Recognizing the drawbacks of VEGA, Goldberg [20] proposed a way of tackling multi-objective problems that would become the standard in MOEA for several years. Pareto-based approaches can be historically divided into two generations. The first is characterized by fitness sharing and niching combined with the concept of Pareto ranking. Keeping in mind the definition of non-dominated individual, the rank of a design corresponds to the number of individuals by which it is dominated. Pareto front element has a rank equal to 1. The most representative algorithms of the first generation are Nondominated Sorting Genetic Algorithm (NSGA), proposed by Srinivas and Deb [50], Niched- Pareto Genetic Algorithm (NPGA) by Horn et al. [21], and Multi-Objective Genetic Algorithm by Fonseca and Fleming [19]. The second generation of MOEAs was born with the introduction of elitism. Elitism refers to the use of an external population to keep track of non-dominated individuals. In such a way, viable solutions are never disregarded in generating offsprings. The second generations of multi-objective algorithms based on GA are: 1. MOGA-II. MOGA-II uses the concept of Pareto ranking. Considering a population of n individuals, if at generation t, individual xi is dominated designs of the current generation, its rank is given by: by p (t) i rank(xi,t)=1+p (t) i . All non-dominated individuals are ranked 1. Fitness assignment is performed by: a) Sort population according to rank; b) Assign fitness to individuals by interpolating from the best to the worst; c) Average the fitness of individuals with the same rank. In this way, the global fitness of the population remains constant, giving quite a selective pressure on better individuals. The modeFRONTIER c○ version of MOGA-II includes the following smart elitism operator [39]:
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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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