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246 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto a) MOGA-II starts with an initial population P of size n and an empty elite set E = ∅ b) For each generation, compute P ′ = P ∪ E c) If cardinality of P ′ is greater than cardinality of P, reduce P ′ randomly removing exceeding points d) Generate P ′′ by applying MOGA algorithm to P ′ e) Calculate P ′′ fitness and copy non-dominated designs to E f) Purge E from duplicated and dominated designs g) If cardinality of E is greater than cardinality of P randomly shrink the set h) Update P with P ′′ andreturntostep(b) 2. NSGA-II. NSGA-II employs a fast non-dominated sorting procedure and uses the crowding distance (which is an estimate of the density of solutions in the objective space) as a diversity preservation mechanism. Moreover, NSGA-II has an implementation of the crossover operator that allows the use of both continuous and discrete variables. The NSGA-II does not use an external memory as MOGA does. Its elitist mechanism consists of combining the best parents with the best offspring obtained. 8.7.3 Multi-Criteria Decision Making (MCDM) As already stated, it is impossible to find out a unique best solution in a multi-objective optimization process, but rather a whole group of designs that dominate the others: the Pareto front or Pareto optimal set.AllPareto optimal solutions can be regarded as equally desirable in a mathematical sense. But from an engineering point of view, the goal is a single solution to be put into practice at the end of an optimization. Hence, the need for a decision maker (DM) who is able to identify the most preferred one among the solutions. The decision maker is a person who is able to express preference information related to the conflicting objectives. Ranking between alternatives is a common and difficult task, especially when several solutions are available or when many objectives or decision makers are involved. Decisions have taken over a limited set of good alternatives mainly due to the experience and competence of the single DM. Therefore, the decision stage can be described as subjective and qualitative rather than objective and quantitative. Multi-Criteria Decision Making (MCDM) refers to the solving of decision problems involving multiple and conflicting goals, coming up with a final solution that represents a good compromise that is acceptable to the entire team. As already underlined, when dealing with a multi-objective optimization, the decision making stage can be done in three different ways [53]:
8 Multi-objective Optimization in Convective Heat Transfer 247 • Decision-making and then search (a priori approach). The preferences for each objective are set by the decision-makers and then, one or various solutions satisfying these preferences have to be found. • Search and then decision-making (a posteriori approach). Various solutions are found and then, the decision-makers select the most adequate. The solutions chosen should represent a trade-off between the various objectives. • Interactive search and decision-making. The decision-makers intervene during the search in order to guide it towards promising solutions by adjusting the preferences in the process. modeFRONTIER c○ implementation of MCDM allows the user to classify all the available alternatives through pair-wise comparisons on attributes and designs. Moreover, modeFRONTIER helps decision makers to verify the coherence of relationships. Thus it is an a posteriori or interactive oriented tool. To be coherent, a set of relationships should be both rational and transitive. To be rational means that if the decision maker thinks that solution A is better than solution B, then solution B is worse than solution A. To be transitive means that if the decision maker says that A is better than B and B is better than C, then solution A should be always considered better than solution C. In this way, the tool allows the correct grouping of outputs into a single utility function that is coherent with the preferences expressed by the user. Four algorithms are implemented in modeFRONTIER: • Linear MCDM. When the number of decision variables is small; • GA MCDM. This algorithm does not perform an exact search so it can be used even when the number of decision attributes is large; • Hurwicz criterion. Used for the uncertain decision problems; • Savage criterion. Used for the uncertain decision problems where both the decision states and their likelihoods are unknown. 8.7.4 Optimization Process The parameters that influence the performances of a heat exchanger, and in particular of the single repeating module under study, are its overall heat transfer coefficient and the pumping power required to supply a desired mass flow. In a dimensionless perspective, these two parameters are represented by the friction factor and the Nusselt number, respectively. Both quantities are function of the geometric variables: [f,Nu] = g(X) (8.42) where vector X is constituted by the degrees of freedom of the chosen parametrization, either linear piecewise, Bézier or NURBS. As stated before,
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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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