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280 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen minimization problem, but if some objective function fi is to be maximized, it is equivalent to considering minimization of −fi. In multi-objective optimization, optimality is understood in the sense of Pareto optimality [4]. A decision vector x ′ ∈ S is Pareto optimal if there does not exist another decision vector x ∈ S such that fi(x) ≤ fi(x ′ )for all i =1,...,nf and fj(x)
9 CFD-based optimization for papermaking 281 9.4.2 Modeling and Optimizing the Complete Papermaking Process Although accurate simulation of the entire paper machine is still far ahead, a virtual papermaking line, as we call it, has been developed [20, 22]. It can be used for papermaking simulation as well as for tailored multi-objective optimization. The virtual papermaking line combines dissimilar unit-process models from different disciplines that include mathematical formulas ranging from simple algebraic equations to CFD models. It also includes models for moisture and heat transfer, and for paper quality properties. Simplifications of computationally expensive models are used, for instance, in water removal which means that an accurate multi-phase flow model of the dewatering phenomenon is replaced with a statistical model based on data produced by the accurate model. The latter models are especially useful in optimization, when tens, hundreds or even thousands of simulation model evaluations are needed. We define the virtual papermaking line model as follows ⎧ ⎪⎨ ⎪⎩ A1(p 1,q 1)=0 A2(p 2,q 1,q 2)=0 . Anm(p nm,q 1,...,q nm)=0 (9.6) where Ai for all i =1,...,nm stand for the unit-process models, p i ∈ R li denotes a vector of the inputs, and q i ∈ R ki is a vector of the outputs (model states) for the i-th unit-process model Ai. We assume here that all the unitprocess models and their outputs are continuously differentiable or if there is nonsmoothness involved, we assume at least H-differentiability [7]. Based on (9.6) multi-objective optimization problems related to papermaking process can be determined. Instead of the problem formulation (9.5) we use the following model-based optimization problem formulation Optimize {f1(x,q 1,...,qnm),...,fnf(x,q 1,...,qnm)} x � (9.6) subject to x ∈ S (9.7) where x ∈ S is a vector of the continuous decision variables (also called control or design variables) which is a selected set of the unit-process model inputs p =(p 1 ,...,p nm ) T . The feasible set S is defined by the box constraints of the decision variables and the linear and non-linear constraint functions similarly to (9.5). In (9.7), we ignore those input parameters that are not chosen as decision variables, because they are constants during the optimization process. By f1(x, q 1,...,q nm),...,fnf(x, q 1,...,q nm) we denote the objectives which are to be optimized, i.e., minimized or maximized. Similarly to
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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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200 Laurent Dumas 7.3.1.1 Genetic A
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202 Laurent Dumas START Random init
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204 Laurent Dumas • if ˜ J(x ng
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206 Laurent Dumas Table 7.3 Compari
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208 Laurent Dumas Fig. 7.9 General
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210 Laurent Dumas Table 7.4 Four ex
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212 Laurent Dumas Fig. 7.13 Blades
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214 Laurent Dumas Table 7.5 Converg
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Chapter 8 Multi-objective Optimizat
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8 Multi-objective Optimization in C
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8 Multi-objective Optimizatio Fig.
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