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284 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen optimization algorithm. This kind of approach has been used in the following kind of paper quality optimization examples where the statistical modeling techniques have been involved [23]. 9.4.3 Numerical Examples Next we present two numerical examples, where the virtual papermaking line was utilized. Both examples were solved using appropriate gradient-based optimizer and gradients were evaluated with the help of the technique described above. 9.4.3.1 Example 1 This example illustrates the advantages of multi-objective optimization compared with trial-and-error simulation. We studied here a conflict between two paper quality properties, formation and tensile strength ratio. Formation is a small scale weight variation of a paper sheet and tensile strength ratio is the machine-directional tensile strength divided by the cross-directional one. Bothofthesepropertiesweretobeminimized, but because they were in conflict they could not reach their optimum at the same time. These conflicting targets were simulated using the above presented virtual papermaking line combining dissimilar unit-process models from different disciplines. The virtual papermaking line used included the CFD models, and there were also models for moisture and heat transfer, and naturally statistical models for paper quality properties were involved. By doing trial-and-error simulations, a number of solutions were obtained as can be seen on the left-hand side of Table 9.1. These solution were obtained by using typical machine control values, which were determined using socalled engineering knowledge. That is, a person with expertise in papermaking provided the control variable values used in simulation. Based on his/her knowledge, she/he tried to find a solution which would fulfill the preferences. As can be seen in Table 9.1, the person made some simulations and obtained different kinds of solutions, but she/he could not be sure if any of these solutions was optimal. Trial-and-error simulations could have continued for as long as the expert had time or a satisfactory solution was obtained. Instead we used a multi-objective optimization method to search the optimal control variable values. In this way, much better solutions were found because the method obtained only Pareto optimal solutions. Figure 9.13 shows all the solutions obtained (simulated and optimized). A few Pareto optimal solutions are presented in Table 9.1 on the right-hand side. As one can see, all the solutions were better than the solutions obtained by simulation.
9 CFD-based optimization for papermaking 285 Table 9.1 Simulated vs. optimized solutions Simulated Optimized Formation Tensile strength Formation Tensile strength (g/m 2 ) ratio (g/m 2 ) ratio 0.388 2.731 0.449 2.000 0.439 2.496 0.300 2.233 0.497 2.414 0.366 2.002 0.563 2.414 0.318 2.037 0.563 2.630 0.540 2.532 0.518 2.234 0.586 2.235 0.479 2.233 0.442 2.233 Tensile strength ratio 2.8 2.6 2.4 2.2 2 1.8 0.3 0.4 0.5 0.6 Formation g/ m2 ( ) Simulated Optimized Fig. 9.13 Optimized and simulated conflicting paper quality properties Thus, there was no need for the time-consuming trial-and-error simulations. Also the number of solutions needed to be generated in optimization before finding the most satisfying compromise was small compared with the simulations. Since all the solutions found in optimization were mathematically equally good Pareto optimal solutions, the papermaking expert’s knowledge could be used in a case-specific way when selecting the most satisfying final solution.
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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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