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282 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen the unit-process models and their outputs, we make an assumption that the objectives are continuously differentiable or H-differentiable [7]. One should note that in (9.7) the objectives fi,i=1,...,nf depend on the decision variables and also, the unit-process model outputs. That is, the virtual papermaking line model (9.6) has to be solved every time the objectives are evaluated. Next we discuss two features related to model-based optimization problems such as (9.7). These features are gradient evaluations and reliability of the used modeling approaches. Optimization methods utilizing gradient information of the objectives (gradient-based optimizers) have often been found computationally efficient in engineering applications. However, gradient information needs to be calculated when using these methods. The finite difference approaches are widely used techniques for gradient evaluations, but a large number of decision variables increases the number of the function evaluations making the approach computationally time-consuming. Instead, more sophisticated techniques can be used. We present briefly a technique based on chain rule approach and implicit differentiation schema. For that we assume that objectives and unitprocess models as well as model outputs are continuously differentiable or at least H-differentiable [7] in nonsmooth cases. Then gradient of fi with respect to the vector x is ∂fi(x,q 1,...,q nm)=∂xfi(x, q 1,...,q nm)+ nm� ∂q fi(x, q j 1,...,qnm)∂xq j(x,q 1,...,qj−1) j=1 (9.8) where differentials ∂xfi(x,q 1 ,...,q nm )and∂q j fi(x, q 1 ,...,q nm ), for all i = 1,...,nf and j =1,...,nm are assumed to be known and ∂xq j(x, q 1,...,q j−1)=−∂q j Aj(x, q 1,...,q j) −1� ∂xAj(x,q 1,...,q j)+ �j−1 � ∂q Aj(x,q k 1 ,...,qj )∂xqk (x,q1 ,...,qk−1 ) , j =1,...,nm. k=1 (9.9) In this way, gradient calculations can be done model-wise and gradients of objective functions can be put together after all the unit-processes have been solved. Thus, different modeling approaches can be easily combined together into the virtual papermaking line model. For more details on this technique, we refer to [21]. The virtual papermaking line (9.6) consists of unit-process models from different disciplines. When statistical or stochastic techniques are used, the unit-process models are based on experimental data and on conditions prevailing during data collection. That is, the models can give reliable results only if there has been enough varying modeling data. Furthermore, ranges of the modeling data should not be exceeded, which can cause problems
9 CFD-based optimization for papermaking 283 Expectation value y Confidence interval Minimum with large confidence interval Fig. 9.12 Example of output and its confidence interval Minimum with small confidence interval when unit-process models are coupled together. In order to use the statistical unit-process models in simulation or optimization, some kind of reliability or uncertainty information related to the models is needed in addition to the primary model output. Moreover, the uncertainties regarding the models need to be brought into the optimization model. In this way, the optimization method used can take them into account and reliability of the results can be guaranteed. There are numerous optimization approaches that are able to utilize uncertainty information such as optimization under uncertainty [6, 19, 25], stochastic programming [2] and robust optimization [8, 30] among others. Figure 9.12 illustrates on a simple example how model uncertainty may affect reliability of the optimization results. In this example for simplicity, we assume that the primary output is a function of only one input parameter. As seen, the output has two minima, which have very different confidence intervals denoted by dotted lines in the figure. The confidence interval gives limits in which the output value has a 99% probability, for example. The left minimum has a wider confidence interval than the right one, that is, the model gives more reliable solution at the right minimum. Hence, from the output value point of view, the solution candidates do not differ, but the latter minimum can be considered as more reliable. Therefore, systematic and efficient control of reliability is a crucial point in industrial optimizations. Nowadays, we can define the multi-objective optimization problem related to papermaking such that it is able to make use of related unit-process uncertainties [23]. Then, we formulate objectives also for uncertainties and minimize them while optimizing the papermaking targets. Thus, the idea is to formulate the originally stochastic optimization problem in such a way that it can be solved efficiently as a deterministic problem using a gradient-based x
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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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