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134 Nicolas R. Gauger Fig. 5.10 Plot of residual (log scale) of flow equation during coupled computation (multigrid is used): AMP wing, M∞ =0.78, α =2.83 ◦ , 2-block structured grid of about 140,000 nodes each Fig. 5.11 Plot of residual (log scale) of adjoint flow equation during coupled computation: AMP wing, M∞ =0.78, α =2.83 ◦ , 2-block structured grid of about 140,000 nodes each. Each 100 iterations, the boundary conditions of the adjoint flow solver are updated The fields (w,Z) are the solution of the system of partial differential equations R(X,w,Z) = 0 (5.36) S(X,w,Z) = 0 (5.37)
5 Efficient Deterministic Approaches for Aerodynamic Shape Optimization 135 Fig. 5.12 Wing structure model Fig. 5.13 Validation of the aero-structural coupled adjoint with finite differences (AMP wing, M∞ =0.78 and α =2.83 ◦ ) being (5.36) the flow and (5.37) the structural equations. We take the first variation of the PDEs. This yields δR = ∂R ∂X δS = ∂S ∂X δX + ∂R ∂w δX + ∂S ∂w ∂R δw + δZ =0, (5.38) ∂Z ∂S δw + δZ =0. (5.39) ∂Z We multiply Eqs. (5.38) and (5.39) with the Lagrange multipliers ψ and φ respectively and add the result to the expression for the differential increment of I in terms of the differentials of the independent set (X,w,Z), obtaining
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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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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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Chapter 9 CFD-based Optimization fo
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9 CFD-based optimization for paperm
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292 Index heat exchanger, 222, 224,
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