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132 Nicolas R. Gauger (a) (b) Fig. 5.9 Comparison between FD and AD gradients on RAE 2822 with (a) Hicks-Henne and (b) cosine functions parameterization 5.6.3 Automatic Differentiation In Fig. 5.9, one can see the comparisons between the FD and AD gradients for the original RAE 2822 airfoil with Hicks-Henne and cosine functions parameterization. This validates the AD approach and shows also that the chosen stepsize for the FD gradient is accurate enough. As with the finite difference method the optimizations have also been done with the AD approach. The optimization histories, the pressure distri-
5 Efficient Deterministic Approaches for Aerodynamic Shape Optimization 133 butions and the surface geometries of the optimum with AD are also shown in Figs. 5.6, 5.7 and 5.8, respectively. This clearly validates the AD approach. 5.7 Adjoint Approch for Aero-Structure Coupling The use of successively performed single disciplinary optimizations in case of a multidisciplinary optimization problem is not only inefficient but in some cases has been shown to lead to faulty, non-optimal designs [24]. Although multidisciplinary optimization is possible with classical approaches for sensitivity evaluation by means of finite differences, this method is extremely expensive in terms of calculation time, requiring the reiterated solution of the coupled problem for every design variable. A new approach that allows the evaluation of the gradient with low computational cost takes advantage of the adjoint formulation of the multidisciplinary optimization problem [23, 24]. Therefore, the FLOWer adjoint option has been coupled with the structure solver MSC Nastran for an efficient coupled aero-structure adjoint solver. The implementation and validation of this approach are described in detail in [4, 5, 6]. 5.7.1 Adjoint Formulation for Aero-Structure Coupling The derivation of the adjoint equations in case of a multidisciplinary problem is similar to what has been carried out for the pure aerodynamic case, with the difference that we will end up with a dual adjoint variable for each set of state variables of the problem. An adjoint formulation is possible for any problem involving the calculation of the gradient of a function of one or more sets of variables obeying one or more constraint equations. We will restrict ourselves to the case of two sets: one that represents the flow variables, the other representing the structure nodal displacement. As already seen I(X,w,Z) denotes the cost function of the optimization problem, dependent now also on the displacement field Z, which is the solution of the structural problem. Then, the gradient takes the form dI dX or, in terms of differentials ∂I ∂I ∂w ∂I ∂Z = + + ∂X ∂w ∂X ∂Z ∂X (5.34) δI = ∂I ∂I ∂I δX + δw + δZ . (5.35) ∂X ∂w ∂Z
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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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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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