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76 H.G. Bock and V. Schulz computed in advance, and have a potential to speed up the final solution process. In the special case of NMPC, a sequence of neighboring optimization problems on a receding time horizon has to be solved, which differ only by (an estimate of) the initial value of the states y(t0) and possibly by a change in the set-point, i.e., parameters and scenarios. Minimal response delays can be achieved by the “real–time iteration” (RTI) in combination with a “perturbation embedding” as developed in [7, 13, 14]. Here, huge savings in computing time are gained by a kind of hot start property because already the first QP solution in a full step inexact SQP method provides a nearly tangential approximation of the exact solution if the problem is initialized with the solution of the previous problem, even in the presence of a change of the active constraints. Furthermore, the calculations can be divided into a preparation phase that can be performed without knowledge of y0, and a much shorter feedback phase that allows to make the delay even shorter than the sampling time. This remaining delay is typically orders of magnitude smaller than the sampling time and thus we can consider the feedback to be instantaneous. So the sampling time is practically only needed to prepare the following real– time iteration. Additionally, cheap feasibility and/or optimality improving subiterations (FOI) can be performed that reuse Jacobians and Hessians of the previous step and require only one forward and additionally one adjoint solution, respectively. For a more detailed description of the real-time iteration scheme and its convergence and nominal stability properties, please refer to [14]. 3.3.4 Sensitivity Driven Multiple Shooting In parameter estimation problems for CFD models, it is possible to use a multiple shooting time-domain decomposition and profit from its superior convergence properties, and on the other hand reduce the optimization space only to the parameters to be estimated. In these problems, one uses a generalized Gauss-Newton approach which achieves rather good convergence already with information of first order only. The linearized least-squares subproblem is condensed to the unknown parameters, which can be achieved while only marching forward in time with a conventional time-integration scheme. This idea was originally developed in [33], and has been successfully applied to multiphase problems, e.g., in [19]. Extensions to Newton-type methods for general objectives, e.g., in control problems were developed in [31, 32].
3 Mathematical Aspects of CFD-based Optimization 77 References 1. Arian, E., Fahl, M., Sachs, E.: Trust-region proper orthogonal decomposition for optimal flow control. NASA/CR-2000-210124 (2000) 2. Battermann, A., Sachs, E.: Block preconditioner for KKT systems in PDE-governed optimal control problems. International Series on Numerical Mathematics (ISNM) 138, 1–18 (2001) 3. Biros, G., Ghattas, O.: Parallel Lagrange-Newton-Krylov-Schur methods for PDEconstrained optimization. Part I: The Krylov-Schur solver. SIAM Journal on Scientific Computing 27(2), 687–713 (2005) 4. Biros, G., Ghattas, O.: Parallel Lagrange-Newton-Krylov-Schur methods for PDEconstrained optimization. Part II: The Lagrange-Newton solver, and its application to optimal control of steady viscous flows. SIAM Journal on Scientific Computing 27(2), 714–739 (2005) 5. Bock, H.: Numerical treatment of inverse problems in chemical reaction kinetics. In: Modelling of Chemical Reaction Systems, vol. 18, pp. 102–125. Springer (1981) 6. Bock, H., Diehl, M., Kostina, E.: SQP methods with inexact Jacobians for inequality constrained optimization. IWR-preprint, Universität Heidelberg, Heidelberg, Germany (2004) 7. Bock, H., Diehl, M., Kostina, E., Schlöder, J.: Constrained optimal feedback control for DAE.In:L.Biegler,O.Ghattas,M.Heinkenschloss,D.Keyes,B.vanBloemenWaanders (eds.) Real-Time PDE-Constrained Optimization, pp. 3–24. SIAM (2007) 8. Bock, H., Egartner, W., Kappis, W., Schulz, V.: Practical shape optimization for turbine and compressor blades. Optimization and Engineering 3, 395–414 (2002) 9. Bock, H., Kostina, E., Schäfer, A., Schlöder, J., Schulz, V.: Multiple set point partially reduced SQP method for optimal control of PDE. In: W. Jäger, R. Rannacher, J. Warnatz (eds.) Reactive Flows, Diffusion and Transport. Springer (2007) 10. Bock, H., Plitt, K.: A multiple shooting algorithm for direct solution of constrained optimal control problems. In: Proceedings 9th IFAC World Congress Automatic Control. Pergamon Press (1984) 11. Bock, H., Schlöder, J., Schulz, V.: Numerik großer Differentiell-Algebraischer Gleichungen – Simulation und Optimierung. In: H. Schuler (ed.) Prozeßsimulation, pp. 35–80. VCH Verlagsgesellschaft mbH, Weinheim (1994) 12. Borzi, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Review (to appear) (2007) 13. Diehl, M.: Real-time optimization for large scale nonlinear processes. Ph.D. thesis, University of Heidelberg, Germany (2001) 14. Diehl,M.,Bock,H.,Schlöder, J.: An iteration scheme for nonlinear optimization in optimal feedback control. SIAM Journal on Control and Optimization 43(5), 1714– 1736 (2005) 15. Gallitzendörfer, J., Bock, H.: Parallel algorithms for optimization boundary value problems in DAE. In: H. Langendörfer (ed.) Praxisorientierte Parallelverarbeitung. Hanser, München, Germany (1994) 16. Gherman, I.: Approximate partially reduced SQP approaches for aerodynamic shape optimization problems. Ph.D. thesis, University of Trier (2007) 17. Griewank, A.: Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation. Frontiers in Applied Mathematics. SIAM (2000) 18. Griewank, A., Walther, A.: Treeverse: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. Transactions on Mathematical Software (TOMS) 26 (2000) 19. Hazra, S., Schulz, V.: Numerical parameter identification in multiphase flow through porous media. Computing and Visualization in Science 5, 107–113 (2002) 20. Hazra, S., Schulz, V.: Simultaneous pseudo-timestepping for aerodynamic shape optimization problems with state constraints. SIAM J. Sci. Comput. 28(3), 1078–1099 (2006)
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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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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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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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