Applied shape optimization for fluids 2ed. Edition Bijan Mohammadi
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- 5. Applied shape optimization for fluids 2ed. Edition Bijan Mohammadi Digital Instant Download Author(s): Bijan Mohammadi, Olivier Pironneau ISBN(s): 9780199546909, 0199546908 Edition: 2ed. File Details: PDF, 6.45 MB Year: 2010 Language: english
- 7. NUMERICAL MATHEMATICS AND SCIENTIFIC COMPUTATION Series Editors A.M. STUART E. SÜLI
- 8. NUMERICAL MATHEMATICS AND SCIENTIFIC COMPUTATION Books in the series Monographs marked with an asterix (*) appeared in the series ‘Monographs in Numerical Analysis’ which has been folded into, and is continued by, the current series. For a full list of titles please visit http://www.oup.co.uk/academic/science/maths/series/nmsc ∗ J. H. Wilkinson: The algebraic eigenvalue problem ∗ I. Duff, A.Erisman, and J. Reid: Direct methods for sparse matrices ∗ M. J. Baines: Moving finite elements ∗ J.D. Pryce: Numerical solution of Sturm–Liouville problems Ch. Schwab: p- and hp- finite element methods: theory and applications to solid and fluid mechanics J.W. Jerome: Modelling and computation for applications in mathematics, science, and engineering Alfio Quarteroni and Alberto Valli: Domain decomposition methods for partial differential equations G.E. Karniadakis and S.J. Sherwin: Spectral/hp element methods for CFD I. Babuška and T. Strouboulis: The finite element method and its reliability B. Mohammadi and O. Pironneau: Applied shape optimization for fluids S. Succi: The Lattice Boltzmann Equation for fluid dynamics and beyond P. Monk: Finite element methods for Maxwell’s equations A. Bellen & M. Zennaro: Numerical methods for delay differential equations J. Modersitzki: Numerical methods for image registration M. Feistauer, J. Felcman, and I. Straškraba: Mathematical and computational methods for compressible flow W. Gautschi: Orthogonal polynomials: computation and approximation M.K. Ng: Iterative methods for Toeplitz systems Michael Metcalf, John Reid, and Malcolm Cohen: Fortran 95/2003 explained George Em Karniadakis and Spencer Sherwin: Spectral/hp element methods for CFD, second edition Dario A. Bini, Guy Latouche, and Beatrice Meini: Numerical methods for structured Markov chains Howard Elman, David Silvester, and Andy Wathen: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics Moody Chu and Gene Golub: Inverse eigenvalue problems: Theory and applications Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre: Mathematical methods for the magnetohydrodynamics of liquid metals Grégoire Allaire: Numerical analysis and optimization Eric Cancès, Claude Le Bris, Yvon Maday, and Gabriel Turinici: An introduction to mathematical modelling and numerical simulation Karsten Urban: Wavelet methods for elliptic partial differential equations B. Mohammadi and O. Pironneau: Applied shape optimization for fluids, second edition
- 9. Applied Shape Optimization for Fluids 2nd Edition Bijan Mohammadi University Montpellier II Olivier Pironneau University Paris VI 1
- 10. 3 Great Clarendon Street, Oxford ox2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Oxford University Press 2010 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2010 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British library catalogue in Publication Data Data available Library of Congress Cataloging-in-Publication Data Data available Typeset by Author using LaTex Printed in Great Britain on acid-free paper by MPG Books, Kings Lynn, Norfolk ISBN 978–0–19–954690–9 1 3 5 7 9 10 8 6 4 2
- 11. We dedicate this second edition to our late master Jacques-Louis Lions. Professor J.-L. Lions passed away in 2001; at the time this book was written he was also the chief scientific advisor to the CEO at Dassault-aviation; our gratitude goes to him for his renewed encouragement and support.
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- 13. PREFACE The first edition of this book was written in 2001 when computers in industry were hardly sufficient to optimize shapes for fluid problems. Since then com- puters have increased twenty fold in power; consequently methods which were not feasible have begun giving results, namely evolutionary algorithms, topolog- ical optimization methods and level set algorithms. While these were mentioned briefly in the first edition, here they now have separate chapters. Yet the book remains mostly a differential shape optimization book and our coverage of these three new methods is still minimal, each requiring in fact a separate book. To our credit, it should also be said that genetic algorithms are not yet capable of solving problems like wing optimization when the number of parameters is bigger than a few dozen without intensive distributed resources; similarly topo- logical optimization is great for structure optimization but only an interesting alternative for fluid flows in most cases. Level sets, on the other hand, are more general but simply another parameterization method; the optimization is done with a gradient or Newton algorithm, so it is within the scope of the book.
- 14. ACKNOWLEDGEMENTS The authors are grateful to F. Alauzet, R. Arina, P. Aubert, A. Baron, R. Bra- hadwaj, L. Debiane, N. Dicesaré, F. Hecht, S. Galera, B. Ivorra, D. Isèbe, G. Medic, N. Petruzzelli, G. Puigt, J. Santiago, M. Stanciu, E. Polak and J. Tuomela for their contributions in the form of scientific material published elsewhere in collaboration with us. For their encouragement and sharing of ideas the authors would like to thank A. Dervieux, H.-G. Bock, C. Farhat, M. Giles, R. Glowinski, M. Gunzburger, W. Habashi, M. Hafez, A. Henrot, D. Hertzog, H. Kawarada, P. Le Tallec, P. Moin, M. Navon, P. Neittanmaaki, J. Periaux, B. Perthame, P. Sagaut, S. Obayashi, M. Wang. We thank also our colleagues at the universities of Montpellier II and Paris VI and at INRIA, for their comments on different points related to this work, namely: H. Attouch, P. Azerad, F. Bouchette, M. O. Bristeau, J. F. Bourgat, M. Cuer, A. Desideri, P. Frey, A. Hassim, P.L. George, B. Koobus, S. Lanteri, P. Laug, E. Laporte, F. Marche, A. Marrocco, F. Nicoud, P. Redont, E. Saltel, M. Vidrascu. We are also very happy to acknowledge the contributions of our industrial part- ners: MM. Duffa, Pirotais, Galais, Canton-Desmeuzes, at CEA-CESTA; MM. Stoufflet, Mallet, Rostand, Rogé, Dinh at Dassault Aviation, MM. Chaput, Cormery and Meaux at Airbus, MM. Chabard, Laurence and Viollet at EDF. MM. Aupoix, Cousteix at Onera. S. Moreau at Valeo. MM. Poinsot and André at Cerfacs. Finally, considerable help was given to us by the automatic differentiation special- ists and especially by C. Bishof, C. Faure, P. Hovland, N. Rostaing, A. Griewank, J.C. Gilbert and L. Hascoet. As this list is certainly incomplete, many thanks and our apologies to colleagues whose name is missing.
- 15. CONTENTS 1 Introduction 1 2 Optimal shape design 6 2.1 Introduction 6 2.2 Examples 7 2.2.1 Minimum weight of structures 7 2.2.2 Wing drag optimization 8 2.2.3 Synthetic jets and riblets 11 2.2.4 Stealth wings 12 2.2.5 Optimal breakwater 15 2.2.6 Two academic test cases: nozzle optimization 16 2.3 Existence of solutions 17 2.3.1 Topological optimization 17 2.3.2 Sufficient conditions for existence 18 2.4 Solution by optimization methods 19 2.4.1 Gradient methods 19 2.4.2 Newton methods 20 2.4.3 Constraints 21 2.4.4 A constrained optimization algorithm 22 2.5 Sensitivity analysis 22 2.5.1 Sensitivity analysis for the nozzle problem 25 2.5.2 Numerical tests with freefem++ 27 2.6 Discretization with triangular elements 28 2.6.1 Sensitivity of the discrete problem 30 2.7 Implementation and numerical issues 33 2.7.1 Independence from the cost function 33 2.7.2 Addition of geometrical constraints 34 2.7.3 Automatic differentiation 34 2.8 Optimal design for Navier-Stokes flows 35 2.8.1 Optimal shape design for Stokes flows 35 2.8.2 Optimal shape design for Navier-Stokes flows 36 References 37 3 Partial differential equations for fluids 41 3.1 Introduction 41 3.2 The Navier-Stokes equations 41 3.2.1 Conservation of mass 41 3.2.2 Conservation of momentum 41 3.2.3 Conservation of energy and and the law of state 42 3.3 Inviscid flows 43
- 16. x Contents 3.4 Incompressible flows 44 3.5 Potential flows 44 3.6 Turbulence modeling 46 3.6.1 The Reynolds number 46 3.6.2 Reynolds equations 46 3.6.3 The k − ε model 47 3.7 Equations for compressible flows in conservation form 48 3.7.1 Boundary and initial conditions 50 3.8 Wall laws 51 3.8.1 Generalized wall functions for u 51 3.8.2 Wall function for the temperature 53 3.8.3 k and ε 54 3.9 Generalization of wall functions 54 3.9.1 Pressure correction 54 3.9.2 Corrections on adiabatic walls for compressible flows 55 3.9.3 Prescribing ρw 56 3.9.4 Correction for the Reichardt law 57 3.10 Wall functions for isothermal walls 58 References 60 4 Some numerical methods for fluids 61 4.1 Introduction 61 4.2 Numerical methods for compressible flows 61 4.2.1 Flux schemes and upwinded schemes 61 4.2.2 A FEM-FVM discretization 62 4.2.3 Approximation of the convection fluxes 63 4.2.4 Accuracy improvement 64 4.2.5 Positivity 64 4.2.6 Time integration 65 4.2.7 Local time stepping procedure 66 4.2.8 Implementation of the boundary conditions 66 4.2.9 Solid walls: transpiration boundary condition 67 4.2.10 Solid walls: implementation of wall laws 67 4.3 Incompressible flows 68 4.3.1 Solution by a projection scheme 69 4.3.2 Spatial discretization 70 4.3.3 Local time stepping 71 4.3.4 Numerical approximations for the k − ε equations 71 4.4 Mesh adaptation 72 4.4.1 Delaunay mesh generator 72 4.4.2 Metric definition 73 4.4.3 Mesh adaptation for unsteady flows 75 4.5 An example of adaptive unsteady flow calculation 77 References 78
- 17. Contents xi 5 Sensitivity evaluation and automatic differentiation 81 5.1 Introduction 81 5.2 Computations of derivatives 83 5.2.1 Finite differences 83 5.2.2 Complex variables method 83 5.2.3 State equation linearization 84 5.2.4 Adjoint method 84 5.2.5 Adjoint method and Lagrange multipliers 85 5.2.6 Automatic differentiation 86 5.2.7 A class library for the direct mode 88 5.3 Nonlinear PDE and AD 92 5.4 A simple inverse problem 94 5.5 Sensitivity in the presence of shocks 101 5.6 A shock problem solved by AD 103 5.7 Adjoint variable and mesh adaptation 104 5.8 Tapenade 106 5.9 Direct and reverse modes of AD 106 5.10 More on FAD classes 109 References 113 6 Parameterization and implementation issues 116 6.1 Introduction 116 6.2 Shape parameterization and deformation 116 6.2.1 Deformation parameterization 117 6.2.2 CAD-based 117 6.2.3 Based on a set of reference shapes 117 6.2.4 CAD-free 118 6.2.5 Level set 122 6.3 Handling domain deformations 127 6.3.1 Explicit deformation 128 6.3.2 Adding an elliptic system 129 6.3.3 Transpiration boundary condition 129 6.3.4 Geometrical constraints 131 6.4 Mesh adaption 133 6.5 Fluide-structure coupling 136 References 138 7 Local and global optimization 140 7.1 Introduction 140 7.2 Dynamical systems 140 7.2.1 Examples of local search algorithms 140 7.3 Global optimization 142 7.3.1 Recursive minimization algorithm 143 7.3.2 Coupling dynamical systems and distributed comput- ing 144
- 18. xii Contents 7.4 Multi-objective optimization 145 7.4.1 Data mining for multi-objective optimization 148 7.5 Link with genetic algorithms 150 7.6 Reduced-order modeling and learning 153 7.6.1 Data interpolation 154 7.7 Optimal transport and shape optimization 158 References 161 8 Incomplete sensitivities 164 8.1 Introduction 164 8.2 Efficiency with AD 165 8.2.1 Limitations when using AD 165 8.2.2 Storage strategies 166 8.2.3 Key points when using AD 167 8.3 Incomplete sensitivity 168 8.3.1 Equivalent boundary condition 168 8.3.2 Examples with linear state equations 169 8.3.3 Geometric pressure estimation 171 8.3.4 Wall functions 172 8.3.5 Multi-level construction 172 8.3.6 Reduced order models and incomplete sensitivities 173 8.3.7 Redefinition of cost functions 174 8.3.8 Multi-criteria problems 175 8.3.9 Incomplete sensitivities and the Hessian 175 8.4 Time-dependent flows 176 8.4.1 Model problem 178 8.4.2 Data mining and adjoint calculation 181 References 183 9 Consistent approximations and approximate gradients 184 9.1 Introduction 184 9.2 Generalities 184 9.3 Consistent approximations 186 9.3.1 Consistent approximation 187 9.3.2 Algorithm: conceptual 187 9.4 Application to a control problem 188 9.4.1 Algorithm: control with mesh refinement 189 9.4.2 Verification of the hypothesis 189 9.4.3 Numerical example 190 9.5 Application to optimal shape design 190 9.5.1 Problem statement 191 9.5.2 Discretization 192 9.5.3 Optimality conditions: the continuous case 192 9.5.4 Optimality conditions: the discrete case 193 9.5.5 Definition of θh 194
- 19. Contents xiii 9.5.6 Implementation trick 195 9.5.7 Algorithm: OSD with mesh refinement 195 9.5.8 Orientation 196 9.5.9 Numerical example 196 9.5.10 A nozzle optimization 197 9.5.11 Theorem 199 9.5.12 Numerical results 200 9.5.13 Drag reduction for an airfoil with mesh adaptation 200 9.6 Approximate gradients 203 9.6.1 A control problem with domain decomposition 204 9.6.2 Algorithm 205 9.6.3 Numerical results 207 9.7 Conclusion 209 9.8 Hypotheses in Theorem 9.3.2.1 209 9.8.1 Inclusion 209 9.8.2 Continuity 209 9.8.3 Consistency 209 9.8.4 Continuity of θ 209 9.8.5 Continuity of θh(αh) 210 9.8.6 Convergence 210 References 210 10 Numerical results on shape optimization 212 10.1 Introduction 212 10.2 External flows around airfoils 213 10.3 Four-element airfoil optimization 213 10.4 Sonic boom reduction 215 10.5 Turbomachines 217 10.5.1 Axial blades 219 10.5.2 Radial blades 222 10.6 Business jet: impact of state evaluations 225 References 225 11 Control of unsteady flows 227 11.1 Introduction 227 11.2 A model problem for passive noise reduction 228 11.3 Control of aerodynamic instabilities around rigid bodies 229 11.4 Control in multi-disciplinary context 229 11.4.1 A model problem 230 11.4.2 Coupling strategies 236 11.4.3 Low-complexity structure models 237 11.5 Stability, robustness, and unsteadiness 241 11.6 Control of aeroelastic instabilities 244 References 245
- 20. xiv Contents 12 From airplane design to microfluidics 246 12.1 Introduction 246 12.2 Governing equations for microfluids 247 12.3 Stacking 247 12.4 Control of the extraction of infinitesimal quantities 249 12.5 Design of microfluidic channels 249 12.5.1 Reduced models for the flow 255 12.6 Microfluidic mixing device for protein folding 255 12.7 Flow equations for microfluids 259 12.7.1 Coupling algorithm 260 References 261 13 Topological optimization for fluids 263 13.1 Introduction 263 13.2 Dirichlet conditions on a shrinking hole 264 13.2.1 An example in dimension 2 264 13.3 Solution by penalty 265 13.3.1 A semi-analytical example 267 13.4 Topological derivatives for fluids 268 13.4.1 Application 268 13.5 Perspective 270 References 270 14 Conclusions and prospectives 272 Index 275
- 21. 1 INTRODUCTION Nowadays the art of computer simulation has reached some maturity; and even for still unsolved problems engineers have learned to extract meaningful answers and trends for their design from rough simulations: numerical simulation is one of the tools on which intuition can rely! Yet for those who want to study trends and sensitivities more rationally the tools of automatic differentiation and op- timization are there. This book deals with them and their application to the design of the systems of fluid mechanics. But brute force optimization is too often an inefficient approach and so our goal is not only to recall some of the tools but also to show how they can be used with some subtlety in an optimal design program. Optimal shape design (OSD) is now a necessity in several industries. In air- plane design, because even a few percent of drag reduction means a lot, aerody- namic optimization of 3D wings and even wing body configurations is routinely done in the aeronautics industry. Applications to the car industry are well un- derway especially for the optimization of structures to reduce weight but also to improve vehicle aerodynamics. Optimization of pipes, heart valves, and even MEMS and fluidic devices, is also done. In electromagnetism stealth objects and antenna are optimized subject to aerodynamic constraints. However, OSD is still a difficult and computer-intensive task. Several chal- lenges remain. One is multi-objective design. In aeronautics, high lift configu- rations are also challenging because the flow needs to be accurately solved and turbulence modelling using DES or LES is still too heavy to be included in the design loop, but also because shape optimization for unsteady flows is still immature. From a mathematical point of view, OSD is also difficult because even if the problem is well posed success is not guaranteed. One should pay attention to the computing complexity and use sub-optimal approaches whenever possible. As we have said, demand is on multi-disciplinary and multi-criteria design and local minima are often present; a good treatment of state constraints is also a numer- ical challenge. Global optimization approaches based on a mix of deterministic and nondeterministic methods, together with surface response model reduction, is necessary to break complexity. Care should also be taken when noise is present in the data and always consider robustness issues. From a theoretical point of view, OSD problems can be studied as infinite dimensional controls with state variables in partial differential equations and constraints. The existence of a solution is guaranteed under mild hypothesis in 2D and under the flat cone property in 3D. Tikhonov regularization is easily
- 22. 2 Introduction done with penalization of the surface of the shape. In variational form results translate without modifications to the discrete cases if discretized by the finite element or finite volume methods. Gradient methods are efficient and convergent even though it is always preferable to use second order methods when possible. Geometric constraints can be handled at no cost but more complex constraints involving the state variables are a real challenge. Multicriteria optimization and Pareto optimality have not been solved in a satisfactory way by differentiable optimization, either because the problems are too stiff and/or there are too many local minima. Evolutionary algorithms offer an expensive alternative. The black box aspect of this solution is a real asset in the industrial context. The consensus seems to go to a mix of stochastic and deterministic approaches using reduced order or surrogate models when possible. Topological optimization is a very powerful tool for optimizing the coefficients of PDEs. It is ideal for structure optimization where the answer can be a composite material or for low Reynolds flows. However, it does not look to be a promising technique for high Reynolds number flow. Different choices can be made for the shape parameter space following the variety of the shapes one would like to reach. If the topology of the target shape is already known and if the available CAD parameter space is thought to be suitable, it should be considered as a control parameter during optimization. On the other hand, one might use a different parameter space, larger or smaller, during optimization having in mind that the final shape should be expressed in a usable CAD format. For some applications it is important to allow for variable topology; then shape parameters can be, for instance, a relaxed characteristic function (level set and immersed boundary approaches belong to this class). The different parameter spaces should be seen as complementary for primary and final stages of optimization. Indeed, the main advantage of a level set over a CAD-based parameter space is in primary optimization where the topology of the target shape is unknown and any a priori choice is hazardous. An important issue in minimization is sensitivity evaluation. Gradients are useful in multi-criteria optimization to discriminate between Pareto equilibrium even when using gradient-free minimization algorithms. Sensitivities also permit us to introduce robustness issues during optimization. Robustness is also central in validation and verification as no simulation or design can be reliable if it is too sensitive to small perturbations in the data. For sensitivity evaluation when the parameter space is large the most effi- cient approach is to use an adjoint variable with the difficulty that it requires the development of specific software. Automatic differentiation brings some sim- plification, but does not avoid the main difficulty of intermediate state storage, even though check-pointing techniques bring some relief. The use of commercial software without the source code is also a limitation for automatic differentia- tion and differentiable optimization in general. Incomplete sensitivity formula- tion and reduced order modelling are therefore preferred when possible to reduce this computational complexity and also because these often bring some extra un-
- 23. Introduction 3 derstanding of the physics of the problem. These techniques are also useful for minimization with surrogate models as mentioned above. Another important issue is that the results may depend on the evolution of the modelling. It is important to be able to provide information in an incremen- tal way, following the evolution of the software. This means that we need the sensitivity of the result with respect to the different independent variables for the discrete form of the model and also that we need to be able to do that without re-deriving the model from scratch. But again use of commercial software puts serious limitations on what can be efficiently done and increases the need for adaptive reduced order modelling. Hence, any design should be seen as constrained optimization. Adding ro- bustness issues implies most likely a change in the solution procedures. From a practical point of view, it is clear that adding such requirements as those men- tioned above will have a prohibitive cost, especially for multi-criteria and multi- physics designs. But answers are needed and even an incomplete study, even a posteriori, will require at least the sensitivity of the solution to perturbation of independent variables. As implied by the title, this book deals with shape optimization problems for fluids with an emphasis on industrial application; it deals with the basic shape optimization problems for the aerodynamics of airplanes and some fluid-structure design problems. All these are of great practical importance in computational fluid dynamics (CFD), not only for airplanes but also for cars, turbines, and many other industrial applications. A new domain of application is also covered: shape optimization for microfluidic devices. Let us also warn the reader that the book is not a synthesis but rather a col- lection of case studies; it builds on known material but it does not present this material in a synthetic form, for several reasons, like clarity, page numbers, and time. Furthermore a survey would be a formidable task, so huge is the literature. So the book begins with a chapter on optimal shape design by local shape vari- ations for simple linear problems, discretized by the finite element method. The goal is to provide tools to do the same with the complex partial differential equations of CFD. A general presentation of optimal shape design problems and of their solution by gradient algorithms is given. In particular, the existence of solutions, sensitivity analysis at the continuous and discrete levels are discussed, and the implementation problems for each case are pointed out. This chapter is therefore an introduction to the rest of the book. It summarizes the current knowhow for OSD, except topological optimization, as well as global optimiza- tion methods such as evolutionary algorithms. In Chapter 3 the equations of fluid dynamics are recalled, together with the k − ε turbulence model, which is used later on for high Reynolds number flows when the topology of the answer is not known. The fundamental equations of fluid dynamics are recalled; this is because applied OSD for fluids requires a good understanding of the state equation: Euler and Navier-Stokes equations in
- 24. 4 Introduction our case, with and without turbulence models together with the inviscid and/or incompressible limits. We recall wall functions also later used for OSD as low complexity models. By wall function we understand domain decomposition with a reduced dimension model near the wall. In other words, there is no universal wall function and when using a wall function, it needs to be compatible with the model used far from the wall. Large eddy simulation is giving a new life to the wall functions especially to simulate high-Reynolds external flows. Chapter 4 deals with the numerical methods that will be used for the flow solvers. As in most commercial and industrial packages, unstructured meshes with automatic mesh generation and adaptation are used together with finite volume or finite element discretization for these complex geometries. The itera- tive solvers and the flux functions for upwinding are also presented here. Then in Chapter 5 sensitivity analysis and automatic differentiation (AD) are presented: first the theory, then an object oriented library for automatic dif- ferentiation (AD) by operator overloading, and finally our experience with AD systems using code generation operating in both direct and reverse modes. We describe the different possibilities and through simple programs give a compre- hensive survey of direct AD by operator overloading and for the reverse mode, the adjoint code method. Chapter 6 presents parameterization and geometrical issues. This is also one of the key points for an efficient OSD platform. We describe different strategies for shape deformation within and without (level set and CAD-Free) computer aided design data structures during optimization. Mesh deformation and remesh- ing are discussed there. We discuss the pros and the cons of injection/suction boundary conditions equivalent to moving geometries when the motion is small. Some strategies to couple mesh adaptation and the shape optimization loop are presented. The aim is to obtain a multi-grid effect and improve convergence. Chapter 7 gives a survey of optimization algorithms seen as discrete forms of dynamical systems. The presentation is not intended to be exhaustive but rather reflects our practical experience. Local and global optimizations are discussed. A unified formulation is proposed for both deterministic and stochastic optimiza- tions. This formulation is suitable for multi-physic problems where a coupling between different models is necessary. One important topic discussed in Chapter 8 is incomplete sensitivity. By incomplete sensitivity we mean that during sensitivity evaluation only the de- formation of the geometry is accounted for and the change of the state variable due to the change of geometry is ignored. We give the class of functionals for which this assumption can be made. Incomplete sensitivity calculations are il- lustrated on several model problems. This gives the opportunity of introducing low-complexity models for sensitivity analysis. We show by experience that the accuracy is sufficient for quasi-Newton algorithms and also that the complexity of the method is drastically reduced making possible real time sensitivity analysis later used for unsteady applications.
- 25. Introduction 5 In Chapter 9 we put forward a general argument to support the use of ap- proximate gradients within optimization loops integrated with mesh refinements, although this does not justify all the procedures that are presented in Chapter 8. We also prove that smoothers are essential. This part was done in collaboration with E. Polak and N. Dicesare. Then follows the presentation of some applications for stationary flows in Chapter 10 and unsteady problems in chapter 11. We gather in Chapter 10 examples of shape optimization in two and three space dimensions using the tools presented above for both inviscid and viscous turbulent cases. Chapter 11 presents applications of our shape optimization algorithms to cases where the flow is unsteady for rigid and elastic bodies and shows that control and OSD problems are particular cases of a general approach. Closed loop control algorithms are presented together with an analogy with dynamical systems. Chapter 12 gathers the application of the ingredients above to the design of microfluidic devices. This is a new growing field. Most of what was made for aeronautics can be applied to these fluids at nearly zero Reynolds and Mach numbers. The book closes with Chapter 13 on topological shape optimization described in simple situations. The selection of this material corresponds to what we think to be a good compromise between complexity and accuracy for the numerical simulation of nonlinear industrial problems, keeping in mind practical aspects at each level of the development, illustrating our proposal, with many practical examples which we have gathered during several industrial cooperations. In particular, the con- cepts are explained more intuitively than with complete mathematical rigor. Thus this book should be important for whoever wishes to solve a practical OSD problem. In addition to the classical mathematical approach, the application of some modern techniques such as automatic differentiation and unstructured mesh adaptation to OSD are presented, and multi-model configurations and some time-dependent shape optimization problems are discussed. The book has been influenced by the reactions of students who have been taught this material at the Masters level at the Universities of Paris and Montpel- lier. We think that what follows will be particularly useful for engineers interested in the implementation and solution of optimization problems using commercial packages, or in-house solvers, graduate and PhD students in applied mathe- matics, aerospace, or mechanical engineering needing, during their training and research, to understand and solve design problems, and research scientists in ap- plied mathematics, fluid dynamics, and CFD looking for new exciting research and development areas involving realistic applications.
- 26. Discovering Diverse Content Through Random Scribd Documents
- 27. But more than ever I want you now; more than ever I need you. Erie, he said earnestly, are you willing to marry me right away—next week? Oh Frank— she began, but he checked her utterance with his lips. The Reverend Reddick is available at any day, any hour, Lighthouse girl; he's conducting revival services in the Valley church. It will all be so simple. Won't you say next week? She gazed into his radiant face with serious eyes. But Frank, she whispered, it may be cold and dismal next week, I—I always thought that I should like our wedding to be—- Her head went down to hide against his arm. Go on, Lighthouse girl. You always thought you would like our wedding to be—when? On a golden, Indian summer day like this, she finished and closed her eyes as his arms went about her. * * * * * And ut's married they were this mornin', whilst the dew still clung to the mosses, and ut's meself was witness to the j'inin' av two av the tinderest hearts in all the wurruld. Old Harry O'Dule, on his rounds to spread the joyful tidings of Frank and Erie's marriage, had met Billy leading a fat bay horse along a sun-streaked forest path. Billy stared at the old man; then his face broke into a grin. O Gee! he sighed, and sinking on a log, closed his eyes. O Gee! he repeated— leaping to his feet and throwing his arms about the neck of the bay and yelling into that animal's twitching ear. Hear that, you Thomas? They're married, Erie an' Teacher Stanhope's married! Billy, is ut clane crazy ye've gone? chided the old man, that ye'd be afther deafenin' the poor steed wid yer yellin'? Listen now, fer ut's more I'll be tellin' ye.
- 28. Billy kicked his hat high in air and turned a handspring. Tell me all about it, Harry. You saw 'em married, did you? Faith and I did, cried Harry. And play 'em a weddin' march on me whistle I did, soft as a spring rain and swate as the very joy they do be feelin' this day. A king he looked, Billy, and his bride a quane, ivery inch av her. But no more av your questions now, he broke off, fer step along I must, singin' me thankfulness from me whistle, and spakin' the good tidings to them I mate along the way. Billy watched the old man move down the path, the wild strains of the Irish tune he was playing falling on his ears long after the player had been swallowed up in the golden haze. Then he too passed on, bay Thomas walking sedately behind. As he rounded a bend he met Maurice Keeler and Jim Scroggie, heads close together and speaking animatedly. Ho, Bill! cried Maurice. Bringin' bay Thomas up to the stable fer winter, eh? Gee! Jim, look at that horse; did you ever see such a change in anythin' in your life? Thomas has sure fattened up, grinned Jim. I guess it would puzzle old Johnston to know our horse now, eh, Bill? You mean your horse, Jim, corrected Billy. No, I don't either; he's only a third mine. One third's yours and the other third's Maurice's. Maurice and Billy stared at him. It was your money paid fer him, Billy asserted. Well, what of it? Maurice found him a soft hidin' place and good pasture on his Dad's farm, didn't he? Sure, but then— And it's you who's gain' to see that he gets cared for all winter, ain't it? You bet it is, cried Billy.
- 29. Well then, I claim he's a company horse an' you an' me an' Maurice is that company. Now, that's settled, let me tell you what Maurice and me was talkin' about when you met us. Billy unsnapped the tie-strap from Thomas' halter so that he might crop the wayside grass without hindrance and sat down on a log opposite the one occupied by his friends. Jim nudged Maurice but Maurice shook his head. You tell him, he said. Bill, Jim cried eagerly. I got a bit of news for you that'll make you want to stand on your head and kick splinters off the trees. Billy grinned. An' I got a piece of news fer you fellers, too, he returned. But go on, your news first, Jim. Teacher Stanhope has made over a deed of Lost Man's Swamp to you, Bill, said Jim. I heard Dad telling Mr. Hinter all about it. Dad was there when Lawyer Maddoc drew up the deed—Maurice, you crazy hyena, will you keep quiet? Maurice had rolled backward off the log, the while he emitted cries that would have done a scalp-hunting Indian credit. Three cheers fer Bill! he yelled. He discovered Lost Man's Swamp oil field. Trigger Finger Tim ain't got nuthin' on our Bill. Billy was standing up now, his perplexed face turned questioningly on his chums. That's right, Bill, cried Jim. You really did discover it, you know. Hinter said he was the only one who knew the oil was there until you rafted out to the ponds and saw the oil-bubbles breakin' on 'em. He says that a fortune likely lies there, so you see— An' Teacher Stanhope, he deeded the swamp to me, said Billy dazedly. He got up from the log and squared his shoulders. Well, he spoke, that was mighty good of him, but I ain't wantin' that swamp.
- 30. But Bill, urged Jim, the oil they've found there'll make you rich. Billy shook his head. I'm as rich as I ever want'a be right now, Jim. Look here, Bill, cried Maurice. You don't want'a hurt Teacher Stanhope's feelin's, do you! Billy glanced at him quickly, a troubled look in his eyes. N-no, he said, you bet I don't. Then that's all there is to it; you keep Lost Man, that's what you do. Billy considered. I ain't sayin' jest what I'll do, he spoke finally. I gotta ask another person's advice on this thing. But if I do take it you, Jim, an' you, Maurice, are goin' to be my partners in Lost Man same's you are in bay Thomas. Here, Maurice, you take Thomas to our stable an' give him a feed. I gotta go somewhere else. And leaving Jim and Maurice sitting, open-mouthed, Billy ducked into the timber. Not until he had put some distance between himself and hia friends did he remember that he had not told them the great and wonderful news that had been imparted to him by old Harry. Well, never mind, they would hear it soon. Harry would see to that. He turned into a path that strayed far up among clumps of red-gold maples and ochre-stained oaks. The whistle of quail sounded from a ridge of brown sumachs. Up the hill, across the deep valley, where wintergreen berries gleamed like drops of blood among the mosses, he passed slowly and on to the beech-crowned ridge. Here he paused and his searching eyes sought the lower sweep of woodland. A clump of tall poplars gleamed silvery-white against the dark green of the beeches; far down at the end of the sweep the yellow tops of hardy willows stood silhouetted against the undying green of massed cedars and pines. Billy gazed down upon it all and his heart swelled with the deep joy of life, his nerves tingled to the tang of the woodland scents. Something deep, stirring, mysterious, had come to him. He did not know what that something was—it was too vague and incomprehensible for definition just yet.
- 31. His arm about the trunk of a tree, he laughed softly, as his eyes, sweeping the checker-board of autumn's glories, rested at last on the grove of coniferous trees. So that was the haunted grove? That dark, silent, spicy bit of isolated loneliness far below was the spot he had so feared! But he feared it no longer. She had cured him of that. She had said that fear of the supernatural was foolish; and of course she was right. A fat red-squirrel frisked down a tree close beside hia and halted, pop- eyed, to gaze upon him. I tell you, Billy addressed it gravely, it takes a good woman to steady a man. The statement was not of his own creation. He had heard it somewhere but he had never understood its meaning before. It seemed the fitting thing to say now and there was nobody to say it to except the squirrel. A blue-jay and a yellow-hammer flashed by him, side by side, racing for the grubbing-fields of the soft woods below, their blue and yellow bodies marking twin streaks against the hazy light. Blue and yellow, truly the most wonderful colors of all the colorful world, thought Billy. The scene faded and in its place grew up a face with blue, laughing eyes and red, smiling lips, above which gleamed a halo of spun gold. Then the woodland picture swam back before him and the squirrel, which with the characteristic patience of its kind had waited to watch this boy who often threw it a nut- kernel, called after him chidingly as he dipped down into the valley. Billy was still thinking of the only girl when he topped the farther ridge and descended into the valley where stood the haunted grove. He wondered what she would say when he told her the great news he had to tell her. He thought he knew. She would put her hand on his arm and say: Billy, I'm glad. Well, he was on his way to hear her say it. As he entered a clump of cedars he saw her. She wore a cloak of crimson; her hat had slipped to her shoulders and her hair glowed softly through the shadowy half lights. She stood beside old man Scroggie's grave, a great bunch of golden-rod in her arms. Billy called and she turned to him with a smile. Oh, I'm so glad you came, Billy, she said. You can help me decorate uncle's grave.
- 32. She dropped the yellow blossoms on the mound and they went out into the sunshine together and gathered more. When they had finished the task they went across to the weedy plot in which stood the tumble-down hut. There, seated side by side beneath a gnarled wild-apple tree, Billy told her all he had to tell her, and heard her say, just as he knew she would say, Billy, I'm glad. Then between them fell silence, filled with understanding and contentment and thoughts that ran parallel the same long track through future promise. Billy spoke, at length: He's goin' to take the school ag'in. An' him an' me are goin' to build that sail-boat we've always wanted—a big broad-beamed, single sticker that'll carry all of us—you, me, teacher, Erie an' anybody wants to come along. Gee! ain't it great? The girl nodded. And what will you name her? she asked. Into Billy's cheeks the blood sprang as into his heart joy ran riot. I aim to call her Lou, he said hesitatingly. That is if you don't mind. The golden head was bowed and when it was raised to him, he saw a deeper color in the cheeks, a softer glow in the eyes. Come, she said softly, we must be getting back. They crossed the sunflecked grass, hand in hand. As they reached the pine grove the girl pointed away above the trees. Look, she whispered. Billy's gaze followed hers. High above the trees a black speck came speeding toward them, a speck which grew quickly into a bird, a big, black bird, who knew, apparently, just where he was going. It's Croaker, Billy whispered. Stand right still, Lou, an' we'll watch an' find out what his game is. He drew her a little further among the pines and they peered out to see Croaker alight on the broken-backed ridge pole of the log hut. Here, with many low croaks, he proceeded to search his surroundings with quick, suspicious eyes, straining forward to peer closely at scrub or
- 33. bush, then cunningly twisting about suddenly as though hoping to take some skulking watcher behind him unawares. Finally he seemed satisfied that he was alone. His harsh notes became soft guttural cooes. He nodded his big head up and down in grave satisfaction, tip-toeing from one end of the ridge-pole to the other and chuckling softly to himself. Then suddenly, he vanished from sight. Where has he gone? whispered Lou. Hush, warned Billy. His heart was pounding. The watchers stood with eyes glued to the ridge-pole. By and by they saw a black tail-feather obtrude itself from a hole just beneath the roof's gable. A black body followed and Croaker came tiptoeing back along the ridge. The girl felt her companion's hand tighten spasmodically on hers. She glanced up to find him staring, wide-eyed at the bird. Billy! she whispered, almost forgetting caution in her anxiety. What is it? He pointed a shaking finger at Croaker. See that shiny thing that old rogue has in his bill, Lou! he asked. What do you 'spose that is? Why, what is it? It's one of the gold pieces your uncle hid away. Come on, now we'll see that Croaker throw a fit. They stepped out into plain view of the crow, who was muttering to the gold-piece which he now held before his eyes in one black claw. Croaker lowered his head and twisted it from side to side in sheer wonder. He could scarcely believe his eyes. Then as Billy stepped forward and called him by name his black neck-ruff arose in anger and, dropping his prized bit of gold, he poured out such a torrent of abuse upon the boy and girl that Lou put her fingers in her ears to stop the sound.
- 34. He's awful mad, grinned Billy. He's been keepin' this find to himself fer a long time. At sound of his master's voice Croaker paused in his harangue and promptly changed his tactics. He swooped down to Billy's shoulder and rubbed the top of his glossy head against the boy's cheek, whispering low and lying terms of endearment. Lou laughed, What's he up to now, Billy? He's tryin' to coax me away from his treasure, Billy answered. Now, jest watch him. What you want'a do, Croaker? he asked, stroking the bird's neck feathers smooth. Kawak! said Croaker, and jumping to the ground he started away, head twisted backward toward the boy and girl, coaxing sounds pouring from his half open beak. No, sir, cried Billy. You don't fool me ag'in. I'm goin' to climb up there an' see jest how much gold is hid in that hole under the gable. Croaker watched him reach for a chink in the logs and raise himself toward the treasure house. Then he became silent and sat huddled up, wings drooping discontentedly, his whole aspect one of utter despair. Lou, bending to caress him, heard Billy give an exclamation, and ran forward. It's here, Lou, he cried excitedly, a tin box an' a shot-bag full of gold in a hollered-out log. The bag has been ripped open by Croaker. I'll have to go inside to get the box out. He dropped to the sward and stepped through an unglazed window into the hut. Nailed to one end was a crude ladder. Billy climbed the ladder and peered closely at the log which held the money. To all appearances it was exactly like its fellows, no door, no latch to be seen. And still, he reasoned, there must be an opening of some kind there. He lit a match and held it close to the log. Then he whistled. What he had mistaken for a pine knot was a small button fixed, as he saw now, in a tiny groove. He moved the button and a small section of the log fell, spraying him with musty dust.
- 35. Another moment and he was outside beside Lou, bag and box in his arms. Croaker was nowhere to be seen; neither was the gold piece which he had dropped in his amazement at sight of Billy and Lou. He went back and got it, said the girl, in answer to Billy's look of amazement. And, Billy, he flew away in an awful grouch. Oh, he'll soon get over it, laughed Billy. We'll find him waitin' fer us farther on. They crossed the lot and went through the pines to the sunny open. There, on a mossy knoll, Lou spread her cloak, and Billy poured the gold from bag and box upon it. Lou started to count the money. Billy sat back, watching her. Yes, sir, he mused, it certainly takes a good woman to steady a man. For ten glorious minutes he built air castles and dreamed dreams. Two thousand nine hundred and forty dollars, Low announced, and Billy jumped up. Whew! he whistled, an' all gold, too. The three pieces that Croaker took make the even three thousand. They placed the money back in the box and bag. Then Billy, picking up the treasure, spoke gently. It'll make 'em a grand weddin' gift, Lou. Yes, she answered, a grand wedding gift, Billy. In silence they passed on through the upland gowned in hazy, golden spray. At the height of land they paused to look down across the sweeping country below them. Then blue eyes sought grey and hand in hand, with a new glad vista of life opening before them, they went on into the valley.
- 36. THE END.
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