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Optimal Shape Design: Lectures given at the joint C.I.M./C.I.M.E. Summer School held in Tróia, Portugal, June 1–6, 1998 PDF

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Lecture Notes in Mathematics 1740 Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris Subseries: Fondazione C. I. M. E., Firenze Adviser: Arrigo Cellina regnirpS Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo B. Kawohl O. Pironneau L. Tartar J.-E Zoldsio lamitpO epahS ngiseD Lectures given at the joint C.I.M./C.I.M.E. Summer School held in Tr6ia, Portugal, June 1-6, 1998 Editor: A. Cellina and A. Ornelas ! enoizadnoF C.I.M.E. r e g ~ n i r p S Author Bernhard Kawohl Olivier Pironneau Mathematisches Institut Universit6 Pierre et Marie Curie Universit~t zu Krln Drpartement de Mathrmatique Weyertal 86-90 4, Place Jussieu 50931 Kfln, Germany 75252 Paris, France E-mail: kawo@hMlL uni-koeln.de E-mail: Olivier.Pironneau @inria.fr Luc Tartar Jean-Paul Zolrsio Carnegie Mellon University INRIA Department of Mathematical Sciences Centre de Mathrmatiques Appliqures Schenley Park 2004 Route de Lucioles, B.P. 93 Pittsburgh, PA, 15213-3890, USA 06902 Sophia Antipolis cedex, France E-mail: tartar @ andrew.cmu.edu E-maih Jean-Paul.Zolesio @ sophia.inria, fr Editors Arrigo Cellina Ant6nio Ornelas Universita di Milano - Bicocca Centro de Investiga¢~o em Dipartimento di Matematica Matemfitica e Aplica¢fes e Applicazioni Universidade de Evora Via Bicocca degli Arcimboldi 8 rua Romeo Ramalho 59 20126 Milano, Italy 7000-671 l~vora Portugal E-maih cellina@ ares.mat .unimi.it E-mail: [email protected] Cataloging-in-Publication Data applied for Die Demsche Bibliothek CIP-Eiltheilsaufnahm~. - Optimaslh ape design : lectures given at the joint CIM, CIME summer school, held in Troia, Portugal, June 1 - 6, 1998 / Fondazione CIME. B. Kawohl ... Ed.: A. Celfina and A. Ornelas. - Berfin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan Paris ; Singapore ; ; Tokyo Springer, : 2000 (Lecture notes in mathematics ; Vol. 1740 Subseries: : Fondazione CIME) ISBN 3-540-67971-5 Mathematics Subject Classification (2000): 49K20, 65K 10, 65N55 ISSN 0075-8434 ISBN 3-540-67971-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproductioonn microfilms or in any other way, and storage in datbaa nks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under theG erman Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-VerlagB erlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10724274 41/3142-543210 - Printed on acid-free paper Table of Contents Optimal Shape Design Introduction .................................................. 1 Some nonconvex shape optimization problems B. Kawohl ................................................... 7 1. Minimizing paths, the opaque square ......................... 7 2. Newton's problem of minimal resistance ...................... 15 3. More on Newton's problem .................................. 25 4. Extremal eigenvalue problems ............................... 37 References ................................................ 44 An Introduction to the Homogenization Method in Optimal Design L. Tartar .................................................... 47 1. Introduction ............................................... 47 2. A counter-example of Francois Murat ......................... 49 3. The independent discoveries of others ......................... 52 4. An elementary model problem ............................... 56 5. H-convergence ............................................. 64 6. Bounds on effective coefficients: first method .................. 80 7. Correctors in Homogenization ............................... 94 8. Bounds on effective coefficients: second method ................ 104 9. Computation of effective coefficients .......................... 114 10. Necessary conditions of optimality: first step .................. 131 11. Necessary conditions of optimality: second step ................ 144 12. Conclusion ................................................ 149 13. Acknowledgements ......................................... 152 References ................................................ 152 Shape Analysis and Weak Flow J-P. Zoldsio .................................................. 157 .1 Introduction ............................................... 157 .2 Large evolution of domains .................................. 158 2.1 Introduction .......................................... 158 2.2 Non cylindrical evolution problem ....................... 159 2.3 Flow Transformation of the Non Autonomous Vector Field V ................................................... 160 2.3.1 Continuous field .................................. 160 2.3.2 Filed in p L ...................................... 161 VI Table of Contents 3. First Variation of the Flow Mapping with respect to the Vector Field ..................................................... 162 3.1 The Tranversal field Z ................................. 162 3.2 Derivatives of the Flow Transformation .................. 165 4. Derivative of an Integral over the Evolution Tube with Respect to the velocity Field ........................................ 165 4.1 The adjoint problem ................................... 167 5. Non cylindrical Large Evolution of an Elastic Domain .......... 167 5.1 Derivative of the Action ................................ 168 5.2 Equations for the Free boundary ........................ 169 6. Weak Convection of characteristic functions ................... 170 6.1 An Uniqueness Result .................................. 170 6.2 The Galerkin Approximation ........................... 171 7. Variational Principle in Euler Problem ........................ 175 8. Differentiability ............................................ 179 8.1 Smooth solutions ...................................... 179 8.1.1 Bounded Velocity Field ............................ 179 8.1.2 Unbounded Velocity Field ......................... 180 8.2 Fields in L4(O,T,H~(D, RN)) ........................... 180 9. Existence results ........................................... 181 10. Existence Results Under Capacitary Constraints ............... 184 10.1 Preliminaries about capacity and domains convergence ..... 185 10.2 Limiting process in the equation ......................... 186 10.3 Continuity under capacitary constraint ................... 188 10.4 Continuity under flat cone condition ..................... 192 10.5 Existence results for extremal domains ................... 193 11. Geometry via the Oriented Distance function .................. 196 11.1 Introduction .......................................... 196 11.2 Topologies generated by distance functions ............... 197 11.2.1 Smoothness of boundary, curvatures, skeletons, convexity ............................................. 199 11.2.2 Sets of bounded curvature and convex sets .......... 201 11.3 Federer's sets of positive reach and curvature measures ..... 212 11.4 The W2,P-case ........................................ 219 11.5 Compactness theorems ................................. 221 11.5.1 Wl,p-topology ................................... 221 11.5.2 Wl,P-complementary topology ..................... 223 11.5.3 Wl,P-oriented distance topology ................... 225 11.6 A continuity of the solution of the Dirichlet boundary value problem .............................................. 226 12 Derivative in a Fluid-Structure problem ....................... 229 Table of Contents VII 12.1 Introduction .......................................... 229 12.2 Definitions and existence results ......................... 230 12.2.1 The dynamical problem .......................... 231 12.2.2 The eigenvalue problem .......................... 233 12.3 The static case ........................................ 236 12.4 Shape derivative of the solution ......................... 236 12.4.1 Boundary Condition ............................. 247 12.5 The Exterior Navier Stokes Problem ..................... 247 13. The Outer Sobolev Space ................................... 248 13.1 Existence Result ...................................... 249 13.2 The coupling with a Potential Flow ...................... 250 13.2.1 The Projected Problem ........................... 252 13.3 Strong Formulation of the Projected Problem ............. 255 13.4 From the Projected problem to the Navier Stokes Flo~ ...... 256 13.5 Uniqueness of Solution for both Navier Stokes flow and 12-projected Problems .................................. 257 13.6 The Weak Formulation of the V-projected Problem ........ 257 13.7 Shell Approach ........................................ 259 13.8 Shell Structures ....................................... 259 13.8.1 Oriented Distance Function ....................... 259 13.9 The Linear Tangent Operator Du ....................... 267 13.10 Weak formulation for the vector field Au ................ 268 13.10.1 The Deformation Tensor ......................... 268 13.11 The pseudo-differential operator K ..................... 270 13.12 Coupling Navier-Stokes and Potential Flows ............. 271 13.12.1 Weak Formulation in the Whole Domain 21 ........ 271 13.12.2 S.N.C. for a global stationary Navier Stokes Flow ... 273 13.13 Potential case in12p .................................. 274 13.14 Shell Representation .................................. 276 13,14.1 Intrinsic Equations ............................. 279 13.15 Intrinsic Shell Form of a(O;u,v) ........................ 280 13.16 Intrinsic Shell Form of b(u,q) .......................... 281 13.17 Intrinsic Shell Form of e..e ............................. 281 13.18 The Fluid Shell Equation ............................. 289 13.19 Intrinsic Equation .................................... 290 13.20 Numerical approximation ............................. 291 13,20.1 The Non stationary Problem ..................... 291 14. MinMax Shape Derivative ................................... 292 14.1 Notation and definitions ................................ 292 14.2 The Navier Stokes problem ............................. 293 14.2.1 The flow generated by u~ ........... ............. 293 14.2.2 The Speed Method .............................. 295 14.2.3 The extractor identity ............................ 296 VIII Table of Contents 14.2.4 The estimate .................................... 003 14.2.5 Dense subspace in 7-/(Y2) .......................... 003 14.2.6 and S star-shaped domains ..................... 203 14.2.7 The pressure approximation ....................... 403 14.2.8 The regularity problem 603 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Shape optimization problem ............................ 703 14.3.1 Shape gradient with linear splitted flow ............. 703 14.3.2 The second design performance functional .......... 823 14.4 Monotone approximations .............................. 133 14.4.1 Definition and existence result ..................... 133 14.4.2 For large viscosity solutions of the relaxed problem are quasi unique ....................................... 335 14.4.3 Upper approximation of the functional ............. 335 14.4.4 Flow with non unique solution .................... 335 Optimal Shape Design by Local Boundary Variations O. Pironneau ................................................. 343 1. Introduction ............................................... 343 2. Examples ................................................. 344 2.1 Two Laboratory Test Cases: Nozzle Optimization ......... 344 2.2 Minimum weight of structures ........................... 345 2.3 Wing design .......................................... 345 2.4 Stealth Wings ......................................... 347 2.4.1 Maxwell equations ................................ 347 2.4.2 Helmholtz equation ............................... 348 2.4.3 Boundary conditions .............................. 348 2.5 Optimal brake water ................................... 349 2.6 Ribblets .............................................. 350 2.7 Sonic boom reduction .................................. 351 3. Existence of Solutions ...................................... 351 3.1 Generalities ........................................... 351 3.2 Sketch of the proof of Sverak's Theorem .................. 353 4. Solution By Optimization Methods ........................... 553 4.1 Gradient Methods ..................................... 355 4.2 Newton Methods ...................................... 356 4.3 Constraints ........................................... 356 5. Sensitivity Analysis ........................................ 357 5.1 Sensitivity Analysis for the nozzle problem ............... 358 5.2 Discretization with Triangular Elements .................. 360 5.3 Discrete Gradients ..................................... 362 5.4 Implementation problems ............................... 364 5.5 Optimal shape design with Stokes flow ................... 364 Table of Contents IX 5.6 OSD for laminar flow .................................. 366 6. Alternative ways ........................................... 367 7. Problems Connected With The Numerical Implementation ...... 368 7.1 Independence from J ................................... 368 7.1.1 Add geometrical constraints ........................ 368 7.1.2 Other discretization methods ....................... 369 7.1.3 Automatic Differentiation of Programs .............. 369 8. Regularity Problems ........................................ 372 8.1 Application ........................................... 373 8.2 Discretization ......................................... 374 8.3 Consequence .......................................... 375 9. Consistent Approximations .................................. 375 9.1 Algorithm ............................................ 376 9.2 Problem Statement .................................... 376 9.3 Discretization ......................................... 377 9.4 Optimality Conditions: the continuous case ............... 378 9.4.1 Definition of 0 .................................... 378 9.5 Optimality Conditions: the discrete case .................. 378 9.6 Definition of Oh ....................................... 379 9.7 Hypothesis of the Theorem ............................. 380 9.8 Algorithm 3 .......................................... 381 9.9 Convergence .......................................... 382 10. Numerical Results ......................................... 382 Introduction These lectures were presented at the .joint C.I.M./C.I.M.E. Summer School on Optimal Shape Design held in Tr6ia (Portugal), June 02 to June 07, 1998. The mathematical problems that can be described by the label "Optimal shape design" form a broad area: it concerns the optimization of some performance criterion where the criterion depends, besides constraints that qualify the problem, on the "shape" of some region. A classical setting is Structural Me- chanics of elastic bodies such as bridges, beams, plates, shells, arches. These structures have to satisfy requirements of load and have to be designed in an optimal way; for example, should be built using the least amount of mate- rial. Alternatively, one might seek the optimal shape of a geometrical object moving in a fluid: B. Kawohl devotes most of his lectures to the classical Newton's problem of minimal resistance. This fascinating problem, studied by Newton in the interest of "Her Majesty's Navy", as reported by Kawohl, goes back about three hundred years and has been a subject of controversy and discussion from the very beginning (the functional to be minimized, al- though rotationally symmetric, is not convex, and this explains the difficulty of the problem). Alternatively, we may think of seeking the optimal shape of a wing. to be designed so as to reduce the drag while keeping a given value for the lift. '1O we might wish to design the optimal shape of a region (a harbor), given suitable constraints on the size of the entrance to the harbor, subject to incoming waves, so as to minimize the height of the waves inside it. Or we might wish to design some electrical device consisting of a (simply connected) region (partially) coated with a conducting material, say copper (the non-covered portion of the region is considered to be a perfect insula- tor): the goal is to minimize the cost of the device, subject to constraints on the performance of the resulting design. Or we might try to design materials obtained layering several materials, with different characteristics: the goal in this case could become that of computing the effective properties of the limit material. A large class of problems of this kind can be reduced to a standard for- mulation of the Calculus of Variations, i.e. that of minimizing an integral functional of the kind / .f(x, ))x(uV xd u(x), subject to boundary conditions and possibly to additional constraints on )x(u )x(uU and (a variant of this formulation would be that of a Control Problem; in this formulation the shape is -in general- the control). In this case one minimizes an energy or work flmctional with respect to the design parameters. However some interesting shape optimization problems do not fall into this fornmlation: a remarkal)le class of problems not of this form are the "opaque square" problem and its generalizations, as presented here by B. t(awohl.

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Optimal Shape Design is concerned with the optimization of some performance criterion dependent (besides the constraints of the problem) on the "shape" of some region. The main topics covered are: the optimal design of a geometrical object, for instance a wing, moving in a fluid; the optimal shape o
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