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

396 Pages·2000·4.77 MB·English

by  Bernhard Kawohl

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
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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.