From Convexity to Nonconvexity

396 Pages·2001·11.74 MB·English

by  Lars-Erik Andersson

From Convexity to Nonconvexity
Nonconvex Optimization and Its Applications
Volume 55
Managing Editor:
Panos Pardalos
University of Florida, U.S.A.
Advisory Board:
l.R. Birge
Northwestern University, U.S.A.
Ding-Zhu Du
University of Minnesota, U.S.A.
C. A. F10udas
Princeton University, U.S.A.
l. Mockus
Lithuanian Academy of Sciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, U.S.A.
G. Stavroulakis
University of Ioannina, Greece
The titles published in this series are listed at the end of this volume.
From Convexity to
N onconvexity
Edited by
R.P. Gilbert
University of Delaware
P.D. Panagiotopoulos
and
P.M. Pardalos
University of Florida
KLUWER ACADEMIC PUBLISHERS
DORDRECHT/BOSTON/LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13:978-1-4613-7979-9 e-ISBN -13: 978-1-4613-0287-2
001: 10.10071978-1-4613-0287-2
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Sold and distributed in North, Central and South America
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved
© 2001 Kluwer Academic Publishers
Soft cover reprint of the hardcover I st edition 200 I
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface xi
1
Frictional contact problems 1
Lars-Erik Andersson, Anders Klarbring
1.1 In trod uction 1
1.2 Elementary example of non-uniqueness and non-existence 2
1.3 Classical formulation of the quasistatic frictional contact problem 4
1.4 The static problem 5
1.5 Steady sliding problem 8
1.6 Existence results for quasistatic friction problems 8
1.7 Conclusion 10
References 11
2
Solutions for quasilinear hemivariational inequalities 15
Siegfried Carl
2.1 Introduction 15
2.2 Notations, hypotheses and the main result 16
2.3 Auxiliary results 20
2.4 Proof of the main result 24
2.4.1 Static problem 25
2.4.2 Example 25
2.4.3 Concluding remarks 26
References 27
3
A Survey on Nonsmooth Critical Point Theory 29
M area Degiovanni
3.1 Introduction 29
3.2 Critical point theory in metric spaces 31
3.3 Subdifferential calculus 33
3.4 Functionals of the calculus of variations 35
3.5 Functionals with quadratic dependence on the gradient 36
3.6 Area-type functionals 38
v
vi FROM CONVEXITY TO NON CONVEXITY
References 39
4
Exhaustive families of approximations revisited 43
V.F.Demyanov A.M.Rubinov
4.1 Directional derivatives and generalizations 43
4.2 Exhaustive families of upper and lower approximations 45
References 49
5
Optimal shape design 51
Zdzislaw Denkowski
5.1 In trod uction 51
5.2 Preliminaries 52
5.3 State relations for physical systems 53
5.4 Abstract OSD and direct method 56
5.5 Mapping method and its applications 57
5.6 Some remarks on other methods for OSD problems 59
5.7 Relaxation in OSD problems 61
5.8 Lower semicontinuity of functionals in OSD problems 63
References 63
6
Duality in Nonconvex Finite Deformation Theory 67
David Yang Cao
6.1 Introduction 67
6.2 Framework and Abstract Boundary Value Problem 69
6.3 Conjugate Stress-Strain Tensors and Gap Functions 71
6.4 Potential Extremum Principle 74
6.5 Classical Complementary Energy Principles 75
6.6 Generalized Variational Principles and Triality Theory 77
6.7 Pure Complementary Energy Principles and Minimax Theory 78
References 80
7
Contact Problems in Multibody Dynamics 85
Friedrich Pfeiffer and Christoph Clocker
7.1 In trod uction 85
7.2 The Evolution of a Theory 86
7.3 Present Mathematical Formulation 99
7.4 Conclusions 107
References 107
8
Hyperbolic Hemivariational Inequality 111
D. Coeleven! and D. Motreanu2
8.1 Introduction and formulation of nonsmooth hyperbolic problem 111
8.2 Finite dimensional approximation 113
Contents Vll
8.3 Main Results 117
References 121
9
Time-integration algorithms 123
Klaus Hackl
9.1 Introduction 123
9.2 An augmented principle of maximum plastic dissipation 124
9.3 The Evolution Problem 126
9.4 Time-Integration Algorithms and their Stability Properties 126
9.5 Algorithms Involving Operator-Split 130
9.6 Conclusion 135
References 135
10
Contact Stress Optimization 137
J. Haslinger
10.1 Introduction 137
10.2 Formulation of the problem 138
References 145
11
Recent results in contact problems with Coulomb friction 147
J. Jarusek and C. Eck
11.1 Introduction 147
11.2 The static case 148
11.3 Dynamic problem with contact condition in displacement and given friction 152
11.4 Dynamic problem with Coulomb friction and contact condition in velocities 155
11.5 Appendix: Thermal aspects of friction 157
11.6 Conclusion 159
References 159
12
Polarization fields in linear piezoelectricity 161
P. Bisegna and F. Maceri
12.1 Introduction 161
12.2 The linear piezoelectric problem 162
12.3 Weak formulations of the linear piezoelectric problem 165
12.4 Hashin-Shtrikman type variational principles 166
12.5 Conclusions 170
References 170
13
Survey of the methods for nonsmooth optimization 177
M. M. Makela
13.1 Introduction 177
viii FROM CONVEXITY TO NONCONVEXITY
13.2 Convex optimization 178
13.3 Nonconvex optimization 185
References 188
14
Hemivariational inequalities and hysteresis 193
M. Miettinen
References 205
15
Non convex a,spects of dynamics with impact 207
L. Paoli and .M. Schatzmann
References 221
16
On Global Properties of D.C.Functions 223
L.N.Polyakova
References 231
17
Variational-Hemivariational Inequalities 233
G.Dinca, G.Pop
References 241
18
Perturbations of Eigenvalue Problems 243
Vicentiu D. Ri1.dulescu
19
Implicit variational inequalities arising in frictional unilateral contact mechanics: 255
analysis and numerical solution of quasistatic problems
Marius Cocu, Michel Raous
19.1 Introduction 255
19.2 Quasistatic contact problems with friction 256
19.3 Extension to a model coupling adhesion and friction 259
19.4 Numerical methods 263
References 265
20
Regularity for variational inequalities 269
Rainer Schumann
20.1 Introduction 269
20.2 Variational inequalities and their applications 270
20.3 Regularity 272
References 280
21
A Survey of 1-D Problems of Dynamic Contact and Damage 283
Contents ix
Meir Shillor
21.1 Introduction 283
21.2 Preliminaries 284
21.3 Dynamic Thermoviscoelastic Contact of a Rod 285
21.4 Vibrations of a Beam between Two Stops 287
21.5 A Beam in Frictional Contact 290
21.6 The Elastic Rod with Damage 292
References 294
22
Nonconvexity in plasticity and damage 297
Georgios E. Stavroulakis
22.1 Introduction 297
22.2 Nonsmooth modeling in mechanics 298
22.3 Elastoplasticity 301
22.4 Damage mechanics 305
References 307
23
Augmented Lagrangian Methods for Contact Problems 311
J6zej Joachim Telega and Andrzej Galka
23.1 Introduction 311
23.2 General results 312
23.3 Ito and Kunisch augmented Lagrangian methods 314
23.4 Contact Problems 318
23.5 Parameter Estimation and Optimal Control 322
23.6 Image Restoration 326
References 328
24
Mountain Pass Theorems 333
Stepan A. Tersian
24.1 Introduction 333
24.2 Deformation theorems and (PS) conditions 334
24.3 Mountain pass theorems 339
References 343
25
Proximal Methods for Variational Inequalities with Set-Valued Monotone Operators 345
A. Kaplan and R. Tichatschke
25.1 Introduction 345
25.2 Multi-step Proximal Regularization Scheme 348
25.3 Convergence Analysis 349
References 358
26
Simons' Problem 363