Introduction to Shape Optimization Shape Sensitivity Analysis
This book is motivated largely by a desire to solve shape optimization prob lems that arise in applications, particularly in structural mechanics and in the optimal control of distributed parameter systems. Many such problems can be formulated as the min
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Editorial Board
R. L. Graham, Murray Hill J. Stoer, WOrzburg R. Varga, Kent (Ohio)
Jan Sokolowski Jean-Paul Zolesio
Introduction to Shape Optimization Shape Sensitivity Analysis
Springer-Verlag Berlin Heidelberg GmbH
Jan Sokolowski Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warszawa Poland Jean-Paul Zolesio Centre National de la Recherche SCientifique et Institut non Lineaire de Nice B. P.71 Faculte des Sciences 06108 Nice Cedex 2 France
Mathematics Subject Classification (1991): 35B30, 49B50, 73C60, 73K40, 73T05
Ubrary 01 Congress Calaloging-in-Publicalion Dala Sokorowski, Jan, 1949- Inlroduction 10 shape optlmization : shape sensilivity analysis I J.Sokotowski, J. P. Zolesio. p. cm . - (Springer se ries In computational mathematics; 16) Includes blbllographlcal relerences and index.c ISBN 978-3-642-63471-0 ISBN 978-3-642-58106-9 (eBook) DOI 10.1007/978-3-642-58106-9
Contents
Chapter 1 Introduction to shape optimization 1.1. Preface Chapter 2 Preliminaries and the material derivative method 2.1. Domains in IRN of class C k 2.2. Surface measures on 2.3. Functional spaces 2.4. Linear elliptic boundary value problems 2.5. Shape functionals 2.6. Shape functionals for problems governed by linear elliptic boundary value problems 2.6.1. Shape functionals for transmission problems 2.6.2. Approximation of homogenuous Dirichlet problems 2.7. Convergence of domains 2.8. Transformations Tt of domains 2.9. The speed method 2.10. Admissible speed vector fields Vk(D) 2.11. Eulerian derivatives of shape functionals 2.12. Non-differentiable shape functionals 2.13. Properties of Tt transformations 2.14. Differentiability of transported functions 2.15. Derivatives for t > 0 2.16. Derivatives of domain integrals 2.17. Change of variables in boundary integrals 2.18. Derivatives of boundary integrals 2.19. The tangential divergence of the field V on r 2.20. Tangential gradients and Laplace-Beltrami operators on 2.21. Variational problems on r 2.22. The transport of differential operators 2.23. Integration by parts on r 2.24. The transport of Laplace-Beltrami operators 2.25. Material derivatives
5 13 14
r
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60 62 65 75
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r
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Contents
2.26. Material derivatives on r 2.27. The material derivative of a solution to the Laplace equation with Dirichlet boundary conditions 2.28. Strong material derivatives for Dirichlet problems 2.29. The material derivative of a solution to the Laplace equation with Neumann boundary conditions 2.30. Shape derivatives 2.31. Derivatives of domain integrals (II) 2.32. Shape derivatives on 2.33. Derivatives of boundary integrals Chapter 3 Shape derivatives for linear problems 3.1. The shape derivative for the Dirichlet boundary value problem 3.2. The shape derivative for the Neumann boundary value problem 3.3. Necessary optimality conditions 3.4. Parabolic equations 3.4.1 Neumann boundary conditions 3.4.2 Dirichlet boundary conditions 3.5. Shape sensitivity in elasticity 3.6. Shape sensitivi
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