Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer Series in Computational Mathematics)

Sokolowski, Jan; Zolesio, Jean-Paul

ISBN 10: 3642634710 / ISBN 13: 9783642634710

Editore: Springer, 2012

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Dati bibliografici

• Titolo: Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer Series in Computational Mathematics)
• Editore: Springer
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 3642634710
• ISBN 13: 9783642634710
• Rilegatura: Brossura
• Condizione: New

Descrizione articolo

Riassunto

Shape Optimization is a classical field of the calculus of variations, optimal control theory and structural optimization. In this book the authors discuss the shape calculus introduced by J. Hadamard and extend it to a broad class of free boundary value problems. The approach is functional analytic throughout and will serve as an excellent basis for the development of numerical algorithms for the solution of shape optimization problems.

Contenuti

1 Introduction to shape optimization.- 1.1. Preface.- 2 Preliminaries and the material derivative method.- 2.1. Domains in ?N of class Ck.- 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 ?.- 2.20. Tangential gradients and Laplace―Beltrami operators on ?.- 2.21. Variational problems on ?.- 2.22. The transport of differential operators.- 2.23. Integration by parts on ?.- 2.24. The transport of Laplace―Beltrami operators.- 2.25. Material derivatives.- 2.26. Material derivatives on ?.- 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.- 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 sensitivity analysis of the smallest eigenvalue.- 3.7. Shape sensitivity analysis of the Kirchhoff plate.- 3.8. Shape derivatives of boundary integrals: the non-smooth case in ?2.- 3.9. Shape sensitivity analysis of boundary value problems with singularities.- 3.10. Hyperbolic initial boundary value problems.- 4 Shape sensitivity analysis of variational inequalities.- 4.1. Differential stability of the metric projection in Hilbert spaces.- 4.2. Sensitivity analysis of variational inequalities in Hilbert spaces.- 4.3. The obstacle problem in H1 (?).- 4.3.1. Differentiability of the Newtonian capacity.- 4.3.2. The shape controlability of the free boundary.- 4.4. The Signorini problem.- 4.5. Variational inequalities of the second kind.- 4.6. Sensitivity analysis of the Signorini problem in elasticity.- 4.6.1. Differential stability of solutions to variational inequalities in Hilbert spaces.- 4.6.2. Shape sensitivity analysis.- 4.7. The Signorini problem with given friction.- 4.7.1. Shape sensitivity analysis.- 4.8. Elasto―Plastic torsion problems.- 4.9. Elasto―Visco―Plastic problems.- References.