Variational approach to shape derivatives

Kazufumi Ito; Karl Kunisch; Gunther H. Peichl

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 3, page 517-539
  • ISSN: 1292-8119

Abstract

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A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems and the Navier-Stokes equations are given.

How to cite

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Ito, Kazufumi, Kunisch, Karl, and Peichl, Gunther H.. "Variational approach to shape derivatives." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2008): 517-539. <http://eudml.org/doc/250272>.

@article{Ito2008,
abstract = { A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems and the Navier-Stokes equations are given. },
author = {Ito, Kazufumi, Kunisch, Karl, Peichl, Gunther H.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Shape derivative; shape derivative},
language = {eng},
month = {2},
number = {3},
pages = {517-539},
publisher = {EDP Sciences},
title = {Variational approach to shape derivatives},
url = {http://eudml.org/doc/250272},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Ito, Kazufumi
AU - Kunisch, Karl
AU - Peichl, Gunther H.
TI - Variational approach to shape derivatives
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/2//
PB - EDP Sciences
VL - 14
IS - 3
SP - 517
EP - 539
AB - A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems and the Navier-Stokes equations are given.
LA - eng
KW - Shape derivative; shape derivative
UR - http://eudml.org/doc/250272
ER -

References

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  10. J. Haslinger and P. Neittaanmaki, Introduction to shape optimization. SIAM, Philadelphia (2003).  Zbl0594.73108
  11. K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl.314 (2006) 126–149.  Zbl1088.49028
  12. F. Murat and J. Simon, Sur le contrôle par un domaine géometrique. Rapport 76015, Université Pierre et Marie Curie, Paris (1976).  
  13. J. Sokolowski and J.P. Zolesio, Introduction to shape optimization. Springer, Berlin (1991).  Zbl0761.73003
  14. R. Temam, Navier Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1979).  Zbl0426.35003
  15. J.T. Wloka, B. Rowley and B. Lawruk, Boundary value problems for elliptic systems. Cambridge Press (1995).  Zbl0836.35042
  16. J.P. Zolesio, The material derivative (or speed method) for shape optimization, in Optimization of Distributed Parameter Structures, Vol. II, E. Haug and J. Cea Eds., Sijthoff & Noordhoff (1981).  Zbl0517.73097

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