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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  2. Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math.79 (1998) 175–202.  
  3. P.G. Ciarlet, Mathematical Elasticity, Vol. 1. North-Holland, Amsterdam (1987).  
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  5. J.C. de los Reyes and K. Kunisch, A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations. Nonlinear Anal.62 (2005) 1289–1316.  
  6. M.C. Delfour and J.P. Zolesio, Shapes and Geometries. SIAM (2001).  
  7. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986).  
  8. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985).  
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  10. J. Haslinger and P. Neittaanmaki, Introduction to shape optimization. SIAM, Philadelphia (2003).  
  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.  
  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).  
  14. R. Temam, Navier Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1979).  
  15. J.T. Wloka, B. Rowley and B. Lawruk, Boundary value problems for elliptic systems. Cambridge Press (1995).  
  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).  

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