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.

@article{COCV_2008__14_3_517_0, author = {Peichl, Gunther H. and Kunisch, Karl and Ito, Kazufumi}, title = {Variational approach to shape derivatives}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {517--539}, publisher = {EDP-Sciences}, volume = {14}, number = {3}, year = {2008}, doi = {10.1051/cocv:2008002}, mrnumber = {2434064}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008002/} }

TY - JOUR AU - Peichl, Gunther H. AU - Kunisch, Karl AU - Ito, Kazufumi TI - Variational approach to shape derivatives JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 517 EP - 539 VL - 14 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008002/ DO - 10.1051/cocv:2008002 LA - en ID - COCV_2008__14_3_517_0 ER -

%0 Journal Article %A Peichl, Gunther H. %A Kunisch, Karl %A Ito, Kazufumi %T Variational approach to shape derivatives %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 517-539 %V 14 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008002/ %R 10.1051/cocv:2008002 %G en %F COCV_2008__14_3_517_0

Peichl, Gunther H.; Kunisch, Karl; Ito, Kazufumi. Variational approach to shape derivatives. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, pp. 517-539. doi : 10.1051/cocv:2008002. http://archive.numdam.org/articles/10.1051/cocv:2008002/

[1] A unified discrete-continuous sensitivity analysis method for shape optimization. Lecture at the Radon Institut, Linz, Austria (2005).

,[2] Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175-202. | MR | Zbl

and , , ,[4]

los Reyes, Constrained optimal control of stationary viscous incompressible fluids by primal-dual active set methods. Ph.D. thesis, University of Graz, Austria (2003). 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] Shapes and Geometries. SIAM (2001). | MR | Zbl

and ,[7] Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986). | MR | Zbl

and ,[8] Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl

,[9] Finite Element Approximation for Optimal Shape, Material and Topological Design. Wiley, Chichester (1996). | MR | Zbl

and ,[10] Introduction to shape optimization. SIAM, Philadelphia (2003). | MR | Zbl

and ,[11] Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314 (2006) 126-149. | MR | Zbl

, and ,[12] Sur le contrôle par un domaine géometrique. Rapport 76015, Université Pierre et Marie Curie, Paris (1976).

and ,[13] Introduction to shape optimization. Springer, Berlin (1991). | MR | Zbl

and ,[14] Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1979). | MR | Zbl

,[15] Boundary value problems for elliptic systems. Cambridge Press (1995). | MR | Zbl

, and ,[16] 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). | MR | Zbl

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