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Variational approach to shape derivatives

Published online by Cambridge University Press:  07 February 2008

Kazufumi Ito ,
Karl Kunisch  and
Gunther H. Peichl
Kazufumi Ito
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA; [email protected]
Karl Kunisch
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens-University Graz, 8010 Graz, Austria; [email protected]; [email protected]
Gunther H. Peichl
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens-University Graz, 8010 Graz, Austria; [email protected]; [email protected]
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Abstract

A general framework for calculating shape derivatives foroptimization problems with partial differential equations asconstraints is presented. The proposed technique allows to obtainthe shape derivative of the cost without the necessity to involvethe shape derivative of the state variable. In fact, the statevariable is only required to be Lipschitz continuous with respectto the geometry perturbations. Applications to inverse interfaceproblems, and shape optimization for elliptic systems and theNavier-Stokes equations are given.

Keywords

Shape derivative
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 517 - 539
Copyright
© EDP Sciences, SMAI, 2008

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References

M. Berggren, A unified discrete-continuous sensitivity analysis method for shape optimization. Lecture at the Radon Institut, Linz, Austria (2005).
Chen, Z. and Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175–202. CrossRef
P.G. Ciarlet, Mathematical Elasticity, Vol. 1. North-Holland, Amsterdam (1987).
J.C. de los Reyes, Constrained optimal control of stationary viscous incompressible fluids by primal-dual active set methods. Ph.D. thesis, University of Graz, Austria (2003).
de los Reyes, J.C. and Kunisch, K., A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations. Nonlinear Anal. 62 (2005) 1289–1316. CrossRef
M.C. Delfour and J.P. Zolesio, Shapes and Geometries. SIAM (2001).
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985).
J. Haslinger and P. Neittaanmaki, Finite Element Approximation for Optimal Shape, Material and Topological Design. Wiley, Chichester (1996).
J. Haslinger and P. Neittaanmaki, Introduction to shape optimization. SIAM, Philadelphia (2003).
Ito, K., Kunisch, K. and Peichl, G., Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl. 314 (2006) 126–149. CrossRef
F. Murat and J. Simon, Sur le contrôle par un domaine géometrique. Rapport 76015, Université Pierre et Marie Curie, Paris (1976).
J. Sokolowski and J.P. Zolesio, Introduction to shape optimization. Springer, Berlin (1991).
R. Temam, Navier Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1979).
J.T. Wloka, B. Rowley and B. Lawruk, Boundary value problems for elliptic systems. Cambridge Press (1995).
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).