Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T08:56:15.637Z Has data issue: false hasContentIssue false
  • English
  • Français

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; kito@unity.ncsu.edu
Karl Kunisch
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens-University Graz, 8010 Graz, Austria; karl.kunisch@uni-graz.at; gunther.peichl@uni-graz.at
Gunther H. Peichl
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens-University Graz, 8010 Graz, Austria; karl.kunisch@uni-graz.at; gunther.peichl@uni-graz.at
Get access

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).