Calculus of Variations/Partial Differential Equations
Dynamical shape gradient for the Navier–Stokes system
Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 183-186.

This Note deals with the sensitivity analysis of a newtonian incompressible fluid driven by the Navier–Stokes equations with respect to the dynamic of the fluid domain boundary. The structure of the gradient with respect to the velocity of the domain for a given cost function is established. This result is obtained using new shape derivation tools for Eulerian functionals and the Min–Max derivation principle.

Dans cette Note, nous nous intéressons à l'analyse de sensibilité de l'évolution d'un fluide newtonien incompressible régi par les équations de Navier–Stokes vis-à-vis de la dynamique de la frontière du domaine fluide. Nous établissons la structure du gradient d'une fonctionnelle coût spécifique par rapport à la vitesse du domaine mobile. Ce résultat est obtenu en utilisant, de façon combinée, des techniques nouvelles de dérivation de forme pour des fonctionnelles eulériennes et le principe de dérivation du Min–Max.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.11.002
Dziri, Raja 1; Moubachir, Marwan 2; Zolésio, Jean-Paul 3

1 LAMSIN/ENIT, faculté des sciences, département de mathématiques, 1060 Tunis, Tunisia
2 Laboratoire central des ponts et chaussées, 58, boulevard Lefebvre, 75732 Paris cedex 15, France
3 INRIA, projet OPALE, BP 93, 2004, route des Lucioles, 06902 Sophia Antipolis, France
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Dziri, Raja; Moubachir, Marwan; Zolésio, Jean-Paul. Dynamical shape gradient for the Navier–Stokes system. Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 183-186. doi : 10.1016/j.crma.2003.11.002. http://www.numdam.org/articles/10.1016/j.crma.2003.11.002/

[1] Delfour, M.C.; Zolésio, J.-P. Shapes and Geometries – Analysis, Differential Calculus and Optimization, Adv. in Design and Control, SIAM, 2001

[2] R. Dziri, M. Moubachir, J.-P. Zolésio, Navier–Stokes dynamical shape control: from state derivative to Min–Max principle, Technical report, INRIA, RR-4610, 2002

[3] Dziri, R.; Zolésio, J.-P. Dynamical shape control in non-cylindrical Navier–Stokes equations, J. Convex Anal., Volume 6 (1999) no. 2, pp. 293-318

[4] Dziri, R.; Zolésio, J.-P. Eulerian derivative for non-cylindrical functionals (Cagnol, J. et al., eds.), Shape Optimization and Optimal Design, Lecture Notes in Pure and Appl. Math., vol. 216, 2001, pp. 87-107

[5] Zolésio, J.-P. Shape differential equation with a non-smooth field, Computational Methods for Optimal Design and Control (Arlington, VA, 1997), Progr. Systems Control Theory, vol. 24, Birkhäuser, Boston, MA, 1998, pp. 427-460

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