Un algorithme d'identification de frontières soumises à des conditions aux limites de Signorini

Free access article

Issue		ESAIM: M2AN Volume 34, Number 3, May-June 2000


Page(s)		707 - 722
DOI		10.1051/m2an:2000163

DOI: 10.1051/m2an:2000163
M2AN, Vol. 34, N $^{\rm o}$ 3, 2000, pp. 707-722

Un algorithme d'identification de frontières
soumises à des conditions aux limites de Signorini

Slim Chaabane
Faculté des Sciences de Sfax & ENIT-L A1. (slim.chaabane@fsm.rnu.tn)

Mohamed Jaoua
ENIT-L A1, BP 37 1002 Tunis-Belvédère, Tunisie. (mohamed.jaoua@enit.rnu.tn)

Reçu : 29 juin 1998. Révisé : 14 janvier 2000.

Abstract: This work deals with a non linear inverse problem of reconstructing an unknown boundary $\gamma$ , the boundary conditions prescribed on $\gamma$ being of Signorini type, by using boundary measurements. The problem is turned into an optimal shape design one, by constructing a Kohn & Vogelius-like cost function, the only minimum of which is proved to be the unknown boundary. Furthermore, we prove that the derivative of this cost function with respect to a direction $\theta$ depends only on the state u⁰, and not on its Lagrangian derivative $u^1(\theta)$ .

Résumé: On s'intéresse dans ce travail à un problème inverse non linéaire d'identification d'une frontière inconnue $\gamma$ par des mesures de surfaces, les conditions aux limites imposées sur cette frontière $\gamma$ étant de type Signorini. Le problème est d'abord transformé en un problème d'optimisation de forme, par la définition d'une fonction de type Kohn-Vogelius, dont nous montrons que le seul minimum est la frontière recherchée, et que le gradient dans une direction donnée $\theta$ ne dépend que du seul état u⁰, et non de sa dérivée lagrangienne $u^1(\theta)$ .

Keywords and phrases: Geometrical inverse problems, identification, Signorini type boundary conditions, unknown boundary, domaine derivatives, Kohn-Vogelius function, optimal shape design.

AMS Subject Classification: 35R30, 35S85, 49Q10, 49Q12, 49M10, 65K10

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