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Détection d'un objet immergé dans un fluide

Abstract : This dissertation takes place in the mathematic field called shape optimization. More precisely, we focus on a detecting inverse problem using shape calculus and asymptotic analysis: the aim is to localize an object immersed in a viscous, incompressible and stationary fluid. This work was motivated by the following main questions: can we localize an obstacle immersed in a fluid from a measurement made on the surface of the fluid? Can we reconstruct numerically this object, i.e. be close to its localization and its shape, from this measure? Can we find how many objects are included in the fluid using this measure? In order to answer to these questions, the inverse problem is studied as an optimization problem by minimizing a cost functional, the variable being the unknown shape. Two different approaches are considered in this work: the geometric optimization (using shape derivatives and shape gradient) and the topological optimization (using an asymptotic expansion and the topological "gradient"). At first, a mathematical framework is introduced in order to prove the existence of the shape derivatives of order one and two in the frame of the detection of inclusions. Then, the considered inverse problem is analyzed using geometric shape opti- mization: an identifiability result is proved, the shape gradient of several shape functionals is characterized and the instability of this inverse problem is proved. These theoretical results are used in order to reconstruct numerically some objets immersed in a fluid using a shape gradient algorithm with a regularization by a projection method. Finally, the detection of small inclusions in a fluid is studied using the topological shape optimization for a Kohn-Vogelius shape functional. This theoretical expression of the topological derivative is used in order to determine numerically the number and the location of some small obstacles immersed in a fluid using a topological gradient algorithm. The limits of this approach are explored: the penetration depth remains poor in this stationary problem.
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Contributor : Fabien Caubet <>
Submitted on : Wednesday, July 11, 2012 - 3:46:43 PM
Last modification on : Tuesday, February 2, 2021 - 2:54:04 PM
Long-term archiving on: : Thursday, December 15, 2016 - 10:19:12 PM

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Fabien Caubet. Détection d'un objet immergé dans un fluide. Optimisation et contrôle [math.OC]. Université de Pau et des Pays de l'Adour, 2012. Français. ⟨tel-00716902⟩

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