Sensibilité de l'équation de la chaleur aux sauts de conductivité

Pantz, Olivier

doi:10.1016/j.crma.2005.07.005

Cited by 44 publications

(57 citation statements)

References 4 publications

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“…Note that (13) is similar to the standard results of eigenvalue optimization (Haug and Rousselet 1980;Seyranian et al 1994;de Gournay 2006), and that (14) incorporates jump terms that are typical of shape sensitivity with discontinuous coefficients (Bernardi and Pironneau 2003;Pantz 2005). Note also that (13) holds for any eigenvalue (and not only the first one) and in the case of a simple eigenvalue dλ is linear w.r.to θ , hence Fréchet differentiability holds.…”

Section: Statement Of the Main Resultssupporting

confidence: 52%

“…2 The sectional geometry of the generalized Graetz problem This requires a more subtle treatment than the cases where the outer boundary of the domain is varying. Shape sensitivity in the case of an interface between two materials was already studied, see Hettlich and Rundell (1998), Bernardi and Pironneau (2003), Pantz (2005), Conca et al (2009), Allaire et al (2009) and Neittaanmäki and Tiba (2012) for a recent review article. To our knowledge the eigenvalue problem for the Graetz operator, where some coefficient of the operator depends on an auxiliary partial differential equation, was not addressed previously.…”

Section: Definitionmentioning

confidence: 99%

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Shape optimization for the generalized Graetz problem

Gournay

Fehrenbach

Plouraboué

2014

Struct Multidisc Optim

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion equation. The problem boils down to the optimization of generalized eigenvalues on a two phases domain. Shape sensitivity analysis is performed with respect to the evolution of the interface between the fluid and solid phase. In particular physical settings, counterexamples where there is no optimal domains are exhibited. Numerical examples of optimal domains with different physical parameters and constraints are presented. Two different numerical methods (level-set and mesh-morphing) are show-cased and compared.

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Section: Statement Of the Main Resultssupporting

confidence: 52%

Section: Definitionmentioning

confidence: 99%

Shape optimization for the generalized Graetz problem

Gournay

Fehrenbach

Plouraboué

2014

Struct Multidisc Optim

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“…Lagrangian methods are commonly used in shape optimization and have the advantage of providing the shape derivative without the need to compute the material derivative of the state; see [4,9,12,37,42,43]. Compared to these known shape-Lagrangian methods, the averaged adjoint method is fairly general due to minimal required conditions.…”

Section: Introductionmentioning

confidence: 99%

Distributed shape derivativeviaaveraged adjoint method and applications

2016

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The structure theorem of Hadamard-Zolésio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. Actually the domain representation, also known as distributed shape derivative, is more general than the boundary expression as it is well-defined for shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithms. In this paper we describe several advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We identify a tensor representation of the distributed shape derivative, study its properties and show how it allows to recover the boundary expression directly. We use a novel Lagrangian approach, which is applicable to a large class of shape optimization problems, to compute the distributed shape derivative. We also apply the technique to retrieve the distributed shape derivative for electrical impedance tomography. Finally we explain how to adapt the level set method to the distributed shape derivative framework and present numerical results.

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“…However, this choice can introduce severe complications in the numerical calculation of a shape derivative [2,3,8] and thus a smooth interpolation scheme is often preferable [3]. Moreover, a regularized Hooke's tensor seems to be more appropriate for some applications [14].…”

Section: Introductionmentioning

confidence: 99%

Structural and Multi-Functional Optimization Using Multiple Phases and a Level-Set Method

Allaire¹,

Jouve²,

Michailidis³

2014

Proceedings of the 3rd South-East European Conference on Computational Mechanics (SEECCM III)

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Abstract. In this work we adress the problem of structural and multi-functional shape and topology optimization using several elastic materials in a fixed working domain. The description and the evolution of the interfaces between the different phases is done using the level-set method. We use a smooth Hooke's tensor, instead of a discontinuous one. The continuous case can be seen as an approximation of the sharp interface case, and in this context, the signed distance function corresponding to each interface is used for modeling the smooth Hooke's tensor. A directional shape derivative is calculated for the objective function to minimize. We show 2d results for compliance minimization, as well as an example of multi-functional optimization, coupling structural and thermal problems. 639

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Sensibilité de l'équation de la chaleur aux sauts de conductivité

Cited by 44 publications

References 4 publications

Shape optimization for the generalized Graetz problem

Shape optimization for the generalized Graetz problem

Distributed shape derivativeviaaveraged adjoint method and applications

Structural and Multi-Functional Optimization Using Multiple Phases and a Level-Set Method

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