Shape Methods for the Transmission Problem with a Single Measurement

Afraites, Lekbir; Dambrine, Marc; Kateb, Djalil

doi:10.1080/01630560701381005

Cited by 45 publications

(52 citation statements)

References 10 publications

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“…Note that our results cover and generalize several results that can be found in the literature of shape optimization approaches for EIT, including [2,22]. For instance taking α 2 = 1, α 1 = 0 in Proposition 5.3 we get p D ≡ 0 which yields the same formula as the one obtained in [2, pp.…”

Section: Shape Derivativessupporting

confidence: 85%

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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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Section: Shape Derivativessupporting

confidence: 85%

“…We consider the particular case where the objective is to reconstruct a piecewise constant conductivity σ which amounts to determine an interface Γ + between some inclusions and the background. We refer the reader to [2,7,8,11,22,25,29] for more details on this approach.…”

Section: Electrical Impedance Tomographymentioning

confidence: 99%

Distributed shape derivativeviaaveraged adjoint method and applications

2016

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“…al. [1]. Our results on the shape derivative of a two-phase eigenvalue problem (see Theorem 1.2 and 1.4) seem to be the first of its kind.…”

Section: Introductionsupporting

confidence: 52%

“…We refer to [1,8] for the details. Thus, (λ(ω t ) , u t ) is an eigenpair of (1.4) if and only if (λ(ω t ) , u t • Φ t ) satisfies the equation (1.5).…”

Section: Shape Derivative Of the First Eigenvalue For Two-phase Condumentioning

confidence: 99%

Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball

2009

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Abstract. In this article we deal with the problem of distributing two conducting materials in a given domain, with their proportions being fixed, so as to minimize the first eigenvalue of a Dirichlet operator. When the design region is a ball, it is known that there is an optimal distribution of materials which does not involve the mixing of the materials. However, the optimal configuration even in this simple case is not known. As a step in the resolution of this problem, in this paper, we develop the shape derivative analysis for this two-phase eigenvalue problem in a general domain. We also obtain a formula for the shape derivative in the form of a boundary integral and obtain a simple expression for it in the case of a ball. We then present some numerical calculations to support our conjecture that the optimal distribution in a ball should consist in putting the material with higher conductivity in a concentric ball at the centre.Résumé. Cet articleétudie le problème de la distribution optimale de deux matériaux conducteurs aux proportions fixes, de manièreà minimiser la première valeur propre d'un opérateur de Dirichlet. Dans le cas d'une boule, on sait qu'il existe une distribution optimale dans laquelle les materiaux ne se melangent pas, mais cette configuration n'est pas connue explicitement. On développe une analyse de dérivée par rapport au domaine pour ce problème spectralà double face. On fournit des arguments analytiques et numériques pour renforcer notre conjecture selon laquelle la distribution optimale dans une boule consisteà placer le matériauà conductivité plus importante dans une autre boule centrée tout au milieu. Les expériences numériques mettent enévidence ce phénomène.

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“…/. Forth step: We deduce that the equation satisfied the shape derivative u 0 D P u ru V. We have the classical identity (for instance, [17]). From Eq.…”

Section: Proofmentioning

confidence: 96%

Stability analysis in the inverse Robin transmission problem

Meftahi

2016

Math Methods in App Sciences

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In this paper, we consider the conductivity problem with piecewise-constant conductivity and Robin-type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non-monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn-Vogelius-type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first-order and second-order shape derivative of the proposed cost functional, which is performed rigorously by means of shape-Lagrangian formulation without using the shape sensitivity of the states variables.

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Shape Methods for the Transmission Problem with a Single Measurement

Cited by 45 publications

References 10 publications

Distributed shape derivativeviaaveraged adjoint method and applications

Distributed shape derivativeviaaveraged adjoint method and applications

Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball

Stability analysis in the inverse Robin transmission problem

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