Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions

Caubet, Fabien; Dambrine, Marc; Kateb, Djalil

doi:10.1088/0266-5611/29/11/115011

Cited by 21 publications

(21 citation statements)

References 44 publications

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“…Notice that when β = 0 we retrieve the Steklov eigenvalues, and we recover the Laplace-Beltrami eigenvalues by considering 1 β λ 1,β and letting β go to +∞, see Section 2.1. Note also that the close but distinct eigenvalue problem − u = λu in Ω u + α∂ n u + γ u = 0 on ∂Ω (7) was considered by J.B. Kennedy in [21]. He transforms this problem into a Robin type problem to prove a Faber-Krahn type inequality when the constants α, γ are nonnegative: the ball is the best possible domain among those of given volume.…”

Section: Introductionmentioning

confidence: 99%

An extremal eigenvalue problem for the Wentzell–Laplace operator

Dambrine

Kateb

Lamboley

2016

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain Ω, involving only geometrical information. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.

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Section: Introductionmentioning

confidence: 99%

An extremal eigenvalue problem for the Wentzell–Laplace operator

Dambrine

Kateb

Lamboley

2016

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…[5,20,22,33]), and obstacle characterized by different limit conditions, such as Neumann or generalized boundary conditions (see e.g. [8,21]). However, since such methods rely on the minimization of a (shape) cost-type functional which in turn needs the resolution of several well-posed direct problems, they need in particular some boundary data on the whole boundary of the domain of study, and therefore cannot be directly used for Problem (1.1).…”

Section: Page 87])mentioning

confidence: 99%

On the data completion problem and the inverse obstacle problem with partial Cauchy data for Laplace’s equation

2019

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We study the inverse problem of obstacle detection for Laplace’s equation with partial Cauchy data. The strategy used is to reduce the inverse problem into the minimization of a cost-type functional: the Kohn–Vogelius functional. Since the boundary conditions are unknown on an inaccessible part of the boundary, the variables of the functional are the shape of the inclusion but also the Cauchy data on the inaccessible part. Hence we first focus on recovering these boundary conditions, i.e. on the data completion problem. Due to the ill-posedness of this problem, we regularize the functional through a Tikhonov regularization. Then we obtain several theoretical properties for this data completion problem, as convergence properties, in particular when data are corrupted by noise. Finally we propose an algorithm to solve the inverse obstacle problem with partial Cauchy data by minimizing the Kohn–Vogelius functional. Thus we obtain the gradient of the functional computing both the derivatives with respect to the missing data and to the shape. Several numerical experiences are shown to discuss the performance of the algorithm.

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“…For example, we can mention the works of Cakoni et al [15,14] and Caubet et al [17] for the Laplace's equation, of Bourgeois et al [12] and Kateb et al [37] for the Helmholtz equation and of Chaulet et al [19] for the Maxwell's equations.…”

Section: Introduction and General Notationsmentioning

confidence: 99%

Shape Sensitivity Analysis for Elastic Structures with Generalized Impedance Boundary Conditions of the Wentzell Type—Application to Compliance Minimization

Caubet

Kateb

Louër

2018

J Elast

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To cite this version:Fabien Caubet, Djalil Kateb, Frédérique Le Louër. Shape sensitivity analysis for elastic structures with generalized impedance boundary conditions of the Wentzell type -Application to compliance minimization. Journal of Elasticity, Springer Verlag, In press, Journal of Elasticity. Abstract This paper focuses on Generalized Impedance Boundary Conditions (GIBC) with second order derivatives in the context of linear elasticity and general curved interfaces. A condition of the Wentzell type modeling thin layer coatings on some elastic structure is obtained through an asymptotic analysis of order one of the transmission problem at the thin layer interfaces with respect to the thickness parameter. We prove the well-posedness of the approximate problem and the theoretical quadratic accuracy of the boundary conditions. Then we perform a shape sensitivity analysis of the GIBC model in order to study a shape optimization/optimal design problem. We prove the existence and characterize the first shape derivative of this model. A comparison with the asymptotic expansion of the first shape derivative associated to the original thin layer transmission problem shows that we can interchange the asymptotic and shape derivative analysis. Finally we apply these results to the compliance minimization problem. We compute the shape derivative of the compliance in this context and present some numerical simulations.

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Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions

Cited by 21 publications

References 44 publications

An extremal eigenvalue problem for the Wentzell–Laplace operator

An extremal eigenvalue problem for the Wentzell–Laplace operator

On the data completion problem and the inverse obstacle problem with partial Cauchy data for Laplace’s equation

Shape Sensitivity Analysis for Elastic Structures with Generalized Impedance Boundary Conditions of the Wentzell Type—Application to Compliance Minimization

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