Detecting perfectly insulated obstacles by shape optimization techniques of order two

Afraites, Lekbir; Dambrine, Marc; Eppler, Karsten; Kateb, Djalil

doi:10.3934/dcdsb.2007.8.389

Cited by 23 publications

(42 citation statements)

References 18 publications

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“…Among all of them, shape optimization methods (see e.g. [3,18,32]) present interesting features: in particular, they are easily adaptable to problem governed by a different partial differential equation, such as Stokes system (see e.g. [5,20,22,33]), and obstacle characterized by different limit conditions, such as Neumann or generalized boundary conditions (see e.g.…”

Section: Page 87])mentioning

confidence: 99%

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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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Section: Page 87])mentioning

confidence: 99%

“…We do so through the minimization of a Kohn-Vogelius functional, which will have both the shape of ω * and the unknown data as variable, since such type of functional has already been used successfully to solve obstacle problems and data completion problem separately (see e.g. [6,3]).…”

Section: Remark 12mentioning

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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“…This method is used e.g. in the work [1] of Afraites et al (where a regularization by parameterization is used). Note that the standard algorithm based on shape derivatives moving the mesh does not provide the opportunity to change the topology of the shape and consequently the number of inclusions has to be known in advance.…”

Section: Flip Procedures In Approximation Of Multiple-component Shapesmentioning

confidence: 99%

Flip procedure in geometric approximation of multiple-component shapes – Application to multiple-inclusion detection

Bonnelie¹,

Bourdin²,

Caubet³

et al. 2016

The SMAI Journal of Computational Mathematics

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Abstract. We are interested in geometric approximation by parameterization of two-dimensional multiplecomponent shapes, in particular when the number of components is a priori unknown. Starting a standard method based on successive shape deformations with a one-component initial shape in order to approximate a multiple-component target shape usually leads the deformation flow to make the boundary evolve until it surrounds all the components of the target shape. This classical phenomenon tends to create double points on the boundary of the approximated shape.In order to improve the approximation of multiple-component shapes (without any knowledge on the number of components in advance), we use in this paper a piecewise Bézier parameterization and we consider two procedures called intersecting control polygons detection and flip procedure. The first one allows to prevent potential collisions between two parts of the boundary of the approximated shape, and the second one permits to change its topology by dividing a one-component shape into a two-component shape.For an experimental purpose, we include these two processes in a basic geometrical shape optimization algorithm and test it on the classical inverse obstacle problem. This new approach allows to obtain a numerical approximation of the unknown inclusion, detecting both the topology (i.e. the number of connected components) and the shape of the obstacle. Several numerical simulations are performed.Math. classification. 68U05, 68W25, 49Q10, 65N21.

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“…Detailed studies including regularity estimates showed that if the domain is C 2,α and the perturbation is a C 2,α diffeomorphism then the Hessian is coercive at the minimum, but with respect to a weaker norm [12,11]. The detection of obstacles for the conductivity equation using second order shape optimization was proposed in a series of work [15,1,2] under various hypotheses regarding the obtacle: perfectly insulating, perfectly conducting, having a different conductivity. Another series of work adressed electromagnetic shaping, and the explicit computation of the Hessian that is affordable using a boundary integral method to estimate the solutions of the equation [22,24].…”

Section: Introductionmentioning

confidence: 99%

Shape optimizationviaa levelset and a Gauss-Newton method

Fehrenbach

Gournay

2019

ESAIM: COCV

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In the context of shape optimization via level-set methods, we propose a general framework for a Gauss-Newton method to optimize quadratic functionals. Our approach provides a natural extension of the shape derivative as a vector field defined in the whole working domain. We implement and discuss this method in two cases: first a least-square error minimization reminiscent of the Electrical Impedance Tomography problem, and second the compliance problem with volume constraints.

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Detecting perfectly insulated obstacles by shape optimization techniques of order two

Cited by 23 publications

References 18 publications

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

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

Flip procedure in geometric approximation of multiple-component shapes – Application to multiple-inclusion detection

Shape optimizationviaa levelset and a Gauss-Newton method

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