On Shape Optimization for an Evolution Coupled System

Leugering, Günter; Novotny, Antônio André; Menzala, G. Perla; Sokołowski, Jan

doi:10.1007/s00245-011-9148-7

Cited by 8 publications

(9 citation statements)

References 11 publications

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“…It follows from Remark 1 applied to problem (59) with respect to the difference u, and from (13), (18) and (54) that u satisfies the estimate (57). We set λ = λ 1 in the inequality (46) written atλ = λ 2 ,û = u 2 , p = p 2 and then set λ = λ 2 in (46) written atλ = λ 1 ,û = u 1 , p = p 1 .…”

Section: Uniqueness and Stability Of Solutions Of Control Problemsmentioning

confidence: 84%

“…Just this idea is used in this paper. (The same goal can be achieved, by shape optimization of the boundary, we refer the reader to [18] for the related results on shape sensitivity analysis for the coupled models). Moreover, unlike cited papers, the cloaking effect in the paper is achieved due choice of surface impedance λ entering into the second boundary condition in (2).…”

Section: Introduction Statement Of Boundary Problemmentioning

confidence: 88%

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Cloaking via impedance boundary condition for the 2-D Helmholtz equation

Алексеев

2013

Applicable Analysis

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We consider control problems for the 2-D Helmholtz equation in an unbounded domain with partially coated boundary. Dirichlet boundary condition is given on one part of the boundary and the impedance boundary condition is imposed on another its part. The role of control in control problem under study is played by boundary impedance. Quadratic tracking-type functionals for the field play the role of cost functionals. Solvability of control problems is proved. The uniqueness and stability of optimal solutions with respect to certain perturbations of both cost functional and incident field are established.

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Section: Uniqueness and Stability Of Solutions Of Control Problemsmentioning

confidence: 84%

Section: Introduction Statement Of Boundary Problemmentioning

confidence: 88%

Cloaking via impedance boundary condition for the 2-D Helmholtz equation

Алексеев

2013

Applicable Analysis

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“…Relation (4.5) can be used to compute the shape derivative from the simpler material derivative, see [18] for an application. In this terminology, the derivatives of boundary functions and operators in (i) and (ii) above would be analogous to material derivatives, whereas the derivatives of potentials in (iii) correspond to shape derivatives.…”

Section: Shape Differentiability Of Boundary Integral Operatorsmentioning

confidence: 99%

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part I: Shape Differentiability of Pseudo-homogeneous Boundary Integral Operators

Costabel

Louër

2012

Integr. Equ. Oper. Theory

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Abstract. In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators.

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“…One also talks about "Eulerian derivative". The difference between the two has the form of a convection term, which can lead to a loss of one order of regularity of the shape derivative on the support of the velocity field [25]. In the context of this terminology, the derivatives of the boundary integral operators that we studied in Part I of this paper [9] for general pseudohomogeneous kernels and in this paper for the boundary integral operators of electromagnetism, correspond to the Lagrangian point of view, because they are obtained by pull-back to a fixed reference boundary.…”

Section: Introductionmentioning

confidence: 99%

“…73 (2012) Shape Derivatives II: Dielectric Scattering 25 We can now define the main boundary integral operators:…”

mentioning

confidence: 99%

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

Costabel

Louër

2012

Integr. Equ. Oper. Theory

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We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity TH − 1 2 (divΓ, Γ). Using Helmholtz decomposition, we can base their analysis on the study of pseudodifferential integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.

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On Shape Optimization for an Evolution Coupled System

Cited by 8 publications

References 11 publications

Cloaking via impedance boundary condition for the 2-D Helmholtz equation

Cloaking via impedance boundary condition for the 2-D Helmholtz equation

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part I: Shape Differentiability of Pseudo-homogeneous Boundary Integral Operators

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

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