Approximate transmission conditions through a weakly oscillating thin layer

Poignard, Clair

doi:10.1002/mma.1045

Cited by 19 publications

(47 citation statements)

References 16 publications

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“…We build approximate transmission conditions in order to replace the rough thin layer by these conditions on the boundary of the interior material. This paper extends previous works [8,7] of the third author, in which the layer had constant or weakly oscillating thickness. …”

supporting

confidence: 88%

“…Then, for any domain ω 0 and ω 1 respectively compactly embedded in Ω \ D 1 and in D 1 and for any s ∈]1, 2] there exists ε 0 > 0 and C > 0 such that for any ε ∈ (0, ε 0 ): [7]. Actually if no oscillation occurs in the thin layer, i.e.…”

Section: Inriamentioning

confidence: 99%

“…In this paper we are definitely interested in approximated transmission conditions, therefore we cannot apply straightforward the previous results to our problem. In a previous article [7], the third author derives the first order approximate boundary condition for a thin weakly oscillating layer. This paper is an extension of [7] to the case of an ε-periodic thin layer with thickness of order ε.…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…In a previous article [7], the third author derives the first order approximate boundary condition for a thin weakly oscillating layer. This paper is an extension of [7] to the case of an ε-periodic thin layer with thickness of order ε. The main idea of the analysis comes from the paper of Abboud et Ammari [1] and is completely different from the analysis of [7], which was based on an appropriate change of variables.…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…This paper is an extension of [7] to the case of an ε-periodic thin layer with thickness of order ε. The main idea of the analysis comes from the paper of Abboud et Ammari [1] and is completely different from the analysis of [7], which was based on an appropriate change of variables. Here the boundary layer corrector are obtained by solving an elliptic partial differential equation in an appropriate infinite band with width equal to 1.…”

Section: Statement Of the Problemmentioning

confidence: 99%

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Approximate transmission conditions through a rough thin layer: The case of periodic roughness

Ciuperca

Jai

Poignard³

2009

Eur. J. Appl. Math

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Abstract:We study the behavior of the steady-state voltage potentials in a material composed by a bidimensional medium surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be ε-periodic, ε beeing the small thickness of the layer. We build approximate transmission conditions in order to replace the rough thin layer by these conditions on the boundary of the interior material. This paper extends previous works [8,7] of the third author, in which the layer had constant or weakly oscillating thickness.

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supporting

confidence: 88%

Section: Inriamentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

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Approximate transmission conditions through a rough thin layer: The case of periodic roughness

Ciuperca

Jai

Poignard³

2009

Eur. J. Appl. Math

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Asymptotic modelling of conductive thin sheets

Schmidt

Tordeux

2009

Z. Angew. Math. Phys.

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We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H 1 -modelling error for an expansion with N terms is bounded by O(ε N +1 ) in the exterior of the sheet and by O(ε N +1/2 ) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results. Mathematics Subject Classification IntroductionMany electric devices contain very thin conducting parts either for electromagnetic shielding [13,16], or as casings, tank walls [9,25] or supply lines [5]. The large aspect ratio of these sheets of about few millimetres or centimetres to metres or hundreds of micrometres to centimetres and the high conductivity causes variations in thickness direction in much smaller scales than in the longitudinal directions. Their discretisation by the finite element method (FEM) is challenging when the thickness ε of the thin sheets is considerably smaller than the size of neighbouring parts for three reasons. First, domains with such thin sheets are difficult to mesh by most mesh generators. Secondly, a discretisation on meshes with cell sizes of different magnitudes can lead to ill-conditioned matrices, and thirdly, meshes of good quality may also contain cells around the sheet with sizes comparable to the sheet thickness which leads to a high number of additional degrees of freedom. By reducing the thin sheet to an interface and by approximating its effect by conditions on this interface, a highly accurate modelling with standard discretisation schemes like the FEM is possible.The so called impedance boundary conditions (IBCs), first proposed by Shchukin [29] and Leontovich [19], are traditionally used for replacing solid conductors, where the domain is artificially confined, by an approximate boundary condition [1][2][3]8,11,15,28]. This technique is proved to be accurate for smooth sheets and can be readily implemented.However, in the context of thin conducting sheets this technique of Shchukin and Leontovich has been seldom applied. Interface conditions for thin sheets are often based on a tensor product ansatz of a set of simple functions in thickness direction and functions defined on the interface. The simplest approaches assume no variation in thickness direction, which leads to a surface quantity [5,22]. Using two functions in thickness direction Krähenbühl and Muller [18] derived a relation between the mean value of the tangential component of the electric or magnetic field on the interfaces of the sheet and the jump of the magnetic or elec...

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On Ventcel-type transmission conditions for a Helmholtz problem with a non-uniform thin layer

Boutarene,

Galleze,

Péron

2024

SeMA

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We study the asymptotic behaviour of the electric field in the transverse magnetic (TM) mode, propagating in a material composed of a two-dimensional object surrounded by a thin layer of non-constant thickness and embedded in an ambient medium. Using an asymptotic expansion of the solution u ε to the Helmholtz problem, we derive Ventcel-type transmission conditions on the limit interface Γ modelling the effect of the thin layer with accuracy up to O(ε 2 ).

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Approximate transmission conditions through a weakly oscillating thin layer

Cited by 19 publications

References 16 publications

Approximate transmission conditions through a rough thin layer: The case of periodic roughness

Approximate transmission conditions through a rough thin layer: The case of periodic roughness

Asymptotic modelling of conductive thin sheets

On Ventcel-type transmission conditions for a Helmholtz problem with a non-uniform thin layer

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