Influence of a Rough Thin Layer on the Potential

Ciuperca, Sorin Ionel; Perrussel, Ronan; Poignard, Clair

doi:10.1109/tmag.2010.2044027

Cited by 3 publications

(3 citation statements)

References 4 publications

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“…4 demonstrates that the convergence rate decreases dramatically for β = 1. This is in accordance with the theory, since the approximate transmission conditions for β = 1 given in [11,12] are very different from the conditions proved in the present paper. Figure 3: Error in the cytoplasm vs ε β for three approximate solutions.…”

Section: Resultssupporting

confidence: 87%

Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

Ciuperca¹,

Perrussel²,

Poignard³

2011

Journal de Mathématiques Pures et Appliquées

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International audienceWe study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. The roughness of the layer is described by a quasi $\eps$--periodic function, $\eps$ being a small parameter, while the mean thickness of the layer is of magnitude $\eps^\beta$, where $\beta\in(0,1)$. Using the two-scale analysis, we replace the very rough thin layer by appropriate transmission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor of Vogelius and Capdeboscq (ESAIM:M2AN. 2003; 37:159-173). This paper extends the previous works Poignard (Math. Meth. App. Sci. 2009; 32:435-453) and Ciuperca \textsl{et al.} (Research report INRIA RR-6812), in which $\beta\geq1$

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Section: Resultssupporting

confidence: 87%

Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

Ciuperca¹,

Perrussel²,

Poignard³

2011

Journal de Mathématiques Pures et Appliquées

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“…For α = 1, the polarization tensor is a full 2 × 2 matrix, while for α > 1, it is diagonal again, but the matrix coefficients involve a mixture of the conductivities (see (45)). Numerical results have been published in [12].…”

Section: Resultsmentioning

confidence: 99%

Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem

Poignard

2012

ESAIM: Proc.

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We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be ε α-periodic, ε being the magnitude of the mean thickness of the layer, and α a positive parameter describing the degree of roughness. For ε tending to zero, we determine the appropriate boundary layer correctors which lead to approximate transmission conditions equivalent to the effect of the rough thin layer. We also provide an explicit characterization of the polarization tensor as defined by Capdeboscq and Vogelius in [9]. The present paper revisits the previous works of the author [11, 13, 16, 17], and it also provides new results for the very rough case α > 1. Résumé. Dans cet article, nous considérons le problème de conduction dans un domaine bidimensionnel composé d'une fine membrane rugueuse entourant un domaine conducteur, le tout plongé dans un milieu ambiant de conductivité différente. La rugosité de la membrane est supposée ε α-périodique, εétant l'épaisseur moyenne de la membrane, et α un paramètre positif décrivant le degré de rugosité. Nous déterminons des correcteurs de couche limite conduisantà la construction de conditions de transmission approchées lorsque le paramètre ε tend vers zero. Nous donnons aussi une caractérisation explicite du tenseur de polarisation défini par Capdeboscq and Vogelius dans [9]. Cet article revisite des résultats précédents de l'auteur obtenus dans [11,13,16,17], et présente de nouveaux résultats pour le cas très rugueux α > 1.

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“…The theoretical proof of the existence and uniqueness of V α as well as the justification that r δ α is of order δ need more than 4 pages, and will be presented in a forthcoming paper. Capturing the singularity of the domain in a profile term is quite natural and has to be linked up similarly to [6], [7].…”

Section: Heuristics Of the Multiscale Expansionmentioning

confidence: 99%

Eddy Currents and Corner Singularities

et al. 2012

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Eddy current problems are addressed in a bidimensional setting where the conducting medium is non-magnetic and has a corner singularity. For any fixed parameter δ linked to the skin depth for a plane interface, we show that the flux density |∇A δ | is bounded near the corner unlike the perfect conducting case. Then as δ goes to zero, the first two terms of a multiscale expansion of the magnetic potential are introduced to tackle the magneto-harmonic problem. The heuristics of the method are given and numerical computations illustrate the obtained accuracy.

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Influence of a Rough Thin Layer on the Potential

Cited by 3 publications

References 4 publications

Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem

Eddy Currents and Corner Singularities

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