Skin-Effect Description in Electromagnetism with a Scaled Asymptotic Expansion (original) (raw)
Related papers
Skin Effect in Electromagnetism and Asymptotic Behaviour of Skin Depth for High Conductivity
We study a three-dimensional model for the skin effect in electromagnetism. The 3-D case of the Maxwell equations in harmonic regime on a domain composed of a dielectric and of a highly conducting material is considered. We derive an asymptotic expansion with respect to a small parameter related to high conductivity. This expansion is theoretically justified at any order. The asymptotic expansion and numerical simulations in axisymmetric geometry exhibit the influence of the geometry of the interface on the skin effect.
Approximate transmission conditions for time-harmonic Maxwell equations in a domain with thin layer
We study the behavior of the electromagnetic field in a biological cell modelized by a medium surrounded by a thin layer and embedded in an ambient medium. We derive approximate transmission conditions in order to replace the membrane by these conditions on the boundary of the interior domain. Our approach is essentially geometric and based on a suitable change of variables in the thin layer. Few notions of differential calculus are given in order to obtain our asymptotic conditions in a simple way. This paper extends to time-harmonic Maxwell equations the previous works presented in . Asymptotic transmission conditions at any order are given in Appendix 1. Conditions de transmission approchées pour leś equations de Maxwell en régime harmonique dans un milieuà couche mince Résumé : Nousétudions le comportement asymptotique du champélectromagnétique dans une cellule biologique plongée dans un milieu ambiant. La cellule est composée d'un cytoplasme entouré d'une fine membrane. Nous obtenons des conditions de transmission sur le bord du cytoplasmeéquivalentesà la couche mince. Notre approche est essentiellement géométrique et basée sur un changement de variables adéquat dans la couche mince. Quelques notions de calcul différentiel sont rappelées afin d'obtenir directement notre développement asymptotique. Par ailleurs des estimations d'erreur sont démontrées. En appendice, nous présentons le développement asymptotiqueà tout ordre. 2 Using the notations of the electrical engineeering community, q = ω 2`ǫ − i σ ω´, where ω is the frequency, ǫ the permittivity and σ the conductivity of the domain [3]. INRIA inria-00347971, version 3 -14 Sep 2009 Péron& Poignard where
Perturbation theory for Maxwell's equations with shifting material boundaries
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
Perturbation theory permits the analytic study of small changes on known solutions, and is especially useful in electromagnetism for understanding weak interactions and imperfections. Standard perturbation-theory techniques, however, have difficulties when applied to Maxwell's equations for small shifts in dielectric interfaces (especially in high-index-contrast, three-dimensional systems) due to the discontinuous field boundary conditions--in fact, the usual methods fail even to predict the lowest-order behavior. By considering a sharp boundary as a limit of anisotropically smoothed systems, we are able to derive a correct first-order perturbation theory and mode-coupling constants, involving only surface integrals of the unperturbed fields over the perturbed interface. In addition, we discuss further considerations that arise for higher-order perturbative methods in electromagnetism.
Asymptotics Methods to Compute Electromagnetic Fields
2007
We provide a rigorous asymptotic method to compute the electromagnetic fields in domains with thin layer. With this method the influency of the membrane on the inner field is replaced by an approximated boundary condition, while in the thin layer, the approximated field is explicitely known. We give error estimates, which validate our asymptotic.
Communications in Partial Differential Equations, 1999
This paper is dedicated to the construction and analysis of so-called Generalized Impedance Boundary Conditions (GIBCs) for electromagnetic scattering problems from imperfect conductors with smooth boundaries. These boundary conditions can be seen as higher order approximations of a perfect conductor condition. We consider here the 3-D case with Maxwell equations in a harmonic regime. The construction of GIBCs is based on a scaled asymptotic expansion with respect to the skin depth. The asymptotic expansion is theoretically justified at any order and we give explicit expressions till the third order. These expressions are used to derive the GIBCs. The associated boundary value problem is analyzed and error estimates are obtained in terms of the skin depth.
The Conductive Boundary Condition for Maxwell’s Equations
SIAM Journal on Applied Mathematics, 1992
The transmission boundary value problem for a perturbed Dirac operator on arbitrary bounded Lipschitz domains in R 3 is formulated and solved in terms of layer potentials of Clifford-Cauchy type. As a byproduct of this analysis, an elliptization procedure for the Maxwell system is devised which allows us to show that the Maxwell and Helmholtz transmission boundary value problems are well-posed as a corollary of the unique solvability of this more general Dirac transmission problem.
The conductive problem for Maxwell's equations at low frequencies
Applied Mathematics Letters, 1997
We study the scattering problem in the case where a low frequency plane electromagnetic wave is incident upon a body covered by a thin layer of very high conductivity. We develop convergent series solutions in powers of the wavenumber for the electric field, and the scattering amplitude. Moreover, we evaluate the leading term of the scattering amplitude and of the scattering croes-section.
Asymptotic expansion of the solution of Maxwell's equations in polygonal plane domains
Abstract:,This paper is mainly,concerned,with the structure of the singular and regular parts of the solution of time-harmonic,Maxwell’s equations in polygonal,plane domains. The asymptotic,behaviour,of the solution near corner points of the domain is studied by means,of discrete Fourier transformation. A detailed functional analysis of the solution shows that the boundary,value problem,does not belong locally to HDepartment of Mathematics, Faculty of Science, University of Buea, Cameroon
Asymptotic Approximation of the Dirichlet to Neumann Map of High Contrast Conductive Media
Multiscale Modeling & Simulation, 2014
We present an asymptotic study of the Dirichlet to Neumann map of high contrast composite media with perfectly conducting inclusions that are close to touching. The result is an explicit characterization of the map in the asymptotic limit of the distance between the particles tending to zero. 1 where σ o is a reference constant conductivity, S(x) is a smooth function with nondegenerate critical points, and ǫ ≪ 1 models the high contrast. An advantage of the model (1.3) is that instead of specializing the analysis in the gaps to various shapes of the inclusions, we can study a generic problem in the vicinity of saddle points of the function S(x).