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This work presents an efficient method based on asymptotic models in order to solve numerically eddy current problems in a bi‐dimensional setting with a high contrast of linear relative magnetic permeability μr ≫ 1 between a conductor and a dielectric subdomain. We describe a magnetic skin effect by deriving a multiscale expansion for the magnetic potential in power series of a small parameter δ which represents the skin depth. We make explicit the first asymptotics up to the order three. Some numerical results based on the finite element method will be presented to illustrate the magnetic skin effect and to validate the performance of the proposed asymptotic models in the dielectric medium. We confirm that the proposed asymptotics provide reduced computational costs for a wide range of physical parameters introduced in our problem. [ABSTRACT FROM AUTHOR]

This work is concerned with the time-harmonic eddy current problem for a medium with a highly conductive thin sheet. We present asymptotic models and impedance conditions up to the second order of approximation for the electromagnetic field. The conditions are derived asymptotically for vanishing sheet thickness ε, where the skin depth is scaled like ε. The first order condition is the perfect electric conductor boundary condition. The second order condition is a Poincaré-Steklov map between tangential components of the magnetic field and the electric field. [ABSTRACT FROM AUTHOR]

We present equivalent conditions and asymptotic models for a diffraction problem of acoustic and elastic waves. The mathematical problem is set with a Robin boundary condition. Elastic and acoustic waves propagate in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. This approach leads to solve only elastic equations. We derive and validate equivalent conditions up to the third order for the elastic displacement. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem. [ABSTRACT FROM AUTHOR]

In this paper, one considers the coupling of a Brinkman model and Stokes equations with jump embedded transmission conditions. In this model, one assumes that the viscosity in the porous region is very small. Then we derive a Wentzel- Kramers-Brillouin (WKB) expansion in power series of the square root of this small parameter for the velocity and the pressure which are solution of the transmission problem. This WKB expansion is justified rigorously by proving uniform errors estimates. [ABSTRACT FROM AUTHOR]

The modeling of eddy currents in conductors is split into a sequence of progressive finite element subproblems. The source fields generated by the inductors alone are calculated at first via either the Biot-Savart law or finite elements. The associated reaction fields for each added conductive region, and in return for the source regions themselves when massive, are then calculated with finite element models, possibly with initial perfect conductor and/or impedance boundary conditions to be further corrected. The resulting subproblem method allows efficient solving of parameterized analyses thanks to a proper mesh for each subproblem and the reuse of previous solutions to be locally corrected. [ABSTRACT FROM AUTHOR]

We present equivalent conditions and asymptotic models for the diffraction problem of elastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. We derive and validate equivalent conditions up to the fourth order for the elastic displacement. These conditions approximate the acoustic waves which propagate in the fluid region. This approach leads to solve only elastic equations. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem. [ABSTRACT FROM PUBLISHER]

In this paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner, and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific nonstandard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials, and further terms are genuine nonsmooth functions generated by the piecewise constant zeroth order term of the operator. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: We study a three-dimensional model for the skin effect in electromagnetism. We first give a multiscale asymptotic expansion for the solution of the harmonic Maxwell equations set on a domain made of two materials, dielectric and highly conducting, with a regular interface between them. To measure the skin effect, we introduce a suitable skin depth function defined on the interface and generalizing the classical scalar quantity. We then prove an asymptotic expansion at high conductivity for this function, which exhibits the influence of the geometry of the interface on the skin depth. [Copyright &y& Elsevier]