Asymptotic Models and Impedance Conditions for Highly Conductive Sheets in the Time-Harmonic Eddy Current Model

International audienceThis 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 $\varepsilon$ where the skin depth is scaled like $\varepsilon$. The first order condition is the perfect electric conductor boundary condition. The second order condition turns out to be a Poincaré-Steklov map between tangential components of the magnetic field and the electric field

Overview on a selection of recent works in asymptotic analysis for wave propagation problems

We give a brief survey of some recent advances of asymptotic analysis methods applied to wave propagation problem

A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model

We introduce tools for a unified analysis and a comparison of impedance transmission conditions (ITCs) for thin conducting sheets within the time-harmonic eddy current model in two dimensions. The first criterion is the robustness with respect to the frequency or skin depth, that means if they give meaningful results for small and for large frequencies or conductivities. As a second tool we study the accuracy for a range of sheet thicknesses and frequencies for a relevant example, and analyse finally their asymptotic order in different asymptotic regimes. For the latter we write all the ITCs in a common form and show how they can be realised within the finite element method. Two new conditions which we call ITC-2-0 and ITC-2-1 are introduced in this article which appear in a symmetric form. They are derived by asymptotic expansions in the asymptotic regime of constant ratio between skin depth and thickness like those in [26]. We analyse these ITCs in comparison with the often used perfect electric boundary condition, the shielding element by Nakata et.al. [23], the thin layer impedance boundary conditions by Mayergoyz and Bedrosian [22] and a family of ITCs derived by asymptotic expansions in the asymptotic regime of constant shielding by Schmidt and Tordeux [28]. Our analysis shows the superiority of the transmission conditions derived by asymptotic expansions where especially the worst-case error level of the ITC-2-1 is remarkably lower than for all the other conditions

Boundary element method for 3D conductive thin layer in eddy current problems

International audiencePurpose-Thin conducting sheets are used in many electric and electronic devices. Solving numerically the eddy current problems in presence of these thin conductive sheets requires a very fine mesh which leads to a large system of equations, and it becomes more problematic in case of higher frequencies. The purpose of this paper is to show the numerical pertinence of equivalent models for 3D eddy current problems with a conductive thin layer of small thickness e based on the replacement of the thin layer by its mid-surface with equivalent transmission conditions that satisfy the shielding purpose, and by using an efficient discretization using the Boundary Element Method (BEM) in order to reduce the computational work. Design/methodology/approach-These models are solved numerically using the BEM and some numerical experiments are performed to assess the accuracy of our models. The results are validated by comparison with an analytical solution and a numerical solution by the commercial software Comsol. Findings-The error between the equivalent models and the analytical and numerical solutions confirms the theoretical approach. In addition to this accuracy, the time consumption is reduced by considering a discretization method that requires only a surface mesh. Originality/Value-Based on an hybrid formulation, we present briefly a formal derivation of impedance transmission conditions for 3D thin layers in eddy current problems where non-conductive materials are considered in the interior and the exterior domain of the sheet. BEM is adopted to discretize the problem as there is no need for a volume discretization

Impedance Transmission Conditions for the Electric Potential across a Highly Conductive Casing

Borehole resistivity measurements are a common procedure when trying to obtaina better characterization of the Earth's subsurface. The use of a casing surrounding the boreholehighly complicates the numerical simulations due to a large contrast between the conductivities ofthe casing and the rock formations. In this work, we consider the casing to be a thin layer of uniformthickness and motivated by realistic scenarios, we assume that the conductivity of such casing isproportional to the thickness of the casing to the power of -3. We derive Impedance TransmissionConditions (ITCs) for the static (zero frequency) electric potential for a 2D configuration. Then,we analyse these models by proving stability and convergence results. Next, we asses the numericalperformance of these models by employing a Finite Element Method. Finally we present presentasymptotic models for similar configurations including the time-harmonic configuration and a 3Daxisymmetric scenario.Les mesures de résistivité en forage sont communément utilisées pour obtenirune meilleure caractérisation du sous-sol de la Terre. Pour obtenir de telles mesures, on utilisetypiquement un tube métallique qui protège le forage, mais cela complique énormément la sim-ulation numérique à cause du fort contraste entre les conductivités du tube et des formationsrocheuses. Dans ce travail , motivé par des configurations réalistes, on considère que la con-ductivité du tube est proportionnelle à l'épaisseur du tube à la puissance -3. On développe desconditions de transmission d'impédance (ITCs en Anglais) pour le potentiel électrique dans lecas statique, dans un domaine bidimensionnel. On présente la construction des modèles asymp-totiques, validés par des résultats de convergence. On illustre les résultats théoriques avec dessimulations numériques obtenues en utilisant une discrétisation par éléments finis. On présenteaussi des modèles asymptotiques pour d'autres problèmes et configurations, à fréquence non-nulleet en 3D

Modélisation par sources ponctuelles multipolaires équivalentes de petites sphères pour le calcul rapide et précis de la diffraction des ondes électromagnétiques

International audienc

Asymptotic Models for the Electric Potential across a Highly Conductive Casing

We analyze a configuration that involves a steel-cased borehole, where the casing that covers the borehole is considered as a highly conductive thin layer. We develop an asymptotic method for deriving reduced problems capable of efficiently dealing with the numerical difficulties caused by the casing when applying traditional numerical methods. We derive several reduced models by employing two different approaches, each of them leading to different classes of models. We prove stability and convergence results for these models. The theoretical orders of convergence are supported by numerical results obtained with the finite element method

Fast Magnetic Flux Line Allocation Algorithm for Interactive Visualization Using Magnetic Flux Line Existence Probability

The visualization of magnetic flux lines is one of the most effective ways to intuitively grasp a magnetic field. The depiction of continuous and smooth magnetic flux lines according to the magnetic field is of paramount importance. Thus, it is important to adequately allocate the distribution of magnetic flux lines in the analyzed space. The authors have already proposed two methods of determining the allocation of magnetic flux lines in 3-D space. However, both methods exhibited a long computation time to determine the allocation of magnetic flux lines. For solving this problem, in this paper, we propose a new improved method for correct allocation of magnetic flux lines in 3-D space with modest computational cost. The main advantages of this method are shorter computation time, correct allocation of the magnetic flux lines, and especially short computation time for visualization of magnetic flux lines when changes in the number of depicted flux lines is requested