Asymptotic expansion for the magnetic potential in the eddy-current problem

Ronan Perrussel (1) , Clair Poignard (2, 3) , Victor Péron (4, 5) , Ruth Sabariego (6) , Patrick Dular (7) , Laurent Krähenbühl (8)

1 LAPLACE-GRE - Groupe de Recherche en Electromagnétisme
2 IMB - Institut de Mathématiques de Bordeaux
3 MONC - Modélisation Mathématique pour l'Oncologie
4 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
5 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
6 KU Leuven - Catholic University of Leuven = Katholieke Universiteit Leuven
7 ACE - Applied and Computational Electromagnetics [Liège]
8 MIS - Ampère, Département Méthodes pour l'Ingénierie des Systèmes

Asymptotics consist in formal series of the solution to a problem which involves a small parameter. When truncated at a certain order, this finite sum provides an approximation of the exact solution with a given accuracy, and the coefficients of this sum are solutions to elementary problems that do not depend on the small parameter. This parameter can be for instance the thickness of the domain or a small or high conductivity coefficient. The asymptotic expansion is a useful tool to obtain approximate expressions of the solution to the so-called Eddy Current problem, which describes the magnetic potential in a material composed by a dielectric material surrounding a conductor.However such expansions are derivatives consuming, in the sense that to go further in the expansion, it is necessary to compute the higher derivatives of the first orders terms, and it also requires a precise knowledge of the geometry, since derivatives of the parameterization of the interface dielectric/conductor are involved. From the numerical point of view, this can lead to instabilities which may restrict or prevent a direct use of the asymptotic expansion. This mathematical approach complements our previous works on “deltaparametrization”.In particular, we will show that several expansions can be involved when considering magnetic conductors depending on the product of the relative permeability of the conductor by the penetration depth. As an example, for the same geometry, boundary conditions and penetration depth we obtain two “very” distinct behaviours (left, f = 10kHz and mur = 1 and right, f = 10Hz and mur = 1000).

Domaines: Modélisation et simulation

Ronan Perrussel, Clair Poignard, Victor Péron, Ruth Sabariego, Patrick Dular, et al.. Asymptotic expansion for the magnetic potential in the eddy-current problem. 10th EMF, Apr 2016, Lyon, France. ⟨hal-01393362⟩