Analysis of eddy current formulations in two-dimensional domains with cracks

Lohrengel, Stephanie; Nicaise, Serge

doi:10.1051/m2an/2014027

Lohrengel, Stephanie ¹ ; Nicaise, Serge ²

¹ Laboratoire de Mathématiques, EA 4535, FR CNRS 3399, UFR Sciences Exactes et Naturelles, Université de Reims Champagne-Ardenne, Moulin de la Housse - B.P. 1039, 51687 Reims cedex 2, France.
² LAMAV, EA 4015, FR CNRS 2956, Université de Valenciennes et du Hainaut Cambrésis, Le Mont Houy, 59313 Valenciennes cedex 9, France.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 141-170.

Résumé

In this paper, the eddy current problem in a two-dimensional conductor containing a crack is studied. The decomposition of the electric field into a piecewise regular part and a singular part deriving from scalar potentials localized at the crack tip and at the crack mouth is proved. At the crack mouth, the electric field is shown to have standard singularities inside the conductor, but presents a singularity outside the conductor that does not belong to the classical $L^{2}$ -space. Well-posedness of the $𝐄$ -based model and the $𝐀 - ψ$ -formulation of combined potentials are proved and an un-gauged discretization of the latter formulation is discussed.

Reçu le : 2013-11-07

MR Zbl

DOI : 10.1051/m2an/2014027

Classification : 35B65, 35Q60, 65N30
Mots clés : Eddy current problems, domains with cracks, singularities of solutions

Affiliations des auteurs :

Lohrengel, Stephanie ¹ ; Nicaise, Serge ²

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     title = {Analysis of eddy current formulations in two-dimensional domains with cracks},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {141--170},
     publisher = {EDP-Sciences},
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     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2014027/}
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Lohrengel, Stephanie; Nicaise, Serge. Analysis of eddy current formulations in two-dimensional domains with cracks. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 141-170. doi : 10.1051/m2an/2014027. http://www.numdam.org/articles/10.1051/m2an/2014027/

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