Corner asymptotics; Calculus of singular functions; Magnetic potential; Eddy current problem
Abstract :
[en] In this paper, we describe the scalar 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 non-standard 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 non-smooth functions generated by the piecewise constant zeroth order term of the operator.
Disciplines :
Mathematics
Author, co-author :
Dauge, Monique; IRMAR CNRS UMR6625, Rennes, France
Dular, Patrick ; Université de Liège - ULiège > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Applied and Computational Electromagnetics (ACE)
Krähenbühl, Laurent; Laboratoire Ampère CNRS UMR5005, Lyon, France
Péron, Victor; LMAP CNRS UMR5142 & Team M3D INRIA, Université de Pau, Pau, France
Perrussel, Ronan; LAPLACE CNRS UMR5213, Toulouse, France
Poignard, Clair; Team MC2, INRIA Bordeaux-Sud-Ouest & CNRS UMR 5251, Bordeaux, France
Language :
English
Title :
Corner asymptotics of the magnetic potential in the eddy-current model
Publication date :
July 2014
Journal title :
Mathematical Methods in the Applied Sciences
ISSN :
0170-4214
eISSN :
1099-1476
Publisher :
John Wiley & Sons, Hoboken, United States - New York
Leontovich MA,. Approximate boundary conditions for the electromagnetic field on the surface of a good conductor. In Investigations on radiowave propagation, Vol. 2. Printing House of the USSR Academy of Sciences: Moscow, 1948; 5-12.
Rytov SM,. Calcul du skin effect par la méthode des perturbations. Journal of Physics 1940; 11 (3): 233-242.
Haddar H, Joly P, Nguyen H-M,. Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations. Mathematical Models and Methods in Applied Sciences 2008; 18 (10): 1787-1827.
Deeley E,. Surface impedance near edges and corners in three-dimensional media. IEEE Transactions on Magnetics 1990; 26 (2): 712-714.
Yuferev S, Proekt L, Ida N,. Surface impedance boundary conditions near corner and edges: rigorous consideration. IEEE Transactions on Magnetics 2001; 37 (5): 3465-3468.
Buret F, Dauge M, Dular P, Krähenbühl L, Péron V, Perrussel R, Poignard C, Voyer D,. Eddy currents and corner singularities. IEEE Transactions on Magnetics 2012; 48 (2): 679-682.
Kondratev VA,. Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskovskogo Matematicheskogo Obshchestva 1967; 16: 209-292.
Kondratev VA, Oleinik OA,. Boundary-value problems for partial differential equations in non-smooth domains. Russian Mathematical Surveys 1983; 38: 1-86.
Grisvard P,. Boundary Value Problems in Non-Smooth Domains. Pitman: London, 1985.
Dauge M,. Elliptic Boundary Value Problems in Corner Domains-Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, Vol. 1341. Springer-Verlag: Berlin, 1988.
Nicaise S,. Polygonal Interface Problems, Methoden und Verfahren der Mathematischen Physik, 39. Verlag Peter D. Lang: Frankfurt-am-Main, 1993.
Kozlov VA, Maz'ya VG, Rossmann J,. Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, 52. American Mathematical Society: Providence, RI, 1997.
Maz'ya VG, Plamenevskii BA,. On the coefficients in the asymptotic of solutions of the elliptic boundary problem in domains with conical points. American Mathematical Society Transactions (2) 1984; 123: 57-88.
Dauge M, Nicaise S, Bourlard M, Lubuma JM-S,. Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques. II. Quelques opérateurs particuliers. RAIRO Modélisation Mathématique et Analyse Numérique 1990; 24 (3): 343-367.
Moussaoui M-A,. Sur l'approximation des solutions du problème de Dirichlet dans un ouvert avec coins. In Singularities and constructive methods for their treatment (Oberwolfach, 1983), Vol. 1121, Lecture Notes in Math. Springer: Berlin, 1985; 199-206.
Szabõ BA, Yosibash Z,. Numerical analysis of singularities in two dimensions. II. Computation of generalized flux/stress intensity factors. International Journal for Numerical Methods in Engineering 1996; 39 (3): 409-434.
Costabel M, Dauge M, Yosibash Z,. A quasidual function method for extracting edge stress intensity functions. SIAM Journal on Mathematical Analysis 2004; 35 (5): 1177-1202.
Shannon S, Yosibash Z, Dauge M, Costabel M,. Extracting generalized edge flux intensity functions by the quasidual function method along circular 3-D edges, 2012. Preprint HAL available at http://hal.archives-ouvertes.fr/hal- 00725928.
Costabel M, Dauge M,. Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems. Mathematische Nachrichten 1993; 162: 209-237.
Dular P, Geuzaine C,. GetDP: a general environment for the treatment of discrete problems. 1997-2012. Source code. http://geuz.org/getdp/.
