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},
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     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/

Bibliographie
Cité par

A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer, Milan (2010). | MR | Zbl

R. Albanese and G. Rubinacci, Magnetostatic field computations in terms of two component vector potentials. Int. J. Numer. Methods Engrg. 29 (1990) 515–532. | DOI | Zbl

H. Ammari, A. Buffa and J.-C. Nédélec, A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60 (2000) 1805–1823. | DOI | MR | Zbl

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. | DOI | MR | Zbl

C. Amrouche and N. El Houda Seloula, $L^{p}$ -theory for vector potentials and Sobolev’s inequalities for vector vields. C.R. Acad. Sci. Paris Ser. I 349 (2011) 529–534. | DOI | MR | Zbl

F. Assous, P. Ciarlet, Jr. and E. Sonnendrücker, Resolution of the Maxwell equations in a domain with reentrant corners. Math. Model. Numer. Anal. 32 (1998) 359–389. | DOI | Numdam | MR | Zbl

A. Bermúdez, R. Rodríguez and P. Salgado, Numerical analysis of electric field formulations of the eddy current model. Numer. Math. 102 (2005) 181–201. | DOI | MR | Zbl

O. Bíró, Edge element formulations of eddy current problems. Comput. Methods Appl. Mech. Engrg. 169 (1999) 391–405. | DOI | MR | Zbl

O. Bíró and A. Valli, The Coulomb gauged vector potential formulation for the eddy-current problem in general geometry: Well-posedness and numerical approximation. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1890–1904. | DOI | MR | Zbl

A.-S. Bonnet-Ben Dhia, C. Hazard and S. Lohrengel, A singular field method for the solution of Maxwell’s equations in polyhedral domains. SIAM J. Appl. Math. 56 (1999) 2028–2044. | DOI | MR | Zbl

J.R. Bowler, Eddy-current interaction with an ideal crack. I. The forward problem. J. Appl. Phys. 75 (1994) 8128–8137. | DOI

J.R. Bowler, S.J. Norton and D.J. Harrison, Eddy-current interaction with an ideal crack. II. The inverse problem. J. Appl. Phys. 75 (1994) 8138–8144. | DOI

J.R. Bowler, Theory of eddy current crack response, Technical Report. Iowa State University, Center for Nondestructive Evaluation, Ames IA (2002).

J.R. Bowler, Thin-skin eddy-current inversion for the determination of crack shapes. Inverse Probl. 18 (2002) 1891–1905. | DOI | MR | Zbl

J.R. Bowler, Y. Yoshida and N. Harfield, Vector-Potential Boundary-Integral Evaluation of Eddy-Current Interaction with a Crack. IEEE Trans. Magn. 33 (1997) 4287–4294. | DOI

A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell equations, Part I: An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24 (2001) 9–30. | DOI | MR | Zbl

M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000) 221–276. | DOI | MR | Zbl

M. Costabel and M. Dauge, Crack singularities for general elliptic systems. Math. Nachr. 235 (2002), pp. 29–49. | MR | Zbl

M. Costabel and M. Dauge, Asymptotics without logarithmic terms for crack problems, Commun. Partial. Differ. Eq. 28 (2003) 869–926. | DOI | MR | Zbl

M. Costabel, M. Dauge and S. Nicaise, Singularities of eddy current problems. ESAIM: M2AN 37 (2003) 807–831. | DOI | Numdam | MR | Zbl

E. Creusé, S. Nicaise, Z. Tang, Y. Le Menach, N. Nemitz and F. Piriou, Residual-based a posteriori estimators for the $𝐀 - ϕ$ magnetodynamic harmonic formulation of the Maxwell system. Math. Models Methods Appl. Sci. 22 (2012) DOI: 10.1142/S021820251150028X. | MR | Zbl

R.W. Freund, Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J. Sci. Statis. Comput. 13 (1992) 425–448. | DOI | MR | Zbl

T.-P. Fries and M. Baydoun, Crack propagation with the extended finite element method and a hybrid explicit-implicit crack description. Int. J. Numer. Methods Engrg. 89 (2012) 1527–1558. | DOI | MR | Zbl

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer, Berlin (1986). | MR | Zbl

P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, London (1985). | MR | Zbl

P. Grisvard, Singularities in boundary value problems. Masson, Paris (1992). | MR | Zbl

N. Harfield and J.R. Bowler, Analysis of eddy-current interaction with a surface-breaking crack. J. Appl. Phys. 76 (1994) 4853–4856. | DOI

C. Hazard and M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell’s equations. SIAM J. Math. Anal. 27 (1996) 1597–1630. | DOI | MR | Zbl

R. Hiptmair, Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40 (2002) 41–65. | DOI | MR | Zbl

M. Křižek and P. Neittaanmäki, Finite element approximation for a div-rot system with mixed boundary conditions in non-smooth domains. Apl. Mat. 29 (1984) 272–285. | MR | Zbl

F. Lefèvre, S. Lohrengel and S. Nicaise, An eXtended Finite Element Method for 2D edge elements. Int. J. Numer. Anal. Model. 8 (2011) 641–666. | MR | Zbl

N. Moës, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing. Int. J. Numer. Methods Engrg. 46 (1999) 131–150. | DOI | MR | Zbl

P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press (2003). | MR | Zbl

P.A. Pinsky, Partial Differential Equations and Boundary-Value Problems with Applications. McGraw-Hill, Singapore (1998). | MR | Zbl

K. Preis, I. Bardi, O. Bíró, C. Magele, G. Vrisk and K.R. Richter, Different finite element formulations of 3D Magnetostatics fields. IEEE Trans. Magn. 28 (1992) 1056–1059. | DOI

Z. Ren, Influence of the R.H.S. on the Convergence Behaviour of the Curl-Curl Equation. IEEE Trans. Magn. 32 (1996) 655–658. | DOI

N. Sukumar, D.L. Chopp and B. Moran, Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Engrg. Fracture Mech. 70 (2003) 29–48. | DOI

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