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 -space. Well-posedness of the -based model and the -formulation of combined potentials are proved and an un-gauged discretization of the latter formulation is discussed.
DOI: 10.1051/m2an/2014027
Keywords: Eddy current problems, domains with cracks, singularities of solutions
@article{M2AN_2015__49_1_141_0, author = {Lohrengel, Stephanie and Nicaise, Serge}, 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}, volume = {49}, number = {1}, year = {2015}, doi = {10.1051/m2an/2014027}, mrnumber = {3342195}, zbl = {1312.35037}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2014027/} }
TY - JOUR AU - Lohrengel, Stephanie AU - Nicaise, Serge TI - Analysis of eddy current formulations in two-dimensional domains with cracks JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 141 EP - 170 VL - 49 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014027/ DO - 10.1051/m2an/2014027 LA - en ID - M2AN_2015__49_1_141_0 ER -
%0 Journal Article %A Lohrengel, Stephanie %A Nicaise, Serge %T Analysis of eddy current formulations in two-dimensional domains with cracks %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 141-170 %V 49 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014027/ %R 10.1051/m2an/2014027 %G en %F M2AN_2015__49_1_141_0
Lohrengel, Stephanie; Nicaise, Serge. Analysis of eddy current formulations in two-dimensional domains with cracks. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 49 (2015) no. 1, pp. 141-170. doi : 10.1051/m2an/2014027. http://www.numdam.org/articles/10.1051/m2an/2014027/
A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer, Milan (2010). | MR | Zbl
Magnetostatic field computations in terms of two component vector potentials. Int. J. Numer. Methods Engrg. 29 (1990) 515–532. | DOI | Zbl
and ,A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60 (2000) 1805–1823. | DOI | MR | Zbl
, and ,Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. | DOI | MR | Zbl
, , and ,-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
and ,Resolution of the Maxwell equations in a domain with reentrant corners. Math. Model. Numer. Anal. 32 (1998) 359–389. | DOI | Numdam | MR | Zbl
, and ,Numerical analysis of electric field formulations of the eddy current model. Numer. Math. 102 (2005) 181–201. | DOI | MR | Zbl
, and ,Edge element formulations of eddy current problems. Comput. Methods Appl. Mech. Engrg. 169 (1999) 391–405. | DOI | MR | Zbl
,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
and ,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
, and ,Eddy-current interaction with an ideal crack. I. The forward problem. J. Appl. Phys. 75 (1994) 8128–8137. | DOI
,Eddy-current interaction with an ideal crack. II. The inverse problem. J. Appl. Phys. 75 (1994) 8138–8144. | DOI
, and ,J.R. Bowler, Theory of eddy current crack response, Technical Report. Iowa State University, Center for Nondestructive Evaluation, Ames IA (2002).
Thin-skin eddy-current inversion for the determination of crack shapes. Inverse Probl. 18 (2002) 1891–1905. | DOI | MR | Zbl
,Vector-Potential Boundary-Integral Evaluation of Eddy-Current Interaction with a Crack. IEEE Trans. Magn. 33 (1997) 4287–4294. | DOI
, and ,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
and ,Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151 (2000) 221–276. | DOI | MR | Zbl
and ,M. Costabel and M. Dauge, Crack singularities for general elliptic systems. Math. Nachr. 235 (2002), pp. 29–49. | MR | Zbl
Asymptotics without logarithmic terms for crack problems, Commun. Partial. Differ. Eq. 28 (2003) 869–926. | DOI | MR | Zbl
and ,Singularities of eddy current problems. ESAIM: M2AN 37 (2003) 807–831. | DOI | Numdam | MR | Zbl
, and ,E. Creusé, S. Nicaise, Z. Tang, Y. Le Menach, N. Nemitz and F. Piriou, Residual-based a posteriori estimators for the magnetodynamic harmonic formulation of the Maxwell system. Math. Models Methods Appl. Sci. 22 (2012) DOI: 10.1142/S021820251150028X. | MR | Zbl
Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J. Sci. Statis. Comput. 13 (1992) 425–448. | DOI | MR | Zbl
,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
and ,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
Analysis of eddy-current interaction with a surface-breaking crack. J. Appl. Phys. 76 (1994) 4853–4856. | DOI
and ,On the solution of time-harmonic scattering problems for Maxwell’s equations. SIAM J. Math. Anal. 27 (1996) 1597–1630. | DOI | MR | Zbl
and ,Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40 (2002) 41–65. | DOI | MR | Zbl
,Finite element approximation for a div-rot system with mixed boundary conditions in non-smooth domains. Apl. Mat. 29 (1984) 272–285. | MR | Zbl
and ,An eXtended Finite Element Method for 2D edge elements. Int. J. Numer. Anal. Model. 8 (2011) 641–666. | MR | Zbl
, and ,A finite element method for crack growth without remeshing. Int. J. Numer. Methods Engrg. 46 (1999) 131–150. | DOI | MR | Zbl
, and ,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
Different finite element formulations of 3D Magnetostatics fields. IEEE Trans. Magn. 28 (1992) 1056–1059. | DOI
, , , , and ,Influence of the R.H.S. on the Convergence Behaviour of the Curl-Curl Equation. IEEE Trans. Magn. 32 (1996) 655–658. | DOI
,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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