# An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions

Alfredo Bermúdez; Bibiana López-Rodríguez; Rodolfo Rodríguez; Pilar Salgado

- Volume: 47, Issue: 3, page 875-902
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBermúdez, Alfredo, et al. "An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.3 (2013): 875-902. <http://eudml.org/doc/273097>.

@article{Bermúdez2013,

abstract = {The aim of this paper is to analyze a formulation of the eddy current problem in terms of a time-primitive of the electric field in a bounded domain with input current intensities or voltage drops as source data. To this end, we introduce a Lagrange multiplier to impose the divergence-free condition in the dielectric domain. Thus, we obtain a time-dependent weak mixed formulation leading to a degenerate parabolic problem which we prove is well-posed. We propose a finite element method for space discretization based on Nédélec edge elements for the main variable and standard finite elements for the Lagrange multiplier, for which we obtain error estimates. Then, we introduce a backward Euler scheme for time discretization and prove error estimates for the fully discrete problem, too. Finally, the method is applied to solve a couple of test problems.},

author = {Bermúdez, Alfredo, López-Rodríguez, Bibiana, Rodríguez, Rodolfo, Salgado, Pilar},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Eddy current problems; time-dependent electromagnetic problems; input current intensities; voltage drops; finite elements; eddy current problems},

language = {eng},

number = {3},

pages = {875-902},

publisher = {EDP-Sciences},

title = {An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions},

url = {http://eudml.org/doc/273097},

volume = {47},

year = {2013},

}

TY - JOUR

AU - Bermúdez, Alfredo

AU - López-Rodríguez, Bibiana

AU - Rodríguez, Rodolfo

AU - Salgado, Pilar

TI - An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 3

SP - 875

EP - 902

AB - The aim of this paper is to analyze a formulation of the eddy current problem in terms of a time-primitive of the electric field in a bounded domain with input current intensities or voltage drops as source data. To this end, we introduce a Lagrange multiplier to impose the divergence-free condition in the dielectric domain. Thus, we obtain a time-dependent weak mixed formulation leading to a degenerate parabolic problem which we prove is well-posed. We propose a finite element method for space discretization based on Nédélec edge elements for the main variable and standard finite elements for the Lagrange multiplier, for which we obtain error estimates. Then, we introduce a backward Euler scheme for time discretization and prove error estimates for the fully discrete problem, too. Finally, the method is applied to solve a couple of test problems.

LA - eng

KW - Eddy current problems; time-dependent electromagnetic problems; input current intensities; voltage drops; finite elements; eddy current problems

UR - http://eudml.org/doc/273097

ER -

## References

top- [1] R. Acevedo, S. Meddahi and R. Rodríguez, An E-based mixed formulation for a time-dependent eddy current problem. Math. Comput.78 (2009) 1929–1949. Zbl1201.78027MR2521273
- [2] A. Alonso Rodríguez, R. Hiptmair and A. Valli, A hybrid formulation of eddy current problems. Numer. Methods Part. Differ. Equ.21 (2005) 742-763. Zbl1079.78023MR2140566
- [3] A. Alonso Rodríguez, R. Hiptmair and A. Valli, Mixed finite element approximation of eddy current problems. IMA J. Numer. Anal.24 (2004) 255–271. Zbl1114.78012MR2046177
- [4] A. Alonso and A. Valli, An optimal decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput.68 (1999) 607–631. Zbl1043.78554MR1609607
- [5] A. Alonso and A. Valli, Eddy Current Approximation of Maxwell Equations: Theory, Algorithms and Applications. Springer–Verlag, Italia (2010). Zbl1204.78001MR2680968
- [6] A. Alonso Rodríguez and A. Valli, Voltage and current excitation for time-harmonic eddy-current problems. SIAM J. Appl. Math.68 (2008) 1477–1494. Zbl1154.35454MR2407134
- [7] 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. Zbl0914.35094MR1626990
- [8] A. Beranúdez, B. López-Rodríguez, R. Rodríguez and P. Salgado, Equivalence between two finite element methods for the eddy current problem. C. R. Math. Acad. Sci. Paris, Series I 34 (2010) 769–774. Zbl1201.78028
- [9] A. Bermúdez, B. López-Rodríguez, R. Rodríguez and P. Salgado, Numerical solution of transient eddy current problems with input current intensities as boundary data. IMA J. Numer. Anal.32 (2012) 1001–1029. Zbl1247.78035MR2954738
- [10] A. Bermúdez, R. Rodríguez and P. Salgado, A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations. SIAM J. Numer. Anal.40 (2002) 1823–1849. Zbl1033.78009
- [11] A. Bermúdez, R. Rodríguez and P. Salgado, Numerical analysis of electric field formulations of the eddy current model. Numer. Math.102 (2005) 181–201. Zbl1084.78004
- [12] A. Bossavit, Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements. Academic Press, San Diego (1998). Zbl0945.78001MR1488417
- [13] A. Bossavit, Most general non-local boundary conditions for the Maxwell equation in a bounded region. COMPEL19 (2000) 239–245. Zbl0966.78002MR1784010
- [14] A. Buffa, M. Costabel and D. Sheen, On traces for H(curl;Ω) in Lipschitz domains. J. Math. Anal. Appl.276 (2002) 845–876. Zbl1106.35304MR1944792
- [15] A. Buffa, Y. Maday and F. Rapetti, Applications of the mortar element method to 3D electromagnetic moving structures. Computational Electromagnetics, edited by C. Carstensen et al., Springer Verlag. Lect. Notes Comput. Sci. Eng. 28 (2003) 35–50. Zbl1065.78017MR1986131
- [16] C.R.I. Emson, and J. Simkin, An optimal method for 3D eddy currents. IEEE Trans. Magn.19 (1983) 2450–2452.
- [17] P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci.7 (1997) 957–991. Zbl0910.35123MR1479578
- [18] P. Fernandes and I. Perugia, Vector potential formulation for magnetostatic and modelling of permanent magnets. IMA J. Appl. Math.66 (2001) 293–318. Zbl0985.78005MR1852930
- [19] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer–Verlag, Berlin (1986). Zbl0585.65077MR851383
- [20] R. Hiptmair and O. Sterz, Current and voltage excitations for the eddy current model. Int. J. Numer. Model.18 (2005) 1–21. Zbl1099.78021
- [21] A. Kameari, Calculation of transient 3D eddy currents using edge elements. IEEE Trans. Magn.26 (1990) 466–469.
- [22] A. Kameari, Three dimensional eddy current calculation using edge elements for magnetic vector potential. Applied Electromagnetic in Materials, Pergamon Press, Oxford (1988) 225–236.
- [23] T. Kang, T. Chen, H. Zhang and K.I. Kim, Improved T − ψ nodal finite element schemes for eddy current problems. Appl. Math. Comput. 218 (2011) 287–302. Zbl1227.78021MR2820491
- [24] C. Ma, The finite element analysis of a decoupled T–Ψ scheme for solving eddy-current problems. Appl. Math. Comput.205 (2008) 352–361. Zbl1169.78005MR2466639
- [25] G. Pichenot, F. Buvat, V. Maillot and H. Voillaume, Eddy current modelling for non destructive testing. Proc. of 16th World Conf. on NDT, Rapport DSR 31. Montreal, August 30 - September 3 (2004).
- [26] B. Weiß and O. Bíró, On the convergence of transient eddy-current problems. IEEE Trans. Magn.40 (2004) 957–960.
- [27] A. Žensíšek, Nonlinear Elliptic and Evolution Problems and their Finite Element Approximations. London, Academic Press (1990). Zbl0731.65090
- [28] W. Zheng, Z. Chen and L. Wang, An adaptive finite element method for the H-ψ formulation of time-dependent eddy current problems. Numer. Math.103 (2006) 667–689. Zbl1099.65092MR2221067

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.