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Thin Layer Models for Electromagnetism

Published online by Cambridge University Press:  03 June 2015

Marc Duruflé ,
Victor Péron  and
Clair Poignard
Marc Duruflé*
Affiliation:
Team Magique-3D, INRIA Bordeaux-Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251 & Université de Bordeaux1, 351 cours de la Libération, 33405 Talence Cedex, France
Victor Péron*
Affiliation:
Team Magique-3D, INRIA Bordeaux-Sud-Ouest, LMAP CNRS UMR 5142 & Université de Pau et des Pays de l’Adour, France
Clair Poignard*
Affiliation:
Team MC2, INRIA Bordeaux-Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251 & Université de Bordeaux1, 351 cours de la Libération, 33405 Talence Cedex, France
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

We present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer. These models appear as first order approximations of the electromagnetic field. They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer, that makes possible to replace the thin layer by approximate conditions. We present the advantages and the drawbacks of several approximations together with numerical validations and simulations. The main motivation of this work concerns the computation of electromagnetic field in biological cells. The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane, which provides a specific behavior of the electromagnetic field at low frequencies.

Keywords

34E0534E1058J37Asymptoticstime-harmonic Maxwell’s equationsfinite element methodedge elements.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 213 - 238
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Caloz, G., Dauge, M., Faou, E., and Péron, V.On the influence of the geometry on skin effect in electromagnetism. Computer Methods in Applied Mechanics and Engineering, 200(9-12): 1053–1068, 2011.Google Scholar
[2]Chun, S., Haddar, H., and Hesthaven, J. S.High-order accurate thin layer approximations for time-domain electromagnetics, Part II: transmission layers. J. Comput. Appl. Math., 234(8): 2587–2608, 2010.Google Scholar
[3]Cohen, C. and Duruflé, M.Non spurious spectral-like element methods for Maxwell’s equations. Journal of Computational Mathematics, 25: 282–304, 2007.Google Scholar
[4]Delourme, B., Haddar, H., and Joly, P.On the Well-Posedness, Stability And Accuracy Of An Asymptotic Model For Thin Periodic Interfaces In Electromagnetic Scattering Problems. Mathematical Models and Methods in Applied Sciences, 2013.Google Scholar
[5]Duruflé, M.Integration numerique et elements finis d’ordre eleve appliques aux equations de Maxwell en regime harmonique. PhD thesis, Universite Paris IX-Dauphine, 2006.Google Scholar
[6]Duruflé, M., Péron, V., and Poignard, C.Thin Layer Models for Electromagnetism. In Waves 2011: The 10th International Conference on Mathematical and Numerical Aspects of Waves, pages 687–690, Vancouver, Canada, July 2011.Google Scholar
[7]Duruflé, M., Péron, V., and Poignard, C.Time-harmonic Maxwell equations in biological cells—the differential form formalism to treat the thin layer. Confluentes Math., 3(2): 325–357, 2011.Google Scholar
[8]Fear, E.C. and Stuchly, M.A.Modelling assemblies of biological cells exposed to electric fields. IEEE Trans Biomed Eng, 45(10): 1259–1271, 0ct 1998.Google Scholar
[9]Fear, E.C. and Stuchly, M.A.A novel equivalent circuit model for gap-connected cells. Phys Med Biol, 43(6): 1439–1448, Jun 1998.Google Scholar
[10]Foster, K.R. and Schwan, H.P.Dielectric properties of tissues and biological materials: a critical review. CRC in Biomedical Engineering, 17(1): 25–104, 1989.Google ScholarPubMed
[11]Kavian, O., Leguebe, M., Poignard, C., and Weynans, L.“Classical” Electropermeabilization Modeling at the Cell Scale. Journal of Mathematical Biology, 2012.Google Scholar
[12]Muñoz, S., Sebastián, J.L., Sancho, M., and Miranda, J.M.Transmembrane voltage induced on altered erythrocyte shapes exposed to RF fields. Bioelectromagnetics, 25(1): 631–633 (electronic), 2004.Google Scholar
[13]Nédélec, J.-C.Acoustic and electromagnetic equations, volume 144 of Applied Mathematical Sciences. Springer-Verlag, New York, 2001. Integral representations for harmonic problems.Google Scholar
[14]Perrussel, R. and Poignard, C.Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer. Applied Mathematics and Computation, to appear, 2013.Google Scholar
[15]Poignard, C.Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer. Math. Methods Appl. Sci., 31(4): 443–479, 2008.Google Scholar
[16]Poignard, C.About the transmembrane voltage potential of a biological cell in time-harmonic regime. ESAIM: Proceedings, 26: 162–179, 2009.CrossRefGoogle Scholar
[17]Poignard, C.Approximate transmission conditions through a weakly oscillating thin layer. Math. Meth. App. Sci., 32(4): 435–453, 2009.CrossRefGoogle Scholar
[18]Poignard, C., Dular, P., Perrussel, R., Krähenbühl, L., Nicolas, L., and Schatzman, M.Approximate conditions replacing thin layer. IEEE Trans. on Mag., 44(6): 1154–1157, 2008.Google Scholar
[19]Pucihar, G., Kotnik, T., Valič, B., and Miklavčič, D.Numerical determination of transmembrane voltage induced on irregularly shaped cells. Ann Biomed Eng, 34(4):642–652, Apr 2006.Google Scholar
[20]Sebastián, J.L., Muñoz, S., Sancho, M., and Miranda, J.M.Analysis of the influence of the cell geometry and cell proximity effects on the electric field distribution from direct rf exposure. Phys. Med. Biol., 46:213–225 (electronic), 2001.Google Scholar
[21]Teissié, J., Golzio, M., and Rols, M.P.Mechanisms of cell membrane electropermeabilization: A minireview of our present (lack of?) knownledge. Biochimica et Biophysica Acta, 1724: 270–280, 2005.Google Scholar