AIMS Mathematics, 2017, 2(2): 269-304. doi: 10.3934/Math.2017.2.269

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Modeling electromagnetism in and near composite material using two-scale behavior of the time-harmonic Maxwell equations

Université de Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France

The main purpose of this article is to study the two-scale behavior of the electromagnetic field in 3D in and near composite material. For this, time-harmonic Maxwell equations, for a conducting two-phase composite and the air above, are considered. Technique of two-scale convergence is used to obtain the homogenized problem.
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1. T. Abboud and I. Terrasse, Modélisation des phénom`enes de propagation d'ondes, Centre Poly-Média de l'école Polytechnique, 2007.

2. Y. Amirat, K. Hamdache and A. Ziani, Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements missibles en milieux poreux, Ann. Inst. H. Poincaré, 6 (1989), 397-417.    

3. G. Allaire, Homogenization and Two-scale Convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.    

4. G. Allaire and M. Briand, Multiscale convergence and reiterated homogenization, Roy.Soc.Edinburgh, 126 (1996), 297-342.    

5. Y. Amirat and V. Shelukhin, Homogenization of time-harmonic Maxwell equations and the frequency dispersion effect, J.Maths.Pures.Appl., 95 (2011), 420-443.    

6. A. Back and E. Frenod, Geometric Two-Scale Convergence on Manifold and Applications to the Vlasov Equation Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 223-241.

7. S. Berthier, Optique des milieux composites, Ed. Polytechnicia, 1993.

8. D. Cionarescu and P. Donato, An introduction to homogenization, Oxford University Press., 1999.

9. M. Costabel, M. Dauge and S. Nicaise, Corner Singularities of Maxwell interface and Eddy current problems, Advances and Applications, 147 (2004), 241-256.

10. N. Crouseilles, E. Frenod, S. Hirstoaga and A. Mouton, Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field, Mathematical Models and Methods in Applied Sciences, 23 (2012), 1527-1559.

11. E. Frénod, P. A. Raviart and E. Sonnendrücker, Asymptotic Expansion of the Vlasov Equation in a Large External Magnetic Field, J. Math. Pures et Appl. 80, (2001), 815-843.

12. S. Guenneau, F. Zolla and A. Nicolet, Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability, Waves in Random and Complex Media, 17 (2007), 653-697.    

13. P.R.P. Hoole and S.R.H. Hoole, Guided waves along an unmagnetized lightning plasma channel, IEEE Transactions on Magnetics, 24 (1998), 3165-3167.

14. P.R.P. Hoole, S.R.H. Hoole, S. Thirukumaran, R. Harikrishnan, K. Jeevan and K. Pirapaharan, Aircraft-lightning electrodynamics using the transmission line model part I: review of the transmission line model, Progress In Electromagnetics Research M, 31, (2013), 85-101.

15. P. Laroche, P. Blanchet, A. Delannoy, and F. Issac, Experimental Studies of Lightning Strikes to Aircraft, JOURNAL AEROSPACELAB, 112 (2012).

16. M. Leboulch, Analyse spectrale VHF, UHF du rayonnement deséclairs, Hamelin, CENT.

17. J.C. Maxwell, A dynamical theory of the Electromagnetic Field, Phisophical transacting of the Royal Society of London, (1885), 459-512.

18. P. Monk, Finite Element Methods for Maxwell's Equations, Oxford Science publication, Numerical Mathematics and scientific computation, Clarendon Press - Oxford, 2003.

19. J.C. Nédélec, Acoustic and electromagnetic equations; integral representations for harmonic problems, Springer-Verlag, Berlin, 2001.

20. M. Neuss-Radu, Some extensions of two-scale convergence, Comptes rendus de l'Academie des sciences, 322 (1996), 899-904.

21. G. Nguetseng. A General Convergence Result for a Functional Related to the Theory of Homogenization, 20 (1989), 608-623.

22. G. Nguetseng, Asymptotic Analysis for a Stiff Variational Problem Arising in Mechanics, SIAM Journal on Mathematical Analysis, 21 (1990), 1394-1414.    

23. S. Nicaise, S. Hassani and A. Maghnouji, Limit behaviors of some boundary value problems with high and/or low valued parameters, Advances in differential equations, 14 (2009), 875-910.

24. O. Ouchetto, S. Zouhdi and A. Bossavit et al., Effective constitutive parameters of periodic composites, Microwave conference, European, 2 (2005).

25. H.E. Pak, Geometric two-scale convergence on forms and its applications to Maxwell's equations, Proceedings of the Royal Society of Edinburgh, European, 135A (2005), 133-147.

26. N. Wellander, Homogenization of the Maxwell equations: Case I. Linear theory, Appl Math, 46 (2001), 29-51.    

27. N. Wellander, Homogenization of the Maxwell equations: Case II. Nonlinear conductivity, Appl Math, 47 (2002), 255-283.    

28. N. Wellander and B. Kristensson, Homogenization of the Maxwell equations at fixed frequency, Technical Report, (2002), 1-37.

29. Pr. Welter, Cours : Matériaux diélectriques, Master Matériaux, Institut Le Bel.

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