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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Keywords Harmonic Maxwell Equations; Two-scale Convergence; Asymptotic Expansion; Asymptotic Analysis; Electromagnetism; Homogenization; E ective Behavior; Frequencies; Composite Material; Electromagnetic Pulses

Citation: Canot Hélène, Frénod Emmanuel. Modeling electromagnetism in and near composite material using two-scale behavior of the time-harmonic Maxwell equations. AIMS Mathematics, 2017, 2(2): 269-304. doi: 10.3934/Math.2017.2.269

References

  • 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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