Dans ce travail, nous nous intéressons à l’homogénéisation des équations de Maxwell harmoniques dans un milieu composite contenant une distribution périodique de petites inclusions de matériau négatif. On désigne ici par matériau négatif un matériau décrit par une permittivité et une perméabilité négatives. En raison du changement de signe des coefficients intervenant dans les équations, il n’est pas évident d’obtenir des estimations d’énergie uniformes et d’appliquer les techniques d’homogénéisation classiques. Le but de cet article est d’indiquer comment on peut néanmoins procéder dans ce contexte. L’analyse des équations de Maxwell est basée sur une étude précise de deux problèmes scalaires : l’un faisant intervenir la permittivité changeant de signe avec des conditions aux limites de Dirichlet, et l’autre la perméabilité changeant de signe avec des conditions aux limites de Neumann. Pour chacun de ces deux problèmes, on obtient un critère portant sur les paramètres physiques garantissant l’inversibilité uniforme des opérateurs associés lorsque la taille des inclusions tend vers zéro. Incidemment, nous expliquons le lien existant avec l’opérateur de Neumann–Poincaré, complétant la littérature existant sur le sujet. Les résultats obtenus pour les problèmes scalaires sont ensuite utilisés pour obtenir des estimations d’énergie uniforme pour le système de Maxwell. A ce stade, il faut contourner une difficulté supplémentaire liée au caractère indéfini induit par le terme fréquentiel. Ceci est réalisé en établissant un résultat de type compacité uniforme.
In this work, we are interested in the homogenization of time-harmonic Maxwell’s equations in a composite medium with periodically distributed small inclusions of a negative material. Here a negative material is a material modelled by negative permittivity and permeability. Due to the sign-changing coefficients in the equations, it is not straightforward to obtain uniform energy estimates to apply the usual homogenization techniques. The goal of this article is to explain how to proceed in this context. The analysis of Maxwell’s equations is based on a precise study of two associated scalar problems: one involving the sign-changing permittivity with Dirichlet boundary conditions, another involving the sign-changing permeability with Neumann boundary conditions. For both problems, we obtain a criterion on the physical parameters ensuring uniform invertibility of the corresponding operators as the size of the inclusions tends to zero. In the process, we explain the link existing with the so-called Neumann–Poincaré operator complementing the existing literature on this topic. Then we use the results obtained for the scalar problems to derive uniform energy estimates for Maxwell’s system. At this stage, an additional difficulty comes from the fact that Maxwell’s equations are also sign-indefinite due to the term involving the frequency. To cope with it, we establish some sort of uniform compactness result.
CC-BY 4.0
@article{AFST_2021_6_30_5_1075_0,
author = {Renata Bunoiu and Lucas Chesnel and Karim Ramdani and Mahran Rihani},
title = {Homogenization of {Maxwell{\textquoteright}s} equations and related scalar problems with sign-changing coefficients},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1075--1119},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 30},
number = {5},
year = {2021},
doi = {10.5802/afst.1694},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1694/}
}
TY - JOUR AU - Renata Bunoiu AU - Lucas Chesnel AU - Karim Ramdani AU - Mahran Rihani TI - Homogenization of Maxwell’s equations and related scalar problems with sign-changing coefficients JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2021 SP - 1075 EP - 1119 VL - 30 IS - 5 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1694/ DO - 10.5802/afst.1694 LA - en ID - AFST_2021_6_30_5_1075_0 ER -
%0 Journal Article %A Renata Bunoiu %A Lucas Chesnel %A Karim Ramdani %A Mahran Rihani %T Homogenization of Maxwell’s equations and related scalar problems with sign-changing coefficients %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2021 %P 1075-1119 %V 30 %N 5 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1694/ %R 10.5802/afst.1694 %G en %F AFST_2021_6_30_5_1075_0
Renata Bunoiu; Lucas Chesnel; Karim Ramdani; Mahran Rihani. Homogenization of Maxwell’s equations and related scalar problems with sign-changing coefficients. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 5, pp. 1075-1119. doi : 10.5802/afst.1694. https://afst.centre-mersenne.org/articles/10.5802/afst.1694/
[1] Remarks on the Neumann–Poincaré integral equation, Pac. J. Math., Volume 2 (1952), pp. 271-280 | DOI | Zbl
[2] Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992) no. 6, pp. 1482-1518 | DOI | MR | Zbl
[3] Homogenization of time harmonic Maxwell equations and the frequency dispersion effect, J. Math. Pures Appl., Volume 95 (2011) no. 4, pp. 420-443 | DOI | MR | Zbl
[4] Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., Volume 21 (1998) no. 9, pp. 823-864 | DOI | MR
[5] Homogenization of periodically varying coefficients in electromagnetic materials, J. Sci. Comput., Volume 28 (2006) no. 2-3, pp. 191-221 | DOI | MR | Zbl
[6] Asymptotic analysis of periodic structures, North-Holland, 1978
[7] On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients, J. Comput. Phys., Volume 322 (2016), pp. 224-247 | DOI | MR | Zbl
[8] Mesh requirements for the finite element approximation of problems with sign-changing coefficients, Numer. Math., Volume 138 (2018) no. 4, pp. 801-838 | DOI | MR | Zbl
[9] -coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM, Math. Model. Numer. Anal., Volume 46 (2012) no. 6, pp. 1363-1387 | Numdam | MR | Zbl
[10] T-coercivity for the Maxwell problem with sign-changing coefficients, Commun. Partial Differ. Equations, Volume 39 (2014) no. 6, pp. 1007-1031 | MR | Zbl
[11] Radiation condition for a non-smooth interface between a dielectric and a metamaterial, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 09, pp. 1629-1662 | DOI | MR | Zbl
[12] Time harmonic wave diffraction problems in materials with sign-shifting coefficients, J. Comput. Appl. Math., Volume 234 (2008) no. 6, pp. 1912-1919 | DOI | MR | Zbl
[13] Erratum to the article: Homogenization of the eigenvalues of the Neumann-Poincaré operator. (2019) (https://ljk.imag.fr/membres/Charles.Dapogny/publis/HomogNP_v4%20corr2.pdf)
[14] Homogenization of the eigenvalues of the Neumann–Poincaré operator, Arch. Ration. Mech. Anal., Volume 234 (2019) no. 2, pp. 777-855 | DOI | Zbl
[15] Pointwise bounds on the gradient and the spectrum of the Neumann–Poincaré operator: the case of 2 discs, Multi-scale and high-contrast PDE. From modelling, to mathematical analysis, to inversion (Contemporary Mathematics), Volume 577, American Mathematical Society, 2012, pp. 81-92 | Zbl
[16] On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 2, pp. 1-27 | Zbl
[17] Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 9-10, pp. 571-576 | DOI | MR | Zbl
[18] Homogenization of materials with sign changing coefficients, Commun. Math. Sci., Volume 14 (2016) no. 4, pp. 1137-1154 | DOI | Zbl
[19] Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator, Waves Random Complex Media, Volume 17 (2007) no. 4, pp. 627-651 | DOI | MR | Zbl
[20] T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients, Numer. Math., Volume 124 (2013) no. 1, pp. 1-29 | DOI | MR | Zbl
[21] On the approximation of electromagnetic fields by edge finite elements. II: a heterogeneous multiscale method for Maxwell’s equations, Comput. Math. Appl., Volume 73 (2017) no. 9, pp. 1900-1919 | DOI | Zbl
[22] An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, 17, Oxford University Press, 1999 | Zbl
[23] Metamaterials: Theory, Design, and Applications, Springer, 2009
[24] On two numerical methods for homogenization of Maxwell’s Equations, J. Electromagn. Waves Appl., Volume 21 (2007) no. 13, pp. 1845-1856 | DOI
[25] The plasmonic eigenvalue problem, Rev. Math. Phys., Volume 26 (2014) no. 3, 1450005, 26 pages | DOI | MR | Zbl
[26] Surface plasmon resonances of an arbitrarily shaped nanoparticle: high-frequency asymptotics via pseudo-differential operators, J. Phys. A, Math. Theor., Volume 42 (2009) no. 13, 135204 | MR | Zbl
[27] Perturbation theory for plasmonic eigenvalues, Phys. Rev. B, Volume 80 (2009) no. 24, 245405 | DOI
[28] A new heterogeneous multiscale method for time-harmonic Maxwell’s equations, SIAM J. Numer. Anal., Volume 54 (2016) no. 6, pp. 3493-3522 | DOI | MR | Zbl
[29] Poincaré’s variational problem in potential theory, Arch. Ration. Mech. Anal., Volume 185 (2007) no. 1, pp. 143-184 | DOI | Zbl
[30] Finite element methods for Maxwell’s, Oxford University Press, 2003
[31] Acoustic and electromagnetic equations, Applied Mathematical Sciences, 144, Springer, 2001 | DOI
[32] A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., Volume 20 (1989) no. 3, pp. 608-623 | DOI | MR | Zbl
[33] Limiting absorption principle and well-posedness for the time-harmonic Maxwell equations with anisotropic sign-changing coefficients, Commun. Math. Phys., Volume 379 (2020) no. 1, pp. 145-176 | DOI | MR | Zbl
[34] A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients, J. Comput. Appl. Math., Volume 235 (2011) no. 14, pp. 4272-4282 | DOI | MR | Zbl
[35] Spectral bounds for the Neumann–Poincaré operator on planar domains with corners, J. Anal. Math., Volume 124 (2014), pp. 39-57 | DOI | Zbl
[36] La méthode de Neumann et le problème de Dirichlet, Acta Math., Volume 20 (1897) no. 1, pp. 59-142 | DOI
[37] The Fredholm eigenvalues of plane domains, Pac. J. Math., Volume 7 (1957), pp. 1187-1225 | DOI
[38] Estimates for Fredholm eigenvalues based on quasiconformal mapping, Numerische, insbesondere approximationstheoretische Behandlung von Funktionalgleichungen (Tagung, Math. Forschungsinst., Oberwolfach, 1972) (Lecture Notes in Mathematics), Volume 333, Springer, 1973, pp. 211-217 | DOI | MR | Zbl
[39] A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization, Multiscale Model. Simul., Volume 4 (2005) no. 1, pp. 149-171 | DOI | MR | Zbl
[40] Metamaterials and Negative Refractive Index, Science, Volume 305 (2004) no. 5, pp. 788-792 | DOI
[41] Homogenization of a stationary periodic Maxwell system in a bounded domain in the case of constant magnetic permeability, St. Petersbg. Math. J., Volume 30 (2019) no. 3, pp. 515-544 | DOI | MR | Zbl
[42] Homogenization of the stationary Maxwell system with periodic coefficients in a bounded domain, Arch. Ration. Mech. Anal., Volume 234 (2019) no. 2, pp. 453-507 | DOI | MR | Zbl
[43] Homogenization error for two scale Maxwell equations (2015) (https://arxiv.org/abs/1512.02788)
[44] A local compactness theorem for Maxwell’s equations, Math. Methods Appl. Sci., Volume 2 (1980), pp. 12-25 | DOI | MR | Zbl
[45] Variational methods for eigenvalue approximation, CBMS-NSF Regional Conference Series in Applied Mathematics, 15, Society for Industrial and Applied Mathematics, 1974 | DOI
[46] Homogenization of the Maxwell equations. Case I. Linear theory, Appl. Math., Praha, Volume 46 (2001) no. 1, pp. 29-51 | MR | Zbl
[47] The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space, Asymptotic Anal., Volume 62 (2009) no. 1-2, pp. 1-40 | DOI | MR | Zbl
[48] Homogenization of the Maxwell equations at fixed frequency, SIAM J. Appl. Math., Volume 64 (2003) no. 1, pp. 170-195 | MR | Zbl
Cité par Sources :