Skip to main content
Log in

Numerical Study of the Plasma-Lorentz Model in Metamaterials

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Since 2000, the study of metamaterial has been a very hot topic due to its potential applications in many areas such as design of invisibility cloak and sub-wavelength imaging. Although several metamaterial models are often used by physicists and engineers, the study of their mathematical properties has lagged behind. In this paper, we initiate our investigation in the plasma-Lorentz model. More specifically, we first discuss the well-posedness of this model, then develop two fully-discrete finite element methods for solving it. Detailed stability and error analysis are carried out, and 3-D numerical results justifying our theoretical analysis are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Banks, H.T., Bokil, V.A., Gibson, N.L.: Analysis of stability and dispersion in a finite element method for Debye and Lorentz media. Numer. Methods Partial Differ. Equ. 25, 885–917 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beck, R., Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B.: Residual based a posteriori error estimators for eddy current computation. Modél. Math. Anal. Numér. 34, 159–182 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boardman, A.D., Marinov, K.: Electromagnetic energy in a dispersive metamaterial. Phys. Rev. B 73, 165110 (2006)

    Article  Google Scholar 

  4. Bochev, P.B., Garasi, C.J., Hu, J.J., Robinson, A.C., Tuminaro, R.S.: An improved algebraic multigrid method for solving Maxwell’s equations. SIAM J. Sci. Comput. 25, 623–642 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonnet-Ben Dhia, A.S., Ciarlet, P. Jr., Zwölf, C.M.: Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234, 1912–1919 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Z., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37, 1542–1570 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ciarlet, P. Jr., Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82, 193–219 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cockburn, B., Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194, 588–610 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cui, T.J., Smith, D.R., Liu, R. (eds.): Metamaterials: Theory, Design, and Applications. Springer, Berlin (2009)

    Google Scholar 

  10. Demkowicz, L., Kurtz, J., Pardo, D., Paszynski, M., Rachowicz, W., Zdunek, A.: Computing with hp-Adaptive Finite Elements. Vol. 2. Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications. CRC Press/Taylor & Francis, Boca Raton (2006)

    Google Scholar 

  11. Donderici, B., Teixeira, F.L.: Mixed finite-element time-domain method for Maxwell equations in doubly-dispersive media. IEEE Trans. Microw. Theory Tech. 56, 113–120 (2008)

    Article  MathSciNet  Google Scholar 

  12. Engheta, N., Ziolkowski, R.W.: Electromagnetic Metamaterials: Physics and Engineering Explorations. Wiley-IEEE Press, New York (2006)

    Google Scholar 

  13. Fernandes, P., Raffetto, M.: Well posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials. Math. Models Methods Appl. Sci. 19, 2299–2335 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fezoui, L., Lanteri, S., Lohrengel, S., Piperno, S.: Convergence and stability of a discontinuous Galerkin time-domain method for the heterogeneous Maxwell equations on unstructured meshes. Math. Model. Numer. Anal. 39, 1149C1176 (2005)

    Article  MathSciNet  Google Scholar 

  15. Frantzeskakis, D.J., Ioannidis, A., Roach, G.F., Stratis, I.G., Yannacopoulos, A.N.: On the error of the optical response approximation in chiral media. Appl. Anal. 82, 839–856 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gopalakrishnan, J., Pasciak, J.E., Demkowicz, L.F.: Analysis of a multigrid algorithm for time harmonic Maxwell equations. SIAM J. Numer. Anal. 42, 90–108 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hao, Y., Mittra, R.: FDTD Modeling of Metamaterials: Theory and Applications. Artech House, Boston (2008)

    Google Scholar 

  18. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)

    MATH  Google Scholar 

  19. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Houston, P., Perugia, I., Schötzau, D.: Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator. Comput. Methods Appl. Mech. Eng. 194, 499–510 (2005)

    Article  MATH  Google Scholar 

  21. Huang, Y., Li, J.: Recent advances in time-domain Maxwell’s equations in metamaterials. In: Zhang, W., et al. (eds.) Lecture Notes in Computer Sciences, vol. 5938, pp. 48–57 (2010)

    Google Scholar 

  22. Hyman, J.M., Shashkov, M.: Mimetic discretizations for Maxwell’s equations. J. Comput. Phys. 151, 881–909 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, J.: Numerical convergence and physical fidelity analysis for Maxwell’s equations in metamaterials. Comput. Methods Appl. Mech. Eng. 198, 3161–3172 (2009)

    Article  MATH  Google Scholar 

  24. Li, J.: Finite element study of the Lorentz model in metamaterials. Comput. Methods Appl. Mech. Eng. 200, 626–637 (2011)

    Article  MATH  Google Scholar 

  25. Li, J., Wood, A.: Finite element analysis for wave propagation in double negative metamaterials. J. Sci. Comput. 32, 263–286 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lu, T., Zhang, P., Cai, W.: Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions. J. Comput. Phys. 200, 549–580 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lin, Q., Yan, N.: Global superconvergence for Maxwell’s equations. Math. Comput. 69, 159–176 (1999)

    Article  MathSciNet  Google Scholar 

  28. Maradei, F., Ke, H., Hubing, T.H.: Full-wave model of frequency-dispersive media with Debye dispersion relation by circuit-oriented FEM. IEEE Trans. Electromagn. Compat. 51, 312–319 (2009)

    Article  Google Scholar 

  29. Markos, P., Soukoulis, C.M.: Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials. Princeton University Press, Princeton (2008)

    MATH  Google Scholar 

  30. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  31. Nédélec, J.-C.: Mixed finite elements in R 3. Numer. Math. 35, 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000)

    Article  Google Scholar 

  33. Pendry, J.B., Holden, A.J., Robbins, D.J., Stewart, W.J.: Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999)

    Article  Google Scholar 

  34. Qiao, Z., Yao, C., Jia, S.: Superconvergence and extrapolation analysis of a nonconforming mixed finite element approximation for time-harmonic Maxwell’s equations. J. Sci. Comput. 46, 1–19 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rieben, R.N., Rodrigue, G.H., White, D.A.: A high order mixed vector finite element method for solving the time dependent Maxwell equations on unstructured grids. J. Comput. Phys. 204, 490–519 (2005)

    Article  MATH  Google Scholar 

  36. Rodriguez-Esquerre, V.F., Koshiba, M., Hernandez-Figueroa, H.E.: Frequency-dependent envelope finite-element time-domain analysis of dispersion materials. Microw. Opt. Technol. Lett. 44, 13–16 (2005)

    Article  Google Scholar 

  37. Scheid, C., Lanteri, S.: Convergence of a discontinuous Galerkin scheme for the mixed time domain Maxwell’s equations in dispersive media. INRIA preprint, No. 7364, May (2011)

  38. Shaw, S.: Finite element approximation of Maxwell’s equations with Debye memory, Advances in Numerical Analysis, Vol. 2010, Article ID 923832, 28 pp. (2010)

  39. Shelby, R.A., Smith, D.R., Nemat-Nasser, S.C., Schultz, S.: Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial. Appl. Phys. Lett. 78(4), 489–491 (2001)

    Article  Google Scholar 

  40. Shelby, R.A., Smith, D.R., Schultz, S.: Experimental verification of a negative index of refraction. Science 292, 489–491 (2001)

    Article  Google Scholar 

  41. Smith, D.R., Kroll, N.: Negative refractive index in left-handed materials. Phys. Rev. Lett. 85, 2933–2936 (2000)

    Article  Google Scholar 

  42. Smith, D.R., Padilla, W.J., Vier, D.C., Nemat-Nasser, S.C., Schultz, S.: Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 84, 4184–4187 (2000)

    Article  Google Scholar 

  43. Solin, P., Dubcova, L., Cerveny, J., Dolezel, I.: Adaptive hp-FEM with arbitrary-level hanging nodes for Maxwell’s equations. Adv. Appl. Math. Mech. 2, 518–532 (2010)

    MathSciNet  Google Scholar 

  44. Stoykov, N.S., Kuiken, T.A., Lowery, M.M., Taflove, A.: Finite-element time-domain algorithms for modeling linear Debye and Lorentz dielectric dispersions at low frequencies. IEEE Trans. Biomed. Eng. 50, 1100–1107 (2003)

    Article  Google Scholar 

  45. Taflove, A., Hagness, S.C.: Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House, Boston (2000)

    MATH  Google Scholar 

  46. Wang, B., Xie, Z., Zhang, Z.: Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media. J. Comput. Phys. 229, 8552–8563 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Science Foundation grant DMS-0810896, and in part by the NSFC Key Project 11031006 and Hunan Provincial NSF project 10JJ7001.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jichun Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, J., Huang, Y. & Yang, W. Numerical Study of the Plasma-Lorentz Model in Metamaterials. J Sci Comput 54, 121–144 (2013). https://doi.org/10.1007/s10915-012-9608-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9608-5

Keywords

Navigation