Skip to main content
SpringerLink
Log in

Developing a Time-Domain Finite Element Method for the Lorentz Metamaterial Model and Applications

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

Abstract

In this paper, we propose a new time-domain finite element method for solving the time dependent Maxwell’s equations coupled with the Lorentz metamaterial model. The Lorentz metamaterial Maxwell’s equations are much more complicated than the standard Maxwell’s equations in free space. Our fully discrete scheme uses edge elements to approximate the unknowns in space, and uses the leap-frog scheme in time discretization. Numerical stability and the optimal error estimate in the \(L^2\) norm are proved for our proposed scheme. Extensive numerical results are presented to confirm the theoretical analysis and applications of our scheme to model many interesting phenomena happened when wave propagates in the Lorentz metamaterials. Examples include the convergence effect happened in the concave lenses formed by the negative refraction index metamatrials, and total reflection and total transmission observed in the zero index metamaterials.

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

Access this article

Log in via an institution

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Ainsworth, M., Coyle, J.: Hierarchic hp-edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Eng. 190, 6709–6733 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. Bao, G., Li, P., Wu, H.: An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures. Math. Comput. 79, 1–34 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Boffi, D., Fernandez, P., Perugia, I.: Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36, 1264–1290 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brenner, S.C., Gedicke, J., Sung, L.-Y.: Hodge decomposition for two-dimensional time-harmonic Maxwell’s equations: impedance boundary condition. Math. Methods Appl. Sci. (2015). doi:10.1002/mma.3398

  7. 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 

  8. Chung, E.T., Ciarlet Jr, P., Yu, T.F.: Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwells equations on Cartesian grids. J. Comput. Phys. 235, 14–31 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. Creusé, E., Nicaise, S., Tang, Z., Le Menach, Y., Nemitz, N., Piriou, F.: Residual-based a posteriori estimators for the \(A-\varphi \) magnetodynamic harmonic formulation of the Maxwell system. Math. Models Methods Appl. Sci. 22, 1150028 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  12. Demkowicz, L., Kurtz, J., Pardo, D., Paszynski, M., Rachowicz, W., Zdunek, A.: Computing with hp finite elements. II. Frontiers: three-dimensional Elliptic and Maxwell problems with applications. Chapman & Hall/CRC, Boca Raton (2007)

    MATH  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. Fu, Y., Xu, L., Hang, Z.H., Chen, H.: Unidirectional transmission using array of zero-refractive-index metamaterials. Appl. Phys. Lett. 104(19), 193509 (2014)

    Article  Google Scholar 

  15. 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 

  16. Grote, M.J., Schneebeli, A., Schötzau, D.: Interior penalty discontinuous Galerkin method for Maxwell’s equations: energy norm error estimates. J. Comput. Appl. Math. 204, 375–386 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hao, J., Yan, W., Qiu, M.: Super-reflection and cloaking based on zero index metamaterial. Appl. Phys. Lett. 96(10), 101109 (2010)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  19. Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 186–221 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Houston, P., Perugia, I., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator: non-stabilized formulation. J. Sci. Comput. 22, 315–346 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang, Y., Li, J., Yang, W.: Modeling backward wave propagation in metamaterials by the finite element time-domain method. SIAM J. Sci. Comput. 35, B248–B274 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Huang, Y., Li, J.: Total reflection and cloaking by triangular defects embedded in zero index metamaterials. Adv. Appl. Math. Mech. 7, 135–144 (2015)

    Article  MathSciNet  Google Scholar 

  24. Lanteri, S., Scheid, C.: Convergence of a discontinuous Galerkin scheme for the mixed time-domain Maxwell’s equations in dispersive media. IMA J. Numer. Anal. 33, 432–459 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. 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 

  27. Li, J., Huang, Y.: Time-Domain Finite Element Methods for Maxwel’s Equations in Metamaterials, vol. 43. Springer, New York (2013)

    Book  Google Scholar 

  28. 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 

  29. 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 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  32. Monk, P., Süli, E.: A convergence analysis of Yee’s scheme on nonuniform grids. SIAM J. Numer. Anal. 32, 393–412 (1994)

    Article  MATH  Google Scholar 

  33. Mu, L., Wang, J., Ye, X., Zhang, S.: A weak Galerkin finite element method for the Maxwell equations. J. Sci. Comput. 65, 363–386 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nguyen, V.C., Chen, L., Halterman, K.: Total transmission and total reflection by zero index metamaterials with defects. Phys. Rev. Lett. 105(23), 233908 (2010)

    Article  Google Scholar 

  35. Sármány, D., Botchev, M.A., van der Vegt, J.J.W.: Dispersion and dissipation error in high-order Runge–Kutta discontinuous Galerkin discretisations of the Maxwell equations. J. Sci. Comput. 33, 47–74 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shi, D., Yao, C.: Nonconforming finite element approximation of time-dependent Maxwell’s equations in Debye medium. Numer. Methods Partial Differ. Equ. 30, 1654–1673 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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 

  38. Wu, Y., Li, J.: Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects. Appl. Phys. Lett. 102(18), 183105 (2013)

    Article  Google Scholar 

  39. Yang, Z., Wang, L.-L.: Accurate simulation of circular and elliptic cylindrical invisibility cloaks. Commun. Comput. Phys. 17, 822–849 (2015)

    Article  MathSciNet  Google Scholar 

  40. Zhong, L., Chen, L., Shu, S., Wittum, G., Xu, J.: Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations. Math. Comput. 81, 623–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, China

    Wei Yang & Yunqing Huang

  2. Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, 89154-4020, USA

    Jichun Li

Authors
  1. Wei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunqing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jichun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jichun Li.

Additional information

This work was supported by NSFC Projects 11401506 and 11271310, NSFC Key Project 91430213, Hunan Education Department Project (15B236), and NSF Grant DMS-1416742.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, W., Huang, Y. & Li, J. Developing a Time-Domain Finite Element Method for the Lorentz Metamaterial Model and Applications. J Sci Comput 68, 438–463 (2016). https://doi.org/10.1007/s10915-015-0144-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0144-y

Keywords

Mathematics Subject Classification

Search

Navigation