Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 


Well-posedness study and finite element simulation of time-domain cylindrical and elliptical cloaks

Authors: Jichun Li, Yunqing Huang and Wei Yang
Journal: Math. Comp. 84 (2015), 543-562
MSC (2010): Primary 78M10, 65N30, 65F10
Published electronically: October 3, 2014
MathSciNet review: 3290954
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The goal of this paper is to prove the well-posedness for the governing equations which are used for cylindrical cloaking simulation. A new time-domain finite element scheme is developed to solve the governing equations. Numerical results demonstrating the cloaking phenomenon with the cylindrical cloak are presented. We finally extend the analysis and simulation to an elliptical cloak model.

References [Enhancements On Off] (What's this?)

  • [1] A. Alú and N. Engheta, Achieving transparency with plasmonic and metamaterial coatings, Phys. Rev. E 72 (2005) 016623.
  • [2] Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Hyundae Lee, and Graeme W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal. 208 (2013), no. 2, 667-692. MR 3035988,
  • [3] Habib Ammari, Josselin Garnier, Vincent Jugnon, Hyeonbae Kang, Hyundae Lee, and Mikyoung Lim, Enhancement of near-cloaking. Part III: Numerical simulations, statistical stability, and related questions, Multi-scale and high-contrast PDE: from modelling, to mathematical analysis, to inversion, Contemp. Math., vol. 577, Amer. Math. Soc., Providence, RI, 2012, pp. 1-24. MR 2985063,
  • [4] Habib Ammari, Hyeonbae Kang, Hyundae Lee, and Mikyoung Lim, Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem, Comm. Math. Phys. 317 (2013), no. 1, 253-266. MR 3010374,
  • [5] Habib Ammari, Hyeonbae Kang, Hyundae Lee, and Mikyoung Lim, Enhancement of near-cloaking. Part II: The Helmholtz equation, Comm. Math. Phys. 317 (2013), no. 2, 485-502. MR 3010192,
  • [6] Habib Ammari, Hyeonbae Kang, Hyundae Lee, Mikyoung Lim, and Sanghyeon Yu, Enhancement of near cloaking for the full Maxwell equations, SIAM J. Appl. Math. 73 (2013), no. 6, 2055-2076. MR 3127003,
  • [7] Gang Bao, Peijun Li, and Haijun Wu, An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures, Math. Comp. 79 (2010), no. 269, 1-34. MR 2552215 (2011a:65427),
  • [8] Rudi Beck, Ralf Hiptmair, Ronald H. W. Hoppe, and Barbara Wohlmuth, Residual based a posteriori error estimators for eddy current computation, M2AN Math. Model. Numer. Anal. 34 (2000), no. 1, 159-182 (English, with English and French summaries). MR 1735971 (2000k:65203),
  • [9] Jean-Pierre Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), no. 2, 185-200. MR 1294924 (95e:78002),
  • [10] S. C. Brenner, J. Cui, Z. Nan, and L.-Y. Sung, Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell's equations, Math. Comp. 81 (2012), no. 278, 643-659. MR 2869031,
  • [11] Susanne C. Brenner, Fengyan Li, and Li-Yeng Sung, A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations, Math. Comp. 76 (2007), no. 258, 573-595. MR 2291828 (2008c:65316),
  • [12] H. Chen, C.T. Chan and P. Sheng, Transformation optics and metamaterials, Nature Materials 9 (2010) 387-396.
  • [13] Zhiming Chen, Qiang Du, and Jun Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal. 37 (2000), no. 5, 1542-1570. MR 1759906 (2001h:78044),
  • [14] Eric T. Chung, Patrick Ciarlet Jr., and Tang Fei Yu, Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell's equations on Cartesian grids, J. Comput. Phys. 235 (2013), 14-31. MR 3017583,
  • [15] P. Ciarlet Jr. and Jun Zou, Fully discrete finite element approaches for time-dependent Maxwell's equations, Numer. Math. 82 (1999), no. 2, 193-219 (English, with English and French summaries). MR 1685459 (2000c:65083),
  • [16] S.A. Cummer, B.-I. Popa, D. Schurig, D.R. Smith and J. Pendry, Full-wave simulations of electromagnetic cloaking structures, Phys. Rev. E 74 (2006) 036621.
  • [17] Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej Paszyński, Waldemar Rachowicz, and Adam Zdunek, Computing with $ hp$-Adaptive Finite Elements. Vol. 2, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2008. Frontiers: three dimensional elliptic and Maxwell problems with applications. MR 2406401 (2009e:65172)
  • [18] Paolo Fernandes and Mirco Raffetto, Well-posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials, Math. Models Methods Appl. Sci. 19 (2009), no. 12, 2299-2335. MR 2599662 (2011b:65222),
  • [19] M. Fridman, A. Farsi, Y. Okawachi and A.L. Gaeta, Demonstration of temporal cloaking, Nature 481 (2012) 62-65.
  • [20] Allan Greenleaf, Matti Lassas, and Gunther Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett. 10 (2003), no. 5-6, 685-693. MR 2024725 (2005f:35316),
  • [21] Allan Greenleaf, Yaroslav Kurylev, Matti Lassas, and Gunther Uhlmann, Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev. 51 (2009), no. 1, 3-33. MR 2481110 (2010b:35484),
  • [22] S. Guenneau, R.C. McPhedran, S. Enoch, A.B. Movchan, M. Farhat and N.-A. P. Nicorovici, The colours of cloaks, J. Opt. 13 (2011) 024014.
  • [23] F. Guevara Vasquez, G.W. Milton and D. Onofrei, Broadband exterior cloaking, Opt. Express 17 (2009) 14800-14805.
  • [24] J. Hao, W. Yan and M. Qiu, Super-reflection and cloaking based on zero index metamaterial, Appl. Phys. Lett. 96, 101109 (2010).
  • [25] Y. Hao and R. Mittra, FDTD Modeling of Metamaterials: Theory and Applications, Artech House Publishers, 2008.
  • [26] Jan S. Hesthaven and Tim Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, analysis, and applications, Texts in Applied Mathematics, vol. 54, Springer, New York, 2008. MR 2372235 (2008k:65002)
  • [27] Paul Houston, Ilaria Perugia, Anna Schneebeli, and Dominik Schötzau, Interior penalty method for the indefinite time-harmonic Maxwell equations, Numer. Math. 100 (2005), no. 3, 485-518. MR 2194528 (2006k:65323),
  • [28] W.X. Jiang, T.J. Cui, G.X. Yu, X.Q. Lin, Q. Cheng, J.Y. Chin, Arbitrarily elliptical-cylindrical invisible cloaking, J. Phys. D: Appl. Phys. 41 (2008), 085504.
  • [29] Robert V. Kohn, Daniel Onofrei, Michael S. Vogelius, and Michael I. Weinstein, Cloaking via change of variables for the Helmholtz equation, Comm. Pure Appl. Math. 63 (2010), no. 8, 973-1016. MR 2642383 (2011j:78004),
  • [30] R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, Cloaking via change of variables in electric impedance tomography, Inverse Problems 24 (2008), no. 1, 015016, 21. MR 2384775 (2008m:78014),
  • [31] Ulf Leonhardt, Optical conformal mapping, Science 312 (2006), no. 5781, 1777-1780. MR 2237569,
  • [32] Ulf Leonhardt and Thomas Philbin, Geometry and Light: The Science of Invisibility, Dover Publications Inc., Mineola, NY, 2010. MR 2798945 (2012f:53001)
  • [33] U. Leonhardt and T. Tyc, Broadband invisibility by non-Euclidean cloaking, Science 323 (2009) 110-112.
  • [34] Jichun Li and Yunqing Huang, Mathematical simulation of cloaking metamaterial structures, Adv. Appl. Math. Mech. 4 (2012), no. 1, 93-101. MR 2876653 (2012m:78015)
  • [35] Jichun Li and Yunqing Huang, Time-domain Finite Element Methods for Maxwell's Equations in Metamaterials, Springer Series in Computational Mathematics, vol. 43, Springer, Heidelberg, 2013. MR 3013583
  • [36] Jichun Li, Yunqing Huang, and Wei Yang, Developing a time-domain finite-element method for modeling of electromagnetic cylindrical cloaks, J. Comput. Phys. 231 (2012), no. 7, 2880-2891. MR 2882105,
  • [37] Jingzhi Li, Hongyu Liu, and Hongpeng Sun, Enhanced approximate cloaking by SH and FSH lining, Inverse Problems 28 (2012), no. 7, 075011, 21. MR 2946799,
  • [38] Z. Liang, P. Yao, X. Sun and X. Jiang, The physical picture and the essential elements of the dynamical process for dispersive cloaking structures, Appl. Phys. Lett. 92, 131118 (2008).
  • [39] Hongyu Liu and Ting Zhou, On approximate electromagnetic cloaking by transformation media, SIAM J. Appl. Math. 71 (2011), no. 1, 218-241. MR 2776835 (2012f:78005),
  • [40] R. Liu, C. Ji, J.J. Mock, J.Y. Chin, T.J. Cui and D.R. Smith, Broadband ground-plane cloak, Science 323 (2009) 366-369.
  • [41] Graeme W. Milton and Nicolae-Alexandru P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2074, 3027-3059. MR 2263683 (2008e:78018),
  • [42] Peter Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447 (2005d:65003)
  • [43] Hoai-Minh Nguyen, Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking, Comm. Pure Appl. Math. 65 (2012), no. 2, 155-186. MR 2855543,
  • [44] V.C. Nguyen, L. Chen and K. Halterman, Total transmission and total reflection by zero index metamaterials with defects, Phys. Rev. Lett. 105, 233908 (2010).
  • [45] N. Okada and J.B. Cole, FDTD modeling of a cloak with a nondiagonal permittivity tensor, ISRN Optics, Article ID 536209, doi:10.5402/2012/536209, 2012.
  • [46] J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science 312 (2006), no. 5781, 1780-1782. MR 2237570,
  • [47] C. Scheid and S. Lanteri, Convergence of a Discontinuous Galerkin scheme for the mixed time domain Maxwell's equations in dispersive media, IMA J. Numer. Anal. (in press). doi:10.1093/imanum/drs008.
  • [48] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, Metamaterial electromagnetic cloak at microwave frequencies, Science 314 (2006) 977-980.
  • [49] Simon Shaw, Finite element approximation of Maxwell's equations with Debye memory, Adv. Numer. Anal. (2010), Art. ID 923832, 28. MR 2747091 (2012c:65207)
  • [50] Bo Wang, Ziqing Xie, and Zhimin Zhang, Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media, J. Comput. Phys. 229 (2010), no. 22, 8552-8563. MR 2719188 (2011f:78011),
  • [51] M. Yan, W. Yan and M. Qiu, Invisibility cloaking by coordinate transformation, Progress in Optics 52 (2009) 261-304.
  • [52] Y. Zhao, C. Argyropoulos and Y. Hao, Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures, Optics Express 16 (2008) 6717-6730.
  • [53] Liuqiang Zhong, Long Chen, Shi Shu, Gabriel Wittum, and Jinchao Xu, Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations, Math. Comp. 81 (2012), no. 278, 623-642. MR 2869030,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 78M10, 65N30, 65F10

Retrieve articles in all journals with MSC (2010): 78M10, 65N30, 65F10

Additional Information

Jichun Li
Affiliation: Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, Nevada 89154-4020

Yunqing Huang
Affiliation: Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China

Wei Yang
Affiliation: Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China

Keywords: Maxwell's equations, invisibility cloak, finite element method, metamaterials
Received by editor(s): January 16, 2013
Received by editor(s) in revised form: July 22, 2013
Published electronically: October 3, 2014
Additional Notes: The first author was supported by NSFC project 11271310 and NSF grant DMS-0810896
The third author was supported by Hunan Education Department Key Project 10A117 and Hunan Provincial Innovation Foundation for Postgraduate (CX2011B243)
This work was supported in part by the NSFC Key Project 11031006 and IRT1179 of PCSIRT
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society