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

Li, Jichun; Huang, Yunqing; Yang, Wei

doi:10.1090/S0025-5718-2014-02911-6

Well-posedness study and finite element simulation of time-domain cylindrical and elliptical cloaks
HTML articles powered by AMS MathViewer

by Jichun Li, Yunqing Huang and Wei Yang PDF

Math. Comp. 84 (2015), 543-562 Request permission

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

A. Alú and N. Engheta, Achieving transparency with plasmonic and metamaterial coatings, Phys. Rev. E 72 (2005) 016623.
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, DOI 10.1007/s00205-012-0605-5
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, DOI 10.1090/conm/577/11460
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, DOI 10.1007/s00220-012-1615-8
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, DOI 10.1007/s00220-012-1620-y
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, DOI 10.1137/120903610
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, DOI 10.1090/S0025-5718-09-02257-1
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, DOI 10.1051/m2an:2000136
Jean-Pierre Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), no. 2, 185–200. MR 1294924, DOI 10.1006/jcph.1994.1159
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, DOI 10.1090/S0025-5718-2011-02540-8
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, DOI 10.1090/S0025-5718-06-01950-8
H. Chen, C.T. Chan and P. Sheng, Transformation optics and metamaterials, Nature Materials 9 (2010) 387–396.
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, DOI 10.1137/S0036142998349977
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, DOI 10.1016/j.jcp.2012.10.019
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, DOI 10.1007/s002110050417
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.
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
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, DOI 10.1142/S0218202509004121
M. Fridman, A. Farsi, Y. Okawachi and A.L. Gaeta, Demonstration of temporal cloaking, Nature 481 (2012) 62–65.
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, DOI 10.4310/MRL.2003.v10.n5.a11
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, DOI 10.1137/080716827
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.
F. Guevara Vasquez, G.W. Milton and D. Onofrei, Broadband exterior cloaking, Opt. Express 17 (2009) 14800–14805.
J. Hao, W. Yan and M. Qiu, Super-reflection and cloaking based on zero index metamaterial, Appl. Phys. Lett. 96, 101109 (2010).
Y. Hao and R. Mittra, FDTD Modeling of Metamaterials: Theory and Applications, Artech House Publishers, 2008.
Jan S. Hesthaven and Tim Warburton, Nodal discontinuous Galerkin methods, Texts in Applied Mathematics, vol. 54, Springer, New York, 2008. Algorithms, analysis, and applications. MR 2372235, DOI 10.1007/978-0-387-72067-8
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, DOI 10.1007/s00211-005-0604-7
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.
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, DOI 10.1002/cpa.20326
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, DOI 10.1088/0266-5611/24/1/015016
Ulf Leonhardt, Optical conformal mapping, Science 312 (2006), no. 5781, 1777–1780. MR 2237569, DOI 10.1126/science.1126493
Ulf Leonhardt and Thomas Philbin, Geometry and light, Dover Publications, Inc., Mineola, NY, 2010. The science of invisibility. MR 2798945
U. Leonhardt and T. Tyc, Broadband invisibility by non-Euclidean cloaking, Science 323 (2009) 110–112.
Jichun Li and Yunqing Huang, Mathematical simulation of cloaking metamaterial structures, Adv. Appl. Math. Mech. 4 (2012), no. 1, 93–101. MR 2876653, DOI 10.4208/aamm.10-m11109
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, DOI 10.1007/978-3-642-33789-5
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, DOI 10.1016/j.jcp.2011.12.026
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, DOI 10.1088/0266-5611/28/7/075011
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).
Hongyu Liu and Ting Zhou, On approximate electromagnetic cloaking by transformation media, SIAM J. Appl. Math. 71 (2011), no. 1, 218–241. MR 2776835, DOI 10.1137/10081112X
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.
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, DOI 10.1098/rspa.2006.1715
Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
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, DOI 10.1002/cpa.20392
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).
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.
J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science 312 (2006), no. 5781, 1780–1782. MR 2237570, DOI 10.1126/science.1125907
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.
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.
Simon Shaw, Finite element approximation of Maxwell’s equations with Debye memory, Adv. Numer. Anal. , posted on (2010), Art. ID 923832, 28. MR 2747091, DOI 10.1155/2010/923832
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, DOI 10.1016/j.jcp.2010.07.038
M. Yan, W. Yan and M. Qiu, Invisibility cloaking by coordinate transformation, Progress in Optics 52 (2009) 261–304.
Y. Zhao, C. Argyropoulos and Y. Hao, Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures, Optics Express 16 (2008) 6717–6730.
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, DOI 10.1090/S0025-5718-2011-02544-5