volume | PIER Journals

Abstract

We report fast and accurate simulations of metamaterial structures constructed with large numbers of unit cells containing split-ring resonators and thin wires. Scattering problems involving various metamaterial walls are formulated rigorously using the electric-field integral equation, discretized with the Rao-Wilton-Glisson basis functions. Resulting dense matrix equations are solved iteratively, where the matrix-vector multiplications are performed efficiently with the multilevel fast multipole algorithm. For rapid solutions at resonance frequencies, convergence of the iterations is accelerated by using robust preconditioning techniques, such as the sparse-approximate-inverse preconditioner. Without resorting to homogenization approximations and periodicity assumptions, we are able to obtain accurate solutions of realistic metamaterial problems discretized with millions of unknowns.

1. Veselago, V. G., "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp., Vol. 47, 509-514, Jan.-Feb. 1968. Google Scholar

2. Smith, D. R., W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity ," Phys. Rev. Lett., Vol. 84, 4184-4187, May 2000.
doi:10.1103/PhysRevLett.84.4184 Google Scholar

3. Shelby, R. A., D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, "Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial," Appl. Phys. Lett., Vol. 78, 489-491, Jan. 2001.
doi:10.1063/1.1343489 Google Scholar

4. Shelby, R. A., D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science, Vol. 292, 77-79, Apr. 2001.
doi:10.1126/science.1058847 Google Scholar

5. Moss, C. D., T. M. Gregorczyk, Y. Zhang, and J. A. Kong, "Numerical studies of left handed metamaterials," Progress In Electromagnetics Research, Vol. 35, 315-334, 2002.
doi:10.2528/PIER02052409 Google Scholar

6. Gokkavas, M., K. Guven, I. Bulu, K. Aydin, R. S. Penciu, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, "Experimental demonstration of a left-handed metamaterial operating at 100 GHz," Phys. Rev. B, Vol. 73, No. 19, 193103-1-193103-4, May 2006.
doi:10.1103/PhysRevB.73.193103 Google Scholar

7. Eleftheriades, G. V. and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications, Wiley-IEEE, 2005.

8. Engheta, N. and R. W. Ziolkowski, "A positive future for double-negative metamaterials," IEEE Trans. Microw. Theory Tech., Vol. 53, No. 4, 1535-1556, Apr. 2005.
doi:10.1109/TMTT.2005.845188 Google Scholar

9. Chen, H., B. I. Wu, and J. A. Kong, "Review of electromagnetic theory in left-handed materials," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 15, 2137-2151, 2006.
doi:10.1163/156939306779322585 Google Scholar

10. Grbic, A. and G. V. Eleftheriades, "Overcoming the diffraction limit with a planar left-handed transmission-line lens," Phys. Rev. Lett., Vol. 92, No. 11, 117403-1-117403-4, Mar. 2004.
doi:10.1103/PhysRevLett.92.117403 Google Scholar

11. Aydin, K., I. Bulu, and E. Ozbay, "Subwavelength resolution with a negative-index metamaterial superlens," Appl. Phys. Lett., Vol. 90, No. 25, 254102-1-254102-3, Jun. 2007.
doi:10.1063/1.2750393 Google Scholar

12. Schurig, D., J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science, Vol. 314, 977-980, Nov. 2006. Google Scholar

13. Weng, Z., N. Weng, Y. Jiao, and F. Zhang, "A directive patch antenna with metamaterial structure," Microw. Opt. Technol. Lett., Vol. 49, No. 2, 456-459, Feb. 2007.
doi:10.1002/mop.22146 Google Scholar

14. Wu, B.-I., W. Wang, J. Pacheco, X. Chen, T. Grzegorczyk, and J. A. Kong, "A study of using metamaterials as antenna substrate to enhance gain," Progress In Electromagnetics Research, Vol. 51, 295-328, 2005.
doi:10.2528/PIER04070701 Google Scholar

15. Poggio, A. J. and E. K. Miller, "Integral equation solutions of three-dimensional scattering problems," Computer Techniques for Electromagnetics, No. 4, R. Mittra (ed.), Chap. 4, Pergamon Press, Oxford, 1973. Google Scholar

16. Rao, S. M., D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propag., Vol. 30, No. 3, 409-418, May 1982.
doi:10.1109/TAP.1982.1142818 Google Scholar

17. Song, J., C.-C. Lu, and W. C. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propag., Vol. 45, No. 10, 1488-1493, Oct. 1997.
doi:10.1109/8.633855 Google Scholar

18. Ergul, O. and L. Gurel, "Hierarchical parallelisation strategy for multilevel fast multipole algorithm in computational electromagnetics," Electron. Lett., Vol. 44, No. 1, 3-5, Jan. 2008.
doi:10.1049/el:20082282 Google Scholar

19. Ergul, O. and L. Gurel, "Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems," IEEE Trans. Antennas Propag., Vol. 56, No. 8, 2335-2345, Aug. 2008.
doi:10.1109/TAP.2008.926757 Google Scholar

20. Saad, Y., Iterative Methods for Sparse Linear Systems, SIAM, 2003.

21. Coifman, R., V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: A pedestrian prescription," IEEE Antennas Propag. Mag., Vol. 35, No. 3, 7-12, Jun. 1993.
doi:10.1109/74.250128 Google Scholar

22. Koc, S., J. M. Song, and W. C. Chew, "Error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem," SIAM J. Numer. Anal., Vol. 36, No. 3, 906-921, 1999.
doi:10.1137/S0036142997328111 Google Scholar

23. Ergul, O. and L. Gurel, "Enhancing the accuracy of the interpolations and anterpolations in MLFMA," IEEE Antennas Wireless Propag. Lett., Vol. 5, 467-470, 2006.
doi:10.1109/LAWP.2006.885010 Google Scholar

24. Chew, W. C., J.-M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, 2001.

25. Ergul, O. and L. Gurel, "Optimal interpolation of translation operator in multilevel fast multipole algorithm," IEEE Trans. Antennas Propag., Vol. 54, No. 12, 3822-3826, Dec. 2006.
doi:10.1109/TAP.2006.886562 Google Scholar

26. Brandt, A., "Multilevel computations of integral transforms and particle interactions with oscillatory kernels ," Comput. Phys. Comm., Vol. 65, 24-38, Apr. 1991.
doi:10.1016/0010-4655(91)90151-A Google Scholar

27. Carpentieri, B., I. S. Duff, and L. Giraud, "Experiments with sparse preconditioning of dense problems from electromagnetic applications ,", Tech. Rep. TR/PA/00/04, CERFACS, Toulouse, France, 1999. Google Scholar

28. Malas, T. and L. Gurel, "Incomplete LU preconditioning with the multilevel fast multipole algorithm for electromagnetic scattering," SIAM J. Sci. Comput., Vol. 29, No. 4, 1476-1494, June 2007.
doi:10.1137/060659107 Google Scholar

29. Paige, C. C. and M. A. Saunders, "LSQR: An algorithm for sparse linear equations and sparse least squares," ACM Trans. Math. Software, Vol. 8, 43-71, Mar. 1982.
doi:10.1145/355984.355989 Google Scholar

30. Ergul, O. and L. Gurel, "Efficient solution of the electric-field integral equation using the iterative LSQR algorithm," IEEE Antennas Wireless Propag. Lett., Vol. 7, 36-39, 2008.
doi:10.1109/LAWP.2007.908008 Google Scholar

31. Sertel, K. and J. L. Volakis, "Incomplete LU preconditioner for FMM implementations," Microw. Opt. Technol. Lett., Vol. 26, No. 4, 265-267, Aug. 2000.
doi:10.1002/1098-2760(20000820)26:4<265::AID-MOP18>3.0.CO;2-O Google Scholar

32. Benzi, M., "Preconditioning techniques for large linear systems: A survey," J. Comput. Phys., Vol. 182, No. 2, 418-477, Nov. 2002.
doi:10.1006/jcph.2002.7176 Google Scholar

33. Gurel, L. and O. Ergul, "Comparisons of FMM implementations employing different formulations and iterative solvers," Proc. IEEE Antennas and Propagation Soc. Int. Symp., Vol. 1, 19-22, 2003. Google Scholar

34. Gurel, L. and O. Ergul, "Extending the applicability of the combined-field integral equation to geometries containing open surfaces," IEEE Antennas Wireless Propag. Lett., Vol. 5, 515-516, 2006.
doi:10.1109/LAWP.2006.887552 Google Scholar

35. Ubeda, E., J. M. Rius, and J. Romeu, "Preconditioning techniques in the analysis of finite metamaterial slabs," IEEE Trans. Antennas Propag., Vol. 54, No. 1, 265-268, Jan. 2006.
doi:10.1109/TAP.2005.861508 Google Scholar

36. Pendry, J. B., A. Holden, J. D. Robbins, and J. W. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microw. Theory Tech., Vol. 47, No. 11, 2075-2084, Nov. 1999.
doi:10.1109/22.798002 Google Scholar

37. Pendry, J. B., A. Holden, J. D. Robbins, and J. W. Stewart, "Low-frequency plasmons in thin wire structures," J. Phys., Condens. Matter, Vol. 10, 4785-4809, Mar. 1998.
doi:10.1088/0953-8984/10/22/007 Google Scholar

38. Smith, D. R., "Negative refractive index in left-handed materials," Phys. Rev. Lett., Vol. 85, 2933-2936, Oct. 2000. Google Scholar

39. Ziolkowski, R. W. and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E, Vol. 64, No. 5, 056625-1-056625-15, Oct. 2001. Google Scholar

40. Gurel, L., T. Malas, and O. Ergul, "Efficient preconditioning strategies for the multilevel fast multipole algorithm," PIERS Proceedings, 1620-1624, 2007. Google Scholar

Dr. Levent Gurel

Dr. Ozgur Ergul

Dr. Alper Unal

Dr. Tahir Malas