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.
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
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
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
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
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
7. Eleftheriades, G. V. and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications, Wiley-IEEE, New Jersey, 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
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
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
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
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.
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
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
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.
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
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
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
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
20. Saad, Y., Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 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
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
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
24. Chew, W. C., J.-M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Boston, MA, 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
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
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.
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
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
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
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
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.
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
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
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
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
38. Smith, D. R., "Negative refractive index in left-handed materials," Phys. Rev. Lett., Vol. 85, 2933-2936, Oct. 2000.
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.
40. Gurel, L., T. Malas, and O. Ergul, "Efficient preconditioning strategies for the multilevel fast multipole algorithm," PIERS Proceedings, 1620-1624, 2007.