PIER
 
Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 95 > pp. 179-198

FAST AND ACCURATE ANALYSIS OF LARGE METAMATERIAL STRUCTURES USING THE MULTILEVEL FAST MULTIPOLE ALGORITHM

By L. Gurel, , A. Unal, and T. Malas

Full Article PDF (636 KB)

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.

Citation:
L. Gurel, , A. Unal, and T. Malas, "Fast and accurate analysis of large metamaterial structures using the multilevel fast multipole algorithm," Progress In Electromagnetics Research, Vol. 95, 179-198, 2009.
doi:10.2528/PIER09060106
http://www.jpier.org/PIER/pier.php?paper=09060106

References:
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

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

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.


© Copyright 2014 EMW Publishing. All Rights Reserved