volume | PIER Journals

Abstract

It is well known tha the conventional edge element method in solving vector Maxwell's eigenvalue problem will lead to the presence of nonphysical zero eigenvalues. This paper uses the mixed finite element method to suppress the presence of these nonphysical zero eigenvalues for 2D vector Maxwell's eigenvalue problem in anisotropic media. We introduce a Lagrangian multiplier to deal with the constraint of divergence-free condition. Our method is based on employing the first-order edge element basis functions to expand the electric field and linear nodal element basis functions to expand the Lagrangian multiplier. Our numerical experiments show that this method can successfully remove all nonphysical zero and nonzero eigenvalues. We verify that when the cavity has a connected perfect electric boundary, then there is no physical zero eigenvalue. Otherwise, the number of physical zero eigenvalues is one less than the number of disconnected perfect electric boundaries.

1. Zhang, K. and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics, Springer, Berlin Heidelberg, 2008.

2. Mira, F., A. A. San Blas, V. E. Boria, L. J. Rogla, and B. Gimeno, "Wideband generalized admittance matrix representation for the analysis and design of waveguide filters with coaxial excitation," Radio Science, Vol. 48, No. 1, 50-60, 2013.
doi:10.1002/rds.20013

3. Dillon, B. M. and A. A. P. Gibson, "Finite element solution of dielectric-ferrite resonators," Radio Science, Vol. 31, No. 5, 1191-1198, 1996.
doi:10.1029/96RS01365

4. Silvester, P., "Finite element solution of homogeneous waveguide problems," Alta Frequenza, Vol. 38, 313-317, 1969.

5. Rahman, B. M. and J. B. Davies, "Penalty function improvement of waveguide solution by finite elements," IEEE Trans. on Microwave Theory and Tech., Vol. 32, 922-928, 1984.
doi:10.1109/TMTT.1984.1132789

6. Winkler, J. R. and J. B. Davies, "Elimination of spurious modes in finite element analysis," J. Comput. Phys., Vol. 56, 1-14, 1984.
doi:10.1016/0021-9991(84)90079-2

7. Kobelansky, A. J. and J. P. Webb, "Eliminating spurious modes in finite-element waveguide problems by using divergence-free fields," Electron. Lett., Vol. 22, 569-570, 1986.
doi:10.1049/el:19860387

8. Chew, W. C., Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York, 1990.

9. Balanis, C. A., Advanced Engineering Electromagnetics, Wiley, New York, 1989.

10. Nedelec, J. C., "Mixed finite elements in R³," Numer. Math., Vol. 35, 315-341, 1980.
doi:10.1007/BF01396415

11. Nedelec, J. C., "A new family of mixed finite elements in R³," Numer. Math., Vol. 50, 57-81, 1986.
doi:10.1007/BF01389668

12. Zhou, X. and G. Pan, "Application of physical spline finite element method (PSFEM) to fullwave analysis of waveguides," Progress In Electromagnetics Research, Vol. 60, 19-41, 2006.
doi:10.2528/PIER05081102

13. Boffi, D., "Finite element approximation of eigenvalue problems," Acta Numerica, Vol. 19, 1-120, 2010.
doi:10.1017/S0962492910000012

14. Bondeson, A., T. Rylander, and P. Ingelstr, Computational Electromagnetics, Springer, New York, 2005.

15. Kikuchi, F., "Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism," Comp. Meth. Appl. Mech. Engng., Vol. 64, 509-521, 1987.

16. Venkatarayalu, N. V. and J. F. Lee, "Removal of spurious DC modes in edge element solutions for modeling three-dimensional resonators," IEEE Trans. on Microwave Theory and Tech., Vol. 54, 3019-3025, 2006.
doi:10.1109/TMTT.2006.877057

17. Goncalves, M. S., F. J. Arnold, L. L. Bravo-Roger, and T. S. Santos, "A novel approach to suppress the DC modes in eigenvalue problems using the finite element method," Microw. Opt. Techn. Lett., Vol. 55, 210-212, 2013.
doi:10.1002/mop.27256

18. Fratalocchi, A. and G. Ruocco, "Single-molecule imaging with X-ray free-electron lasers: Dream or reality?," Phys. Rev. Lett., Vol. 106, 105504, 2013.

19. Gentilini, S., A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, "Ultrashort pulse propagation and the Anderson localization," Opt. Lett., Vol. 34, No. 2, 130-132, 2009.
doi:10.1364/OL.34.000130

20. Fratalocchi, A., C. Conti, and G. Ruocco, "Three-dimensional ab initio investigation of light-matter interaction in Mie lasers," Physical Review A, Vol. 78, 013806, 2008.
doi:10.1103/PhysRevA.78.013806

21. Brenner, S. C., F. Li, and L. Sung, "Nonconforming Maxwell eigensolvers," J. Sci. Comput., Vol. 40, 51-85, 2009.
doi:10.1007/s10915-008-9266-9

22. Ciarlet, P. G., "Basic error estimates for elliptic problem," Handbook of Numerical Analysis, Volume II, Finite Element Methods (Part 1), Elsevier Science Publishers, North-Holland, 1991.

23. Boffi, D., P. Fernandes, L. Gastaldi, and I. Perugia, "Computational models of electromagnetic resonators: Analysis of edge element approximation," SIAM J. Numer. Anal., Vol. 36, 1264-1290, 1999.
doi:10.1137/S003614299731853X

24. Costabel, M. and M. Dauge, "A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains," Math. Methods Appl. Sci., Vol. 12, 365-368, 1990.
doi:10.1002/mma.1670120406

25. Costabel, M. and M. Dauge, "Maxwell and Lame eigenvalues on polyhedra," Math. Methods Appl. Sci., Vol. 22, 243-258, 1999.
doi:10.1002/(SICI)1099-1476(199902)22:3<243::AID-MMA37>3.0.CO;2-0

26. Costabel, M. and M. Dauge, "Singularities of electromagnetic fields in polyhedral domains," Arch. Ration. Mech. Anal., Vol. 151, 221-276, 2000.
doi:10.1007/s002050050197

Dr. Wei Jiang

Dr. Na Liu

Dr. Yifa Tang

Prof. Qing Huo Liu