A three-dimensional (3D) time domain approach can be particularly valuable for the analysis of many different types of practical structures. In this regard, the finite difference time domain (FDTD) method is a popular technique, being used successfully to analyze the electromagnetic properties of many structures, including a range of optical or photonic devices. This FDTD method offers several major advantages: a minimum level of calculation is required for each of the cells into which the structure is divided, as well as data parallelism and explicit and easy implementation: however, the use of a cuboid grid makes the method very resource intensive for large simulations, especially those in 3D. Although the finite element (FE) approach is superior for the discretization of two-dimensional (2D) and 3D structures, most of the FE-based time domain approaches reported so far suffer from limitations due to the implicit or iterative form or the mass matrix formulation, for example. This paper presents an FE based time domain technique for 3D structures which uses a unique perforated mesh system. It calculated the numerical dispersion characteristics for the FDTD and the proposed method and compared. This paper finally discusses how to utilize the improved numerical dispersion characteristics of the proposed method to increase the simulation speed beyond the FDTD 3D method by using Intel micro-processors.
2. Taflove, A. and M. Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," IEEE Transactions on Microwave Theory and Techniques, Vol. 23, No. 8, 623-630, 1975.
3. Sun, G. and C. Trueman, "Some fundamental characteristics of the one-dimensional alternate-direction-implicit finite-difference time-domain method," IEEE Transactions on Microwave Theory and Techniques, Vol. 52, No. 1, 46-52, 2004.
4. Lee, J., R. Lee, and A. Cangellaris, "Time-domain finite-element methods," IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, 430-442, 1997.
5. Yee, K., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, No. 3, 302-307, 1966.
6. Gedney, S. and U. Navsariwala, "An unconditionally stable finite element time-domain solution of the vector wave equation," IEEE Microwave and Guided Wave Letters, Vol. 5, No. 10, 332-334, 1995.
7. Cangellaris, A., C. Lin, and K. Mei, "Point-matched time domain finite element methods for electromagnetic radiation and scattering," IEEE Transactions on Antennas and Propagation, Vol. 35, No. 10, 1160-1173, 1987.
8. Hesthaven, T. W. J. S., "High-order/spectral methods on unstructured grids i. Time-domain solution of Maxwell's equations,", Tech. Rep. 2001-6, ICASE NASA Langley Research Center, Hampton, Virginia, March 2001.
9. Songoro, H., M. Vogel, and Z. Cendes, "Keeping time with Maxwell's equations," IEEE Microwave Magazine, Vol. 11, No. 2, 42-49, 2010.
10. Raiyan Kabir, S. M., B. M. A. Rahman, A. Agrawal, and K. T. V. Grattan, "Elimination of numerical dispersion from electromagnetic time domain analysis by using resource efficient finite element technique," Progress In Electromagnetics Research, Vol. 137, 487-512, 2013.
11. Courant, R., K. Friedrichs, and H. Lewy, "Über die partiellen differenzengleichungen der 568 mathematischen physik," Mathematische Annalen, Vol. 100, No. 1, 32-74, 1928.
12. Cangellaris, A., "Time-domain finite methods for electromagnetic wave propagation and scattering," IEEE Transactions on Magnetics, Vol. 27, No. 5, 3780-3785, 1991.
13. Leung, D., N. Kejalakshmy, B. M. A. Rahman, and K. Grattan, "Rigorous modal analysis of silicon strip nanoscale waveguides," Optics Express, Vol. 18, No. 8, 8528-8539, 2010.
14. Kirby, E., J. Hamm, K. Tsakmakidis, and O. Hess, "FDTD analysis of slow light propagation in negative-refractive-index metamaterial waveguides," Journal of Optics A: Pure and Applied Optics, Vol. 11, No. 11, 114027, 2009.
15. Rahman, B. M. A. and J. B. Davies, "Finite-element solution of integrated optical waveguides," Journal of Lightwave Technology, Vol. 2, No. 5, 682-688, 1984.
16. Rahman, B. M. A. and J. B. Davies, "Finite-element analysis of optical and microwave waveguide problems," IEEE Transactions on Microwave Theory and Techniques, Vol. 32, No. 1, 20-28, 1984.
17. Themistos, C. and B. M. A. Rahman, "Design issues of a multimode interference-based 3-dB splitter," Applied Optics, Vol. 41, No. 33, 7037-7044, 2002.
18. Juntunen, J. and T. Tsiboukis, "Reduction of numerical dispersion in FDTD method through artificial anisotropy," IEEE Transactions on Microwave Theory and Techniques, Vol. 48, No. 4, 582-588, 2000.
19. Intel Corporation, Intel® 64 and IA-32 Architectures Optimization Reference Manual, 2014.
20. Jamieson, L. H., P. T. Mueller, and H. J. Siegel, "FFT algorithms for simd parallel processing systems," Journal of Parallel and Distributed Computing, Vol. 3, No. 1, 48-71, 1986.
21. Agaian, S. and D. Gevorkian, "Synthesis of a class of orthogonal transforms. Parallel simd-algorithms and specialized processors," Pattern Recognition and Image Analysis, Vol. 2, No. 4, 394-408, 1992.
22. Ben-Asher, Y., D. Egozi, and A. Schuster, "2-D simd algorithms for perfect shuffle networks," Journal of Parallel and Distributed Computing, Vol. 16, No. 3, 250-257, 1992.
23. Apostolakis, J., P. Coddington, and E. Marinari, "New simd algorithms for cluster labeling on parallel computers," International Journal of Modern Physics C, Vol. 4, No. 4, 749-763, 1993.
24. Chen, H., N. S. Flann, and D. W. Watson, "Parallel genetic simulated annealing: A massively parallel simd algorithm," IEEE Transactions on Parallel and Distributed Systems, Vol. 9, No. 2, 126-136, 1998.
25. Hong, I., S. Chung, H. Kim, Y. Kim, Y. Son, and Z. Cho, "Ultra fast symmetry and simd-based projection-backprojection (ssp) algorithm for 3-D pet image reconstruction," IEEE Transactions on Medical Imaging, Vol. 26, No. 6, 789-803, 2007.
26. Goualard, F., "Fast and correct simd algorithms for interval arithmetic," PARA'08, Springer, 2010.
27. Berenger, J., "A perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics, Vol. 114, No. 2, 185-200, 1994.
28. Berenger, J., "Perfectly matched layer for the FDTD solution of wave-structure interaction problems," IEEE Transactions on Antennas and Propagation, Vol. 44, No. 1, 110-117, 1996.