volume | PIER Journals

Abstract

In this paper, we present a new model using a Four-dimensional (4D) Element-Oriented physical concepts based on a topological approach in electromagnetism. Its general finite formulation on dual staggered grids reveals a flexible Finite-Difference Time-Domain (FDTD) method with reasonable local approximating functions. This flexible FDTD method is developed without recourse to the traditional Taylor based forms of the individual differential operators. This new formulation generalizes both the standard FDTD (S-FDTD) and the nonstandard FDTD (NS-FDTD) methods. Moreover, it can be used to generate new numerical methods. As proof, we deduce a new nonstandard scheme more accurate than the S-FDTD and the known nonstandard NS-FDTD methods. Through some numerical examples, we validate this proposal, and we show the power and the advantage of this Element-Oriented Model.

1. Sadiku, M. N. O., Numerical Techniques in Electromagnetics, 2nd Ed., CRC Press, 2001.

2. Salon, S. and M. V. K. Chari, Numerical Methods in Electromagnetism, Academic Press, 1999.

3. Rjasanow, S. and O. Steinbach, The Fast Solution of Boundary Integral Equations, Springer, 2007.

4. Baranger, J., J. F. Maitre, and F. Oudin, "Connection between finite volume and mixed finite element methods," RAIRO, Modelisation Math. Anal. Numer., Vol. 30, 445-465, 1996.

5. De La Bourdonnay, A. and S. Lala, "Duality between finite elements and finite volumes and Hodge operator," Numerical Methods in Engineering'96, 557-561, Wiley & Sons, Paris, 1996.

6. Bossavit, A. and L. Kettunen, "Yee-like schemes on staggered cellular grids: A synthesis between FIT and FEM approaches," IEEE Trans. Magn., Vol. 36, No. 4, 861-867, 2000.
doi:10.1109/20.877580

7. Teixeira, F. L., "Geometric aspects of the simplicial discretization of Maxwell's equations," Progress In Electromagnetics Research, Vol. 32, 171-188, 2001.
doi:10.2528/PIER00080107

8. Tonti, E., "Finite formulation of the electromagnetic field," Progress In Electromagnetics Research, Vol. 32, 1-44, 2001.
doi:10.2528/PIER00080101

9. Mattiussi, C., "An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology," J. Comp. Phys., Vol. 9, 295-319, 1997.

10. Gross, P. W. and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, Cambridge University Press, 2004.
doi:10.1017/CBO9780511756337

11. Maxwell, J. C., A Treaties on Electricity and Magnetism, Clarendon Press, 1892, Reprinted in 2002.

12. Stern, A., Y. Tong, M. Desbrun, and J. E. Marsden, "Computational electromagnetism with variational integrators and discrete differential forms," arXiv: 0707.4470 [math.NA], 2007.

13. Hiptmair, R., "Discrete Hodge-operators: An algebraic perspective," Progress In Electromagnetics Research, Vol. 32, 247-269, 2001.
doi:10.2528/PIER00080110

14. Hiptmair, R., "Discrete Hodge operators," Numer. Math., Vol. 90, No. 2, 65-289, 2001.
doi:10.1007/s002110100295

15. Auchmann, B. and S. Kurz, "A geometrically defined discrete Hodge operator on simplicial cells," IEEE Trans. Magn., Vol. 42, No. 4, 643-646, 2006.
doi:10.1109/TMAG.2006.870932

16. Mickens, R. E., Applications of Nonstandard Finite Difference Schemes, World Scientific, 2000.
doi:10.1142/9789812813251

17. Marrone, M., "Computational aspects of the cell method in electrodynamics," Progress In Electromagnetics Research, Vol. 32, 317-356, 2001.
doi:10.2528/PIER00080113

18. Garcia, S. G. and T.-W. Lee, "On the accuracy of the ADI-FDTD method," IEEE Antennas and Wireless Propagation Letters, Vol. 1, No. 1, 2002.

19. Ahmed, I., E. K. Chua, E. P. Li, and Z. Chen, "Development of the three dimensional unconditionally stable LOD-FDTD method," IEEE Trans. Antennas Propag., Vol. 56, No. 11, 3596-3600, 2008.
doi:10.1109/TAP.2008.2005544

20. Zheng, F., Z. Chen, and J. Zhang, "A finite-difference time-domain method without the Courant stability conditions," IEEE Microw. Guided Wave Lett., Vol. 9, No. 11, 441-443, 1999.
doi:10.1109/75.808026

21. Anguelov, R. and S. Lubuma, "On non-standard finite difference models of reaction-diffusion equations," Journal of Applied Mathematics, Vol. 175, No. 1, 2005.

22. Bossavit, A., "Generalized finite differences' in computational electromagnetics," Progress In Electromagnetics Research, Vol. 32, 45-64, 2001.
doi:10.2528/PIER00080102

23. Teixeira, F. L. and W. C. Chew, "Lattice electromagnetic theory from a topological viewpoint," Journal of Mathematical Physics, Vol. 40, No. 1, 1999.
doi:10.1063/1.532767

24. Alotto, P., F. Freschi, and M. Repetto, "Multiphysics problems via the cell method: The role of Tonti diagrams," IEEE Trans. Magn., 2959-2962, Aug. 2010.
doi:10.1109/TMAG.2010.2044487

25. Tarhasaari, T., L. Kettunen, and A. Bossavit, "Some realizations of a discrete Hodge operator: A reinterpretation of finite element techniques," IEEE Trans. Magn., Vol. 35, No. 3, 1494-1497, 1999.
doi:10.1109/20.767250

26. Tarhasaari, T. and L. Kettunen, "Topological approach to computational electromagnetism," Progress In Electromagnetics Research, Vol. 32, 189-206, 2001.
doi:10.2528/PIER00080108

27. Truesdell, C. and R. A. Toupin, The Classical Field Theories, Harrdbuch der Physik, Vol. 311, 226-793, edited by S. Flugge, 1960.

28. Tarhasaari, T. and L. Kettunen, "Topological approach to computational electromagnetism," Progress In Electromagnetic Research, Vol. 32, 189-206, 2001.
doi:10.2528/PIER00080108

29. Kirawanich, P., et al. "Methodology for interference analysis using electromagnetic topology techniques," Applied Physics Letters, Vol. 84, 2004.

30. Lindel, I. V., Differential Forms in Electromagnetics, IEEE Press, 2004.
doi:10.1002/0471723096

31. Lindel, I. V., "Electromagnetic wave equation in differential-form representation," Progress In Electromagnetics Research, Vol. 54, 321-333, 2005.
doi:10.2528/PIER05021002

32. Lindell, I. V., "Electromagnetic fields in self-dual media in differential-form representation," Progress In Electromagnetics Research, Vol. 58, 319-333, 2006.
doi:10.2528/PIER05072201

33. Mattiussi, C., "The geometry of time-stepping," Progress In Electromagnetics Research, Vol. 32, 123-149, 2001.
doi:10.2528/PIER00080105

34. Anguelov, R. and S. Lubuma, "Nonstandard dfinite-difference methods by nonlocal approwimations," Mathematics and Computer in Simulation, 2003.

35. Magrez, H. and A. Ziyyat, "Modélisation orientée objet en electromagntisme," Congrés Méditerranen des Télécommunications CMT, Casablanca, 2010.

36. Balanis, C. A., Advanced Engineering Electromagnetics, Wiley, 1989.

37. Pinheiro, H., J. P. Webb, and I. Tsukerman, "Flexible local approximation models for wave scattering in photonic crystal devices," IEEE Trans. Magn., Vol. 43, No. 4, 1321-1324, 2007.
doi:10.1109/TMAG.2006.891004

38. Tsukerman, I., "A class of difference schemes with flexible local approximation," The Journal of Computational Physics, Vol. 211, No. 2, 659-699, 2006.
doi:10.1016/j.jcp.2005.06.011

Prof. Hamid Magrez

Dr. Abdelhak Ziyyat