A new discontinuous Galerkin Finite Element Time Domain (DG-FETD) method for Maxwell's equations is developed. It can suppress spurious modes using basis functions based on polynomials with the same order of interpolation for electric field intensity and magnetic flux density (EB scheme). Compared to FETD based on EH scheme, which reqires different orders of interpolation polynomials for electric and magnetic field intensities, this method uses fewer unknowns and reduces the computation load. The discontinuous Galerkin method is employed to implement domain decomposition for the EB scheme based FETD. In addition, a well-posed time-domain perfectly matched layer (PLM) is extended to the EB scheme to simulate the unbounded problem. Leap frog method is utilized for explicit time stepping. Numerical results demonstrate that the above proposed methods are effective and efficient for 2D time domain TMz multi-domain problems.
Luis E. Tobon,
Qing Huo Liu,
"A New 2D Non-Spurious Discontinuous Galerkin Finite Element Time Domain (DG-FETD) Method for Maxwell's Equations," Progress In Electromagnetics Research,
Vol. 143, 385-404, 2013. doi:10.2528/PIER13100901
1. Jin, J., The Finite Element Method in Electromagnetics, Wiley, New York, 2007.
3. Silvester, P. and R. Ferrari, "Finite Elements for Electrical Engineers," Cambridge University Press, 1996.
4. Andersen, L. and J. Volakis, "Development and application of a novel class of hierarchical tangential vector finite elements for electromagnetics," IEEE. Trans. Antennas Propagat., Vol. 47, No. 1, 112-120, 1999. doi:10.1109/8.753001
5. Volakis, J., T. Ozdemir, and J. Gong, "Hybrid finite-element methodologies for antennas and scattering," IEEE. Trans. Antennas Propagat., Vol. 45, No. 3, 493-507, 1997. doi:10.1109/8.558664
6. Tonti, E., "Finite formulation of the electromagnetic fiel," Progress In Electromagnetics Research, Vol. 32, 1-44, 2001. doi:10.2528/PIER00080101
7. Polycarpou, A., P. Tirkas, and C. Balanis, "The finite-element method for modeling circuits and interconnects for electronic packaging," IEEE Trans. Microw. Theory Techn., Vol. 45, No. 10, 1868-1874, 1997. doi:10.1109/22.641784
8. Lee, J., R. Lee, and A. Cangellaris, "Time-domain finite-element methods," IEEE. Trans. Antennas Propagat., Vol. 45, No. 3, 430-442, 1997. doi:10.1109/8.558658
9. Jiao, D. and J. Jin, "A general approach for the stability analysis of the time-domain finite-element method for electromagnetic simulations," IEEE. Trans. Antennas Propagat., Vol. 50, No. 11, 1624-1632, 2002. doi:10.1109/TAP.2002.803965
10. Petersson, L. and J. Jin, "A three-dimensional time-domain finite-element formulation for periodic structures," IEEE. Trans. Antennas Propagat., Vol. 54, No. 1, 12-19, 2006. doi:10.1109/TAP.2005.861547
11. Vaseghi, B., N. Takorabet, and F. Meibody-Tabar, "Transient finite element analysis of induction machines with stator winding turn fault," Progress In Electromagnetics Research, Vol. 95, 1-18, 2009. doi:10.2528/PIER09052004
12. Faghihi, F. and H. Heydari, "A combination of time domain finite element-boundary integral and with time domain physical optics for calculation of electromagnetic scattering of 3-D structures ," Progress In Electromagnetics Research, Vol. 79, 463-474, 2008. doi:10.2528/PIER07110206
13. Mur, G., "The finite-element modeling of three-dimensional time-domain electromagnetic fields in strongly inhomogeneous media," IEEE Trans. Magn., Vol. 28, No. 2, 1130-1133, 1992. doi:10.1109/20.123883
14. Gedney, S. and U. Navsariwala, "An unconditionally stable finite element time-domain solution of the vector wave equation," IEEE Microw. Guided Wave Lett., Vol. 5, No. 10, 332-334, 1995. doi:10.1109/75.465046
15. Tsai, H., Y. Wang, and T. Itoh, "An unconditionally stable extended (USE) finite-element time-domain solution of active nonlinear microwave circuits using perfectly matched layers," IEEE Trans. Microw. Theory Techn., Vol. 50, No. 10, 2226-2232, 2002. doi:10.1109/TMTT.2002.803442
16. Jiao, D., J. Jin, E. Michielssen, and D. Riley, "Time-domain finiteelement simulation of three-dimensional scattering and radiation problems using perfectly matched layers ," IEEE. Trans. Antennas Propagat., Vol. 51, No. 2, 296-305, 2003. doi:10.1109/TAP.2003.809096
17. Cangellaris, A. and Point-matched time, "Point-matched time domain finite element methods for electromagnetic radiation and scattering," IEEE. Trans. Antennas Propagat., Vol. 35, No. 10, 1160-1173, 1987. doi:10.1109/TAP.1987.1143981
18. Wong, M., O. Picon, and V. Fouad Hanna, "A finite element method based on whitney forms to solve Maxwell equations in the time domain," IEEE Trans. Magn., Vol. 31, No. 3, 1618-1621, 1995. doi:10.1109/20.376343
19. Feliziani, M. and F. Maradei, "An explicit-implicit solution scheme to analyze fast transients by finite elements," IEEE Trans. Magn., Vol. 33, No. 2, 1452-1455, 1997. doi:10.1109/20.582533
20. Donderici, B. and F. Teixeira, "Mixed finite-element time-domain method for transient Maxwell equations in doubly dispersive media," IEEE Trans. Microw. Theory Techn., Vol. 56, No. 1, 113-120, 2008. doi:10.1109/TMTT.2007.912217
21. Donderici, B. and F. Teixeira, "Conformal perfectly matched layer for the mixed finite element time-domain method," IEEE. Trans. Antennas Propagat., Vol. 56, No. 4, 1017-1026, 2008. doi:10.1109/TAP.2008.919215
22. Yioultsis, T., N. Kantartzis, C. Antonopoulos, and T. Tsiboukis, "A fully explicit whitney element-time domain scheme with higher order vector ¯nite elements for three-dimensional high frequency problems ," IEEE Trans. Magn., Vol. 34, No. 5, 3288-3291, 1998. doi:10.1109/20.717772
23. Guillouard, K., M. Wong, V. Fouad Hanna, and J. Citerne, "A new global time-domain electromagnetic simulator of microwave circuits including lumped elements based on finite-element method," IEEE Trans. Microw. Theory Techn., Vol. 47, No. 10, 2045-2049, 1999. doi:10.1109/22.795085
24. Sekine, T. and H. Asai, "Mixed finite element time domain method based on iterative leapfrog scheme for fast simulations of electromagnetic problems," IEEE International Symposium on Electromagnetic Compatibility (EMC), 2011, 596-601, 2011. doi:10.1109/ISEMC.2011.6038381
25. Cohen, G. and M. Durufle, "Non spurious spectral-like element methods for Maxwell's equations," J. Comput. Math., Vol. 25, 282-300, 2007.
26. Winkler, J. R. and J. B. Davies, "Elimination of spurious modes in finite element analysis," J. Computat. Phys., Vol. 56, 1-14, 1984. doi:10.1016/0021-9991(84)90079-2
27. Tobon, L., J. Chen, and Q. H. Liu, "Spurious solutions in mixed finite element method for Maxwell's equations: Dispersion analysis and new basis functions," J. Computat. Phys., Vol. 230, No. 19, 7300-7310, 2011. doi:10.1016/j.jcp.2011.05.035
28. Chen, J., L. Tobon, M. Chai, J. Mix, and Q. H. Liu, "Effcient implicit-explicit time stepping scheme with domain decomposition for multiscale modeling of layered structures ," IEEE Trans. Compon. Packag. Manuf. Technol., Vol. 1, No. 9, 1438-1446, 2011. doi:10.1109/TCPMT.2011.2162726
29. Chen, J. and Q. H. Liu, "A non-spurious vector spectral element method for Maxwell's equations," Progress In Electromagnetics Research, Vol. 96, 205-215, 2009. doi:10.2528/PIER09082705
30. Cangellaris, A. and H. Wu, "Domain decomposition and multi-scale finite elements for electromagnetic analysis of integrated electronic systems ," IEEE International Symposium on Electromagnetic Compatibility (EMC), 2005, Vol. 3, 817-822, 2005.
31. Gedney, S., T. Kramer, C. Luo, J. Roden, R. Crawford, B. Guernsey, J. Beggs, and J. Miller, "The discontinuous Galerkin finite element time domain method (DGFETD)," IEEE International Symposium on Electromagnetic Compatibility (EMC), 2008 , 1-4, 2008. doi:10.1109/ISEMC.2008.4652146
32. Lu, T., W. Cai, and P. Zhang, "Discontinuous Galerkin time domain method for gpr simulation in dispersive media," IEEE Trans. Seosci. Remote Sens., Vol. 43, No. 1, 72-80, 2005. doi:10.1109/TGRS.2004.838350
33. Gan, H. and D. Jiao, "A time-domain layered finite element reduction recovery (LAFE-RR) method for high-frequency VLSI design," IEEE. Trans. Antennas Propagat., Vol. 55, No. 12, 3620-3629, 2007. doi:10.1109/TAP.2007.910473
34. Canouet, N., L. Fezoui, and S. Piperno, "Discontinuous Galerkin time-domain solution of Maxwell's equations on locally-refined nonconforming cartesian grids ," COMPEL: Int. J. for Computation and Maths. in Electrical and Electronic Eng., Vol. 24, No. 4, 1381-1401, 2005. doi:10.1108/03321640510615670
35. Shi, Y. and C.-H. Liang, "Simulations of the left-handed medium using discontinuous Galerkin method based on the hybrid domains ," Progress In Electromagnetics Research, Vol. 63, 171-191, 2006. doi:10.2528/PIER06050803
36. Nedelec, J., "A new family of mixed finite elements in R3," Numerische Mathematik, Vol. 50, No. 1, 57-81, 1986. doi:10.1007/BF01389668
37. Nedelec, J., "Mixed finite elements in R3," Numerische Mathematik, Vol. 35, No. 3, 315-341, 1980. doi:10.1007/BF01396415
38. Peterson, A., S. Ray, and R. Mittra, , Computational Methods for Electromagnetics, Vol. 24, IEEE Press, New York, 1998.
39. Fan, G.-X. and Q. H. Liu, "A strongly well-posed PML in lossy media," IEEE Antennas Wireless Propagat. Lett., Vol. 2, No. 7, 97-100, 2003.