Vol. 168

Back:[PDF file]
Latest Volume
All Volumes
All Issues
2020-10-03

Fundamental Implicit FDTD Schemes for Computational Electromagnetics and Educational Mobile Apps (Invited Review)

By Eng Leong Tan
Progress In Electromagnetics Research, Vol. 168, 39-59, 2020
doi:10.2528/PIER20061002

Abstract

This paper presents an overview and review of the fundamental implicit finite-difference time-domain (FDTD) schemes for computational electromagnetics (CEM) and educational mobile apps. The fundamental implicit FDTD schemes are unconditionally stable and feature the most concise update procedures with matrix-operator-free right-hand sides (RHS). We review the developments of fundamental implicit schemes, which are simpler and more efficient than all previous implicit schemes having RHS matrix operators. They constitute the basis of unification for many implicit schemes including classical ones, providing insights into their inter-relations along with simplifications, concise updates and efficient implementations. Based on the fundamental implicit schemes, further developments can be carried out more conveniently. Being the core CEM on mobile apps, the multiple one-dimensional (M1-D) FDTD methods are also reviewed. To simulate multiple transmission lines, stubs and coupled transmission lines efficiently, the M1-D explicit FDTD method as well as the unconditionally stable M1-D fundamental alternating direction implicit (FADI) FDTD and coupled line (CL) FDTD methods are discussed. With the unconditional stability of FADI methods, the simulations are fast-forwardable with enhanced efficiency. This is very useful for quick concept illustrations or phenomena demonstrations during interactive teaching and learning. Besides time domain, many frequency-domain methods are well-suited for further developments of useful mobile apps as well.

Citation


Eng Leong Tan, "Fundamental Implicit FDTD Schemes for Computational Electromagnetics and Educational Mobile Apps (Invited Review)," Progress In Electromagnetics Research, Vol. 168, 39-59, 2020.
doi:10.2528/PIER20061002
http://jpier.org/PIER/pier.php?paper=20061002

References


    1. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, MA, 2005.

    2. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag., Vol. 14, No. 3, 302-307, 1966.

    3. Zheng, F., Z. Chen, and J. Zhang, "Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method," IEEE Trans. Microw. Theory Tech., Vol. 48, No. 9, 1550-1558, 2000.

    4. Namiki, T., "3-D ADI-FDTD method --- Unconditionally stable time-domain algorithm for solving full vector Maxwell's equations," IEEE Trans. Microw. Theory Tech., Vol. 48, 1743-1748, 2000.

    5. Fu, W. and E. L. Tan, "Development of split-step FDTD method with higher order spatial accuracy," Electron. Lett., Vol. 40, No. 20, 1252-1254, 2004.

    6. Fu, W. and E. L. Tan, "Compact higher-order split-step FDTD method," Electron. Lett., Vol. 41, No. 7, 397-399, 2005.

    7. Tan, E. L., "Unconditionally stable LOD-FDTD method for 3-D Maxwell's equations," IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 2, 85-87, 2007.

    8. Tan, E. L., "Fundamental schemes for e±cient unconditionally stable implicit finite-difference time-domain methods," IEEE Trans. Antennas Propag., Vol. 56, No. 1, 170-177, 2008.

    9. Yang, Z. and E. L. Tan, "A microstrip circuit tool kit app with FDTD analysis including lumped elements," IEEE Microw. Mag., Vol. 16, No. 1, 74-80, 2015.

    10. Yang, Z. and E. L. Tan, "A microwave transmission line courseware based on multiple 1-D FDTD method on mobile devices," Asia-Pacific Conf. Antennas Propag., 251-252, Bali, 2015.

    11. Tan, E. L. and D. Y. Heh, "Demonstration of electromagnetic polarization app on iPad," IEEE Int. Conf. Comput. Electromagn., 196-197, Kumamoto, 2017.

    12. Tan, E. L. and D. Y. Heh, "Mobile device aided teaching and learning of electromagnetic polarization," IEEE Int. Conf. Teaching, Assessment, and Learning for Engineering, 52-55, Hong Kong, 2017.

    13. Tan, E. L. and D. Y. Heh, "Teaching and learning electromagnetic polarization using mobile devices," IEEE Antennas Propag. Mag., Vol. 60, No. 4, 112-121, 2018.

    14. Tan, E. L. and D. Y. Heh, "Teaching and learning electromagnetic plane wave reflection and transmission using 3-D TV," IEEE Antennas Propag. Mag., Vol. 61, No. 2, 101-108, 2019.

    15. Peaceman, D., H. Rachford, and Jr., "The numerical solution of parabolic and elliptic differential equations," J. Soc. Ind. Appl. Math., Vol. 3, No. 1, 28-41, 1955.

    16. Mitchell, A. R. and D. F. Griffths, The Finite-Difference Method in Partial Differential Equations, Wiley, New York, 1980.

    17. Thomas, J. W., Numerical Partial Differential Equations: Finite Difference Methods, Springer-Verlag, New York, 1998.

    18. Tan, E. L., "Efficient algorithm for the unconditionally stable 3-D ADI-FDTD method," IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 1, 7-9, 2007.

    19. Tan, E. L., "Concise current source implementation for efficient 3-D ADI-FDTD method," IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 11, 748-750, 2007.

    20. Douglas, J., "Alternating direction methods for three space variables," Numerische Mathematik, Vol. 4, No. 1, 94-102, 1962.

    21. Sun, G. and C. W. Trueman, "Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method," IEEE Trans. Microw. Theory Tech., Vol. 54, No. 5, 2275-2284, 2006.

    22. Tan, E. L., "Efficient algorithms for Crank-Nicolson-based finite-difference time-domain methods," IEEE Trans. Microw. Theory Tech., Vol. 56, No. 2, 408-413, 2008.

    23. Fu, W. and E. L. Tan, "Stability and dispersion analysis for higher order 3-D ADI-FDTD method," IEEE Trans. Antennas Propag., Vol. 53, No. 11, 3691-3696, 2005.

    24. Fu, W. and E. L. Tan, "A compact higher-order ADI-FDTD method," Microwave Opt. Technol. Lett., Vol. 44, No. 3, 273-275, 2005.

    25. Fu, W. and E. L. Tan, "A parameter optimized ADI-FDTD method based on the (2,4) stencil," IEEE Trans. Antennas Propag., Vol. 54, No. 6, 1836-1842, 2006.

    26. Fu, W. and E. L. Tan, "Stability and dispersion analysis for ADI-FDTD method in lossy media," IEEE Trans. Antennas Propag., Vol. 55, No. 4, 1095-1102, 2007.

    27. Singh, G., E. L. Tan, and Z. N. Chen, "Efficient tensor based FDTD scheme for modeling sloped interfaces in lossy media," Microwave Opt. Technol. Lett., Vol. 51, No. 6, 1530-1537, 2009.

    28. Heh, D. Y. and E. L. Tan, "Dispersion analysis of FDTD schemes for doubly lossy media," Progress In Electromagnetics Research B, Vol. 17, 327-342, 2009.

    29. Heh, D. Y. and E. L. Tan, "Generalized stability criterion of 3-D FDTD schemes for doubly lossy media," IEEE Trans. Antennas Propag., Vol. 58, No. 4, 1421-1425, 2010.

    30. Heh, D. Y. and E. L. Tan, "Lyapunov and matrix norm stability analysis of ADI-FDTD schemes for doubly lossy media," IEEE Trans. Antennas Propag., Vol. 59, No. 3, 979-986, 2011.

    31. Heh, D. Y. and E. L. Tan, "Efficient implementation of 3-D ADI-FDTD method for lossy media," IEEE MTT-S Int. Microwave Symp., 313-316, Boston, Massachusetts, 2009.

    32. Heh, D. Y. and E. L. Tan, "Unified efficient fundamental ADI-FDTD schemes for lossy media," Progress In Electromagnetics Research B, Vol. 32, 217-242, 2011.

    33. Fu, W. and E. L. Tan, "Effective permittivity scheme for ADI-FDTD method at the interface of dispersive media," Appl. Comput. Electromag. Soc. J., Vol. 23, No. 2, 120-125, 2008.

    34. Heh, D. Y. and E. L. Tan, "Modeling Lorentz dispersive media in FDTD using the exponential time differencing method," Asia-Pacific Microwave Conf., Hong Kong, 2008.

    35. Heh, D. Y. and E. L. Tan, "FDTD modeling for dispersive media using matrix exponential method," IEEE Microw. Wireless Compon. Lett., Vol. 19, No. 2, 53-55, 2009.

    36. Tan, E. L. and D. Y. Heh, "Corrected impulse invariance method for dispersive media using FDTD," Asia-Pacific Symp. Electromag. Compat., 56-59, Singapore, 2008.

    37. Heh, D. Y. and E. L. Tan, "Corrected impulse invariance method in Z-transform theory for frequency-dependent FDTD methods," IEEE Trans. Antennas Propag., Vol. 57, No. 9, 2683-2690, 2009.

    38. Heh, D. Y. and E. L. Tan, "Modeling Debye dispersive media using efficient ADI-FDTD method," IEEE AP-S Int. Symp. Antennas Propag., Charleston, 2009.

    39. Heh, D. Y. and E. L. Tan, "Fundamental ADI-FDTD method for multiple-pole Debye dispersive media," Asia-Pacific Conf. Antennas Propag., 9-10, Singapore, 2012.

    40. Heh, D. Y. and E. L. Tan, "Stable formulation of FADI-FDTD method for multiterm, doubly, second-order dispersive media," IEEE Trans. Antennas Propag., Vol. 61, No. 8, 4167-4175, 2013.

    41. Heh, D. Y. and E. L. Tan, "Unconditionally stable fundamental alternating direction implicit FDTD method for dispersive media," Computational Electromagnetics --- Retrospective and Outlook, Chapter 4, 85-116, Springer, 2015.

    42. Heh, D. Y. and E. L. Tan, "Modeling hemoglobin at optical frequency using the unconditionally stable fundamental ADI-FDTD method," Biomedical Opt. Expr., Vol. 2, No. 5, 1169-1183, 2011.

    43. Heh, D. Y. and E. L. Tan, "Modeling the interaction of terahertz pulse with healthy skin and basal cell carcinoma using the unconditionally stable fundamental ADI-FDTD method," Progress In Electromagnetics Research B, Vol. 37, 365-386, 2012.

    44. Fu, W. and E. L. Tan, "ADI-FDTD method including linear lumped networks," Electron. Lett., Vol. 42, No. 13, 728-729, 2006.

    45. Fu, W. and E. L. Tan, "Unconditionally stable FDTD technique including passive lumped elements," 2006 Int. RF Microwave Conf., Putrajaya, Malaysia, 2006.

    46. Fu, W. and E. L. Tan, "Unconditionally stable ADI-FDTD method including passive lumped elements," IEEE Trans. Electromagn. Compat., Vol. 48, No. 4, 661-668, 2006.

    47. Gan, T. H., Z. Yang, and E. L. Tan, "A polarization-reconfigurable filtering antenna system," IEEE Antennas Propag. Mag., Vol. 55, No. 6, 198-219, 2013.

    48. Yang, Z. and E. L. Tan, "A de-embedding technique for diode-incorporated reconfigurable antenna simulation," IEEE AP-S Int. Symp. Antennas Propag., 1437-1438, Farjardo, Puerto Rico, 2016.

    49. Yang, Z. and E. L. Tan, "A fundamental ADI-FDTD method with implicit update for magnetic fields in the second procedure," Asia-Pacific Conf. Antennas Propag., 6-7, Bali, 2015.

    50. Tay, W. C. and E. L. Tan, "Implementation of Mur first order absorbing boundary condition in efficient 3-D ADI-FDTD," IEEE AP-S Int. Symp. Antennas Propag., Charleston, 2009.

    51. Tay, W. C. and E. L. Tan, "Split-field PML implementation for the efficient fundamental ADI-FDTD method," Asia-Pacific Microwave Conf., 1553-1556, Singapore, 2009.

    52. Singh, G., E. L. Tan, and Z. N. Chen, "Analytic fields of a focused beam with higher-order compensations for FDTD TF/SF formulation," IEEE AP-S Int. Symp. Antennas Propag., 2278-2281, Spokane, 2011.

    53. Singh, G., E. L. Tan, and Z. N. Chen, "Analytic fields with higher-order compensations for 3-D FDTD TF/SF formulation with application to beam excitations," IEEE Trans. Antennas Propag., Vol. 59, No. 7, 2588-2598, 2011.

    54. Gan, T. H. and E. L. Tan, "An efficient total-field/scattered-field technique for the fundamental ADI-FDTD method," IEEE AP-S Int. Symp. Antennas Propag., 159-160, Memphis, 2014.

    55. Singh, G., E. L. Tan, and Z. N. Chen, "Efficient complex envelope ADI-FDTD method for the analysis of anisotropic photonic crystals," IEEE Photon. Technol. Lett., Vol. 23, No. 12, 801-803, 2011.

    56. Singh, G., E. L. Tan, and Z. N. Chen, "Modeling magnetic photonic crystals with lossy ferrites using efficient complex envelope ADI-FDTD method," Opt. Lett., Vol. 36, No. 8, 1494-1496, 2011.

    57. D'Yakonov, Ye. G., "On some difference schemes for solutions of boundary problems," U.S.S.R. Comput. Math. and Math. Phys., Vol. 2, 55-77, 1962.

    58. Shibayama, J., T. Hirano, J. Yamauchi, and H. Nakano, "Efficient implementation of frequency-dependent 3D LOD-FDTD method using fundamental scheme," Electron. Lett., Vol. 48, No. 13, 774-775, 2012.

    59. Yang, Z., E. L. Tan, and L. Wang, "Upgrading LOD-FDTD method to efficient method with second-order accuracy," Asia-Pacific Microwave Conf., Nanjing, 2015.

    60. Gan, T. H. and E. L. Tan, "Unconditionally stable fundamental LOD-FDTD method with second-order temporal accuracy and complying divergence," IEEE Trans. Antennas Propag., Vol. 61, No. 5, 2630-2638, 2013.

    61. Gan, T. H. and E. L. Tan, "Convolutional perfectly matched layer (CPML) for fundamental LOD-FDTD method with 2nd order temporal accuracy and complying divergence," Asia-Pacific Microwave Conf., 839-841, Seoul, Korea, 2013.

    62. Gan, T. H. and E. L. Tan, "Application of the fundamental LOD2-CD-FDTD method for antenna modeling," Asia-Pacific Conf. Antennas Propag., 445-446, Bali, 2015.

    63. Gan, T. H. and E. L. Tan, "Current source implementations for fundamental SS2-FDTD method," Asia-Pacific Microwave Conf., 1292-1294, Kaohsiung, 2012.

    64. Heh, D. Y. and E. L. Tan, "Split-step finite-difference time-domain method with fourth order accuracy in time," Asia-Pacific Symp. Electromag. Compat., 68-71, Singapore, 2008.

    65. Tan, E. L. and D. Y. Heh, "ADI-FDTD method with fourth order accuracy in time," IEEE Microw. Wireless Compon. Lett., Vol. 18, No. 5, 296-298, 2008.

    66. Heh, D. Y. and E. L. Tan, "Further reinterpretation of multi-stage implicit FDTD schemes," IEEE Trans. Antennas Propag., Vol. 62, No. 8, 4407-4411, 2014.

    67. Garcia, S. G., T. W. Lee, and S. C. Hagness, "On the accuracy of the ADI-FDTD method," IEEE Antennas Wireless Propag. Lett., Vol. 1, 31-34, 2002.

    68. Kong, Y. and Q. Chu, "High-order split-step unconditionally-stable FDTD methods and numerical analysis," IEEE Trans. Antennas Propag., Vol. 59, No. 9, 3280-3289, 2011.

    69. Tan, E. L., "Acceleration of LOD-FDTD method using fundamental scheme on graphics processor units," IEEE Microw. Wireless Compon. Lett., Vol. 20, No. 12, 648-650, 2010.

    70. Tay, W. C., D. Y. Heh, and E. L. Tan, "GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer," Progress In Electromagnetics Research M, Vol. 14, 177-192, 2010.

    71. Tay, W. C. and E. L. Tan, "Mur absorbing condition for efficient fundamental 3D LOD-FDTD," IEEE Microw. Wireless Compon. Lett., Vol. 20, No. 2, 61-63, 2010.

    72. Tay, W. C. and E. L. Tan, "Implementations of PMC and PEC boundary conditions for efficient fundamental ADI and LOD-FDTD," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 4, 565-573, 2010.

    73. Singh, G., E. L. Tan, and Z. N. Chen, "Implementation of total-field/scattered-field technique in the 2-D LOD-FDTD method," Asia-Pacific Microwave Conf., 1505-1508, Singapore, 2009.

    74. Singh, G., E. L. Tan, and Z. N. Chen, "A split-step FDTD method for 3-D Maxwell's equations in general anisotropic media," IEEE Trans. Antennas Propag., Vol. 58, No. 11, 3647-3657, 2010.

    75. Heh, D. Y. and E. L. Tan, "Complex-envelope LOD-FDTD method for ionospheric propagation," IEEE AP-S Int. Symp. Antennas Propag., 2027-2028, Farjardo, Puerto Rico, 2016.

    76. Yang, Z. and E. L. Tan, "Efficient 3-D fundamental LOD-FDTD method with lumped elements," IEEE MTT-S Int. Microwave Symp., Tampa, 2014.

    77. Yang, Z. and E. L. Tan, "3-D unified FLOD-FDTD method incorporated with lumped elements," IEEE AP-S Int. Symp. Antennas Propag., San Diego, 2017.

    78. Yang, Z. and E. L. Tan, "Two finite-difference time-domain methods incorporated with memristor," Progress In Electromagnetics Research M, Vol. 42, 153-158, 2015.

    79. Yang, Z. and E. L. Tan, "Efficient 3-D fundamental LOD-FDTD method incorporated with memristor," IEICE Trans. Electronics, Vol. E99-C, No. 7, 788-792, 2016.

    80. Yang, Z. and E. L. Tan, "3-D non-uniform time step locally onedimensional FDTD method," Electron. Lett., Vol. 52, No. 12, 993-994, 2016.

    81. Yang, Z. and E. L. Tan, "Stability analyses of non-uniform time-step schemes for ADI- and LOD-FDTD methods," IEEE Int. Conf. Comput. Electromagn., 312-313, Kumamoto, 2017.

    82. Tan, E. L. and D. Y. Heh, "Stability analyses of nonuniform time-step LOD-FDTD methods for electromagnetic and thermal simulations," IEEE J. Multiscale Multiphys. Comput. Tech., Vol. 2, 183-193, 2017.

    83. Heh, D. Y. and E. L. Tan, "Some recent developments in fundamental implicit FDTD schemes," Asia-Pacific Symp. Electromag. Compat., 153-156, Singapore, 2012.

    84. Tay, W. C. and E. L. Tan, "Efficient algorithm for 3-D thermal alternating-direction-implicit method," IEEE Electrical Design Adv. Packag. Syst. Symp., 177-180, Taipei, 2012.

    85. Tay, W. C., E. L. Tan, and D. Y. Heh, "Fundamental locally one-dimensional method for 3-D thermal simulation," IEICE Trans. Electronics, Vol. E97-C, No. 7, 636-644, 2014.

    86. Heh, D. Y., E. L. Tan, and W. C. Tay, "Fast alternating direction implicit method for e±cient transient thermal simulation of integrated circuits," Int. J. Numer. Model. Electron. Networks Devices Fields, Vol. 29, No. 1, 93-108, 2016.

    87. Tay, W. C. and E. L. Tan, "Pentadiagonal alternating-direction-implicit finite-difference time-domain method for two-dimensional Schrodinger equation," Computer Phys. Comm., Vol. 185, No. 7, 1886-1892, 2014.

    88. Cooke, S. J., M. Botton, T. M. Antonsen, and B. Levush, "A leapfrog formulation of the 3D ADI-FDTD algorithm," Int. J. Numer. Model, Vol. 22, No. 2, 187-200, 2009.

    89. Gan, T. H. and E. L. Tan, "Stability and dispersion analysis for three-dimensional (3-D) leapfrog ADI-FDTD method," Progress In Electromagnetics Research M, Vol. 23, 1-12, 2012.

    90. Gan, T. H. and E. L. Tan, "Divergence of electric field for the two-dimensional (2-D) leapfrog ADI-FDTD method," IEEE AP-S Int. Symp. Antennas Propag., Chicago, 2012.

    91. Gan, T. H. and E. L. Tan, "Analysis of the divergence properties for the three-dimensional leapfrog ADI-FDTD method," IEEE Trans. Antennas Propag., Vol. 60, No. 12, 5801-5808, 2012.

    92. Gan, T. H. and E. L. Tan, "Unconditionally stable leapfrog ADI-FDTD method for lossy media," Progress In Electromagnetics Research M, Vol. 26, 173-186, 2012.

    93. Gan, T. H. and E. L. Tan, "On the field leakage of the leapfrog ADI-FDTD method for nonpenetrable targets," Microwave Opt. Technol. Lett., Vol. 56, No. 6, 1401-1405, 2014.

    94. Heh, D. Y. and E. L. Tan, "Divergence-preserving alternating direction implicit scheme for multi-pole Debye dispersive media," IEEE Microw. Wireless Compon. Lett., Vol. 24, No. 2, 69-71, 2014.

    95. 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.

    96. Saxena, A. K. and K. V. Srivastava, "A three-dimensional unconditionally stable five-step LOD-FDTD method," IEEE Trans. Antennas Propag., Vol. 62, No. 3, 1321-1329, 2014.

    97. Yang, Z., E. L. Tan, and D. Y. Heh, "Second-order temporal-accurate scheme for 3-D LOD-FDTD method with three split matrices," IEEE Antennas Wireless Propag. Lett., Vol. 14, 1105-1108, 2015.

    98. Yang, Z., E. L. Tan, and D. Y. Heh, "Variants of second-order temporal-accurate 3-D FLOD-FDTD schemes with three split matrices," IEEE Int. Conf. Comput. Electromagn., 265-267, Guangzhou, 2016.

    99. Tan, E. L. and D. Y. Heh, "M1-D FDTD methods for mobile interactive teaching and learning of wave propagation in transmission lines," IEEE Antennas Propag. Mag., Vol. 61, No. 5, 119-126, 2019.

    100. Yang, Z. and E. L. Tan, "Interconnected multi-1-D FADI- and FLOD-FDTD methods for transmission lines with interjunctions," IEEE Trans. Microw. Theory Tech., Vol. 65, No. 3, 684-692, 2017.

    101. Tan, E. L. and D. Y. Heh, "Demonstration of electromagnetic waves propagation along transmission lines on iPad," 2018 Joint IEEE Int. Symp. Electromag. Compat. and Asia-Pacific Symp. Electromag. Compat., 599-601, Singapore, 2018.

    102. Yang, Z. and E. L. Tan, "Non-uniform time-step FLOD-FDTD method for multiconductor transmission lines including lumped elements," IEEE Trans. Electromagn. Compat., Vol. 59, No. 6, 1983-1992, 2017.

    103. Yang, Z. and E. L. Tan, "Multiple one-dimensional FDTD method for coupled transmission lines and stability condition," IEEE Microw. Wireless Compon. Lett., Vol. 26, No. 11, 864-866, 2016.

    104. Yang, Z. and E. L. Tan, "Multiple one-dimensional finite-difference time-domain method for asymmetric coupled transmission lines," IEEE Int. Conf. Comput. Electromagn., Chengdu, 2018.

    105. Paul, C. R., Analysis of Multiconductor Transmission Lines, 2nd Ed., Wiley, New York, 2008.

    106. Mongia, R. K., I. J. Bahl, P. Bhartia, and J. Hong, RF and Microwave Coupled-line Circuits, 2nd Ed., Artech House, Norwood, MA, USA, 2007.

    107. Pozar, D. M., Microwave Engineering, 4th Ed., Wiley, New York, 2011.

    108. Heh, D. Y. and E. L. Tan, "Unconditionally stable multiple one-dimensional ADI-FDTD method for coupled transmission lines," IEEE Trans. Antennas Propag., Vol. 66, No. 12, 7488-7492, 2018.

    109. Heh, D. Y. and E. L. Tan, "Numerical stability analysis of M1-D ADI-FDTD method for coupled transmission lines," IEEE Int. Conf. Comput. Electromagn., Shanghai, 2019.

    110. Tan, E. L. and D. Y. Heh, "Source-incorporated M1-D FADI-FDTD method for coupled transmission lines," 11th Int. Conf. Microw. Millimeter Wave Techn., Guangzhou, 2019.

    111. Tan, E. L. and D. Y. Heh, "Multiple 1-D fundamental ADI-FDTD method for coupled transmission lines on mobile devices," IEEE J. Multiscale Multiphys. Comput. Tech., Vol. 4, 198-206, 2019.

    112. Tan, E. L. and D. Y. Heh, "Mobile teaching and learning of coupled-line structures," IEEE Antennas Propag. Mag., Vol. 62, No. 4, 62-69, 2020.

    113. Heh, D. Y. and E. L. Tan, "Numerical stability analysis of M1-D LOD-FDTD method for inhomogeneous coupled transmission lines," IEEE AP-S Int. Symp. Antennas Propag., 1657-1658, Atlanta, 2019.

    114. Heh, D. Y. and E. L. Tan, "Multiple LOD-FDTD method for inhomogeneous coupled transmission lines and stability analyses," IEEE Trans. Antennas Propag., Vol. 68, No. 3, 2198-2205, 2020.

    115. Tan, E. L. and S. Y. Tan, "Spectral-domain dyadic Green's functions for surface current excitation in planar stratified bianisotropic media," IEE Proc. Microw. Antennas Propag., Vol. 146, No. 6, 394-400, 1999.

    116. Tan, E. L. and S. Y. Tan, "Unbounded and scattered field representations of the dyadic Green's functions for planar stratified bianisotropic media," IEEE Trans. Antennas Propag., Vol. 49, No. 8, 1218-1225, 2001.

    117. Tan, E. L., "Note on formulation of the enhanced scattering- (transmittance-) matrix approach," J. Opt. Soc. Am. A, Vol. 19, No. 6, 1157-1161, 2002.

    118. Tan, E. L., "Recursive asymptotic impedance matrix method for electromagnetic waves in bianisotropic media," IEEE Microw. Wireless Compon. Lett., Vol. 16, No. 6, 351-353, 2006.

    119. Ning, J. and E. L. Tan, "Hybrid matrix method for stable analysis of electromagnetic waves in stratified bianisotropic media," IEEE Microw. Wireless Compon. Lett., Vol. 18, No. 10, 653-655, 2008.

    120. Ning, J. and E. L. Tan, "Generalized eigenproblem of hybrid matrix method for stable analysis of periodic multilayered bianisotropic media," Asia-Pacific Microwave Conf., Hong Kong, 2008.

    121. Tan, E. L. and S. Y. Tan, "Singularities and discontinuities in the eigenfunction expansions of the dyadic Green's functions for biisotropic media," Progress In Electromagnetics Research, Vol. 19, 301-318, 1998.

    122. Tan, E. L. and S. Y. Tan, "A unified representation of the dyadic Green's functions for planar, cylindrical and spherical multilayered biisotropic media," Progress In Electromagnetics Research, Vol. 20, 75-100, 1998.

    123. Tan, E. L. and S. Y. Tan, "Dyadic Green's functions for circular waveguides filled with biisotropic media," IEEE Trans. Microw. Theory Tech., Vol. 47, No. 7, 1134-1137, 1999.

    124. Tan, E. L., "Unified solutions of static Green's functions for open and covered planar two-layered anisotropic media," IEEE AP-S Int. Symp. Antennas Propag., 892-895, Salt Lake City, 2000.

    125. Tan, E. L., "Electrostatic Green's functions for planar multilayered anisotropic media," IEE Proc. Microw. Antennas Propag., Vol. 149, No. 1, 78-83, 2002.

    126. Ning, J. and E. L. Tan, "Simple and stable analysis of multilayered anisotropic materials for design of absorbers and shields," Mater. Des., Vol. 30, No. 6, 2061-2066, 2009.

    127. Tan, E. L. and S. Y. Tan, "Cylindrical vector wave function representations of electromagnetic fields in gyrotropic bianisotropic media," Journal of Electromagnetic Waves and Applications, Vol. 13, No. 11, 1461-1476, 1999.

    128. Tan, E. L. and S. Y. Tan, "Cylindrical vector wave function representations of the dyadic Green's functions for cylindrical multilayered gyrotropic bianisotropic media," Progress In Electromagnetics Research, Vol. 26, 199-222, 2000.

    129. Tan, E. L. and S. Y. Tan, "On the eigenfunction expansions of the dyadic Green's functions for bianisotropic media," Progress In Electromagnetics Research, Vol. 20, 227-247, 1998.

    130. Tan, E. L. and S. Y. Tan, "Coordinate-independent dyadic formulation of the dispersion relation for bianisotropic media," IEEE Trans. Antennas Propag., Vol. 47, No. 12, 1820-1824, 1999.

    131. Tan, E. L. and S. Y. Tan, "Concise spectral formalism in the electromagnetics of bianisotropic media," Progress In Electromagnetics Research, Vol. 25, 309-331, 2000.

    132. Tan, E. L., "Vector wave function expansions of dyadic Green's functions for bianisotropic media," IEE Proc. Microw. Antennas Propag., Vol. 149, No. 1, 57-63, 2002.

    133. Tan, E. L., "Reduced conditions for the constitutive parameters of lossy bi-anisotropic media," Microwave Opt. Technol. Lett., Vol. 41, No. 2, 133-135, 2004.

    134. Tan, E. L., "Enhanced R-matrix algorithms for multilayered diffraction gratings," Appl. Opt., Vol. 45, No. 20, 4803-4809, 2006.

    135. Tan, E. L., "Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings," J. Mod. Opt., Vol. 53, No. 4, 417-428, 2006.

    136. Ning, J. and E. L. Tan, "Generalized eigenproblem of hybrid matrix for Bloch-Floquet waves in one-dimensional photonic crystals," J. Opt. Soc. Am. B, Vol. 26, No. 4, 676-683, 2009.

    137. Tan, E. L. and Y. W. M. Chia, "Green's function and network analysis of quasi-2D SAW ID-tags," IEEE Ultrasonics Symp., 55-58, San Juan, Puerto Rico, 2000.

    138. Tan, E. L., "A robust formulation of SAW Green's functions for arbitrarily thick multilayers at high frequencies," IEEE Trans. Ultrason., Ferroelec., Freq. Contr., Vol. 49, No. 7, 929-936, 2002.

    139. Tan, E. L., "A concise and efficient scattering matrix formalism for stable analysis of elastic wave propagation in multilayered anisotropic solids," Ultrasonics, Vol. 41, No. 3, 229-236, 2003.

    140. Tan, E. L., "Stiffness matrix method with improved efficiency for elastic wave propagation in layered anisotropic media," J. Acoust. Soc. Am., Vol. 118, No. 6, 3400-3403, 2005.

    141. Tan, E. L., "Hybrid compliance-stiffness matrix method for stable analysis of elastic wave propagation in multilayered anisotropic media," J. Acoust. Soc. Am., Vol. 119, No. 1, 45-53, 2006.

    142. Tan, E. L., "Generalized eigenproblem for acoustic wave propagation in periodically layered anisotropic media," J. Comput. Acoustics, Vol. 16, No. 1, 1-10, 2008.

    143. Tan, E. L., "Generalized eigenproblem of hybrid matrix for Floquet wave propagation in one- dimensional phononic crystals with solids and fluids," Ultrasonics, Vol. 50, No. 1, 91-98, 2010.

    144. Tan, E. L., "Recursive asymptotic hybrid matrix method for acoustic waves in multilayered piezoelectric media," Open J. Acoustics, Vol. 1, 27-33, 2011.

    145. Tan, E. L., "Simple derivation and proof of geometrical stability criteria for linear two-ports," Microwave Opt. Technol. Lett., Vol. 40, No. 1, 81-83, 2004.

    146. Tan, E. L., "Simplified graphical analysis of linear three-port stability," IEE Proc. Microw. Antennas Propag., Vol. 152, No. 4, 209-213, 2005.

    147. Tan, E. L., J. Ning, and K. S. Ang, "Geometrical stability criteria for two-port networks in invariant immittance parameters representation," Asia-Pacific Microwave Conf., Hong Kong, 2008.

    148. Tan, E. L., "Comments on `Distance from unconditional stability boundary of a two-port network'," IET Microw. Antennas Propag., Vol. 8, No. 1, 64, 2014.

    149. Tan, E. L., "Rollett-based single-parameter criteria for unconditional stability of linear two-ports," IEE Proc. Microw. Antennas Propag., Vol. 151, No. 4, 299-302, 2004.

    150. Tan, E. L., X. Sun, and K. S. Ang, "Unconditional stability criteria for microwave networks," Progress In Electromagnetics Research Symposium, 1524-1528, Beijing, China, March 23-27, 2009.

    151. Tan, E. L., "A Quasi-invariant single-parameter criterion for linear two-port unconditional stability," IEEE Microw. Wireless Compon. Lett., Vol. 14, No. 10, 487-489, 2004.

    152. Tan, E. L., "Quasi-invariant single-parameter criterion for unconditional stability: Review and application," Asia-Pacific Microwave Conf., 429-432, Yokohama, 2006.

    153. Tan, E. L. and S. Z. Fan, "Graphical analysis of stabilization loss and gains for three-port networks," IEEE Trans. Microw. Theory Tech., Vol. 60, No. 6, 1635-1640, 2012.

    154. Tan, E. L. and D. Y. Heh, "Application of Belevitch theorem for pole-zero analysis of microwave filters with transmission lines and lumped elements," IEEE Trans. Microw. Theory Tech., Vol. 66, No. 11, 4669-4676, 2018.

    155. Tan, E. L. and D. Y. Heh, "Analysis and determination of microwave filter order," Asia-Pacific Microwave Conf., 1360-1362, Kyoto, 2018.

    156. Tan, E. L. and D. Y. Heh, "Pole-zero analysis of microwave filters using contour integration method exploiting right-half plane," Progress In Electromagnetics Research M, Vol. 78, 59-68, 2019.

    157. Smunyahirun, R. and E. L. Tan, "Derivation of the most energy-efficient source functions by using calculus of variations," IEEE Trans. Circuits Syst. I: Regul. Pap., Vol. 63, No. 4, 494-502, 2016.

    158. Smunyahirun, R. and E. L. Tan, "Optimum lowest input energy for first-order circuits in transient state," Int. Conf. Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology, 143-146, Phuket, 2017.

    159. Smunyahirun, R. and E. L. Tan, "Most energy-efficient input voltage function for RC delay line," 2018 Joint IEEE Int. Symp. Electromag. Compat. and Asia-Pacific Symp. Electromag. Compat., 1022-1026, Singapore, 2018.