Electromagnetic numerical modelling using conformal and dispersive

FDTD method

The finite-difference time-domain (FDTD) technique [1] is well suited and widely applied for the computation of electromagnetic problems. Different FDTD schemes have been devised to perform the various electromagnetic computations efficiently. Traditionally, conformal modelling of electromagnetic structures using staircase Finite-Difference Time-Domain method often produces inaccurate results even with the help of more powerful computer resources [2]. Thanks to Holland 's effort [3], the classic Yee's FDTD algorithm can be expressed in the generalized non-orthogonal coordinates and it is identified as Nonorthogonal FDTD (NFDTD). In NFDTD, the electric and magnetic fields are analysed using their covariant and contravariant components respectively. Recently, Hao et al [4] modified the conventional NFDTD scheme within the underlying Cartesian coordinate system and only those cells closed to the curved boundaries are distorted. Namely, it is Local-distorted Nonorthogonal FDTD (LNFDTD), in which a Cartesian grid is used for the majority of the problem space and therefore less CPU time and memory are needed than the NFDTD. However, such a scheme suffers later time numerical instability even when the Courant criterion is satisfied. Gedney [5] and Schuhmann [6] et al demonstrated that the projection operators of NFDTD scheme must be symmetric positive definite. In our research, it is shown theoretically that the source of numerical instability in LNFDTD method is due to the inherent dissatisfaction of divergence free condition for the electric field in a source-free space. Conventionally, a unique time step is chosen for NFDTD simulation, and this approach is less efficient, particularly in LNFDTD. An efficient 'time sub-griding' scheme is proposed to reduce the late time instability of LNFDTD method when long time simulation is required.

  

  GUI for QMUL in-house conformal FDTD program

 

Comparison on VSWRs of horn antenna obtained from conformal FDTD simulation,

staircase FDTD approximation and measurement.

 

Dispersive FDTD modelling on Left-Handed Meta-Materials (LHMs)

 

A realistic LHM must be dispersive and such a medium suggested in [10] can be defined in equations (1). It is an isotropic, lossy and cold plasma medium model widely used in many literatures such as [11].

Details on how to convert the above frequency domain Maxwell's Equations into the second-order time-domain differential representations are well discussed in [10] and [11]. The time-domain equations are discretised using the second-order central differences on the standard Yee's lattice in FDTD method. Some sample FDTD iteration equations are given as the followings:

 

 

More information on LHMs can be found here .

References

  1. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302-307, May 1966.
  2. A. C. Cangellaris and D. B. Wright, "Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulation of electromagnetic phenomena," IEEE Trans. Antennas Propagat., vol. 39, pp. 1518-1525, Oct. 1991.
  3. R. Holland, "Finite difference solutions of Maxwell's equations in generalized non-orthogonal coordinates", IEEE Trans. Nuc. Sci., vol. NS-30, no6, pp 4689-4591, Dec 1983.
  4. Y. Hao, C.J. Railton, "Analyzing Electromagnetic Structures With Curved Boundaries on Cartesian FDTD Meshes", IEEE Trans. on Microwave Theory and Techniques, vol. 46, pp. 82-88, Jan.1998.
  5. S. D. Gedney, J. A. Roden, "Numerical Stability of Nonorthogonal FDTD methods", IEEE Trans. Antennas Propagat., vol. 48, pp. 231-239, Feb.2000.
  6. R. Schuhmann, T. Weiland, "Stability of the FDTD Algorithm on Nonorthogonal Grids Related to the Spatial Interpolation Scheme", IEEE Trans. on Magnetics., vol. 34, pp. 2751-2754, Sep.1998.
  7. A. Taflove, Computational Electrodynamics: the finite-difference time-domain method, (Artech House, 2000).
  8. J. A. Stratton, Electromagnetic Theory, (New York: McGraw-Hill, 1941).
  9. J. F. Lee, R. Palandech, and R. Mittra, "Modelling three-dimensional discontinuities in waveguides using nonorthogonal FDTD algorithm," IEEE Trans. Microwave Theory Tech. Vol. 40, pp. 346-352, Feb. 1992.
  10. J. B. Pendry, "Negative Refraction Makes a Perfect Lens", Phys. Rev. Lett., 85, 3966 (2000).
  11. Y. Hao, L. Lu and C. G.Parini , "Time domain modelling on wave propagation through single/multi-layer left-handed meta-materials slabs", The 12th International Conference on Antennas and Propagation, 2003.
  12. Richard W. Ziolkowski and Ehud Heyman, "Wave Propagation in media having negative permittivity and permeability", Phys. Rev .E ,Volume 64,056625 2001
  13. Jeffrey L. Yong and Ronald O. Nelson, " A Summary and Systematic Analysis of FDTD Algorithms for Linearly Dispersive Media", IEEE Antennas and Propagation Magazine, Vol.43, No.1, Feb.2001

For more information on our research, please contact Prof. Y. Hao .