1. 1 EEE 431 Computational Methods in Electrodynamics Lecture 8 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

2. 2 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

3. 3 Basic FDTD Requirements Space Cell Sizes Time Step Size

4. 4 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Space Cell Sizes Determination of the cell sizes and the time step size are very important aspects of the FDTD method. Cell sizes must be small enough to achieve accurate results at the highest frequency of interest and must be large enough to be handled by the computer resources.

5. 5 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Space Cell Sizes The cell sizes must be much less than the smallest possible wavelength (which corresponds to the highest frequency of interest) to achieve accurate results. Usually the cell sizes are taken to be smaller than,.

6. 6 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Time Step Size The time step size, required for FDTD algorithm, has to be bounded relative to the space sizes. This bound is necessary to prevent numerical instability.

7. 7 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Time Step Size For a 3-Dimensional rectangular grid, with v the maximum velocity of propagation in any medium the following well-known stability criterion

8. 8 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Excitations At t=0 all fields are assumed to be identically 0 throughout the computational domain. The system can be excited either by using a single frequency excitation (i.e. sine wave) or a wideband frequency excitation (i.e. Gaussian Pulse)

9. 9 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Single Frequency Plane Wave A desirable plane wave source condition, for the three dimensional case, applied at plane (near y=0) is :

10. 10 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Analytical expression of the Gaussian pulse: Where, T is the Gaussian half-width and is the time delay. Then the computer code:

11. 11 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Absorbing Boundary Conditions The FDTD method has been applied to different types of problems successfully in electromagnetics, including the open region problems.

12. 12 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) In order to model open region problems, absorbing boundary conditions (ABCs) are often used to truncate the computational domain since the tangential components of the electric field along the outer boundary of the computational domain cannot be updated by using the basic Yee algorithm.

13. 13 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Differential based ABCs are generally obtained by factoring the wave equation and by allowing a solution, which permits only outgoing waves. Material-based ABCs, on the other hand, are constructed so that fields are dampened as they propagate into an absorbing medium.

14. 14 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) We will consider ABC’s developed by A. Taflove. The conditions relate the values of the field components at the truncation planes to the field components at points one or more space steps within the solution region (lattice)

15. 15 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) For One dimensional Wave Propagation: Assume that waves have only Ez and Hx components and propagate in the +ve and –ve y directions, then; When the lattice extends form y=0 to y=j

16. 16 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) One dimensional free space formulation: Assume a plane wave with the electric filed having Ex, magnetic field having Hy components and traveling in the z direction.

17. 17 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Maxwell’s Equations become:

18. 18 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Taking central difference approximation for both temporal and spectral derivatives: