EEE 431 Computational Methods in Electrodynamics

1
EEE 431Computational Methods in Electrodynamics

• Lecture 8
• By
• Dr. Rasime Uyguroglu
• Rasime.uyguroglu_at_emu.edu.tr

2
FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)
3
FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Basic FDTD Requirements
• Space Cell Sizes
• Time Step Size

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
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
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
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
FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Excitations
• At t0 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
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 y0) is

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
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
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
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
FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• We will consider ABCs 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
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 y0 to yj

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
FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Maxwells Equations become

18
FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

• Taking central difference approximation for both
  temporal and spectral derivatives