1. Finite Difference Time Domain Method (FDTD)

2. FDTD: The Basic Algorithm
Maxwell’s Equations in the TIME Domain:

3. Equate Vector Components: Six E and H-Field Equations

4. 2-D Equations: Assume that all fields are uniform in y direction (i. e
2-D Equations: Assume that all fields are uniform in y direction (i.e. d/dy = 0)
2D - TE
2D - TM

5. 1-D Equations: Assume that all fields are uniform in y and x directions (i.e. d/dy =d/dx= 0)
1D - TE
1D - TM

6. Discretize Objects in Space using Cartesian Grid
2D Discretization
3D Discretization z x
1D Discretization

7. Define Locations of Field Components: FDTD Cell called Yee Cell
Finite-Difference
Space is divided into small cells
One Cell: (dx)(dy)(dz)
E and H components are distributed in space around the Yee cell (note: field components are not collocated)
FDTD: Yee, K. S.: Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Transactions on Antennas Propagation, Vol. AP-14, pp , 1966.

8. Replace Continuous Derivatives with Differences
Derivatives in time and space are approximated as DIFFERENCES

9. Solution then evolves by time-marching difference equations
Time is Discretized
One Time Step: dt
E and H fields are distributed in time
This is called a “leap-frog” scheme.

10. Yee Algorithm Definitions of spatial meshing and time step
Initialization
Iteration n
Calculation of En from Hn-1/2 and En-1
Calculation of Hn+1/2 from En and Hn-1/2
n=n+1
N=Nbiteration
End

11. 1-D FDTD
Assuming that field values can only vary in the z-direction (i.e. all spatial derivatives in x and z direction are zero), Maxwell’s Equations reduce to: z Hy Ex
e(z), m(z)

12. 1-D FDTD – Staggered Grid in Space
Interleaving of the Ex and Hy field components in space and time in the 1-D FDTD formulation
Time plane

13. Replace all continuous derivatives with central finite differences
1-D FDTD
Replace all continuous derivatives with central finite differences
Note: finite differences are 2nd order in time and space

14. Replace all continuous derivatives with finite differences
1-D FDTD
Replace all continuous derivatives with finite differences
Solve for

15. 1-D FDTD
Replace all continuous derivatives with finite differences and increment time by one half time step
Solve for

16. After some simple algebra:
1-D FDTD
After some simple algebra:

17. 1-D FDTD – Staggered Grid in Space
Interleaving of the Ex and Hy field components in space and time in the 1-D FDTD formulation
i=1
i=2
i=3
i=4
z=0
z=dz
z=2*dz
z=3*dz
i=1
i=2
i=3
i=4
z=dz/2
z=3dz/2
z=5*dz/2
z=7*dz/2

18. 1-D FDTD – Staggered Grid in Space
Interleaving of the Ex and Hy field components in space and time in the 1-D FDTD formulation
i=1
i=2
i=3
i=4
z=0
z=dz
z=2*dz
z=3*dz
i=1
i=2
i=3
i=4
z=dz/2
z=3dz/2
z=5*dz/2
z=7*dz/2

19. 1-D FDTD – Basic Core of Code
for n=2:Nt % all time steps
for i=2:Nx-1 % all interior nodes
Loop through all
of the E -grid
end
Loop through all time steps
Note: H uses the new values of E. This is
Equivalent to incrementing by ½ a time step
for i=2:Nx-1 % all interior nodes
Loop through all
of the H -grid
end
end

20. SOME OPEN QUESTIONS??
How do we determine what Dt and Dz should be?
How do we implement real sources?
How do we simulate open boundaries?
How accurate is the solution?

21. Numerical Stability
Like all iterative algorithms FDTD has the possibility of not converging on a solution. This usually results in the algorithm going unstable and producing ever increasing field values over time. When does this happen?
1-D
2-D
3-D

22. Numerical Dispersion
In real life plane waves traveling in a homogenous medium propagate at the speed of light in that medium independent of frequency or propagation direction. However, due to our approximation of continuous derivatives with finite differences if we launch a plane wave in FDTD it will actually propagate at a slightly different speed than that of light. Moreover, and more disturbing, the propagation velocity will depend on the frequency of the plane wave and its direction of propagation with respect to the FDTD grid. This effect is known as numerical dispersion. The effect has been well studied and mathematically quantified.

23. Numerical Dispersion Continuous Plane Wave 1-D FDTD Magic time step!
At optimal time step

24. Implementation of Boundary Conditions 1D
Boundary condition for a perfectly electrically conducting (PEC) material
Absorbing/open boundary condition in 1-D
Space-time-extrapolation of the first order
For the magic time-step
a plane wave needs one time steps to travel over one grid cell with the size ∆z
Space-time-extrapolation of the first order

25. (MATLAB DEMO)

26. 2-D Equations: Assume that all fields are uniform in y direction (i. e
2-D Equations: Assume that all fields are uniform in y direction (i.e. d/dy = 0)
2D - TM
2D - TE

27. 2D - TM E-GRID H-GRID Ex Ez Hy dz dx (i-1,j-1) (i,j-1) (i+1,j-1)

28. 2D – TM: Derive FDTD EquationsEx Ez Hy dz
(i-1,j-1)
(i,j-1)
(i+1,j-1)
(i-1,j)
(i,j)
(i+1,j)
(i,,j+1)
(i+1,j+1)
(i-1,j+1)

29. 2D – TM: Derive FDTD Equations
H-GRID dx Ex Ex Hy Hy Ez Ez Ez
(i,j)
(i-1,j) Ex
(i+1,j)
(i-1,j) Ex Hy Hy dz Ez Ez Ez
(i-1,j-1)
(i,j-1) Ex Ex
Solve for

30. 2D – TM: Derive FDTD Equations

31. 2-D FDTD – Basic Core of Code
for n=2:Nt % all time steps
for i=2:Nx-1 % all interior nodes
Loop through all
of the E -grid
end
for i=2:Nx-1 % all interior nodes
Loop through all
of the H -grid
end
end

32. 2D – TE: Derive FDTD Equations

33. Numerical Dispersion 2D case
Using same procedure as for the 1D case we obtain:

34. Source Modeling
1. We can implement a “hard” source by forcing the fields to predefined values at specific nodes in the FDTD grid.
For example:

35. Source Modeling: Soft Sources Scattered Field Formulation
1. We can implement a “soft” source by first reformulating Maxwell’s equations for only the scattered field.
(1)
(2)
(3)
(1)-(2):
Use (3) in above
and a little algebra:

36. Six E and H-Field Equations Scattered Field

37. 2D Scattered Field TM TE

38. 1D Scattered Field TM TE

39. 1-D FDTD Scattered Fields (TE)

40. 1-D FDTD Scattered Fields (TE)
EXAMPLE INCIDENT FIELD
let
In FDTD

41. 1-D FDTD – Basic Core of 1D Scattered Field Code (non-magnetic)
for n=2:Nt % all time steps
for i=2:Nx-1 % all interior nodes
If tau<0
Loop through all
of the E -grid
else
end
end
for i=2:Nx-1 % all interior nodes
Loop through all
of the H -grid
end
end

42. 2-D FDTD Scattered Fields (TE)

43. Implementation of Boundary Conditions 2D
Boundary condition for a perfectly electrically conducting (PEC) material
Absorbing/open boundary condition in 1-D
Many different methods
Interpolation/Extrapolation in time and space (Liaos)
One way way equation (Mur)
Artificial absorbing boundaries (PMLs)

44. One Way Wave Equation

45. One Way Wave Equation

46. One Way Wave Equation
If we could implement the one way equations on a FDTD boundary we would have the perfect ABC (i.e. zero reflection. Unfortunately we can’t do that since we don’t know how to implement the square root operator. So we need to approximate.
First order Taylor series expansion

47. One Way Wave Equation

48. Absorbing Boundary Conditions 2D: Mur 1st order
One-way Wave Equations: They approximately represent waves traveling in only one direction.
y=h
y=0
x=0
x=w

49. Absorbing Boundary Conditions 2D: Mur
One-way Wave Equations: They approximately represent waves traveling in only one direction.
y=h
y=0
x=0
x=w

50. Absorbing Boundary Conditions 2D: 1st Order Mur
2D 1st Order Mur Equation
y=h
y=0
x=0
x=w

51. Absorbing Boundary Conditions 2D: Mur 2nd Order
One-way Wave Equations: They approximately represent waves traveling in only one direction.
y=h
y=0
x=0
x=w

52. (MATLAB DEMO)