THE FINITE-DIFFERENCE TIME-DOMAIN (FDTD) METHOD – PART I

25 Pages·2011·0.12 MB·English

by  Natalia K Nikolova

Numerical Techniques in Electromagnetics ECE 757
THE FINITE-DIFFERENCE TIME-DOMAIN
(FDTD) METHOD – PART I
NATALIA K. NIKOLOVA
[email protected]
ITB-A220, ext. 27141
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1. Outline
• finite differences for derivative approximation
• the wave equation in 1-D
initial/boundary conditions and excitation sources
• generalization to 2-D and 3-D
• Maxwell’s equations; 2-D problems: TM and TE modes
• Yee’s algorithm in 3-D space
• Yee’s algorithm in 2-D space
• introduction to absorbing boundary conditions
• PROJECT: determine the modes of a rectangular
waveguide
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2. References and recommended further reading
[1] R.C. Booton, Computational Methods for Electromagnetics and
Microwaves, Wiley, 1992, pp. 59-73
[2] M.N.O. Sadiku, Numerical Techniques in Electromagnetics, CRC
Press, 2001, pp. 159-192
[3] W. Yu et al., Parallel Finite Difference Time Domain Method, Artech,
2006
[3] A. Taflove, Computational Elertodynamics: the Finite-Difference Time-
Domain Method, Artech, 1995
[4] A. Taflove and S.C. Hagness, same as above, 2nd ed., Artech, 2000
[5] K. Kunz and R. Luebbers, Finite-Difference Time-Domain Method for
Electromagnetics, CRC Press, 1993
[6] K.S. Yee, “Numerical solution of initial boundary value problems
involving Maxwell’s equations in isotropic media,” IEEE Trans.
Antennas Propagat., vol. AP-14, No. 3, pp. 302-307, May 1966
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3. Finite differences for derivative approximation
1st order derivatives
forward FD
f (x)
∆x ∆x
df (x ) df f − f f
i = i ≈ i+1 i i−1
dx dx ∆x
f
backward FD
i
f
df (x ) df f − f i+1
i i i i−1
= ≈
dx dx ∆x xi−1 xi xi+1 x
central FD
df (x ) df f − f
i i i+1 i−1
= ≈
dx dx 2∆x
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3. Finite differences for derivative approximation – cont.
accuracy
Taylor expansions
at x + ∆x
i
2 3
df 1 d f 1 d f
f (x + ∆x) = f = f + ∆x i + ∆x2 i + ∆x3 i + O4
i i+1 i
2 3
dx 2 dx 6 dx
df f − f
i = i+1 i + O(∆x)1
dx ∆x
at x − ∆x
i
2 3
df 1 d f 1 d f
f (x − ∆x) = f = f − ∆x i + ∆x2 i − ∆x3 i + O4
i i−1 i
2 3
dx 2 dx 6 dx
df f − f
i = i i−1 + O(∆x)1
dx ∆x
forward and backward FDs have 1st order accuracy
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3. Finite differences for derivative approximation – cont.
accuracy: central FDs have 2nd order accuracy
df f − f
combine both expansions to obtain: i = i+1 i−1 + O2
dx 2∆x
central FD at half steps
f (x)
∆x ∆x
f df (x + ∆x / 2)
i−1 i
df =
i−1/2
dx
dx
df
i+1/2 df f − f
f = i+1/2 = i+1 i + O2
i dx
dx ∆x
f
i+1
x x x x
i−1 i i+1
∆x ∆x
x − x +
i i
2 2
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3. Finite differences for derivative approximation – cont.
second-order accurate backward/forward approximations
of 1st order derivatives
df −3 f + 4 f − f df 3 f − 4 f + f
i i i+1 i+2 i i i−1 i−2
≈ ≈
dx ∆x dx 2∆x
2nd order derivatives
2 3
df 1 d f 1 d f
f = f + ∆x i + ∆x2 i + ∆x3 i + O4
i+1 i
2 3
dx 2 dx 6 dx
+
2 3
df 1 d f 1 d f
f = f − ∆x i + ∆x2 i − ∆x3 i + O4
i−1 i
2 3
dx 2 dx 6 dx
d 2 f f − 2 f + f
= i−1 i i+1 + O2
dx2 ∆x2
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3. Finite differences for derivative approximation – cont.
Laplace operator in 2-D space: (x, y)
∂2 f ∂2 f
∇2 f = +
xy
∂x2 ∂y2
f + f − 2 f f + f − 2 f
∇2 f ≈ i−1, j i+1, j i, j + i, j−1 i, j+1 i, j
xy
∆x2 ∆y2
y
f
i, j+1
f f f
i−1, j i, j i+1, j
if ∆x = ∆y = ∆h x
f + f + f + f − 4 f
∇2 f ≈ i−1, j i+1, j i, j−1 i, j+1 i, j
xy f
∆h2 i, j−1
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3. Finite differences for derivative approximation – cont.
Laplace operator in 3-D space (x, y, z)
∂2 f ∂2 f ∂2 f
∇2 f = + +
z
∂x2 ∂y2 ∂z2 f
i, j,k+1
f
i−1, j,k
f + f − 2 f
∇2 f ≈ i−1, j,k i+1, j,k i, j,k +
∆x2 f f
i, j−1,k
i, j,k
y
f + f − 2 f f
i, j−1,k i, j+1,k i, j,k + i, j+1,k
∆y2 f
i+1, j,k
x
f + f − 2 f
i, j,k−1 i, j,k+1 i, j,k
f
∆z2 i, j,k−1
if ∆x = ∆y = ∆z = ∆h
f + f + f + f + f + f − 6 f
∇2 f ≈ i−1, j,k i+1, j,k i, j−1,k i, j+1,k i, j,k−1 i, j,k+1 i, j
∆h2
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4. The wave equation in 1-D space
∂2 f 1 ∂2 f
− = −g(x,t)
∂x2 c2 ∂t2
general solution
+ −
f (x,t) = f (x − ct) + f (x + ct)
wave traveling in the +x direction wave traveling in the -x direction
to determine the particular solution, we need
• 2 boundary conditions:
at x = 0 at x = x
max
∂f ∂f
f (0,t) or f (x ,t) or
max
∂x ∂x
x=0 x=x
max
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