Numerical Techniques in Electromagnetics
THE FINITE-DIFFERENCE TIME-DOMAIN
(FDTD) METHOD – PART III
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Yee Discrete Algorithm - Advanced Device Simulation - Lecture Slides, Slides of Computer Science
These are the Lecture Slides Advanced Device Simulation which includes Discretized Numerical Method, Quantum Point Contact, Total Wave Function, Real Quantum Point, Quantum Dot, Transmission Coefficient, Transfer-Scattering Matrix Formalism etc. Key important points are: Yee Discrete Algorithm, Finite-Difference Time-Domain Method, Maxwell’s Equations, Discretization Steps, Numerical Constant, Yee’s Cell, Absorbing Boundary Conditions, Liao Extrapolation
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Numerical Techniques in Electromagnetics
THE FINITE-DIFFERENCE TIME-DOMAIN
(FDTD) METHOD – PART III
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11. Yee’s discrete algorithm^ Maxwell’s equations are discretized using central FDs. Weset the magnetic loss equal to zero. Then,^ 0,
i^0 m^
m
σ^ =^
= J
,^ 1,^
, ,^
, ,^1
, ,
, ,^
, ,
0.^ 0.^
i j^ k^
i j k^
i j k^
i j k
i j k^
i j k
n^
n^
n^
n
z^
z^
y^
y
n^
n x^
x
E^
E^
E^
E
t H^
H^
y^
z
+^
+^
−^
−^
⎡^
=^
−^ ⋅^
⎢^
⎢^
⎣^
+^
, ,^1
, ,^
1, ,^
, ,
, ,^
, ,
0.^ 0.^
i j k^
i j k^
i^ j k^
i j k
i j k^
i j k
n^
n^
n^
n
x^
x^
z^
z
n^
n y^
y
E^
E^
E^
E
t H^
H^
z^
x
μ
+^
+^
−^
−^
⎡^
=^
−^ ⋅^
⎢^
⎢^
⎣^
+^
1, ,^
, ,^
,^ 1,^
, ,
, ,^
, ,
0.^ 0.^
i^ j k^
i j k^
i j^ k^
i j k
i j k^
i j k
n^
n^
n^
n
y^
y^
x^
x
n^
n z^
z
E^
E^
E^
E
t H^
H^
x^
y
μ
+^
+^
−^
−^
⎡^
=^
−^ ⋅^
⎢^
⎢^
⎣^
+^
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11. Yee’s discrete algorithm – cont. The above coefficients are obtained by averaging the
E -
field, which appears in the loss term. For example,^ , ,^
, ,^
, ,^
, , , ,^
,^ 1,^
, ,^
, ,^1
, ,
1
1
0.^ 0.^ 0.^
2
i j k^
i j k^
i j k^
i j k i j k^
i j^ k^
i j k^
i j k
i j k
y^ i
x^
z e^ x^
ex
n^
n^
n^
n
x^
x^
x^
x e n^
n^
n^
n
z^
z^
y^
y^
i ex
H E^
HE
J
t^
y^ z E^
E^
E^
E
t H^
H^
H^
H^
J
y^
z
ε^
σ −^
−
+^
+^
+^
+^
∂ ∂^
+^
=^ −
−^
∂^
∂^ ∂
−^
+^
−^
−^
+^
The discretization steps in time and in space, as well asthe numerical constant
are determined as
for the wave equation.
/c t h
α^ =^ +^
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11. Yee’s discrete algorithm – cont. Yee’s cell^ ( , , ) E^ x i j k
E^ y ( , , )i j k H^ y ( , , )i j k
H^ z ( , , )i j k E^ x ( , ,^ 1)i j k+ E^ z ( , , )i j k
H^ x ( , , )i j k
x^ ( )i^ (^ z^ (^ )k y)j E^ x ( , 1, )i j^ k^ +
E^ x ( , 1,^ 1)i j k +^ + (^ 1, ,^ E^ y ( , ,^ 1)i j k+ E^ y 1)i j k + + E^ y ( 1, , )i j k +
E^ z ( 1,^ 1, )i j^ k + + E^ z ( ,^ 1, )i j^ k^ +
E^ z ( 1, , )i j k +
H^ y ( ,^ 1, )i j^ k^ + H^ x ( 1, , )i j k +
H^ z ( , ,^ 1)i j k+
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12. Absorbing (radiation) boundary conditions^ ABCs constitute a special type of BCs, which simulatereflection-free propagation out of the computational domain.ABCs are necessary in open (radiation/scattering)problems, as well as in guided-wave problems wherematched port terminations are needed.^ The simplest ABCs are associated with variousapproximations of a one-way plane wave propagation.^ • One-way wave equation (Mur’s ABC)^ • Liao extrapolation^ • Perfectly Matched Layers – basics^ • Others: Higdon operator, Bayliss-Turkel annihilatingoperators, etc.
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12. ABCs – cont. A.The one-way wave equation
(B.Engquist and A.Majda, “Absorbing
boundary conditions for the numerical simulation of waves,”
Mathematics of Computation
, vol.
31, 1977, pp. 629-
This is an equation which permits wave propagation in onlyone direction. Consider the 3-D scalar wave equation
2 2
2
2
2 2
2
f^ f
f^
f
x^ y
z^
c^ t ∂^ ∂
∂^
+^ +
−^
∂^ ∂
∂^
∂^
(^0) Lf =
The partial derivative operator is defined as
2 2
2
2 2
2 2
2 2
2 2
2
x^ y^
z^
t
L^
c
x^ y
z^
c^ t
−
∂^ ∂
∂^
=^ +
+^
−^
= ∂^ + ∂
+ ∂^ −
∂^ ∂
∂^
We wish to simulate one-way propagation along –
x^ at^ x =0. Docsity.com
12. ABCs – cont. This becomes obvious in the case of a plane wavepropagating along
+/-^ x.
1
1 , x^
t^
x^
t
L^
c^
L^
c
−^
−^
+^
−
⇒^ = ∂
−^ ∂
+^ ∂
0,^
z^
y^
S
∂^ =^
∂^ =^ ⇒
f^ f L f^
x^ c^
t ∂^ −
=^ −^
∂^
∂^
Solution:
(^
f^ x^ ct+
(^1) f f^0 L f^
x^ c^
t ∂^ +
=^ +^
∂^
∂^
Solution:
(^
f^ x^ ct−
The radical appearing in
+^ L and
-^ L can be expanded using
Taylor series
2 1/ 2^
2
4 1 (1^ )
(^ )
S^
S^ O S
−^
=^ −^
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12. ABCs – cont.^2 If^ S is very small, then
(^2)
1/ 2(1 )^1 S− ≈
The above is a first-order approximation of
S. This means
that the partial derivatives with respect to
y^ and^
z^ are very
small when compared with the partial derivative with respectto time scaled by the velocity of propagation
c. 2
2
2
1
1 y^
z t^
t
S^ c
c −^
− ∂^
⎛^ ⎞
⎛^
=^
⎜^ ⎟
⎜^
∂^
⎝^ ⎠
⎝^
This happens when the wave is incident upon the
x =const.
plane almost normally. The
-^ L operator then becomes 1
1 1
x^
t x t L^
c L f f^ c
f −^
− − − = ∂^ −^
⇒^
= ∂^ −
∂^
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12. ABCs – cont.
(^
2
2
2
2 1
xt^
tt^
yy^
zz c
L f^
f^
f^
f^ f c −^ = ∂^
−^ ∂^
+^ ∂^
+ ∂^
at= 0 x^ =
(^
2
2
2
2 1
xt^
tt^
yy^
zz c
L f^
f^
f^
f^ f c +^ = ∂^
+^ ∂^
−^ ∂^
+ ∂^
=^
max at^ x^
x=
(^
2
2
2
2 1
yt^
tt^
xx^
zz c
L f^
f^
f^
f^ f c −^ = ∂^
−^ ∂^
+^ ∂^
+ ∂^
=^
at^
(^0) y =
(^
2
2
2
2 1
yt^
tt^
xx^
zz c
L f^
f^
f^
f^ f c +^ = ∂^
+^ ∂^
−^ ∂^
+ ∂^
=^
max at^ y^
y= etc.
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12. ABCs – cont. Mur’s ABC of 2
nd^ order
(G. Mur, “Absorbing boundary conditions for the finite-
difference approximation of the time-domain electromagnetic field equations,”
IEEE Trans.
Electromagnetic Compatibility
, vol. 23, 1981, pp. 377-382.
Mur implemented the above approximate expressions intofinite-difference equations. Mur expands the partialderivatives in the
+^ L and
-^ L operators using central finite
differences of the field component about an auxiliary gridpoint displaced half a step along the direction of absorptionand along time. Consider propagation along –
x , at the
x =0 boundary. We
assume that the scalar function
f^ is evaluated at integer
spatial grid positions (
i , j , k ) and time positions
n.
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12. ABCs – cont. Substitute all the FD approximations above in
(^
2
2
2 2 1
0 2 xt^
tt^ c yy^ zz L f^
f^
f^
f^ f c −^ =^ ∂^
−^ ∂^
+^ ∂^
- ∂^
=
The result is
(^
)^ (^
(^
(^
1
1
1
1 1
2
0, ,^
0, ,^
1, ,^
0, ,^
0, ,^
1, ,
3 0,^
1,^
0, ,^
0,^ 1,^
1,^ 1,^
1, ,^
1,^ 1,
3 0, ,
1
0, ,^
0, ,^1
1, ,^1
1, ,^
1, ,^1
n^
n^
n^
n^
n^
n
j k^
j k^
j k^
j k^
j k^
j k
n^
n^
n^
n^
n^
n
y^ j^
k^
j k^
j^ k^
j^ k^
j k^
j^ k
n^
n^
n^
n^
n^
n
z^ j k^
j k^
j k^
j k^
j k^
j k
f^
f^ k
f^
f^
k^ f^
f
k^ f^
f^
f^
f^
f^
f
k^ f^
f^
f^
f^
f^
f
+^
−^
+^
− −^
+^
−^
−^
+^
−^
= −^
+^
+^
+^
+^
+^
−^
+^
+^
−^
+^
+^
−^
+^
+^
−^
c t^1 x k^ c t
− + += x++ +^
x k^ c t
- = x++ +
2 3
(^ )^22 (^
y
c t^ x k^
y^ c t^
x =^
+^ + +
+^ +
3
(^ )^22 (^
z
c t^ x k^
z^ c t^
x =^
+^ + +
+^ +
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12. ABCs – cont.^ Mur’s ABC of 1
st^ order To obtain Mur’s approximation of
(^1 0) x t L f^
f^ c^
f −^
− =^ ∂^ −
∂^
=
simply remove the 2nd order
y - and
z -derivatives from
the formula above:
(^
)^ (^
1
1
1
1 1
2
0, ,^
0, ,^
1, ,^
0, ,^
0, ,^
1, ,
n^
n^
n^
n^
n^
n
j k^
j k^
j k^
j k^
j k^
j k
f^
f^ k
f^
f^
k^ f^
f
+^
−^
+^
−
= −^
+^
+^
+^
c t^1 x k^ c t
− + += x++ +^
x k^ c t
- = x++ +
In Yee’s algorithm, the
E -field components tangential to
the boundary are evaluated at this boundary. Forexample, at an
x =0 boundary wall, the
E and y^
E fieldz^
components define the boundary values of the EM fieldproblem. Mur’s ABC is applied to them.
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12. ABCs – cont.^ The field values used for the approximation are obtainedby a simultaneous shift in space-time:
(^
(^
(^
(^
1
max 2 max 3
max max 1
3 ,^
2 , (^
N m^
f^ f^
x^
c t t
m^
f^ f^
x^
c t t^
t
m^
f^ f^
x^
c t t^
t
m^ N^
f^ f^
x^ N
c t t^
N^
t
α^ α α α =^ =
=^
=^
−^
=^
=^
−^
=^
=^
−^
−^ −
+ +^
+^
Notice that such representation corresponds to a wavepropagating in the +x direction:
(^
f^ x^ ct−
We aim at finding
0
(^ ,^ max
f^ f^
x^ t^
t =^
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12. ABCs – cont.^ We now define backward finite-difference approximationth^ of^ p
order at the point
(^1 max
x^
c t t ξ
α =^
−^ +
1
1 1 1 1
2 2
2
1 1 1
1
1 2 3
3
2
2 1
1
1
2 1 1
1
1
1
2
(^ ) (^ )^
(^ )^
N^ (^ )
N^
N^
N
D f^
f^ f^
f
D^ f^
f^
f^ f
D f^
f^
f^
f
D^ f^
f^
f^
f
ξ^ ξ ξ ξ^
−^
−
≡ ∆^
=^ − ≡ ∆ = ∆^
≡ ∆^
= ∆^
≡ ∆^
= ∆^
1 2 f^ f^ f^2 ∆^ =^
2
1
1 2
2
3 f^
f^
f ∆^ = ∆
1 3 3 f^ f^ f^4 ∆^ =^
(^
)
max (^ )^
,^ (^
m^
m f^ f^
f^ x^
m^ c t t
m^
t
ξ
α
=^
=^
−^
−^ −+
We denote the discrete functions, which go back in spaceand time as (see previous slide)
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