finite difference approximation of maxwell’s equations on
TRANSCRIPT
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Advanced Computation:
Computational Electromagnetics
Finite‐Difference Approximation of Maxwell’s Equations on a Yee Grid
Outline
• The Starting Point
• Finite‐Difference Approximation of ∇ 𝐸 𝜇 𝐻
• Finite‐Difference Approximation of ∇ 𝐻 𝜀 𝐸• Summary of Finite‐Difference Equations on a Yee Grid
Slide 2
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Slide 3
The Starting Point
Normalize the Grid Coordinates 𝑥, 𝑦 and 𝑧
Slide 4
0
0
0
x k x
y k y
z k z
yzxx x
x zyy y
y xzz z
yzxx x
x zyy y
y xzz z
EEH
y z
E EH
z xE E
Hx y
HHE
y z
H HE
z x
H HE
x y
Normalizethe grid
This “absorbs” the k0 term into the spatial derivatives and simplifies Maxwell’s equations.
0
0
0
0
0
0
yzxx x
x zyy y
y xzz z
yzxx x
x zyy y
y xzz z
EEk H
y z
E Ek H
z xE E
k Hx y
HHk E
y z
H Hk E
z x
H Hk E
x y
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The Starting Point
Slide 5
yz
x
z
xx
y
x
z
xz
yy
zy
EE
E E
E E
y z
z x
x
H
H
yH
xx
y
z
yz
x z
x
y
yz
y
z
x
HH
H H
y z
z x
E
H
y
HE
x
E
Here, only diagonally anisotropic material tensors were retained. This will be needed to incorporate a uniaxial perfectly matched layer (UPML) absorbing boundary condition.
These equations are valid independent of the chosen sign convention.
Slide 6
Finite‐Difference Approximation of
𝐫
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Finite‐Difference Equation for 𝐻
Slide 7
x xyz
xy z
EEH
, , ,, ki j k ixx
jxH
Finite‐Difference Equation for 𝐻
Slide 8
x xyz
xy z
EEH
, 1, , ,i j k i j kz zE
y
E
, , ,, ki j k ixx
jxH
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Finite‐Difference Equation for 𝐻
Slide 9
x xyz
xy z
EEH
, 1, , ,i j k i j kz zE
y
E
, , ,, ki j k ixx
jxH
, , 1 , ,i j k i j ky yE
z
E
Finite‐Difference Equation for 𝐻
Slide 10
y yx z
yz x
E EH
, , ,, ki j k iyy
jyH
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Finite‐Difference Equation for 𝐻
Slide 11
y yx z
yz x
E EH
, , 1 , ,i j k i j kx xE
z
E
, , ,, ki j k iyy
jyH
Finite‐Difference Equation for 𝐻
Slide 12
y yx z
yz x
E EH
, , 1 , ,i j k i j kx xE
z
E
, , ,, ki j k iyy
jyH
1, , , ,i j k i j kz zE
x
E
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Finite‐Difference Equation for 𝐻
Slide 13
z zy x
zx y
E EH
, , ,, ki j k izz
jzH
Finite‐Difference Equation for 𝐻
Slide 14
z zy x
zx y
E EH
1, , , ,i j k i j ky yE
x
E
, , ,, ki j k izz
jzH
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Finite‐Difference Equation for 𝐻
Slide 15
z zy x
zx y
E EH
1, , , ,i j k i j ky yE
x
E
, , ,, ki j k izz
jzH
, 1, , ,i j k i j kx xE
y
E
Slide 16
Finite‐Difference Approximation of
𝐫
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Finite‐Difference Equation for 𝐸
Slide 17
xxxyz
H
yE
z
H
, , ,, ki j kx
i jxx E
Finite‐Difference Equation for 𝐸
Slide 18
xxxyz
H
yE
z
H
, , , 1,i j k i j kz z
y
H H
, , ,, ki j kx
i jxx E
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Finite‐Difference Equation for 𝐸
Slide 19
xxxyz
H
yE
z
H
, , , 1,i j k i j kz z
y
H H
, , ,, ki j kx
i jxx E
, , , , 1i j k i j ky yH H
z
Finite‐Difference Equation for 𝐸
Slide 20
yyx z
y
H H
xE
z
, , ,, ki j ky
i jyy E
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Finite‐Difference Equation for 𝐸
Slide 21
yyx z
y
H H
xE
z
, , ,, ki j ky
i jyy E
, , , , 1i j k i j kx x
z
H H
Finite‐Difference Equation for 𝐸
Slide 22
yyx z
y
H H
xE
z
, , ,, ki j ky
i jyy E
, , 1, ,i j k i j kz zH H
x
, , , , 1i j k i j kx x
z
H H
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Finite‐Difference Equation for 𝐸
Slide 23
zzy x
z
H H
yE
x
, , ,, ki j kz
i jzz E
Finite‐Difference Equation for 𝐸
Slide 24
zzy x
z
H H
yE
x
, , ,, ki j kz
i jzz E
, , 1, ,i j k i j ky y
x
H H
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Finite‐Difference Equation for 𝐸
Slide 25
zzy x
z
H H
yE
x
, , , 1,i j k i j kx xH H
y
, , ,, ki j kz
i jzz E
, , 1, ,i j k i j ky y
x
H H
Slide 26
Summary ofFinite‐Difference Equations
on a Yee Grid
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Summary of Finite‐Difference Approximations of Maxwell’s Equations
Slide 27
x
y
z
yz
x z
y
xx
yy
zz
xx
yy
zz
yz
x z
y
x
x
x
y
z
EE
E E
E E
E
E
E
y z
z x
x y
y z
H
H
H
HH
H H
H H
z x
x y
, , 1, ,
, ,
,
, ,
,
, ,, 1, , ,
, , 1 , , 1, , , ,
1, , , , , 1,
,
,
, ,
, , ,
, ,
1
,
i j kx
i j ky
i j kz
i j k i
i j k i j ki j ki j kxx
i j kyy
i j ky yz z
i j k i j k i j k i j kx x z z
i j k i j k i j k i j ky y x x i j k
j
z
kz z
z
y
E E
y z
E E
E E E E
H
H
H
HH
z x
x
E E E
y
H
y
E
, , , , 1
, , , , 1
, ,
, ,
, ,
, , 1, ,
, , 1, , , ,
, ,
, ,
,, 1
,,
i j kx
i j ky
i
i j k i j ky
i j k i j k i j k i j k
i j kxx
i jx x z z
i j k i j k i j k i j k
kyy
i j kyz
kz
xz
jy x
H
H H H H
H H
z
z x
x y
H
E
E
HE