electromagnetic wave propagation lecture 5: finite differences in … · 2010. 9. 13. · the nite...
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![Page 1: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/1.jpg)
Electromagnetic Wave PropagationLecture 5: Finite differences in the time
domain
Daniel Sjoberg
Department of Electrical and Information Technology
September 14, 2010
![Page 2: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/2.jpg)
Outline
1 Finite differences
2 Generalization to 3D
3 Dispersion analysis in 3D
4 Example applications
5 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 3: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/3.jpg)
Outline
1 Finite differences
2 Generalization to 3D
3 Dispersion analysis in 3D
4 Example applications
5 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 4: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/4.jpg)
Numerical methods
Some problems are too complicated to do by hand, such as apacemaker in the human body. Numerical methods help.
Daniel Sjoberg, Department of Electrical and Information Technology
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Finite difference approximation
To make numerical simulations, we approximate the derivativeswith finite differences (where the field is evaluated in grid pointsE|nr = E(r∆z, n∆t):
∂E
∂z
∣∣∣∣nr
=E|nr+1/2 − E|
nr−1/2
∆z
∂2E
∂z2
∣∣∣∣nr
=
∂E∂z
∣∣nr+1/2
− ∂E∂z
∣∣nr−1/2
∆z=
E|nr+1−E|nr∆z − E|nr−E|nr−1
∆z
∆z
Thus, the wave equation ∂2E∂z2− 1
c2∂2E∂t2
= 0 becomes
E|nr+1 − 2E|nr + E|nr−1(∆z)2
− 1
c2E|n+1
r − 2E|nr + E|n−1r
(∆t)2= 0
We expect the error in the approximation to decrease as ∆z and∆t become small.
Daniel Sjoberg, Department of Electrical and Information Technology
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Time stepping
The finite difference approximation of the wave equation is
E|nr+1 − 2E|nr + E|nr−1(∆z)2
− 1
c2E|n+1
r − 2E|nr + E|n−1r
(∆t)2= 0
By solving for the field at time step n+ 1, we find
E|n+1r = 2E|nr − E|n−1r +
(c∆t
∆z
)2
(E|nr+1 − 2E|nr + E|nr−1)
Thus, the solution at time (n+ 1)∆t can be found if the solutionat two previous time steps is known.
Daniel Sjoberg, Department of Electrical and Information Technology
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Leap frog stencil
-
6
0
0
z
t
∆z
∆t
r∆z
n∆t
b b b b b b bb b b b b b bb b b b b b bb b b b b b bb b b b b b b
jj
j jj
Initial conditions: E|0r = fr and E|1r = gr (known functions).Boundary conditions: E|n0 = 0 and E|nR = 0 (metal walls).
Daniel Sjoberg, Department of Electrical and Information Technology
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Leap frog stencil
-
6
0
0
z
t
∆z
∆t
r∆z
n∆t
b b b b b b bb b b b b b bb b b b b b bb b b b b b bb b b b b b b
j
jj j
j
Initial conditions: E|0r = fr and E|1r = gr (known functions).Boundary conditions: E|n0 = 0 and E|nR = 0 (metal walls).
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 9: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/9.jpg)
Leap frog stencil
-
6
0
0
z
t
∆z
∆t
r∆z
n∆t
b b b b b b bb b b b b b bb b b b b b bb b b b b b bb b b b b b b
jj
j jj
Initial conditions: E|0r = fr and E|1r = gr (known functions).Boundary conditions: E|n0 = 0 and E|nR = 0 (metal walls).
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 10: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/10.jpg)
Leap frog stencil
-
6
0
0
z
t
∆z
∆t
r∆z
n∆t
b b b b b b bb b b b b b bb b b b b b bb b b b b b bb b b b b b b
jj
j j
j
Initial conditions: E|0r = fr and E|1r = gr (known functions).Boundary conditions: E|n0 = 0 and E|nR = 0 (metal walls).
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 11: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/11.jpg)
Leap frog stencil
-
6
0
0
z
t
∆z
∆t
r∆z
n∆t
b b b b b b bb b b b b b bb b b b b b bb b b b b b bb b b b b b b
jj
j jj
Initial conditions: E|0r = fr and E|1r = gr (known functions).Boundary conditions: E|n0 = 0 and E|nR = 0 (metal walls).
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 12: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/12.jpg)
Leap frog stencil
-
6
0
0
z
t
∆z
∆t
r∆z
n∆t
b b b b b b bb b b b b b bb b b b b b bb b b b b b bb b b b b b b
jj
j jj
Initial conditions: E|0r = fr and E|1r = gr (known functions).Boundary conditions: E|n0 = 0 and E|nR = 0 (metal walls).
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 13: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/13.jpg)
Dispersion relation
A harmonic wave propagating through the lattice can be written
ejωn∆t−jkr∆z
For a “real” wave we should have ω = ck . Inserting theexponential function into the difference approximation we obtain
ejωn∆t−jkr∆z(e−jk∆z − 2 + ejk∆z
(∆z)2− 1
c2ejω∆t − 2 + e−jω∆t
(∆t)2
)= 0
which can be rewritten as(sin
ω∆t
2
)2
=
(c∆t
∆z
)2(sin
k∆z
2
)2
This relates the spatial frequency k to the temporal frequency ωon the numerical grid, and is called the dispersion relation.
Daniel Sjoberg, Department of Electrical and Information Technology
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Time step and stability: R = c∆t/∆z
R = 1: Magic time step, ω = ±ck, no dispersion.
R < 1: Stable solution, ω 6= ±ck, different frequencies travel withdifferent speed (lower than c).
R > 1: Unstable solution, growing exponentially.
Daniel Sjoberg, Department of Electrical and Information Technology
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Example: square wave
R = 1
R < 1
R > 1
Daniel Sjoberg, Department of Electrical and Information Technology
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Example: smooth wave
Upper graph: 12 points across 1/e width.Lower graph: 6 points across 1/e width.
Daniel Sjoberg, Department of Electrical and Information Technology
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Computation of resonance frequencies 1
Even though FDTD solves problems in the time domain, it can beused to compute resonance frequencies.In a rectangular box with metal walls (cavity), only waves withparticular resonance frequencies can exist.
Lx
Lz
Ly
fmnp =c
2
[(m
Lx
)2
+
(n
Ly
)2
+
(p
Lz
)2]1/2
Daniel Sjoberg, Department of Electrical and Information Technology
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Computation of resonance frequencies 2
To compute the resonance frequencies in a cavity using FDTD, dothe following:
I Discretize the cavity with finite differences.
I Set up PEC boundary conditions.
I Give random data as initialization in order to excite allfrequencies.
I Run simulation for a long time.
I Fourier transform the field, sampled at a point.
I The resulting spikes correspond to the resonances of thecavity.
Explicit matlab code in Bondeson, Rylander, Ingelstrom.
Daniel Sjoberg, Department of Electrical and Information Technology
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Outline
1 Finite differences
2 Generalization to 3D
3 Dispersion analysis in 3D
4 Example applications
5 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 20: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/20.jpg)
One dimension
Maxwell’s equations are
∂Ex∂z
= −µ∂Hy
∂t∂Hy
∂z= −ε∂Ex
∂t
Instead of discretizing Ex and Hy at the same grid points, shiftone field half a grid point in space and time:
Ex|nr+1 − Ex|nr∆z
= −µHy|n+1/2
r+1/2 −Hy|n−1/2r+1/2
∆t
Hy|n+1/2r+1/2 −Hy|n+1/2
r−1/2
∆z= −εEx|
n+1r − Ex|nr∆t
This makes it possible to use central differences for all derivatives.
Daniel Sjoberg, Department of Electrical and Information Technology
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Graphical point of view
Daniel Sjoberg, Department of Electrical and Information Technology
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Generalization to 3D
E-field along edges, H-field along sides. Also staggered in time.Daniel Sjoberg, Department of Electrical and Information Technology
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Integral interpretation
∮CE·dr =
∫S∇×E·ndS = −
∫Sµ0∂H
∂t·ndS
= −∆x∆yµ0∂
∂tHz
∮CE · dr = Ex|p+1/2,q∆x+ Ey|p+1,q+1/2∆y
− Ex|p+1/2,q+1∆x− Ey|p,q+1/2∆y
Hz can be updated from knowledge of surrounding Exy-fields.Similarly for ∇×H = ∂D/∂t.
Daniel Sjoberg, Department of Electrical and Information Technology
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Integral interpretation
∮CE·dr =
∫S∇×E·ndS = −
∫Sµ0∂H
∂t·ndS = −∆x∆yµ0
∂
∂tHz
∮CE · dr = Ex|p+1/2,q∆x+ Ey|p+1,q+1/2∆y
− Ex|p+1/2,q+1∆x− Ey|p,q+1/2∆y
Hz can be updated from knowledge of surrounding Exy-fields.Similarly for ∇×H = ∂D/∂t.
Daniel Sjoberg, Department of Electrical and Information Technology
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Integral interpretation
∮CE·dr =
∫S∇×E·ndS = −
∫Sµ0∂H
∂t·ndS = −∆x∆yµ0
∂
∂tHz
∮CE · dr = Ex|p+1/2,q∆x+ Ey|p+1,q+1/2∆y
− Ex|p+1/2,q+1∆x− Ey|p,q+1/2∆y
Hz can be updated from knowledge of surrounding Exy-fields.Similarly for ∇×H = ∂D/∂t.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 26: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/26.jpg)
Integral interpretation
∮CE·dr =
∫S∇×E·ndS = −
∫Sµ0∂H
∂t·ndS = −∆x∆yµ0
∂
∂tHz
∮CE · dr = Ex|p+1/2,q∆x+ Ey|p+1,q+1/2∆y
− Ex|p+1/2,q+1∆x− Ey|p,q+1/2∆y
Hz can be updated from knowledge of surrounding Exy-fields.Similarly for ∇×H = ∂D/∂t.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 27: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/27.jpg)
Fields on the boundary
From http://fdtd.wikispaces.com. For PEC boundary, n×E = 0and n ·H = 0.
Daniel Sjoberg, Department of Electrical and Information Technology
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The staggered grid
I Invented by K. S. Yee in 1966.
I Boundary conditions involve only half of the field components(usually n×E and n ·H)
I Completely dominating in commercial codes
I Based on decoupling of electric and magnetic fields:∇×E = −∂B/∂t and ∇×H = ∂D/∂t: time differences ofE depend only on space differences of H and vice versa
I For bianisotropic materials, where D and B depend on bothE and H, the fields may need to be evaluated at the samepoints in space and time
Daniel Sjoberg, Department of Electrical and Information Technology
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Staircasing
The rectangular grid does not conform to curved surfaces:
Daniel Sjoberg, Department of Electrical and Information Technology
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Staircasing
The rectangular grid does not conform to curved surfaces:
Daniel Sjoberg, Department of Electrical and Information Technology
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Grid refinement
Simple method: refinealong planes where thegeometry has small scale
Advanced method: implicit time step-ping in regions with unstructured grid(Rylander & Bondeson 2000)
Daniel Sjoberg, Department of Electrical and Information Technology
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Outline
1 Finite differences
2 Generalization to 3D
3 Dispersion analysis in 3D
4 Example applications
5 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 33: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/33.jpg)
Harmonic waves and difference approximations
When considering harmonic waves
u = ej(ωt−kxx−kyy−kzz)
the difference approximation of the derivatives become
∂u
∂t≈ u|n+1/2
r − u|n−1/2r
∆t= u|nr
ejω∆t/2 − e−jω∆t/2
∆t
= u|nr2j
∆tsin
ω∆t
2
∂u
∂x≈u|nr+1/2 − u|
nr−1/2
∆x= u|nr
e−jkx∆x/2 − ejkx∆x/2
∆x
= u|nr−2j∆x
sinkx∆x
2
Daniel Sjoberg, Department of Electrical and Information Technology
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Dispersion relation
This means derivatives can be replaced by algebraic expressions as
∂
∂t→ Dt =
2j
∆tsin
ω∆t
2∂
∂x→ Dx =
−2j∆x
sinkx∆x
2
and so on. Eliminating the magnetic field, Maxwell’s equationsreduce to the wave equation ∇2E − 1
c2∂2
∂t2E, or(
D2x +D2
y +D2z −
1
c2D2t
)E = 0
Using the expressions for Dx, Dy, Dz, and Dt, we find
sin2(ω∆t/2)
(c∆t)2=
sin2(kx∆x/2)
(∆x)2+
sin2(ky∆y/2)
(∆y)2+
sin2(kz∆z/2)
(∆z)2
Daniel Sjoberg, Department of Electrical and Information Technology
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Time step limit
In order to have a stable scheme, we need
sin2(ω∆t/2) ≤ 1
for all wave vectors k = kxx+ kyy + kzz. The dispersion relationthen implies
c∆t ≤[
1
(∆x)2+
1
(∆y)2+
1
(∆z)2
]−1/2For a cubic grid with ∆x = ∆y = ∆z = h,
∆t ≤ h
c√3≈ 0.577
h
c
Thus, the maximum time step is almost half as short in threedimensions than in one.
Daniel Sjoberg, Department of Electrical and Information Technology
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Discretization and error
From the dispersion relation
sin2(ω∆t/2)
(c∆t)2=
sin2(kx∆x/2)
(∆x)2+
sin2(ky∆y/2)
(∆y)2+
sin2(kz∆z/2)
(∆z)2
we can compute ω(k) and hence the phase and group velocities byTaylor series (allowing 1% error):
vp =ω
k= c
(1− (kh)2
36+ O
((kh)4
))10.5 cells per λ
vg =∂ω
∂k= c
(1− (kh)2
12+ O
((kh)4
))18 cells per λ
The total phase error in domain of diameter L is
ephase = (knum−k)L =
(ω
c(1− (kh)2/36 + · · · )− ω
c
)L ≈ k3h2L
36
meaning the cell size must scale as ω−3/2 to control the totalphase error.
Daniel Sjoberg, Department of Electrical and Information Technology
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Outline
1 Finite differences
2 Generalization to 3D
3 Dispersion analysis in 3D
4 Example applications
5 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
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Typical flow chart for an FDTD code
I PreprocessingI Load grid and geometryI Memory allocationI Declare constantsI Compute parameters
I Main loop (iterate the following steps)I Evaluate sourcesI Update internal H fieldsI Update internal E fieldsI Update boundary conditionsI Save some data, typically on the boundary
I PostprocessingI Fourier transform to frequency domainI Near-to-farfield transformationI Scattering coefficientsI . . .
Daniel Sjoberg, Department of Electrical and Information Technology
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Typical complexity
I Size of computational region in wavelengths
I At least 10–15 points per wavelength
I Small geometries may require high discretization
I Number of time steps (becomes large at resonances)
Daniel Sjoberg, Department of Electrical and Information Technology
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Wireless communication with a pacemaker
I Frequency?
f = 402–405MHz
I Permittivity?
εr ≈ 64
I Conductivity?
σ ≈ 0.94 S/m
I Wavelength?
λ = c/f ≈ c0/(f√εr) =
3·108400·106·8 m ≈ 0.09m
I Geometry?
1 · 1 · 2m3 = 2m3
I Sampling?
λ/15 ≈ 0.006m
I Number of unknowns?
6 20.0063
= 55 · 106 Refined mesh for antennaand organs
The number 6 stems from 3 components in E and 3 components inH. With double precision floats, this means 55 · 106 · 8 = 0.4GB.
Daniel Sjoberg, Department of Electrical and Information Technology
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Wireless communication with a pacemaker
I Frequency? f = 402–405MHz
I Permittivity?
εr ≈ 64
I Conductivity?
σ ≈ 0.94 S/m
I Wavelength?
λ = c/f ≈ c0/(f√εr) =
3·108400·106·8 m ≈ 0.09m
I Geometry?
1 · 1 · 2m3 = 2m3
I Sampling?
λ/15 ≈ 0.006m
I Number of unknowns?
6 20.0063
= 55 · 106 Refined mesh for antennaand organs
The number 6 stems from 3 components in E and 3 components inH. With double precision floats, this means 55 · 106 · 8 = 0.4GB.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 42: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/42.jpg)
Wireless communication with a pacemaker
I Frequency? f = 402–405MHz
I Permittivity? εr ≈ 64
I Conductivity? σ ≈ 0.94 S/m
I Wavelength?
λ = c/f ≈ c0/(f√εr) =
3·108400·106·8 m ≈ 0.09m
I Geometry?
1 · 1 · 2m3 = 2m3
I Sampling?
λ/15 ≈ 0.006m
I Number of unknowns?
6 20.0063
= 55 · 106 Refined mesh for antennaand organs
The number 6 stems from 3 components in E and 3 components inH. With double precision floats, this means 55 · 106 · 8 = 0.4GB.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 43: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/43.jpg)
Wireless communication with a pacemaker
I Frequency? f = 402–405MHz
I Permittivity? εr ≈ 64
I Conductivity? σ ≈ 0.94 S/m
I Wavelength?λ = c/f ≈ c0/(f
√εr) =
3·108400·106·8 m ≈ 0.09m
I Geometry?
1 · 1 · 2m3 = 2m3
I Sampling?
λ/15 ≈ 0.006m
I Number of unknowns?
6 20.0063
= 55 · 106 Refined mesh for antennaand organs
The number 6 stems from 3 components in E and 3 components inH. With double precision floats, this means 55 · 106 · 8 = 0.4GB.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 44: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/44.jpg)
Wireless communication with a pacemaker
I Frequency? f = 402–405MHz
I Permittivity? εr ≈ 64
I Conductivity? σ ≈ 0.94 S/m
I Wavelength?λ = c/f ≈ c0/(f
√εr) =
3·108400·106·8 m ≈ 0.09m
I Geometry? 1 · 1 · 2m3 = 2m3
I Sampling?
λ/15 ≈ 0.006m
I Number of unknowns?
6 20.0063
= 55 · 106 Refined mesh for antennaand organs
The number 6 stems from 3 components in E and 3 components inH. With double precision floats, this means 55 · 106 · 8 = 0.4GB.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 45: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/45.jpg)
Wireless communication with a pacemaker
I Frequency? f = 402–405MHz
I Permittivity? εr ≈ 64
I Conductivity? σ ≈ 0.94 S/m
I Wavelength?λ = c/f ≈ c0/(f
√εr) =
3·108400·106·8 m ≈ 0.09m
I Geometry? 1 · 1 · 2m3 = 2m3
I Sampling? λ/15 ≈ 0.006m
I Number of unknowns?
6 20.0063
= 55 · 106
Refined mesh for antennaand organs
The number 6 stems from 3 components in E and 3 components inH. With double precision floats, this means 55 · 106 · 8 = 0.4GB.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 46: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/46.jpg)
Wireless communication with a pacemaker
I Frequency? f = 402–405MHz
I Permittivity? εr ≈ 64
I Conductivity? σ ≈ 0.94 S/m
I Wavelength?λ = c/f ≈ c0/(f
√εr) =
3·108400·106·8 m ≈ 0.09m
I Geometry? 1 · 1 · 2m3 = 2m3
I Sampling? λ/15 ≈ 0.006m
I Number of unknowns?6 20.0063
= 55 · 106 Refined mesh for antennaand organs
The number 6 stems from 3 components in E and 3 components inH. With double precision floats, this means 55 · 106 · 8 = 0.4GB.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 47: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/47.jpg)
Wireless communication with a pacemaker
I Frequency? f = 402–405MHz
I Permittivity? εr ≈ 64
I Conductivity? σ ≈ 0.94 S/m
I Wavelength?λ = c/f ≈ c0/(f
√εr) =
3·108400·106·8 m ≈ 0.09m
I Geometry? 1 · 1 · 2m3 = 2m3
I Sampling? λ/15 ≈ 0.006m
I Number of unknowns?6 20.0063
= 55 · 106 Refined mesh for antennaand organs
The number 6 stems from 3 components in E and 3 components inH. With double precision floats, this means 55 · 106 · 8 = 0.4GB.
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 48: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/48.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?
λ0 ≈ 0.632µm
I Refractive index plasma?
np ≈ 1.34
I Refractive index in cell?
nc ≈ 1.41
I Minimum wavelength?
λ = λ0/nc ≈ 0.632µm1.41 ≈ 0.45µm
I Diameter?
7.76µm ≈ 17λ
I Height?
2.50µm ≈ 6λ
I Geometry?
82 · 3 (µm)3 = 192 (µm)3
I Sampling?
λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 49: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/49.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma?
np ≈ 1.34
I Refractive index in cell?
nc ≈ 1.41
I Minimum wavelength?
λ = λ0/nc ≈ 0.632µm1.41 ≈ 0.45µm
I Diameter?
7.76µm ≈ 17λ
I Height?
2.50µm ≈ 6λ
I Geometry?
82 · 3 (µm)3 = 192 (µm)3
I Sampling?
λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 50: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/50.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma? np ≈ 1.34
I Refractive index in cell?
nc ≈ 1.41
I Minimum wavelength?
λ = λ0/nc ≈ 0.632µm1.41 ≈ 0.45µm
I Diameter?
7.76µm ≈ 17λ
I Height?
2.50µm ≈ 6λ
I Geometry?
82 · 3 (µm)3 = 192 (µm)3
I Sampling?
λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 51: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/51.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma? np ≈ 1.34
I Refractive index in cell? nc ≈ 1.41
I Minimum wavelength?
λ = λ0/nc ≈ 0.632µm1.41 ≈ 0.45µm
I Diameter?
7.76µm ≈ 17λ
I Height?
2.50µm ≈ 6λ
I Geometry?
82 · 3 (µm)3 = 192 (µm)3
I Sampling?
λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 52: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/52.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma? np ≈ 1.34
I Refractive index in cell? nc ≈ 1.41
I Minimum wavelength?λ = λ0/nc ≈ 0.632µm
1.41 ≈ 0.45µm
I Diameter?
7.76µm ≈ 17λ
I Height?
2.50µm ≈ 6λ
I Geometry?
82 · 3 (µm)3 = 192 (µm)3
I Sampling?
λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 53: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/53.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma? np ≈ 1.34
I Refractive index in cell? nc ≈ 1.41
I Minimum wavelength?λ = λ0/nc ≈ 0.632µm
1.41 ≈ 0.45µm
I Diameter? 7.76µm ≈ 17λ
I Height?
2.50µm ≈ 6λ
I Geometry?
82 · 3 (µm)3 = 192 (µm)3
I Sampling?
λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 54: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/54.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma? np ≈ 1.34
I Refractive index in cell? nc ≈ 1.41
I Minimum wavelength?λ = λ0/nc ≈ 0.632µm
1.41 ≈ 0.45µm
I Diameter? 7.76µm ≈ 17λ
I Height? 2.50µm ≈ 6λ
I Geometry?
82 · 3 (µm)3 = 192 (µm)3
I Sampling?
λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 55: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/55.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma? np ≈ 1.34
I Refractive index in cell? nc ≈ 1.41
I Minimum wavelength?λ = λ0/nc ≈ 0.632µm
1.41 ≈ 0.45µm
I Diameter? 7.76µm ≈ 17λ
I Height? 2.50µm ≈ 6λ
I Geometry? 82 · 3 (µm)3 = 192 (µm)3
I Sampling?
λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 56: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/56.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma? np ≈ 1.34
I Refractive index in cell? nc ≈ 1.41
I Minimum wavelength?λ = λ0/nc ≈ 0.632µm
1.41 ≈ 0.45µm
I Diameter? 7.76µm ≈ 17λ
I Height? 2.50µm ≈ 6λ
I Geometry? 82 · 3 (µm)3 = 192 (µm)3
I Sampling? λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?
6 192(0.03)2
≈ 42 · 106
Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 57: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/57.jpg)
Scattering of light against a blood cell
I Wavelength in vacuum?λ0 ≈ 0.632µm
I Refractive index plasma? np ≈ 1.34
I Refractive index in cell? nc ≈ 1.41
I Minimum wavelength?λ = λ0/nc ≈ 0.632µm
1.41 ≈ 0.45µm
I Diameter? 7.76µm ≈ 17λ
I Height? 2.50µm ≈ 6λ
I Geometry? 82 · 3 (µm)3 = 192 (µm)3
I Sampling? λ/15 ≈ 0.4515 µm ≈ 0.03µm
I Number of unknowns?6 192(0.03)2
≈ 42 · 106 Refined mesh formembrane
Daniel Sjoberg, Department of Electrical and Information Technology
![Page 58: Electromagnetic Wave Propagation Lecture 5: Finite differences in … · 2010. 9. 13. · The nite di erence approximation of the wave equation is Ejn r+1 2Ejnr+E jn r r1 (z )2 1](https://reader036.vdocuments.mx/reader036/viewer/2022071610/6149724f080bfa6260149d95/html5/thumbnails/58.jpg)
What about larger problems?
Sometimes the problem is too large to fit in one computer. Howdoes FDTD perform on clusters?
Platon cluster in Lund,216 × 2 × 4 = 1728 pro-cessors, 64 bit 2.26GHz3GB per core, startednov 2009.
I Each cell “communicates” only with its nearest neighbors; ablock of N3 cells only exchanges N2 data with others.
I Well suited for parallelization!I Current trend: computation on GPU:s (hardware adapted to
parallel operations).
Daniel Sjoberg, Department of Electrical and Information Technology
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Outline
1 Finite differences
2 Generalization to 3D
3 Dispersion analysis in 3D
4 Example applications
5 Conclusions
Daniel Sjoberg, Department of Electrical and Information Technology
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Conclusions
I FDTD is a simple numerical scheme for time domain.
I It is vital to choose a smaller discretization in time than infrequency. The relation between time and space determinesthe numerical dispersion.
Some pros and cons:
+ Easy to implement
+ Can handle advanced material models (next lecture!)
+ Easy analysis
+ Broad band
+ Parallelizable
− Stair case approximation of geometry
− Global time step (but local implicit time stepping is possible)
− Numerical dispersion
Daniel Sjoberg, Department of Electrical and Information Technology