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Electromagnetic Wave Propagation Lecture 5: Finite differences in the time domain Daniel Sj¨ oberg Department of Electrical and Information Technology September 14, 2010

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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

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

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

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

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

Page 5: 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

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

Page 6: 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

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

Page 7: 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

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 8: 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

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

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

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

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

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

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

Page 14: 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

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

Page 15: 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

Example: square wave

R = 1

R < 1

R > 1

Daniel Sjoberg, Department of Electrical and Information Technology

Page 16: 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

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

Page 17: 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

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

Page 18: 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

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

Page 19: 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

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

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

Page 21: 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

Graphical point of view

Daniel Sjoberg, Department of Electrical and Information Technology

Page 22: 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

Generalization to 3D

E-field along edges, H-field along sides. Also staggered in time.Daniel Sjoberg, Department of Electrical and Information Technology

Page 23: 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

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 24: 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

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 25: 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

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

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

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

Page 28: 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

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

Page 29: 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

Staircasing

The rectangular grid does not conform to curved surfaces:

Daniel Sjoberg, Department of Electrical and Information Technology

Page 30: 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

Staircasing

The rectangular grid does not conform to curved surfaces:

Daniel Sjoberg, Department of Electrical and Information Technology

Page 31: 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

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

Page 32: 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

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

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

Page 34: 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

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

Page 35: 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

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

Page 36: 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

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 =

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

Page 37: 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

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 38: 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

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

Page 39: 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

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

Page 40: 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

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 41: 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Page 59: 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

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 60: 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

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