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Spring 2003

Prof. Tim Warburtontimwar@math.unm.edu

MA557/MA578/CS557Lecture 29

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Maxwell’s 2D TM Equations

• We established Maxwell’s TM equations as:

• We added PEC boundary conditions (say suitable for a domain bounded by a superconducting material):

0

0

0

0y

x z

y z

yxz

x

H E

t y

H E

HH

x y

t xHHE

t y x

0

0

x x y y

z

H n H n

E

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Exterior Domain

• Now suppose we wish to consider the case of an electromagnetic wave bouncing off a scatterer:

PEC

Incidentfield Scatte

red

field

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Exterior Domain

• We know which boundary conditions to apply at the PEC scatterer.• We do not know what to do at an artificial domain boundary.• We do suppose that there are no waves incident from infinity.

PEC

Incidentfield Scatte

red

field

Artificial radiation boundary condition

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Perfectly Matched Layer (PML)

• We will consider the PML approach proposed by Berenger (J. Comp. Phys. 114:185-200, 1994).

• Berenger’s approach:– Create a region at the border of the domain where

dissipative terms are introduced into the pde.

– Make sure that plane waves entering the absorbing region are not reflected at the interface between the absorbing region and the free space region.

– Make sure that there is no reflection at the interface for any angle of incident and any frequency.

– Try to make the fields decay by orders of magnitude in the PML region (reduces backscatter into the domain).

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TE Mode Maxwell’s

• The subset of Maxwell’s equations considered in Berenger’s paper involve (Ex,Ey,Hz):

• Note the dissipative terms on the rhs.• If the spatial terms were not there then the solutions would

decay as:

*

x zx

y zy

yxzz

E HE

t y

E HE

t xEEH

Ht y x

*

, , , ,0

, , , ,0

, , , ,0

tx x

ty y

tz z

E x y t e E x y

E x y t e E x y

H x y t e H x y

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Stage 1: Split the magnetic field Hz

• First Berenger split the Hz field into two coefficients and made the dissipative terms anisotropic:

z

*

*

, are defined so that H

:

zx zy zx zy

zx zyxy x

zx zyyx y

yzxx zx

zy xy zy

H H H H

and

H HEE

t y

H HEE

t xEH

Ht xH E

Ht y

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Wave Structures

• In Matrix form:

*

*

0 0 0 0 0 0 1 1

0 0 1 1 0 0 0 0...

0 1 0 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

x x x

y y y

zx zx zx

zy zy zy

y x

x y

x zx

y zy

E E E

E E E

H H Ht x y

H H H

E

E

H

H

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Wave Speeds

• In the previous notation we looked at eigenvalues of linear combination of the flux matrices:

• The eigenvalues computed by Matlab:

• i.e. 0,0,1,-1 under constraint on(alpha,beta)

• So for Lax-Friedrichs wetake

0 0 0 0 0 0 1 1 0 0

0 0 1 1 0 0 0 0 0 0,

0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0

A B C

1

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Structure of PML Regions

* * 0x y x y

* *0, , 0x x y y

* *0, , 0x x y y

*

*

0

, 0

y y

x x

*

*

0

, 0

y y

x x

* *, , , 0y y x x

* *, , , 0y y x x * *, , , 0y y x x

* *, , , 0y y x x

PEC

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*

*

zx zyxy x

zx zyyx y

yzxx zx

zy xy zy

H HEE

t y

H HEE

t xEH

Ht xH E

Ht y

0

0

0

0

sin

cos

i t x yx

i t x yy

i t x yzx zx

i t x yzy zy

E E e

E E e

H H e

H H e

Split EquationsLook for traveling plane wave solutions,Where alpha,beta are unknown complex

numbers. E0, phi, and frequency all specified

We now seek plane wave solutions for the split equations (parameterized by direction)

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0 0 0 0

0 0 0 0

*

0 0 0

*

0 0 0

sin sin

cos cos

cos

sin

yzx zy

yzx zy

xzx zx

yzy z

iE E H H

iE E H H

iH H E

iH H E

*

*

zx zyxy x

zx zyyx y

yzxx zx

zy xy zy

H HEE

t y

H HEE

t xEH

Ht xH E

Ht y

0

0

0

0

sin

cos

i t x yx

i t x yy

i t x yzx zx

i t x yzy zy

E E e

E E e

H H e

H H e

+

Conditions on alpha, beta, Hzx0, Hzy0

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Looking for waves traveling into the domain

0 0 0 0

0 0 0 0

*

0 0 0

*

0 0 0

sin sin

cos cos

cos

sin

yzx zy

yzx zy

xzx zx

yzy z

iE E H H

iE E H H

iH H E

iH H E

2 2

*

*

11 cos

11 sin

where:

cos sin

1

1

1

1

x

y

x y

x

x

x

y

y

y

i

G

i

G

G w w

iw

i

i

w

i

Basic

algebra

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Special Case

• We did not specify yet how the fields are defined. • Suppose we set:

*

*

*

*

2 2

11

1

1

1

1

1 since cos sin

x

x x x

x

y

y y y

y

x y

iw

i

i

w

i

G G w w

* *,x y

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Special Case

0

0

0

0

sin

cos

i t x yx

i t x yy

i t x yzx zx

i t x yzy zy

E E e

E E e

H H e

H H e

Recall form of plane wave.

11 cos

11 sin

x

y

i

G

i

G

cos sin sincos

0 sinyx

x yi t x yG G G G

xE E e e e

+

* *Setting: , 1x x y y G

cos sin sincos0 sin yxi t x y yx

xE E e e e

We can repeat this for the other 3 fields.

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Interpretation of Solutions

• We derived the solutions of the TE Maxwell equations with dissipative terms assuming there is a plane wave solution.

• What we find is for solutions traveling into the PML layer that the influence of the absorption terms only modifies the plane wave inside the layer.

sincocos sin

0

cos sin

0

2 cos sin

0

2 cos sin

0

s

sincos

sincos

sincos

sin

cos

cos

sin

yx

yx

yx

yx

x

y

z

i t x y

i t x y

i t x y

i t

yx

yx

yx

zyyx

x

xy

e e

e

E

E

H

E e

E e

E e

EH ee

e

e e

e

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sincocos sin

0

cos sin

0

2 cos sin

0

2 cos sin

0

s

sincos

sincos

sincos

sin

cos

cos

sin

yx

yx

yx

yx

x

y

z

i t x y

i t x y

i t x y

i t

yx

yx

yx

zyyx

x

xy

e e

e

E

E

H

E e

E e

E e

EH ee

e

e e

e

Traveling wave solution

Dissipation in layer

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How Thick Does the PML Region Need To Be

• Suppose we consider the region in blue [a,a+delta] • And we set:

PEC

x=a

* *, 0n

x x y y

x aC

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Effectiveness of PML

• All plane waves travelling in the direction (cos(phi),sin(phi)) decay as:

• After traversing the pml region from a to a+delta the field will decay by a factor of:

• For the sigma we chose:

cos cossin

0 sin xi t x

xxy

eE E e

cosa

a

x dx

e

1

coscos cos 1

cos1

ana a n

n

aa a

x aCx ax dx C dx n

C

n

e e e

e

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Terminating the PML

• If we terminate the far boundary of the PML with a reflecting PEC boundary condition then the solution will decay on the way to the boundary and back into the domain.

• i.e. the reflection coefficient for the PML will be:

• i.e. if you want the solution to decay by a factor of 100 for an n’th order PML region then the width delta is determined approximately by:

• Notice if phi = pi/2 (i.e. the wave is traveling parallel to the region) then the solution does not decay in the pml region.

• However, it will decay when it reaches the upper pml region..

2 cos1

C

ne

2 cos

11 1 1ln

100 2 cos 100

C

n ne

C

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Papers and Web Notes

• Prime source:

A Perfectly Matched Layer for the Absorption of Electromagnetic Waves, Journal of Computational Physics Volume 114, Pages 185-200, 1994.

• Notes:http://www.duke.edu/~gs16/prelim.pdf

• Some related papers:http://www.lions.odu.edu/~fhu/reprints.htm

• Misc:• Optimizing the perfectly matched layer, Computer Methods

in Applied Mechanics and Engineering, Volume 164, Issues 1-2, October 1998, Pages 157-171

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Solving with the USEMe PDE Wizard

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Comparing with and without PML

• Demo..

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Next Class

• We will discuss possible instabilities created by splitting the magnetic field.

• Investigate unsplit alternative PMLs.

• Visit PML in the context of DG.