MA557/MA578/CS557
Lecture 29
Spring 2003
Prof. Tim Warburton
timwar@math.unm.edu
1
Maxwell’s 2D TM Equations
• We established Maxwell’s TM equations as:
H x Ez

t
y
H y Ez

t
x
Ez H x H y


t
y
x
H x H y

x
y
0
0
0
0
• We added PEC boundary conditions (say suitable for
a domain bounded by a superconducting material):
H x nx  H y n y  0
Ez  0
2
Exterior Domain
• Now suppose we wish to consider the case of an
electromagnetic wave bouncing off a scatterer:
PEC
3
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.
Artificial radiation boundary condition
PEC
4
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
5
PML region (reduces backscatter into the domain).
TE Mode Maxwell’s
• The subset of Maxwell’s equations considered in Berenger’s
Ex H z
paper involve (Ex,Ey,Hz):

  Ex
t
y
E y H z

  E y
t
x
H z Ex E y


  * H z
t
y
x
• Note the dissipative terms on the rhs.
• If the spatial terms were not there then the solutions would
decay as:
 t
E x  x, y , t   e
E x  x, y,0 
E y  x, y, t   e  t E y  x, y,0 
H z  x, y , t   e
 *t
H z  x, y ,0 
6
Stage 1: Split the magnetic field Hz
• First Berenger split the Hz field into two coefficients
and made the dissipative terms anisotropic:
H zx , H zy are defined so that H z   H zx  H zy 
and :
Ex   H zx  H zy 

  y Ex
t
y
E y
t

  H zx  H zy 
x
  x E y
H zx E y

  x* H zx
t
x
H zy Ex

  *y H zy
t
y
7
Wave Structures
• In Matrix form:
 Ex   0 0 0 0   Ex   0

 
 
 
  Ey   0 0 1 1    Ey   0






H
H
 0 1 0 0  x
0
t
zx
zx

 
  H  
H
0
0
0
0
  zy   1
 zy  
0
0  Ex 
 y 0
 0 
 E 
0
0
x
 y 

 0
0  x* 0  H zx 

* 
 H 
0
0
0

y 
zy 

0 1 1  Ex 


0 0 0    Ey 

 ...


H
0 0 0  y
zx



0 0 0   H zy 
8
Wave Speeds
• In the previous notation we looked at eigenvalues of linear
combination of the flux matrices:
0
0
A
0

0
0
0
1
0
0
1
0
0
0
0
0
1
, B  
0
0


0
 1
0 1 1
 0
 0
0 0 0
C
 0
0 0 0


0 0 0
 
0
0


0
0
0

 
 
0 

0 
• The eigenvalues computed by Matlab:
• i.e. 0,0,1,-1 under constraint on
(alpha,beta)
1
• So for Lax-Friedrichs
we
take
9
Structure of PML Regions
 y , *y , x , x*  0
 x   x*  0, y , *y  0
 y , *y , x , x*  0
 x   y   x*   *y  0
 y   *y  0
 y   *y  0
 x , x*  0
 x , x*  0
 y , *y , x , x*  0
PEC
 x   x*  0, y , *y  0
*
 y , *y , 10
,

x
x 0
We now seek plane wave solutions for the split
equations (parameterized by direction)
Split Equations
Ex   H zx  H zy 
  y Ex

y
t
E y
t

  H zx  H zy 
x
  x E y
H zx E y
  x* H zx

x
t
H zy Ex
  *y H zy

y
t
Look for traveling plane wave solutions,
Where alpha,beta are unknown complex
numbers. E0, phi, and frequency all specified
Ex   E0 sin   e
E y  E0 cos   e
H zx  H zx 0 e
i  t  x   y 
i  t  x   y 
i  t  x   y 
H zy  H zy 0 e
i  t  x   y 
11
Ex   H zx  H zy 

  y Ex
t
y
E y
t

  H zx  H zy 
x
  x E y
H zx E y

  x* H zx
t
x
H zy Ex

  *y H zy
t
y
+
Ex   E0 sin   e
E y  E0 cos   e
H zx  H zx 0e
i  t  x   y 
i  t  x   y 
i  t  x   y 
H zy  H zy 0e
Conditions on alpha, beta, Hzx0, Hzy0
E0 sin   
i y
E0 sin      H zx 0  H zy 0 

i y
E0 cos   
E0 cos      H zx 0  H zy 0 

i x*
H zx 0 
H zx 0   E0 cos  

i *y
H zy 0 
H z 0   E0 sin  

i  t  x   y 
12
Looking for waves traveling into the domain
E0 sin   
i y
E0 sin      H zx 0  H zy 0 

i y
E0 cos   
E0 cos      H zx 0  H zy 0  Basic

i x*
algebra
H zx 0 
H zx 0   E0 cos  

i *y
H zy 0 
H z 0   E0 sin  


1  i x 
1 
 cos  
G
 

1  i y
1 
G


 sin  

where:
G  wx cos    wy sin  
2
 
1 i x 
 
wx 
  x* 
1 i 
 
 
1 i y 
 
wy 
  *y 
1 i 
 
13
2
Special Case
• We did not specify yet how the  x* , *y fields are defined.
• Suppose we set:
 
1 i x 
   1
 x   x*  wx 
  x* 
1 i 
 
 
1 i y 
   1
 y   *y  wy 
  *y 
1 i 
 
 G  1 since G  wx cos    wy sin  
2
2
14
Special Case
Recall form of plane wave.
i t  x   y 
Ex   E0 sin   e 
i t  x   y 
E y  E0 cos   e 
+
H zx  H zx 0e
i  t  x   y 
H zy  H zy 0e
i  t  x   y 
Ex   E0 sin   e

1  i x 
1 
 cos  
G
 

1  i y
1 
G


 sin  

 cos  x sin  y   cos 
 y sin 
x
i  t 


x
y
G
G


G
G
e
e
Setting:  x   x* , y   *y  G  1
Ex   E0 sin   e
i  t cos  x sin  y   x cos  x  y sin  y
e
e
We can repeat this for the other 3 fields.
15
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.
i t cos  x sin   y   x cos  x  y sin   y
Ex   E0 sin   e 
e
e
i t cos  x sin   y   x cos  x  y sin   y
E y  E0 cos   e 
e
e
H zx  E0 cos   e
2
i  t cos  x sin   y   x cos  x  y sin   y
e
e
2 i t cos  x sin   y   x cos  x  y sin   y
H zy  E0 sin   e 
e
e
16
Dissipation in layer
i t cos  x sin   y   x cos  x  y sin   y
Ex   E0 sin   e 
e
e
i t cos  x sin   y   x cos  x  y sin   y
E y  E0 cos   e 
e
e
H zx  E0 cos   e
2
i  t cos  x sin   y   x cos  x  y sin   y
e
e
2 i t cos  x sin   y   x cos  x  y sin   y
H zy  E0 sin   e 
e
e
Traveling wave solution
17
How Thick Does the PML Region Need To Be
• Suppose we consider the region in blue [a,a+delta]
n
x

a
• And we set:


*
 x   x*  C 
,



y
y 0

  

PEC
18
x=a
Effectiveness of PML
• All plane waves travelling in the direction (cos(phi),sin(phi))
decay as:
i  t cos  x sin  y   x cos  x
Ex   E0 sin   e
e
• After traversing the pml region from
a to a+delta the field will decay by a factor of:
 cos 
e
a 
   x dx
a
• For the sigma we chose:
e

 cos 

a 

a

  x dx 

e
e

 cos 

a 

a

 xa 
C
 dx 




n
e

n 1  a  

C  xa 


 
 cos  n 
  n 1  

a 

C 

 cos 

n 1 

19
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:
e
C 

2cos 

n 1 

• 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
C 

approximately by: 1
2cos 

n 1
1
100
e

n1


 
ln 

2C cos    100 
• 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..20
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
21
Solving with the USEMe PDE Wizard
22
Comparing with and without PML
• Demo..
23
Next Class
• We will discuss possible instabilities created by
splitting the magnetic field.
• Investigate unsplit alternative PMLs.
• Visit PML in the context of DG.
24