Efficient numerical solution methods for Maxwell's equations
by separations of near field and far field interaction
Prof. Dr. Frank Gronwald
Chair „Electromagnetic Compatibility“
Institute of Electromagnetic Theory
Hamburg University of Technology
gronwald@tu-harburg.de
Hamburg University of Technology
Institute of Electromagnetic Theory
The Institute of Electromagnetic Theory at the
University of Technology Hamburg (TUHH)
around 1930
Prof. Dr. Christian Schuster
Chair
“Electromagnetic Theory”
Hamburg University of Technology
today
Dr. Heinz-D. Brüns
Senior Scientist
“Numerical Field Computation
Frank Gronwald
aerial view
Prof. Dr. Frank Gronwald
Chair
“Electromagnetic Compatibility”
Transparency 2
Institute of Electromagnetic Theory
Overview of the talk
1.
Introduction and Motivation:
Electromagnetic Engineering Applications and Numerical Field Computation
2.
Aspects of efficient numerical field computation in electromagnetic theory
3.
Electromagnetic near fields and far fields
(Coulomb fields and radiation fields)
4.
Separations of near field and far field interactions
5.
–
Method of analytical regularization
–
Hybrid representations of Green‘s functions
–
Multilevel Fast Multipole Algorithm
Conclusion
Hamburg University of Technology
Frank Gronwald
Transparency 3
Institute of Electromagnetic Theory
1. Introduction and Motivation - interference analysis
of avionic systems
transmitting antennas
receiving
antennas
interference matrix
some important avionic systems :
TCAS = Traffic Collision Avoidance System
ATC = Air Traffic Control
GPS = Global Positioning System
SATCOM = Satellite Communication
ADF = Automatic Direction Finder
VOR = VHF Omnidirectional Radio Range
DME = Distance Measuring Equipment
ILS = Instrument Landing System
Hamburg University of Technology
Frank Gronwald
common avionic systems operate in the frequency range
100 kHz to 15 GHz
corresponding wavelengths are in the range of
3 km to 2 cm
analysis of (unwanted) antenna couplings requires to
calculate both near and far fields
necessity to numerically solve large scale
electromagnetic boundary value problem to obtain
interference matrix
Transparency 4
Institute of Electromagnetic Theory
1. Introduction and Motivation – interior problems of
Electromagnetic Compatibility
a low cost electronic power meter
complex electric/electronic systems contain various
electric/electronic components which might interfer
with each other
electromagnetic coupling between various
electric/electronic components needs to be analyzed
and estimated
electromagnetic coupling often takes place within a
resonating environment (e.g. within a metallic housing,
as given by common computer housings)
Hamburg University of Technology
Frank Gronwald
on a more abstract level: modelling of antenna
coupling within a cavity
antennas carry electric charges and currents that
generate electric and magnetic near fields
cavity supports electromagnetic modes that
correspond to free electromagnetic (far) fields
(i.e., solutions of sourceless Helmholtz equations)
necessity to solve electromagnetic boundary
value problem with near and far field
characteristics
Transparency 5
Institute of Electromagnetic Theory
2. Aspects of efficient numerical field computation
in electromagnetic theory
Usually, the solution f of an electromagnetic boundary value problem is given by an element of an infinite
dimensional function space (such as a Hilbert or Soboloev space):

f   n fn
n 1
Remark: Often the explicit solution for f is found as the solution of a linear operator equation, see, e.g.:
Hanson, G.W. and Yakolev, A.: Operator Theory for Electromagnetics, (Springer, New York, 2002).
A numerical solution is a finite approximation f of the form
~ N ~ ~
f  f   n f n
with numerically calculated coefficients
~
n 1
n
An efficient numerical solution requires:
number N of approximating basis functions
fast numerical calculation of the coefficients
Hamburg University of Technology
Frank Gronwald
~
f n not too large
~n
Transparency 6
Institute of Electromagnetic Theory
2. Aspects of efficient numerical field computation
in electromagnetic theory
~
Important observation: Basis functions f n that are suitable to approximate a near field (=Coulomb field)
often are not suitable to approximate a far field (=radiation field) and vice versa
This observation is illustrated by the complementary properties of „rays“ (propagator functions) and
„modes“ (eigenfunctions of a compact and self-adjoint operator) that both are often used as approximating
basis functions (Felsen, 1984):
Rays
Modes
Scattering processes yield local information of a
system
Oscillations yield global infornation of a system
Characterize early response in time domain
Characterize late response in time domain
Advantageous for high-frequency regime where
the mode-density is high and rays of
geometrical optics characterize the field
Advantageous for low-frequency regime where
the mode-density is low and a small number of
modes characterizes the field
Advantageous to model Coulomb singularities
Advantageous to model resonances
It follows that in order to numerically solve electromagnetic boundary value problems it is often necessary
to separately analyze near field and far field interactions in order to find approximating basis functions
which are suitable to approximate both near fields and far fields
Note: Efficient numerical computation schemes often are necessary because computer memory and
computation time are limited.
Hamburg University of Technology
Frank Gronwald
Transparency 7
Institute of Electromagnetic Theory
3. Electromagnetic near fields and far fields
- fields generated by a point charge





2

 
q  er ',r   1  
q

E (r , t ) 
  3   2 
4 0  1    er ',r | r  r ' | 
4 0

 ret


near field (Coulomb)






far field (radiation)


 e   e   e       

 r ',r  r ',r  r ',r




3



2
c
1



e
|
r

r
'
|


r ', r

 ret
near field (Coulomb)
Hamburg University of Technology

 e   e      
 r ',r  r ',r


3


  
 c 1    er ',r | r  r ' | 

 ret
 
2

 
q    er ',r 1  
q

B(r , t ) 
  3   2 
4 0  c 1    er ',r | r  r ' | 
4 0

 ret


Advantage: Near field and far field
contributions can be written as separate
terms

Disadvantage: Motion of point charge
must be known
Disadvantage: Model of point charge
not useful for most engineering
applications where charge and current
distributions are needed

far field (radiation)
Frank Gronwald
Transparency 8
Institute of Electromagnetic Theory
3. Electromagnetic near fields and far fields
- fields generated by continuous sources

 
r  r'

J
observer
Advantage: Near field and
far field contributions can
be written as separate
terms
charge and current distribution
 
E (r , t ) 
 
 
 
 d 3r '

r  r'
1
r  r ' 1  
3

 (r ' , t 'ret )    J (r ' , t 'ret )  
   (r ' , t 'ret ) d r ' 
4 0  | r  r ' |3
4 0 c  
| r  r '| c
 | r  r '|
1
near field (Coulomb)


B(r , t )  0
4
far field (radiation)
 
 
 
 
J (r ' , t 'ret )  r  r ' 3
 0 J (r ' , t 'ret )  (r  r ' ) 3
d r' 
d r'
 3
 2


4

c
r  r'
r  r'
near field (Coulomb)
Hamburg University of Technology
Disadvantage: Current
and charge distributions
must be known
Remark: To obtain current
and charge distributions it is
often required to solve an
integral equation where
near and far field
contributions still are
coupled
far field (radiation)
Frank Gronwald
Transparency 9
Institute of Electromagnetic Theory
3. Electromagnetic near fields and far fields
- contrasted to longitudinal and transverse fields
Remark: Quantization of the electromagnetic field requires to quantize true dynamical degrees of freedom
only – these are not part of the near (Coulomb) field but part of the far (radiation) field
It is then a standard approach to split electromagnetic
fields in their longitudinal and tranverse part – for a

F
general vector field F , its longitudinal part || and transverse part F are defined by
  
     
F  F||  F ,   F||  0 ,   F  0
corresponding split of Maxwell‘s equations:
 
  D||  

 
D 
  H 
 J
t

  B 
  E 
0
t
longitudinal part
transverse part
Separation of longitudinal and transverse parts does not correspond to a separation of near (Coulomb)
and far (radiation) fields - entangled near and far fields still have to be taken care of in the quantization
process, but are there any alternatives?
Hamburg University of Technology
Frank Gronwald
Transparency 10
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
There appears to be no canonical method to separate, for the general solution of electromagnetic
boundary value problems, near (Coulomb) fields from far (radiation) field
To nevertheless employ efficient numerical solution schemes it is nevertheless required to separate near
field and far field interactions in some way
In the following, three methods for separating near field and far field interactions are introduced:
method of analytical regularization
(conversion of an integral equation of the first kind to an integral equation of the second kind)
hybrid representation of Green‘s function
(transforming a canonical Green‘s function into a ray-mode representation)
multilevel fast multipole algorithm
(effective grouping and translating of near field interactions)
Hamburg University of Technology
Frank Gronwald
Transparency 11
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Method of analytical regularization
Electromagnetic boundary value problems often are formulated as electric field integral equations of the
form
E
G
 ( z, z ' ) I ( z ' )dz '   E ( z )
or, if written as linear operator equation,
L( I )   E
Idea: Split L in two parts L0 and L1 where L0 contains the Coulomb singularity
Then:
L0 ( I )  L1 ( I )  E
First, construct the „near field solution“ L0-1
1
I 0  L0 (E)
Second, solve the remaining integral equation of the second kind with the Coulomb singularity
removed
1
I  I 0  L0 L1 (I )
Solution often possible by iteration:
I
Hamburg University of Technology
I0
1
1  L0 L1
Frank Gronwald
Transparency 12
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Method of analytical regularization
Example: Electrically small antenna inside a cavity
(Tkachenko & Gronwald, 2003)
Consider first a linear wire antenna (length L, radius a, directed along z-axis) in free space which is excited
by an incoming wave
Ezinc ( z)  E0 sin i exp(jkz cosi )
Approximate solution for the induced current:

sin(kz) 
I 0 ( z,  )  20 ln(jL4/aE)0k sin i cos(kz)  exp( jkz cosi   exp( jkL cosi )  cos(kL)
sin(kL) 

For a small antenna
kL  1 this solution can be written in the factorized form
I 0 ( z,  )  K 0 ( ) f ( z ) E0 , with
Hamburg University of Technology
Frank Gronwald
4z 2
f(z)  1  2
L
Transparency 13
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Method of analytical regularization
It follows with
L1 ( I ) 
E
G
1
 ( z, z ' ) I ( z ' )dz '
antenna
that the solution for the current along the small antenna within the cavity is given by
I ( z,  ) 
K 0 ( ) f ( z )
1  K 0 ( ) 
antenna
G1 ( z , z ' ) f ( z ' )dz '
E0
This result can be used, for example, to calculate the coupling between two antennas within a rectangular
cavity
For a specific configuration the current transfer ratio is characterized by sharp resonance peaks
Hamburg University of Technology
Frank Gronwald
Transparency 14
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Hybrid representations of Green’s functions
General idea: Construct Green‘s functions with both ray and mode properties
Illustration by example: Consider the Green‘s function of the Helmholtz equation for the vector potential A
 
 
 
2
A(r )  k A(r )   J (r )

 
  3
A  
A(r )   G (r , r ' ) J (r ' ) d r '
inside a three-dimensional rectangular cavity
Application of the mirror principle yields:
G (r , r ' ) 
A


7
 Ci
exp( jkRi ,mnp (r, r ' ))
m, n , p   i 0
4Ri ,mnp (r , r ' )
„ray representation“
Hamburg University of Technology
Frank Gronwald
Transparency 15
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Hybrid representations of Green’s functions
Now consider the three-dimensional Poisson transformation

 f (2m ,2n ,2 p ) 
m , n , p  
1


2 3 m,n
, p     ,
f ( 1 , 2 , 3 ) exp( j (m 1  n 2  p 3 )) d 1d 2 d 3
Application of the Poisson transformation to the ray representation yields the mode representation
(Wu & Chang 1987)
0p
1 
G (r , r ' ) 

8 m ,n , p   l x l y l z
A
 mx   mx'   ny   ny '   pz   pz 
 sin 
 cos
 sin 
 sin 
 cos

sin 





 lx   lx   l y   l y   lz   lz 
 m

 lx
2
  n   p 
 
  
  k 2



  l y   lz 
2
2
This is not exactly what we want: We have turned rays into modes but we want to have both ray and mode
contributions!
Hamburg University of Technology
Frank Gronwald
Transparency 16
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Hybrid representations of Green’s functions
By the application of an Ewald-transformation (Ewald, 1921) it can be shown that (Gronwald, 2005)
G A (r, r ' )  G Aray (r, r ' )  G Amode (r, r ' )
where
G A ray (r , r ' ) 
1
8
 exp( jkRi ,mnp erfc(Ri ,mnp E  jk / 2 E ) exp( jkRi ,mnp erfc(Ri ,mnp E  jk / 2 E ) 
C




i
R
R
m , n , p   i  0


i , mnp
i , mnp

7
and
2
 k mnp
 k2 

exp 
2


7
4E
1

 exp j k X  k Y  k Z 
A
G mode (r , r ' ) 
Ci


x i
y i
z i
2
2
8l xl y l z m,n , p  i 0
k mnp  k
This hybrid representation has very good convergence both in the source region and at resonance!
Hamburg University of Technology
Frank Gronwald
Transparency 17
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Hybrid representations of Green’s functions
Example: Calculation of the Green‘s function GAzz within a canonical cavity
cuboidal cavity with source point
r ' (0.25,0.25,0.25) L
number of terms
accuracy
Ray sum
108
no convergence
10-5
Mode sum
106
10-5
10-8
Ewald sum
102
10-8
number of terms
accuracy
Ray sum
106
10-2
Mode sum
108
Ewald sum
102
convergence properties in source region
(x=y=z=0.26L)
Hamburg University of Technology
values of the Green‘s function GAzz for varying
observation point r and fixed wavenumber k=9.42/L
Frank Gronwald
convergence properties at resonance
(k=9.42/L)
Transparency 18
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Hybrid representations of Green’s functions
Example: Calculation of the mutual coupling Z12 between two antennas inside a rectangular cavity
Mutual coupling is expressed by the mutual impedance which is calculated by a formula of the form
Z12  

antenna 2
b  a  3
E (r )  J (r )d r
I 2a I1b
and obtained by the numerical solution an integral equation, involving the cavity‘s Green‘s function
rectangular cavity with dimensions lx=6m ly=7m lz=3m
Hamburg University of Technology
Frank Gronwald
G A (r , r ' )
absolute value of the mutual impedance
Transparency 19
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Hybrid representations of Green’s functions
The results of the previous slide have been obtained by the Method of Moments
The Method of Moments converts a linear operator equation into an algebraic system of equations:
First, the original equation
L( f )  g
is approximated by
~
L( f )  g~, with
~ N
f   k k ,
k 1
it follows
N
g~   g , w j w j
j 1
N
~
L( f )    k ( L( k ))
k 1
N

   k L( k ), w j w j
k 1 j 1
N
N
   k L( k ), w j w j
k 1 j 1
and the result is a linear algebraic equation for the unknown coefficients
N

k 1
Hamburg University of Technology
k
k
,
L( k ), w j  g , w j
Frank Gronwald
Transparency 20
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Multilevel fast multipole algorithm
The standard Method of Moments: Usually applied to the conversion of an integral equation to a linear
system of equations which contains the unknown electric current elements as primary variables
Linear system of equation characterized by the interaction between all electric current elements
Standard Method of Moments: Interaction between all
current elements is taken into account
Multilevel Fast Multipole Algorithm:
(Rokhlin, 1990; Lu & Chew, 1993)
Interaction between regions that are
not within each other near field regions
can be approximated by a smaller
number of interactions
(i.e., less interactions need to be computed
and stored!)
Hamburg University of Technology
Frank Gronwald
Multilevel Fast Multipole Algorithm:
Far field interactions are effectively
grouped and translated
Transparency 21
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Multilevel fast multipole algorithm
The multilevel fast multipole algorithm treats near field and far field interactions differently
Near field interactions: Only interactions between neighboring current elements are taken into account,
the corresponding interaction matrix becomes sparse
Far field interactions: Less interactions due to Aggregation, Translation, and Disaggregation
Translation is mathematically performed by the use of „addition theorems“ that arise from the addition of
angular momentum in quantum mechanics
Hamburg University of Technology
Frank Gronwald
Transparency 22
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Multilevel fast multipole algorithm
Both the standard Method of Moments and the Multilevel Fast Multipole Algorithm are implemented in the
program CONCEPT-II which has been developed at the Institute of Electromagnetic Theory, Hamburg
University of Technology, since the mid 1980ies
CONCEPT-II used as
platform to incorporate research results
electromagnetic tool to work on academic and industrial projects
Screenshots of CONCEPT-II graphical user interface
(compare: http:www.tet.tu-harburg.de/concept/)
Hamburg University of Technology
Frank Gronwald
Transparency 23
Institute of Electromagnetic Theory
4. Separations of near field and far field interactions
- Multilevel fast multipole algorithm
Example: Calculation of the surface currents on a ship which are generated
by a monopole antenna, operating at f = 150 MHz (λ = 2m)
Ship dimensions: 120m length
21m height
15m width
Discretisation yields 424158 unknowns (=edge currents)
Standard Method of Moments would require 2.6 TeraByte of memory
MLFMA requires 5.6 GigaByte of memory, on a single workstation the
problem can be solved in 2.8 hours
Hamburg University of Technology
Frank Gronwald
Transparency 24
Institute of Electromagnetic Theory
5. Conclusion
Efficient numerical solution methods for electromagnetic boundary value problems often require to
separate near field interactions (Coulomb fields) from far field interactions (radiation fields)
There appears to be no general and complete method to achieve this separation
But it is often possible to isolate the Coulomb singularity in a way such that numerical computations
both in the source region and at resonance become possible.
Thank you for your attention!
Hamburg University of Technology
Frank Gronwald
Transparency 25
Institute of Electromagnetic Theory