Finite Elements in
Electromagnetics
2. Static fields
Oszkár Bíró
IGTE, TU Graz
Kopernikusgasse 24Graz, Austria
email: [email protected]
Overview
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Maxwell‘s equations for static fields
Static current field
Electrostatic field
Magnetostatic field
Maxwell‘s equations for static fields
curlH  J
divJ  0
curlE  0
divB  0
divD  
B  H, H  B; J  E, E  J; D  E
Static current field (1)
curlE  0
divJ  0
J  E 
or

E  J
n voltages between the
electrodes are given:
En
n
J
In
 0
Un
J
Ei
J
 E  dl  U
i
Ci
Ii
Ui
I2
E2

 0
U2
J
or
Cn
Ci
C2
C1
I1
E1
U1
J
E0
I0=I1+I2+...+Ii+...+In
n currents through the
electrodes are given:
 J  n d   I
i
Ei
i = 1, 2, ..., n
En  0
on n+1 electrodes E E0+E1+E2+ ...+ Ei+ ...+ En
J n  0
on the interface J to the nonconducting region
Static current field (2)
Symmetry
E0 may be a symmetry
plane
A part of J may be a
symmetry plane
E0
I1
J 2I1
I1
2U1
U2- U1

I2
U1
I2
U2- U1

U2

U1

U2
I2
2(I1+I2)
I2
Static current field (3)
Interface conditions
E1  n  E2  n Tangential E is continuous
J1  n  J 2  n
Normal J is continuous
2>1
2>1
1

J2
n J
1
E2
1

n
E1
Static current field (4)
Network parameters (n>0)
U1
n=1: R 
U1 is prescribed and I1    J  nd
I1
E 1
I1 is prescribed and U1   E  dl
or
C1
n
n
j 1
j 1
n>1: U   r I or I   g U i = 1, 2, ..., n
i
ij j
i
ij
j
Ui
rij 
Ij
I k 0 , k  j
Ii
g ij 
Uj U
i = 1, 2, ..., n
k 0 , k 
j
Static current field (5)
Scalar potential V
curlE  0 
E   gradV
divJ  0, J  E   div(gradV )  0 in 
En  0 
J n  0 
V  constant on Ei
0 auf E 0 ,
V 
V  U 0 on E
U i auf Ei .
V
gradV  n  
 0 on J
n
Static current field (6)
Boundary value problem for the scalar potential V
 div (gradV )  0
in ,
(1)
V  U0
on E,
(2)
on J.
(3)
gradV  n  
V
0
n
V  VD  V 
VD  U 0
on E , otherwise arbitrary
 div (gradV  )  div (gradV D )
in ,
V0
auf E,

V
V 
  D
n
n
auf J.
Static current field (7)
Operator for the scalar potential V

A   div (grad )    
n
J
DA  V   DA :V   0 on E 
VD
AV    div (gradVD )    
n
J
Static current field (8)
Finite element Galerkin equations for V
n
V (r )  V ( n ) (r )  VD (r )  Vk N k (r )
k 1
VD (r ) 
nn
V N
k  n 1
k
n
V  gradN  gradN d   gradN  gradV
k 1
k
i

k
i
D
k
(r )
d,

i = 1, 2, ..., n
 AV   b  A is positive definite
High power bus bar
Finite element discretization
Current density represented by arrows
Magnitude of current density represented by colors
Static current field (9)
Current vector potential T
divJ  0 
J  curlT
curlE  0, E  J  curl ( curlT)  0 in 
J n  0 
curlT  n  0  n  T  t on J
divt  0  divn  T   n  curlT  0
 (t  n)  dl   I
i
Ei
En  0 
curlT  n  0 on E
Static current field (10)
Boundary value problem for the vector potential T
curl ( curlT)  0
in ,
(1)
nT  t
on J,
(2)
curlT  n  0
on E.
(3)
T  TD  T
n  TD  t on J , otherwise arbitrary
curl ( curlT)  curl ( curlTD ) in ,
n  T  0
on J,
curlT  n   curlTD  n
on E.
Static current field (11)
Operator for the vector potential T
A  curl ( curl)   E curl  n
DA  T  DA : n  T  0 on J 
AT  curl ( curlTD )   E curlTD  n
Static current field (12)
Finite element Galerkin equations forT
n
T(r )  T( n ) (r )  TD (r )   tk N k (r )
k 1
TD (r ) 
ne
t N
k  n 1
k
k
(r )
n
 t  curlN  curlN d   curlN  curlT d
k 1
k
i

k
i
D

i = 1, 2, ..., n
 AT   b  A is positive semi definite
Current density represented by arrows
Magnitude of current density represented by colors
Electrostatic field (1)
curlE  0
divD  
D  E
n voltages between the
electrodes are given:
En
n
D
Un

Ui
Ci
D
Q2
E2
C2
C1
U2
D
or
Cn


i
Ci
Qi
Ei
 E  dl  U
D
Qn
Q1
E1
U1
E0
Q0=-Q1-Q2-...-Qi-...-Qn
D
n charges on the
electrodes are given:
 D  n d   Q
i
Ei
i = 1, 2, ..., n
En  0
on n+1 electrodes E E0+E1+E2+ ...+ Ei+ ...+ En
D n  
on the boundary D
Electrostatic field (2)
Symmetry
E0 may be a symmetry
plane
A part of D (=0) may
be a symmetry plane
E0
-Q1
-Q2
2U1
U2- U1 U2- U1
   
D 2Q1
Q1

Q2
U2
U1

U1 

U2
Q2
-2(Q1+Q2)
Q2
Electrostatic field (3)
Interface conditions
E1  n  E2  n Tangential E is continuous
D2  n  D1  n   Special case =0:
D1  n  D2  n
Normal D is continuous

 1
=0
D2
n D
1

2>1
2>1

 1
=0
E2
n
E1


  0
D1
D2
n
Electrostatic field (4)
Network parameters (n>0)
U1 is prescribed and Q1    D  nd
Q1
n=1: C 
U1
E 1
Q1 is prescribed and U1   E  dl
or
C1
n
n
j 1
j 1
n>1: U   p Q or Q   c U i = 1, 2, ..., n
i
ij j
i
ij
j
Ui
pij 
Q j Q 0 , k  j
k
Qi
cij 
UjU
i = 1, 2, ..., n
k 0 , k 
j
Electrostatic field (5)
Scalar potential V
curlE  0 
E   gradV
divD   , D  E   div (gradV )  
En  0 
D  n   
in 
V  constant on Ei
0 auf E 0 ,
V 
V  U 0 on E
U i auf Ei .
V
gradV  n  
  on D
n
Electrostatic field (6)
Boundary value problem for the scalar potential V
 div (gradV )  
in ,
(1)
V  U0
on E,
(2)
(3)
gradV  n  
V

n
on D.
V  VD  V 
VD  U 0
on E , otherwise arbitrary
 div (gradV )    div (gradVD )
in ,
V0
on E,

V 
V
  D
n
n
on D.
Electrostatic field (7)
Operator for the scalar potential V

A    div (grad )    
n
D
DA  V   DA :V   0 on E 
VD
AV    [   div(gradVD )]    (  
)
n
D
Electrostatic field (8)
Finite element Galerkin equations for V
n
V (r )  V ( n ) (r )  VD (r )  Vk N k (r )
k 1
VD (r ) 
nn
V N
k  n 1
k
k
(r )
n
V  gradN  gradN d   N d   N d
k 1
k
i
k

i

i
D
  gradNi  gradVD d, i = 1, 2, ..., n

 AV   b  A is positive definite
380 kV transmisson line
380 kV transmisson line, E on ground
380 kV transmisson line, E on ground in
presence of a hill
Magnetostatic field (1)
curlH  J
divB  0
B  H 
or
H  B 
n magnetic voltages between
magnetic walls are given:
Hn
n
Umn
B
Hi
 H  dl  U
B
n
i

Umi
J
Ci
or
Cn

Ci
B/T
B
2.0
1.8
2
Iron
1.6
1.4
H2
1.2
1.0
Um2
B
C2
C1
1
H1
Um1
mi
H0
n fluxes through the
magnetic walls are given:
0=1+2+...+i+...+n
B
0.8
0.6
 B  n d   
i
Hi
0.4
0.2
0.0
0
Air
20
40
60
80
100
120
140
-1
H/Am
i = 1, 2, ..., n
H  n  K on n+1 magn. walls E E0+E1+E2+ ...+ Ei+ ...+ En
B n  b on the boundary B
Magnetostatic field (2)
Symmetry
H0 (K=0) may be a
symmetry plane
A part of B (b=0) may
be a symmetry plane
H0
1
Um2- Um1
2
1
2Um1
Jx
Jz Jz
Jx
Jy Jy


Um2- Um1
2
Um2
2
B 21
Um1
Um1
Jy
Jz
2
Jx
Jx
Jz
Jy Um2


2( 1+ 2)
Magnetostatic field (3)
Interface conditions
H1  n  H 2  n  K Special case K=0:
H1  n  H 2  n Tangential H is continuous
B1  n  B 2  n
Normal B is continuous

 1
=0
B2
n B
1

2>1
2>1

 1
=0
H2
n
H1


H2
K  0
n
H1
Magnetostatic field (4)
Network parameters (n>0), J=0
U m1
n=1: Rm 
Um1 is prescribed and 1    B  nd
1
H 1
1 is prescribed and U m1   H  dl
or
C1
n
n
j 1
j 1
n>1: U   r  or    g U i = 1, 2, ..., n
mi
mij
j
i
mij mj
rmij
U mi
i

g mij 
 j  0 , k  j
U mj U
k
i = 1, 2, ..., n
mk  0 , k 
j
Magnetostatic field (5)
Network parameter (n=0), b=0, K=0, J0
Inductance:
1
1
2
2
L  2   H d  2  B d
I 
I 
Magnetostatic field (6)
Scalar potential F, differential equation
curlH  J 
H  T0  gradF
T0 : curlT0  J, otherwise arbitrary
1 J (Q)  eQP
e.g. : T0 ( P)  H S ( P) 
dQ
2

4  rQP
divB  0, B  H   div( gradF)  div( T0 )
Magnetostatic field (7)
Scalar potential F, boundary conditions
Hn  K 
F  F 0 on H
F 0 P   Fi 
 n  K  T  n  ds on 
0
CP
0 on H 0 ,
Fi  
U mi on Hi .
B  n  b 
F

 b  T0  n on B
n
Hi
Magnetostatic field (8)
Boundary value problem for the scalar potential F
 div ( gradF )   div ( T0 )
in ,
(1)
F  F0
on H,
(2)
on B.
(3)
gradF  n  
F
 b  T0  n
n
Full analogy with the electrostatic field
 ,
F  V , F0  U0 ,
 div( T0 )   , b  T0  n   ,
Magnetostatic field (9)
Finite element Galerkin equations for F
n
F (r )  F ( n ) (r )  F D (r )   F k N k (r ) F D (r ) 
k 1
nn
F
k  n 1
k
N k (r )
n
 F  gradN  gradN d   gradN  T d   bN d
k 1
k
i
k

i

0
i
B
  gradNi  gradF D d, i = 1, 2, ..., n

 AF  b  A is positive definite
Magnetostatic field (10)
In order to avoid cancellation errors in computing
H  T0  gradF (n )
T0 should be represented by means of edge elements:
ne
T0   ti N i
ti 
 T  dl
0
edgei
i 1
ne
since gradN i   cik N k and hence T0 and gradF (n)
k 1
are in the same function
space
Magnetostatic field (11)
Magnetic vector potential A
divB  0 
B  curlA
curlH  J, H  B  curl (curlA)  J in 
B  n  b  curlA  n  b  n  A  a on B
diva  b  divn  A   n  curlA  b
 (a  n)  dl  
i
Hi
H  n  K  curlA  n  K on H
Magnetostatic field (12)
Boundary value problem for the vector potential A
curl (curlA )  J
in ,
(1)
n A  a
on B,
(2)
curlA  n  K
on H.
(3)
A  A D  A n  A D  a on B , otherwise arbitrary
curl (curlA)  J  curl (curlA D ) in ,
n  A  0
on B,
curlA  n  K  curlA D  n
on H.
Magnetostatic current field (13)
Operator for the vector potential A
A  curl (curl)   Hcurl  n
DA  A  DA : n  A  0 on B 
AA   (J  curl (curlA D )   E (K curlA D  n)
Magnetostatic field (14)
Finite element Galerkin equations for A
n
A (r )  A ( n ) (r )  A D (r )   ak N k (r ) A D (r ) 
k 1
ne
a N
k  n 1
k
k
(r )
n
 a  curlN curlN d   N  Jd   N  Kd
k 1
k
i
k

i

i
H
  curlN i curlA D d i = 1, 2, ..., n

 AA  b  A is positive semi definite
Magnetostatic field (15)
Consistence of the right hand side of the
Galerkin equations
Introduce T0 as curlT0  J in , T0  n  K on H .
bi   N i  curlT0 d   N i  (T0  n ) d
H

n  Ni  0 on B
  curlN i curlA D d

 N  (T
i
H
0
 n) d   (n  Ni )  T0d   ( Ni  T0 )  nd 

H
  div (Ni  T0 )d   T0  curlN i d   N i  curlT0 d .
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bi   T0  curlN i d   curlN i rotA D d
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