Finite Elements in Electromagnetics by yurtgc548

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```									    Finite Elements in
Electromagnetics
1. Introduction
Oszkár Bíró
IGTE, TU Graz
Kopernikusgasse 24, Graz, Austria
email: biro@igte.tu-graz.ac.at
Overview

•   Maxwell‘s equations
•   Boundary value problems for potentials
•   Nodal finite elements
•   Edge finite elements
Maxwell‘s equations
D
curlH  J 
t
B
curlE  
t
divB  0
divD  
B  H, H  B; J  E, E  J; D  E
Potentials
• Continuous functions
• Satisfy second order differential equations
• Neumann and Dirichlet boundary conditions
E.g. magnetic vector and electric scalar potential (A,V formulation):

B  curlA
A        V
t        t
Differential equations
E
in a closed domain        
D
curlH  J       0:                      H
t
A         V    2A        2V
curl (curlA)       grad      2  grad 2  0
t         t    t        t
D
div (J  )  0 :
t
A          V    2A        2V
 div(      grad      2  grad 2 )  0
t         t    t        t
Dirichlet boundary conditions
Prescription of tangential E (and normal B) on E:
A          V
E  n    n  grad     n, B  n  n  curlA
t          t

B      E
A  n  a0 ,           n          E
V  V0


n is the outer unit normal at the boundary H
Neumann boundary conditions
Prescription of tangential H (and normal J+JD) on H:
H  n  curlA  n,
D           A      V            2A       2V
(J  )  n   (  grad    )  n   ( 2  grad 2 )  n
t          t      t            t       t
curlA  n  K ,
A     V          A 2     V 2
 (  grad )  n   ( 2  grad 2 )  n   j
t     t          t       t                            E
H

J+JD
H       n
General boundary value problem
Differential equation:                     D
u       2u                  
L2u  Lt1  Lt 2 2  f in 
t      t
N
Boundary conditions:
LD u  0 on D Dirichlet BC   D  N  

u       2u
LN u  LNt1  LNt 2 2  g on N   Neumann BC
t      t
Nonhomogeneous Dirichlet
boundary conditions
LD u  u0 on D      u  u D  u
u D : arbitrary so that LD u D  u0 on D
u  : new unknown function w ith LD u   0 on D

u        2u                u D        2u D
L2u  Lt1      Lt 2 2  f  L2u D  Lt1       Lt 2 2
t        t                    t         t
u         2u                  u D          2u D
LN u  LNt1      LNt 2 2  g  LN u D  LNt1       LNt 2 2
t          t                     t           t
Formulation as an operator equation (1)
Scalar product for ordinary functions: u, v    uvd
3
Characteristic function of a domain : Q
Q , w   wd, w                  1, if P  ,
Q ( P)  

0, if P  .
Dirac function of a surface : 

  , w   wd, w        n   gradQ 

Formulation as an operator equation (2)
Define the operators A, B and C as
Au  Q L2u   N LN u
(with the definition set
Bu  Q Lt1u   N LNt1u      DABC  {u : LDu  0 on D })
Cu  Q Lt 2u   N LNt 2u
Equivalent operator equation:
u   2u
Au  B  C 2  Q f   N g
t  t
Formulation as an operator equation (3)
Properties of the operators:

Symmetry:  Au, w  u , Aw,
 Bu, w  u, Bw,
Cu , w  u, Cw , u, w  DABC .

Positive property:  Au , u   0, u  DABC
Operators of the A,V formulation (1)
A             Q  curl (curl )   Hcurl  n 0
u           A
V                              0                  0

A 
DA  {  : A  n  0, V  0 on E }
V 
          Q                             Q grad                 
B
  Q  div( )   H n    Q  div(grad )   H n  grad  
          Q                            Q grad                
C
  Q  div( )   H n    Q  div(grad )   H n  grad 

A,V formulation: symmetry of A
 A u  A w  
 A ,      Q curl(curlA u )    curlA u  n A w d
 V
  u   Vw   
         3
H

  A w  curl (curlA u )d   A w  (curlA u  n)d
                                H

  curlA w curlA u d   ( A w curlA u )  nd
                          
  A w  (curlA u  n)d
H

  curlA w curlA u d   ( A w  n) curlA u d
                          E

0 , since A w n 0 on E

  curlA w curlA u d

A,V formulation: positive property of A

 A  A  
 A ,      curlA curlAd
 V V 
     

  curlA d  0
2


A,V formulation: symmetry of B and C
 A u  A w         Q ( A u  gradVu )  A w                                   
 B  ,                                                                          d
 V                        Q div( ( A u  gradVu ))   H n  ( A u  gradVu ) w 
  u   Vw       3
V

  A w   ( A u  gradVu )  Vw div[ ( A u  gradVu )]d


  Vw ( Au  gradVu )  nd
H

  A w   ( A u  gradVu )  gradVw   ( A u  gradVu )d


  Vw ( Au  gradVu )  nd   Vw ( Au  gradVu )  nd
                                H



  Vw ( A u  gradVu )  nd
E
      
  
 0 , since V w  0 on E


Weak form of the operator equation

       u   2u 
 Au  B  C 2 , w   Q  f   N g , w, w  DABC
       t  t     
Galerkin’s method:
discrete counterpart of the weak form
n
u (r, t )  u ( n )   uk (t ) f k (r )   f k  D ABC
k 1

 f k , k  1,2,...: basis functions forming an entire set in DABC
 (n)    u     u          
, f i   Q  f   N g , f i ,
(n)         2   (n)
 Au  B     C
         t     t 2

i  1, 2, ..., n
Set of ordinary differential equations
Galerkin equations
 Au  Bu  C u  b
         
 A  aik ,   aik   Af k , f i    f k , Af i   aki
[A] is a symmetric positive matrix
B   bik , bik   Bf k , f i    f k , Bf i   bki
C   cik ,   cik  Cf k , f i    f k , Cf i   cki
[B] and [C] are symmetric matrices
b  bi , bi  Q f   N g , f i 
Finite element discretization
Nodal finite elements (1)
17

16
18

11
15
19
20         14
13
5
12                                10
4
6
9

7                                3

8
2
1

Shape functions:
1 in node i,
N i (r )                                        i = 1, 2, ..., nn
0 in all other nodes.
Nodal finite elements (2)
Shape functions

Corner node              Midside node
Nodal finite elements (3)
Basis functions for scalar quantities (e.g. V):
Shape functions

Number of nodes: nn, number of nodes on D: nDn
n  nn  nDn , nodes on D: n+1, n+2, ..., nn
n
V (r, t )  V ( n )  Vk (t ) N k (r )
k 1
Nodal finite elements (4)
Linear independence of nodal shape functions
nn

N
i 1
i   1

nn

i 1
i   0

The number of linearly independent gradients of the
shape functions is nn-1 (tree edges)
Edge finite elements (1)
20

8
19
32         12
7
24           36
35 18                                         6
30    11       31    17             5                       28
23
26
33                            34
4        22     16
10             27
29
3                    9                      15
21
14                   25                          2
13             1

Edge basis functions:
1 , if i  j ,
 Ni (r)  dl  0 , if i  j.
Edge j            
i = 1, 2, ..., ne
Edge finite elements (2)
Basis functions

Side edge              Across edge
Edge finite elements (3)
Basis functions for vector intensities (e.g. A):
Edge basis functions

Number of edges: ne, number of edges on D: nDe
n  ne  nDe , edges on D: n+1, n+2, ..., ne

n
A(r, t )  A   (n)
  ak (t )N k (r )
k 1
Edge finite elements (4)
Linear independence of edge basis functions
ne          ne
gradN i   cik N k ;         cik  0 i=1,2,...,nn-1.
2

k 1        k 1

Taking the curl:
ne

c
k 1
ik   curlN k  0, i=1,2,...,nn-1.

The number of linearly independent curls of the
edge basis functions is ne-(nn-1) (co-tree edges)

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