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
                     E     grad
                         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

                       A w  A u  A w  gradVu  gradVw  A u  gradVw  gradVu d
                       

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



   A w  A u  A w  gradVu  gradVw  A u  gradVw  gradVu d
   
 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

Taking the gradient:
 nn

 grad N
 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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