Finite Elements in Electromagnetics1. 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

B

BE

DJH

div

divt

curl

tcurl

0

EDJEEJBHHB ;,;,

Potentials

tV

gradt

curl

A

E

AB

• Continuous functions

• Satisfy second order differential equations

• Neumann and Dirichlet boundary conditions

E.g. magnetic vector and electric scalar potential (A,V formulation):

Differential equations

:0D

JH t

curl

0AA

A

2

2

2

2

)(tV

gradtt

Vgrad

tcurlcurl

:0)( t

divD

J

0)( 2

2

2

2

tV

gradtt

Vgrad

tdiv AA

E

H

in a closed domain

Dirichlet boundary conditions

AnnBnnA

nE curltV

gradt

,

Prescription of tangential E (and normal B) on E:

0

,

VV 0anA

n is the outer unit normal at the boundary

E

H

nEB

Neumann boundary conditions Prescription of tangential H (and normal J+JD) on H:

j)()(

,

2

2

2

2

nA

nA

KnA

tV

gradtt

Vgrad

t

curl

nA

nA

nD

J

nAnH

)()()(

,

2

2

2

2

tV

gradtt

Vgrad

tt

curl

E

H n

H

J+JD

General boundary value problem

in 2

2

212 ftu

Ltu

LuL tt

Differential equation:

Boundary conditions:

DDuL on 0

NNtNtN gtu

Ltu

LuL

on 2

2

21

Dirichlet BC

Neumann BC

D

N

ND

Nonhomogeneous Dirichlet boundary conditions

uuu D DD uuL on 0

ithfunction wunknown new :uDDDD uuLu on that soarbitrary : 0

2

2

2122

2

212 tu

Ltu

LuLftu

Ltu

LuL Dt

DtDtt

2

2

212

2

21 tu

Ltu

LuLgtu

Ltu

LuL DNt

DNtDNNtNtN

DDuL on 0

Formulation as an operator equation (1)

Characteristic function of a domain

. if ,0

, if ,1)(

P

PP wwdw

,,

wwdw

,,

Dirac function of a surface

gradn

Scalar product for ordinary functions:

3

, uvdvu

Formulation as an operator equation (2)

uLuLAu NN 2

gftu

Ctu

BAuN

2

2

Define the operators A, B and C as

(with the definition set})on 0:{ DDABC uLuD

Equivalent operator equation:

uLuLCu Ntt N 22 uLuLBu Ntt N 11

Formulation as an operator equation (3)

Properties of the operators:

Symmetry: ,,, AwuwAu ,,, BwuwBu .,,,, ABCDwuCwuwCu

Positive property: ABCDuuAu ,0,

Operators of the A,V formulation (1)

V

uA

00

0)( ncurlcurlcurlA H

gradgraddivdiv

gradB

HHnn

)()(

}on 0,:{ EA VV

D

0nAA

gradgraddivdiv

gradC

HHnn

)()(

A,V formulation: symmetry of A

w

w

u

u

VVA

AA,

3

)( dcurlcurlcurl wuu HAnAA

H

dcurldcurlcurl uwuw )()( nAAAA

dcurldcurlcurl uwuw nAAAA )(

H

dcurl uw )( nAA

Ew

E

dcurldcurlcurl uwuw

on since ,0

)(

0nA

AnAAA

dcurlcurl uw AA

A,V formulation: positive property of A

dcurlcurlVV

A AAAA

,

02

dcurlA

A,V formulation: symmetry of B and C

w

w

u

u

VVB

AA,

3 )())((

)(d

VgradVgradVdiv

gradV

wuuuu

wuu

HAnA

AA

dgradVdivVgradV uuwuuw )]([)( AAA

H

dgradVV uuw nA )(

dgradVgradVgradV uuwuuw )()( AAA

H

dgradVVdgradVV uuwuuw nAnA )()(

dgradVgradVgradVgradV uwuwuwuw AAAA

Ew

E

V

uuw dgradVV

on 0 since ,0

)( nA

dgradVgradVgradVgradV uwuwuwuw AAAA

Weak form of the operator equation

ABCDwwgfwtu

Ctu

BAuN

,,,2

2

Galerkin’s method:discrete counterpart of the weak form

n

kkk

n ftuutu1

)( )()(),( rr ABCk Df

ABCk Dkf in set entirean forming functions basis: ,...2,1,

,,,2

)(2)()(

ii

nnn fgff

tu

Ctu

BAuN

ni ..., 2, 1,

Set of ordinary differential equations

Galerkin equations

buCuBuA

kiikikikik aAfffAfaaA ,,,

[A] is a symmetric positive matrix kiikikikik bBfffBfbbB ,,,

kiikikikik cCfffCfccC ,,,

[B] and [C] are symmetric matrices iii fgfbbb

N,,

Finite element discretization

Nodal finite elements (1)

12

3

4

5

6

7

8

9

10

11

12

1314

15

16

17

18

19

20

nodes.other allin 0

, nodein 1)(

iN i r i = 1, 2, ..., nn

Shape functions:

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,Dnn nnn nodes on D: n+1, n+2, ..., nn

n

kkk

n NtVVtV1

)( )()(),( rr

Nodal finite elements (4)Linear independence of nodal shape functions

11

nn

iiN

Taking the gradient:

01

nn

iiNgrad

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

Edge finite elements (1)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1718

19

20

22

23

24

25

26

27

28

29

30 31

32

33 34

35

36

21

Edge basis functions:

. if , 0

, if , 1)(

ji

jid

jEdge

i lrN 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,Dee nnn edges on D: n+1, n+2, ..., ne

n

kkk

n tat1

)( )()(),( rNArA

Edge finite elements (4)Linear independence of edge basis functions

Taking the curl:

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

;1

en

kkiki cgradN N 0

1

2

en

kikc i=1,2,...,nn-1.

,1

0N

en

kkikcurlc i=1,2,...,nn-1.