ee757 numerical techniques in electromagnetics lecture 13mbakr/ece757/lecture13_2016.pdf · lecture...

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EE757

Numerical Techniques in Electromagnetics

Lecture 13

2 EE757, 2016, Dr. Mohamed Bakr

2D FEM

We consider a 2D differential equation of the form

subject to

),(, yx f

yyxxyx

1on p

x, y and are functions associated with the physical

parameters and f is the excitation

Notice that the boundary conditions may be a Dirichlet,

Neuman or mixed Dirichlet and Neuman.

2on. q

yxyx

nji

where = 1 2 is the contour enclosing the domain and

n is the unit outward normal


2D FEM (Cont’d)

2

2

2

22

2

5.0)(

dqd f-

d yx

F yx

The functional associated with this problem is

(Prove it)!


i

e

1

2

3

1 2 4 1

2 5 4 2

3 3 5 2

4 5 6 4

2D FEM Analysis

Each node has both a local and a

global index

e

1 2

3 The computational domain is divided

into triangles (elements)

A connectivity array n(i,e), i=1, 2, 3 and e=1, 2, , M

stores the global indices of the nodes

5

1

2 3 4

1

2

3

4

5 6


2D FEM Analysis (Cont’d)

We assume that there are Ms line segments on 2

2

s=1

s=2

s=3

We store the index array ns(i,s), i=1, 2 and s=1, 2, , Ms

of global indices of nodes on 2


Input Data to the 2D FEM Analysis

The coordinates of the nodes ri=(xi,yi), i=1, 2, , N, where

N is the total number of nodes

The values of x, y, and f for each element

The value of p for each node residing on 1

The value of and q for each segment with nodes on 2

The two arrays n(i,e), i=1, 2, 3 and e=1, 2, , M and

ns(i,s), i=1, 2 and s=1, 2, , Ms


Elemental Interpolation

Over the eth element we utilize the linear approximation

eeeee

yx ycxbayx ),(,),(

The three nodes of the eth element must satisfy the linear interpolation relation

ycxbaeeeeee,

111 ycxba

eeeeee,

222

ycxbaeeeeee

333

Solving for ae, be and ce we obtain

where

e

jj

ej

eyxNyx

3

1

),(),(

3,2,1),(2

1),( j ycxba

AyxN

ej

ej

ej

e

ej

xxc yyb xyyxaeeeeeeeeeee23132132321 ,,




Elemental Interpolation (Cont’d)

Ae is the area of the eth element and is given by

yx

yx

yx

Aee

ee

ee

e

33

22

11

1

1

1

2

1

The interpolation functions satisfy

ji

ji y xN ij

e

j

ej

ei

0,

,1),(


The Homogenous Neuman BC case

We first consider the case ( = q = 0)

The functional is expressed as a sum of elemental

subfunctions

M

e

eeFF

1

)()(

where

e

e

e

e

e

y

e

x

ee df-d dy

d

dx

dF

)(5.0)(

2

22

Substituting with the linear interpolation expression, we

write


The Homogenous Neuman BC case

e i

e

i

ei

ei j

e

j

ej

ei

e

i

e

j

ej

eie

iy

e

j

ej

ei

i j

e

ix

ee

dNfdNN

d dy

Nd

dy

Nd

dx

Nd

dx

NdF

e3

1

3

1

3

1

3

1

3

1

5.0)(

e

e

i

ej

e

i

e

ej

ei

y

ej

ei

xj

e

je

i

e

dfN-

dNNdy

Nd

dy

Nd

dx

Nd

dx

NdF

3

1

i=1, 2, 3

bKeee

e

eF

131333

13


The Homogenous Neuman BC case (Cont’d)

dNNdy

Nd

dy

Nd

dx

Nd

dx

NdK

ej

e

i

e

ej

ei

y

ej

ei

xeij

i=1, 2, 3 and j=1, 2, 3 e

e

i

ei dfNb

If x, y , and f are taken as constants over each element,

and utilizing the property

!!!

!!!2321

nml

nmlAdNNN e

e ne m

e

e l

)1(124

1 ij

eeej

ei

ey

ej

ei

ex

e

eij

Accbb

AK

we get

fA

beee

i3


The Process of Assembly

The process of assembly involves storing the local

elemental components into their proper location in the

global system of equations

0

F bK 11

NNNN

The element is added to

Keij K ejnein ),(),,(

The element is added to

bei b ein ),(


An Assembly Example

5

1

2 3 4

1

2

3

4

5 6

i

e

1

2

3

1 2 4 1

2 5 4 2

3 3 5 2

4 5 6 4

There are six nodes

1666 and bK

We initialize K and b with zeros

Evaluate K(1) and b(1) an add them to the proper locations

to get


An Assembly Example (Cont’d)

0

0

0,

000000

000000

000

000000

000

000

)1(2

)1(1

)1(

3

)1(

22

)1(

21

)1(

32

)1(

12

)1(

11

)1(

31

)1(

23

)1(

13

)1(

33

b

b

b

KKK

KKK

KKK

bK

Evaluate K(2) and b(2) an add them to the proper locations

0

0,

000000

000

00

000000

00

000

)2(1

)2(2

)1(2

)2(

3

)1(1

)1(

3

)2(

11

)2(

12

)2(

31

)2(

21

)2(

22

)1(

22

)2(

32

)1(

21

)1(

32

)2(

13

)2(

23

)1(

12

)2(

33

)1(

11

)1(

31

)1(

23

)1(

13

)1(

33

b

bb

bb

b

KKK

KKKKKK

KKKKKK

KKK

bK

Repeat the same steps for all elements


Incorporating a Boundary Condition of the 3rd Kind

In this case and q are not zeroes

The extra subfunctional is added to

the functional F

2

2

2)( dqFb

Because 2 is comprised of Ms line segments, we may write

)()(1

M s

s

ssbb FF

We approximate the function over the segment s by the

linear expression

N N Nsss

jj

sj

s

21

2

1

,1,

is the normalized distance from node 1 to node 2

s 2

=1

=0


Incorporating a 3rd Kind BC (Cont’d)

s

sb dqF

2

2)(

Use the expansion

s

s

ii

si

s

ji j

sj

si

s

i

sb dNq-NNF

2

1

2

1

2

12)(

Differentiate and use d =ls d

21,2

1

1

0

1

0

,i dlNq-dlNNF ss

ij

ssj

si

s

js

i

sb

bKsss

s

sbF

121222

in matrix form

21and21,,1

0

1

0

,j ,i dlNqb dlNNKss

isi

ssj

si

sij

Assembly is then applied to store these coefficients


The Dirichlet Boundary Condition

The Dirichlet boundary conditions are imposed by

eliminating the known nodes by substituting for their

values

b

b

KK

KK

u

p

u

p

uupu

uppp

ppuuuuu KbK

Original system

Reduced system


An Example: A Shielded Microstrip Line

=0 o

r

=1

o

r

=1

/n =0

The microstrip is kept at potential =1 while the external

shielding box is kept at potential =0

Symmetry may be employed to reduce the computational

domain by one half

The governing BVP is


An Example: A Shielded Microstrip Line (Cont’d)

o

crr

yyxx

with = 0 on the outer conductor, =1 on the microstrip

and / n=0 on the plane of symmetry

It follows that we have x= y=r, =0, f = 0

The electric field is obtained through E=-. But over

each element is approximated by

,),(),(3

1

e

jj

ej

eyxNyx

3,2,1),(2

1),( j ycxba

AyxN

ej

ej

ej

e

ej

3

1

)(2

1

j

e

j

ej

ej

e

cbAyx

jijiE Over the eth element



The Finite Element Method in Electromagnetics, Jianming Jin



The equi-potential lines

The Finite Element Method

in Electromagnetics,

Jianming Jin

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