fem: 2-d problems

FEM: Two Dimensional Elements

Mohammad Tawfik #WikiCourses

http://WikiCourses.WikiSpaces.com

Introduction to the Finite

Element Method

Two Dimensional Elements




2-D Elements

• In this section, we will be introduced to two

dimensional elements with single degree

of freedom per node.

• Detailed attention will be paid to

rectangular elements.




For the 2-D BV Problem

• Let’s consider a problem with a single

dependent variable

• We may set one degree of freedom to

each node; say fi.

• Further, let’s only consider a rectangular

element that is aligned with the physical

coordinates




A Rectangular Element

• For the approximation

of a general function

f(x,y) over the element

you need a 2-D

interpolation function

xyayaxaayxf 4321,




Let’s follow the same

procedure!




2-D Interpolation Function

ayxHyxf ,, xyayaxaayxf 4321),(

aHff 0,00,0 1

aT

a

a

a

a

bH

baH

aH

H

f

f

f

f

4

3

2

1

4

3

2

1

0

,

0,

0,0

aaHfaf ,00, 2

abaHfbaf ,, 3 abHfbf ,0,0 4





4

3

2

1

4

3

2

1

001

1

001

0001

a

a

a

a

b

abba

a

f

f

f

f

4

3

2

1

4

3

2

1

1111

100

1

0011

0001

f

f

f

f

abababab

bb

aa

a

a

a

a





efyxNayxHyxf ,,,

ab

xy

b

yab

xyab

xy

a

xab

xy

b

y

a

x

yxNyxNT

1

,,




How does this look like?




2-D Interpolation Functions

0

0.3

0.6

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

N1

x 0

0.3

0.6

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

N2

x




2-D Interpolation Functions

0

0.3

0.6

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

N3

x 0

0.3

0.6

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

N4

x




Example: Laplace Equation

02

02

2

2

2

yx

ei

ii yxNyxN ,,4

1




Example: Laplace Equation

ei

ii yxNyxN ,,4

1

0e

Area

yyxx dANNNN

Applying the Galerkin method and integrating by parts,

the element equation becomes




The Element Equaiton

0

222

222

222

222

6

1

22222222

22222222

22222222

22222222

e

babababa

babababa

babababa

babababa

ab




The Logistic Problem!




The Logistic Problem

• In the 2-D problems, the numbering

scheme, usually, is not as straight forward

as the 1-D problem




1-D Example

• Element #1 is associated with nodes 1&2

• Element #2 is associated with nodes 2&3, etc…




2-D Example




For Element #5

Global Node Number Local Node Number

5 1

6 2

9 3

8 4




Contribution of element #5 to global

matrix 12 11 10 9 8 7 6 5 4 3 2 1

1

2

3

4

1,3 1,4 1,2 1,1 5

2,3 2,4 2,2 2,1 6

7

4,3 4,4 4,2 4,1 8

3,3 3,4 3,2 3,1 9

10

11

12




A Solution for the Logistics’

Problem

• One solution of the logistic problem is to

keep a record of elements and the

mapping of the local numbering scheme to

the global numbering scheme in a table!




Elements Register: Global

Numbering

Node Number Element

Number 4 3 2 1

4 5 2 1 1

7 8 5 4 2

10 11 8 7 3

5 6 3 2 4

8 9 6 5 5

11 12 9 8 6




Algorithm for Assembling Global

Matrix 1. Create a square matrix “A”;

N*N (N=Number of nodes)

2. For the ith element

3. Get the element matrix “B”

4. For the jth node

5. Get its global number k

6. For the mth node

7. Get its global number n

8. Let Akn=Akn+Bjm

9. Repeat for all m

10. Repeat for all j

11. Repeat for all i

Node Number Element

Number 4 3 2 1

4 5 2 1 1

7 8 5 4 2

10 11 8 7 3

5 6 3 2 4

8 9 6 5 5

11 12 9 8 6

12 11 10 9 8 7 6 5 4 3 2 1

1

2

3

4

5

6

7

8

9

10

11

12

fem: 2-d problems

Education

dimensional elements

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d elements

rectangular elements

y n2 x fem

y n4 x fem

physical coordinates

d interpolation functions