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FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
Introduction to the Finite
Element Method
Two Dimensional Elements
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
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.
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
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
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
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,
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
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Let’s follow the same
procedure!
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
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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
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
2-D Interpolation Function
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
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
2-D Interpolation Function
efyxNayxHyxf ,,,
ab
xy
b
yab
xyab
xy
a
xab
xy
b
y
a
x
yxNyxNT
1
,,
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
How does this look like?
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
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
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
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
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
Example: Laplace Equation
02
02
2
2
2
yx
ei
ii yxNyxN ,,4
1
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
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Example: Laplace Equation
ei
ii yxNyxN ,,4
1
0e
Area
yyxx dANNNN
Applying the Galerkin method and integrating by parts,
the element equation becomes
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
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The Element Equaiton
0
222
222
222
222
6
1
22222222
22222222
22222222
22222222
e
babababa
babababa
babababa
babababa
ab
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
The Logistic Problem!
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
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The Logistic Problem
• In the 2-D problems, the numbering
scheme, usually, is not as straight forward
as the 1-D problem
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
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1-D Example
• Element #1 is associated with nodes 1&2
• Element #2 is associated with nodes 2&3, etc…
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
2-D Example
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
2-D Example
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
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For Element #5
Global Node Number Local Node Number
5 1
6 2
9 3
8 4
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
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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
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
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!
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
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
FEM: Two Dimensional Elements
Mohammad Tawfik #WikiCourses
http://WikiCourses.WikiSpaces.com
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