FEM: 2-D Problems

FEM: Two Dimensional Elements
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
Introduction to the Finite
Element Method
Two Dimensional Elements

Mohammad Tawfik
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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.

Mohammad Tawfik
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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

Mohammad Tawfik
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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, 

Mohammad Tawfik
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Let’s follow the same
procedure!

Mohammad Tawfik
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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 

Mohammad Tawfik
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








































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
1
00
1
00
11
0001
f
f
f
f
abababab
bb
aa
a
a
a
a

Mohammad Tawfik
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         e
fyxNayxHyxf ,,, 
     


























ab
xy
b
y
ab
xy
ab
xy
a
x
ab
xy
b
y
a
x
yxNyxN
T
1
,,

Mohammad Tawfik
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How does this look like?

Mohammad Tawfik
#WikiCourses
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

Mohammad Tawfik
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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

Mohammad Tawfik
#WikiCourses
Example: Laplace Equation
02
 
02
2
2
2






yx

     e
i
ii yxNyxN  ,,
4
1
 

Mohammad Tawfik
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Example: Laplace Equation
     e
i
ii yxNyxN  ,,
4
1
 
         0
e
Area
yyxx dANNNN 
Applying the Galerkin method and integrating by parts,
the element equation becomes

Mohammad Tawfik
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The Element Equaiton
 
 
 
 
   0
222
222
222
222
6
1
22222222
22222222
22222222
22222222



















e
babababa
babababa
babababa
babababa
ab


Mohammad Tawfik
#WikiCourses
The Logistic Problem!

Mohammad Tawfik
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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

Mohammad Tawfik
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1-D Example
• Element #1 is associated with nodes 1&2
• Element #2 is associated with nodes 2&3, etc…

Mohammad Tawfik
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2-D Example

Mohammad Tawfik
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For Element #5
Global Node NumberLocal Node Number
51
62
93
84

Mohammad Tawfik
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Contribution of element #5 to global
matrix
121110987654321
1
2
3
4
1,31,41,21,15
2,32,42,22,16
7
4,34,44,24,18
3,33,43,23,19
10
11
12

Mohammad Tawfik
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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!

Mohammad Tawfik
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Elements Register: Global
Numbering
Node NumberElement
Number 4321
45211
78542
1011873
56324
89655
1112986

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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 NumberElement
Number
4321
45211
78542
1011873
56324
89655
1112986
121110987654321
1
2
3
4
5
6
7
8
9
10
11
12

FEM: 2-D Problems

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FEM: 2-D Problems