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# FEM: 2-D Problems

Hoe to model 2-D problems using the finite element method?

#WikiCourses #Num001
https://wikicourses.wikispaces.com/TopicX+2-D+Problems

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### FEM: 2-D Problems

1. 1. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Introduction to the Finite Element Method Two Dimensional Elements
2. 2. 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.
3. 3. 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
4. 4. 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, 
5. 5. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Let’s follow the same procedure!
6. 6. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 
7. 7. 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 1 00 1 00 11 0001 f f f f abababab bb aa a a a a
8. 8. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 2-D Interpolation Function          e fyxNayxHyxf ,,,                                  ab xy b y ab xy ab xy a x ab xy b y a x yxNyxN T 1 ,,
9. 9. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com How does this look like?
10. 10. 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
11. 11. 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
12. 12. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example: Laplace Equation 02   02 2 2 2       yx       e i ii yxNyxN  ,, 4 1  
13. 13. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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
14. 14. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Element Equaiton            0 222 222 222 222 6 1 22222222 22222222 22222222 22222222                    e babababa babababa babababa babababa ab 
15. 15. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Logistic Problem!
16. 16. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Logistic Problem • In the 2-D problems, the numbering scheme, usually, is not as straight forward as the 1-D problem
17. 17. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 1-D Example • Element #1 is associated with nodes 1&2 • Element #2 is associated with nodes 2&3, etc…
18. 18. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 2-D Example
19. 19. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 2-D Example
20. 20. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com For Element #5 Global Node NumberLocal Node Number 51 62 93 84
21. 21. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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
22. 22. 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!
23. 23. FEM: Two Dimensional Elements Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Elements Register: Global Numbering Node NumberElement Number 4321 45211 78542 1011873 56324 89655 1112986
24. 24. 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 NumberElement Number 4321 45211 78542 1011873 56324 89655 1112986 121110987654321 1 2 3 4 5 6 7 8 9 10 11 12
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Aug. 6, 2017

Hoe to model 2-D problems using the finite element method? #WikiCourses #Num001 https://wikicourses.wikispaces.com/TopicX+2-D+Problems

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