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An Introduction to the Finite Element Method
1.
Introduction to the
Finite Element Method Spring 2010
2.
3.
4.
5.
6.
Numerical Solution of
Boundary Value Problems Weighted Residual Methods
7.
8.
9.
10.
11.
12.
13.
14.
15.
Collocation Method
16.
Example Problem
17.
The bar tensile
problem
18.
Bar application Applying
the collocation method
19.
In Matrix Form
Solve the above system for the “generalized coordinates” a i to get the solution for u(x)
20.
21.
22.
23.
24.
25.
The Subdomain Method
26.
Bar application Applying
the subdomain method
27.
In Matrix Form
Solve the above system for the “generalized coordinates” a i to get the solution for u(x)
28.
29.
The Galerkin Method
30.
Bar application Applying
Galerkin method
31.
In Matrix Form
Solve the above system for the “generalized coordinates” a i to get the solution for u(x)
32.
33.
Substituting with the
approximate solution:
34.
Substituting with the
approximate solution: (Int. by Parts) Zero!
35.
36.
37.
38.
39.
40.
Exact Solution
41.
The Finite Element
Method 2 nd order DE’s in 1-D
42.
43.
44.
45.
46.
47.
48.
Polynomial Approximation
49.
50.
51.
52.
53.
What happens for
adjacent elements?
54.
55.
56.
57.
58.
Two–Element example
59.
60.
Performing Integration: Note
that if the integration is evaluated from 0 to h e , where h e is the element length, the same results will be obtained .
61.
Two–Element bar example
62.
Applying Boundary Conditions
63.
64.
65.
66.
67.
68.
69.
Bars and Trusses
70.
Objectives
71.
72.
73.
74.
75.
76.
77.
78.
79.
Equation of Motion
80.
Transformation Matrix
81.
82.
Recall Where:
83.
Element Stiffness Matrix
in Global Coordinates
84.
Element Stiffness Matrix
in Global Coordinates
85.
86.
Element Equations
87.
Assembly Procedure
88.
Global Force Vector
Remember! NO distributed load is applied to a truss
89.
Boundary Conditions Remove
the corresponding rows and columns Continue! (as before)
90.
Results
91.
Postcomputation
92.
Postcomputation
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
Beams and Frames
110.
111.
112.
113.
114.
Beam Interpolation Function
115.
Beam Interpolation Function
116.
Beam Interpolation Function
117.
Beam Interpolation Function
118.
Beam Interpolation Function
119.
Interpolation Functions
120.
121.
122.
123.
Use of Symbolic
Manipulator Beam Example
124.
125.
Two Dimensional Elements
126.
127.
128.
129.
Let’s follow the
same procedure!
130.
2-D Interpolation Function
131.
2-D Interpolation Function
132.
2-D Interpolation Function
133.
How does this
look like?
134.
2-D Interpolation Functions
135.
2-D Interpolation Functions
136.
Example: Laplace Equation
137.
Example: Laplace Equation
Applying the Galerkin method and integrating by parts, the element equation becomes
138.
The Element Equaiton
139.
The Logistic Problem!
140.
141.
142.
2-D Example
143.
2-D Example
144.
For Element #5
Global Node Number Local Node Number 5 1 6 2 9 3 8 4
145.
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
146.
147.
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
148.
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