FEM: Introduction and Weighted Residual Methods

What are weighted residual methods?
How to apply Galerkin Method to the finite element model?

#WikiCourses #Num001
https://wikicourses.wikispaces.com/TopicX+Approximate+Methods+-+Weighted+Residual+Methods

1. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Introduction to the Finite
Element Method
Mohammad Tawfik

2. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
References
•J.N. Reddy, “An Introduction to the Finite
Element Method” 3rd ed., McGraw Hill, ISBN
007-124473-5
•D.V. Hutton, “Fundamentals of Finite Element
Analysis” 1st ed., McGraw Hill, ISBN 007-
121857-2
•K. Bathe, “Finite Element Procedures,” Prentice
Hall, 1996. (in library)
•T. Hughes, “The finite Element Method: Linear
Static and Dynamic Finite Element analysis,”
Dover Publications, 2000. (in library)

3. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Numerical Solution of
Boundary Value Problems
Weighted Residual Methods

4. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Objectives
•In this section we will be introduced to the
general classification of approximate
methods
•Special attention will be paid for the
weighted residual method
•Derivation of a system of linear equations
to approximate the solution of an ODE will
be presented using different techniques

5. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Why Approximate?
•Ignorance
•Readily Available Packages
•Need to Develop New Techniques
•Good use of your computer!
•In general, the problem does not have an
analytical solution!

6. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Classification of Approximate
Solutions of D.E.’s
•Discrete Coordinate Method
–Finite difference Methods
–Stepwise integration methods
•Euler method
•Runge-Kutta methods
•Etc…
•Distributed Coordinate Method

7. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Distributed Coordinate Methods
•Weighted Residual Methods
–Interior Residual
•Collocation
•Galrekin
•Finite Element
–Boundary Residual
•Boundary Element Method
•Stationary Functional Methods
–Reyligh-Ritz methods
–Finite Element method

8. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Basic Concepts
•A linear differential equation may be written in the form: xgxfL 
•Where L(.) is a linear differential operator.
•An approximate solution maybe of the form:  


n
i
iixaxf
1


9. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Basic Concepts
•Applying the differential operator on the approximate
solution, you get:  
0
1
1












xgxLa
xgxaLxgxfL
n
i
ii
n
i
ii

 xRxgxLa
n
i
ii 
1

Residue

10. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Handling the Residue
•The weighted residual methods are all
based on minimizing the value of the
residue.
•Since the residue can not be zero over the
whole domain, different techniques were
introduced.

11. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Collocation Method
•The idea behind the collocation method is
similar to that behind the buttons of your
shirt!
•Assume a solution, then force the residue
to be zero at the collocation points

12. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Collocation Method 0
j
xR 
 0
1




j
n
i
jii
j
xFxLa
xR


13. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example Problem

14. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The bar tensile problem 0
2
2



xF
x
u
EA

15. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Bar application 0
2
2



xF
x
u
EA  


n
i
iixaxu
1
 
xRxF
dx
xd
aEA
n
i
i
i

1
2
2

Applying the collocation method 
0
1
2
2


j
n
i
ji
i
xF
dx
xd
aEA


16. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
In Matrix Form 










































nnnnnn
n
n
xF
xF
xF
a
a
a
kkk
kkk
kkk

2
1
2
1
21
22212
12111
...
...
...
Solve the above system for the “generalized
coordinates” a
i to get the solution for u(x) 
jxx
i
ij
dx
xd
EAk


2
2


17. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Notes on the trial functions
•They should be at least twice
differentiable!
•They should satisfy all boundary
conditions!
•Those are called the “Admissibility
Conditions”.

18. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Using Admissible Functions
•For a constant forcing function, F(x)=f
•The strain at the free end of the bar should
be zero (slope of displacement is zero).
We may use:  






l
x
Sinx
2



19. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Using the function into the DE:
•Since we only have one term in the series,
we will select one collocation point!
•The midpoint is a reasonable choice! 













l
x
Sin
l
EA
dx
xd
EA
22
2
2
2
 faSin
l
EA 





















1
2
42


20. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Solving:
•Then, the approximate
solution for this problem is:
•Which gives the maximum
displacement to be:

•And maximum strain to be:  EA
fl
EA
fl
SinlEA
f
a
2
2
2
21 57.0
24
42

  






l
x
Sin
EA
fl
xu
2
57.0
2
   5.057.0
2
 exact
EA
fl
lu   0.19.00  exact
EA
lf
u
x

21. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The Subdomain Method (free
reading)
•The idea behind the
subdomain method is
to force the integral
of the residue to be
equal to zero on an
subinterval of the
domain

22. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The Subdomain Method 0
1

j
j
x
x
dxxR  0
11
1



j
j
j
j
x
x
n
i
x
x
ii
dxxgdxxLa

23. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Bar application 0
2
2



xF
x
u
EA  


n
i
iixaxu
1
 
xRxF
dx
xd
aEA
n
i
i
i

1
2
2

Applying the subdomain method 




11
1
2
2 j
j
j
j
x
x
n
i
x
x
i
i
dxxFdx
dx
xd
aEA


24. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
In Matrix Form 
 




















 11
2
2 j
j
j
j
x
x
i
x
x
i
dxxFadx
dx
xd
EA

Solve the above system for the “generalized
coordinates” a
i to get the solution for u(x)

25. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The Galerkin Method
•Galerkin suggested that the residue
should be multiplied by a weighting
function that is a part of the suggested
solution then the integration is performed
over the whole domain!!!
•Actually, it turned out to be a VERY
GOOD idea

26. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The Galerkin Method 0
Domain
j
dxxxR  0
1

 Domain
j
n
i Domain
iji
dxxgxdxxLxa 

27. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Bar application 0
2
2



xF
x
u
EA  


n
i
iixaxu
1
 
xRxF
dx
xd
aEA
n
i
i
i

1
2
2

Applying Galerkin method 



 Domain
j
n
i Domain
i
ji
dxxFxdx
dx
xd
xaEA 


1
2
2

28. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
In Matrix Form 

 




















Domain
ji
Domain
i
j
dxxFxadx
dx
xd
xEA 


2
2
Solve the above system for the “generalized
coordinates” a
i to get the solution for u(x)

29. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Same conditions on the functions
are applied
•They should be at least twice
differentiable!
•They should satisfy all boundary
conditions!
•Let’s use the same function as in the
collocation method:  






l
x
Sinx
2



30. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Substituting with the approximate
solution: 



 Domain
j
n
i Domain
i
ji
dxxFxdx
dx
xd
xaEA 


1
2
2 



























l
l
fdx
l
x
Sin
dx
l
x
Sin
l
x
Sina
l
EA
0
0
1
2
2
222

 
 ll
a
l
EA
2
22
1
2






 EA
fll
EA
f
a
2
3
2
1
52.0
16



31. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Substituting with the approximate
solution: (Int. by Parts) 



 Domain
j
n
i Domain
i
ji
dxxFxdx
dx
xd
xaEA 


1
2
2 
 ll
a
l
EA
2
22
1
2






 EA
fll
EA
f
a
2
3
2
1
52.0
16

 


 



Domain
ij
l
i
j
Domain
i
j
dx
dx
xd
dx
xd
dx
xd
x
dx
dx
xd
x




0
2
2
Zero!

32. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
What did we gain?
•The functions are required to be less
differentiable
•Not all boundary conditions need to be
satisfied
•The matrix became symmetric!

33. Introduction to the Finite Element Method
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Summary
•We may solve differential equations using a
series of functions with different weights.
•When those functions are used, Residue
appears in the differential equation
•The weights of the functions may be determined
to minimize the residue by different techniques
•One very important technique is the Galerkin
method.