introduction to the finite element method__2010
TRANSCRIPT
7/24/2019 Introduction to the Finite Element Method__2010
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Introduction to the Finite
Element Method
Spring 2010
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Course Objectives
• The student should be capable of writing
simple programs to solve different
problems using finite element method.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Assessment
• 10% Assignments (1 per week)
• 20% Quizzes (best 2 out of 3)
– Week of 12/11/2006
– Week of 20/12/2006
– Week of 17/1/2006
• 20% Course Project
• 25% Midterm exam (Week of 2/12/2006)• 25% Final exam (starting 3/2/2007)
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Fundamental Course Agreement
• Homework is sent in electronic format (Nohardcopies are accepted)
• Computer programs have to written in
MATLAB or Mathematica script• No late homework is accepted
• No excuses are accepted for missing a
quiz• Best two out of three quizzes are counted
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
References
• J.N. Reddy, “An Introduction to the FiniteElement Method” 3rd ed., McGraw Hill, ISBN007-124473-5
• D.V. Hutton, “Fundamentals of Finite Element
Analysis” 1st ed., McGraw Hill, ISBN 007-121857-2
• K. Bathe, “Finite Element Procedures,” PrenticeHall, 1996. (in library)
• T. Hughes, “The finite Element Method: LinearStatic and Dynamic Finite Element analysis,”Dover Publications, 2000. (in library)
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Numerical Solution of
Boundary Value Problems
Weighted Residual Methods
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Objectives
• In this section we will be introduced to thegeneral classification of approximatemethods
• Special attention will be paid for theweighted residual method
• Derivation of a system of linear equations
to approximate the solution of an ODE willbe presented using different techniques
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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!
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Basic Concepts
• A linear differential equation may be written in the form:
x g x f L
• Where L(.) is a linear differential operator.• An approximate solution maybe of the form:
n
i
ii xa x f 1
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Basic Concepts
• Applying the differential operator on the approximatesolution, you get:
01
1
x g x La
x g xa L x g x f L
n
i
ii
n
i
ii
x R x g x La
n
i
ii 1
Residue
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Collocation Method
0 j x R
01
j
n
i
jii
j
x F x La
x R
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Example Problem
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The bar tensile problem
02
2
x F
x
u EA
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bar application
02
2
x F
x
u EA
n
i
ii xa xu1
x R x F dx
xd a EA
n
i
ii
12
2 Applying the collocation method
01
2
2
j
n
i
ji
i x F dx
xd a EA
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
In Matrix Form
nnnnnn
n
n
x F
x F
x F
a
a
a
k k k
k k k
k k k
2
1
2
1
21
22212
12111
...
...
...
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
j x x
iij
dx
xd EAk
2
2
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Notes on the trial functions
• They should be at least twice
differentiable!
• They should satisfy all boundary
conditions!
• Those are called the “Admissibility
Conditions”.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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
xSin x
2
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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
xSin
l EA
dx
xd EA
22
2
2
2
f aSinl
EA
1
2
42
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Solving:
• Then, the approximate
solution for this problem is:
• Which gives the maximum
displacement to be:
• And maximum strain to be:
EA
f l
EA
f l
Sinl EA
f a
2
2
2
21 57.024
42
l
xSin EA
f l xu 257.0
2
5.057.02
exact EA
f l l u
0.19.00 exact EA
lf u x
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Subdomain Method (free
reading)
• The idea behind the
subdomain method is
to force the integral
of the residue to beequal to zero on an
subinterval of the
domain
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Subdomain Method
01
j
j
x
x
dx x R
0
11
1
j
j
j
j
x
x
n
i
x
x
ii dx x g dx x La
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bar application
02
2
x F
x
u EA
n
i
ii xa xu1
x R x F dx
xd a EA
n
i
ii
12
2 Applying the subdomain method
11
12
2 j
j
j
j
x
x
n
i
x
x
ii dx x F dx
dx
xd a EA
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
In Matrix Form
11
2
2 j
j
j
j
x
x
i
x
x
i dx x F adxdx
xd EA
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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 performedover the whole domain!!!
• Actually, it turned out to be a VERY
GOOD idea
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Galerkin Method
0 Domain
j dx x x R
0
1
Domain
j
n
i Domain
i ji dx x g xdx x L xa
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bar application
02
2
x F
x
u EA
n
i
ii xa xu1
x R x F dx
xd a EA
n
i
ii
12
2 Applying Galerkin method
Domain
j
n
i Domain
i ji dx x F xdx
dx
xd xa EA
12
2
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
In Matrix Form
Domain
ji
Domain
i j dx x F xadx
dx
xd x EA
2
2
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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
xSin x 2
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Substituting with the approximate
solution:
Domain
j
n
i Domain
i ji dx x F xdx
dx
xd xa EA
12
2
l
l
fdxl
xSin
dxl
xSin
l
xSina
l EA
0
01
2
2
222
l l a
l EA
2
22 1
2
EA
fl l
EA
f a
2
3
2
1 52.016
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Substituting with the approximate
solution: (Int. by Parts)
Domain
j
n
i Domain
i ji dx x F xdx
dx
xd xa EA
12
2
l l a
l EA
2
22 1
2
EA
fl l
EA
f a
2
3
2
1 52.016
Domain
i j
l
i j
Domain
i j
dxdx
xd
dx
xd
dx
xd x
dx
dx
xd x
0
2
2
Zero!
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
What did we gain?
• The functions are required to be less
differentiable
• Not all boundary conditions need to be
satisfied
• The matrix became symmetric!
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
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.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
NOTE
• Next Sunday 5/11 (No lecture)
• Following week 12/11, Quiz #1 will be held
covering all the material up-to this lecture
• Homework #1 is due next week (Electronic
submission of report and code is
mandatory.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Report Should Include …
• Cover page
• Introduction section indicating the
procedure you used with the equations as
implemented in your code
• Results section
• Observations and Conclusions if any
according to the output of your program.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Homework #1
• Solve the beam bendingproblem, for beamdisplacement, for a simplysupported beam with a loadplaced at the center of the
beam using – Collocation Method
– Subdomain Method
– Galerkin Method
• Use three term Sin series thatsatisfies all BC’s
• Write a program that producesthe results for n-term solution.
)(4
4
x F dx
wd
0)()0(
0)()0(
2
2
2
2
dx
l wd
dx
wd
l ww
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Exact Solution
12/110
3
15
7
412
2/1060
13
12)(
23
3
x x x x
x x x
xw
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Finite Element Method
2nd order DE’s in 1-D
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Objectives
• Understand the basic steps of the finite
element analysis
• Apply the finite element method to second
order differential equations in 1-D
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Mathematical Model
• Solve:
• Subject to:
L x
f cudx
dua
dx
d
0
0
00 ,0 Qdx
du
auu L x
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Step #1: Discretization
• At this step, we divide
the domain into
elements .
• The elements areconnected at nodes .
• All properties of the
domain are defined at
those nodes.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Step #2: Element Equations
• Let’s concentrate ourattention to a singleelement.
• The same DE applieson the element level,hence, we may followthe procedure forweighted residual
methods on theelement level!
21
0
x x x
f cu
dx
dua
dx
d
21
2211
21
,
,,
QdxduaQ
dxdua
u xuu xu
x x x x
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Polynomial Approximation
• Now, we may propose an approximate
solution for the primary variable, u(x),
within that element.
• The simplest proposition would be a
polynomial!
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Polynomial Approximation
• Interpolating the values
of displacement
knowing the nodal
displacements, we maywrite:
01 b xb xu
01111 b xbu xu
2
12
11
12
2 u x x x xu
x x x x xu
02122 b xbu xu
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Polynomial Approximation
eu xu
uuu
u x x
x xu
x x
x x xu
2
1
212211
2
12
11
12
2
St #2 El t E ti
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Step #2: Element Equations
(cont’d)
• Assuming constant
domain properties:
• Applying the
Galerkin method:
21
2
2
0
x x x
f cudx
ud a
02
2
Domain
jii jii
j dx f xu x xcudx
xd xa
St #2 El t E ti
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Step #2: Element Equations
(cont’d)
• Note that:
• And:
ee hdx
xd
hdx
xd 1
,
1 21
Domain
i j
x
x
i
j
Domain
i j
dxdx
xd
dx
xd a
dx
xd xa
dxdx
xd xa
2
1
2
2
St #2 El t E ti
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Step #2: Element Equations
(cont’d)
• For i=j=1: (and ignoring boundary terms)
• Which gives:
012
1
21
2
2
2
x
x eee
dxh
x x f u
h
x xc
h
a
023
1
ee
e
fhuchha
St #2 El t E ti
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Step #2: Element Equations
(cont’d)
• Repeating for all terms:
• The above equation is called the element
equation .
1
1
221
12
611
11
2
1 ee
e
fh
u
uch
h
a
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
What happens for adjacent
elements?
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Homework #2
• Derive the element equation without
ignoring the boundary terms.
• What are differences in the element
equation.
• The solution should be handed using the
same report format (use equation editor to
write your report).
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Finite Element Procedure
1. Connecting Elements
2. Boundary Conditions3. Solving Equations
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Objectives
• Learn how the finite element model for the
whole domain is assembled
• Learn how to apply boundary conditions
• Solving the system of linear equations
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Recall
• In the previous lecture, we obtained the
element equation that relates the element
degrees of freedom to the externally
applied fields
• Which maybe written:
1
1
221
12
611
11
2
1 ee
e
fh
u
uch
h
a
2
1
2
1
43
21
f f
uu
k k k k
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Two –Element example
1
2
1
1
1
2
1
1
1
4
1
3
1
2
1
1
f f
uu
k k k k
2
2
2
1
2
2
2
1
2
4
2
3
2
2
2
1
f
f
u
u
k k
k k
3
2
1
3
2
1
3
2
1
2
4
2
3
2
2
2
1
1
4
1
3
1
2
1
1
0
0
Q
f f
f
uu
u
k k k k k k
k k
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Illustration: Bar application
1. Discretization: Divide the bar into N number of
elements. The length of each element will be(L/N)
2. Derive the element equation from the differentialequation for constant properties an externally
applied force:
02
2
x F xu EA
02
1
2
x
x
i ji j
e
dx f udx
d
dx
d
h
EA
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Performing Integration:
1
1
211
11
2
1 e
e
e
e
fh
u
u
h
EA
Note that if the integration is evaluated from 0 to he,where he is the element length, the same results
will be obtained.
02
1
2
x
x
i ji j
e
dx f udx
d
dx
d
h
EA
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Two –Element bar example
1
2
1
1
1
2
1
1
1111
f f
uu
h EA
e
2
2
2
1
2
2
2
1
11
11
f
f
u
u
h
EA
e
00
12
1
2110121
011
3
2
1 R fh
uu
u
h
EA e
e
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Applying Boundary Conditions
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Applying BC’s
• For the bar with fixed left side and free
right side, we may force the value of the
left-displacement to be equal to zero:
0
0
1
2
1
2
0
110
121
011
3
2
R fh
u
uh
EA e
e
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Solving
• Removing the first row and column of the
system of equations:
• Solving:
1
2
211
12
3
2 e
e
fh
u
u
h
EA
4
3
2
2
3
2
EA
fh
u
ue
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Secondary Variables
• Using the values of the displacements
obtained, we may get the value of the
reaction force:
0
0
1
2
1
2
2
42
30
110
121
011 R fh
fh
fh e
e
e
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Secondary Variables
• Using the first equation, we get:
• Which is the exact value of the reactionforce.
R fh fh ee 22
3
e fh R 2
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Summary
• In this lecture, we learned how to
assemble the global matrices of the finite
element model; how to apply the boundary
conditions, and solve the system ofequations obtained.
• And finally, how to obtain the secondary
variables.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Homework #3
• Problems #3.9 & 3.13 from the text book
• Write down a computer code that solves
the problem for N elements.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bars and Trusses
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Objectives
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bar Example (Ex. 4.5.2, p. 187)
• Consider the bar shown in the above figure.
• It is composed of two different parts. One steel tapered
part, and uniform Aluminum part.• Calculate the displacement field using finite elementmethod.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bar Example
• The bar may be represented by two
elements.
• The stiffness matrices of the two elements
may be obtained using the followingintegration:
2
1
2
122
22
21
2
1
11
11 x
x
ee
ee
x
x
e dx
hh
hh x EAdxdx
d
dx
d
dx
d dx
d
x EA K
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bar Example
• For the Aluminum bar: E=107 psi, and A=1
in2. we get:
• For the Steel bar: E=38107 psi, and
A=(1.5-0.5x/96) in2. we get:
11
11
120
10
11
11
120
10 7
2
7 2
1
x
x Al dx K
11
11
96
10.75.4
11
11
96
5.05.1
96
10.3 7
2
7 2
1
x
x
Fe dx x
K
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bar Example
• Assembling the Stiffness matrix and
utilizing the external forces, we get:
• The boundary conditions may be applied
and the system of equations solved.
0
0
10
10.2
0
33.833.80
33.88.575.49
05.495.49
105
5
3
2
1
4
R
u
u
u
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Bar Example
• Solving, we get:
• For the secondary
variables:
inu
u
181.0
061.0
3
2
lb R 30000
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Reading Task
• Please read and understand examples,4.5.1 & 4.5.3.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Trusses
• A truss is a set of bars that are connectedat frictionless joints.
• The Truss bars are generally oriented in
the plain.
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Trusses
• Now, the problem lies in thetransformation of the local displacements
of the bar, which are always in the
direction of the bar, to the global degreesof freedom that are generally oriented in
the plain.
E i f M i
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Equation of Motion
0
0
0000
01010000
0101
2
1
2
2
1
1
F
F
v
uv
u
h EA
T f ti M t i
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Transformation Matrix
DOF d Transforme DOF Local v
u
v
u
CosSin
SinCos
CosSin
SinCos
v
u
v
u
2
2
1
1
2
2
1
1
00
00
00
00
DOF
d Transforme DOF Local T
Th E i f M i B
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Equation of Motion Becomes
• Substituting into theFEM:
• Transforming the
forces:
• Finally:
F T K
F T T K T T T
F K
R ll
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Recall
T K T K T
CosSin
SinCos
CosSin
SinCos
T
00
00
00
00
Where:
0000
0101
0000
0101
h
EA
K
Element Stiffness Matrix in Global
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Element Stiffness Matrix in Global
Coordinates
CosSinSinCos
CosSin
SinCos
CosSin
SinCos
CosSin
SinCos
h
EA K
0000
00
00
0000
0101
0000
0101
00
00
00
00
Element Stiffness Matrix in Global
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Element Stiffness Matrix in Global
Coordinates
22
22
22
22
22
1
22
1
22
12
2
1
22
1
22
1
22
12
2
1
SinSinSinSin
SinCosSinCos
SinSinSinSin
SinCosSinCos
h
EA K
E l 4 6 1 196 201
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Example: 4.6.1 pp. 196-201
• Use the finite element analysis to find thedisplacements of node C.
El t E ti
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Element Equations
0000
0101
0000
0101
1
L
EA K
1010
0000
1010
0000
2
L
EA K
3536.03536.03536.03536.03536.03536.03536.03536.0
3536.03536.03536.03536.0
3536.03536.03536.03536.0
3
L
EA K
A bl P d
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Assembly Procedure
3536.13536.0103536.03536.0
3536.03536.0003536.03536.0
101000000101
3536.03536.0003536.03536.0
3536.03536.0013536.03536.1
L EA K
Gl b l F V t
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Global Force Vector
P
P
F F
F F
F
F
F
F
F F
F y
x
y
x
y
x
y
x
y
x
2
2
2
1
1
3
3
2
2
1
1
Remember!
NO distributed loadis applied to a truss
B d C diti
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Boundary Conditions
02211 V U V U Remove the corresponding rows and columns
P
P
V
U
L
EA
23536.13536.0
3536.03536.0
3
3
Continue! (as before)
R lt
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Results
EA
PLV
EA
PLU
3 ,828.5 33
P F F
P F P F
y x
y x
3 ,0
, ,
22
11
Postcomp tation
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Postcomputation
e
e
e
ee
A
P
A
P 21
e
e
eee
e
u
u L E A
P
P
2
1
2
1
1111
2
2
1
1
2
2
1
1
00
00
0000
v
u
vu
CosSin
SinCos
CosSinSinCos
v
u
v
u
Postcomputation
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Postcomputation
A
P
A
P 2 ,
3 ,0 )3()2()1(
Summary
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Summary
• In this lecture we learned how to apply thefinite element modeling technique to bar
problems with general orientation in a
plain.
Homework #5
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Homework #5
• Problem 4.27, – Due 13/12/2006 before 9:00am
• Problem 4.44,
– Due 20/12/2006 before 9:00am
Announcements
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Announcements
• Compensation Tutorial for E15: – Next Sunday 17/12/2006 3rd Period in H6
• Next Lecture:
– Wednesday 20/12/2006 3rd Period in H6
• Next Quiz:
– Wednesday 20/12/2006 3rd Period in H6
– (This Lecture is included)
Term Projects
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Term Projects
• A problem has got to be solved using thefinite element method
• A report is going to be presented by each
group presenting the problem and itssolution
The Report should contain:
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Report should contain:
• Cover page – Project Title
– Names of team members
• Table of contents
• Introduction and literature survey – Introduction to the problem
– Historical background and relevance of the problem
– Papers and books that presented the problem
– Latest achievements in the problem
The Report should contain:
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Report should contain:
• The finite element derivation – Governing equation
– Derivation of the element matrices
• Using Glerkin method• Application of Symbolic manipulator to derive the
matrix equations will be appreciated
– Solution procedure
The Report should contain:
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Report should contain:
• The numerical results and verification – Program results
– Verification of results compared to published results
– Parametric study
• Discussion – Observations of the results
– Further work that may be performed with the problem
– Future developments of the model
• References
Evaluation
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Evaluation
• Report (50%)
• Code (30%)
– Structured: Functions built, easily modified
– Readability: Organization, remarks
– Length: The shorter the better
• Results (20%)
Projects
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Projects
• Heat transfer in a 2-D heat sink
• 2-D flow around a blunt body in a wind
tunnel
• Vibration characteristics of a pipe with
internal fluid flow
• Panel flutter of a beam
• Rotating Timoshenko beam/blade
Heat transfer in a 2 D heat sink
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Heat transfer in a 2-D heat sink
• The heat sink will have heat flowing fromone side
• Convection transfer on the surfaces
• Different boundary conditions on the otherthree sides
• Plot contours of temperature distribution
with different boundary conditions
2-D flow around a blunt body in a
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
wind tunnel
• Potential flow in a duct
• Rectangular body with different
Dimensions
• Study the effect of the body size on theflow speed on both sides
• Plot contours of potential function,
pressure, and velocity potential
Vibration characteristics of a pipe
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
with internal fluid flow
• Study the change of the naturalfrequencies with the flow speed under
different boundary conditions and fluid
density• Indicate the flow speeds at which
instabilities occur
Panel flutter of a beam
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Panel flutter of a beam
• A fixed-fixed beam is subjected to flowover its surface
• Plot the effect of the flow speed on the
natural frequencies of the beam• Indicate the speed at which instability
occurs
Rotating Timoshenko beam/blade
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Rotating Timoshenko beam/blade
• Rotating beams undergo centrifugaltension that results in the change of its
natural frequencies
• Study the effect of rotation speed on thebeam natural frequencies and frequency
response to excitations at the root
Teams
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Teams
• 2-3 Students teams
• Names and selected projects should be
submitted before 4PM on Thursday
21/12/2006
Work Progress
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Work Progress
• A report should be submitted By 4PM every Wednesday• 27/12/2006
– The report should contain a preliminary literature survey
– Problem statement
– Governing equations
• 10/1/2007
– The report should contain a deeper literature survey – The preliminary derivations of the finite element model
• 17/1/2007 – A more mature version of the report should be presented
– Preliminary results of the code
– List of the program script should be included
• 24/1/2007 – Final version of the report should be presented together with the code
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beams and Frames
Beams and Frames
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beams and Frames
• Beams are the most-used structuralelements.
• Many real structures may be approximated
as beam elements• Two main beam theories:
– Euler-Bernoulli beam theory
– Timoshenko beam theory
Euler-Bernoulli Beam Theory
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Euler Bernoulli Beam Theory
• The main assumption in the Euler-Bernoulli beam theory is that the beam’s
thickness is too small compared to the
beam length• That assumption resulted in that the sheer
deformation of the beam may be
neglected without much error in theanalysis
Governing Equation
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Governing Equation
• The equation governing the deformation ofand E-B beam under transverse loading
may be written in the form:
)(2
2
2
2
x F dxwd x EI
dxd
The Thin-Beam Elements
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Thin Beam Elements
• The thin beam element has a special feature,namely, the two degrees of freedom at each
node are related.
Beam Interpolation Function
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beam Interpolation Function
a x H xw
34
2321)( xa xa xaa xw
a x x x xw 321
a x xa x H adx
xdH
dx
xdw x
2
3210
Beam Interpolation Function
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beam Interpolation Function
a H ww 00 1
aT
a
a
a
a
l H
l H
H
H
w
w
w
w
x
x
4
3
2
1
2
2
1
1
00
'
'
a H ww x 0'0' 1 al H wl w 2
al H wl w x 2''
Beam Interpolation Function
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beam Interpolation Function
4
3
2
1
2
32
2
2
1
1
3210
1
0010
0001
'
'
a
a
a
a
l l
l l l
w
w
w
w
2
2
1
1
2323
22
4
3
2
1
'
'
1212
13230010
0001
w
w
w
w
l l l l
l l l l
a
a
a
a
Beam Interpolation Function
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beam Interpolation Function
ee w x N wT x H a x H xw 1
ewT a 1
4
1i
ii w x N xw
Beam Interpolation Function
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beam Interpolation Function
2
32
3
3
2
2
3
32
3
3
2
2
23
2
231
l
x
l
xl
x
l
x l
x
l
x x
l
x
l
x
x N x N T
Interpolation Functions
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Interpolation Functions
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
X
N ( x
)
N1
N2
N3
N4
Beam Stiffness Matrix
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beam Stiffness Matrix
• The governing equation is:
• Using the series solution
)(2
2
2
2
x F dx
wd x EI
dx
d
4
1i
ii w x N xw
Beam Stiffness Matrix
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beam Stiffness Matrix
• The governing equation becomes
• Applying Galerkin method:
)()(4
12
2
2
2
x R x F wdx
N d x EI
dx
d
i
ii
ee l
j
i
ii
l
j dx N x F wdx
N d x EI
dx
d dx N x R
0
4
12
2
2
2
0
)()(
Beam Stiffness Matrix
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Beam Stiffness Matrix
• Using integration by parts, twice, andignoring the boundary terms, we get:
• In matrix form: 0)(0
4
12
2
2
2
el
ji
i
ji
dx N x F wdx
N d
dx
N d
x EI
ee l
xxe
l
xx xx dx N x F wdx N N x EI 00
)(
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Use of Symbolic Manipulator
Beam Example
Optional Homework #6
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Optional Homework #6
• Derive the expression for the interpolationfunction for a beam in terms of nodal
displacements and slopes.
• Try to use a symbolic manipulator togenerate the expressions.
)(4
4
2
2
x F dxwd EI
dt wd A
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Two Dimensional Elements
2-D Elements
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
• In this section, we will be introduced to twodimensional elements with single degree
of freedom per node.
• Detailed attention will be paid torectangular elements.
For the 2-D BV Problem
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
• Let’s consider a problem with a single dependent variable
• We may set one degree of freedom to
each node; say f i.• Further, let’s only consider a rectangular
element that is aligned with the physical
coordinates
A Rectangular Element
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
g
• For the approximationof a general function
f(x,y) over the element
you need a 2-D
interpolation function
xya ya xaa y x f 4321,
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Let’s follow the same
procedure!
2-D Interpolation Function
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
p
a y x H y x f ,, xya ya xaa y xf 4321),( a H f f 0,00,0 1
aT
aa
a
a
b H ba H
a H
H
f f
f
f
4
3
2
1
4
3
2
1
0,
0,
0,0
aa H f a f ,00, 2
aba H f baf ,, 3 ab H f b f ,0,0 4
2-D Interpolation Function
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
p
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
aa
2-D Interpolation Function
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
p
e f y x N a y x H y x f ,,,
ab xy
b y
ab
xyab
xy
a
xab
xy
b
y
a
x
y x N y x N T
1
,,
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
How does this look like?
2-D Interpolation Functions
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
p
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
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
N1
x
y
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
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
N2
x
y
2-D Interpolation Functions
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
p
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
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
N3
x
y
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
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
N4
x
y
Example: Laplace Equation
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
p p q
02
02
2
2
2
y x
ei
ii y x N y x N ,,4
1
Example: Laplace Equation
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
p p q
ei
ii y x N y x N ,,4
1
0 e
Area
y y x x dA N N N N
Applying the Galerkin method and integrating by parts,
the element equation becomes
The Element Equaiton
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
0222
222
222
222
6
1
22222222
22222222
22222222
22222222
e
babababa
babababa
babababa
babababa
ab
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
The Logistic Problem!
The Logistic Problem
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
• In the 2-D problems, the numberingscheme, usually, is not as straight forward
as the 1-D problem
1-D Example
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
• Element #1 is associated with nodes 1&2• Element #2 is associated with nodes 2&3, etc…
2-D Example
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
2-D Example
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
For Element #5
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Global Node NumberLocal Node Number
51
62
93
84
Contribution of element #5 to global
matrix
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
matrix121110987654321
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
A Solution for the Logistics’
Problem
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Problem
• One solution of the logistic problem is tokeep a record of elements and the
mapping of the local numbering scheme to
the global numbering scheme in a table!
Elements Register: Global
Numbering
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Introduction to the Finite Element Method
Dr. Mohammad Tawfik
Numbering
Node NumberElementNumber 4321
45211
785421011873
56324
896551112986
Algorithm for Assembling Global
Matrix
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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 k6. For the mth node
7. Get its global number n
8. Let Akn=Akn+B jm
9. Repeat for all m
10. Repeat for all j11. Repeat for all i
Node NumberElement
Number4321
45211
78542
1011873
56324
89655
1112986
121110987654321
1
2
3
4
5
6
7
8
9
10
11
12