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finite element introTRANSCRIPT
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Introduction to the Finite Element Method
Spring 2010
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Course Objectives
• The student should be capable of writing simple programs to solve different problems using finite element method.
Introduction to the Finite Element MethodDr. 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)
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Fundamental Course Agreement
• Homework is sent in electronic format (No hardcopies 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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)
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Numerical Solution of Boundary Value Problems
Weighted Residual Methods
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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
Introduction to the Finite Element MethodDr. 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!
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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
iii xaxf
1
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Basic Concepts• Applying the differential operator on the approximate
solution, you get:
01
1
xgxLa
xgxaLxgxfL
n
iii
n
iii
xRxgxLan
iii
1
Residue
Introduction to the Finite Element MethodDr. 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.
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Collocation Method
0jxR
0
1
j
n
ijii
j
xFxLa
xR
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Example Problem
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The bar tensile problem
02
2
xFx
uEA
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Bar application
02
2
xFx
uEA
n
iii xaxu
1
xRxFdx
xdaEA
n
i
ii
12
2Applying the collocation method
01
2
2
j
n
i
jii xF
dx
xdaEA
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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” ai to get the solution for u(x)
jxx
iij dx
xdEAk
2
2
Introduction to the Finite Element MethodDr. 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”.
Introduction to the Finite Element MethodDr. 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
xSinx
2
Introduction to the Finite Element MethodDr. 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
lEA
dx
xdEA
22
2
2
2
faSinl
EA
1
2
42
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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
fa
2
2
2
21 57.024
42
l
xSin
EA
flxu
257.0
2
5.057.02
exactEA
fllu
0.19.00 exactEA
lfux
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The Subdomain Method
01
j
j
x
x
dxxR
011
1
j
j
j
j
x
x
n
i
x
x
ii dxxgdxxLa
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Bar application
02
2
xFx
uEA
n
iii xaxu
1
xRxFdx
xdaEA
n
i
ii
12
2Applying the subdomain method
11
12
2 j
j
j
j
x
x
n
i
x
x
ii dxxFdx
dx
xdaEA
Introduction to the Finite Element MethodDr. Mohammad Tawfik
In Matrix Form
11
2
2 j
j
j
j
x
x
i
x
x
i dxxFadxdx
xdEA
Solve the above system for the “generalized coordinates” ai to get the solution for u(x)
Introduction to the Finite Element MethodDr. 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 performed over the whole domain!!!
• Actually, it turned out to be a VERY GOOD idea
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The Galerkin Method
0Domain
j dxxxR
01
Domain
j
n
i Domain
iji dxxgxdxxLxa
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Bar application
02
2
xFx
uEA
n
iii xaxu
1
xRxFdx
xdaEA
n
i
ii
12
2Applying Galerkin method
Domain
j
n
i Domain
iji dxxFxdx
dx
xdxaEA
12
2
Introduction to the Finite Element MethodDr. Mohammad Tawfik
In Matrix Form
Domain
ji
Domain
ij dxxFxadx
dx
xdxEA
2
2
Solve the above system for the “generalized coordinates” ai to get the solution for u(x)
Introduction to the Finite Element MethodDr. 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
xSinx
2
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Substituting with the approximate solution:
Domain
j
n
i Domain
iji dxxFxdx
dx
xdxaEA
12
2
l
l
fdxl
xSin
dxl
xSin
l
xSina
lEA
0
0
1
2
2
222
ll
al
EA2
22 1
2
EA
fll
EA
fa
2
3
2
1 52.016
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Substituting with the approximate solution: (Int. by Parts)
Domain
j
n
i Domain
iji dxxFxdx
dx
xdxaEA
12
2
ll
al
EA2
22 1
2
EA
fll
EA
fa
2
3
2
1 52.016
Domain
ijl
ij
Domain
ij
dxdx
xd
dx
xd
dx
xdx
dxdx
xdx
0
2
2
Zero!
Introduction to the Finite Element MethodDr. 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!
Introduction to the Finite Element MethodDr. 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.
Introduction to the Finite Element MethodDr. 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.
Introduction to the Finite Element MethodDr. 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.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Homework #1
• Solve the beam bending problem, for beam displacement, for a simply supported beam with a load placed at the center of the beam using– Collocation Method– Subdomain Method– Galerkin Method
• Use three term Sin series that satisfies all BC’s
• Write a program that produces the results for n-term solution.
)(4
4
xFdx
wd
0)()0(
0)()0(
2
2
2
2
dx
lwd
dx
wd
lww
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Exact Solution
12/110
3
15
7
412
2/1060
13
12)(
23
3
xxxx
xxx
xw
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The Finite Element Method
2nd order DE’s in 1-D
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The Mathematical Model
• Solve:
• Subject to:
Lx
fcudx
dua
dx
d
0
0
00 ,0 Qdx
duauu
Lx
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Step #1: Discretization
• At this step, we divide the domain into elements.
• The elements are connected at nodes.
• All properties of the domain are defined at those nodes.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Step #2: Element Equations
• Let’s concentrate our attention to a single element.
• The same DE applies on the element level, hence, we may follow the procedure for weighted residual methods on the element level!
21
0
xxx
fcudx
dua
dx
d
21
2211
21
,
,,
Qdx
duaQ
dx
dua
uxuuxu
xxxx
Introduction to the Finite Element MethodDr. 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!
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Polynomial Approximation
• Interpolating the values of displacement knowing the nodal displacements, we may write: 01 bxbxu
01111 bxbuxu 2
12
11
12
2 uxx
xxu
xx
xxxu
02122 bxbuxu
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Polynomial Approximation
euxu
uuu
uxx
xxu
xx
xxxu
2
1212211
212
11
12
2
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Step #2: Element Equations (cont’d)
• Assuming constant domain properties:
• Applying the Galerkin method:
21
2
2
0
xxx
fcudx
uda
02
2
Domain
jiijii
j dxfxuxxcudx
xdxa
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Step #2: Element Equations (cont’d)
• Note that:
• And:
ee hdx
xd
hdx
xd 1,
1 21
Domain
ijx
x
ij
Domain
ij
dxdx
xd
dx
xda
dx
xdxa
dxdx
xdxa
2
1
2
2
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Step #2: Element Equations (cont’d)
• For i=j=1: (and ignoring boundary terms)
• Which gives:
012
1
21
2
22
x
x eee
dxh
xxfu
h
xxc
ha
023 1
ee
e
fhu
ch
h
a
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
What happens for adjacent elements?
Introduction to the Finite Element MethodDr. 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).
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Finite Element Procedure
1. Connecting Elements
2. Boundary Conditions
3. Solving Equations
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. 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
u
u
kk
kk
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Two–Element example
1
2
11
12
11
14
13
12
11
f
f
u
u
kk
kk
2
2
21
22
21
24
23
22
21
f
f
u
u
kk
kk
3
2
1
3
2
1
3
2
1
24
23
22
21
14
13
12
11
0
0
Q
Q
Q
f
f
f
u
u
u
kk
kkkk
kk
Introduction to the Finite Element MethodDr. 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 differential equation for constant properties an externally applied force:
02
2
xFx
uEA
02
1
2
x
x
ijij
e
dxfudx
d
dx
d
h
EA
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Performing Integration:
1
1
211
11
2
1 ee
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
ijij
e
dxfudx
d
dx
d
h
EA
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Two–Element bar example
1
2
11
12
11
11
11
f
f
u
u
h
EA
e
2
2
21
22
21
11
11
f
f
u
u
h
EA
e
0
0
1
2
1
2110
121
011
3
2
1 Rfh
u
u
u
h
EA e
e
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Applying Boundary Conditions
Introduction to the Finite Element MethodDr. 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
Rfh
u
uh
EA e
e
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. 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 Rfh
fh
fh e
e
e
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Secondary Variables
• Using the first equation, we get:
• Which is the exact value of the reaction force.
Rfhfh ee 22
3
efhR 2
Introduction to the Finite Element MethodDr. 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 of equations obtained.
• And finally, how to obtain the secondary variables.
Introduction to the Finite Element MethodDr. 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.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Bars and Trusses
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Objectives
Introduction to the Finite Element MethodDr. 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 element
method.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Bar Example
• The bar may be represented by two elements.
• The stiffness matrices of the two elements may be obtained using the following integration:
2
1
2
122
2221
2
1
11
11x
x
ee
ee
x
x
e dx
hh
hhxEAdx
dx
d
dx
d
dx
ddx
d
xEAK
Introduction to the Finite Element MethodDr. 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 dxK
11
11
96
10.75.4
11
11
96
5.05.1
96
10.3 7
2
7 2
1
x
x
Fe dxx
K
Introduction to the Finite Element MethodDr. 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
14
R
u
u
u
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Bar Example
• Solving, we get:
• For the secondary variables:
inu
u
181.0
061.0
3
2
lbR 30000
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Reading Task
• Please read and understand examples, 4.5.1 & 4.5.3.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Trusses
• A truss is a set of bars that are connected at frictionless joints.
• The Truss bars are generally oriented in the plain.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Trusses
• Now, the problem lies in the transformation of the local displacements of the bar, which are always in the direction of the bar, to the global degrees of freedom that are generally oriented in the plain.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Equation of Motion
0
0
0000
0101
0000
0101
2
1
2
2
1
1
F
F
v
u
v
u
h
EA
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Transformation Matrix
DOFdTransforme
DOFLocalv
u
v
u
CosSin
SinCos
CosSin
SinCos
v
u
v
u
2
2
1
1
2
2
1
1
00
00
00
00
DOF
dTransformeDOFLocal T
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The Equation of Motion Becomes
• Substituting into the FEM:
• Transforming the forces:
• Finally:
FTK
FTTKT TT
FK
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Recall
TKTK T
CosSin
SinCos
CosSin
SinCos
T
00
00
00
00
Where:
0000
0101
0000
0101
h
EAK
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Element Stiffness Matrix in Global Coordinates
CosSin
SinCos
CosSin
SinCos
CosSin
SinCos
CosSin
SinCos
h
EAK
00
00
00
00
0000
0101
0000
0101
00
00
00
00
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Element Stiffness Matrix in Global Coordinates
22
22
22
22
22
12
2
1
22
12
2
1
22
12
2
1
22
12
2
1
SinSinSinSin
SinCosSinCos
SinSinSinSin
SinCosSinCos
h
EAK
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Example: 4.6.1 pp. 196-201
• Use the finite element analysis to find the displacements of node C.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Element Equations
0000
0101
0000
0101
1
L
EAK
1010
0000
1010
0000
2
L
EAK
3536.03536.03536.03536.0
3536.03536.03536.03536.0
3536.03536.03536.03536.0
3536.03536.03536.03536.0
3
L
EAK
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Assembly Procedure
3536.13536.0103536.03536.0
3536.03536.0003536.03536.0
101000
000101
3536.03536.0003536.03536.0
3536.03536.0013536.03536.1
L
EAK
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Global Force Vector
P
P
F
F
F
F
F
F
F
F
F
F
Fy
x
y
x
y
x
y
x
y
x
2
2
2
1
1
3
3
2
2
1
1
Remember!
NO distributed load is applied to a truss
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Boundary Conditions
02211 VUVU
Remove the corresponding rows and columns
P
P
V
U
L
EA
23536.13536.0
3536.03536.0
3
3
Continue! (as before)
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Results
EA
PLV
EA
PLU
3 ,828.5 33
PFF
PFPF
yx
yx
3 ,0
, ,
22
11
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Postcomputation
e
e
e
e
e
A
P
A
P 21
e
e
eee
e
u
uL
EA
P
P
2
1
2
1
11
11
2
2
1
1
2
2
1
1
00
00
00
00
v
u
v
u
CosSin
SinCos
CosSin
SinCos
v
u
v
u
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Postcomputation
A
P
A
P2 ,
3 ,0 )3()2()1(
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Summary
• In this lecture we learned how to apply the finite element modeling technique to bar problems with general orientation in a plain.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Homework #5
• Problem 4.27, – Due 13/12/2006 before 9:00am
• Problem 4.44,– Due 20/12/2006 before 9:00am
Introduction to the Finite Element MethodDr. 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)
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Term Projects
• A problem has got to be solved using the finite element method
• A report is going to be presented by each group presenting the problem and its solution
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Evaluation
• Report (50%)
• Code (30%)– Structured: Functions built, easily modified– Readability: Organization, remarks– Length: The shorter the better
• Results (20%)
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Heat transfer in a 2-D heat sink
• The heat sink will have heat flowing from one side
• Convection transfer on the surfaces
• Different boundary conditions on the other three sides
• Plot contours of temperature distribution with different boundary conditions
Introduction to the Finite Element MethodDr. Mohammad Tawfik
2-D flow around a blunt body in a wind tunnel
• Potential flow in a duct
• Rectangular body with different Dimensions
• Study the effect of the body size on the flow speed on both sides
• Plot contours of potential function, pressure, and velocity potential
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Vibration characteristics of a pipe with internal fluid flow
• Study the change of the natural frequencies with the flow speed under different boundary conditions and fluid density
• Indicate the flow speeds at which instabilities occur
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Panel flutter of a beam
• A fixed-fixed beam is subjected to flow over its surface
• Plot the effect of the flow speed on the natural frequencies of the beam
• Indicate the speed at which instability occurs
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Rotating Timoshenko beam/blade
• Rotating beams undergo centrifugal tension that results in the change of its natural frequencies
• Study the effect of rotation speed on the beam natural frequencies and frequency response to excitations at the root
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Teams
• 2-3 Students teams
• Names and selected projects should be submitted before 4PM on Thursday 21/12/2006
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beams and Frames
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beams and Frames
• Beams are the most-used structural elements.
• Many real structures may be approximated as beam elements
• Two main beam theories:– Euler-Bernoulli beam theory– Timoshenko beam theory
Introduction to the Finite Element MethodDr. 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 the analysis
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Governing Equation
• The equation governing the deformation of and E-B beam under transverse loading may be written in the form:
)(2
2
2
2
xFdx
wdxEI
dx
d
Introduction to the Finite Element MethodDr. 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.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beam Interpolation Function
axHxw
34
2321)( xaxaxaaxw
axxxxw 321
axxaxHadx
xdH
dx
xdwx
23210
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beam Interpolation Function
aHww 00 1
aT
a
a
a
a
lH
lH
H
H
w
w
w
w
x
x
4
3
2
1
2
2
1
1
0
0
'
'
aHww x 0'0' 1
alHwlw 2
alHwlw x 2''
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beam Interpolation Function
4
3
2
1
2
32
2
2
1
1
3210
1
0010
0001
'
'
a
a
a
a
ll
lll
w
w
w
w
2
2
1
1
2323
22
4
3
2
1
'
'
1212
13230010
0001
w
w
w
w
llll
llll
a
a
a
a
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beam Interpolation Function
ee wxNwTxHaxHxw 1
ewTa 1
4
1iii wxNxw
Introduction to the Finite Element MethodDr. 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
xl
x
l
xx
l
x
l
x
xNxN T
Introduction to the Finite Element MethodDr. 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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beam Stiffness Matrix
• The governing equation is:
• Using the series solution
)(2
2
2
2
xFdx
wdxEI
dx
d
4
1iii wxNxw
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beam Stiffness Matrix
• The governing equation becomes
• Applying Galerkin method:
)()(4
12
2
2
2
xRxFwdx
NdxEI
dx
d
ii
i
ee l
ji
ii
l
j dxNxFwdx
NdxEI
dx
ddxNxR
0
4
12
2
2
2
0
)()(
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Beam Stiffness Matrix
• Using integration by parts, twice, and ignoring the boundary terms, we get:
• In matrix form:
0)(0
4
12
2
2
2
el
ji
iji dxNxFw
dx
Nd
dx
NdxEI
ee l
xxe
l
xxxx dxNxFwdxNNxEI00
)(
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Use of Symbolic Manipulator
Beam Example
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Optional Homework #6
• Derive the expression for the interpolation function for a beam in terms of nodal displacements and slopes.
• Try to use a symbolic manipulator to generate the expressions.
)(4
4
2
2
xFdx
wdEI
dt
wdA
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Two Dimensional Elements
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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.
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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,
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Let’s follow the same procedure!
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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
100
1
0011
0001
f
f
f
f
abababab
bb
aa
a
a
a
a
Introduction to the Finite Element MethodDr. Mohammad Tawfik
2-D Interpolation Function efyxNayxHyxf ,,,
ab
xy
b
yab
xyab
xy
a
xab
xy
b
y
a
x
yxNyxN T
1
,,
Introduction to the Finite Element MethodDr. Mohammad Tawfik
How does this look like?
Introduction to the Finite Element MethodDr. Mohammad Tawfik
2-D Interpolation Functions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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 0.5 0.6 0.7 0.8 0.9 10
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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
2-D Interpolation Functions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
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 0.5 0.6 0.7 0.8 0.9 10
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
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Example: Laplace Equation
02
02
2
2
2
yx
ei
ii yxNyxN ,,4
1
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Example: Laplace Equation
ei
ii yxNyxN ,,4
1
0 e
Area
yyxx dANNNN
Applying the Galerkin method and integrating by parts, the element equation becomes
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The Element Equaiton
0
222
222
222
222
6
1
22222222
22222222
22222222
22222222
e
babababa
babababa
babababa
babababa
ab
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The Logistic Problem!
Introduction to the Finite Element MethodDr. Mohammad Tawfik
The Logistic Problem
• In the 2-D problems, the numbering scheme, usually, is not as straight forward as the 1-D problem
Introduction to the Finite Element MethodDr. Mohammad Tawfik
1-D Example
• Element #1 is associated with nodes 1&2• Element #2 is associated with nodes 2&3, etc…
Introduction to the Finite Element MethodDr. Mohammad Tawfik
2-D Example
Introduction to the Finite Element MethodDr. Mohammad Tawfik
2-D Example
Introduction to the Finite Element MethodDr. Mohammad Tawfik
For Element #5
Local Node NumberGlobal Node Number
15
26
39
48
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Contribution of element #5 to global matrix
123456789101112
1
2
3
4
51,11,21,41,3
62,12,22,42,3
7
84,14,24,44,3
93,13,23,43,3
10
11
12
Introduction to the Finite Element MethodDr. Mohammad Tawfik
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!
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Elements Register: Global Numbering
Element Number
Node Number
1234
11254
24587
3781110
42365
55698
6891211
Introduction to the Finite Element MethodDr. Mohammad Tawfik
Algorithm for Assembling Global Matrix
1. Create a square matrix “A”; N*N (N=Number of nodes)
2. For the ith element3. Get the element matrix “B”4. For the jth node5. Get its global number k6. For the mth node7. Get its global number n
8. Let Akn=Akn+Bjm
9. Repeat for all m10. Repeat for all j11. Repeat for all i
Element Number
Node Number
1234
11254
24587
3781110
42365
55698
6891211
123456789101112
1
2
3
4
5
6
7
8
9
10
11
12