finite element analysis introduction _pres.ppt

fem ppt

1. Presented by
Niko Manopulo
An Introduction to the
Finite Element Analysis

2. Joint Advanced Student School
St.Petersburg 2005
Agenda
PART I
Introduction and Basic Concepts
1.0 Computational Methods
1.1 Idealization
1.2 Discretization
1.3 Solution
2.0 The Finite Elements Method
2.1 FEM Notation
2.2 Element Types
3.0 Mechanichal Approach
3.1 The Problem Setup
3.2 Strain Energy
3.3 External Energy
3.4 The Potential Energy Functional

3. Joint Advanced Student School
St.Petersburg 2005
Agenda
PART II
Mathematical Formulation
4.0 The Mathematics Behind the Method
4.1 Weighted Residual Methods
4.2 Approxiamting Functions
4.3 The Residual
4.4 Galerkin’s Method
4.5 The Weak Form
4.6 Solution Space
4.7 Linear System of Equations
4.8 Connection to the physical system

4. Joint Advanced Student School
St.Petersburg 2005
Agenda
PART III
Finite Element Discretization
5.0 Finite Element Discretization
4.1 The Trial Basis
4.2 Matrix Form of the Problem
4.3 Element Stiffness Matrix
4.4 Element Mass Matrix
4.5 External Work Integral
4.6 Assembling
4.7 Linear System of Equations
6.0 References
7.0 Question and Answers

5. PART I
Introduction and Basic Concepts

6. Joint Advanced Student School
St.Petersburg 2005
1.0 Computational Methods

7. Joint Advanced Student School
St.Petersburg 2005
1.1 Idealization
•Mathematical Models
•“A model is a symbolic device built to
simulate and predict aspects of behavior of
a system.”
•Abstraction of physical reality
•Implicit vs. Explicit Modelling
•Implicit modelling consists of using existent
pieces of abstraction and fitting them into
the particular situation (e.g. Using general
purpose FEM programs)
•Explicit modelling consists of building the
model from scratch

8. Joint Advanced Student School
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1.2 Dicretization
1.Finite Difference Discretization
•The solution is discretized
•Stability Problems
•Loss of physical meaning
2.Finite Element Discretization
•The problem is discretized
•Physical meaning is conserved on elements
•Interpretation and Control is easier

9. Joint Advanced Student School
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1.3 Solution
1.Linear System Solution Algorithms
•Gaussian Elimination
•Fast Fourier Transform
•Relaxation Techniques
2.Error Estimation and Convergence
Analysis

10. Joint Advanced Student School
St.Petersburg 2005
2.0 Finite Element Method
•Two interpretations
1.Physical Interpretation:
The continous physical model is divided
into finite pieces called elements and laws
of nature are applied on the generic
element. The results are then recombined
to represent the continuum.
2.Mathematical Interpretation:
The differetional equation reppresenting the
system is converted into a variational form,
which is approximated by the linear
combination of a finite set of trial functions.

11. Joint Advanced Student School
St.Petersburg 2005
2.1 FEM Notation
Elements are defined by the following
properties:
1.Dimensionality
2.Nodal Points
3.Geometry
4.Degrees of Freedom
5.Nodal Forces
(Non homogeneous RHS of the DE)

12. Joint Advanced Student School
St.Petersburg 2005
2.2 Element Types

13. Joint Advanced Student School
St.Petersburg 2005
3.0 Mechanical Approach
•Simple mechanical problem
•Introduction of basic mechanical concepts
•Introduction of governing equations
•Mechanical concepts used in mathematical
derivation

14. Joint Advanced Student School
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3.1 The Problem Setup

15. Joint Advanced Student School
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•Hooke’s Law:
where
•Strain Energy Density:
3.2 Strain Energydx
du
x)( )()( xEx )()(
2
1
xx

16. Joint Advanced Student School
St.Petersburg 2005
•Integrating over the Volume of the Bar:
•All quantities may depend on x.
3.2 Strain Energy (cont’d)dxEAuuU
dxuEAudxpdVU
L
LL
V




0
00
''
2
1
')'(
2
1
2
1
2
1


17. Joint Advanced Student School
St.Petersburg 2005
3.3 External Energy
•Due to applied external loads
1.The distributed load q(x)
2.The point end load P. This can be
included in q.
•External Energy:

L
dxquW
0

18. Joint Advanced Student School
St.Petersburg 2005
3.4 The Total Potential
Energy Functional
•The unknown strain Function u is found by
minimizing the TPE functional described below:)]([)]([)]([ xuWxuUxu
or
WU



19. PART II
Mathematical Formulation

20. Joint Advanced Student School
St.Petersburg 2005
4.0 Historical Background
•Hrennikof and McHenry formulated a 2D
structural problem as an assembly of bars
and beams
•Courant used a variational formulation to
approximate PDE’s by linear interpolation
over triangular elements
•Turner wrote a seminal paper on how to
solve one and two dimensional problems
using structural elements or triangular and
rectangular elements of continuum.

21. Joint Advanced Student School
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4.1 Weighted Residual Methods
The class of differential equations containing also the one
dimensional bar described above can be described as
follows :.0)1()0(
)1(10),()())((][


uu
xxquxz
dx
du
xp
dx
d
uL

22. Joint Advanced Student School
St.Petersburg 2005
It follows that:
Multiplying this by a weight function v and integrating over
the whole domain we obtain:
For the inner product to exist v must be “square integrable”
Therefore:
Equation (2) is called variational form0][ quL
4.1 Weighted Residual Methods)2(0)][,()][(
1
0
 quLvdxvquL )1,0(
2
Lv

23. Joint Advanced Student School
St.Petersburg 2005
4.2 Approximating Function
•We can replace uand vin the formula with their approximation
function i.e.
•The functionsf
jandy
j are of our choice and are meant to be
suitable to the particular problem. For example the choice of
sine and cosine functions satisfy boundary conditions hence it
could be a good choice.





N
j
jj
N
j
jj
xdxVxv
xcxUxu
1
1
)()()(
)()()(
y
f

24. Joint Advanced Student School
St.Petersburg 2005
4.2 Approximating Function
•U is called trial functionand V is called test function
•As the differential operator L[u] is second order
•Therefore we can see U as element of a finite-diemnsional
subspace of the infinite-dimensional function space C
2
(0,1)
•The same way)1,0()1,0(
22
CUCu  )1,0()1,0(
2
CSU
N
 )1,0()1,0(
ˆ
2
LSV
N


25. Joint Advanced Student School
St.Petersburg 2005
4.3 The Residual
•Replacing vand uwith respectively Vand U(2) becomes
•r(x) is called the residual (as the name of the method suggests)
•The vanishing inner product shows that the residual is orthogonal to
all functions V in the test space.qULxr
SVrV
N


][)(
ˆ
,0),(

26. Joint Advanced Student School
St.Petersburg 2005
•Substituting into
and exchanging summations and integrals we obtain
•As the inner product equation is satisfied for all choices of V in
S
N
the above equation has to be valid for all choices of d
j
which implies that
4.3 The Residual


N
j
jjxdxV
1
)()( y 0)][,( qULV NjdqULd
j
N
j
jj ,...,2,1,0)][,(
1


y NjqUL
j
,...,2,10)][,( y

27. Joint Advanced Student School
St.Petersburg 2005
•One obvious choice of would be taking it equal to
•This Choice leads to the Galerkin’s Method
•This form of the problem is called the strong form of the
problem. Because the so chosen test space has more
continuity than necessary.
•Therefore it is worthwile for this and other reasons to convert
the problem into a more symmetrical form
•This can be acheived by integrating by parts the initial strong
form of the problem.
4.4 Galerkin’s Methodj
y j
f NjqUL
j
,...,2,10)][,( f

28. Joint Advanced Student School
St.Petersburg 2005
•Let us remember the initial form of the problem
•Integrating by parts
4.5 The Weak FormdxqzupuvquLv 

1
0
])''([)][,( .0)1()0(
10),()())((][


uu
xxquxz
dx
du
xp
dx
d
uL 0')''(])''([
1
0
1
0
1
0
 
vpudxvqvzupuvdxqzupuv

29. Joint Advanced Student School
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4.5 The Weak Form
•The problem can be rewritten as
where
•The integration by parts eliminated the second derivatives
from the problem making it possible less continouity than the
previous form. This is why this form is called weak formof the
problem.
•A(v,u)is called Strain Energy.0),(),( qvuvA dxvzupuvuvA )''(),(
1
0



30. Joint Advanced Student School
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4.6 Solution Space
•Now that derivative of v comes into the picture v needs to have more
continoutiy than those in L
2
. As we want to keep symmetry its
appropriate to choose functions that produce bounded values of
•As p and z are necessarily smooth functions the following restriction is
sufficient
•Functions obeying this rule belong to the so called Sobolev Space
and they are denoted by H
1
. We require v and u to satisfy boundary
conditions so we denote the resulting space asdxzuupuuA ))'((),(
1
0
22

 1
0
1
0
22
)'(
H
dxuu


31. Joint Advanced Student School
St.Petersburg 2005
•The solution now takes the form
•Substituting the approximate solutions obtained earlier in the
more general WRM we obtain
•More explicitly substituting U and V (remember we chose
them to have the same base) and swapping summations and
integrals we obtain
4.7 Linear System of Equations1
0),(),( HvqvuvA  N
N
SVqVUVA
HSVU
0
1
00
),(),(
,

 


N
k
jkjk
NjqAc
1
,...,2,1,),(),( fff

32. Joint Advanced Student School
St.Petersburg 2005
4.8 Connection to the Physical System
Mechanical Formulation Mathematical FormulationdxEAuuU
L


0
''
2
1 

L
dxquW
0 0 WU dxvzupuvuvA )''(),(
1
0

 0),(),( qvuvA ),(qv

33. PART III
Finite Element Discretization

34. Joint Advanced Student School
St.Petersburg 2005
•Let us take the initial value problem with constant coefficients
•As a first step let us divide the domain in N subintervals with
the following mesh
•Each subinterval is called finite element.
5.0 Finite Element Discretization Njxx
jj
:1),,(
1

 1...0
10 
Nxxx .0)1()0(
0,
10),(''



uu
zp
xxqzupu

35. Joint Advanced Student School
St.Petersburg 2005
•Next we select as a basis the so called “hat function”.
5.1 The Trial Basis





















otherwise
xxx
xx
xx
xxx
xx
xx
x
jj
jj
j
jj
jj
j
j
0
)(
1
1
1
1
1
1
f

36. Joint Advanced Student School
St.Petersburg 2005
•With the basis in the previous slide we construct our
approximate solution U(x)
•It is interesting to note that the coefficients correspond to the
values of U at the interior nodes
5.1 The Trial Basis

37. Joint Advanced Student School
St.Petersburg 2005
•The problem at this point can be easily solved using the
previously derived Galerkin’s Method
•A little more work is needed to convert this problem into
matrix notation
5.1 The Trial Basis


N
k
jkjk
NjqAc
1
,...,2,1,),(),( fff

38. Joint Advanced Student School
St.Petersburg 2005
•Restricting U over the typical finite element we can write
•Which in turn can be written as
in the same way
5.2 Matrix Form of the Problem],[)]()([
)(
)(
][)(
1
1
1
1
1 jj
j
j
jj
j
j
jj xxx
c
c
xx
x
x
ccxU




 











 ff
f
f ],[)()()(
111 jjjjjj
xxxxcxcxU

 ff ],[)]()([
)(
)(
][)(
1
1
1
1
1 jj
j
j
jj
j
j
jj xxx
d
d
xx
x
x
ddxV




 











 ff
f
f

39. Joint Advanced Student School
St.Petersburg 2005
•Taking the derivative
•Derivative of V is analogus
5.2 Matrix Form of the Problem1
1
1
1 ],[]/1/1[
/1
/1
][)('


















jjj
jj
j
j
jj
j
j
jj
xxh
xxx
c
c
hh
h
h
ccxU ],[]/1/1[
/1
/1
][)('
1
1
1 jj
j
j
jj
j
j
jj xxx
d
d
hh
h
h
ddxV


 













40. Joint Advanced Student School
St.Petersburg 2005
•The variational formula can be elementwise defined as
follows:
5.2 Matrix Form of the Problem












j
j
j
j
j
j
x
x
x
x
M
j
x
x
S
j
M
j
S
jj
N
j
jj
dxVqqV
dxzVUUVA
dxUpVUVA
UVAUVAUVA
qVUVA
1
1
1
),(
),(
''),(
),(),(),(
0]),(),([
1

41. Joint Advanced Student School
St.Petersburg 2005
•Substituting U,V,U’ and V’ into these formulae we obtain
4.3 The Element Stiffnes Matrix




































































11
11
][
11
11
][),(
]/1/1[
/1
/1
][),(
1
1
1
21
1
1
1
1
j
j
j
j
jjj
j
j
x
x
j
jj
S
j
j
j
x
x
jj
j
j
jj
S
j
h
p
K
c
c
Kdd
c
c
dx
h
p
ddUVA
c
c
dxhh
h
h
pddUVA
j
j
j
j

42. Joint Advanced Student School
St.Petersburg 2005
4.4 The Element Mass Matrix
•The same way









































21
12
6
][),(
][][),(
1
1
1
1
1
1
1
j
j
j
j
jjj
S
j
j
j
x
x
jj
j
j
jj
M
j
zh
M
c
c
MddUVA
c
c
dxzddUVA
j
j
ff
f
f

43. Joint Advanced Student School
St.Petersburg 2005
•The external work integral cannot be evaluated for every
function q(x)
•We can consider a linear interpolant of q(x) for simplicity.
•Substituting and evaluating the integral
4.5 External Work
Integral


j
j
x
x
dxVqqV
1
),( ],[),()()(
111 jjjjjj
xxxxqxqxq

 ff 





























jj
jjj
j
jjj
j
j
jj
x
x
j
j
jjj
qq
qqh
l
ldddx
q
q
ddqV
j
j
2
2
6
],[],[],[),(
1
1
1
1
1
1
1
1
ff
f
f
Element load vector

44. Joint Advanced Student School
St.Petersburg 2005
•Now the task is to assemble the elements into the whole
system in fact we have to sum each integral over all the
elements
•For doing so we can extend the dimension of each element
matrix to N and then put the 2x2 matrix at the appropriate
position inside it
•With all matrices and vectors having the same dimension the
summation looks like
4.6 Assembling


N
j
S
jA
1
Kcd
T 

























21
121
.........
121
121
12
h
p
K 

























 1
2
1
1
2
1
......
NN d
d
d
d
c
c
c
c

45. Joint Advanced Student School
St.Petersburg 2005
•Doing the same for the Mass Matrix and for the Load Vector
4.6 Assembling


N
j
M
jA
1
Mcd
T 




















41
141
.........
141
141
14
6
zh
M 


N
j
jqV
1
),( ld
T 















 NNN qqq
qqq
qqq
h
12
321
210
4
...
4
4
6
l

46. Joint Advanced Student School
St.Petersburg 2005
•Substituting this Matrix form of the expressions in
we obtain the following set of linear equations
•This has to be satisfied for all choices of d therefore
4.7 Linear System of Equations0lcMKd
T
 ])[( 


N
j
jj qVUVA
1
0]),(),([ 0lcMK )(

47. Joint Advanced Student School
St.Petersburg 2005
References
•Carlos Felippa
http://caswww.colorado.edu/courses.d/IFEM.d/IFE
M.Ch06.d/IFEM.Ch06.pdf
•Joseph E Flaherty,Amos Eaton Professor
http://www.cs.rpi.edu/~flaherje/FEM/fem1.ps
•Gilbert Strang, George J. Fix
An Analysis of the Finite Element Method
Prentice-Hall,1973