8/2/2019 San Geeta 1
1/37
FINITE ELEMENT METHOD
8/2/2019 San Geeta 1
2/37
Objective
The purpose of the present course is to describe the use of
Finite Element method for the solution of problems of fluid
flow or seepage through porous media.
8/2/2019 San Geeta 1
3/37
What is FEM
Powerful computational technique for the solution ofdifferential equtions.
too complicated to be solved satisfactorily by classical
analytical methods.
Problems-Solid mechanics, Heat transfer, fluid mechanics,acoustics, electromagnetism, coupled interaction of thesephenomena
Commercial packages
8/2/2019 San Geeta 1
4/37
Course coverage
Finite element Basic concepts
Formulation of Finite elements
Steps in the finite element method
Examples
8/2/2019 San Geeta 1
5/37
WHY FEM ?
8/2/2019 San Geeta 1
6/37
Simple Example
Simple 1-D problem
fdx
du
dx
d=
0 1x on
Boundary conditions
0)0( uu = (1)du qdx
=
( Dirichlet ) ( Neumann )
:
0 :
1 :
Strong form
8/2/2019 San Geeta 1
7/37
Analytical solution
2
02
q f fu u x x
= + +
d duf
dx dx =
Integrating both sides
2
1 22
f xu C x C = + +
Imposing the boundary conditions we get
1 2 0C q f C u= =
8/2/2019 San Geeta 1
8/37
Finite difference formulation
is approximated in the mid-point of the intervaldx
du
1
1/2
l l
l
u udu
dx x
+
+
1
1/2
l l
l
u udu
dx x
1/2 1 1/2 1
2
( ) ( )l l l l l l
l
u u u uf
x
+ + =
Finite difference approximation
llll f
x
uuu=
+ +
2
11 2
For constant
1..l L=
0 0x = 1x lx 1Lx =1lx +x
1 1( , ) ( , )
l l l lx x and x x by+
8/2/2019 San Geeta 1
9/37
Finite difference formulation
Neumann boundary condition
1/2 1 1/2 1( ) ( )1 1
2 2
L L L L L Lu u u u
qx x
+ + + =
Discretisation at the end l=L
1/2 1 1/2 1
2
( ) ( )L L L L L L
L
u u u uf
x
+ + =
Combining
11/2
1
2
L LL L
u uq f x
x
=
8/2/2019 San Geeta 1
10/37
Finite difference formulation
Discrete problem-algebraic equations
KU F=
2 1
1 2 1
...
1 2 11 1
K
=
2
0 1
2
2
2
1
2
2
L
xu f
xf
F
x
f
x xq f
=
M
where,
1
2
1L
L
u
u
U
uu
=
M
8/2/2019 San Geeta 1
11/37
Finite difference -solution
2
01
221
2 1
1
2
xu f
u
u x xq f
=
For a three noded mesh
1u
0u 1u 2u
8/2/2019 San Geeta 1
12/37
Finite difference formulation
Grid is structured .
Equation is discretized.
Proper treatment of boundary conditions is vitally important.
Gives solution at only discrete points in the domain grid
points.
8/2/2019 San Geeta 1
13/37
FEM- Building blocks
Form of the approximate solution is assumed apriori
Approximate solution0
0 1
N N
k k k k
k k
u u u = =
= = +
on substitution leaves a residual
d dur f
dx dx
=
Optimizing criterion- the weighted integral of the residual is zero.
0lw r= 1, 2...,l n=
8/2/2019 San Geeta 1
14/37
Trial solution procedure
Trial solution u
uis undetermined
Optimizing criterion
Determines best values of ui
Approximate solution
Accuracy
Accuracy unacceptable repeat cycle with a different trial function
8/2/2019 San Geeta 1
15/37
Galerkins weighted residual method
weight functions are the basis functions
The optimization criteria transforms the original differential
equation into a set of algebraic equations.
Can be solved to determined the undetermined parameters
Can be applied to any differential equation
l lw =
8/2/2019 San Geeta 1
16/37
Galerkin method
2
0 1 2( )u x u u x u x= + +
Approximating function of the form
are to be determined
Imposing the boundary conditions
'i
u s
0 0 1 22q
u u u u
= =
2
0 2 2
2
0 2 2
( ) 2
2
qu x u u x u x
qu x u x u x
= + +
= + +
8/2/2019 San Geeta 1
17/37
Galerkins method
xx =)(12
2 )(xx
=
2( : ) 2R x u u f=
2
02
q f fu u x x
= + +
From the weighted residual statements we get 2 2u
f
=
2
0 2 2( ) 2
qu x u x u x u x
= + +
0 0
q
u x = +
8/2/2019 San Geeta 1
18/37
Galerkin method-limitations
Approximating field is defined over the entire domain.
Major disadvantage is the construction of approximation
functions that satisfy the boundary conditions of the problem
Approximating fields must be admissible and easy to use.
Only polynomials and sine and cosine functions are simple
enough to be practicable.
The undetermined parameters have no physical meaning.iu
8/2/2019 San Geeta 1
19/37
Finite Element Method
Approximating field is defined in a piecewise fashion bydividing the entire region over subregions
Undetermined parameters are the nodal values of the field
The approximation functions can be generated systematically
over these subregions
FEM is the piecewise (or elementwise )application of the
weighted residual method.
We get different finite element approximations depending on
the choice of the weighted residual method.
iu
8/2/2019 San Geeta 1
20/37
Steps in the finite element method
Discretization of the domain into a set of finite elements.
Defining an approximate solution over the element.
Weighted integral formulation of the differential equation.
Substitute the approximate solution and get the algebraicequation
?
8/2/2019 San Geeta 1
21/37
Steps in the finite element method
1. Discretization of the domain into a set of finite elements (mesh generation).--- the domain is subdivided into non-overlapping subdomains e calledelements of simple geometrical form.
---for this 1-D problem an obvious choice for an element e is the interval
1e e
x x x
0 0x = 1x 1ex 1Lx =exex
8/2/2019 San Geeta 1
22/37
Steps in the finite element method
Step 2: To set up a weak formulation of the differential equation.
(i) Multiply the equation by a weight function and integrate the equation over thedomain e.
0e
e
l
d duw f dx
dx dx
=
(ii) Move the differentiation to the weight function by by doing integration by parts
1
0
e
e e
e
xe e
ll l
x
dwdu duw dx w fdx
dx dx dx
=
1
e
e e
e
xe e
ll l
x
dw du dudx w fdx wdx dx dx
= + Weak Form
8/2/2019 San Geeta 1
23/37
Steps in the finite element method
1
1
e e
e e
x x
du duq qdx dx
= =
1
1 1( ). ( ).
e
e
xe
l l e e l e e
x
duw w x q w x q
dx
=
Weak Form
1 1( ). ( ).e e
e
ll l e e l e e
dw dudx w f dx w x q w x q
dx dx
= +
Equivalent to both the governing differential equation and the associated natural boundary condition
8/2/2019 San Geeta 1
24/37
Steps in the development of weak form
Multiply the equation by a weight function and integrate the
equation over the domain e.
Distribute the differentiation among the weight function w
and ue
by doing integration by parts
Use the definition of the natural boundary condition in the
weak form
ew
8/2/2019 San Geeta 1
25/37
Steps in the finite element method
Step 3:FEM model from the weak form
Approximation over a finite elemente
2
1
( )e e e
j j
j
u x u =
=
1
e e
e
x xx
=
where
12
e e
e
x xx
=
From the weak form
l lw =
2
1 1
1
( ). ( ).e e
e
jl e
ele l e ej l
j
ddw u d w x q w x qx w fdxdx dx
=
= +
8/2/2019 San Geeta 1
26/37
Steps in the finite element method
The choice 1 1 2 2e e
w and w gives = =2
11
1
1 1 1 1( ) ( )e e
eeje e
j
e e
e e e e
j
ddu dx x q x qfdx
dx dx
=
= +
Now,
22
2
1
2 2 1 1( ) ( )e e
eeje e
j
e e
e e e e
j
ddu dx x q x qfdxdx dx
=
= +
1 1 1 1 1 1( ) ( )e e e
e e e e eq x q q q = =
2 2 1 1 2( ) ( )e e e
e e e e ex q x q q q = =
8/2/2019 San Geeta 1
27/37
Steps in the finite element method
The choice 1 1 2 2e e
w and w gives = =2
11 1
1e e
eeje e e
j
j
ddu dx fdx q
dx dx
=
= +
e e e e
ij j i iK u f q= +
which takes the form
e
eeje i
ij
ddK
dx dx
= ee ei if fdx=
1
2
ee
i e
q
=
22
2 21
e e
eeje e e
jj
dd
u dx fdx qdx dx
=
= +
8/2/2019 San Geeta 1
28/37
Steps in the finite element method
The element equations are
1 1
2 2
1 1 2
1 12
ee e
e eee
xf
u q
xx u qf
= +
Diagram here
2
e
u
1
1
eu
2
2
eq1eq
8/2/2019 San Geeta 1
29/37
Results
Draw a diagram for assembly here
1 1
1 1
1 1
2 2
21 1 0
1 1 1 02
0 0 0 0 00
e
e
e
x f
u qx
forelement u f qx
= +
First we rewrite the element matrices in terms of the global numbering scheme
2
1
2
2
2
1
2
2
00 0 0 0
2 0 1 1
1
2
0
20 1
e
e
e
xfor element f q
xu
xf
u
q
= +
1
1u2
1u12u
2
2u
1 2 3
1 2 2 3
1 2
1 2
8/2/2019 San Geeta 1
30/37
Steps in the finite element method
Step 4:Assemble the element equations to obtain the globalsystem
1 1
1 1 1
1 2
2 2 1
2 2
3 2 2
1 1 0 2
1 2 1 0
0 1 1
e
e
e
e
x fu q q
u x f q qx
u x q qf
= + + =
8/2/2019 San Geeta 1
31/37
Steps in the finite element method
STEP 5:Imposition of boundary conditions
Dirichlet condition
1, , , 0
s s
ss s s i i is s is si
u u
k f u f f k u k k
=
= = = = =
Neumann condition
Replace the corresponding component of the right hand column bythe specified value.
8/2/2019 San Geeta 1
32/37
Steps in the finite element method
STEP 5:Imposition of boundary conditions
( )
( )
( )
2
121 1
2 0
23
21 0 0
0 2 1 0
0 1 1 0
2
e
e
e
e
xf
u qx
u f u
q xu
xf
= + +
+
8/2/2019 San Geeta 1
33/37
Steps in the finite element method
Step 6: Solution of the algebraic system of equations
This is a standard matrix equation can be solved by direct or
indirect(iterative) method.
Step 7: Postprocessing
This final operation displays the solution to system equations in
tabular graphical or pictorial form. Other meaningful quantities
may be derived from the solution and also displayed
8/2/2019 San Geeta 1
34/37
Results
1.51.51.5u3(@ x=1.0)
0.6250.6250.625u2
(@ x=0.5)
000
u1
(@ x=0)
Finite
Difference
Finite
ElementExact
8/2/2019 San Geeta 1
35/37
Results
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1 1.2
spatial co-ordinate, x
displacement,u
Exact Solution
Finite Element solution
Finite Element solution
8/2/2019 San Geeta 1
36/37
FEM and FDM
FDM is a direct method- replacesdifferential equations by a differenceapproximations
Based on rectangular discretisation
Treatment of boundary conditions
needs extra care
Making higher order approximation istedious
FEM is an indirect method- works on aweak form
Arbitrary geometry can be modelled
(can be rectangular, quadrilateral,
triangle)
Boundary conditions is more
systematic
Making higher order approximation iseasy
8/2/2019 San Geeta 1
37/37
FEM and FDM
No closed form solution that permits analytical study of the
effects of changing various parameters.
Good engineering judgement is required
- type of element- type of difference approximation
Many input data are required and voluminous output must
be sorted and understood
Top Related