1 Finite Element Method THE FINITE ELEMENT METHOD for readers of all backgrounds G. R. Liu and S. S....
-
Upload
brandon-kent -
Category
Documents
-
view
234 -
download
4
Transcript of 1 Finite Element Method THE FINITE ELEMENT METHOD for readers of all backgrounds G. R. Liu and S. S....
1
FFinite Element Methodinite Element Method
THE FINITE ELEMENT
METHOD
for readers of all backgroundsfor readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 3:
2Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS STRONG AND WEAK FORMS OF GOVERNING EQUATIONS HAMILTON’S PRINCIPLE FEM PROCEDURE
– Domain discretization– Displacement interpolation– Formation of FE equation in local coordinate system– Coordinate transformation– Assembly of FE equations– Imposition of displacement constraints– Solving the FE equations
STATIC ANALYSIS EIGENVALUE ANALYSIS TRANSIENT ANALYSIS REMARKS
3Finite Element Method by G. R. Liu and S. S. Quek
STRONG AND WEAK STRONG AND WEAK FORMS OF GOVERNING FORMS OF GOVERNING
EQUATIONSEQUATIONS System equations: strong form, difficult to solve. Weak form: requires weaker continuity on the
dependent variables (u, v, w in this case). Weak form is often preferred for obtaining an
approximated solution. Formulation based on a weak form leads to a set of
algebraic system equations – FEM. FEM can be applied for practical problems with
complex geometry and boundary conditions.
4Finite Element Method by G. R. Liu and S. S. Quek
HAMILTON’S PRINCIPLEHAMILTON’S PRINCIPLE
“Of all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional a minimum.”
An admissible displacement must satisfy:– The compatibility equations– The essential or the kinematic boundary conditions
– The conditions at initial (t1) and final time (t2)
5Finite Element Method by G. R. Liu and S. S. Quek
HAMILTON’S PRINCIPLEHAMILTON’S PRINCIPLE
Mathematically
02
1 dtL
t
t
where L=T+Wf
VUUT T
V
d2
1
VcVΠ T
V
T
V
dd εε2
1σε
2
1
fsT
Sb
T
Vf SfUVfUW
f
dd
(Kinetic energy)
(Potential energy)
(Work done by external forces)
6Finite Element Method by G. R. Liu and S. S. Quek
FEM PROCEDUREFEM PROCEDURE
Step 1: Domain discretization Step 2: Displacement interpolation Step 3: Formation of FE equation in local coordinate
system Step 4: Coordinate transformation Step 5: Assembly of FE equations Step 6: Imposition of displacement constraints Step 7: Solving the FE equations
7Finite Element Method by G. R. Liu and S. S. Quek
Step 1: Domain discretizationStep 1: Domain discretization
The solid body is divided into Ne elements with proper connectivity – compatibility.
All the elements form the entire domain of the problem without any overlapping – compatibility.
There can be different types of element with different number of nodes.
The density of the mesh depends upon the accuracy requirement of the analysis.
The mesh is usually not uniform, and a finer mesh is often used in the area where the displacement gradient is larger.
8Finite Element Method by G. R. Liu and S. S. Quek
Step 2: Displacement interpolationStep 2: Displacement interpolation
Bases on local coordinate system, the displacement within element is interpolated using nodal displacements.
eii
n
i
zyxzyxzyxd
dNdNU ),,( ),,(),,(1
1
2
displacement compenent 1
displacement compenent 2
displacement compenent f
i
n f
d
d
d n
d
1
2
displacements at node 1
displacements at node 2
displacements at node d
e
n dn
d
dd
d
9Finite Element Method by G. R. Liu and S. S. Quek
Step 2: Displacement interpolationStep 2: Displacement interpolation
N is a matrix of shape functions
1 2( , , ) ( , , ) ( , , ) ( , , )
for node 1 for node 2 for node
dn
d
x y z x y z x y z x y z
n
N N N N
fin
i
i
i
N
N
N
000
000
000
000
2
1
Nwhere
Shape function for each displacement component at a node
10Finite Element Method by G. R. Liu and S. S. Quek
Displacement interpolationDisplacement interpolation
Constructing shape functions– Consider constructing shape function for
a single displacement component– Approximate in the form
1
( ) ( ) ( )dn
hi i
i
Tu p
x x p x α
1 2 3 ={ , , , ......, }d
Tn α
pT(x)={1, x, x2, x3, x4,..., xp} (1D)
11Finite Element Method by G. R. Liu and S. S. Quek
Pascal triangle of monomialsPascal triangle of monomials: : 2D2D
xy x2
x3
x4
x5
y2
y3
y4
y5
x2y
x3y
x4y x3y2
xy2
xy3
xy4 x2y3
x2y2
Constant terms: 1
x y
1
Quadratic terms: 3
Cubic terms: 4
Quartic terms: 5
Quintic terms: 6
Linear terms: 2 3 terms
6 terms
10 terms
15 terms
21 terms
2 2( ) ( , ) 1, , , , , ,..., ,T T p px y x y xy x y x y p x p
12Finite Element Method by G. R. Liu and S. S. Quek
Pascal pyramid of monomialsPascal pyramid of monomials : : 3D3D
x
x2
x3
x4
y
y2
y3
y4
xy
z
xz yz
x2y xy2
x2z zy2
z2
xz2 yz2
xyz
z3
x3y
x3z
x2y2
x2z2 x2yz
xy3
zy3
z2y2
xy2z xyz2
xz3
z4 z3y
1 Constant term: 1
Linear terms: 3
Quadratic terms: 6
Cubic terms: 10
Quartic terms: 15
4 terms
10 terms
20 terms
35 terms
2 2 2( ) ( , , ) 1, , , , , , , , , ,..., , ,T T p p px y z x y z xy yz zx x y z x y z p x p
13Finite Element Method by G. R. Liu and S. S. Quek
Displacement interpolationDisplacement interpolation
– Enforce approximation to be equal to the nodal displacements at the nodes
di = pT(xi) i = 1, 2, 3, …,nd
or
de=P
where
1
2=
d
e
n
d
d
d
d
T1
T2
T
( )
( )
( )dn
p x
p xP
p x
,
14Finite Element Method by G. R. Liu and S. S. Quek
Displacement interpolationDisplacement interpolation
– The coefficients in can be found by
e 1α P d
– Therefore, uh(x) = N( x) de
1 2
1 1 1 11 2
( ) ( ) ( )
1 2
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
n
T T T Tn
N N N
nN N N
x x x
N x p x P p x P p x P p x P
x x x
15Finite Element Method by G. R. Liu and S. S. Quek
Displacement interpolationDisplacement interpolation
Sufficient requirements for FEM shape functions
1 , 1,2, ,
0 , , 1, 2, ,d
i j ijd
i j j nN
i j i j n
x
1. (Delta function
property)
1
( ) 1n
ii
N
x2. (Partition of unity property – rigid body movement)
1
( )dn
i ii
N x x x
3. (Linear field reproduction property)
16Finite Element Method by G. R. Liu and S. S. Quek
Step 3: Formation of FE equations in local Step 3: Formation of FE equations in local coordinatescoordinates
Since U= Nde
Therefore, = LU = L N de= B de
Strain matrix
eTeΠ kdd
2
1or where
(Stiffness matrix)
eT
Ve
Tee
TTe
Ve
T
Ve
VcVcVcΠ ddBBddBdBdd )(2
1
2
1εε
2
1
VcT
Ve
e dBBk
17Finite Element Method by G. R. Liu and S. S. Quek
Step 3: Formation of FE equations in local Step 3: Formation of FE equations in local coordinatescoordinates
Since U= Nde eU Nd
or eeTeT dmd
2
1 where
(Mass matrix)
1 1 1d d ( d )
2 2 2e e e
T T T T Te e e e
V V V
T V V V U U d N Nd d N N d
de
Te
V
Vm N N
de
Te
V
Vm N N
18Finite Element Method by G. R. Liu and S. S. Quek
Step 3: Formation of FE equations in local Step 3: Formation of FE equations in local coordinatescoordinates
eTes
Teb
TefW FdFdFd
sbe FFf (Force vector)
d d ( d ) ( d )e e e e
T T T T T T T Tf e b e s e b e s
V S V S
W V S V S d N f d N f d N f d N f
de
Tb b
V
VF N f de
Ts s
S
SF N f
19Finite Element Method by G. R. Liu and S. S. Quek
Step 3: Formation of FE equations in local Step 3: Formation of FE equations in local coordinatescoordinates
0d)(2
1
te
Teee
Teee
Te
t
tFddkddmd
)(d
d)
d
d( T
e
TeT
e ttd
dd
ttt ee
t
t
Teee
t
t
Te
t
teeTeee
t
t
Te ddd
2
1
2
1
2
1
2
1
dmddmddmddmd
0d)(2
1
teeee
Te
t
tFkddmd
0d)2
1
2
1(
2
1
te
Teee
Teee
Te
t
tFddkddmd
eeeee fdmdk
FE Equation
(Hamilton’s principle)
20Finite Element Method by G. R. Liu and S. S. Quek
Step 4: Coordinate transformationStep 4: Coordinate transformation
eeee fdmkd
x
y
x'y'
y'
x'
Local coordinate systems
Global coordinate systems
ee TDd
eeeee FDMDK
TkTK eT
e TmTM eT
e eT
e fTF , ,
where
(Local)
(Global)
21Finite Element Method by G. R. Liu and S. S. Quek
Step 5: Assembly of FE equationsStep 5: Assembly of FE equations
Direct assembly method– Adding up contributions made by elements
sharing the node
FDMKD
FKD (Static)
22Finite Element Method by G. R. Liu and S. S. Quek
Step 6: Impose displacement constraintsStep 6: Impose displacement constraints
No constraints rigid body movement (meaningless for static analysis)
Remove rows and columns corresponding to the degrees of freedom being constrained
K is semi-positive definite
23Finite Element Method by G. R. Liu and S. S. Quek
Step 7: Solve the FE equationsStep 7: Solve the FE equations
Solve the FE equation,
for the displacement at the nodes, D
The strain and stress can be retrieved by using = LU and = c with the interpolation, U=Nd
FDMKD
24Finite Element Method by G. R. Liu and S. S. Quek
STATIC ANALYSISSTATIC ANALYSIS
Solve KD=F for D
– Gauss elmination– LU decomposition– Etc.
25Finite Element Method by G. R. Liu and S. S. Quek
EIGENVALUE ANALYSISEIGENVALUE ANALYSIS
0 DMKD (Homogeneous equation, F = 0)
Assume )exp( tiD
0][ 2 MK
Let 2 0][ MK
0]det[ MKMK
[ K i M ] i = 0 (Eigenvector)
(Roots of equation are the eigenvalues)
26Finite Element Method by G. R. Liu and S. S. Quek
EIGENVALUE ANALYSISEIGENVALUE ANALYSIS
Methods of solving eigenvalue equation– Jacobi’s method– Given’s method and Householder’s method– The bisection method (Sturm sequences)– Inverse iteration– QR method– Subspace iteration– Lanczos’ method
27Finite Element Method by G. R. Liu and S. S. Quek
TRANSIENT ANALYSISTRANSIENT ANALYSIS
Structure systems are very often subjected to transient excitation.
A transient excitation is a highly dynamic time dependent force exerted on the structure, such as earthquake, impact, and shocks.
The discrete governing equation system usually requires a different solver from that of eigenvalue analysis.
The widely used method is the so-called direct integration method.
28Finite Element Method by G. R. Liu and S. S. Quek
TRANSIENT ANALYSISTRANSIENT ANALYSIS
The direct integration method is basically using the finite difference method for time stepping.
There are mainly two types of direct integration method; one is implicit and the other is explicit.
Implicit method (e.g. Newmark’s method) is more efficient for relatively slow phenomena
Explicit method (e.g. central differencing method) is more efficient for very fast phenomena, such as impact and explosion.
29Finite Element Method by G. R. Liu and S. S. Quek
Newmark’s method (Implicit)Newmark’s method (Implicit)
Assume that
2 1
2t t t t t t tt t
D D D D D
1t t t t t tt D D D D
KD CD MD FSubstitute into
2 1
2
1
t t t t t
t t t t t t t t
t t
t
K D D D D
C D D D MD F
30Finite Element Method by G. R. Liu and S. S. Quek
Newmark’s method (Implicit)Newmark’s method (Implicit)residual
cm t t t t K D F
where
2
cm t t K K C M
2residual 11
2t t t t t t t t tt t t
F F K D D D C D D
Therefore, 1cm
residualt t t t
D K F
31Finite Element Method by G. R. Liu and S. S. Quek
Newmark’s method (Implicit)Newmark’s method (Implicit)
Start with D0 and 0D
Obtain 0D KD CD MD Fusing
1cm
residualt t t t
D K FObtain tD using
Obtain Dt and tD using
2 1
2t t t t t t tt t
D D D D D
1t t t t t tt D D D D
March forward in time
32Finite Element Method by G. R. Liu and S. S. Quek
Central difference method (explicit)Central difference method (explicit)int residual MD F CD KD F F F
residual 1D M F (Lumped mass – no need to solve matrix equation)
2t t t t tt D D D
2t t t t tt D D D
2
12t t t t t t
t
D D D D
2
2t t t t t
tt
D D D D
33Finite Element Method by G. R. Liu and S. S. Quek
Central Central difference difference
method method (explicit)(explicit)
D,
t
x
x
x x
x
t0t-t -t/2 t/2
Find average velocity at time t = -t/2 using
Find using the average acceleration at time t = 0.
Find Dt using the average velocity at time t =t/2
Obtain D-t using
D0 and are prescribed and
can be obtained from
Use to obtain assuming . Obtain using
Time marching in half the time step
0D
0D
residual 1D M F
2
2t t t t t
tt
D D D D
/ 2t D
/ 2 / 2t t t t tt D D D
/ 2tD
/ 2 / 2t t t t tt D D D
/ 2 / 2t t t t tt D D D
/ 2 / 2t t t t tt D D D
tD / 2 0t D D
tD residual 1D M F
34Finite Element Method by G. R. Liu and S. S. Quek
REMARKSREMARKS In FEM, the displacement field U is expressed by
displacements at nodes using shape functions N defined over elements.
The strain matrix B is the key in developing the stiffness matrix.
To develop FE equations for different types of structure components, all that is needed to do is define the shape function and then establish the strain matrix B.
The rest of the procedure is very much the same for all types of elements.