دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 1
Finite Element
Analysis of Boundary
Value Problem
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 2
FE Analysis of 1D Bars
00 ), ()0(
0)(
Qdx
duEAuu
qdx
duEA
dx
d
Lx
The DE is in the form of
q is the distributed load and
Q0 is the axial force.
Q0
q(x)
x L
Physical Model
FE Model 1 2 3 4 5 6
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 3
In FE analysis, we seek an approximation solution over each element.
x x
A B
1 2 Axxdx
duEAQ
1
Bxxdx
duEAQ
2
0)()(
BBAA
x
x
QxwQxwdxwqdx
du
dx
dwEA
B
A
BBAA
x
x
x
x
QxwQxwdxwqwl
dxdx
du
dx
dwEAuwB
B
A
B
A
)()()(
),(
)(),( wluwB
Weak form
FE Analysis of 1D Bars
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 4
Approximation of the solution
1- The approximation solution should be continuous and differentiable
as required by the weak form. (nonzero coefficient matrix)
2- It should be a complete polynomial (capture all possible States, e.g.
constant, linear, ….)
3- It should be an interpolant of variables at the nodes (satisfy EBCs)
2211 )(,)(, uxUuxUbxaU
/, /1 21
2211
xNxN
uNuNU
First order
332211
2 )(,)(,)(, uxUuxUuxUcxbxaU
( / )( / ), / ( / ), / ( / )1 2 3N 1 x 1 2x N 4x 1 x N x 1 2x Second Order
1 2
1 2
3 1 1 2 2 3 3U N u N u N u
FE Analysis of 1D Bars
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 5
FE Model
j
n
j
jj
Nw
NuUu
1
0)()(
BBAA
x
x
QxwQxwdxwqdx
du
dx
dwEA
B
A
and
0)(
0)(
0)(
11
1
22
1
2
1
11
1
1
n
j
jjn
x
x
n
n
j
j
jn
n
j
jj
x
x
n
j
j
j
n
j
jj
x
x
n
j
j
j
QxNdxqNdx
dNu
dx
dNEA
QxNdxqNdx
dNu
dx
dNEA
QxNdxqNdx
dNu
dx
dNEA
B
A
B
A
B
A
If n > 2 then the above integral should modify to
include interior nodal forces
ni
QfuKn
j
iijij
,,2,1
01
Stiffness matrix Force vector
Primary nodal
DOF
Secondary nodal
DOF
FE Analysis of 1D Bars
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 6
FE Model
where
n
j
ijij
i
x
x
ii
ji
x
x
jiij
QQxN
NldxqNf
NNBdxdx
dN
dx
dNEAK
B
A
B
A
1
)(
)(
),(
Note
Note that the problem has 2n unknowns for each element, i.e. ui
and Qi, so it cannot be solved without having another n
conditions. Some of these will be provided by BCs and the
remainder by balance of the secondary variables (forces) at node
common to several element. (assembling process)
FE Analysis of 1D Bars
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 7
FE Model (Linear Element)
/, /1 21
2211
xNxN
uNuNU
/)/1)(/1(
/)/1)(/1(
/)/1)(/1(
0
22
0
12
0
11
AEdxEAK
AEdxEAK
AEdxEAK
qdxxqf
qdxxqf
2/1)/(
2/1)/1(
0
2
0
1
1
1
2}; {
11
11][
qf
AEK eEventually for
Linear shape function
FE Analysis of 1D Bars
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 8
FE Model (Quadratic Element)
)/21(/), /1(/4), /21)(/1( 321
332211
xxNxxNxxN
uNuNuNU
3/8)/8/4)(/4/3(
3/7)/4/3)(/4/3(
0
2
2112
0
22
11
AEdxxxEAKK
AEdxxxEAK
qdxxxqf
qdxxxqf
6/4)/1()/4(
6/1))/(2/31(
0
2
0
2
1
FE Analysis of 1D Bars
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 9
FE Model (Quadratic Element)
For quadratic Shape function
1
4
1
6}; {
781
8168
187
3][
qf
AEK e
FE Analysis of 1D Bars
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 10
Assembly (or connectivity) of elements
In driving the element equation
-Isolate the element from mesh
-Formulate weak form (variational form)
-Developed its finite element model
To solve the total problem
-put the element in its original position
-Impose continuity of PVs at nodal points
-Balance of SVs at connecting nodes
1
1
ee
n uu
.
e e 1
n 1
0 0
0 if no external point source is appliedQ Q
Q if an external point source of Q is applied
1 2 1 2
e 1e
eQ1 eQ2
1
1
eQ 1
2
eQ
FE Analysis of 1D Bars
(*)
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 11
Assembly (or connectivity) of elements (For linear element n=2)
The interelement continuity of the primary variables is imposed by
renaming the two variable une and u1
e+1 at x=xN as one and same,
namely the value of u at the global node N: e e 1
n 1 Nu u U
FE Analysis of 1D Bars
( )N n 1 e 1 where
For a mesh of E linear finite elements (n=2): 1
1 1
1 2
2 1 2
2 3
2 1 3
E 1 E
2 1 E
E
2 E 1
u U
u u U
u u U
u u U
u U
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 12
Assembly (or connectivity) of elements (For linear element n=2)
To enforce balance of secondary variables Qie , eq. (*), we must
add nth equation of the element e to the first equation of the
element e+1 :
FE Analysis of 1D Bars
n
e e e e
nj j n n
j 1
K u f Q
and
n
e 1 e 1 e 1 e 1
1 j j 1 1
j 1
K u f Q
to give
( ) ( )n
e e e 1 e 1 e e 1 e e 1
nj j 1 j j n 1 n 1
j 1
e e 1
n 1 0
K u K u f f Q Q
f f Q
This process reduces the number of equations from 2E to E+1.
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 13
Assembly (or connectivity) of elements (For linear element n=2)
The first equation of the first element and the last equation of the
last element will remain unchanged, except for renaming of the
primary variables. The left-hand of the equation can be written in
terms of the global nodal values as
FE Analysis of 1D Bars
( )
( ) ( )
( )
( )
( )
e e e e e e e 1 e 1 e 1 e 1 e 1 e 1
n1 1 n2 2 nn n 11 1 12 2 1n n
e e e
n1 N n2 N 1 nn N n 1
e 1 e 1 e 1
11 N n 1 12 N n 1n N 2n 2
e e e
n1 N n2 N 1 n n 1 N n 2
e e 1
nn 11 N
K u K u K u K u K u K u
K U K U K U
K U K U K U
K U K U K U
K K U
e 1 e 1
n 1 12 N n 1n N 2n 2K U K U
where ( )N n 1 e 1
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 14
Assembly (or connectivity) of elements (For linear element n=2)
FE Analysis of 1D Bars
( )
( )
( )
( )
(
1 1 1 1
11 1 12 2 1 1
1 1 2 2 1 2 1 2
21 1 22 11 2 12 3 2 1 2 1
2 2 3 3 2 3 2 3
21 2 22 11 3 12 4 2 1 2 1
E 1 E 1 E E E 1 E E 1 E
21 E 1 22 11 E 12 E 1 2 1 2 1
E E E E
21 E 22 E 1 2 2
K U K U f Q unchanged
K U K K U K U f f Q Q
K U K K U K U f f Q Q
K U K K U K U f f Q Q
K U K U f Q un
)changed
For a mesh of E linear finite elements (n=2):
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 15
Assembly (or connectivity) of elements (For linear element n=2)
FE Analysis of 1D Bars
0
.................................................................................
0
1 1
11 12 1
1 1 2 2
12 22 11 12 2
2 2 3321 22 11
E 1 E EE22 11 12
EE 1
22
1
1
2
K K U
K K K K U
UK K K
UK K K
UK
f
f
... ...
1
1
1 2 1 2
1 2 1
2 3 2 3
2 1 2 1
E 1 E E 1 E
2 1 2 1
E E
2 2
Q
f Q Q
f f Q Q
f f Q Q
f Q
In matrix form
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 16
FE Analysis of BEAM
Lxxfdx
wdb
dx
d 0) ()(
2
2
2
2
The DE is in the form of
f(x)
L
M0
F0
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 17
FE Analysis of BEAM
Weak form
0)(1
2
2
2
2
dxfdx
wdb
dx
dv
e
e
x
x
0)()( 431212
2
2
2
1
1
e
x
e
e
e
x
e
e
x
x
Qdx
dvQxvQ
dx
dvQxvdxvf
dx
wd
dx
vdb
ee
e
e
or
e
x
e
e
e
x
e
e
x
x
x
x
Qdx
dvQxvQ
dx
dvQxvdxvfvl
dxdx
wd
dx
vdbwvB
ee
e
e
e
e
43121
2
2
2
2
1
1
1
)()()(
),(
where
11
2
2
42
2
3
2
2
22
2
1
;
;
ee
ee
x
e
x
e
x
e
x
e
dx
wdbQ
dx
wdb
dx
dQ
dx
wdbQ
dx
wdb
dx
dQ
BCs
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 18
FE Analysis of BEAM
Approximation of the solution
1- The approximation solution should be continuous and differentiable
as required by the weak form. (nonzero coefficient matrix)
2- It should be a complete polynomial (capture all possible States, e.g.
constant, linear, ….)
3- It should be an interpolant of variables at the nodes (satisfy EBCs)
211211
3
4
2
321
)(,)(,)(,)(
eeee xxwxwwxw
xcxcxccw
First order
1 2
1
41321
3
4
2
321
),(;),(
ee x
e
e
e
x
e
e
e
dx
dwuxwu
dx
dwuxwu
xcxcxccw
or
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 19
FE Analysis of BEAM
Shape Functions
4
1
)(j
j
e
j
e Nuxw
Calculating Ci and substituting in the equation for w
The interpolation functions in term of local coordinates are
h
x
h
xxN
h
x
h
xN
h
xxN
h
x
h
xN
2
4
32
3
2
2
32
1
;23
1;231
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 20
FE Analysis of BEAM
Hermite cubic interpolation function
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 21
FE Analysis of BEAM
FE Model
0)(114
12
2
2
2
e
i
x
x
ij
j
x
x
ji QfdxNudxdx
Nd
dx
Ndb
e
e
e
e
or 04
1
ij
j
ij FuK
For b=EI constant and also a constant f over the element.
4
3
2
1
22
22
3 6
6
12}; {
233
3636
323
3636
2][
Q
Q
Q
Q
h
hfhF
hhhh
hh
hhhh
hh
h
EIK
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 22
FE Analysis of 1D FIN
Model Boundary Value Problem
00 , )0(
)(
Qx
TkAQTT
TPAqTPdx
dTkA
dx
d
The DE is in the form of
k is thermal conductivity
is convection heat transfer coefficient
is the ambient temperature
q is the heat energy generated per unit volume
Physical Model
FE Model 1 2 3 4 5 6
A, cross section
P, Perimeter
T
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 23
FE Analysis of 1D FIN
Weak form
BA
B
A
B
A
x
e
x
e
x
x
ii
x
x
ji
jiij
dx
dTkAQ
dx
dTkAQ
dxTPqANf
dxNNPdx
dN
dx
dNkAK
21 ;
)(
Assume the lateral surfaces of the bar are isolated and the BCs
21 )(, )0(
)(
TLTTT
Aqdx
dTkA
dx
d
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 24
FE Analysis of 1D FIN
Approximation of the solution 1- the approximation solution should be continuous and differentiable
as required by the weak form. (nonzero coefficient matrix)
2- it should be a complete polynomial (capture all possible States, e.g.
constant, linear, ….)
3- it should be an interpolant of variables at the nodes (satisfy EBCs)
2211 )(,)(, TxTTxTbxaT
/, /1 21
2211
xNxN
TNTNT
First order
332211
2 )(,)(,)(, TxTTxTTxTcxbxaT
)/21(/), /1(/4), /21)(/1( 321
332211
xxNxxNxxN
TNTNTNT
Second Order
1 2
1 2
3
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 25
FE Analysis of 1D FIN
FE Model
Evaluating the integral using linear shape function
e
e
e
e
eeee
Q
QAq
T
TkA
QfTK
2
1
2
1
1
1
211
11
}{}{}]{[
For a uniform mesh and after assembling NL /
N
N Q
Q
Aq
T
T
T
kA
1
2
1
1
2
1
1
1
2
1
1
2
1
2
1100
121
011
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 26
FE Analysis of 1D FIN
FE Model
11
11
NN TT
TT
NeforQQ ee ,,3,2 01
1
2
Heat balance at global nodes 2,3,….,N
Boundary conditions at nodes 1 and N+1
N
n Q
Q
Aq
T
T
T
kA
1
1
1
1
2
1
0
1
2
1
2
1100
121
011
After applying the above conditions:
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 27
Virtual work as the „weak form‟ of equilibrium
equations for analysis of solids
In a general three-dimensional continuum the equilibrium
equations of an elementary volume can be written in terms of the
components of the symmetric cartesian stress tensor as
1
2
3
0
0 ,
0
xyx xz
x
xy y yz
y
yzxz z
z
bx y z
L
L bx y z
L
bx y z
u x 0L
T
x x xb b bb The body forces acting per unit volume
T
u v wu The displacement vector
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 28
Virtual work as the „weak form‟ of equilibrium
equations for analysis of solids
The weighting function vector defined as
2 3( ) v ( ) ( ) v
0
xyT x xz
xV Vd u b v L w L d
x y z
u uL
T
u v w u
Integrating each term by parts and rearranging we can write this as
We can now write the integral statement of equilibrium equations as
v
+ ( ) (..) (..)
x xy x y zV
x x xy y xz z
u u vub vb wb d
x y x
u n n n v w d
(*)
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 29
Virtual work as the „weak form‟ of equilibrium
equations for analysis of solids
where
are tractions acting per unit area
of external boundary surface
of the solid
x x x xy y xz z
y xy x y y yz z
xz x yz y z zz
t n n n
t n n n
n n nt
t
In the first set of bracketed terms in eq. (*) we can recognize
immediately the small strain operators acting on δu, which can be
termed a virtual displacement.
We can therefore introduce a virtual strain defined as
, , , , ,
T
Tu v w u v v w v w
x y z y x z y z y
D u
Arranging the six stress components in a vector σ in an order
corresponding to that used for δε, we can write Eq. (*) simply as
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 30
Virtual work as the „weak form‟ of equilibrium
equations for analysis of solids
we see from the above that the virtual work statement is precisely
the weak form of equilibrium equations and is valid for non-linear
as well as linear stress–strain (or stress–strain rate) relations.
v vT T T
V Vd d d
u b u t
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 31
Appendix
From “Energy Principles and Variational Methods in Applied Mechanics”, by: J. N. Reddy,
2nd ed., John Wiley (2002). (pp. 441--447)
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 32
Appendix
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 33
Appendix
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 34
Appendix
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 35
Appendix
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 36
Appendix
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 37
Appendix
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 38
Galerkin‟s Method in Elasticity
FE Model
Governing equations
Interpolated Displ Field Interpolated Weighting Function
ii uzyxNu ,,
jj uzyxNv ,,
kk uzyxNw ,,
ixix zyxN ,,
jyjy zyxN ,,
kzkz zyxN ,,
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 39
Galerkin‟s Method in Elasticity
Integrate by part…
0
dVfzyx
fzyx
fzyx
zzzzyxz
yy
yzyxy
V
xxxzxyx
0~ V
dVPuL
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 40
Galerkin‟s Method in Elasticity
Galerkin’s Method in Elasticity Virtual Work
Compare to Total Potential Energy
Virtual Total Potential Energy
i
i
T
iS
T
V
T
V
T dSdVdV PuTufuεσ2
1
دانشكده مكانيك -دانشگاه صنعتي اصفهان روش اجزاي محدود 41
Galerkin‟s Method in Elasticity
0
0
0 ,
0
on .
ˆ on
xyx xz
x
xy y yz
y
yzxz z
z
u
bx y z
bx y z
bx y z
A
A
E u x 0
u u 0B u x 0
σ n t 0