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CONTINUUM & FINITE ELEMENT METHOD
Variational Approach in FEM Part II
Prof. Seong Jin ParkMechanical Engineering, POSTECH
Find to minimize .)(x )(x
One wants to obtain an approximate solution to minimize a functional . One of the
historically famous approximate methods for this kind of problem is Rayleigh-Ritz Method, and
the other modern method is the Finite Element Method.
Finite Element Method vs. Rayleigh-Ritz Method
)(x
: approximate solution satisfying the essential B.C.
: trial functions (defined over the whole domain)where
i) Rayleigh-Ritz Method:
n
i
iic1
)()(~
xx
)(xi
Then, ),,,()~
( 21 nccc to be minimized w.r.t. ci.
Therefore,ni
ci
,,1,0)
~(
: n equations for n unknown ci’s.
This method is very simple and easy to understand. However, it is not easy to find a family of trial functions forthe entrie domain satisfying the essential boundary conditions when geometry is complicated. The solution tothis troublesome point can be found in the Finite Element Method.
Finite Element Method vs. Rayleigh-Ritz Method
Then,
iiN )()(~
xx
),,,()~
( 21 n
nii
,,1,0
to be minimized w.r.t. i
: n equations for n unknown i’s.
functions shape :
values nodal :
)(xi
i
N
Finite Element Method vs. Rayleigh-Ritz Method
ii) Finite Element Method
In this case, the shape functions can be found more easily than the trial functions without having to worryabout satisfying the essential boundary conditions, which makes FEM much more useful than Ralyleigh-RitzMethod. In this regard, the Finite Element Method is as modernized approximation method suitable forcomputer environment.
Example: Tight string problem via two methods
minimized. be to dxwydx
dyTxy
l
0
2
2
1)(
* Special case: w(x)=w (constant)
w(x)
y(x)
x
020
02
120
2
2
2
2
1
1
lTA
A
lTA
lw-
A
0 ,4
23
2
1
AT
wlA
l
x
T
wly
sin
4~3
2
T
wly
T
wly
22
125.02
1129.0
2
1~
vs. (Note: )
i) Rayleigh-Ritz Method
l
xA
l
xAy
2sinsin~
21 (Note: Trial functions satisfy essential B.C.)
Example: Tight string problem via two methods
dxwydx
dyTxy
l
0
2
2
1)( to be minimized is equivalent to
.00
δydxywdx
dy
dx
dyT
l
any for
Introduce elements to the system as depicted below.
1 ① 2 ② 3 ③ 4 x
y e
y1 ey2lx
el
①,② ,③ : element number1, 2, 3, 4 : node number
: local coordinate lx
ii) Finite Element Method
Example: Tight string problem via two methods
Introduce the approximate solution via interpolation functions (or shape functions) for each element.
)()( e
ii
e yxNxy
Then,
e
ii
e
ii
e
ii
e
ydx
dNy
dx
d
dx
dy
yNxy
ydx
dN
dx
dy
)(
and the variation of functional over each element is summed to result in the variation of thewhole system, i.e.,
)3()2()1( e
e
Example: Tight string problem via two methods
Let us consider forany element
el
lee dxyw
dx
dy
dx
dyT
0 (henceforth xl → x for convenience)
form) matrix a (in
form) indicial an (in
e
e
e
ee
T
e
e
e
i
e
j
e
ij
e
i
le
i
e
j
leije
i
lee
ii
e
i
ie
j
je
f
f
y
yK
y
y
fyKy
dxwNydxdx
dN
dx
dNTy
dxywNydx
dNy
dx
dNT
2
1
2
1
2
1
00
0
Example: Tight string problem via two methods
where
force nodal equivalent- work:
matrix stiffnesselement :
0
0
le
i
e
i
leije
ij
dxwNf
dxdx
dN
dx
dNTK
summation
iijiji
e
e
yFyKy
any for 0
FyK
Example: Tight string problem via two methods
Linear element for simplicity
eee yxNyxNxy 2211 )()()(
)(
1)(
2
1
N
N)(
e
l
l
x
d
dN
ldx
d
d
dN
dx
dN i
e
ii 1
ee ldx
dN
ldx
dN 1 ,
1 21
)(1 N )(2 N
1 1
0 1
Example: Tight string problem via two methods
11
11
e
e
l
TK
1
1
202
1 ele wl
dxN
Nwf
e
2/1
1
1
2/1
3
1100
1210
0121
0011
3
4
3
2
1
wl
y
y
y
y
l
T
Global matrix equation
B.C. y1 = y4 =0T
wlyy
2
329
1
Example: Tight string problem via two methods
Example: Tight string problem via two methods
Notes:
1. 1st and 4th equations are not to be used (or obtained more precisely) since and are notarbitrary, but zero. As a matter of fact, however, introduction of boundary conditions replacesthose equations. The reaction force and can be obtained from the 1st and 4th equations,respectively.
2. Advantages of variational approach over the direct one: 1) Use of scalar quantity (energy) versusvectors, 2) Ease in treatment of distributed load
3. Treatment of concentrated loads:
1y 4y
1F 4F
cF
1
y cy
FEM for Second-Order Elliptic Partial Differential Equation
* Steady state heat conduction, flow though porous media, torsion, etc.
2S
1S
x
y
z
n̂
in ),,( zyxf
zk
zyk
yxk
xzyx
B.C.
2
1
on 0),,(),,(
on ),,(
Szyxhzyxgnz
kny
knx
k
Szyx
zzyyxx
.any for 0Ji.e.
FEM for Second-Order Elliptic Partial Differential Equation
Minimize
The above partial differential equation with boundary conditions is equivalent to the following variational principle:
With many elements introduced, one can sum the contributions of each element to the functionals as described below:
e
e
e
e
JJ
JJ
e
ii
e
e
ii
e
N
N
Introduce the approximate solution in terms of shape functions and nodal values of over each element:
e
e
2S
FEM for Second-Order Elliptic Partial Differential Equation
FEM for Second-Order Elliptic Partial Differential Equation
e
j
S
ij
e
i
S
i
e
i
i
e
i
e
jij
zij
yij
x
e
ie
ee
e
e
dSNhNdSN
dVfN
dVz
N
z
Nk
y
N
y
Nk
x
N
x
NkJ
22
g
e
Si
e
fi
e
j
e
Sij
e
j
e
Cij
e
ie RRKKJ
FEM for Second-Order Elliptic Partial Differential Equation
where
e
dVz
N
z
Nk
y
N
y
Nk
x
N
x
NkK
jiz
jiy
jix
e
Cij
2S
ij
e
Sij
e
dSNhNK
e
dVfNR i
e
fi
2
gS
i
e
Si
e
dSNR
(stiffness matrix due to conduction)
(forcing matrix due to convection)
(forcing matrix due to distributed heat sink)
(forcing matrix due to distributed heat outflux)
FEM for Second-Order Elliptic Partial Differential Equation
After the assembly procedure, one can obtain
1
0 for any arbitray , with 0 for nodes on .
ee
i Cij j Sij j fi Si
i
J J
K K R R
S
0 SfSC RRKK
FK
SC KKK
Sf RRF forcing matrix due to heat source and heat flux
FEM for Second-Order Elliptic Partial Differential Equation
For a given statically admissible stress field , consider any kinematically admissible virtual displacement and external work due to the virtual displacement.
ij
iu
A. Principle of Virtual Displacement (Work)
Variational Principle for Deformation
jiji nt
0, ijij f
ntdisplaceme virtual: iu
state mequilibriu
dV
dVufx
dVx
u
dVufdSun
dVufdSutW
ijij
ii
j
ij
j
iij
iiijij
iiiiext
where
0 ,
2
1 ,
2
1
ijijijij
j
i
i
j
j
i
ij
i
j
j
i
ij
x
u
x
u
x
u
x
u
x
u
Variational Principle for Deformation
dVdVufdSutW ijijiiiiext
0
dVufdSutdV iiiiijij
Physical meaning
iuFind Any virtual displacement iiij ft ,,if are in equilibrium
Principle of virtual velocity (power)
dVt
dVt
ufdS
t
ut
t
W ij
ij
i
i
i
i
ext
Principle of virtual displacement
Variational Principle for Deformation
( can be replaced with respectively.)
Notes:
1. Kinematically admissible virtual displacement where prescribed.
2. Statically admissible stress field satisfies not only the equilibrium equation but also theprescribed traction boundary condition (i.e., natural boundary condition), i.e.
3. Principle of virtual velocity (power)
4. If inertial term is included in the body force term, the principle of virtual displacement canbe extended to fictitious equilibrium state.
Variational Principle for Deformation
0 ii uu ii uu
ijiji tnt
dVt
dVt
ufdS
t
ut
t
W ij
ij
i
i
i
i
ext
ijii δdvv
iu
dVdVuvfdSut ijijiiiii ijiu ,iji dv ,
B. Principle of Minimum Potential Energy
Principle of virtual displacement
dVdVufdSutW ijijiiiiext
For an elastic body, there exists strain energy density, Uo, such thatij
oij
U
then UdVUdVUdVU
dV ooij
ij
oijij
Variational Principle for Deformation
ii ρft and
where
dVUU o: strain energy
UWext
Define the potential energy V as
dVufdSutV iiii with fixed
VWext Then
Therefore 0VU
Variational Principle for Deformation
Defining the total potential energy asp
VUp 0 p
With and ij
o
ij
U
ijjiij uu ,,
2
1
Variational Principle for Deformation
yields
In summary, the deformation of any elastic body (linear or nonlinear) is governed by minimizing the functional, the total potential energy,
dVufdSutdVUu iiiioip
Note: Principle of Virtual Work is valid for any material, whereas Principle of Minimum Potential Energy is valid only for elastic materials.
C. Principle of Complementary Virtual Work
)( on
in
Stn
f
ijij
ijij
0,
dV
dVufdVu
dVfudSnu
dVfudStuW
ijij
iijijijji
iijiji
iiii
,,
*
d
d
*W
extW
Variational Principle for Deformation
Consider a variation of statically admissible stress field and external forces while keeping kinematically admissible displacement .
ijii tf ,
iu
Define a complementary virtual work as*W
ij
o
ij
U
*
(a constitutive equation)
Variational Principle for Deformation
Now as a counterpart to the Principle of Minimum Potential Energy for an elastic material, consider the case for an elastic material for which a complementary strain energy density exists as below:
Then we can rewrite the right hand side of the expression for as*W
***
*
* UdVUdVUdVU
dVW ooij
ij
o
ijij
and the left hand side can be rewritten in terms of the complementary potential energy as follows
dVfudStuV iiii
*
dVfudStuV iiii
*with iu(keeping fixed)
*** ,, VUft iiij
0*
Therefore, one can have
<Total Complementary potential energy>
For any statically admissible stress, force system
Variational Principle for Deformation
Note:
ip u
iiij ft ,,*
: leads to displacement-based FEM yielding a stiffness matrix
: leads to equilibrium-based FEM yielding a flexibility matrix
Principle of Virtual Displacement (Work): 0
dVufdSutdV iiiiijijp
fixed) (with iiijiiiiijijp ftdVufdSutdV ,,
Let us apply the principle of virtual displacement (or principle of minimum potential energy) to the two-dimensional elastic deformation problem.
Principle of Minimum Potential Energy:
dVufdSutdVUu iiiioip
ij
o
ij
U
ijjiij uu ,,
2
1
0)()(
k
i
k
k
iiiiiiijijp uFdVuufdSutdV
where
Variational expression:
Displacement-based FEM for Elasticity
0)()(
k
i
k
k
iiiiiiijijp uFdVuufdSutdV
Displacement-based FEM for Elasticity
In this section, let us include the inertia force and concentrated forces as the most generalFEM formulation for elasticity. So, consider the following variational expression as thestarting form.
VNyxv
yxuu
),(
),(
3
3
2
2
1
1
321
321
000
000
v
u
v
u
v
u
NNN
NNN
v
uu
VNv
uu
Displacement approximation via shape functions
Acceleration
1
2
3
Displacement-based FEM for Elasticity
VN
xy
y
x
x
N
y
N
x
N
y
N
x
N
y
N
y
N
y
N
y
Nx
N
x
N
x
N
N
332211
321
321
000
000
xy
y
x
xy
y
x
,
C
Strain matrix:
e.g.
Stress-Strain relation(constitutive law):
Displacement-based FEM for Elasticity
stress plane for
2
100
01
01
1 2
EC
strain plane for
2
2100
01
01
)21)(1(
EC
dVCUT
2
1
)()(
2
1 k
i
k
k
iiiiii
T
p uFdVuufdSutdVC
0)()(
k
i
k
k
iiiiii
T
p uFdVuufdSutdVC
and the total potential energy becomes
Displacement-based FEM for Elasticity
Note: With this notation, the strain energy can be represented as
y
x
t
tt
y
x
f
ff
)(
)(
)(
k
y
k
xk
F
FF
Define the force matrices as follows:
: traction force
: body force
: concentrated force applied at k-th position
Displacement-based FEM for Elasticity
Note: In case of initial strain , is to be replaced with . In case of initial stress,
init init
initC
VNyxv
yxuu
),(
),(
VN
xy
yy
xx
k
kTT
e
e
p
k
k
T
k
k
e
e
pp
FNV
Fv
u
k
)(
)(
)(
)(
)(
x
eeee
eee
dVuudVfudStudVC
dVuufdSutdVC
TT
S
TT
iii
S
ii
Te
p
2
2
Virtual displacement and corresponding strain:
Variation of total potential energy:
For an element:
Displacement-based FEM for Elasticity
eee
b
e
d
eeTe
TT
TT
S
TTTTe
p
VmFFVKV
VdVNNV
dVfNVdStNVVdVNCNV
e
eee
2
VVMFFFVKV cbd
T
p any for 0
Introducing the matrix notations for approximated displacement field andconstitutive law and so on into the above equation gives:
After assembly, one finally obtains:
Displacement-based FEM for Elasticity
dVNNmTe
e
dVNCNKTe
e
2S
Te
d
e
dStNF
dVfNFTe
b
e
ek
kTe
C FNFc
)(
x
FFFFVKVM cbd
where : element mass matrix
: element stiffness matrix
: work equivalent nodal force
: work equivalent nodal force
: concentrated force
Displacement-based FEM for Elasticity
is the functional to be minimized. results in .)
FVVKVTT
p 2
1
0 p FVK
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