v. finite element method - afdex · 5.1 introduction to finite element method. ... basic idea of...
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V. Finite Element Method
5.1 Introduction to Finite Element Method
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5.1 Introduction to FEM
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⊙ Boundary value problem⊙ Problem definition
* 41( )
12x x x
⊙ Variational principle
Prob. 1
Prob. 2
1k ○ Exact :
22
2, 0 1
0 0, 1 0
dx x
dx
212
0
1( 2 ( )
2
(0) 0, (1) 0
dExtremize F x x dx
dx
subject to
2 4
1 2
1( ) ( )
12x x x x C x C
Ritz method to differential equation
0 0 , 1 0
* x
*F F x
2
1x x
1x x
2 1x x
41
12x x
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( 1)(2 1) 0y x x x 2 2( 1)Extremize y x x
22
2, 0 1, (0) 0, (1) 0
d Tx x T T
dx
212
0
( ) 2 ( )dT
Extremize F T x T x dxdx
(0) 0, (1) 0T T
Trial function:
1
( ) ( ), (0) 0, (1) 0n
i ii
T x C f x T T
(≡ Prob. A)
(≡ Prob. B)
Ritz method to differential equation
⊙ Extremization of a function and its related algebraic equation
⊙ Extremization of a functional and its related differentiation equation
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Ritz method to differential equation
○ Necessary condition for to be extreme :
⊙ Approximation
○ 2
1 2( ) 1 1x C x x C x x
1 2
2 3 4
1 2 1 20
1( ) 1 2 2 3 2 1 2 1
2F C x C x x C x x C x x dx
1 122 2 2 2
1 200
1 11 2 2 3
2 2x dx C x x dx C
1 1 1
2 3 4
1 2 1 20 0 0
1 2 2 3 1 1x x x dx C C x x dx C x x dx C
1 2
0, 0F F
C C
2 2
1 2 1 2 1 2 1 2
1 1 1 1 1( , ) ( )
6 15 6 20 30F C C F C C C C C C ○
○ Linear equation: 11 2
2
1 1 1 110 5 3 ,
5 4 230 60 15 6
CC C
C
2 31( ) 2 3 - 5
30x x x x ○ Approximate solution:
○ Transformation: Function space ⇒ Finite dimensional vector space
* 41( )
12x x x Exact
solution
21
2
0
1( 2 ( )
2
(0) 0, (1) 0
dExtremize F x x dx
dx
subject to
Trial function
1 2( , )F C C
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12
0[ ( ) ( ) ( )] 0
(0) 0, (1) 0
( )
(0) 0 (1) 0
where is arbitrary exc
x x x x
ept t
d
h
x
a
an
tx
d
22
20, 0 1
0 0, 1 0
dx x
dx
0 0 , 1 0
* x
2
1x x
1x x
2 1x x
41
12x x
21
2
20( ) 0
0 1 0
dx x dx
dx
11 2
0 0( ) ( ) ( ) ( ) ( ) 0
(0) (1) 0
Assume (0) (1) 0
x x x x x x dx
( ) is arbitrary.x
Weak form
sin x
tan x
cos x
logxe x
2x
0 0 , 1 0
2
1x x
1x x
2 1x x
41
12x x
sin x
tan x
cos x
logxe x
2x
b bb
aa au v uv uv
where is arbitrary except that
Weighted residual approach to differential equation
Prob. 1 Prob. 2
Set of functions
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⊙ Linear equations
2 31( ) 2 3 5
30x x x x ⊙ Approx. solution :
2
1 2 1 2( ) 1 1 ; and are unknownx C x x C x x C C
2
1 2 1 2( ) 1 1 ; and are arbitraryx W x x W x x W W
1 2 2 3 4
1 2 1 2 1 20(1 2 ) (2 3 ) (1 2 ) (2 3 ) (1 ) (1 ) 0C x C x x W x W x x dx W x x W x x dx
1 1 12 3
1 1 20 0 0
1 1 12 2 2 4
2 1 20 0 0
(1 2 )(1 2 ) (1 2 )(2 3 ) (1 )
(2 3 )(1 2 ) (2 3 )(2 3 ) (1 ) 0
W x x dx C x x x dx C x x dx
W x x x dx C x x x x dx C x x dx
2
1 15 4
0 0
1 1, , etc.
6 5x dx x dx
1
12
2
( ) (1 )
( ) (1 )
x x x
x x x
1 1 2 2 0W W
W1 are W2 are arbitrary
1x x
2 1x x
41
12x x
sin x
logxe x
0 0 , 1 0
2
1x x
Galerkin approach
11 2
2
1 1 1 110 5 3 ,
5 4 230 60 15 6
CC C
C
1
2
0
0
Trial function
Basic function
Approximate weighting function
Set of approximate
weighting functions
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⊙ Characteristics of solution convergence
○ Accuracy Accuracy
○
⊙ Requirements on basic function
○ Linearly independent
○ or
1
1 1
( ( )) ( ( ))n n
i i i ii i
C x C x
*
1
lim ( ) ( )n
i ini
x C x x
i x
p
i x H 1( ) ( )p
i x C
<Comparison of approximate
and exact solutions>
○ ⇒ Error 2.2%
○ ⇒ Error 20% * 1 10 , 0 ,
12 15 * 1 7
1 , 14 30
*
max max0.029165, 0.029799 x x
⊙ Comparison between approximate and exact solutions
Exact
Approximate
2 31( ) 2 3 5
30x x x x
* 41( )
12x x x
maxx
Accuracy of the approximate solution
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<Basic function = Interpolation function>
⊙ Trial function : 1
1
1
12 for 01 21 2
12 2 (1 ) for 12
C x xx C x
C x x x
7 7 1( )
192 96 2x x ⊙ Approximate solution :
Eaxct
FE solution
<Comparison of FE solution
with exact solution>
1
11 2
2x x
1 x
Superconvergence
Basic idea of Finite Element Method (FEM)
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Exact
FE solutions
⊙ FEM = Ritz or Galerkin method + FE discretization and interpolation (approximation)
⊙ FE discretization and interpolation function: ( )iN x
⊙ Finite element solutions
3( )N x
1 2 3 4① ② ③
2 ( )N x
2 2 3 3( ) ( ) ( ) in Finite Element Methodx N x N x
1 2 2( ) ( ) ( ) in Ritz or Galerkin methodx C x C x
Technique of making basic functions
FE solutions with different FEA models
( )iN x( )iN x
1 1
( )iN x
1
0 1 01
ElementNode
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⊙ Definition of a beam deflection problem
4 2* 5( )
12 2 12
x xv x
○ Solution:
22
21 , 0 1
dx x
dx
(0) 0, (1) 0v v
⊙ Variational principle:21
2
0
( ) 2( 1) ( )dv
Extremize F v x v x dxdx
(1) 0subject to v
⊙ Ritz method
○ It should satisfy essential boundary condition, (1)=0 v
○ [Ex. 2.1]1( ) )v x C x = (1-
○ [Ex. 2.2] 2
1 2( ) ) )v x C x C x = (1- (1-
○ [Ex. 2-3] 2 2
1 2( ) ) )v x C x x C x= (1- (1-
○ BVP based on beam theory:
V
( )V x
x
0 1
2
x
0 1
1
( )bM x
( ) ( )bEIv x M x , (0) 0, ( ) 0v v L
2( ) 1bM x x
Ritz method to a beam deflection
01, 2, 1EI w L
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⊙ [Ex. 2.3] :2 2
1 2( ) (1 ) (1 )v x C x x C x
12 2 2 2 3 2
1 2 1 20
( ) { (2 3 ) ( 2 )} 2( 1){ ( ) (1 )}F v C x x C x x C x x C x dx 2 2
1 2 1 2 1 2 1 2
5 4 1 1 16( , )
12 3 3 10 15C C C C C C F C C
1
2
4 1 1
15 3 101 8 16
3 3 15
C
C
34 153 113
27 270 270v x x x
4 2* 5( )
12 2 12
x xv x Exact:
Ritz method to a beam deflection
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5.2 FEM of Partial Differential
Equations
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Poisson’s equation
Poisson’s equation
TT T on S
( , ) 0
k k f x yx x x y
22
2, 0 1
0 0, 1 0
dx x
dx
212
0
1( 2 ( )
2
(0) 0, (1) 0
dExtremize F x x dx
dx
subject to
221
( ) 2 ( , ) ( , )2
Extremize F k k f x y x y dxdy
x y
subject toTT T on S
1 1 2 2( , ) ( , ) ( , ) J J
J
x y N x y N x y N ?, ? J J
N
2D
1D
= 3D
Nodal value
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Mesh
Mesh system
J
Interpolation function ( , )JN x y
1
1
-1
-1ξ
η
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Intelligent remeshing of quadrilaterals
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Intelligent remeshing of tetrahedrals
Accurate description
with minimum elements!
As number of elements increases,that of remeshings does so much,which deteriorates solution accuracy.
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2D mesh systems with higher quality
Initial
Final
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KSTP Fall 2016.
3D mesh systems with higher quality
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4.5 Finite element formulation
Equation of equilibrium
Equation of heat conduction
T
,ij ij
g ij ijq , , T
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Coordinate axis
𝑦
𝑥
𝑥2
𝑧𝑥3
𝑥1𝐞𝟏
𝐞𝟐
𝐞𝟑
𝐣
𝐢
𝐤
𝑥, 𝑦, 𝑧 − 𝑎𝑥𝑖𝑠 ⟹ 𝑥1, 𝑥2, 𝑥3 − axis
Mechanical quantities
𝑢𝑥, 𝑢𝑦 ⟹ 𝑢1, 𝑢2𝜎𝑥𝑦 𝑜𝑟 𝜏𝑥𝑦 ⟹ 𝜎12
Unit vector
𝐢, 𝐣, 𝐤 ⟹ 𝐞𝟏, 𝐞𝟐, 𝐞𝟑
Kronecker delta 𝜹𝒊𝒋
𝛿𝑖𝑗0 𝑖𝑓 𝑖 ≠ 𝑗1 𝑖𝑓 𝑖 = 𝑗
First and second order
𝑢𝑖 = 𝐮 𝑜𝑟 𝑢
𝜎𝑖𝑗 = 𝜎𝑖𝑗 =
𝜎11 𝜎12 𝜎13𝜎21 𝜎22 𝜎23𝜎31 𝜎32 𝜎33
=
𝜎𝑥𝑥 𝜎𝑥𝑦 𝜎𝑥𝑧𝜎𝑦𝑥 𝜎𝑦𝑦 𝜎𝑦𝑧𝜎𝑧𝑥 𝜎𝑧𝑦 𝜎𝑧𝑧
Partial differentiation
𝜙,𝑖 =𝜕𝜙
𝜕𝑥𝑖, 𝜙,𝑖𝑗 =
𝜕2𝜙
𝜕𝑥𝑖𝜕𝑥𝑗, 𝑣𝑖,𝑗 =
𝜕𝑣𝑖
𝜕𝑥𝑖, 𝜎𝑖𝑗,𝑗 =
𝜕𝜎𝑖𝑗
𝜕𝑥𝑗
Summation
𝑗=1
3
𝜎𝑖𝑗,𝑗 + 𝑓𝑖 = 0 ⟹ 𝜎𝑖𝑗,𝑗 + 𝑓𝑖 = 0
𝐅𝐫𝐞𝐞 𝐢𝐧𝐝𝐞𝐱: 𝐨𝐧𝐜𝐞 𝐢𝐧 𝐚 𝐭𝐞𝐫𝐦 (𝐜𝐚𝐧𝐧𝐨𝐭 𝐛𝐞 𝐜𝐡𝐚𝐧𝐠𝐞)𝐃𝐮𝐦𝐦𝐲 𝐢𝐧𝐝𝐞𝐱: 𝐭𝐰𝐢𝐜𝐞 𝐢𝐧 𝐚 𝐭𝐞𝐫𝐦
Divergence theorem
𝑉
𝑄𝑗𝑘,𝑚..𝑚,𝑖𝑑𝑉 = 𝑆
𝑄𝑗𝑘,𝑚..𝑚𝑛𝑖𝑑𝑆
𝒏𝒊: 𝐎𝐮𝐭𝐰𝐚𝐫𝐝𝐥𝐲 𝐝𝐢𝐫𝐞𝐜𝐭𝐞𝐝 𝐮𝐧𝐢𝐭 𝐧𝐨𝐫𝐦𝐚𝐥 𝐯𝐞𝐜𝐭𝐨𝐫
Permutation symbol 𝜺𝒊𝒋𝒌
휀𝑖𝑗𝑘 01−1
𝑖𝑓𝑖𝑓𝑖𝑓
𝑖 = 𝑗 𝑜𝑟 𝑗 = 𝑘 𝑜𝑟 𝑘 = 𝑖
𝑖, 𝑗, 𝑘 = 1,2,3 𝑜𝑟 2,3,1 = 3,1,2
𝑖, 𝑗, 𝑘 = 1,3,2 𝑜𝑟 2,1,3 = 3,2,1
Examples
𝑐 = 𝐚 × 𝐛 ⟹ 𝑐𝑖 = 휀𝑖𝑗𝑘𝑎𝑗𝑏𝑘
𝑑𝑖𝑣 𝐯 ⟹ 𝑣𝑖.𝑖
𝑔𝑟𝑎𝑑 𝜙 ⟹ 𝜙,𝑖
𝑐𝑢𝑟𝑙 𝐯 ⟹ 휀𝑖𝑗𝑘𝑣𝑘,𝑗
𝛻2𝜙 ⟹ 𝜙,𝑖𝑖
𝑊 = 𝐚 𝐛⟹𝑊 = 𝑎𝑖𝑏𝑖
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Finite element analysis of 3D elastic problems of isotropic materials
0ti
ij ij i i i iV S V
dV t dS f dV , 0 on
ii i i uu u S
Weak form
S
P
S
VV
2 2 2ij ij xx xx yy yy zz zz xy xy yz yz zx zx
i i
Finite element equations
T
, , , , ,xx yy zz xy yz zx
1, 1
2, 2
3, 3
1, 2 2, 1
2, 3 3, 2
3, 1 1, 3
2
2
2
xx
yy
zz
i
xy
yz
zx
i ij jD
1, 1
2, 2
3, 3
1, 2 2, 1
2, 3 3, 2
3, 1 1, 3
2
2
2
xx
yy
zz
i
xy
yz
zx
u
u
u
u u
u u
u u
1 0 0 0
1 0 0 0
1 0 0 0(1 )
0 0 0 0 0(1 )(1 2 )
0 0 0 0 0
0 0 0 0 0
E
D
/(1 ), (1 2 ) / 2(1 )
3
1
N
i iI I
I
u N U
i iI IB U
1 , 1
2 , 2
3 , 3
1 , 2 2 , 1
2 , 3 3 , 2
3 , 1 1 , 3
I
I
I
i I iI I
I I
I I
I I
N
N
NW B W
N N
N N
N N
i ij jJ JD B U
( )( )
T TW B D B U
ij ij i i
ij jJ J iI I
I jI ij jJ J
D B U B W
W B D B U
0ti
I iI ij jJ J i iI i iIV S V
W B D B dV U t N dS f N dV
IJ J IK U F
IJ iI ij jJV
K B D B dV ti
I i iI i iIS V
F t N dS f N dV
, ,
1( )
2ij i j j i
Galerkinapproximation
Weighted residual method
i iI IN W
Shape function matrix
Stiffness matrix
Force vector
Strain-displacement matrix
Nodal displacement
Elastic matrix
1, 1 2, 1 , 1
1, 2 2, 2 , 2
1, 3 2, 3 , 3
1, 2 1, 1 2, 2 2, 1 , 2 , 1
1, 3 1, 2 2, 3 2, 2 , 3 , 2
1, 3 1, 1 2, 3 2, 1 , 3 , 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
N
N
N
iI
N N
N N
N N
N N N
N N N
N N N
N N N N N N
N N N N N N
N N N N N N
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Rigid-plastic finite element method
Mixed formulation: knowns of velocity and pressure
Weak form –Penalty method
Weak form – Lagrange multiplier method
Lagrange multiplier
Penalty constant
( )( )
T TW B D B U
ij ij i i
ij jJ J iI I
I iI ij jJ J
D B U B W
W B D B U
0t ci
ij ij ii jj i i t tV V S SdV K dV t dS dS
, 0t ci
ij ij ii i i i i i i t tV V V V S SdV p dV f dV v qdV t dS dS
,
, 0
ti
I iI ij jJ J iI i M M i iI i iIV V S V
M iJ i M JV
W B D B dV U N H dV P t N dS f N dV
Q N H dV U
2 2
3 3 2
3
ij ij ij
kl kl
Non-linear
0
ij IQ J I
MJ MQ Q M
K C U F
C P G
IJ iI ij jJV
K B D B dV
,IM iI i MV
C N H dV
Numerical problem occurs in the elastic region. Minimum allowable effective strain rate is needed.
Finite element equations
-Reduced integration-MINI-element
Over-constrained problem
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Finite element equations of heat conduction
4 4
, ,( ) ( ) ( )
( ) ( ) 0
c q
e
VS S
eS
i i f c c q w
e e
t
Tc kT Q dV q h T T dS h T T dS
T T h T T dS
Weak form
Finite element equations
g ij ijQ C
I IT N T
I IN W
I I
TN T
t
, ,
4 4( ) 0
[
]
c
q c
I J J I J i I i J I f c J J c I
V S
q J J w I J J e e J J e I
S S
W cN T N kN N T QN dV q h N T T N dS
h N T T N dS N T T h N T T N dS
0 1 2 3 4 1 2 3 4( )IJ J IJ IJ IJ IJ IJ J I I I IC T K K K K K T Q Q Q Q
1
I IV
Q Q N dV
2 ( )c
I f c c IS
Q q h T N dS 3
qI q w I
SQ h T N dS
4 4( )e
I e e e IS
Q T h T N dS
1i i i i
i I I I I
I
T T T TT
t t
0, 1, 2, 3, 4,
1, 2, 3, 4,
( /( ) )
/( )
i i i i i i i
IJ IJ IJ IJ IJ IJ J
i i i i i i
I I I I IJ J
C t K K K K K T
Q Q Q Q C T t
i it t t
1i i i
I I IT T T t
IJ I JV
C cN N dV 0
, ,IJ I i J iV
K k N N dV 1
cIJ c I J
SK h N N dS
2
qIJ q I J
SK h N N dS
3
eIJ e I J
SK h N N dS
4 3( )e
IJ I J I JS
K N N N N dV
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Coupled analysis
T
,ij ij
g ij ijq , , T
Equation of equilibrium
Equation of heat conduction