Transcript Finite Element Method CHAPTER 10: SPECIAL PURPOSE ELEMENTS

Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 10:
SPECIAL PURPOSE
ELEMENTS
1
CONTENTS

CRACK TIP ELEMENTS
 METHODS FOR INFINITE DOMAINS
– Infinite elements formulated by mapping
– Gradual damping elements
– Coupling of FEM and BEM
– Coupling of FEM and SEM

FINITE STRIP ELEMENTS
 STRIP ELEMENT METHOD
Finite Element Method by G. R. Liu and S. S. Quek
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CRACK TIP ELEMENTS

Fracture mechanics – singularity point at
crack tip.

Conventional finite elements do not give
good approximation at/near the crack tip.
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CRACK TIP ELEMENTS
From fracture mechanics,

3 

1

sin
sin

2
2
 xx 


KI


3 
 
 xy   2 r cos 2  sin 2 sin 2 
 yy 


 

3

1  sin sin 
2
2 

 
2 
cos
(


1

2sin
)
u 
KI r 
2
2



v 


  2G 2 sin (  1  2 cos 2 ) 
 2
2 
(Near crack tip)
y, v
r

(Mode I fracture)
x, u
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CRACK TIP ELEMENTS
Special purpose crack tip element
with middle nodes shifted to quarter
position:









H/4

H


L/4








L

Finite Element Method by G. R. Liu and S. S. Quek

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CRACK TIP ELEMENTS
x = 0.5 (1-)x1 + (1+)(1-)x2 + 0.5 (1+) x3
u = 0.5 (1-)u1 + (1+)(1-)u2 + 0.5 (1+) u3
y
6
7
5
4 r
(Measured from node 1)
8
Move node 2 to L/4 position
x1 = 0, x2 = L/4, x3 = L, u1 = 0

x = 0.25(1+)(1-)L + 0.5 (1+)L
u = (1+)(1-)u2+0.5 (1+) u3
Finite Element Method by G. R. Liu and S. S. Quek
1

x
3
2
L/4
L
-1
0

1
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CRACK TIP ELEMENTS
Simplifying,
x = 0.25(1+)2L
u= (1+)[(1-)u2+0.5u3]
Along x-axis, x = r
r = 0.25(1+)2L

or
r
(1   )  2
L
u = 2(r/L) [(1-)u2 + 0.5u3]
Note: Displacement is
proportional to r
 u  u 
x

 0.5(1   ) L = r L
where
 x   x

u
1 1
1

[

2

u

(
  )u3 ]
Therefore,
2
x
2
r L
Finite Element Method by G. R. Liu and S. S. Quek
Note: Strain (hence
stress) is proportional to
1/r
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CRACK TIP ELEMENTS

Therefore, by shifting the nodes to quarter
position, we approximating the stress and
displacements more accurately.
 Other crack tip elements:
L






L



 







L/4
Triangular crack tip
elements

L/4



 





A 3-D, wedge crack tip
element
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METHODS FOR INFINITE DOMAIN

Infinite elements formulated by mapping
(Zienkiewicz and Taylor, 2000)
 Gradual damping elements
 Coupling of FEM and BEM
 Coupling of FEM and SEM
Finite Element Method by G. R. Liu and S. S. Quek
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Infinite elements formulated by mapping
Use shape functions to approximate
decaying sequence:
C1 C2 C3
 2  3  ...
r r
r
P
In 1D:

O
x
x



xO  1 
1 
 1 

 xQ


r  x  xO    1 


-1
y
R
Q
+1
Map
P

Q


R at 
(Coordinate interpolation)
xQ  xO
x  xO
 1
xQ  xO
r
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Infinite elements formulated by mapping
If the field variable is approximated by
polynomial,
u   0  1   2 2   3 3  ...
Substituting  will give function of decaying form,
For 2D (3D):
x



xO1  1 
1 
 1 
C1 C2 C3
 2  3  ...
r r
r

 xQ1


 
yO1  1 
 yQ1
1 
 1  


 
z
zO  1 
 zQ
1  1  1   1
y

Finite Element Method by G. R. Liu and S. S. Quek
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Infinite elements formulated by mapping
y
x
Element
PP1QQ1RR1 :

O

P
Q



O1
 


x  N1 ( )  
xO 1 
 1  1
 
 xQ 
 
 


 N 0 ( )  
xO1 
xQ1 
1
 1

with N1 ( ) 

P1
R at 
Q1
R1 at 
Q
P

R
1



-1

P1
-1
Q1
Map
1

R1
1
1 
, N 0 ( ) 
2
2
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Infinite elements formulated by mapping
Infinite elements are attached to
conventional FE mesh to simulate
infinite domain. 
























Finite Element Method by G. R. Liu and S. S. Quek
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Gradual damping elements

For vibration problems with infinite domain
 Uses conventional finite elements, hence great
versatility
 Study of lamb wave propagation
Finite Element Method by G. R. Liu and S. S. Quek
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Gradual damping elements

Attaching additional damping elements outside
area of interest to damp down propagating waves
Additional damping element sections
Area of interest
of analysis
Gradual increase in structural damping
Finite Element Method by G. R. Liu and S. S. Quek
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Gradual damping elements
Structural damping is defined as
c  
H

Equation of motion with damping under
harmonic load:
(Since the energy
dissipated by
damping is usually
independent of )
mu  cu   k u  f exp( i t )
Since, damping force  
H

u
 iHu
Therefore,
k  iHu  mu  f exp( i t )
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Gradual damping elements
k  iHu  mu  f exp( i t )
Complex stiffness
Replace E with E(1 + i) where  is the
material loss factor.
Therefore, k  iH  k  ik 
Hence,
 ce u e  

k e u e   k e u

Finite Element Method by G. R. Liu and S. S. Quek
17
Gradual damping elements
For gradual increase in damping,
E k  E  i o k E
Complex modulus
for the kth damping
element set
Initial modulus
Constant factor
Initial
material loss
factor
•
Sufficient damping such that the effect of the
boundary is negligible.
•
Damping is gradual enough such that there is no
reflection cause by a sudden damped condition.
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Coupling of FEM and BEM

The FEM used for interior and the BEM for exterior which
can be extended to infinity [Liu, 1992]
Coupling of FEM and SEM

The FEM used for interior and the SEM for exterior which
can be extended to infinity [Liu, 2002]
Finite Element Method by G. R. Liu and S. S. Quek
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FINITE STRIP ELEMENTS

Developed by Y. K. Cheung, 1968.
 Used for problems with regular geometry and
simple boundary.
 Key is in obtaining the shape functions.
y
z
x
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20
FINITE STRIP ELEMENTS
r
w   f m  x Ym
(Approximation of displacement
function)
(Polynomial)
(Continuous series)
m 1
Polynomial function must represent state of
constant strain in the x direction and continuous
series must satisfy end conditions of the strip.
Together the shape function must satisfy compatibility
of displacements with adjacent strips.
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FINITE STRIP ELEMENTS
Y(0) = 0, Y’’(0) = 0, Y(a)
= 0 and Y’’(a) = 0
 y
Ym  y   sin  m 
 a 
m = , 2, 3, …, m
f m ( x)  C1 C2
C3
d1m 
 m
d 2 
C4   m 
d3 
d 4m 
y
a
Satisfies
z
x
1
2
d1, d2
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d3, d4
x
b
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FINITE STRIP ELEMENTS
2 x3 3x 2
C1  x   3  2  1
b
b
x3 2 x 2
C2  x   2 
x
b
b
2 x3 3x 2
C3  x    3  2
b
b
x3 x 2
C4 ( x )  2 
b
b
1
2
d1, d2
d3, d4
x
b
Therefore,
r


w  x, y    C1  x  d1m  C2  x  d 2m  C3  x  d3m  C4  x  d 4m Ym ( y )
m 1
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23
FINITE STRIP ELEMENTS
r


w  x, y    C1  x  d1m  C2  x  d 2m  C3  x  d3m  C4  x  d 4m Ym ( y )
m 1
or
r
w  x, y     N1m
m 1
where
N 2m
N 3m
d1m 
 m
m d 2 
N 4   m 
d3 
d 4m 
Nim  x, y   Ci  x  Ym  y 
i = 1, 2, 3 ,4
The remaining procedure is the same as the FEM.
The size of the matrix is usually much smaller and
makes the solving much easier.
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STRIP ELEMENT METHOD (SEM)





Proposed by Liu and co-workers [Liu et al., 1994,
1995; Liu and Xi, 2001].
Solving wave propagation in composite laminates.
Semi-analytic method for stress analysis of solids
and structures.
Applicable to problems of arbitrary boundary
conditions including the infinite boundary
conditions.
Coupling of FEM and SEM for infinite domains.
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