Finite Element Method

1
Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 10

• SPECIAL PURPOSE ELEMENTS

2
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

3
CRACK TIP ELEMENTS

• Fracture mechanics singularity point at crack
  tip.
• Conventional finite elements do not give good
  approximation at/near the crack tip.

4
CRACK TIP ELEMENTS
From fracture mechanics,
(Near crack tip)
(Mode I fracture)
5
CRACK TIP ELEMENTS
Special purpose crack tip element with middle
nodes shifted to quarter position
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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
(Measured from node 1)
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-?)u20.5? (1?) u3
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CRACK TIP ELEMENTS
Simplifying,
x 0.25(1?)2L
u (1?)(1-?)u20.5?u3
Along x-axis, x r
r 0.25(1?)2L or
Note Displacement is proportional to ?r
?
u 2(?r/?L) (1-?)u2 0.5?u3
where
Note Strain (hence stress) is proportional to
1/?r
Therefore,
8
CRACK TIP ELEMENTS

• Therefore, by shifting the nodes to quarter
  position, we approximating the stress and
  displacements more accurately.
• Other crack tip elements

9
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

10
Infinite elements formulated by mapping
Use shape functions to approximate decaying
sequence
In 1D
(Coordinate interpolation)
?
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Infinite elements formulated by mapping
If the field variable is approximated by
polynomial,
Substituting ? will give function of decaying
form,
For 2D (3D)
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Infinite elements formulated by mapping
Element PP1QQ1RR1
with
13
Infinite elements formulated by mapping
Infinite elements are attached to conventional FE
mesh to simulate infinite domain.
14
Gradual damping elements

• For vibration problems with infinite domain
• Uses conventional finite elements, hence great
  versatility
• Study of lamb wave propagation

15
Gradual damping elements

• Attaching additional damping elements outside
  area of interest to damp down propagating waves

16
Gradual damping elements
(Since the energy dissipated by damping is
usually independent of ?)

• Structural damping is defined as

Equation of motion with damping under harmonic
load
Since,
Therefore,
17
Gradual damping elements
Complex stiffness
Replace E with E(1 i?) where ? is the material
loss factor.
Therefore,
Hence,
18
Gradual damping elements
For gradual increase in damping,
Constant factor
Complex modulus for the kth damping element set
Initial modulus
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

20
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.

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FINITE STRIP ELEMENTS
(Approximation of displacement function)
(Polynomial)
(Continuous series)
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
a
Satisfies
?m ?, 2?, 3?, , m?
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FINITE STRIP ELEMENTS
Therefore,
24
FINITE STRIP ELEMENTS
or
where
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.
25
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.