1. SPECIAL PURPOSE ELEMENTS
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:

6. CRACK TIP ELEMENTS Move node 2 to L/4 position 
x = -0.5 (1-)x1 + (1+)(1-)x  (1+) x3
u = -0.5 (1-)u1 + (1+)(1-)u  (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-)u2+0.5 (1+) u3

7. CRACK TIP ELEMENTS Simplifying, Along x-axis, x = r  where Therefore,
x = 0.25(1+)2L
u= (1+)[(1-)u2+0.5u3]
Along x-axis, x = r
r = 0.25(1+)2L or 
Note: Displacement is proportional to r
u = 2(r/L) [(1-)u 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) 

11. Infinite elements formulated by mapping
If the field variable is approximated by polynomial,
Substituting  will give function of decaying form,
For 2D (3D):

12. 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.

19. Coupling of FEM and BEM Coupling of FEM and SEM
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.

21. 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.

22. FINITE STRIP ELEMENTS Y(0) = 0, Y’’(0) = 0, Y(a) = 0 and Y’’(a) = 0
Satisfies
m = , 2, 3, …, m

23. 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.