1 finite element method special purpose elements for readers of all backgrounds g. r. liu and s. s....

1

FiFinite Element Methodnite Element Method

SPECIAL PURPOSE ELEMENTS

for readers of all backgroundsfor readers of all backgrounds

G. R. Liu and S. S. Quek

CHAPTER 10:

2Finite Element Method by G. R. Liu and S. S. Quek

CONTENTSCONTENTS

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


CRACK TIP ELEMENTSCRACK TIP ELEMENTS

Fracture mechanics – singularity point at crack tip.

Conventional finite elements do not give good approximation at/near the crack tip.



x, u

y, v

r

From fracture mechanics,

31 sin sin

2 23

cos sin sin2 2 22

31 sin sin

2 2

xx

Ixy

yy

K

r

2

2

cos ( 1 2sin )2 2

2 2 sin ( 1 2cos )2 2

Iu K r

v G

(Near crack tip)

(Mode I fracture)



H

L

L/ 4

H /4

Special purpose crack tip element with middle nodes shifted to quarter position:



1 2 3

4

5 6

8

7

- 1 0 1

x

r

y

L L/4

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-)u2+0.5 (1+) u3


CRACK TIP ELEMENTSCRACK TIP ELEMENTSSimplifying,

x = 0.25(1+)2L

u= (1+)[(1-)u2+0.5u3]

Along x-axis, x = r

r = 0.25(1+)2L or (1 ) 2r

L

u = 2(r/L) [(1-)u2 + 0.5u3] Note: Displacement is proportional to r

u u

x x

0.5(1 ) = x

L r L

where

Therefore, 2 3

1 1 1[ 2 ( ) ]

2

uu u

x r L

Note: Strain (hence stress) is proportional to 1/r


CRACK TIP ELEMENTSCRACK TIP ELEMENTS Therefore, by shifting the nodes to quarter

position, we approximating the stress and displacements more accurately.

Other crack tip elements:

L

L /4

L L /4

Triangular crack tip elements

A 3 - D, wedge crack tip element


METHODS FOR INFINITE DOMAINMETHODS FOR INFINITE DOMAIN

Infinite elements formulated by mapping

(Zienkiewicz and Taylor, 2000)Gradual damping elementsCoupling of FEM and BEMCoupling of FEM and SEM


Infinite elements formulated by mappingInfinite elements formulated by mapping

Use shape functions to approximate decaying sequence:

31 22 3

...CC C

r r r

In 1D:

P Q R - 1 +1

R at P Q x

y

O

Map

11 1O Qx x x

(Coordinate interpolation)

Or x x 1 1Q O Q O

O

x x x x

x x r



If the field variable is approximated by polynomial,

2 30 1 2 3 ...u

Substituting will give function of decaying form,

For 2D (3D):1 1

1 1

1 1

11 1

11 1

11 1

O Q

O Q

O Q

x x x

y y y

z z z

31 22 3

...CC C

r r r



R at P Q

x

y

O

O 1 P 1 Q 1 R 1 at

P Q

R

P 1 Q 1 R 1

1

1

- 1

- 1

Map

Element PP1QQ1RR1 :

1 1

1

0

( ) 11 1

( )1 1

O Q

O Q

x N x x

N x x

with 1 0

1 1( ) , ( )

2 2N N



Infinite elements are attached to conventional FE mesh to simulate infinite domain.


Gradual damping elementsGradual damping elements

For vibration problems with infinite domain Uses conventional finite elements, hence great

versatility Study of lamb wave propagation



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

Gradual increase in structural damping Area of interest of analysis

Additional damping element sections



Structural damping is defined as

Hc

Equation of motion with damping under harmonic load:

) exp( tifukucum

iHu

uH

force dampingSince,

Therefore, ) exp( tii fumuHk

(Since the energy dissipated by damping is usually independent of )



) exp( tii fumuHk

Complex stiffness

Replace E with E(1 + i) where is the material loss factor.

Therefore, kkHk ii

ukukuc eeeee

Hence,



EiEE kok

For gradual increase in damping,

Complex modulus for the kth damping element set

Initial modulusInitial material loss factor

Constant 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.


Coupling of FEM and BEMCoupling 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 SEMCoupling of FEM and SEM

The FEM used for interior and the SEM for exterior which can be extended to infinity [Liu, 2002]


FINITE STRIP ELEMENTSFINITE STRIP ELEMENTS

Developed by Y. K. Cheung, 1968. Used for problems with regular geometry and

simple boundary. Key is in obtaining the shape functions.

x

y

z



1

r

m mm

w f x Y

(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.


x

y

z


Y(0) = 0, Y’’(0) = 0, Y(a) = 0 and Y’’(a) = 0 a

sin mm

yY y

a

m = , 2, 3, …, m

b

1 2

x d1, d2 d3, d4 1

21 2 3 4

3

4

( )

m

m

m m

m

d

df x C C C C

d

d

Satisfies



3 2

1 3 2

3 2

2 2

3 2

3 3 2

3 2

4 2

2 31

2

2 3

( )

x xC x

b b

x xC x x

b b

x xC x

b b

x xC x

b b

b

1 2

x d1, d2 d3, d4

Therefore,

1 1 2 2 3 3 4 41

, ( )r

m m m mm

m

w x y C x d C x d C x d C x d Y y


FINITE STRIP ELEMENTSFINITE STRIP ELEMENTS 1 1 2 2 3 3 4 4

1

, ( )r

m m m mm

m

w x y C x d C x d C x d C x d Y y

or 1

21 2 3 4

1 3

4

,

m

mrm m m m

mm

m

d

dw x y N N N N

d

d

where ,mi i mN x y C x Y 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.


STRIP ELEMENT METHOD (SEM)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.

1 finite element method special purpose elements for readers of all backgrounds g. r. liu and s. s....

Documents

conventional finite

finite strip elements

crack tip elements x

additional damping elements

r slide

crack tip elements simplifying

infinity liu

sem slide