Transcript PPT file (~1.1 MB) - Mechanical Engineering at IISc

Lecture 4: follow-up
Some results and discussion
What happens to topology when the volume is reduced?
What happens if the desired direction of the output is
changed?
What are the things to watch out for?
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.1
Your specifications for a stiff structure
Distributed ramp force
Fixed
Use 40 % material that can fit into
this rectangle
Fixed
Point force
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.2
40% volume
30% volume
20% volume
10% volume
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.3
Your specifications for the
compliant mechanism
Fixed
Output
deflection
Hole
Input force
Fixed
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.4
30% volume
40% volume
20% volume
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.5
Effect of changing desired
direction of output deflection
Use 20 % material
Fixed
Output
deflection
Hole
Input force
Fixed
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.6
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.7
Some things to watch out for
• Mesh dependency
• Non-convexity
• Numerical artifacts:
– Checker-board pattern in structures
– Point flexures in compliant mechanisms
A checker-board pattern is artificially
stiff.
A point flexure is artificially flexible while
minimizing the strain energy.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.8
Ways to avoid the checker-board
pattern
• Perimeter constraint
Haber, R.B., Bendsoe, M.P., and Jog, C., “A new approach to variable-topology shape
design using a constraint on the perimeter,” Structural Optimization, 11, 1996, pp. 1-12.
*


dV

P


• Global constraint on artificial density
variation
    dV  G
2
2
*
Bendsoe, M.P., Optimization of Structural Topology, Shape,
and Material, Springer, Berlin, 1995.

• Local constraints on artificial density
variations

 c (i  1,2)
xi
Petersson, J. and Sigmund, O., “Slope constrained Topology
Optimization,” Int. J. Numer. Meth. In Engineering, 41,
1998, pp. 1417-1434.
• Filters
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.9
Filters to avoid the checkerboard
pattern
Compute sensitivity as a weighted
r
average of sensitivities of elements
within a prescribed radius.

f 
 (r  dist (i, k ))  k

 k 
k 1 
f

n
 i
 (r  dist (i, k ))  k 
n
k 1
Sigmund, O., Design of Material Structures Using Topology Optimization, Ph.D. Thesis, Dept. Solid
Mechanics, Technical University of Denmark.
( x  x ) ( y  y )

Distributed interpolation of

N


E
(
x
,
y
)

E

e


0
i
the material properties
 i 1
2
i
i
2

2





Bruns, T.E. and Tortorelli, D., “Topology Optimization of Nonlinear Elastic Structures and Compliant
Mechanisms,” Comp. Meth. In App. Mech. And Engrg., 190 (26-27), 2001, pp. 3443-3459.
See for a more mathematical treatment of filters:
Bourdin, B., “Filters in Topology Optimization,” Int. J. for Numer. Meth. In Engrg., 50(9), 2001, pp. 21432158.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.10
Ways to avoid point flexures
Force
Desired
disp.
• Restraining relative
rotation at all material
points
4
1
3
d
iv
c
a
b
ii
2
cos  a  c / a c  and cos   b  d / b d 
cos0  cos  1  cos
Relative rotations at a node cos 0  cos   1  cos 
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.11
A way to avoid point flexures
 1  cos  i   n1  cos  
n
Noting that
Minimize 
i 1
MSE
   i   iii 1  cos k     ii   iv 1  cos  k 
n
k 1
Subject to
 SEi   SE *  0
N
i 1
Equilibrium equations
 2 
    1  exp  2 
  
Yin, L. and Ananthasuresh, G.K., “A Novel Formulation for the Design of Distributed Compliant
Mechanisms,” Mechanics Based Design of Structures and Machines, Vol. 31, No. 2, 2003, pp. 151-179.
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh
Slide 4f.12