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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?. Your specifications for a stiff structure. Distributed ramp force. Fixed. - PowerPoint PPT Presentation
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Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.1
Lecture 4: follow-upSome 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.2
Your specifications for a stiff structure
Distributed ramp force
Point forceFixed
Fixed
Use 40 % material that can fit into this rectangle
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.3
40% volume
10% volume20% volume
30% volume
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.4
Your specifications for the compliant mechanism
Hole
Fixed
Fixed
Input force
Output deflection
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.5
40% volume 30% 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.6
Effect of changing desired direction of output deflection
Hole
Fixed
Fixed
Input force
Output deflection
Use 20 % material
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.7
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.8
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.9
Ways to avoid the checker-board pattern
• Perimeter constraintHaber, 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.
• Global constraint on artificial density variation
• Local constraints on artificial density variations
• Filters
*22 GdV
)2,1(
icxi
Petersson, J. and Sigmund, O., “Slope constrained Topology Optimization,” Int. J. Numer. Meth. In Engineering, 41, 1998, pp. 1417-1434.
Bendsoe, M.P., Optimization of Structural Topology, Shape, and Material, Springer, Berlin, 1995.
*PdV
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.10
Filters to avoid the checkerboard pattern
Sigmund, O., Design of Material Structures Using Topology Optimization, Ph.D. Thesis, Dept. Solid Mechanics, Technical University of Denmark.
r
n
kk
n
k kk
i kir
fkir
f
1
1
)),(dist(
)),(dist(
Compute sensitivity as a weighted average of sensitivities of elements within a prescribed radius.
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.
N
i
yyxx
i
ii
eEyxE1
)()(
0
2
22
),( Distributed interpolation of the material properties
Bourdin, B., “Filters in Topology Optimization,” Int. J. for Numer. Meth. In Engrg., 50(9), 2001, pp. 2143-2158.
See for a more mathematical treatment of filters:
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.11
Ways to avoid point flexures
• Restraining relative rotation at all material points
Force
Desired disp.
a
d c
b
ii
iv
1 2
4 3
dbdbcaca /cos/cos and
cos1coscos 0 cos1coscos 0 Relative rotations at a node
Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4f.12
A way to avoid point flexures
cos1cos11
nn
ii
n
kkk
MSE
1iviiiiii cos1cos1
2
2
exp1
0*
1
SESEN
ii
Minimize
Subject to
Noting that
Equilibrium equations
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.
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