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.