1. Slide 4f.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh 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?

2. Slide 4f.2 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Your specifications for a stiff structure Distributed ramp force Point force Fixed Use 40 % material that can fit into this rectangle

3. Slide 4f.3 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh 40% volume 10% volume20% volume 30% volume

4. Slide 4f.4 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Your specifications for the compliant mechanism Hole Fixed Input force Output deflection

5. Slide 4f.5 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh 40% volume 30% volume 20% volume

6. Slide 4f.6 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Effect of changing desired direction of output deflection Hole Fixed Input force Output deflection Use 20 % material

7. Slide 4f.7 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

8. Slide 4f.8 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh 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.

9. Slide 4f.9 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh 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. Global constraint on artificial density variation Local constraints on artificial density variations Filters 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.

10. Slide 4f.10 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh 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. 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. 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:

11. Slide 4f.11 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Ways to avoid point flexures Restraining relative rotation at all material points Force Desired disp. a dc b ii iv 12 43 Relative rotations at a node

12. Slide 4f.12 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh A way to avoid point flexures 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.