1. Slide 4a.1 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

2. Slide 4a.2 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Stiff structure for your specifications

3. Slide 4a.3 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 Use 30 % material

4. Slide 4a.4 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Compliant mechanism to your specifications

5. Slide 4a.5 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Lecture 4a Design parameterization in structural optimization Various ways of defining design variables for size, shape, and topology optimization schemes.

6. Slide 4a.6 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Contents Hierarchical description of the physical form of a structure –Topology –Shape –Size Size (dimensional, parameter) optimization Shape optimization Topology optimization –Ground structure method –Homogenization method –Power law, and SIMP methods –Micro-structure based models –“peak” function –Level-set methods

7. Slide 4a.7 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Hierarchical description of the physical form of a structure Topology or layout Connectivity among portions of interest force support

8. Slide 4a.8 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Topology or layout (contd.) Number of holes in the design domain also determine the connectivity force support Topology or layout design

9. Slide 4a.9 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Hierarchical description of a physical form of structure: Shape Shape design

10. Slide 4a.10 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Hierarchical description of a physical form of structure: Size = thickness When the topology and shape are selected, one can optimize by varying size related parameters such as dimensions. Dimensional or parametric or size design

11. Slide 4a.11 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Stiffest structure for these specifications for a given volume 60x40=2400 120x80=9600 30x20=600 elements Results given by PennSyn program for… Volume = 40%

12. Slide 4a.12 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Design parameterization In order to optimize topology (layout), shape, or size, we need to identify optimization variables. This is called the “design parameterization”. Size optimization Thickness, widths, lengths, radii, etc. Shape optimization Polynomials Splines Bezier curves, etc. Topology optimization We will discuss in detail

13. Slide 4a.13 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Ground structure with truss elements Define a grid of joint locations and connect them in all possible ways with truss elements so that all the lements lie within the design region. Associated with each truss element, define a c/s area variable. This leads to N optimization variables. Each variable has lower (almost zero) and upper bounds. Ground structure A possible solution Kirsch, U. (1989). Optmal Topologies of Structures. Applied Mechanics Reviews 42(8):233-239.

14. Slide 4a.14 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Ground structures with beam elements Overlapping beam elements are avoided because they create complications in practical realization of the designs. Realizable slopes are limited but it does not matter in most cases. Again, each element has a design variable related to its cross- section. Saxena, A., Ananthasuresh, G.K., “On an optimal property of compliant topologies,” Structural and Multidisciplinary Optimization, Vol. 19, 2000, pp. 36-49.

15. Slide 4a.15 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Continuum modeling: the homogenization-based method At each point, we need to interpolate the material between 0 and 1 in order to do optimization. Three optimization variables per element: , , and .    Each element is imagined to be made of a composite material with microstructural voids. Bendsøe, M.P., and Kikuchi, N. (1988). Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering 71:197-224.

16. Slide 4a.16 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Homogenization-based method (contd.) Material with microstructure Homogeneous material with equivalent properties Homogenization  Homogenized property Relevant homogenized properties are pre-computed and fitted to smooth polynomials for ready interpolation.

17. Slide 4a.17 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Another microstructure based method The original homogenization-based method used three variables to get some anisotropicy (orthotropy, in particular). But practical considerations mostly need isotropic materials. Assume isotropic (spherical inclusions) Volume fraction = Gea, H. C., 1996, Topology Optimization: A New Micro-Structural Based Design Domain Method, Computers and Structures, Vol. 61, No. 5, pp. 781 – 788. Young’s modulus =

18. Slide 4a.18 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Fictitious density method; power law model Fictitious density approach SIMP (Solid Isotropic Material with Penalty) p is the penalty parameter to push densities to black (1) and white (0). For optimization, there will be as many as the number of elements in the discretized model. Rozvany, G.I.N., Zhou, M., and Gollub, M. (1989). Continuum Type Optimality Criteria Methods for Large Finite Element Systems with a Displacement Connstraint, Part 1. Structural Optimization 1:47-72.

19. Slide 4a.19 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Penalty parameter in the SIMP method: some justification Therefore, Hashin-Shtrikman bounds Bendsøe, M.P. and Sigmund, O., “Material Interpolation Schemes in Topology Optimization,” Archives in Applied Mechanics, Vol. 69, (9-10), 1999, pp. 635-654.

20. Slide 4a.20 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Microstructure for intermediate densities Bendsøe, M.P. and Sigmund, O., “Material Interpolation Schemes in Topology Optimization,” Archives in Applied Mechanics, Vol. 69, (9-10), 1999, pp. 635-654.

21. Slide 4a.21 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Multiple-material interpolation 00.51 0 1 0 1 For two-materials, in the SIMP method, two variables are needed. Alternatively…with just one variable, many materials can be interpolated. Yin, L. and Ananthasuresh, G.K., “Topology Optimization of Compliant Mechanisms with Multiple Materials Using a Peak Function Material Interpolation Scheme,” Structural and Multidisciplinary Optimization, Vol. 23, No. 1, 2001, pp. 49-62.

22. Slide 4a.22 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Advantages of the peak function based probabilistic material interpolation Begin with large  ’s and gradually decrease to get peaks eventually. No bounds on the variables!

23. Slide 4a.23 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Peak function method for embedding objects Embedded objects Connecting structure Traction forces on  T Fixed boundary   Z. Qian and G. K. Ananthasuresh, “Optimal Embedding in Topology Optimization,” CD-ROM proc. of the IDETC-2002, Montreal, CA, Sep. 29-Oct. 2, 2002, paper #DAC-34148. Contours (level set curves)

24. Slide 4a.24 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Level-set method A very powerful method for topology optimization. The boundary defined as the level set of a surface defined on the domain of interest. “Zero” level set curve defines the boundary, while positive surface values define the interior of the region. Interior Boundary Exterior M. Y. Wang, X. M. Wang, and D. M. Guo, “A Level Set Method for Structural Topology Optimization,” Computer Methods in Applied Mechanics and Engineering, 192 (1), pp. 227-246, 2003.

25. Slide 4a.25 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Level set method for multiple materials Multiple materials can be dealt with more level set surfaces. With level set surfaces, materials can be exclusively chosen. Two level sets and four materials Three level sets and eight materials M. Y. Wang, personal communication, 2003. 3 1 2 4 5 6 7 8 2 3 4 1

26. Slide 4a.26 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Main points Topology, shape, and size provide a hierarchical description of the geometry of a structure. Different “smooth” interpolations techniques for topology optimization SIMP is widely used Peak function based probabilistic interpolation method can easily handle multiple materials with few variables Level-set method provides a larger design space

27. Slide 4a.27 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

28. Slide 4a.28 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Stiff structure for your specifications

29. Slide 4a.29 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Optimal synthesis solution Solved with 96x48 = 4608 variables in the optimization problem. Actual time taken on this laptop = ~10 minutes

30. Slide 4a.30 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Designs with different mesh sizes 96x48 = 4608 elements 72x36 = 2592 elements 48x24 = 1152 elements 24x12 = 288 elements

31. Slide 4a.31 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 Use 30 % material

32. Slide 4a.32 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Compliant mechanism to your specifications

33. Slide 4a.33 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh A rigid-body mechanism (if you want)

34. Slide 4a.34 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Optimal compliant mechanism to your specifications

35. Slide 4a.35 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Compliant designs for different mesh sizes Rough mesh Medium mesh Fine mesh