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 4a.1

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

Your specifications for the compliant mechanism Use 30 % material 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 4a.3

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

Lecture 4a Design parameterization in structural optimization

Various ways of defining design variables for size, shape, and topology optimization schemes.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.5

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 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.6

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

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

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

Hierarchical description of a physical form of structure: Size When the topology and shape are selected, one can optimize by varying size related parameters such as dimensions.

R

1

R

2

w

2

w

1

R

1

R

1

t

= thickness Dimensional or parametric or size design Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.10

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

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 Slide 4a.12

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.

Ground structure A possible solution 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.

Kirsch, U. (1989). Optmal Topologies of Structures. Applied Mechanics Reviews 42(8):233-239.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.13

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.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.14

Continuum modeling: the homogenization-based method At each point, we need to interpolate the material between 0 and 1 in order to do optimization.

q b a Three optimization variables per element: a , b , and q .

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.

Slide 4a.15

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

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

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 =   2   

E

0 Gea, H. C., 1996, Topology Optimization: A New Micro-Structural Based Design Domain Method, Computers and Structures, Vol. 61, No. 5, pp. 781 – 788.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.17

Fictitious density method; power law model Fictitious density approach

E

 

E

0 with 0    1 SIMP (Solid Isotropic Material with Penalty)

E

 

p E

0 with 0    1

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.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.18

Penalty parameter in the SIMP method: some justification Hashin-Shtrikman bounds 0 

E

 

E

0 3  2  Therefore , 

p E

0  

E

0 3  2  

p

 3 Bendsøe, M.P. and Sigmund, O., “Material Interpolation Schemes in Topology Optimization,” Archives in Applied Mechanics, Vol. 69, (9-10), 1999, pp. 635-654.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.19

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.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.20

Multiple-material interpolation For two-materials, in the SIMP method, two variables are needed.

E

  2   1

E

1  ( 1   1 )

E

2  Alternatively…with just one variable, many materials can be interpolated.

E



E

1

e

   2    1 1 2  2 

E

2

e

   2    2 2 2  2

E E

0 0  0.5

1 0 0.5

 1 0 0.5

 1 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.

Slide 4a.21

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

E E

1

E



i N

  1

E i e

   2   

i i

2  2 

E void E



E

1

e

   2    1 1 2  2 

E

2

e

   2    2 2 2  2 Begin with large  ’s and gradually decrease to get peaks eventually.

No bounds on the variables!

E

2   1  1 Slide 4a.22

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 Fixed boundary G Connecting structure Traction forces on G T W

E

(

x

, 0 

y

)

E

0  exp   0   

i



n

  1  ˆ

i

  2  2   ˆ

i



E i

exp      

i

2

xi



i

2

yi

 Contours (level set curves)      ~

i i

       cos sin 

i



i

sin 

i

cos 

i

   

x y

 

x y i i

  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.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.23

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.

 (

x

)   (

x

)  0 0  (

x

)  0 

x

 W \

d

W 

x



d

W  G 

x



D

\ W Interior Boundary Exterior W

D

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.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.24

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.

2

n

3 1 2 4 5 4 7 1 2 3 6 8 Two level sets and four materials Three level sets and eight materials M. Y. Wang, personal communication, 2003.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.25

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 Slide 4a.26

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 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 4a.27

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

Optimal synthesis solution Solved with 96x48 = 4608 variables in the optimization problem.

Actual time taken on this laptop = ~10 minutes Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 4a.29

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

Your specifications for the compliant mechanism Use 30 % material 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 4a.31

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

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

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

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