your specifications for a stiff structure

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

Your specifications for a stiff structure

Distributed ramp force

Point forceFixed

Fixed

Use 40 % material that can fit into this rectangle


Stiff structure for your specifications


Your specifications for the compliant mechanism

Hole

Fixed

Fixed

Input force

Output deflection

Use 30 % material


Compliant mechanism to your specifications


Lecture 4aDesign parameterization in structural optimization Various ways of defining design variables for size, shape, and topology optimization schemes.


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


Hierarchical description of the physical form of a structureTopology or layout

Connectivity among portions of interest

force

force

support


Topology or layout (contd.)Number of holes in the design domain also determine the connectivityforce

force

support

Topology or layout design


Hierarchical description of a physical form of structure: Shape

Shape design


Hierarchical description of a physical form of structure: Size

1R

1w

2R

1R

1R

2w

t= 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


Stiffest structure for these specifications for a given volume

60x40=2400

120x80=9600

30x20=600 elements

Results given by PennSyn program for…

Volume = 40%


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


Ground structure with truss elementsDefine 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.


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.


Continuum modeling:the homogenization-based method

At each point, we need to interpolate the materialbetween 0 and 1 in order to do optimization.

Three optimizationvariables 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.


Homogenization-based method (contd.)

Material with microstructure Homogeneous material with equivalent properties

Homogenization

Hom

ogeniz

ed p

ropert

y

Hom

ogeniz

ed p

ropert

y

Hom

ogeniz

ed p

ropert

y

Relevant homogenized properties are pre-computed and fitted to smooth polynomials for ready interpolation.


Another microstructure based methodThe 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.

02EE

Young’s modulus =


Fictitious density method; power law model

Fictitious density approach

10with0 EE

SIMP (Solid Isotropic Material with Penalty)

10with0 EE p

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.

s'

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.


Penalty parameter in the SIMP method: some justification

230

0

EE

23

00

EEp

Therefore,

3 p

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.


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.


Multiple-material interpolation

22

22

21

21

22

21

eEeEE

0E

E

0 0.5 1 0 0.5 1 0 0.5 1

21112 )1( EEE 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.


Advantages of the peak function based probabilistic material interpolation

22

22

21

21

22

21

eEeEE

1E

2E

E

1 1

Begin with large ’s and graduallydecrease to get peaks eventually.

voidi

N

i

EeEE i

i

2

2

2

1

No bounds on the variables!


Peak function method for embedding objects

Embedded objects

Connecting structure

Traction forces on T

Fixed boundary

n

iiEEyxE

10

ˆˆ),(

i

i

ii

ii

i

i

yi

i

xi

iii

yy

xx

y

x

yxEE

EE

cossin

sincos~

~

~~expˆ

expˆ

22

2

2

00

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)


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.

\0)(

0)(

\0)(

Dxx

dxx

dxx

Interior

Boundary

Exterior

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.


Level set method for multiple materials

Multiple materials can be dealt with more level set surfaces.

n n2With 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.

312

4

56

78

2

34

1


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


Your specifications for a stiff structure

Distributed ramp force

Point forceFixed

Fixed

Use 40 % material that can fit into this rectangle


Stiff structure for your specifications


Optimal synthesis solution

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

Actual time taken on this laptop = ~10 minutes


Designs with different mesh sizes

96x48 = 4608 elements





Your specifications for the compliant mechanism

Hole

Fixed

Fixed

Input force

Output deflection

Use 30 % material


Compliant mechanism to your specifications


A rigid-body mechanism (if you want)


Optimal compliant mechanism to your specifications


Compliant designs for different mesh sizes

Rough mesh Medium mesh

Fine mesh

your specifications for a stiff structure

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