toptology optimixation

8/10/2019 toptology optimixation

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Slide 4a.1Stiff 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 forceFixed

FixedUse 40 % material that can fit intothis rectangle


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Stiff structure for your specifications


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Your specifications for thecompliant mechanism

Hole

Fixed

Fixed

Input force

Outputdeflection

Use 30 % material


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Compliant mechanismto your specifications


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Lecture 4aDesign parameterization instructural optimization

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


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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


7/35Slide 4a.7Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Hierarchical description of the physicalform of a structure Topology 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 orlayout design



Hierarchical description of a physicalform of structure: Shape

Shape design



Hierarchical description of a physicalform of structure: Size

1R

1w

2R

1R

1R

2w

t = thickness

When the topology and shape are selected, one canoptimize by varying size related parameters such asdimensions.

Dimensional orparametric orsize design



Stiffest structure for thesespecifications 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 allpossible ways with truss elements so that all the lementslie within the design region.

Associated with each truss element, define a c/s areavariable. 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 Reviews42(8):233-239.



Ground structures with beam elements

Overlapping beam elements are avoided because they createcomplications 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 andMultidisciplinary Optimization, Vol. 19, 2000, pp. 36-49.

d l



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:a, b, and q.

ab

q

Each element is imagined to bemade of a composite materialwith microstructural voids.

Bendse, M.P., and Kikuchi, N. (1988). Generating optimal topologies in structural design using a

homogenization method. Computer Methods in Applied Mechanics and Engineering71:197-224.



Homogenization-based method (contd.)

Material with microstructure Homogeneous material withequivalent properties

Homogenization

a b q

Homogenizedpr

operty

Homogenizedpr

operty

Homogenizedpr

operty

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

h i b d



Another microstructure basedmethodThe original homogenization-based method used three

variables to get some anisotropicy (orthotropy, inparticular). But practical considerations mostly needisotropic materials.

Assume isotropic (sphericalinclusions)

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.

02

EE

Youngs modulus =

Fi i i d i h d



Fictitious density method; powerlaw model

Fictitious density approach

10with0 EE

SIMP (Solid Isotropic Material with Penalty)

10with0 EE p

pis the penalty parameter to push densitiestoblack (1) andwhite (0).

For optimization, there will be as many as thenumber of elements in the discretized model.

s'

Rozvany, G.I.N. , Zhou, M., and Gollub, M. (1989). Continuum Type Optimality Criteria Methods for LargeFinite Element Systems with a Displacement Connstraint, Part 1. Structural Optimization1:47-72.

P l i h SIMP



Penalty parameter in the SIMPmethod: some justification

230

0

E

E

23

00

EEp

Therefore,

3 p

Hashin-Shtrikman bounds

Bendse, M.P. and Sigmund, O., Material Interpolation Schemes in Topology

Optimization, Archives in Applied Mechanics, Vol. 69, (9-10), 1999, pp. 635-654.

Mi t t f i t di t



Microstructure for intermediatedensities

Bendse, 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

2

2

2

1

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.

Alternativelywith just one variable, manymaterials can be interpolated.

Yin, L. and Ananthasuresh, G.K., Topology Optimization of Compliant Mechanisms with Multiple MaterialsUsing a Peak Function Material Interpolation Scheme, Structural and Multidisciplinary Optimization, Vol.

23, No. 1, 2001, pp. 49-62.

Ad t f th k f ti b d



Advantages of the peak function basedprobabilistic material interpolation

22

22

21

21

2

2

2

1

eEeEE

1E

2E

E

1

1

Begin with large s and graduallydecrease to get peaks eventually.

voidi

N

i

EeEE ii

2

2

2

1

No bounds on the variables!

P k f ti th d f b ddi


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Peak function method for embeddingobjects

Embedded

objects

Connecting structure

Traction forceson GT

Fixedboundary

W

G

n

iiEEyxE

10

),(

i

i

ii

ii

i

i

yi

i

xi

iii

yy

xx

y

x

yxEE

EE

cossin

sincos

~

~

~~exp

exp

22

2

2

00

Z. Qian and G. K. Ananthasuresh, Optimal Embedding inTopology Optimization, CD-ROM proc. of the IDETC-2002,

Montreal, CA, Sep. 29-Oct. 2, 2002, paper #DAC-34148.

Contours (level set curves)


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Level-set method

A very powerful method for topology optimization.The boundary defined as the level set of a surface definedon the domain of interest. Zero level set curve definesthe boundary, while positive surface values define the

interior of the region.

W

GW

WW

\0)(

0)(

\0)(

Dxx

dxx

dxx

Interior

Boundary

Exterior

D W

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.


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Level set method for multiple materials

Multiple materials can be dealt with more level set surfaces.

n n2With level set surfaces, materials can be exclusivelychosen.

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


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Main points

Topology, shape, and size provide ahierarchical description of the geometry ofa 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


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Your specifications for a stiff structure

Distributed ramp force

Point forceFixed

FixedUse 40 % material that can fit intothis rectangle


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Stiff structure for your specifications


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Optimal synthesis solution

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

Actual time taken on this laptop = ~10 minutes


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Designs with different mesh sizes

96x48 = 4608 elements




Your specifications for the


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Your specifications for thecompliant mechanism

Hole

Fixed

Fixed

Input force

Outputdeflection

Use 30 % material

Compliant mechanism


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Compliant mechanismto your specifications


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A rigid-body mechanism (if you want)

Optimal compliant mechanism


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Optimal compliant mechanismto your specifications

Compliant designs for different


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Compliant designs for differentmesh sizes

Rough mesh Medium mesh

Fine mesh

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