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Topology OptimizationMathematics for Design

Homogenization Design Method(HMD)

Why topology ?

Change in shape & size may not lead our design criterion for reduction of

structural weight.

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Structural Design3 Sets of Problems

Sizing Optimization thickness of a plate or

membrane height, width, radius of the

cross section of a beam

Shape Optimization outer/inner shape

Topology Optimization number of holes configuration

Shape of the Outer Boundary

Location ofthe ControlPoint of aSpline

thickness distribution

hole 2hole 1

Sizing OptimizationStarting of Design Optimization

1950s : Fully Stressed Design

1960s : Mathematical Programming ( L. Schmit at UCLA )

= allowable in a structure

minmax

allowableu

Total Weightu

Design Sensitivity Analysis

3

Ku = f

1

Design VelocitySensitivity

Dg g gD

Dg g gD

= +

= + =

= + +

ud d d u

K u fKu f u Kd d d

K fK ud d d d u

Equilibrium : State Equation

Design SensitivityPerformanceFunctions g

Typical Performance FunctionsStrain Energy Density

For Structural Design (This must be constant !)

Mises Equivalent StressFor Strength Design and Failure Analysis

Mean Compliance & Maximum DisplacementFor Stiffness Design

Maximum StrainFor Formability Study of Sheet Metals

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Hemp in 1950sSize to Topology

Eliminate unnecessary bars by designing the cross sectional area.

An Optimization Algorithm

P1 P2

E, Amin

max

max

Ku f

u

=

=

e allowablei u

e e ee

N

A L1

K u K u f

D B uD B u

u uu

u

=

+

=

=

=

A A A

A A A

A A

e e e

e

e

e e e

ee e

e

e

i

e

i

i

i

e

b g

Design Sensitivity

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Prager in 1960sDesign Optimization Theory

Maximizing the minimum total potential energy

1 1

12

e eN NT T

e e e e e ee e= =

= = d K d d f

max mine

e

designA

Leads Equilibrium

d

Why Total Potential ?Maximizing the Global Stiffness

Minimizing the mean compliance (Prager) when forces are applied

Maximizing the mean compliancewhen displacement is specified

min T sdesign u f

max Tsdesign u f

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1 1

Weight Constraint

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E EN NT T

e e e e e e e ee e

Total Potential Energy

L A L W = =

= +

d K d d f

( )1

1

12

E

E

NT T e

e e e e e e e e ee e

N

e e ee

L L AA

A L W

=

=

= +

+

Kd K d f d d

Lagrangian

Variation

1

1 02

0E

e e e

T ee e e e

eN

e e ee

LA

A L W

=

=

+ =

K d f

Kd d

Optimality Condition

1 12

T ee e

e e eL A

= Kd d

Something must beConstant !

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Physical MeaningStrain Energy Density Must be Constant

1 12

1 12

T ee e

e e e

Te e e

e e e

Weight Average of the Stiffness

L A

A L

=

=

Kd d

d K d

Pragers Condition

Example 1

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Example 2

(a) Single Loading

(b) Multiple Loading

Design Domain

Example 3

Applying Torque

Design Domain

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TOPODANUKIA Topology Optimization Soft

Toyota Central R&D Labs.

Making up a grand-structure

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Set up support and load conditions

Only a bending load is applied

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Two Loads are applied

Further DevelopmentFirst Order Analysis in Toyota Central R&D

Microsoft EXCEL Based Software

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13

14

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Extension to ContinuumCharacteristic Function

D

= unknown optimum domainD = specified fixed domain

xxx x D

a f =

RST10

ifif i.e. \

What can we get from this ?Optimal Material Distribution

1 12 2

new

T T

DU d dD

= =

= = D

D D

Strain Energy of a Body

Shape Design

Find the best Find the best Dnew

Material Design

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Homogenization Design Method Shape and Topology Design of Structures is

transferred to Material Distribution Design(Bendsoe and Kikuchi, 1988)

X

Y

b

t

t

g

Unit cell

Unit cell

Review Under the assumption of periodic microstructures which can be

represented by unit cells.

Using the asymptotic expansion of all variables and the average technique to determine the homogenized material properties and constitutive relations of composite materials.

Homogenization Method : Mathematics

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HDM Test Problem

P

Design Domain

Nondesign Domain

support

40

55

R10

15

20

20

Starting from Uniform Perforation

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19

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Design ProcessStructural Formation Process

2.800

3.000

3.200

3.400

3.600

3.800

4.000

4.200

4.400

0 10 20 30 40 50Ieration

Convergence History of Iteration

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Mesh Refinement

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23

Change Volumes

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Design Constraint

Design area

No design area

(Full material)

Load case 1

Load case 2

Load case 3

Displacement fixed along circle

Design area

No design area

(Full material)

No design area

(No material)

(Same boundary condition)

100

40

20

102

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Result of Design Constraint

Influence of Design Domain

Non-design Domain

Design Domain

Design Domain

1.25

1.25

5

2

2

1.25

1.25

5

0.5

12

12

0.5

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Different Topology

Shape Design Example

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10

60

30

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Shape to Topology

Extension to ShellsRib Formation

P

20

20

20

10

30

h 0 =0.1 h 1 =1.0

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Commercialization of HMDFrom University to Industry

Three-dimensionalshaping of a structure forOptimum without any spline functions

OPTISHAPE Development1986~1989

AcceptanceTopology Optimization Methods

Commercial Codes have been developed in USA, Europe, and Pacific Regions

OPTISHAPE@Quint Corporation, Tokyo, Japan, 1989

OPTISTRUCT@Altair Computing, Troy, USA, 1996

MSC/CONSTRUCT@MSC German, 1997 And Others (OPTICON, ANSYS, ..)

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MSC/NASTRAN-OPTISHAPE

Quint/OPTISHAPE + MSC/NASTRAN Shape and Topology Optimization

Static Global Stiffness Maximization Maximizing the Mean Eigenvalues

Frequency Control for Free Vibration Increase of the Critical Load

MSC/PATRAN integration Developed by MSC Japan and Quint Corp.

Static/Dynamic Stiffness Maximization

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MSC/PATRANGUI Environment

MSC/NASTRAN Solver

Design Example by MSC.NASTRAN-OPTISHAPE

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Integration withShape Optimization

Prof. Azegamis Method

Shape Design Optimization by MSC.NASTRAN-OPTISHAPE

Optimized

Initial Design

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Compliant Mechanism Design by QUINT/OPTISHAPE

Application of QUINT/OPTISHAPE @ Kanto Automotive

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Surface GeometrySurface GeometryGenerationGeneration

TopologyTopologyOptimizationOptimization

Package SpacePackage Space

Parametric Shape VectorsParametric Shape VectorsSize and Shape Size and Shape

OptimizationOptimization

System Level RequirementsSystem Level Requirements

Finite ElementFinite ElementModelingModeling

Control Arm Development ExampleControl Arm Development Example

Altair: Concept Design Environment Product Design Synthesis

Altair: Concept Design Environment Product Design Synthesis

Altair/OptiStruct Input:

FE model of design space Load cases, frequencies, constraints Mass target

Output: Optimal material distribution via density plot CAD geometry interpretation : using OSSmooth

Thenuse to create optimal design

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OptiStruct Version 3.4 Expanded Objective function

Minimize Mass, Stiffness or Frequency Constraints on Mass, Stiffness, Freq, Disp

Now available on Windows NT FE improvements, faster solution time HTML/Windows on-line documentation Improved integration with HyperMesh3.0

OptiStruct Case StudyVolkswagen Bracket

Minimize Mass of Engine Bracket Subject to stiffness/frequency constraints

7 loadcases: operating, pulley, transport

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OptiStruct Case StudyVolkswagen Bracket Results

Mass reduced by 23% Original mass 950g ; Final mass 730g

Performance targets were met

OptiStruct: Topography Designfor Future Automotive Body Engineering

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ALTAIR/OPTISTRUCT Results

Extension of HDMTopology Optimization Method

Structural Design Static and Dynamic Stiffness Design Control Eigen-Frequencies Design Impact Loading Elastic-Plastic Design

Material Microstructure Design Youngs and Shear Moduli, Poissons Ratios Thermal Expansion Coefficients

Flexible Body Design (MEMS application) Piezocomposite and Piezoelectric Actuator Design

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QUINT/OPTISHAPE Application to Contro Frequencies

Material DesignSpecial Mechanism : Negative

Special Mechanism

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Compliant Mechanism DesignProfessor S. Kota @ UM

Negative Thermal ExpansionBing-Chung Chens Design

=

=

9.58007.52

89.70001.8

H

H

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