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Topology Optimization
Mathematics for Design Homogenization Design Method (HMD)

Why topology ?
Change in shape & size may not lead our design criterion for reduction of structural weight.

1

Structural Design
3 Sets of Problems
? Sizing Optimization
? thickness of a plate or membrane ? height, width, radius of the cross section of a beam
Shape of the Outer Boundary Location of the Control Point of a Spline hole 2 hole 1

? Shape Optimization
? outer/inner shape

? Topology Optimization
? number of holes ? configuration

thickness distribution

Sizing Optimization
Starting of Design Optimization
1950s : Fully Stressed Design

σ = σ allowable in a structure
1960s : Mathematical Programming ( L. Schmit at UCLA )
σ ≤ σ allowable u ≤ umax

min

Total Weight
Design Sensitivity Analysis

2

Equilibrium : State Equation

Ku = f
Design Sensitivity

Dg ?g = + Dd ?d

Design Velocity Sensitivity

?u ?d N

?g ?u

Performance Functions g

Ku = f ?

?K ?u ?f = u+K ?d ?d ?d

?f ? ?g Dg ?g ? ?K = + K ?1 ? ? u+ ? ?d ? ?u Dd ?d ? ?d

Typical Performance Functions
Strain Energy Density For Structural Design (This must be constant !) Mises Equivalent Stress For Strength Design and Failure Analysis Mean Compliance & Maximum Displacement For Stiffness Design Maximum Strain For Formability Study of Sheet Metals

3

Hemp in 1950s
Size to Topology

Eliminate unnecessary bars by designing the cross sectional area.

An Optimization Algorithm
N max
E, A

Ku = f σ e ≤ σ allowable ui ≤ umax

min

∑ρ A L
e =1 e e

e

Design Sensitivity
K
P1 P2

?u ?K ?f =? u+ ?Ae ?Ae ?Ae

?u ?σ e ? De Be ue = De Be e = ?Ae ?Ae ?Ae ? ui u ?u = i ? i ?Ae ui ?Ae

b

g

4

Prager in 1960s
Design Optimization Theory
Maximizing the minimum total potential energy

1 T T Π = ∑ Π e = ∑ d e K ed e ? d e f e e =1 e =1 2

Ne

Ne

max
design Ae

min Π de 

Leads Equilibrium

Why Total Potential ?
Maximizing the Global Stiffness
Minimizing the mean compliance (Prager) when forces are applied

min uT f s
design

Maximizing the mean compliance when displacement is specified

max u s f
design

T

5

Lagrangian

? NE ? 1 NE T T L = ∑ d e K ed e ? d e fe + λ ? ∑ ρ e Ae Le ? W ? 2 e =1 e =1 ? ? 


Total Potential Energy Weight Constraint

Variation

δ L = ∑ δ de
e =1

NE

T

(K e d e ? fe ) + ?

? 1 T ?K e ? de d e ? λρ e Le ? δ Ae ?Ae ?2 ?

? NE ? + δλ ? ∑ ρ e Ae Le ? W ? ? e =1 ?

Optimality Condition
K ed e = fe 1 T ?K e de d e + λρ e Le = 0 ?Ae 2
Something must be Constant !

∑ρ A L
e =1 e e

NE

e

?W ≤ 0 1 T 1 ?K e de d e = ?λ ρ e Le ?Ae 2

6

Physical Meaning
Strain Energy Density Must be Constant

1 T 1 ?K e de d e = ?λ ρ e Le ?Ae 2 ? 1 T de 2 1 Ke ρ e Ae Le 

Weight Average of the Stiffness

Prager’s Condition

d e = ?λ

Example 1

7

Example 2
160

Design Domain

(a) Single Loading

Example 3
200 100

100

(b) Multiple Loading

Design Domain
0 10

Applying Torque

8

TOPODANUKI
A Topology Optimization Soft Toyota Central R&D Labs.

Making up a grand-structure

9

Set up support and load conditions

Only a bending load is applied

10

Two Loads are applied

Further Development
First Order Analysis in Toyota Central R&D

Microsoft EXCEL Based Software

11

12

13

14

Extension to Continuum
Characteristic Function
?

? = unknown optimum domain D = specified fixed domain

D

χ? x =

1 af R S T0

if if

x ∈? x ? ? i.e. x ∈ D \ ?

What can we get from this ?
Optimal Material Distribution
Strain Energy of a Body

U=

1 1 T T ? = χ ? D εdD ε D ε ε d N ∫ ∫ N D ? 2 2 =σ =D
new

Shape Design Find the best ?

Material Design Find the best Dnew

15

Homogenization Design Method
? Shape and Topology Design of Structures is transferred to Material Distribution Design (Bendsoe and Kikuchi, 1988)

Homogenization Method : Mathematics
t
Γt

Y

Unit cell

?

b

Γ

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.

X

Γg

Unit cell

16

HDM Test Problem
Design Domain Nondesign Domain R10 20 15 support 20

40

P

55

Starting from Uniform Perforation

17

18

19

Design Process
Structural Formation Process
Convergence History of Iteration
4.400 4.200 4.000 3.800 3.600 3.400 3.200 3.000 2.800 0 10 20 Ieration 30 40 50

20

Mesh Refinement

21

22

Change Volumes

23

Design Constraint
Displacement fixed along circle Design area Load case 1

Load case 2 No design area (Full material) Load case 3

100
Design area

2

10

20
No design area (Full material) (Same boundary condition)

40

No design area (No material)

24

Result of Design Constraint

Influence of Design Domain
12 2 2 1.25 Design Domain 5

1.25

Non-design Domain

1.25

Design Domain

5

1.25 0.5 12 0.5

25

Different Topology

Shape Design Example

20

60 30 10

26

Shape to Topology

Extension to Shells
Rib Formation
P

20 30 20 10 h 0=0.1 h 1=1.0

20

27

Commercialization of HMD
From University to Industry
Three-dimensional shaping of a structure for Optimum without any spline functions

OPTISHAPE Development 1986~1989

Acceptance
Topology 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, …..)

28

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

29

MSC/PATRAN GUI Environment

MSC/NASTRAN Solver

Design Example by MSC.NASTRAN-OPTISHAPE

30

Integration with Shape Optimization

Prof. Azegami’s Method

Initial Design

Optimized

Shape Design Optimization by MSC.NASTRAN-OPTISHAPE

31

Compliant Mechanism Design by QUINT/OPTISHAPE

Application of QUINT/OPTISHAPE @ Kanto Automotive

32

Altair: Altair: Concept Concept Design Design Environment Environment Product Product Design Design Synthesis Synthesis

System Level Requirements Package Space Topology Optimization

Control Arm Development Example
Surface Geometry Generation

Size and Shape Optimization

Parametric Shape Vectors

Finite Element Modeling

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

? Then…use to create optimal design

33

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 Study
Volkswagen Bracket ? Minimize Mass of Engine Bracket
? Subject to stiffness/frequency constraints

? 7 loadcases: operating, pulley, transport

34

OptiStruct Case Study
Volkswagen Bracket Results ? Mass reduced by 23%
? Original mass 950g ; Final mass 730g

? Performance targets were met

OptiStruct: Topography Design
for Future Automotive Body Engineering

35

ALTAIR/OPTISTRUCT Results

Extension of HDM
Topology Optimization Method

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

? Material Microstructure Design
? Young’s and Shear Moduli, Poisson’s Ratios ? Thermal Expansion Coefficients

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

36

QUINT/OPTISHAPE Application to Contro Frequencies

Special Mechanism : Negative ν

Material Design

Special Mechanism

37

Compliant Mechanism Design
Professor S. Kota @ UM

Negative Thermal Expansion
Bing-Chung Chen’s Design

0 ? ?? 8.01 βH = ? ? 7.89? ? 0 ? 0 ? ?? 52.7 αH = ? ? 58.9? ? 0 ?

38

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