CAE-Based Design Optimization

CAE-Based Design Optimization Dong-Hoon Choi [email protected] Director, the Center of Innovative Design Optimization Technology (iDOT) Professor, ...

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CAE-Based Design Optimization

Dong-Hoon Choi [email protected] Director, the Center of Innovative Design Optimization Technology (iDOT) Professor, School of Mechanical Engineering Hanyang University, Seoul, Korea September 1, 2006

Outline Brief Introduction of iDOT

Brief Introduction of iDOT

Sequential Approximate Optimization

Applications

2

Motivation Brief Introduction of iDOT

Why MDO ?

Automation Integration

Development of Computing Technology

Optimization

Multidisciplinary Design Optimization

Shorter Design Cycle

Advance in Optimization Technology

Improved Product Quality Cost Reduction

3

iDOT Mission Brief Introduction of iDOT

MDO Research Research and development of multidisciplinary design optimization methods

Transfer of promising MDO technology to industry

Train industry designers and educate students on MDO methods and design procedures

Industrial Application

iDOT

International Cooperation

Training and Education

4

The Center of

innovative Design Optimization Technology Brief Introduction of iDOT

Selected as one of the Engineering Research Centers of excellence by Korean government in 1999 Supported for 9 years by the KOrean Science and Engineering Foundation (KOSEF) Located at Hanyang University in Seoul, Korea A research alliance with ASDL since 1999 14 4 4 2 81

Professors from 7 universities Research Staffs Computer Programmers Administrators Graduate Students

5

Research Areas Brief Introduction of iDOT

DB

Research Topics Application Technology

Optimization Methods

CAD Electromagnetic Analysis

Fluids

Design Process Management

Dynamics

MDO

Computing Infrastructure Optimization Formulation

Kernel

Structures

Integrated Design Users MDO Methods

Global Optimization Approximate Optimization

PIDO Tool PIDO : Process Integration and Design Optimization

“The Ultimate Design Machine”

6

EMDIOS Architecture Brief Introduction of iDOT

Non Linear Analysis

Visual Modeling Resources

USER INTERFACE

Crash Analysis

Optimization Schedule Template Manager Component Manager

PROCESS Manager

DATABASE Manager

MDO Kernel CO

MDF

IDF

Component Abstract Layer

Component Abstract Layer

I/O Manager Script Message Parser Manager

Script Manager

Local Optimizer(s) Global Optimizer(s) Approximation Module(s) DOE Module(s) Other Resource(s)



Experimental Results

7

Applications (2001-2005): 74 Design Optimizations Brief Introduction of iDOT

Aerospace 2 Automotives 15 Biotechnology 2 Electronics 13 Information Technology 2 Marine Technology 2 Materials 6 Mechanical 26 MEMS Devices 2 Nuclear 2 2 Railway Vehicles

8

FRAMAX: a Spin-off Company of iDOT Brief Introduction of iDOT

Research & Development

PIDO Technology Transfer

Commercialization & Maintenance Engineering Consulting

FRAMAX is a spin-off company of iDOT, having the world-class PIDO technologies (Founded in June, 2003).

Customization of Design S/W Development of PIDO Tool

9

Outline Sequential Approximate Optimization

Brief Introduction of iDOT

Sequential Approximate Optimization

Applications

10

Motivation Sequential Approximate Optimization

How to Enhance the Computational Performance?

HARDWARE

Higher Efficiency

Sequential Approximation Optimization Technique

Distributed Parallel Computing Technique

SOFTWARE

11

Why Approximate Optimization ? Sequential Approximate Optimization

Before Schmit and Farshi (1974)

After Schmit and Farshi (1974)

Iterative Numerical Optimization Loop

Iterative Numerical Optimization Loop

New Design Exact Function & Gradient Values

New Design Approximate Function & Gradient Values

Approximation

High Fidelity Model

Low Fidelity Model Exact Function & Gradient Values

Approximate Optimum

Decouple an Expensive Analysis from Iterative Optimization Process High Fidelity Model

12

Efficiency of Sequential Approximate Optimization (SAO) Sequential Approximate Optimization

SAO

Exact Optimization

Optimizer Optimizer 1=2sec 2=4sec 40 x 0.05sec x 3=6sec 40 x 2hr = 80 hr

24 hr 6 sec Low Fidelity Model

High Fidelity Model

2=16hr 1=8hr 4 x 2hr x 3=24hr

High Fidelity Model requires 2hr per analysis Low Fidelity Model requires 0.05 sec per analysis

High Fidelity Model

13

AO vs. SAO Sequential Approximate Optimization

f

Approximate Optimization

f

real function

Quadratic

x

real function

Simple Cubic

f

x

Sequential Approximate Optimization

Using Quadratic Function real function

x 14

SAO Framework of iDOT Sequential Approximate Optimization

Min. f x s.t. g x d 0 h x 0 x L d x d xU

Approximate Model Manager

SAO Manager

x *

Define approximate optimization problem ~

*

Min. f x s.t. ~g x d 0

Approximate Model Developer

x

~~~ f , g, h

~ h x 0 x0  x d *

x

Local/Global Optimizers

Convergence Checking Analysis or Experimental Data

~~~ f , g, h

15

Approximate Model Developer Sequential Approximate Optimization

Typical Local Approximation Methods One-Point Approximation ƒ Linear, Reciprocal, Conservative

Approximate Model Developer n™ˆ‹Œ•›GiˆšŒ‹ h——™–Ÿ”ˆ›–•G tŒ›–‹š

Two-Point Approximation ƒ TPEA, TANA, TANA1, TANA2, TANA3

New Two-Point Approximation Method STDQAO Typical RSM based on Experimental Design Quadratic Approximation Modeling ƒ CCD, SCD and BBD

mœ•Š›–•GiˆšŒ‹ h——™–Ÿ”ˆ›–• tŒ›–‹š

Alphabetic Optimality Criteria ƒ D-optimal Design

New RSM based on Experimental Design` Augmented D-optimal Design Subspace CCD/SCD PQRSM Kriging RBF (Radial Basis Function) SVR (Support Vector Regression) MARS (Multivariate Adaptive Regression Splines) 16

SAO using Trust Region Concept Sequential Approximate Optimization

x2

START START

: Trust region

x2U

determine

ī

*k

1

x k0 xk*

Build Build Approximate Approximate Model Model

L 2

x

x1L

k=k+1

x1

x1U

Approximate Approximate Optimization Optimization x2

Evaluate Evaluate the the exact exact function function value value at the approximate at the approximate optimum optimum

Build Build Approximate Approximate Model Model in the in the new new Trust Trust region region

x2U * k 1

x k01

Update Update Trust Trust Region Region

ī

k+1

=Ȗī

J *k

k

x2L

Converge Converge ??

No

x1L

x1U

x1

: Previous Trust region

Yes

: New Trust region

STOP STOP

17

Trust Region Concepts (1/2) Sequential Approximate Optimization

f

approximate function f x

f x - f x 0 k

* k

0



f x0



Trust region ratio

f x*

f xk0 - f x*k f x*

x0

f xk0 - f x*k ȡ = f xk0 - f x*k

original function

x*

x

R. Flecher, Practical Methods of Optimization, 1987

Ȗ0 = 0.25

k

if

ȡ k d 0, k

approximation is bad

if

ȡ | 1,

if

ȡ > 1 or ȡ k  0,1 ,

approximation is excellent

k

moving in the right direction.

Trust radius

Ȗ1 = 1

if

0 < ȡ k d İ1 ,

ī k+1 = Ȗ0 ī k

­2 if x* - x0 = ī s ° Ȗ2 = ® * 0 s °¯ 1 if x - x < ī İ1 = 0.25,İ2 = 0.75

if if

İ1 < ȡ k d İ2 , ȡ k t İ2 ,

ī k+1 = Ȗ1 ī k ī k+1 = Ȗ2 ī k

where,

0 < İ1 < İ2

18

Trust Region Concepts (2/2) Sequential Approximate Optimization

ȡk =

f xk0 - f x*k f xk0 - f x*k

previous trust region new trust region x2

x2

xk0

x

x k0

x k0

Ω * k

x2

x2

x

* k

xk0 x

* k

x1

x1

x2

x k0 x

x k*

* k

x1

x1

ī k + 1 = Ȗ0 ī k

ī k + 1 = Ȗ0 ī k

ī k + 1 = Ȗ1 ī k

ī k+1 = Ȗ2 ī k

xk0 + 1 = xk0

xk0 + 1 = x*k

xk0 + 1 = x*k

xk0 + 1 = x*k

(reject)

right direction

bad

0

excellent

good

Ș1 = 0.25

Ș2 = 0.75

K = 0.25, K = 0.75,

right direction

good

1 = 1.33 Ș2 Ȗ 0 = 0.25,

U

1 = 4.00 Ș1 Ȗ 1 = 1.0,

­ 2 if ° Ȗ2 = ® °¯ 1 if

x* - x 0 = ī k x* - x 0 < ī k 19

SAO - Function Based Approximation Methods Sequential Approximate Optimization

START

DOE

Build Approximate Model

DOE

Approximate

Build Approximate Model

Approximate Optimization

Model Management Model Management

f x

f x

f x

x

x

f x

x

x

: Design Region Converged?

f x

f x

x

f x

x

f x

x

x

: Trust Region END

20

Concept of the Progressive Quadratic Response Surface Method Sequential Approximate Optimization

PQRSM

Conventional Quadratic Modeling jjkGOX`\XPSGzjkGOX`\`PSGiikGOX`]WP y

y DOE

x3

y

kvl

y y y

y y

y y

y yy

x1 y

H

x2

y

h“Ž–™›”Šˆ““ GŠ–”‰•ŒG›ž–GŠ–•ŠŒ—›šU

ª H11 « « H21 ¬« H31

H13 º » H23 » H33 ¼»

H12 H22 H32

Mathematical Programming

H k 1 

Hk

H k 1į k H k 1į k T T k

į H k 1į k

H

H y



ª H11 « « H21 «¬ H31

H22

º » » H33 »¼

y x3

kmwGOX`]ZPSGimnzGOX`^WP

ª H11 « « «¬

y

y

H12 H22 H32

H13 º » H23 » H33 ¼»

x2

y

y k y Tk y Tk į k

H

y

ª H11 « « H21 ¬« H31

x1 y

H

H13 º » H23 » H33 »¼

H12 H22 H32

ª « « H21 «¬ H31

H12 H32

H13 º » H23 » »¼

imnz 21

PQRSM Procedure Sequential Approximate Optimization

DOE

Make Full Quadratic Approximate Model

x2

: Trust region

x2U

“2n+1” points

Response Surface Method

x22

x11

x0

f x

x12

x21

x2L

x1L

number of unknowns

x1

x1U

ndv

ndv

ndv ndv

i 1

i 1

i 1 j !i

n

n

n(n-1)/2

At least (n+1)(n+2)/2 Experimental points are required!!

E 0  ¦ E i xi  ¦ Eii xi2  ¦¦ Eij xi x j 1

Least Square Method

Update Trust Region f x

f(x) f x

f(x)

f(x) f x

f(x)

f(x)

x

x

: Trust region

ndv

ndv

ndv ndv

i 1

i 1

i 1 j !i

n

n

f x D 0  ¦ D i xi  ¦ D ii xi2  ¦¦ D ij xi x j

f(x)

x

PQRSM

number of unknowns

1

Only 2n+1 Experimental points are required!!

0

3-points polynomial interpolation

Quasi-Newton method

Approximate Optimization

Hessian term H

ªD11 º « » D 22 « » «¬ D 33 »¼

+

3-points polynomial interpolation

H

D12 D13 º ª «D D 23 »» « 21 «¬D 31 D 32 »¼

Quasi-Newton method

H

ªD11 D12 D13 º «D » « 21 D 22 D 23 » «¬D 31 D 32 D 33 »¼ 22

Noisy Mathematical Function Sequential Approximate Optimization

Minimize

f x

¦ > x n 1

@

 xk2  1  xk ^1  0.1sin 2 70 xk ` 2

k 1

2

k 1

Total Number of Function Evaluations

Initial Values: xk

1.2,

xk 1 1.0, k 1, 2,!, n  1

Optimum Values:

f x 0.0, at x*

>1.0, 1.0,!,[email protected] ,

Number of Cumulative Function Evaluations

2500 2122

2051

PQRSM D-optimal CCD

2000

1500 1201 1038

1000

859 559

500

345

316

66,91,81 196

0 0

5

10

15

20

25

30

35

40

45

50

Number of Design Variables

23

Test Problems Sequential Approximate Optimization

1. Gear Reducer Design

3. Two-DOF linear dynamic absorber design 4. Elasto-plastic Ten-member Truss Design 5. Design of a Tracked Vehicle

Relative Number of Analyses

2. Rosen-Suzuki Problem

3 2.61

PQRSM

2.5

D-opt. CCD

2 1.5

1.24 1

1 0.5

1.24

1 0.75

1

1

1

0.75

0.27

0.28

0.17

0 1

2

3

4

5

Problem Number

Performance comparison between PQRSM, D-optimal, and CCD

24

Two-Point Approximation Sequential Approximate Optimization

{ž–Tw–•› lŸ—–•Œ•›ˆ“ h——™–Ÿ”ˆ›–• O{wlhP g~TPEA y

pi is determined to match

wg y 2 yi  yi ,2 wyi 1

n

ª wg x1 wg x 2 º pi 1  ln « wxi »¼ ¬ wxi

g y 2  ¦ i

g x g x1

xipi

yi

wg x1 wxi

{™ˆ•š–™”ˆ›–•

g x 2

x1

x2

wg~ y1 wyi

§x · ln¨¨ i ,1 ¸¸ © xi , 2 ¹

g y wg y1 wyi

g y1 wg x 2 wxi

wg y1 wyi

g~ y1

h——™–Ÿ”ˆ›–•G Œ™™–™

wg y 2 wyi

g y 2

xi

y1

y2

yi

p•G–™‹Œ™G›–G”•”¡ŒG›šGŒ™™–™SG{huhGšŒ™ŒšGžŒ™ŒG‹ŒŒ“–—Œ‹G

1990

1994

1998

1995

2001

Fadel et al. Wang & Grandhi Wang & Grandhi Xu & Grandhi Kim et al. TPEA TANA TANA-1 & -2 TANA-3 TDQA 25

A New Two-Point Approximation Sequential Approximate Optimization

Two-Point Diagonal Quadratic Approximation n

g~ y g y 2 

¦ i 1

wg y 2 yi  yi , 2  K wyi

Intervening Variable with Shifting Constant ci

n

¦G y  y i

i,2

2

i 1

Diagonal Hessian having different signed values Gi g (y )

g

To define an intervening variable when the design variable value is near zero or negative y2

1

i

Curvatures of different signs along each variable axis

y1

xi

yi xi  ci ­if xi d H , ci ® ci otherwise, ¯

pi

Correction Coefficient K L i

x 1 0

To match the function at x1 g x g~ x 1

Gi

§ wg y1 wg y 2 · 1 ¨ ¸  wyi ¸¹ 2 yi ,1  yi , 2 ¨© wyi

1

26

STDQAO: Sequential Two-point Diagonal Quadratic Approximate Optimization Sequential Approximate Optimization

4㇦ Ꮊ♪㰒 5㇦ Ꮊ♪㰒

㌚⶿ⷎ⮎ ⟖ᷲ

㌚⶿ⷎ⮎ ⟖ᷲ

⛢ኂⷎ ≾㰒

⛢ኂⷎ ≾㰒

⢢ⷚ FDH#㬲⛛

Ꮊ♪ ⁦ᤶ

Ꮾ⸲ ㌚⶿㰒

⢢ⷚ FDH#㬲⛛

STDQAO Ꮾ≓ⴂ ⴲⱧ㬚 ㌚⶿㰒 27

Test Problems Sequential Approximate Optimization

1. Welded Beam Design

ͧ͢ ͥ͢

3. Rosen-Suzuki Problem 4. Piston Design Problem 5. Three-bar truss design problem

Number of Anlayses

2. Gear Reducer Design

TDQA TANA3

ͣ͢ ͢͡ ͩ ͧ ͥ

6. Ten-bar truss design problem Case 1a

ͣ ͡

7. Ten-bar truss design problem Case 1b 8. Ten-bar truss design problem Case 2a 9. Ten-bar truss design problem Case 2b

͢

ͣ

ͤ

ͥ

ͦ

ͧ

ͨ

ͩ

ͪ

Problem Number

Performance comparison between TDQA and TANA3 : Convergence failed : Prematurely converged

28

Outline Sequential Approximate Optimization

Brief Introduction of iDOT

Sequential Approximate Optimization

Applications

29

Applications Sequential Approximate Optimization

Aerospace Automotives Biotechnology Electronics Information Technology

2 15 2 13 2

Marine Technology Materials Mechanical MEMS Devices Nuclear Railway Vehicles

2 6 26 2 2 2

Go

Heat Sink for a 40A-Drive Package System of an Elevator

(FLUENT)

Go

Air Bearing Surface of HDD

(in-house code)

Go

ABS Controller

(Simulink & CarSim)

Go

Switched Reluctance Motor

(SRM Analyzer)

Go

Hydro-Pneumatic Suspension Unit of a Tracked Vehicle

(RecurDyn-alpha)

Go

Automotive Body Structure

(MSC/NASTRAN)

Go

Drum Washer Suspension System

(DADS & ANSYS) NEXT 30

Application 1:

40A-Drive Package System Sequential Approximate Optimization

Optimization of Heat Sink ඞ Objective

IGBT_2

Design variables

IGBT_1

- To obtain the optimum design variables (B1, B2, and t ) by minimizing (1) the pressure drop (2) the temperature rise, simultaneously

Reactor

Objective functions

Heat Sink

Duct

ඞ Model : 40A Drive Package System

Schematics of Drive Package System

- Heat Sink (188 x 400 x 60 mm) - IGBTs (46,007 / 42,438 W/m2) - Fan (Model : 3112KL-05W-B50)

7.52 7 (t)

2.0 (B1)

- FLUENT : Predict Flow and Thermal Fields in System - SQP (Sequential Quadratic Programming Method) : Propose the optimal variables numerically - Batch-process : Integrate CFD(FLUENT) and CAO (SQP)

53

ඞ Numerical Methodology

1.5 (B2) 94

Schematics and baseline geometry of heat sink

31

Application 1:

40A-Drive Package System Sequential Approximate Optimization

Results ඞ Comparison with Experimental Data

- For the temp. rise, difference is only 1.2 K ˧ Good agreement with the experimental data ඞ Optimum variables ( Temp. rise is less than 35 K) Unit : [mm]

B1

B2

t

Temp. Rise

Pre. Drop

Baseline

2

1.5

7

38.66 K (˦)

43.3 Pa

Optimal

2.6

1.9

10.7

34.93 K (˨)

50.3 Pa

Vortex flow

Pathlines for understanding the flow fields

• Ambient temp. : 318 K (45 oC) • Max. temp. of baseline geometry : 356.66K, Max. temp. of optimization : 352.93 K

ඞ Comments

- Optimization is strongly needed to guarantee the thermal stability of IGBTs - It is easily applicable to the other specification of heat sink - As shown the above table, pressure drop increases while temperature rise decreases. - Note that, in optimum model, the value of pressure drop is ranged in the characteristic curve of fan.

Max. Temp : 352.93 K

Isotherms for understanding the temp. fields HOME

32

Applications

Application 2:

Head-Disk Interface (HDI)

• Reduced flying height (as low as 10 nm) • Complicated geometry of ABS

For higher areal density …… 33

Applications

Application 2:

Design Requirements h*

Uniform FH (Fly Height) (target FH = 9 nm) radius

DU

Limited Pitching (between 250 and 300 Ɇrad)

DL radius

Limited Rolling (between -5 and 5 Ɇrad)

EU EL radius

Altitude Insensitivity (80% FH @10k ft) halt radius

Fast TOV (80% FH @ra=15 mm/skew=mm/skew=-25.54, rpm=2.5K)

hglide

rotational vel.

34

Applications

Application 2:

Design Variables zXW

z^

z\

z]

zX

zY z[

zZ z`

z_

• Design variables: S1~S10, recess depth, and shallow step • Using SAE’s shape function for design variable linking -> [shape.in] 35

Applications

Application 2:

Mathematical Formulation

find

si , i 1 ~ 8, recess, shallow

min.

F

subject to

g1 :250 P rad d

0.5*(hINTmax  9 nm) 2  0.5*( hINTmin  9 nm) 2 Pitchmin ,

g2 : Pitchmax d 300 µrad , g3 :  5 P rad d

Rollmin ,

g4 : Rollmax d 5 µrad , g5 : 9nm  hINTALT

max

d (1  0.8) *9 nm,

g6 : 0.8*9 nm d hINTFAST , where

-0.05 mm d s1 d 0.02 mm -0.05 mm d s2 d 0.02 mm -0.05 mm d s3 d 0.02 mm -0.02 mm d s4 d 0.05 mm -0.05 mm d s5 d 0.05 mm -0.05 mm d s6 d 0.05 mm -0.02 mm d s7 d 0.05 mm -0.02 mm d s8 d 0.05 mm 1 P m d recess d 2 P m 0.1 P m d shallow d 0.2 P m

36

Applications

Application 2:

Altitude Insensitivity

Uniform FH , Limited Pitching , Limited Rolling Pitch angle [Prad]

350

™ŒŠŒššGš›Œ—GaGXU[WWŒT] OGRWUYGLP šˆ““–žGš›Œ—aGXUX]\ŒT^ OTX^GLP

Initial Optimum 300

250

200 20

30

40

50

Disk radius [mm]

15

Initial Optimum

Initial Optimum

10

Roll angle [Prad]

Flying height [nm]

20

15

10

5

5 0 -5 -10 -15

0 20

30

40

20

50

30

40

50

Disk radius [mm]

Disk radius [mm]

37

Applications

Application 2:

Optimization Results

9 iteration number : 17 9 total function calls : 221 50

Constraint value

1

Cost value

40

30

20

10

0

G1

G2

G3

G4

G5

G6

0

-1

-2

0

5

10

Iteration number

15

20

0

5

10

15

20

Iteration number

HOME

38

Applications

h——“Šˆ›–•GZaGhizGj–•›™–““Œ™GkŒšŽ•Gœš•ŽGz”œ“•’ MGjˆ™z” „

„

„

p•›ˆ“Vv—Œ™ˆ›•ŽGj–•‹›–•š …

p•›ˆ“Gz—ŒŒ‹aGXYW’”V

…

z—’ŒGi™ˆ’•ŽG–GX\”wˆGgGWUYš

…

z—“›Gtœ OWUYsVWU\yPGgGZ”

kŒšŽ•G}ˆ™ˆ‰“Œš …

i™ˆ’ŒGhŠ›œˆ›–™Gnˆ•GsyGMGyy

…

hizGj–•›™–““Œ™Gmy{Gv•Gzž›ŠG}ˆ“œŒ

…

hizGj–•›™–““Œ™Gmy{GvGzž›ŠG}ˆ“œŒ

…

hizGj–•›™–““Œ™GyyGv•Gzž›ŠG}ˆ“œŒ

…

hizGj–•›™–““Œ™GyyGvGzž›ŠG}ˆ“œŒ

…

hizGj–•›™–““Œ™Gzž›ŠG{™Œš–“‹

Design Path X Road

v‰‘ŒŠ›ŒGmœ•Š›–•š …

z˜œˆ™ŒGzœ”G–G€ˆž

…

z›ˆ›–• X Design Requirements

39

Applications

Application 3: Simulink Model (Including CarSim S-Function) ABS Controller Switch Threshold

ABS Controller FRT On Switch Value ABS Controller FRT Off Switch Value ABS Controller RR On Switch Value ABS Controller RR Off Switch Value

X Simulink Model

Brake Actuator Gain LR & RR

40

Applications

Application 3:

CarSim Model

Initial/Operating Conditions (CarSim)

X CarSim GUI

41

Applications

h——“Šˆ›–•GZaGp•—œ›Vvœ›—œ›G~™ˆ——•ŽGœš•ŽGmyhth

42

Applications

Application 3:

Integration

Ghvljq#Yduldeohv/ Ghvljq#Udqjh/ Ghvljq#Uhtxluhphqwv

FRAMAX

Rswlpdo#Ghvljq Vdwlvi|lqj Doo#ghvljq#Uhtxluhphqwv

Parametric Study Design of Experiment Approximation Optimization Sequential Approximate Opt.

Simulink + CarSim

Uncertainty Analysis

Design under Uncertainty

43

Applications

Application 3:

Pareto Optimum (1) Initial Optimum

X Station(m) vs. Time(s)

X Yaw(deg) vs. Time(s)

Multiobjective Problem •Weight Factor(Station) : 0.75 •Weight Factor(Yaw) : 0.25

44

Applications

Application 3:

Pareto Optimum (2) Optimum Initial

X Station(m) vs. Time(s)

X Yaw(deg) vs. Time(s)

Multiobjective Problem •Weight Factor(Station) : 0.5 •Weight Factor(Yaw) : 0.5

45

Applications

Application 3:

Pareto Optimum (3) Optimum

Initial

X Station(m) vs. Time(s)

X Yaw(deg) vs. Time(s)

Multiobjective Problem •Weight Factor(Station) : 0.25 •Weight Factor(Yaw) : 0.75

46

Applications

Application 3:

Pareto Optimum Set Pareto Set Weighting Ratio (Staion/SquareSumYaw=0.75/0.25)

Square Sum of Yaw

700 600 500 400

Weighting Ratio (Staion/SquareSumYaw=0.5/0.5)

300

Weighting Ratio (Staion/SquareSumYaw=0.25/0.75)

200 100 0 150

160

170

180

190

200

210

Station X Pareto Optimum Set

HOME

Application 4:

47

SRM (Switched Reluctance Motor) Applications

Design of a Switched Reluctance Motor v‰‘ŒŠ›Œ

Maximize Average Torque

Tave

j–•š›™ˆ•›š



Torque ripple Trip d 20% Maximum Current Phase I max d 6 A kŒšŽ•G}ˆ™ˆ‰“Œš

Switching on Angle T on

E r

Switching off Angle T off Rotor Pole Arc

48

Application 4:

SRM (Switched Reluctance Motor) Applications

Parameter Studies of Switching On/Off Angles Switching on Angle

Switching off Angle 0.24

0.6 0.23

Average Torque (N-m)

Average Torque (N-m)

0.5 0.4 0.3 0.2

0.22 0.21 0.20 0.19 0.18 0.17

0.1

0.16 0.15

0.0

45.0

0.0

5.0

10.0

15.0

20.0

25.0

47.0

49.0

51.0

53.0

55.0

57.0

59.0

55.0

57.0

59.0

30.0

Angle (deg.)

Angle (deg.)

105.0

100.0

Torque Ripple (%)

95.0

Torque Ripple (%)

95.0

90.0

85.0

75.0 65.0 55.0

80.0

45.0 45.0

75.0 0.0

5.0

10.0

15.0

20.0

25.0

47.0

49.0

51.0

53.0

Angle (deg.)

30.0

Angle (deg.)

4.40

Maximum Current Phase (A)

11.0

Maximum Current Phase (A)

85.0

10.0 9.0 8.0 7.0 6.0 5.0 4.0

4.35 4.30 4.25 4.20 4.15 4.10 4.05 4.00

3.0 0.0

5.0

10.0

15.0

20.0

25.0

45.0

30.0

47.0

49.0

51.0

Application 4:

53.0

55.0

57.0

59.0

49

Angle (deg.)

Angle (deg.)

SRM (Switched Reluctance Motor) Applications

Optimization Results {–™˜œŒ

j–•Œ™ŽŒ•ŠŒGoš›–™  0.6

Initial Design Optimum Design

0.4

0.3

0.4

Torque (N-m)

Average Torque

0.5

0.3 0.2

0.2

0.1

0.1 0

0

Maximum Constraint Violation

0

1

2

3

4

5

6

7

8

9

0

10

20

30

5.0

40

50

60

70

80

Trip

I max

90

Rotation Angle (deg.)

Iteration

4.0 3.0

T on

2.0 1.0

0

1

2

3

4

5

Iteration

6

7

8

Er

Tave

Initial

25.2 45.1 30.0 0.17 98.60 4.02

Optimum

22.4 55.7 38.0 0.30 19.98 4.91

0.0 -1.0

T off

9 HOME

50

Application 5:

Tracked Vehicle Applications

Objective Minimize the vertical acceleration at the CG of the hull 14 inch

Constraints (15)

Design Variables (9)

z wheel travels during jounce(6)

z static track tension

z equally distributed static forces for the wheels(6)

z charging pressures of the 1st, 2nd, 5th and 6th HSU’s

z track tension(1)

z length of gas chambers

z charging pressures of the 3rd and 4th HSU’s(2)

z pre-load for Belleville springs z choking flow rate z inner diameter of orifice

51

Application 5:

Tracked Vehicle Applications

Vertical Acceleration (m/s^2)

20.0

Noisy Cost Function

15.0

Noise ? 10.0

5.0

0.0

-5.0

-10.0

-15.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time

Is the acceleration a nosy function ? • Gradient based optimization algorithm may cannot be converged. • Function based approximate optimization algorithm should be used. 52

Application 5:

Tracked Vehicle Applications

Optimization Result - Acceleration

53

Application 5:

Tracked Vehicle Applications

Optimization Result – Wheel Travel

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54

Automotive Body Structure

Application 6:

Applications

v‰‘ŒŠ›Œ

t1

1

t•”¡ŒG~ŒŽ› Xš› ‰Œ•‹•Ž ™Œ˜U

vw{ptply |š•ŽG{kxh

t3

m™Œ˜œŒ•Š Gj–•š›™ˆ•›š Y•‹ ‰Œ•‹•Ž ™Œ˜U

t2

Xš› ›–™š–•ˆ“ ™Œ˜U

y

SV2

z z '

T

SV1

Hz

kŒšŽ•Gˆ™ˆ‰“Œš

kŒšŽ• }ˆ™ˆ‰“Œš

y'

zŒŠ›–•G{Š’•ŒššŒšGOZ]P zŒŠ›–•Gzˆ—ŒG}ˆ™ˆ‰“ŒšGOXXYP

lwTzlj{

h•ˆ“ ššGM kzhGkˆ›ˆ

tzjVuhz{yhu pu{lymhjl

p•—œ›Gm“Œ

t–‹ŒT{™ˆŠ’

vœ›—œ›G“Œš

p•›Œ™”Œ‹ˆ›ŒG}ˆ™ˆ‰“Œš zŒŠ›–•Gw™–—Œ™›ŒšGOhSGpSGqP q–•›Gz›•ŒššGOrP

tzjVuhz{yhu

ŒPGz‹ŒGy––Gyˆ“

ˆPGm™–•›Gy––Gyˆ“

‹PGjGw““ˆ™

‰PGhGw““ˆ™

ŠPGiGw““ˆ™ PGy–Š’Œ™

ihyOZ]YPSG lshzXO\W^PSG x|hk[OYZ\^PSG yvkO[^PSG khtwXO[PSG olhOZ[[PSGjvutYOX]`PSGwlu{hO`YPSG{yphZO]X]P

55

Application 6:

Automotive Body Structure Applications

Convergence History

Optimization Results TDQA IT

Objective function

TPCA Max. Violation

Objective function

Max. Violation

1

1.38883E+3

4.30770E-2

1.38883E+3

4.30770E-2

2

1.37034E+3

6.29008E-2

1.38829E+3

3.78596E-2

3

1.36478E+3

8.19064E-2

1.38771E+3

3.69611E-2

4

1.36717E+3

6.08585E-2

1.38227E+3

2.35630E-2

5

1.36501E+3

5.48596E-2

1.38161E+3

1.61079E-2

6

1.36567E+3

3.80981E-2

1.37600E+3

2.27294E-2

7

1.36642E+3

2.94570E-2

1.36875E+3

5.50377E-2

8

1.36823E+3

2.15811E-2

1.36701E+3

4.39808E-2

9

1.36953E+3

1.54264E-2

1.37201E+3

1.19377E-2

10

1.37084E+3

9.90981E-3

1.37340E+3

3.77472E-3

1.37371E+3

2.53623E-3

11

1.37171E+3

6.22981E-3

12

1.37245E+3

3.49547E-3

13

1.37259E+3

2.98415E-3

| 16 kg

| 15 kg HOME

56

CAE-Based Design Optimization

Dong-Hoon Choi [email protected] Director, the Center of Innovative Design Optimization Technology (iDOT) Professor, School of Mechanical Engineering Hanyang University, Seoul, Korea September 1, 2006