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INDUSTRIAL OPTIMIZATION: STATUS AND PROSPECTS

G. Vanderplaats

Vanderplaats Research & Development, Inc.

1767 S. 8th StreetColorado Springs, CO 80906

Ph. (719) 473-4611

Copyright VR&D 2004

www. vrand.com

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• Minimize F(X) Objective Function

• Subject to (Such That);

Inequality Constraints

Equality Constraints

Side Constraints

F(X), gj(X) and hk(X) May be Linear, Nonlinear, Explicit, Implicit, but Should be Continuous with Continuous First Derivatives

GENERAL OPTIMIZATION GENERAL OPTIMIZATION PROBLEM STATEMENTPROBLEM STATEMENT

NiXXX

LkXh

MjXg

Uii

Li

k

j

,1

,10)(

,10)(

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• Given Xq

• Update the Design by

Xq = Xq-1 + Sq Xq-1 + X

• Note that this is Very Close to the Traditional Design Process of Beginning with a Design and Modifying it

THE OPTIMIZATION PROCESSTHE OPTIMIZATION PROCESS

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• 1948: SIMPLEX Method for Linear Programming

• 1950’s: Various Random Methods. Gradient Based Methods Developed in the Late 1950’s

• 1960’s: Sequential Unconstrained Minimization Techniques, Sequential Linear Programming, Feasible Directions Methods

• 1970’s: Enhanced Feasible Directions Methods, Multiplier Methods, Reduced Gradient Methods, Response Surface Approximations

• 1980’s: Variable Metric Methods, Sequential Quadratic Programming Methods

OPTIMIZATION ALGORITHMSOPTIMIZATION ALGORITHMS

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• 1990’s: Genetic Algorithms, Simulated Annealing, New Interest in Sequential Unconstrained Minimization Techniques

• 2000’s: Particle Swarming, Advanced Sequential Unconstrained Minimization Techniques

• Largest Known Test Example– 500,000 Variables With 500,000 Active Constraints

• Largest Known Real Structural Optimization Problem– 250,000+ Thickness Variables with Frequency Constraints

– 2,000,000+ Topology Variables

OPTIMIZATION ALGORITHMSOPTIMIZATION ALGORITHMS

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OPTIMIZATION PROBLEM SIZEOPTIMIZATION PROBLEM SIZE

0

100

1,000

10,000

1960 1970 1980 1990 2000 2010

# Des. Var.

Year

100,000

BIGDOT500,000VARIABLES

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STRUCTURAL OPTIMIZATIONSTRUCTURAL OPTIMIZATION

• 1960 - Schmit combined optimization and analysis– 2Variables; 1/2 hour on IBM 653

• 1973 - Schmit et al introduced physics based approximations• 1986 - Vanderplaats et al developed 2nd generation

approximations• 1975 - 1989 Optimization added to commercial structural

analysis programs• 1984 - 2000 General purpose engineering optimization

software• Optimization software used by engineers is usually created by

engineers

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G . N . V a nd erp l aats

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• 1975

OPTIMIZATION WORKSOPTIMIZATION WORKS

C O M B AT

M IS SIO N

I N ITIA L O PT IM U M

SUPERSO NIC CRUISE AIRCRAFT

S OL VED B Y T H E AC SY N T PR O GR A M

5 D ES IGN V AR IA BLE S, 2 PER F O R M AN C E C ON S T R AIN T S

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• 1975

OPTIMIZATION WORKSOPTIMIZATION WORKS

SU PER SON IC CRU ISE AIR CRAFT

TRAD E - OFF STU DY

{1.0

0.9

0.8

0

0.6

0.8

1.0

1.2

1.4

3 4 5 6 7 8

S UST AINE D LOA D FAC TO R AT M = 0 .9

RE

LA

TIV

E M

AS

S

NOMINALDESIG N

TECHNO LO GYFACTO R

CO NVENTIONAL

ADVANCED

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• 1976: A Two Hour Optimization Study

OPTIMIZATION WORKSOPTIMIZATION WORKS

STO L AIRCRAFT TAKEO FF

CO N VEN TIO N AL: W = WG0

VARIATIO N AL CALCU LU S: W = 2.5WG0

N U M ERICAL O PTIM IZATIO N : W = 1.2WG0

20gFlig htSp e e d

2g

500 ft/m inC lim b

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• 1978: Today Called “Response Surface Method”

OPTIMIZATION WORKSOPTIMIZATION WORKS

HIGH SPEED AIRFOIL OPTIMIZATION

INIT IAL SHAP E

O PT IMU M: M AXIM IZE LIF T WIT H DRAG & M O ME NT CO NST RAINT S

O PT IMUM : MINIM IZE D RAG W ITH LIF T & M O ME NT CO NS TRAINT S

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• It Has Been Working For Many Years– The Above Examples are 25-30 Years Old!

• The Aircraft Example was a 1 Man Month Study, Verified by a One Year, $250,000 Study by a Commercial Aircraft Company

• The Aircraft Take-off Example Solved a Ph.D. Problem that Took Over a Year and Got the Wrong Answer

• The Airfoil Example Produced a Design Almost Identical to a Multi Year Wind Tunnel Study

• It is Not Debatable that Optimization is Useful

OPTIMIZATION WORKSOPTIMIZATION WORKS

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• Use Approximations to Avoid Many Calls to the FEA– Optimizer Never Actually Calls the Finite Element Analysis

MODERN STRUCTURAL MODERN STRUCTURAL OPTIMIZATIONOPTIMIZATION

CONTROLPROGRAM

SENSITIVITYANALYSIS

OPTIMIZER

APPROXIMATEPROBLEM

GENERATORAPPROXIMATE

ANALYSIS

FEMANALYSIS

CONSTRAINTSCREENING

INNER LOOP

OUTER LOOP

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• Criteria– Find a Very Good Optimum Quickly

– Use as Few Full Finite Element Analyses as Possible

• Basis for Criteria– Each Analysis Requires a Full Finite Element Solution

• This Can be Very Expensive

• Cost– About 10-15 Times the Cost of One Analysis

• This Estimate Assumes Analytic Gradients are Calculated

• It Also Assumes 2nd Generation Approximation Techniques are Used

THE COST OF STRUCTURAL THE COST OF STRUCTURAL OPTIMIZATIONOPTIMIZATION

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• Modern Structural Optimization Converts the Original Design Problem to an Approximate Form Before Calling the Optimizer– Optimizer Calls Approximate Analysis Many Times

– Usually About Ten Detailed Finite Element Analyses are Needed• 95% of CPU Time is Analysis and Gradient (Sensitivity) Calculations

• Finite Element Models of the Order of 1,000,000 Degrees of Freedom are Becoming Common

• Problems in Excess of 250,000 Design Variables Have Been Solved by the GENESIS Program

STRUCTURAL OPTIMIZATIONSTRUCTURAL OPTIMIZATION

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Rocket Curved Stiffened Panel

• Minimize mass of the aluminum curved stiffened panel

• Eight design variables:– Thickness of skin and stiffeners– Stiffener web height– Stiffener flange widths

• Frequency constraint > 45 Hz (Initially = 23 HZ)

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Panel Optimization Results

• Frequency constraint is satisfied

• 30% mass reduction

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Spinning Disk

• Axi-symmetric structure w.r.t. the Y axis• Centrifugal load resulting from a 12 Hz rotation• Two material structure

– Outer part is aluminum– Inner part is steel

x

y

Aluminum

Steel

100 mm

30 mm

2 mm

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Spinning Disk Results

• 26% 26% Mass reduction

• Lowest natural frequency increased• Maximum stress reduced

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Shape Optimization of a Pin

• Pin must carry a specified load

• Nonlinear contact problem solved using ABAQUS

• Three materials: pin, adhesive,solid base

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Shape Optimization of a Pin

• Minimize maximum stress inthe solid base

• Constraints: displacement, stress

• Nine shape design variables

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Shape Optimization of a Pin

• Maximum stress reduction: 11%11%

• Improved stress distribution

• Small changes in the initial shape

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Optimize Heat Sink Shape (for PC processor)

Minimize: Mass

Subject To:

– min heat dissipation into the air

– max tO in thyristor

– max tO in chassis

Analysis:Flux2D - FE based package

for the analysis of

electromagnetic and thermal

devices and processes

VisualDOC/FLUX2D

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Initial design Final design

Design variables:height of the baseheight and width of fins

Result:Result:47% mass reduction47% mass reductionall constraints satisfiedall constraints satisfied

Initial design was choseninfeasible for demo

Final design looks likenormal heat sink in PC

VisualDOC/FLUX2D

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CORE+

CO

IL

- C

OIL

GAP

C-Shaped Magnetic Circuit FLUX2D Model

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CORE

GAP

X

Y

Flux Density in GAP of Initial Design

- Initial geometry gives a non-uniform magnetic field (flux) in the air gap- Optimize the geometry of the gap to give a prescribed point flux, or uniform flux along the length of the gap

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Pt. 1

Pt. 2Pt. 3 Pt. 4

Pt. 5

Pt. 6GAP

Change the Y coordinates of points 1-6 and X coordinates of points 2-5 in order to produce a uniform flux density of 0.6 Tesla within the gap. Note: Symmetry Imposed

Case 3: Optimum Flux Density in GAP

Minimize the sum of the squares of the error (SSE) at 200 points

CORE

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Flux Density Variation in GAP of Optimized Designs

Designing all X and Y coordinates produces the flattest flux density as shown in case 3 above

Magnetic Flux Density in Gap

0.57

0.62

0.67

0.72

0.77

0 2 4 6 8 10

X (mm)

Flu

x (

Te

sla

)

Case 1

Case 2

Case 3

Orig

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Transport Aircraft Wing

• Multilevel Optimization– System level: configuration

design variables

– Disciplinary level: aerodynamic analysis and structural analysis / Optimization

• Multidisciplinary Optimization with both aerodynamics & structures components

• Maximize the range for constant gross weight

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Disciplinary Issues

• Interaction between system level and structural sub-optimization is complexcomplex

• Must converge on loads and displacements– Changes in aerodynamic shape at the system level

affect the structural geometry

Aerodynamic loads deform the structure

Structural deformations affect aerodynamic loads

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Transport Aircraft Wing

initial

final

Parameter Initial Value Direct Optimization

Optimization using RS

Range (n.mi.) 5,000 6,342 6,403 25% increase Depth to chord ratio

0.12 0.14 0.14 Aspect ratio 6.86 5.92 5.88

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Airfoil Optimization

• NACA 4-digit airfoil• Design variables:

– maximum mean line camber as fraction of chord (m)

– chordwise position of maximum camber (p)– maximum thickness as fraction of chord (t)– Angle of attack ()

• Maximize ratio of Lift/Drag.• Use GAMBIT/FLUENT for geometric/flow

modeling.

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Optimization ResultsPressure Distribution

Initial design

Final design

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Equivalent Material Properties

• Reduce the size of the Heat Exchanger FE model by replacing air fin shell elements with equivalent anisotropic solid elements(not able to run the modal analysis on available computers)

Match the frequency anddisplacement responses

Analysis:Genesis - FE analysis and

Optimization code

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Equivalent Material ...

Original Configuration

Equivalent ConfigurationOverall mode shapes not changed

3% error in frequencies5% error in displacements1 million DOF reduction in FE model size1 million DOF reduction in FE model size

1st twist mode

Diagonal elements of 6x6material property matrix [G] were adjusted:

}{ ] [ }{ G

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Number of Elements=60,704 Number of Elements=60,704

Number of Design Variables = 60,704 Number of Design Variables = 7936

Traditional Results Casting Results

Design variables reduced by 87%

Manufacturing Constraints

No constraints added

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Topometry Optimization Example:Where to Reinforce?

•Objective:

– Maximize Natural frequencies

•Constraints:

– Mass

•Design Variables: 34,560

– Each Element thickness

Added Mass (Kg)

Increased Frequency (Hz)

Maximizig First

Torsion Frequency

Maximizig First

BendingFrequency

Maximizig Averageof two

Frequencis

2.64 4.81 6.42 4.24

7.32 7.56 9.89 6.41

15.06 9.66 12.15 12.58

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COMMERCIAL SOFTWARECOMMERCIAL SOFTWARE

Company Web AddressGeneral

OptimizationStructural

OptimizationAltair Engineering www.altair.com HyperOpt OptiStruct

Ansys www.ansys.com - - - Ansys-CADOE

Engineous Software www.engineous.com iSIGHT - - -

MSC Software www.mscsoftware.com - - - MSC.Nastran

Noesis www.noesissolutions.com Optimus - - -

Opttek www.optteck.com OptQuest - - -

Oculus Technologies www.oculustech.com CO - - -

Phoenix Integration www.phoenix-int.com Model Center - - -

UGS PLM www.ugs.com - - - NX.Nastran

Vanderplaats R&D www.vrand.com VisualDOC, DOT, BIGDOT

GENESIS

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FUTURE PROSPECTSFUTURE PROSPECTS

• Schmit – 1980s– “I believe optimization has a future because people

think they can make money on it”

• Just as Spreadsheets are Routinely Used by Accounts

• Just as Word Processors are Routinely Used by Secretaries

• So Should Optimization be Used by Engineers

• Optimization Will be Widely Used when Management Understands the Enormous Benefit

• Progress Is Made One Retirement at a Time

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SUMMARYSUMMARY

• Optimization Technology is Well Developed• For General Applications

– We Can Couple Almost Any Analysis With Optimization

• For Structural Optimization– Technology is Very Advanced

– Find an Optimum Using Only About 10 Finite Element Analyses

• Optimization is the Most Powerful Design Improvement Tool Available Today