INTRODUCTION TO
DESIGN OPTIMIZATION
Ranjith Dissanayake
Structures Laboratory
Dept. of Civil of Engineering
Faculty of Engineering
University of Peradeniya
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OPTIMIZATION
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G. N. V.
REDUCE DESIGN
TIME
TASK 1
TASK 2
DEADLINE
IMPROVE DESIGN
QUALITY
FREE THE ENGINEER
FOR CREATIVE WORK
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THE PHYSICAL PROBLEM
BET I CAN
FIND THE
TOP OF THE
HILL!
YOU CAN
TRY, BUT
STAYINSIDE
THE FENCES
G. N. Vanderplaats
OBJECTIVE: FIND THE HIGHEST POINT
DESIGN VARIABLES: LONGITUDE AND LATITUDE
CONSTRAINTS: STAY INSIDE THE FENCES
YOU MAY START OUTSIDE THE FENCES
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THE ENGINEERING PROBLEM
G. N. V.
FENCE NO. 1
F1 = f1(X1, X2)
< 0 INSIDE
> 0 OUTSIDE
FENCE NO. 2
HILL
Y = f (X1, X2)
F2 = f2(X1, X2)
< 0 INSIDE
> 0 OUTSIDE
THE PHYSICAL PROBLEM IS NOW DEFINED
MATHEMATICALLY
BY CONVENTION, < MEANS INSIDE THE FENCES
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TH E OPTIMIZ ATION PROBLEM
MAX IMIZ E Y = f ( X 1, X 2)
OBJECTIVE
SUBJECT TO :
F1 = f1( X 1, X 2) <_ 0
CON STRAIN TS
F2 = f2( X 1, X 2) <_ 0
DESIGN VARIABLES
OPTIMIZATION IS A VERY SIMPLE EXTENSION
OF THE ENGINEERING PROBLEM
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THE OPTIMIZATION PROCESS
S3
X2
S2
X1
S1
G. N. Vanderplaats
FIND A SEARCH DIRECTION THAT WILL IMPROVE THE
OBJECTIVE WHILE STAYING INSIDE THE FENCES
SEARCH IN THIS DIRECTION UNTIL NO MORE
IMPROVEMENT IS MADE
REPEAT TO CONVERGENCE
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ANALYSIS VERSUS DESIGN
ANALYSIS: Given a Component or
System, Together With Loads, Materials,
etc.
Calculate the Responses to See if They Satisfy the Requirements
DESIGN: Given A Set of Requirements
Find the Component or System that Satisfies the Requirements with
Minimum Mass or Cost, Maximum Reliability, Maximum
Performance, etc.
ANALYSIS IS A SUB-PROBLEM OF DESIGN
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DESIGN
NONLINEAR, CONSTRAINED, OPTIMIZATION TASK
Find the Set of Design Variables, X, that will
Minimize F(X)
Subject to (Such That);
g j(X ) 0
X iL X i X iU
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j = 1, M
i = 1, n
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DESIGN
For Example
Minimize Structural Mass
Subject to Stress Limits;
ijk
g j(X ) =
0
Objective Function
Inequality Constraints
i = Load Condition
j = Stress Calculation Point
k = Stress Component
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OPTIMIZATION ALGORITHMS
1948: SIMPLEX Method for Linear Programming
1950s: Various Random Methods. Gradient Based
Methods Developed in the Late 1950s
1960s: Sequential Unconstrained Minimization
Techniques, Sequential Linear Programming, Feasible
Directions Methods
1970s: Enhanced Feasible Directions Methods, Multiplier
Methods, Reduced Gradient Methods
1980s: Variable Metric Methods, Sequential Quadratic
Programming Methods
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OPTIMIZATION ALGORITHMS
1990s: Genetic Algorithms, Simulated Annealing, New
Interest in Sequential Unconstrained Minimization
Techniques
2000s: Particle Swarming, Advanced Sequential
Unconstrained Minimization Techniques
Largest Known Test Example
250,000 Variables With 250,000 Active Constraints
Largest Known Real Structural Optimization Problem
190,000 Thickness Variables with Frequency Constraints
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OPTIMIZATION PROBLEM SIZE
BIGDOT
250,000
VARIABLES
100,000
10,000
# Des. Var.
1,000
100
0
1960
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1970
1980
1990
Year
2000
2010
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GENERAL OPTIMIZATION
PROBLEM STATEMENT
Minimize F(X)
Objective Function
Subject to (Such That);
g j (X ) 0
j = 1, M
Inequality Constraints
hk ( X ) = 0
k = 1, L
Equality Constraints
X iL X i X iU
i = 1, N
Side Constraints
F(X), gj(X) and hk(X) May be Linear, Nonlinear, Explicit,
Implicit, but Should be Continuous with Continuous First
Derivatives
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GENERAL STRATEGY
Given X0
At Iteration q, Update X by
Xq = Xq-1 + Sq
Sq = Vector Search Direction
= Step Size
Calculation of Sq Requires Gradients
Calculation of (One-Dimensional Search)
Requires Several Function Evaluations
Some Methods Dont Use a One-Dimensional Search
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THE OPTIMIZATION PROCESS
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
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THE OPTIMIZATION PROCESS
The Best Search Direction Is One That Will
Minimize
F ( X q )T S q
Sq is Usable
Subject to;
g j ( X )T S q 0
jJ
Sq is Feasible
where J = the Set of Active (Critical) Constraints
This Requires Calculating the Gradient Vector of
the Objective and Active Constraints
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PHYSICAL INTERPRETATION
X2
F
F = CONSTANT
FEASIBLE
S
INFEASIBLE
g =0
X1
During the Search in Direction S, We Adjust the Variables
X1 and X2 to Move Back to the Feasible Region
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GRADIENT CALCULATIONS
By First Forward Finite Difference
F ( X + X1 ) F ( X )
X1
F(X + X ) F(X )
2
X2
......
......
F
(
X
+
X
)
F
(
X
)
N
XN
The Finite Difference Step Size Must be Chosen to Avoid
Errors From Noise and/or Round-Off
Central Difference is More Accurate but Twice as Costly
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GRADIENT CALCULATIONS
Analytically in Structural Optimization
KU = P
U
1 P K
=K
U
X
X X
K
X
Is the Sum of Derivatives of Element Matrices, ki
K
X
May be Calculated by Finite Difference
Gradients of Stress are Calculated From This
Gradients of Other Responses (Eigenvalues, Dynamic
Response, etc.) May Also be Calculated in This Way
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USEFUL DENINITIONS
Design Variables: Those Parameters to be Changed to
Improve the Design
Objective Function: The Function of the Design
Variables to be Minimized or Maximized
Inequality Constraints: One Sided Conditions that Must
be Satisfied for the Design to be Acceptable
Equality Constraints: Precise Conditions that Must be
Satisfied for the Design to be Acceptable
Side Constraints: Bounds on the Design Variables that
Limit the Region of Search for the Optimum
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USEFUL DENINITIONS
Feasible Design: One that Satisfies All Constraints
Infeasible Design: One that Violates One or More
Constraints
Optimum Design: The Set of Design Variables and the
Corresponding Minimum (Maximum) Objective Satisfying
All Constraints
Kuhn-Tucker Conditions: Necessary Mathematical
Conditions that Must be Satisfied for a Design to be
Optimum
Two-Variable Function Space: Geometric
Representation of a Two-Variable Design Problem
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EXAMPLE
Minimize
F ( X ) = X1 + X 2
Subject to;
g(X ) =
1
1
+
0
X1 X 2
X1 0
X2 0
X2
4
F=4
3
FEASIBLE REGION
3
2
2
OPTIMUM
1
1
g=0
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X1
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THE KUHNKUHN-TUCKER
CONDITIONS FOR OPTIMALITY
X* is Feasible
jg j = 0
F ( X ) =
j = 1, M
M
M +L
j =1
k = M +1
jg j ( X ) + k hk ( X ) = 0
j 0, j = 1, M
k Unrestricted in Sign, k=M+1,L
This States that the Vector Sum of the Gradients of the
Objective and the Critical Constraints (Scaled by Their
Lagrange Multipliers) Must Add to Zero at the Optimum
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KUHN--TUCKER CONDITIONS
KUHN
Geometric Interpretation
X2
g3 = 0
3g3
F
1g1
F = CONSTANT
g1
g2 = 0
g 3
g1 = 0
X1
Note that Constraint Number 2 is Not Active, so it is Not
Included in the Vector Sum
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TYPICAL APPLICATIONS
Structures
Trusses, Panels (Isotropic, Composite), Shells, Pressure Vessels,
Frames
Automotive, Aerospace, Space, Ship, Rail
Mechanical Components
Shafts, Gears, Vibration Isolation, Piping Systems, Flyweels
Thermal Systems, Injection Molding, Heat Exchangers,
Steam Condensers, Heat Sinks, Combustion Cycles
Systems
Aircraft, Space, Automobile, Ships
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TYPICAL APPLICATIONS
Computational Fluid Dynamics, Combustion, Acoustics
Airfoils, Ducting, Noise Minimization, etc.
Other Applications
Control Systems
System Identification
Curve Fits
Lens Optics
Portfolio
Other
Applications are Limited Only by Our Ingenuity
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THE COST OF OPTIMIZATION
(BLACK BOX)
Criteria
Find a Near Optimum Quickly
Use as Few Function Evaluations as Possible
Basis for Criteria
Precise Optimum is Seldom Meaningful
Loads, Material Properties, Heat Transfer Coefficients, etc. are Seldom
Known to Within a Few Percent
Each Analysis can be Expensive
Cost
About 10N + 40 Times the Cost of One Analysis
This Estimate Assumes Finite Difference Gradients
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OPTIMIZATION WORKS
1975
SUPERSONIC CRUISE AIRCRAFT
COMBAT
MISSION
INITIAL
OPTIMUM
SOLVED BY THE ACSYNT PROGRAM
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5 DESIGN VARIABLES, 2 PERFORMANCE CONSTRAINTS
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OPTIMIZATION WORKS
1975
SUPERSONIC CRUISE AIRCRAFT
1.4
RELATIVE
E MASS
CONVENTIONAL
1.2
NOMINAL
DESIGN
1.0
1.0
TECHNOLOGY
0.9
FACTOR
0.8
ADVANCED
0.8
0.6
SUSTAINED LOAD FACTOR AT M = 0.9
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TRADE - OFF STUDY
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OPTIMIZATION WORKS
1976: A Two Hour Study
STOL AIRCRAFT TAKEOFF
CONVENTIONAL: W =
G W
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SKI JUMP: W =
G 1.2W
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OPTIMIZATION WORKS
1978: Today Called Response Surface Method
HIGH SPEED AIRFOIL OPTIMIZATION
INITIAL SHAPE
OPTIMUM: MAXIMIZE LIFT WITH DRAG & MOMENT CONSTRAINTS
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OPTIMUM: MINIMIZE DRAG WITH LIFT & MOMENT CONSTRAINTS
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OPTIMIZATION WORKS
It Has Been Working For Many Years
The Above Examples are 25-28 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 IS THE MOST POWERFUL
DESIGN PRODUCTIVITY TOOL AVAILABLE
TODAY
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STRUCTURAL OPTIMIZATION
HISTORY
1960 Schmit: Structural Optimization by Systematic
Synthesis
Combined Finite Element Analysis with Numerical Optimization
1974 Schmit & Farshi: Basic Approximation Concepts
Create an Approximation of the Responses and Use this During
Optimization
The Approximation is Based on Physics, Not Just Linearization
1975 Schmit & Miura: Approximation Concepts Refined
Rod and Membrane Elements
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STRUCTURAL OPTIMIZATION
HISTORY
1985+ Optimization Added to Commercial Finite Element
Programs
ANSYS, MSC/Nastran, UAI/Nastran, CSA/Nastran, COSMOS, etc.
None used 2nd Generation Approximation Concepts
1986+ Vanderplaats et al: 2nd Generation Approximation
Concepts
Intermediate Variables
Intermediate Responses
Rod Elements, Beam Elements, Plate/Shell Elements, Solid
Elements, Composites, etc.
1992 GENESIS Version 1.0 Released
Full Use of 2nd Generation Methods
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PERSPECTIVE
1960 Optimization of the Three-Bar Truss Require Hour
on an IBM 653 Computer. Today it takes Under 1 Second.
1980 Schmit: Only a Congenial Optimist Could
Conclude that Optimization had a Future with Run Times
Like That
A2
A1
A 3=A1
MINIMIZE WEIGHT
2 LOAD CASES
STRESS LIMITS
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OPTIMUM IS NOT
P1
P2
FULLY STRESSED!
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STRUCTURAL OPTIMIZATION
BEFORE 1974
CONTROL
PROGRAM
FEM
ANALYSIS
OPTIMIZER
SENSITIVITY
ANALYSIS
The Optimizer Often Required Hundreds of Finite Element
Analyses
Too Expensive for Real Design Applications
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MODERN STRUCTURAL
OPTIMIZATION
FEM
ANALYSIS
OUTER LOOP
CONSTRAINT
SCREENING
CONTROL
PROGRAM
SENSITIVITY
ANALYSIS
APPROXIMATE
PROBLEM
GENERATOR
APPROXIMATE
ANALYSIS
INNER LOOP
OPTIMIZER
Use Approximations to Avoid Many Calls to the FEA
Optimizer Never Actually Calls the Finite Element Analysis
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THE COST OF STRUCTURAL
OPTIMIZATION
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
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STRUCTURAL OPTIMIZATION
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
99% 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 100,000 Design Variables Have Been
Solved by the GENESIS Program
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RECENT EXAMPLES USING
MODERN SOFTWARE
DOT: Basic Optimizer
GENESIS: Structural Analysis & Optimization
VISUALDOC: General Multidiscipline Optimization
VISUALSCRIPT: Integration Tool
BIGDOT: Very Large Scale Optimization
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DOT
1987: VR&Ds First Commercial Product
3rd Generation Optimization Software by Vanderplaats
CONMIN 1972, ADS - 1984
Solves General Nonlinear Constrained Optimization
Tasks
Used by VisualDOC, GENESIS, MSC.Nastran,
COSMOS/M, FEM5, LMS/CADSI, BEASY,
MODEL CENTER, TBCAD, SINDA/FLUINT,
MERCURY, DAKOTA, Other
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GENESIS
1992: Fully Integrated Finite Element Analysis
and Optimization
Topology, Member Sizing & Shape Optimization
NASTRAN Compatible Analysis Data
Currently About 90% of NASTRAN Capabilities
Solves About 99% of Daily Analysis Tasks
Second Generation Approximation Methods
Typically 10 Finite Element Analyses Optimize
Have Optimized with Almost 200,000 Variables
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GENESIS
Truck Frame Topology (10,910 Variables)
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PACCAR
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GENESIS
STEERING KNUCKLE
8 SHAPE VARIABLES
REDUCED MASS 13%
NO INCREASE IN MAXIMUM STRESS
FORD
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GENESIS
Fuel Tank Bead Design (99 Shape Variables)
Increased Stiffness > 50%
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VISTEON
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GENESIS
Air Cleaner Design
Reduced Radiated Noise 10 db
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DELPHI
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Femb
Interfaces
EDS/IDEAS
MSC.Patran
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VisualDOC/VisualScript
1998: Graphics Based Design Environment
General Optimization (Gradient & Non-Gradient)
Design Of Experiments
Response Surface Optimization
Allows User to Interface Optimization Tools with
Almost Any Analysis
Modules Available as APIs For Easy Integration
with other Control Programs
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VisualDOC
Hybrid Electric Vehicle
Optimize Control Strategy
Maximize Economy
Minimize Emissions
Limits on Acceleration
and Grade
Results
3.3% Economy Increase
36% Nox Reduction
14% Particulate Mass Reduction
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NATIONAL RENEWABLE ENERGY LAB
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VisualDOC
Aircraft Wing MDO
Maximize Range for Fixed Gross Weight
Call Aerodynamic Analysis
Call GENESIS for Structural Mass Sub-Optimization
25% Range Increase
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NASA/VR&D
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Cantilevered Beam
Number of Design Variables, NDV
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10,000
50,000
100,000
250,000
CONTINUOUS
OPTIMUM
53,744
(233/43)
[9,995/12]
53,744
(243/46)
[49,979/46]
53,720
(209/38)
[99,927/150]
53,755
(262/49)
[249,919/211]
DISCRETE OPTIMUM
54,864
(80/14)
54,864
(92/38)
54,848
(96/25)
54,887
(143/24)
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Summary
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
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