Computational Multidisciplinary Design Optimization IL YONG KIM, PhD Dept of Mechanical and Materials Engineering Queen’s University November 1, 2005 2 Presentation Agenda What is Design Optimization? MDO: Multidisciplinary Design Optimization MOO: Multiobjective Optimization Optimization Methods Applications Summary What is Design Optimization? 4 What is Design Optimization? Selecting the “best” design within the available means 1. What is our criterion for “best” design? 2. What are the available means? 3. How do we describe different designs? Objective function Constraints (design requirements) Design Variables 5 J(x) : Objective function to be minimized g(x) : Inequality constraints h(x) : Equality constraints x : Design variables Optimization Statement 0 0 Subject to ) ( Minimize ≤ ≤ h(x) g(x) x J 6 Improve Design Computer Simulation START Converge ? Y N END Optimization Procedure Evaluate J(x), g(x), h(x) Change x Determine an initial design (x 0 ) Does your design meet a termination criterion? 0 0 Subject to ) ( Minimize ≤ ≤ h(x) g(x) x J
Transcript
1
Computational Multidisciplinary
Design Optimization
IL YONG KIM, PhDDept of Mechanical and Materials Engineering
Queen’s University
November 1, 2005 2
Presentation AgendaWhat is Design Optimization?
MDO: Multidisciplinary Design Optimization
MOO: Multiobjective Optimization
Optimization Methods
Applications
Summary
What is
Design Optimization?
4
What is Design Optimization?
Selecting the “best” design within the available means
1. What is our criterion for “best” design?
2. What are the available means?
3. How do we describe different designs?
Objective function
Constraints (design requirements)
Design Variables
5
J(x) : Objective function to be minimizedg(x) : Inequality constraintsh(x) : Equality constraintsx : Design variables
Optimization Statement
00Subject to
)(Minimize
≤≤
h(x)g(x)
xJ
6
Improve DesignComputer Simulation
START
Converge ?Y
N
END
Optimization Procedure
Evaluate J(x), g(x), h(x)
Change x
Determine an initial design (x0)
Does your design meet a termination
criterion?
00Subject to
)(Minimize
≤≤
h(x)g(x)
xJ
2
7
Examples
H
L
H
Topology Optimization by DSO
MDO:Multidisciplinary Design
Optimization
9
MDO Definition
What is MDO ?
Optimal design of complex engineering systems that requires analysis that accounts for interactions amongst the disciplines
• But there can be more than two disciplines interacting• Some can be non-technical, e.g. cost estimation
Progress often occurs within disciplines and at the intersection of traditional disciplines
Traditional Pairings
3
13
Multidisciplinary Aspects of Design
Emphasis is on the multidisciplinary nature of thecomplex engineering systems design process.
Structures
Aerodynamics
ControlEmphasis in recent years has
been on advances that can be achieved due to the inter-
action of two or more disciplines.
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 14
MDO Framework
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
MOO:Multiobjective Optimization
16
J(x) : Objective function to be minimizedg(x) : Inequality constraintsh(x) : Equality constraintsx : Design variables
Optimization Statement
00Subject to
)(Minimize
≤≤
h(x)g(x)
xJ
17
Multiobjective Optimization Problem Formal Definition
When multiple objectives (criteria) are present
( ) ( )[ ]
1
2
1
1
1
1
where
( ) ( )
( ) ( )
=
=
=
=
Tz
Ti n
T
m
T
m
J J
x x x
g g
h h
J x x
x
g x x
h x x
00s.t.
min
≤≤
h(x)g(x)J(x)
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Multiple Objectives
1
2
3
cost [$]- range [km]weight [kg]
- data rate [bps]
- ROI [%]
i
z
JJJJ
J
= =
J
The objective can be a vector J of z system responsesor characteristics we are trying to maximize or minimize
Often the objective is ascalar function, but forreal systems often we attempt multi-objectiveoptimization:
x J(x)Objectives often
conflict with each other!
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
4
19
1 2[ ]Tnx x x
x1
x2
Jx (x)
n-dimension 1-dimension
J
Single objective
x J(x)
m-dimension
x1
x2
1 2[ ]Tnx x xn-dimension
J1
J2
Multiobjective
Mapping
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Pareto Frontier
1
J1: Manufacturing cost
J2:
Weight
2
5
4
3
Pareto frontier
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Pareto FrontierIn a two-dimensional trade space (i.e. two decision criteria), the Pareto Optimal set represents the boundary of the most design efficient solutions.
0 500 1000 1500 2000 2500 3000 3500 4000
800
1000
1200
1400
1600
1800
2000
2200
Performance (total # of images)
Life
cycl
e C
ost (
$M)
TPF System Trade Space Pareto-Optimal Front
Dominated Solutions Non-Dominated Solutions
$2M/Image
$1M/Image
$0.5M/Image
$0.25M/Image
SSI
SCI
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 22
Pareto Optimal means …..
“Take from Peter to pay Paul”
Pareto Frontier
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MDO and MOO
Zang, Thomas and Green, Lawrence, “Multidisciplinary Design Optimization Techniques: Implications and Opportunities for Fluid Dynamics Research,” 30th AIAA Fluid Dynamics Conference Norfolk, VA June 28 -July 1, 1999 24
single discipline multiple disciplines
sing
le o
bjec
tive
mul
tiple
obj
.
single discipline multiple disciplines
Minimize displacements.t. mass and loading constraint
F
δl
mcantilever beam support bracket
Minimize stamping costs (mfg) subject
to loading and geometryconstraint
F
D
$
airfoilα (x,y)
Maximize CL/CD and maximizewing fuel volume for specified α, vo
Vfuelvo
Minimize cost and maximize cruisespeed s.t. fixed range and payload
commercial aircraft
MDO and MOO
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
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Computational Optimization Methods
(1) Gradient-based Methods
(2) Heuristic Methods
Computational Optimization Methods
(1) Gradient-based Methods
(2) Heuristic Methods
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Optimum Solution – Graphical Representation
J(x)
xNo active constraints
You do not know this function before optimization
Optimum solution (x*)
Start
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Gradient-based Methods
J(x)
x
Start
Check gradient
Move
Check gradient
Gradient=0
No active constraints
Stop!
You do not know this function before optimization
Optimum solution (x*)
(Termination criterion: Gradient=0)
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Gradient-based Methods
Two steps are repeated until a local optimum is found.
(1) Sensitivity Analysis: Which direction to go?
(2) Line Search: How much to go?(to the direction that was determined by sensitivity analysis)
30
Global vs. Local Optimum
J(x)
xNo active constraints
Global Optimum
Local Optimum
Local Optimum
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Gradient-based Methods
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 32
• Create a quadratic approximation to the Lagrangian
• Solve the quadratic problem to find the search direction, S
• Perform the 1-D search
• Update the approximation to the Lagrangian
Sequential Quadratic Programming
Computational Optimization Methods
(1) Gradient-based Methods
(2) Heuristic Methods
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A Heuristic is simply a rule of thumb that hopefully will find a good answer.
Why use a Heuristic?- Heuristics are typically used to solve complex optimization
problems that are difficult to solve to optimality.
Heuristics are good at dealing with local optimawithout getting stuck in them while searching for the global optimum.
Heuristic Methods
Schulz, A.S., “Metaheuristics,” 15.057 Systems Optimization Course Notes, MIT, 1999.
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Most Common Heuristic Techniques
• Genetic Algorithms
• Simulated Annealing
• Tabu Search
• Particle Swarm Method
Heuristic Methods
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Genetic AlgorithmsPrinciple by Charles Darwin - Natural Selection
7
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Genetic Algorithms: Procedures
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Genetic AlgorithmsGradient search
- Treats one design at one time
Genetic algorithms
- Treats a set of designs at one time
60
40
20
80
Gradient search
60
40
20
80
Genetic algorithms
60
40
20
80
Genetic algorithms
60
40
20
80
Genetic algorithms
60
40
20
80
Genetic algorithms
60
40
20
80
Genetic algorithms
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• Linear/nonlinear
• Type of design variables (real/integer, continuous/discrete)
• Equality/inequality constraints
• Discontinuous feasible spaces
• Initial solution feasible/infeasible
• Simulation code runtime
How To Choose an Algorithm?
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GASA, Tabu SearchPSO
Branch-and-Bound
Discretex (at least one)
SQP (constrained)Newton(unconstrained)
SimplexBarrier Methods
Continuous, realx (all)
NonlinearJ or g or h
LinearJ and g and h
Algorithm Selection Matrix
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Feasible Improved Local Optimum
Range of Objectives
Global Optimum
Do you really need to obtain the global optimum?
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Part vs. System
Parts System
MultidisciplineSingle-discipline
MultiobjectiveSingle-objective
Difficult to optimizeEasy to optimize
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Applications
MDO is a useful tool in the design of virtually all complex, multidisciplinary
systems…
44
Glass (Pyrex)
Fluid
Si
PZT Outlet choke
(heating)
Inlet choke
Net flow
Valveless MicropumpMIT, AIST*
* AIST (Japan): National Institute of Advanced Science and Technology
45 46
FE modeled region
Structure of Valveless Micropump
47Unit: g/mm-sec
right choke (outlet)
t=0 ms
t=0.5 ms
t=1.5 ms
t=2.5 ms
t=3.5 ms
Result – Viscosity Change
Prototype by Dr. Mastumura at AIST
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0 2 4 6 8 100.00
0.05
0.10
0.15
0.20
0.25
Rect
ifica
tion
effic
ienc
y
Time (ms)
Result – Rectification efficiency
−+
−+
+−
=mm
mm
uuuuε
Rectification efficiency
)1(0:
)0(:
outletatvelocityMean
inletatvelocityMean
==
==
=
=
−
−+
−
+
ε
ε
m
mm
m
m
u
uu
u
u
IICase
ICase
9
49
3,,1,
0),,(0),,(tosubject
),,(maximizeI]ion [Optimizat
0321max
3210
321
=≤≤
≤−≤−
ibbb
TbbbTbbb
bbbQ
upperii
loweri
εε
3,,1,
0),,(0),,(tosubject
),,(maximizeII]ion [Optimizat
0321max
3210
321
=≤≤
≤−≤−
ibbb
TbbbTbbbQQ
bbb
upperii
loweri
ε
Optimization Formulation
50Determine dynamic thermal distortions and pattern blur.
Reduce distortion due to change in the gravity direction
Semiconductor equipment: X-ray mask
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Precision Machine Design
Reduce structural and thermal distortions.
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[2]
Multi-Disciplinary Design Optimization considers:– Application of practical dynamic loading conditions– Reduction in overall mass (weight savings)– Increase in maximum joint angle (performance gain)– Decrease of manufacturing cost (economics)– Monitor Von-Mises stress and Strain Energy Density– Modify up to 10 design variables per part
MDO is a design tool that help create advanced and complex engineering systems that are competitive not only in terms of performance, but also in terms of manufacturability, serviceability, and overall life-cycle cost effectiveness.
