# Document 241919

```Presentation Agenda
Computational Multidisciplinary
 What is Design Optimization?
 MDO: Multidisciplinary Design Optimization
Design Optimization
 MOO: Multiobjective Optimization
 Optimization Methods
 Applications
IL YONG KIM, PhD
Dept of Mechanical and Materials Engineering
Queen’s University
 Summary
November 1, 2005
2
What is Design Optimization?
Selecting the “best” design within the
available means
What is
1. What is our criterion for “best” design?
Objective function
Design Optimization?
2. What are the available means?
Constraints (design requirements)
3. How do we describe different designs?
Design Variables
4
Optimization Statement
Optimization Procedure
Minimize J (x)
Subject to g(x) ≤ 0
h(x) ≤ 0
Minimize J (x)
START
Subject to g(x) ≤ 0
h(x) ≤ 0
Improve Design
J(x) : Objective function to be minimized
g(x) : Inequality constraints
h(x) : Equality constraints
x
: Design variables
Determine an initial
design (x0)
Computer
Simulation
Evaluate J(x), g(x), h(x)
Converge ?
a termination
criterion?
Change x
N
Y
END
5
6
1
Examples
Topology Optimization by DSO
MDO:
L
H
Multidisciplinary Design
Optimization
H
7
MDO Definition
Definition of “Discipline”
What is MDO ?
 Optimal design of complex engineering
systems that requires analysis that accounts
for interactions amongst the disciplines
A discipline can often be defined by a
set of equations that govern the
underlying physical processes of
interest.
 “How to decide what to change, and to what
extent to change it, when everything
influences everything else.”
Ref: AIAA MDO website http://endo.sandia.gov/AIAA_MDOTC/main.html
9
Engineering Design Disciplines
Aircraft:
Spacecraft:
Automobiles:
Aerodynamics
Propulsion
Structures
Controls
Avionics/Software
Manufacturing
others
Astrodynamics
Thermodynamics
Communications
Structures
Optics
Guidance & Control
Engines
Body/chassis
Aerodynamics
Electronics
Hydraulics
Industrial design
others
10
• Aerodynamics + Structures = Aeroelasticity
• Optics + Controls = Adaptive Optics
• Thermodynamics + Structures = Thermostructures
• Acoustics + Structures = Acoustic Structures
•Finance + Manufacturing = Lean Production
• Scheduling + Manufacturing = Just-in-Time
Progress often occurs within disciplines and
at the intersection of traditional disciplines
• But there can be more than two disciplines interacting
• Some can be non-technical, e.g. cost estimation
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
11
12
2
Multidisciplinary Aspects of Design
MDO Framework
Emphasis is on the multidisciplinary nature of the
complex engineering systems design process.
Structures
Control
Aerodynamics
Emphasis in recent years has
be achieved due to the interaction of two or more
disciplines.
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
13
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
14
Optimization Statement
Minimize J (x)
MOO:
Subject to g(x) ≤ 0
h(x) ≤ 0
Multiobjective Optimization
J(x) : Objective function to be minimized
g(x) : Inequality constraints
h(x) : Equality constraints
x
: Design variables
16
Multiobjective Optimization Problem
Formal Definition
Multiple Objectives
The objective can be a vector J of z system responses
or characteristics we are trying to maximize or minimize
When multiple objectives (criteria) are present
min J(x)
s.t. g(x) ≤ 0
h(x) ≤ 0
Often the objective is a
where J =  J1 ( x ) " J z ( x ) 
 J1   cost [\$]  scalar function, but for
 J   - range [km]  real systems often we
 2 
 attempt multi-objective
 J 3   weight [kg] 
J= =
 optimization:
 J i   - data rate [bps]
x 6 J(x)
# 

#
Objectives often
  

 J z   - ROI [%]  conflict with each other!
T
x = [ x1 " xi " xn ]
T
g =  g1 ( x ) " g m1 ( x ) 
T
h =  h1 (x ) " hm2 ( x ) 
T
17
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
18
3
Mapping
Single objective
n-dimension T
Pareto Frontier
1-dimension
[ x1 x2 " xn ]
J2:
x2
Pareto frontier
Weight
x 6 J (x)
J
1
3
x1
Multiobjective
n-dimension
J2
[ x1 x2 " xn ]T
m-dimension
4
2
x2
5
x 6 J(x)
J1
x1
J1: Manufacturing cost
19
20
Pareto Frontier
Pareto Frontier
In a two-dimensional trade space (i.e. two decision criteria),
the Pareto Optimal set represents the boundary of the most
design efficient solutions.
TPF System Trade Space Pareto-Optimal Front
2200
Dominated Solutions
Non-Dom inated S olutions
2000
“Take from Peter to pay Paul”
\$1M/Im age
1800
\$2M /Im age
1600
\$0.5M /Im age
SSI
1400
1200
1000
SCI
800
0
500
\$0.25M/Im age
1000
1500
2000
2500
3000
3500
4000
Performance (total # of images)
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
21
22
MDO and MOO
single objective
MDO and MOO
single discipline
cantilever beam
m
F
l
\$
δ
Minimize displacement
single discipline
multiple obj.
Lifecycle Cost (\$M)
Pareto Optimal means …..
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
23
vo
α
multiple disciplines
support bracket
F
Minimize stamping
costs (mfg) subject
constraint
D
multiple disciplines
commercial aircraft
airfoil V
fuel
(x,y)
Maximize CL/CD and maximize
Minimize cost and maximize cruise
wing fuel volume for specified α, vo speed s.t. fixed range and payload
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
24
4
Computational
Computational
Optimization Methods
Optimization Methods
(2) Heuristic Methods
(2) Heuristic Methods
Optimum Solution
– Graphical Representation
You do not know this function before optimization
You do not know this function before optimization
J(x)
J(x)
Start
Start
Move
No active constraints
Optimum solution (x*)
x
No active constraints
Stop!
Optimum solution (x*)
x
27
28
Global vs. Local Optimum
Two steps are repeated until a local optimum is found.
J(x)
(1) Sensitivity Analysis
: Which direction to go?
Local Optimum
(2) Line Search
Local Optimum
: How much to go?
(to the direction that was determined by sensitivity analysis)
Global Optimum
No active constraints
29
x
30
5
• 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
Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004
31
32
Heuristic Methods
Computational
 A Heuristic is simply a rule of thumb that hopefully
Optimization Methods
 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 optima
without getting stuck in them while searching for
the global optimum.
(2) Heuristic Methods
Schulz, A.S., “Metaheuristics,” 15.057 Systems Optimization Course Notes, MIT, 1999.
34
Genetic Algorithms
Heuristic Methods
Principle by Charles Darwin - Natural Selection
Most Common Heuristic Techniques
• Genetic Algorithms
• Simulated Annealing
• Tabu Search
• Particle Swarm Method
35
36
6
Genetic Algorithms: Procedures
Genetic Algorithms
- Treats one design at one time
Genetic algorithms
- Treats a set of designs at one time
Genetic algorithms
20
40
60
80
37
How To Choose an Algorithm?
38
Algorithm Selection Matrix
Linear
J and g and h
• Linear/nonlinear
• Type of design variables
(real/integer, continuous/discrete)
Nonlinear
J or g or h
Continuous, real Simplex
SQP (constraine
x (all)
Barrier Methods d)
Newton
(unconstrained)
Discrete
Branch-and-Bou GA
nd
x (at least one)
SA, Tabu Search
PSO
• Equality/inequality constraints
• Discontinuous feasible spaces
• Initial solution feasible/infeasible
• Simulation code runtime
39
40
Range of Objectives
Part vs. System
Do you really need to obtain the global optimum?
Feasible
Improved
Local
Optimum
Global
Optimum
41
Single-discipline
Multidiscipline
Single-objective
Multiobjective
Easy to optimize
Difficult to optimize
Parts
System
42
7
Valveless Micropump
MIT, AIST*
Applications
Net flow
Glass
(Pyrex)
Fluid
Si
MDO is a useful tool in the design of
Inlet
choke
virtually all complex, multidisciplinary
Outlet
choke
(heating)
PZT
systems…
* AIST (Japan): National Institute of Advanced Science and Technology
44
Structure of Valveless Micropump
FE modeled region
45
Result – Viscosity Change
46
Result – Rectification efficiency
right choke (outlet)
Rectification
efficiency
t=0.5 ms
ε=
t=1.5 ms
t=2.5 ms
u m+ − u m−
u m+ + u m−
u m+ = Mean velocity at inlet
t=3.5 ms
u m− = Mean velocity at outlet
Case I : u m+ = u m− (ε = 0)
Prototype by Dr.
Mastumura at AIST
Case
Unit: g/mm-sec
47
II : u m−
= 0 (ε = 1)
Rectification efficiency
t=0 ms
0.25
0.20
0.15
0.10
0.05
0.00
0
2
4
6
8
10
Time (ms)
48
8
Semiconductor equipment:
Optimization Formulation
[Optimization I]
maximize Q(b1 , b2 , b3 )
subject to ε 0 − ε (b1 , b2 , b3 ) ≤ 0
Tmax (b1 , b2 , b3 ) − T0 ≤ 0
bilower ≤ bi ≤ biupper , i = 1,",3
[Optimization II]
maximize ε (b1 , b2 , b3 )
subject to Q0 − Q (b1 , b2 , b3 ) ≤ 0
Tmax (b1 , b2 , b3 ) − T0 ≤ 0
Reduce distortion due to change in the gravity direction
bilower ≤ bi ≤ biupper , i = 1,",3
Determine dynamic thermal distortions and pattern blur.
49
50
Universal Joint
Precision Machine Design

Universal Joints transmit rotation between shafts whos
e axis are coplanar, but not coinciding

Assembly comprised of 3 components:
[1]
– Flange yoke, Weld yoke, Cross Trunion
– Conduct optimization on individual parts & assembly

At non-zero joint angles, the output shaft will experienc
e both acceleration & deceleration every revolution, lea
ding to dynamic instability

Multi-Disciplinary Design Optimization considers:
–
–
–
–
–
–
Reduce structural and thermal distortions.

51
[2]
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
Employ ‘Adaptive Weighted Sum’ to construct a Pareto
Surface representing optimal designs
52
[1] http://www.nrg.com.au/~hemp/bigjoint/bigjoint.htm&h=225&w=300&sz=12&tbnid=IfuOOO7gop0J:&tbnh=83&tbnw=111&start=10&prev=/images%3Fq%3Duniversal%2Bjoint%26hl%3Den%26lr%3D%26client%3Dfirefoxa%26rls%3Dorg.mozilla:en-US:official_s%26sa%3DG
[2] Parker, Sybil P. “Encyclopedia of Engineering.” McGraw-Hill Book Company, New York, United States of America, 1983 pp 1151-1153.
Cementless hip prosthesis

Motivation:
–
–

Un-cemented implants should be used for younger patient
s since younger patients are more active, and more bone s
tock is preserved for revision surgery
Un-cemented implants with a longer life reduce the numbe
r times the patient has to undergo revision surgery
Design Objectives:
–
Minimize wear of the bearing and acetabulum surfaces
–
Minimize relative motion at the implant-bone interface
•
•
–

Micro-movements at implant-bone interface inhibit bone in-grow
th into the implant surface which is necessary for long term impl
ant fixation
Minimize cost
Femoral Osteotomy
Implant shape/size
Bearing shape/size
Acetabular component shape/size
Chromosome length change
–
–

Wear debris can cause an inflammatory reaction eventually lead
ing to bone degradation at the implant-bone interface
Design Variables
–
–
–
–

Variable Chromosome Length GA
Normal Walking
Stair climbing
refinement
Contact Analysis:
–
–
–
Bearing-acetabulum
Taper-bearing
Implant-bone
53
54
9
Design for Flexibility
Design for Flexibility
Bridge
Customers such as the military would
like a simply-designed bridge that can be
used for various applications
- Various span lengths
Short creek
Large river
Support tanks, trucks, cars, etc.
Flexibility, Changeability, Extensibility, Reconfigurability, Modularity, etc.
55
Optimization for Manufacturability
56
Design Under Uncertainty
J(x)
f ( x1 ) < f ( x2 )
But, ∆f ( x1 ) > ∆f ( x2 ) for the same ∆x
Determine x2 using design optimization
∆J1
∆J 2
x1
∆x
x2
x
Design variable
∆x
57
58
Summary
Reference
 Il Yong Kim, “MECH465: Computer Aided Design,” Lecture notes,
min J(x)
s.t. g(x) ≤ 0
h(x) ≤ 0
Queen’s University, 2005
 Panos Y. Papalambros and Douglass J. Wilde, Principles of
Optimal Design, 2nd edition, Cambridge University Press
 Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary
MDO is a design tool that help create advanced and
System Design Optimization,” Lecture notes, MIT, 2004
complex engineering systems that are competitive
 AIAA MDO website http://endo.sandia.gov/AIAA_MDOTC/main.html
not only in terms of performance, but also in terms of
 Jasbir S. Arora, Introduction to Optimum Design, 2nd edition,
manufacturability, serviceability, and overall life-cycle
McGraw-Hill
cost effectiveness.
59
60
10
```