Multidisciplinary System
Design Optimization (MSDO)
Decomposition and Coupling
Lecture 4
Anas Alfaris
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© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Today’s Topics
• Information Flow and Coupling
• MDO frameworks
– Single-Level (Distributed analysis)
– Multi-Level (Distributed design)
• Collaborative Optimization
• Analytical Target Cascading
• (Hierarchical Decomposition & Multi-Domain Formulation)
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© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Standard Optimization Problem
Given
x
n
J:
n
g:
n
*
x0
x
z
m
Optimization Engine
J ( x)
g ( x)
x
Solve the problem
min J ( x)
s.t. g ( x) 0
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That is, find x* s.t. J( x* )
Function Evaluator
f ( x),
x dom( J )
dom( g )
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Information Flow
A
A
B
B
Dependent Tasks (Series)
Independent Tasks (Parallel)
Interdependent Tasks (Coupled)
A B
A A
B
B
A B
A A
B
B
A B
A A
B
B
A
B
Image by MIT OpenCourseWare.
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B C A K L J F I E D H G
B •
C
•
A
•
K
•
L
•
J
•
F
•
I
•
E
•
D
•
H
•
G
•
Feed Backward
Feed Forward
Information Flow
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Information Flow
B C A K L J F I E D H G
B •
Sequential
C
•
Parallel
A
•
K
•
L
•
Coupled
J
•
F
•
I
•
E
•
D
•
H
•
G
•
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Advantages of Decoupling
Computation of g(x) can be very time consuming, want
to divide the work and compute in parallel.
For example, if x ( x1 , x2 ), where x1 Rn1 , x2 R n 2
and g( x) ( g1 ( x1 ), g 2 ( x2 ))
Then g1 and g2 can be computed in parallel. Graphically,
Optimizer
x1
g1g 2
SS1
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Optim
x2
SS2
g1
g2
x1
x2
SS1
SS2
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Coupling
The decoupled constraints assumption is not general. Subsystems
can be coupled and loops can arise. For example,
Optim
Optimizer
x1
SS1
u1
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x2
w1 w2
SS2
u2
w1
w2
x1
SS1
u1
x2
u2
SS2
Loop
x: decision variables
vline: SS input
w: SS outputs (constraint, cost)
hline: SS output
u: SS input (dependent)
Computation of w1 and w2 requires an iterative method.
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w1
w2
Coupling
• An example where such a loop happens is as follows:
min J ( x1 , x2 )
w1
s.t.
w2
g1 ( x1 , g 2 ( x2 , w1 )) 0
g 2 ( x2 , g1 ( x1 , w2 )) 0
where x1 R n1 , x2
R n 2 , gi : xi ui
wi , i 1, 2
• w1 and w2 satisfy coupled relations at each optimization iteration.
At each constraint evaluation, nonlinear equations must be solved
(e.g. by Newton’s method) in order to obtain w1 and w2, which can
be time consuming.
Want a way to return to the situation of decoupled constraints.
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Surrogate Variables (“Tearing”)
Information loop can be broken by introducing surrogate variables.
min J ( x1 , x2 )
min J ( x1 , x2 )
s.t.
w1
s.t.
w2
g1 ( x1 , u1 ) 0
g1 ( x1 , g 2 ( x2 , w1 )) 0
g 2 ( x2 , g1 ( x1 , w2 )) 0
g 2 ( x2 , u2 ) 0
u2
g1 ( x1 , u1 )
0
u1 g 2 ( x2 , u2 )
0
• u1 and u2 are decision variables acting as the inputs to
g1(SS1) and g2 (SS2). Introducing surrogate variables
breaks information loop but increases the number of
decision variables.
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© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Numerical Example
min J1
J2
s.t. w1
0
w2
0
where J1
decoupled
2
1
x
2
2
x
2
J2
( x3 3)
( x4 4)
w1
x13
x23 2w2
w2
x33
x43 2w1
2
coupled
min x12
x22 ( x3 3)2 ( x4 4)2
s.t. w1
g1 ( x1 , x2 , x3 , x4 ) 0
w2
g 2 ( x1 , x2 , x3 , x4 ) 0
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min x12
x22 ( x3 3) 2 ( x4 4) 2
s.t. w1
x13
x23 2 x5
0
w2
x33
x43 2 x6
0
x13
x23 2 x5
x6
0
x33
x43 2 x6
x5
0
Solution:
x (0,0, 4,3,12 13 , 24 2 3)
MATLAB® 5.3
coupled: 356,423 FLOPS 4.844s
uncoupled: 281,379 FLOPS 0.453s
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Single-level and Multi-Level Frameworks
Single-level
(Distributed Analysis)
-disciplinary models provide
analysis
-all optimization done at
system level
Multi-level
(Distributed Design)
-provide disciplinary
models with design tasks
-optimization at
subsystem and system
levels
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non-hierarchical
decomposition
hierarchical
decomposition
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Single-level (Distributed Analysis)
• Disciplinary models provide analysis
• Optimization is controlled by some overseeing code
or database
e.g.
ISight (Optimizer)
iSight
Optimizer
design variables
constraints
System
Optimizer
Shared data
x
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Structures
Aero
Local data
Local data
J(x),g(x),h(x)
subsystem
analyses
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Single-level (Distributed Analysis)
Optimizer
objective
design variables
constraints
x
g(x)
h(x)
aerodynamic
analysis
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x
J(x)
performance
analysis
x
g(x)
h(x)
structural
analysis
• During the optimization, the overseeing code keeps track of the
values of the design variables and objective
• The values of the design variables are changed according to
the optimization algorithm
• Disciplinary models are asked to evaluate constraints/objective
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Multi-level (Distributed Design)
• Multi-level Optimization methods distribute decision
making throughout the system
• Subsystem level models are provided with design
tasks
• Optimization is performed at a subsystem level in
addition to the system level
• Provide some autonomy to design groups and
reduces communication requirements.
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Multi-level (Distributed Design)
System level optimizer
command/result
SS1
optimizer
command/result
SS2
optimizer
command/result
SSN
optimizer
……
SS1
analyzer
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SS2
analyzer
SSN
analyzer
Subsystem
black
box (BB)
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Collaborative Optimization
Collaborative Optimization (CO)
• disciplinary teams satisfy local constraints while
trying to match target values specified by a
system coordinator
• preserves disciplinary-level design freedom.
• CO is used typically to solve discipline-based
decomposed system optimization problems.
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Collaborative Optimization
OPTIMIZER
Coupled
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TARGET STATE
Uncoupled
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Collaborative Optimization
Two levels of optimization:
• A system-level optimizer provides a set of targets.
– These targets are chosen to optimize the system-level
objective function
• A subsystem optimizer finds a design that minimizes the
difference between current states and the targets.
– Subject to local constraints
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Collaborative Optimization
min Jsys
wrt: x 0
s.t. Jk
x0
min J1
x
local variables
s.t. local constraints
x
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computed
results
analysis for
subsystem 1
0
subproblemsk Jsys
2
performance
analysis
Jk
x0
J1
target
local
variables variables
target variables
x0
min Jk
x
target
local
variables variables
local variables
s.t. local constraints
x
computed
results
analysis for
subsystem k
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
2
CO – Subsystem Level
min J1
x
target
local
variables variables
2
local variables
s.t. local constraints
• The subsystem optimizer modifies local variables to
achieve the best design for which the set of local
variables and computed results most nearly matches the
system targets
• The local constraints must also be satisfied
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CO – System Level
min Jsys
wrt: x 0
s.t. Jk
target variables
0
subproblemsk
• System-level optimizer changes target variables to
improve objective and reduce differences Jk
– Jk=0 are called compatibility constraints
– compatibility constraints are driven to zero, but may
be violated during the optimization
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CO Example: Aircraft Design
Consider a simple aircraft design problem:
maximize range for a given take-off weight by choosing
wing area, aspect ratio, twist angle, L/D, and wing weight.
wing area, S
aspect ratio, AR
L/D
aero
twist angle,
modified from Kroo et al. AIAA 94-4325
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struct
wing weight,
W
perf
range, R
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CO Example: Aircraft Design
max R0
x0 = [R0 S0 AR0 0 L/D0 W0]T
s.t. J1=0, J2=0, J3=0
x0
J1
min J1
J1= (S-S0)2 + (AR-AR0)2+
( - 0)2 + (L/D-L/D0)2
x = [AR ]T
x
L/D
aero analysis
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x0
J2
min J2
J2= (S-S0)2 + (AR-AR0)2 +
( - 0)2 + (W-W0)2
x = [S AR]T
x
struct analysis
W
J3
x0
min J3
J3= (L/D-L/D0)2+ (W-W0)2
+ (R-R0)2
x = [L/D W]T
x
R
perf analysis
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Collaborative Optimization
min Jsys
x0
wrt: x 0
target variables
s.t. Jk 0
subproblemsk Jsys
x0
J1
target
local
variables variables
min J1
coupling
local
variables variables
x
local variables
s.t. local constraints
x
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computed
results
analysis for
subsystem 1
2
Jk
x0
2
min Jk
y1k
yk 1
performance
analysis
target
local
variables variables
coupling
local
variables variables
x
local variables
s.t. local constraints
x
computed
results
analysis for
subsystem k
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
2
2
Collaborative Optimization
x0 = system-level target variable values
x = subsystem local variables
yij = coupling functions
• yij =outputs of subsystem j which are needed as inputs to
subsystem i.
• Coupling equations must also be satisfied, so coupling
variables are included in subsystem objective.
• Used to reduce the number of system-level parameters.
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Analytical Target Cascading
• ATC was initially developed as a product development
tool to cascade system-level product targets through a
hierarchy of design groups
• ATC is typically used to solve object-based decomposed
system optimization problems
• The ATC paradigm is based on hierarchical organizational
and analysis structures
• ATC approach is to take a high-level system analysis and
use more detailed subsystem analyses at the lower levels.
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Analytical Target Cascading
R11
Y11
SS1
Level 1
X11
R21
Y21
SS2
Level 2
.
.
.
R22
R31
.
.
.
SS3
X21
R32
R33
.
.
.
Y22
.
.
.
X22
R34
.
.
.
Level NL
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Image by MIT OpenCourseWare.
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Analytical Target Cascading
• ATC’s mathematical formulation is similar to CO
although they were developed with different motivations.
• Bottom level problems have the most design freedom.
Many possible solutions can exist that both match
targets while satisfying local design constraints.
• At higher levels design freedom is progressively
reduced, until it is a minimum at the top level.
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Analytical Target Cascading
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Analytical Target Cascading
• Linking variables y: Quantities that are input to more than one
subspace. These could be either shared variables or coupling
variables.
• Local decision variables x: Variables that a particular subspace
determines the value of.
• Responses R: Values generated by subspaces required as inputs
to respective parent subspaces.
• Targets T: Values set by parent subspaces to be matched by the
corresponding quantities from child subspaces.
• ƐR and Ɛy: allowable compatibility tolerance.
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© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Hierarchical Decomposition &
Multi-Domain Formulation
Decomposition
Courtesy of Anas Alfaris. Used with permission.
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© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Hierarchical Decomposition &
Multi-Domain Formulation
Decomposition
L1
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Courtesy of Anas Alfaris.Used with permission.
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Hierarchical Decomposition &
Multi-Domain Formulation
Decomposition
Building System
34
Courtesy of Anas Alfaris. Used with permission.
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Hierarchical Decomposition &
Multi-Domain Formulation
Formulation
35
Courtesy of Anas Alfaris. Used with permission.
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Hierarchical Decomposition &
Multi-Domain Formulation
Formulation
0 0 0 0 0
1 0 0 0 0
A
An
n 1
36
A2
1 0 0 0 0
1 0 0 0 0
0 0 0 0 0
1 0 0 0 0
4
V
1 0 0 0 0
0 0 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
j
1 0 0 0 0
1 0 1 0 0
1 1 0 0 0
i
1
N
1
N
N
ai j
i 1
N
ai j
j 1
A3
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Where:
j is an indicator of the fraction of
total elements to which element j
provides input,
i is the fraction of total elements
on which element i depends, and
aij is an element of a matrix that
can be the DSM, a power of the
DSM, or the V matrix.
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Hierarchical Decomposition &
Multi-Domain Formulation
Formulation
Courtesy of Anas Alfaris. Used with permission.
37
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Hierarchical Decomposition &
Multi-Domain Formulation
Formulation
38
Courtesy of Anas Alfaris. Used with permission.
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
References
I.P. Sobieski and I.M. Kroo. Collaborative Optimization Using Response Surface
Estimation. AIAA Journal Vol. 38 No. 10. Oct 2000.
Erin J. Cramer et al. Problem Formulation for Multidisciplinary Optimization.
SIAM Journal of Optimization. Vol. 4, No. 4 pp. 754-776, Nov 1994.
Kim, H.M., Michelena, N.F., Papalambros, P.Y., and Jiang, T., "Target Cascading
in Optimal System Design," Transaction of ASME: Journal of Mechanical
Design, Vol. 125, pp. 481-489, 2003
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© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
MIT OpenCourseWare
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ESD.77 / 16.888 Multidisciplinary System Design Optimization
Spring 2010
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