Start
RsU1 , y Us1
Project Overview
Ps1
J sys  f (x)
Minimize:
x  x1 , x2 , ... , xn
Subject To:
J1  0
|| Rs1  RsU1 ||  || ys1  yUs1 ||  R   y
~
x , y , y , R , ,
Minimize:
With Respect To:
Subject To:
s1
 || R
ss, k
s1
ss
ss
x
J1
R ,y
J 1   ( xi  x )
i 1
With Respect To: x  x , x ,  , x
*
Subject To:
*
1
*
2
U
ss1
R
L
ss1
R ,y
n
n
Minimize:
*
n
hs1 ( Rs1 , ~
xs1 , ys1 )  0
y
min
s1
 ys1  y
Minimize:
J 2   ( xi  x )
i 1
With Respect To: x  x , x ,  , x
*
*
1
*
2
local variables
g1m ( x )  0
g 2m ( x )  0
Subject To:
,y
*
n
h1m ( x * )  0
h2 m ( x * )  0
xi* min  xi*  xi* max
xi* min  xi*  xi* max
2. Update subsystem boundary conditions using stochastic
variations.
max
s1
R ss ,
~
x s1 , y s1
U
ss 2
L
ss1
R
L
ss 2
,y
Rs1
Rs1  rs1 ( Rss , ~
x s1 , y s1 )
L
ss 2
3. Perform design optimization using subsystem model and
most recent stochastic subsystem boundary conditions.
f (x )
Minimize:
Pss1
Minimize:
|| Rss1  R
U
ss1
||  || y ss1  y
U
ss1
||
~
xss1 , yss1
With Respect To:
g ss1 ( Rss1 , ~
xss1 , yss1 )  0
Subject To:
h (R , ~
x ,y )0
*
i  1 to n
U
ss 2
* 2
i
local variables
*
 yssL , k ||   y
J2
U
ss1
* 2
i
ss, k
k 2
g s1 ( Rs1, ~
xs1, ys1 )  0
~
xsmin
~
xs1  ~
xsmax
1
1
x
y
 || y
 RssL , k ||   R
k 2
J2  0
R
1. Evaluate system model using initial subsystem design
and extract boundary conditions for subsystem model.
ss1
ss1
~
~
~ max
xssmin
1  xss1  xss1
i  1 to n
ss1
ss1
max
y ssmin
1  y ss1  y ss1
~
xss1 , y ss1
R ss1
Pss 2
Minimize:
|| Rss 2  R
U
ss 2
With Respect To:
Subject To:
g ss 2
||  || yss 2  y
~
xss 2 , yss 2
(R , ~
x ,y
ss 2
ss 2
ss 2
U
ss 2
||
4. Evaluate system model using updated subsystem design
and extract new boundary conditions for subsystem model.
)0
Max cycle
number
exceeded?
y ssmin2  y ss 2  y ssmax2
~
xss2 , y ss2
Update
subsystem
boundary
conditions?
No
g ( x)  0
h( x )  0
ximin  xi  ximax
i  1 to n
Subject To:
No
hss 2 ( Rss 2 , ~
xss 2 , yss 2 )  0
~
xssmin2  ~
xss 2  ~
xssmax
2
x  x1 , x2 , ... , xn
With Respect To:
Yes
Rss2
Yes
Disciplinary
Analysis 2
Disciplinary
Analysis 1
Rss1  rss1 ( ~
xss1 , yss1 )
Collaborative Optimization
• Nested optimization
• Bi-level hierarchy
• Sends copies of variables to
lower level
• Sometimes difficult to locate
global minimum
Beam Bending Example Problem
Stop
Analytical Target Cascading
COMPOSE Algorithm
All-At-Once
• Sequence of subproblems at
each level
• Two or more level hierarchy
• Variable sharing occurs between
related elements
• Computationally expensive, but
efficient
• COMPonent Optimization within a
System Environment
• Global analysis followed by local
optimization
• Updates local boundary conditions
• Complex problems require
stochastic BC updates
• Optimizes entire system at
the same time
• Non-hierarchy
• Used to locate global
minimum
• Computationally most
expensive
Problem Formation
Two I-beams, bounded at the ends, contain a single
load of 100 kN applied at the junction point B. The
objective is to minimize the volume of section 1
with respect to the I-beam design variables h1, l1,
t1, and f1. The problem is subject to a minimum
moment constraint of 300,000 N-m at point A. Each
design variable is subject to upper and lower
boundary values.
Rss 2  rss 2 ( ~
xss 2 , yss 2 )
Results and Discussion
A
1
100 kN
B
2
C
Beam Diagram
Optimal Volume of Beam Section 1
This problem was used to demonstrate the
COMPOSE algorithm and the AAO method. The
AAO method proved to find the better optimal
design and converged faster. However, the
global analysis of this problem was simple. For
a complex global analysis, the required
computational cost is dramatically increased.
The COMPOSE method reduces the number of
global analysis, thus decreasing the cost.
0.35
0.30
AAO
0.25
Volume (m3)
Decomposing a problem into a
global/local model reduces the
convergence time for complex
optimization problems. This
research focused on comparing
the common methods of
multidisciplinary structural
optimization by decomposition.
Further exploration included a
detailed study of the COMPOSE
algorithm, specifically its
performance and application to
various problems. This involved
solving a simple beam bending
problem using COMPOSE and the
All-At-Once (AAO) method,
followed by comparing the results.
With Respect To:
RsL1 , y sL1
COMPOSE
0.20
0.15
0.10
0.05
0.00
0
2
4
6
8
10
12
14
Optimization Cycle
Objective Function Values
Real Life Application and Future Research
Certain problems, like the design of rails in a truck chassis for
crashworthiness, can contain hundreds of design variables. The
computational expense of solving problems of this magnitude is
substantial using AAO. When strong coupling exists between the
global and local model, decomposing the problem can largely
decrease the analysis time of the optimization process. For future
research, one can explore the range of solvable problems by
decomposition, specifically with the COMPOSE algorithm. The
addition of reliability based design optimization would further
benefit the use of these methods.
David Wyrembelski
Ron C. Averill (Advisor)
Department of Mechanical
Engineering
Rails (Local Model)
Global Model of Truck for Crashworthiness
Michigan State University