(problem decomposition)

7/27/2019 (Problem Decomposition)

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1 Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

Anas Alfaris

Decomposition and Coupling

Lecture 4

Multidisciplinary System

Design Optimization (MSDO)


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Todays 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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Standard Optimization Problem

Given

min ( )s.t. ( ) 0

J xg x

Solve the problem

* *

That is, find s.t. J( ) ( ), dom( ) dom( )x x f x x J g

Optimization Engine

Function Evaluator

x( )J x

( )g x

0x*

x

mn

zn

n

g

J

x

:

:


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Information Flow

A

B

A B

A

B

A

B

A B

A

B

AA

B

A B

B

A

B

A B

A

B

Dependent Tasks (Series) Independent Tasks (Parallel) Interdependent Tasks (Coupled)

Image by MIT OpenCourseWare.


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Information Flow

C

D

A

B

G

H

E

F

K

L

I

J

C DAB GHEFK L IJ

FeedForward

FeedBackward


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Information Flow

C

D

A

B

G

H

E

F

K

L

I

J

C DAB GHEFK L IJ

Sequential

Parallel

Coupled


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Advantages of Decoupling

Computation ofg(x) can be very time consuming, want

to divide the work and compute in parallel.

For example, if 1 21 2 1 2

1 1 2 2

( , ), where ,

and g( ) ( ( ), ( ))

n nx x x x x

x g x g x

Then g1 and g2can be computed in parallel. Graphically,

Optimizer

SS1 SS2

1x 1g 2g 2x SS1

SS2

Optim1x 2x

1g

2g

RR


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Coupling

The decoupled constraints assumption is not general. Subsystems

can be coupled and loops can arise. For example,

Optimizer

SS1 SS2

1x 2x

1u 2u

2w1w

x: decision variables

w: SS outputs (constraint, cost)

u: SS input (dependent)

SS1

SS2

Optim

vline: SS input

hline: SS output

1w

2w1u

1x

2u

2x

1

w

2w

Loop

Computation ofw1 and

w2 requires an iterative method.


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Coupling

An example where such a loop happens is as follows:

1 2

1 1 1 2 2 1

2 2 2 1 1 2

min ( , )

( , ( , )) 0s.t.

( , ( , )) 0

J x x

w g x g x w

w g x g x w

1 21 2where , , : , 1,2n n i i i ix x g x u w i

w1

and w2

satisfy coupled relations at each optimization iteration.

At each constraint evaluation, nonlinear equations must be solved

(e.g. by Newtons method) in order to obtain w1

and w2, which can

be time consuming.

Want a way to return to the situation of decoupled constraints.

R R


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Surrogate Variables (Tearing)

Information loop can be broken by introducing surrogate variables.

1 2

1 1 1 2 2 1

2 2 2 1 1 2

min ( , )

( , ( , )) 0s.t.

( , ( , )) 0

J x x

w g x g x w

w g x g x w

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.

1 2

1 1 1

2 2 2

2 1 1 1

1 2 2 2

min ( , )s.t.

( , ) 0

( , ) 0

( , ) 0

( , ) 0

J x x

g x u

g x u

u g x u

u g x u


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Numerical Example

1 2

1

2

2 2

1 1 2

2 2

2 3 4

3 3

1 1 2 2

3 3

2 3 4 1

min

s.t. 0

0

where

( 3) ( 4)

2

2

J J

w

w

J x x

J x x

w x x w

w x x w

2 2 2 2

1 2 3 4

1 1 1 2 3 4

2 2 1 2 3 4

min ( 3) ( 4)

s.t. ( , , , ) 0

( , , , ) 0

x x x x

w g x x x x

w g x x x x

coupled

2 2 2 2

1 2 3 4

3 3

1 1 2 5

3 3

2 3 4 6

3 3

1 2 5 6

3 33 4 6 5

min ( 3) ( 4)

s.t. 2 0

2 0

2 0

2 0

x x x x

w x x x

w x x x

x x x x

x x x x

decoupled

Solution:

1 23 3(0,0, 4,3,12 , 24 )x

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

non-hierarchical

decomposition

Multi-level

(Distributed Design)

-provide disciplinary

models with design tasks-optimization at

subsystem and system

levels

hierarchical

decomposition


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Single-level (Distributed Analysis)

Disciplinary models provide analysis

Optimization is controlled by some overseeing codeor database

e.g. ISight (Optimizer)

SystemOptimizer

Shared data

Local data

Structures

Local data

Aero

Optimizer

design variables

constraints

iSight

x J(x),g(x),h(x)

subsystemanalyses


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Optimizer

objectivedesign variables

constraints

x J(x)

performance

analysis

aerodynamic

analysisstructural

analysis

x g(x) h(x) x

g(x)

h(x)

During the optimization, the overseeing code keeps track of thevalues 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

Single-level (Distributed Analysis)


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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.

Multi-level (Distributed Design)


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Multi-level (Distributed Design)

System level optimizer

SS1

optimizer

SS2

optimizer

SSN

optimizer

SS1

analyzer

SS2

analyzer

SSN

analyzer

command/resultcommand/result

command/result

Subsystem

black box (BB)


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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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OPTIMIZER

TARGET STATECoupled

Uncoupled


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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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0

k

min

wrt: target variabless.t. 0 subproblems

sys

k

J

J

x

2

1

target local

variables variables

local variables

local constraints

min

s.t.

-J

x

0x

1

J

2target local

variables variables

local variables

local constraints

min

s.t.

-kJ

x

0xkJ

analysis for

subsystem 1

analysis for

subsystem k

x xcomputedresults

computed

results

performance

analysis

0x

sysJ


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CO Subsystem Level

2

1 target localvariables variables

local variables

local constraints

min

s.t.

-J

x

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

System-level optimizer changes target variables toimprove objective and reduce differences Jk

Jk=0are called compatibility constraints

compatibility constraints are driven to zero, but may

be violated during the optimization

0

k

min

wrt: target variables

s.t. 0 subproblems

sys

k

J

J

x


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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.

aero

struct

perfmodified from Kroo et al. AIAA 94-4325

wing area, S

aspect ratio,AR

twist angle,

range, R

L/D

wing weight,

W


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0

k

min

wrt: target variables

s.t. 0 subproblems

sys

k

J

J

x

1

2

2

coupling local

variables variable

target local

variables variables

local variables

local constraints

s

min

s.t.

-

-J

x

0x

1J 0xkJ

analysis for

subsystem 1

x xcomputedresults

computed

results

performance

analysis

0x

sysJ

1ky

1ky

analysis for

subsystem k

2

2coupling local

variables variab

target local

variables variables

local variables

local constra

le

in s

s

t

min

s.t.

-

-kJ

x


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x0 = system-level target variable values

x = subsystem local variables

yij = coupling functions

yij =outputs of subsystemjwhich 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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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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SS1

SS2 SS3

R11

Y11

X11

R21 R22

R34R33R32R31

Level 2

Level NL

Level 1

Y22

X22

Y21

X21

.

.

..

.

.

.

.

.

.

.

.

.

.

.

Image by MIT OpenCourseWare.


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ATCs 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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Linking variables y: Quantities that are input to more than one

subspace. These could be either shared variables or couplingvariables.

Local decision variablesx: 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.

R and y: allowable compatibility tolerance.

Hierarchical Decomposition &


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Multi-Domain Formulation

Decomposition

Courtesy of Anas Alfaris. Used with permission.



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33

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

Decomposition

L1



Courtesy of Anas Alfaris.Used with permission.



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34


Building System

Decomposition






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35


Formulation






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A

0 0 0 0 0

1 0 0 0 0

1 0 0 0 0

0 0 1 0 0

0 1 0 0 0

A2

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 0 0 0 0

1 0 0 0 0

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

V An

n 1

4

0 0 0 0 0

1 0 0 0 0

1 0 0 0 0

1 0 1 0 0

1 1 0 0 0

j1

Nai j

i 1

N

i

1

Nai j

j 1

N

Where:

j is an indicator of the fraction oftotal elements to which element j

provides input,

i is the fraction of total elements

on which element idepends, and

aij is an element of a matrix that

can be the DSM, a power of the

DSM, or the V matrix.

Formulation





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Formulation






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Formulation





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References

I.P. Sobieski and I.M. Kroo. Collaborative Optimization Using Response SurfaceEstimation. 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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