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• Multidisciplinary SystemMultidisciplinary System Design Optimization (MSDO)Design Optimization (MSDO)

Decomposition and Coupling

Lecture 4

17 February 2004

Olivier de Weck

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

• Todays TopicsTodays Topics

Last time discussed standard approach:

Sequential modular analysis (Lecture 3).

Modules are executed sequentially with or without feedback loops.

MDO frameworks

Other Approaches: Distributed analysis

Distributed design

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 2

• Fundamentally different approaches in MDOFundamentally different approaches in MDO

Distributed Analysis

-disciplinary models provide analysis

-all optimization done at system level

non-hierarchical

decomposition

hierarchical

decomposition

Distributed Design -provide disciplinary models with design tasks CSSO

-optimization at subsystem and system levels CO

BLISS

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 3

• Standard Optimization ProblemStandard Optimization Problem

Given *xx 0

( )J x x ( )g x

Optimization Engine

Function Evaluator

nx ! nJ : !

!

n mg : ! !

Solve the problem

(min J x) (s.t. g x) 0

* *That is, find x s.t. J( x ) f x J( x), dom( ) dom( )g Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

4

• Distributed AnalysisDistributed Analysis Disciplinary models provide analysis

Optimization is controlled by some overseeing code or database

e.g. GenIE database system (Stanford) ISight (Optimizer) iSight

GenIE NPSol

Shared data

Local data

Structures

Local data

Aero

Optimizer design variables

constraints

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

subsystem analyses

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 5

• Distributed AnalysisDistributed AnalysisOptimizer objective

design variables constraints

x J(x)

performance analysis

aerodynamic analysis

structural analysis

x g(x) h(x) x

g(x) h(x)

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

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 6

• Distributed DesignDistributed Design

System level optimizer

SS1 optimizer

SS2 optimizer

SSN optimizer

SS1 analyzer

SS2 analyzer

SSN analyzer

command/result command/result

command/result

Subsystem black box (BB)

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

Computation of g(x) can be very time consuming, want to divide the work and compute in parallel.

n2For example, if x = ( ,x x2 ), where x !n1, x !1 1 2

and g(x) = (g x g x ))( ), (1 1 2 2

Then g1 and g2 can be computed in parallel. Graphically,

Optimizer

SS1 SS2

1x 1g 2g x2 g g2

SS1

SS1

Optim 1x 2x

1

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 8

• Coupled SituationCoupled Situation 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 SS1

SS2

Optim

1w

2w 1u

1x

2u 2x

1w

2w

Loop

x: decision variables vline: SS inputw: SS outputs (constraint, cost) hline: SS outputu: SS input (dependent)Computation of w1 and w2 requires an iterative method.

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 9

• Information Flow Loop (2)Information Flow Loop (2)

An example where such a loop happens is as follows:

( 1,min J x x2 )

s.t. 1 = ( , 2 ( 2 , 1w g x g x w )) 01 1

( , ( 1,w2 = g x g x w )) 02 2 1 2

n2 i ,where x1 !

n1, x ! , g : x i " w i = 1, 22 i 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 w , which can2be time consuming.

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 10

• Surrogate Variables (Tearing)Surrogate Variables (Tearing)Information loop can be broken by introducing surrogate variables.

( 1,min J x x2 ) ( 1,min J x x2 ) s.t.

s.t. 1 = ( , 2 ( 2 , 1w g x g x w )) 0 ( ,g x u ) 01 1 1 1 1

( ,g x u ) 0= ( , ( 1,w g x g x w )) 02 2 2 1 2 2 2 2 ( ,u g x u ) = 02 1 1 1 ( ,u g x u1 2 2 2 ) = 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.

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 11

• Numerical ExampleNumerical Example

1 + 2 + 2min J J2 decoupled min x x2 + (x 3)2 + (x 4)2 1 3 4

s.t. w1 0 3s.t. w x x23 + 2x5 0= 1 1w2 0 3

= 2 + 2 w2 = x x4 + 2x6 0

3

where J x x23

3 5 1J21

= (x 1

3)2 + (x 4)2 x x23 + 2x x6 = 0

3 4 3

= 3 3 2 x x4

3 + 2x x5 = 03w x x2 + 2w 6

1 1

3 3w2 = x x4 + 2w3 1 Solution:

coupled x = (0, 0, 4, 3,12 , 2413 )2 3 2 + 2 2 2 MATLAB 5.3min x x2 + (x 3) + (x 4)1 3 4

s.t. w g x x x x ) 0 coupled: 356,423 FLOPS 4.844s= ( 1, 2 , 3,1 1 4 = ( 1, 2 , 3, 4 ) 0

uncoupled: 281,379 FLOPS 0.453s w g x x x x2 2

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 12

• Distributed Design MethodsDistributed Design Methods Disciplinary models are provided with design tasks

Optimization is performed at a subsystem level in addition to the system level

Concurrent Subspace Optimization (CSSO)

divide the design problem into several discipline-

related subspaces each subspace shares responsibility for satisfying

constraints while trying to reduce a global objective

Collaborative Optimization (CO)

disciplinary teams satisfy local constraints while

trying to match target values specified by a system coordinator

preserves disciplinary-level design freedom

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 13

• Collaborative OptimizationCollaborative Optimization

OPTIMIZER

TARGET STATE Coupled

Uncoupled14

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

• Collaborative OptimizationCollaborative 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

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 15

• Collaborative OptimizationCollaborative Optimizationmin Jsys {x0} wrt: x0 = {target variables}

syss.t. Jk = 0 subproblems J

performance analysis

k

{x0} J1 {x0}

Jk

2 2local localmin J1 =

target - {variables variables}{variables variables} min Jk = target -x = {local variables} x = {local variables} s.t. {local constraints} s.t. {local constraints}

analysis for subsystem 1

analysis for subsystem k

{x} computed {x} computed results results

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 16

• COCO Subsystem LevelSubsystem Level

2localmin J1 =

target -{variables variables} x = {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

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 17

• COCO System LevelSystem Levelmin Jsys wrt: x0 = {target variables} s.t. Jk = 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

CO may therefore discover parts of the design space

that cannot be reached by sequential optimization

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 18

• CO Example: Aircraft DesignCO Example: Aircraft DesignConsider 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

Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 19

• x

CO Example: Aircraft DesignCO Examp