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 3

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

8

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

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. 10

© 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 12

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

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

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. 15

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

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 18

TARGET STATE Uncoupled © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

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

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

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 23

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 24

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

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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

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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. 26

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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. 27

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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 28

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

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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 Formulation

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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 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.

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

Hierarchical Decomposition & Multi-Domain Formulation Formulation

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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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ESD.77 / 16.888 Multidisciplinary System Design Optimization Spring 2010

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