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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

  • Advantages of DecouplingAdvantages of Decoupling

    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.

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

    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

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