MDO Algorithms 1

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MDO Architechtures
Bi-Level Formulations
Prof. P.M. Mujumdar Dept. of Aerospace
Engineering, IIT Bombay
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OUTLINE

• Classification of MDO Architectures Summary
  of
• Single level Architectures / formulations
• Bi-level Architectures / formulations
• Collaborative Optimization
• Concurrent Subspace Optimization
• Bi-Level Integrated System Synthesis

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CLASSIFICATION OF MDO ARCHITECTURES
Based whether the optimization is
carried out at Single level
Bi-level One optimizer
System Optimizer - controls all
design - System variables
variables
Disciplinary Optimizer
-
Disciplinary variables
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CLASSIFICATION OF MDO ARCHITECTURES

• Based on manner in which the Inter-Disciplinary
  Feasibility
• and Disciplinary Analysis is carried out.
• Interdisciplinary Consistent Solution
  implies NAND at system Level.
• Otherwise SAND
• Disciplinary Consistent solution implies
  NAND at discipline level
• Otherwise SAND
• NAND ? (Nested ANalysis and Design)
• SAND ? (Simultaneous ANalysis and Design)

Basic Single Level Formulations NAND-NAND
SAND-NAND SAND-SAND (MDF)
(IDF) (AAO)
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SINGLE LEVEL MDO ARCHITECTURES
Multi-Disciplinary Feasible (MDF or NAND-NAND)
Individual Discipline Feasible (IDF or SAND-NAND)
All At Once (AAO or SAND-SAND)
Optimizer
Optimizer
Optimizer
Interface
Interface
Interface
Analysis 1 Iterations till convergence
Analysis 2 Iterations till convergence
Analysis 1 Iterations till convergence
Analysis 2 Iterations till convergence
Evaluator 1 No iterations
Evaluator 2 No iterations
Iterative coupled
Non-iterative Uncoupled
Uncoupled
Multi-Disciplinary Analysis (MDA)
Disciplinary Evaluation
Disciplinary Analysis
1. Optimizer load increases tremendously 2. No
useful results are generated till the end of
optimization 3. Parallel evaluation 4. Evaluation
cost relatively trivial

• 1. Minimum load on optimizer
• 2. Complete interdisciplinary consistency
  is assured at each optimization call
• 3. Each MDA
• i Computationally expensive
• ii Sequential

1. Complete interdisciplinary consistency
is assured only at successful termination of
optimization 2. Intermediate between MDF and
AAO 3. Analysis in parallel
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BI-LEVEL FORMULATIONS

• Industry design environment
• Distributed approach many groups/work centers,
  
• partitioning into sub-tasks
• Disciplines retain control over their respective
  design
• tasks through Disciplinary Experts
• Coordination - through Project Office/Chief
  Designer

Bi-level formulations attempt to incorporate such
features in the mathematical definition of the
problem statement
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BI-LEVEL FORMULATIONS
How to reconcile need for subtask/disciplinary
autonomy With System Level challenge of
everything influences everything else in
formal mathematical optimization of complicated
engineering systems through decomposition ???
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BI-LEVEL FORMULATIONS

• GSE - Global Sensitivity Equations (Sobieski
  1990, Olds 1992)
• Sensitivity Analysis partitioned through GSEs
• System Optimization All-in-One
• CSSO - Concurrent SubSpace Optimization
  (Sobieski 1988, Renaud 1991,93,94, Batill 1998)
• Separate optimizations in sub-tasks/disciplines
• Co-ordination problem all design variables
  (shared disciplinary)

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BI-LEVEL FORMULATIONS

• CO Collaborative Optimization (Braun Kroo
  1996, Sobieski Kroo 1998)
• Separate optimizations in sub-tasks/disciplines
• Co-ordination problem system design variables
  augmented with coupling variables
• BLISS Bi-Level Integrated System Synthesis
  (Sobieski et al 1998, 2000, Kodiyalam 2002)
• Separate optimizations in sub-tasks/disciplines
• Co-ordination problem system design variables

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Collaborative Optimization(CO)
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COLLABORATIVE OPTIMIZATION

• CO Vocabulary (System Level)
• ZS ? zSi (zSi ? zSj ? 0) shared design
  variables
• YC ? yci (yci ? ycj ? 0) coupling
  variables
• Z ZS ? YC - system variables
• zSi shared design variables required in
  discipline i
• yci coupling variables associated with
  discipline i
• ycIi coupling variables required as input to
  discipline i
• ycOi coupling variables output from discipline
  i
• yci ycIi ? ycOi
• zi zsi ? yci
• ycIi subset (? ycOj j ? i)
• ycOi subset (? ycIj j ? i)

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COLLABORATIVE OPTIMIZATION

• CO Vocabulary (at ith discipline level)
• zsi system level target values of shared
  design variables associated
• with discipline i (held fixed)
• yci system level target values of input
  output coupling variables
• associated with ith discipline (held
  fixed)
• xLi disciplinary design variables of discipline
  i (engineering design
• variables)
• xsi local counterparts of zsi treated as design
  variables within i
• xcIi local counterparts of ycIi treated as
  design variables within i
• wcOi disciplinary analysis output of i coupled
  to other disciplines
• xi xLi ? xsi ? xcIi - augmented design
  variables set in
• discipline i

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COLLABORATIVE OPTIMIZATION FORMULATION
System level Optimizer Min f(Z) s.t. rj (Z) 0
j 1, n
zn
z1

xn
x1
g1 , wcO1
gn , wcOn
Analysis 1
Analysis n
zSi shared variables ycIi ycOi coupling
variables (targets) xsi , xcIi local copies of
system targets at discipline level
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COLLABORATIVE OPTIMIZATION System level
Optimization Problem Find Z (shared and
coupling variables) which Minimize F (ZS) s.t.
r (Z) 0 F objective
function Z design variable vector (targets
issued to sub-spaces) r non-linear
constraint vector, whose elements are
discrepancy functions returned from solution
of the subspace optimization
problems The system-level solution is defined
as, F F and Z Z and XL XL
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COLLABORATIVE OPTIMIZATION
Discipline / Subspace Optimization Problem For a
n discipline problem, there will be n
sub-space optimization problems. Mathematical
statement for an ith sub-space Find xi Min
ri(xi) ??xsi - zsi ?? ??xcIi - ycIi ??
??wcOi - ycOi ?? s.t gi (xi ) ? 0
hi(xi ) 0
ri ri xi xi is
the optimized disciplinary soln. The norm in the
objective function ri (xi ) is generally,
calculated as L2 norm.
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Concurrent SubSpace Optimization(CSSO)
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CONCURRENT SUB-SPACE OPTIMIZATION
Convergence
System Level Coordination
?
Sensitivities
RSM
System Analysis
Approximation Model
System Analysis
Subspace Optimizations
Process flow
Information flow
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CONCURRENT SUB-SPACE OPTIMIZATION

• Step 1 System Analysis at initial system
  design vector,
• Local Sensitivities Analysis
• Step 2 Total System Sensitivities using GSE
• Step 3 Concurrent Subspace Optimizations
• Each Subspace solves the system level
  optimization problem (same
• objective and constraints)
• Subspace design vector is a subset of the system
  design vector
• local to the subspace. Non-local variables
  kept fixed
• Non-local states approximated linearly using
  sensitivities (First
• Order Taylor series) with move limits
• Local states obtained from disciplinary analysis
• Each subspace return different optima

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CONCURRENT SUB-SPACE OPTIMIZATION

• Step 4 Design database updated after subspace
  optimizations
• System Analysis at each optima
• Step 5 System level co-ordination for
  compromise/trade-off
• Database used to create second order response
  surfaces for
• objective and constraints
• System optimization based on these
  approximations with all
• design variables used to direct system
  convergence
• The approximate system optimum generated by the
  co-
• ordination process is used as the next design
  iterate in Step 1.
• Repeat Step 1 to Step 5 till convergence.

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CSSO iSIGHT EXAMPLE
http//www.crd.ge.com/cooltechnologies/pdf/1997crd
186.pdf
CA1
CA1
DP
SA
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CSSO iSIGHT EXAMPLE
SSO1
SSO2
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CSSO iSIGHT EXAMPLE
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CSSO iSIGHT EXAMPLE
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Bi-Level Integrated System Synthesis(BLISS)
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Bi-Level Integrated System Synthesis - BLISS
Opportunity for Concurrent Processing
BLISS Philosophy
Discipline 1 Optimization and Optm. Sensitivity
Analysis
initialize X Z
System Analysis and Sensitivity Analysis (GSE)
Discipline 2 Optimization and Optm. Sensitivity
Analysis
System Level Optimization
Update Variables
Z Z0 DZOPT
X X0 DXOPT
X X0 DXOPT Z Z0 DZOPT
Discipline k Optimization and Optm. Sensitivity
Analysis
Human Intervention
BLISS CYCLE (Builds a gradient-guided path)
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Bi-Level Integrated System Synthesis - BLISS

• BLISS Vocabulary
• Yr state/coupling variables, output of
  discipline r, element yri
• Yrs input to discipline r from discipline s
• Z system level design variables (shared)
• ?r local objective fn in discipline r
• system objective (one of the elements yri ,
  say y1i)
• Xr design variables local to discipline r
• Gr constraints in discipline r
• P fixed parameters
• d(ab) partial derivative ?a/?b
• D(ab) total derivative da/db

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Bi-Level Integrated System Synthesis - BLISS
BLISS Vocabulary DA Disciplinary Analysis
(Yr Yr (Z, Xr)) DOPT Disciplinary Optimization
(Min.?r, Find Xr , s.t. Gr ) DSA
Disciplinary Sensitivity Analysis D (Yr (Z,
Xr, Yrs)) DOSA Disciplinary Optimum Sensitivity
Analysis d(Xr(opt)(Z, Yrs)) SA System
Analysis Y Y(Z, X, P) SOPT System
Optimization (Min ?, Find Z) SSA System
Sensitivity Analysis D(Y(Z,X))
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Bi-Level Integrated System Synthesis - BLISS
Y
N
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Bi-Level Integrated System Synthesis - BLISS

• BLISS Algorithm
• Step 1 System Analysis
• Highly Problem Dependent
• Iterative if nonlinear
• Given Znew Zprev ?Z(opt)
• Xnew Xprev ?X(opt)
• Solve for Ynew

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Bi-Level Integrated System Synthesis - BLISS

• BLISS Algorithm
• Step 2 System convergence check
• Znew - Zprev lt ?
• Xnew - Xprev lt ?

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Bi-Level Integrated System Synthesis - BLISS

• BLISS Algorithm
• Step 3 Disciplinary Sensitivity Analysis
  
• At Z Znew , X Xnew , Y Ynew compute
• d(Yr Xr)
• d(Yr Z)
• d(Yr Ys)
• Derivatives may be computed by any of the several
  methods. Immaterial how they are computed.

Local sensitivity Jacobians
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Bi-Level Integrated System Synthesis - BLISS

• BLISS Algorithm
• Step 4 System Sensitivity Analysis
• At Z Znew , X Xnew , Y Ynew compute
• D (Y Xr)
• Using d(Yr Xr), d(Yr Z) , d(Yr Ys)
• With GSE
• By Solving
• A D (Y Xr) d(Y Xr)
• Elements of A are formed from d(Yr Ys)

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Bi-Level Integrated System Synthesis - BLISS

• BLISS Algorithm
• Step 5 Discipline Optimizations
• Basis formulation of unique obj. fn. for each
  discipline such that minimization of these fns.
  Results in minimization of system obj. fn.
• System obj. fn. (?) chosen as one of the outputs
  Y say Y1i
• First Order Taylor series expansion
• ? ?o D(Y1i , X1)T ?X1 D(Y1i , X2)T ?X2
  ...
• Local optimization problem
• Given (X, Z, Y) Find ?Xr(opt) that
  minimize ?r s.t. Gr ? 0 including
  side constraints

?2
?1
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Bi-Level Integrated System Synthesis - BLISS

• BLISS Algorithm
• Step 6 Optimum Sensitivity Analysis
• System level optimization requires D (Y Z) to
  obtain D (? Z)
• Modified GSE (GSE Optimized Subsystems)
• - Optimization of a discipline causes Xr(opt)
  to become a function of Y Z which are
  parameters in DOPT
• M11 D(YZ) M12 D(XZ) d(YZ)
• M21 D(YZ) M22 D(XZ)
  d(XoptZ)
• Mij matrices consist of terms d(XrYs),
  d(YrXr), d(YrYs)

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Bi-Level Integrated System Synthesis - BLISS

• BLISS Algorithm
• Step 6 Optimum Sensitivity Analysis
  (contd.)
• d(XZ) d(XrYs) - derivatives of the optimum
  w r t to parameters. D(Xopt , P)
• Obtained from the following approximate process
• - Parameters perturbed by small increment one
  
• by one
• - Using DSA to linearly approximate ?r and Gr
  
• around the optimal point with move
  limits
• - Discipline optimization repeated by Linear
• Programming
• - D(Xopt , P) ?X(opt) / ?P

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Bi-Level Integrated System Synthesis - BLISS

• BLISS Algorithm
• Step 7 System Level Optimization
• Given Z and ?o
• Find ?Z
• Minimize ? ?o D(Y1i , Z)T ?Z
• s.t. side constraints and move limits.

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List of References
  Robert Braun Collaborative Optimization An
Architecture for Large Scale Distributed Design
Ph.D Dissertation, Stanford University
May,1996 Ian P. Sobieski Multidisciplinary
Design Using Collaborative Optimization Ph.D.
Dissertation , Stanford University, August 1998
R.S. Sellar, S.M. Batill and J.E. Renaud
Response Surface Based, Concurrent Subspace
Optimization for Multi-disciplinary System
Design AIAA 96-0714   J. Sobieszczanski-Sobieski
Bi-level Integrated System Synthesis NASA
Langley Research Center. AIAA Journal Vol38 No1,
Jan2000   N. M. Alexandrov and R. M. Lewis
Analytical and Computational Properties of
Distributed Approaches to MDO AIAA2000-4718 N.
M. Alexandrov and R.M. Lewis Comparative
Properties of Collaborative Optimization and
Other Approaches to MDO First ASMO UK/ISSMO
Conference on Engineering Design Optimization,
July,1999 N. M. Alexandrov and R. M. Lewis
Analytical and Computational properties of
Collaborative Optimization NASA-TM-2000-210104  
R.J. Balling and C.A. Wilkinson Execution of
Multidisciplinary Design Optimization Approaches
on Common Test Problems Brigham Young
University, Utah. AIAA Journal Vol.35 No1,
Jan1997  
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Thank You Visithttp//www.casde.iitb.ac.in/MDO/
4th Meeting of SIG-MDO March 2004