MDO Algorithms 1

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System Design New Paradigms
Part II MDO Architechtures
Prof. P.M. Mujumdar, Prof. K. Sudhakar Dept. of
Aerospace Engineering, IIT Bombay Umakant
Joysula, DRDL, Hyderabad
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OUTLINE

• Coupled System
• Engg. Design Optimization Problem Statement
• Analyzer Evaluator
• Classification of MDO Architectures
• Single level Architectures / formulations
• Bi-level Architectures / formulations

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COUPLED SYSTEM

• Comprises of several modules or components or
  disciplines
• Output of one module affects another module and
  vice-
• versa
• Analysis of one discipline requires information
  from
• analysis of another discipline

MULTI DISCIPLINARY ANALYSIS (MDA)
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MDO ARCHITECTURE / FORMULATION

• Stating the design problem as a Formal
• Engineering Optimization problem
• Integration of Optimization and Analysis of
• Coupled Systems - MDAO
• MDAO can be accomplished in several ways
• leading to different MDO architectures

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ENGINEERING DESIGN PROBLEM
Min f (Z , S(Z) ) subject to h(Z,S(Z))
0 g(Z,S(Z)) ? 0 S(Z) is a solution of A (Z,
S(Z)) 0 A(Z,S) 0 Non-linear , Iterative,
Fully Converged Coupled Multi- Disciplinary
Analysis (MDA) Time Intensive
Nested ANalysis and Design (NAND)
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ANALYSIS AND EVALUATOR
Closed Analysis Nested ANalysis and Design (NAND)
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ALTERNATE STATEMENT

• Optimizer searches for solution
• Evaluator light on time
• Converged analysis not sought
• when far away from optimum?

• Analysis Open
• Analysis feasible only at optimum
• Design Constraint vectors are
• augmented
• Simultaneous ANalysis Design
• (SAND)

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Analysis v/s Evaluators
3. Calculates
Solving pushed to optimization level
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SYSTEM AND DISCIPLINE LEVEL
SYSTEM LEVEL (DISCIPLINE COORDINATOR) ZS
ZS1
ZS2
Y
DISCIPLINE 2 ZL2
DISCIPLINE 1 ZL1
Z (? ZLi) ? (ZSi ) ZL Local to
discipline (Disciplinary Variables) ZS
Shared by more than one discipline (System
Variables) Y Coupling functions
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CLASSIFICATION OF MDO ARCHITECTURES
Based on the fact whether the optimization is
carried out at Single level
Bi-level One optimizer
System Optimizer - controls all
- System variables
design variables
Disciplinary
Optimizer
- Disciplinary variables
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CLASSIFICATION OF MDO ARCHITECTURES

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

Basic Single Level Formulations NAND-NAND
SAND-NAND SAND-SAND (MDF)
(IDF) (AAO)
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NAND-NAND FORMULATION (MDF)
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NAND-NAND FORMULATION
MATHEMATICAL STATEMENT Find Z
which Minimize f (Z ) subject to g 0 ? 0
(System Design
Constraints) g1 ? 0 g2 ? 0 g3 ? 0
(Disciplinary Design Constraints)
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SAND-NAND FORMULATION (IDF)
Zaug design variables Z, coupling variables
Y y13 -y13 0
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SAND-NAND FORMULATION (IDF)
Augmented Design Variable Vector Zaug ( Z ,
y12 , y13, y21, y23, y31, y32 )
Design Constraints (DC) g0 ? 0
( system design constraints) g1 ? 0
g2 ? 0 g3 ? 0 (disciplinary design
constraints) Auxiliary Constraints
( Inter disciplinary Consistency Constraints)
y21 - y21 0 y31 - y31
0 y12 - y12 0 y32 -
y32 0 ( ICC) y13
- y13 0 y23 - y23 0
Min f (Zaug ) subject to constraints DC and
ICC
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SAND-SAND FORMULATION
Zaug design variables Z, coupling variables
Y, state variables S
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SAND-SAND FORMULATION (AAO)
Augmented Design Variable Vector Zaug ( Z ,
S, y12 , y13, y21, y23, y31, y32 )
Design Constraints (DC) g0 ? 0
( system design constraints) g1 ? 0 g2
? 0 g3 ? 0 (disciplinary design
constraints) Auxiliary Constraints
y21 - y21 0 y31 - y31 0
y12 - y12 0 y32 - y32
0 ( ICC) y13 - y13
0 y23 - y23 0
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SAND-SAND FORMULATION
Auxiliary Constraints(Disciplinary Analysis
Constraints) r1 s1 E1(
z1, y21 ,y31) 0 r2 s2
E2( z2, y12 ,y32) 0 (DAC)
r3 s3 E3( z3, y13 ,y23)
0 Optimization problem statement Find Zaug
which Minimize f (Zaug ) Subject to DC ,
ICC and DAC as stated above.
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Single Level MDO Architectures
Multi-Disciplinary Feasible (MDF)
Individual Discipline Feasible (IDF)
All At Once (AAO)
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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COMPARISON OF SINGLE LEVEL FORMULATIONS
NAND - NAND SAND-NAND SAND-SAND Z Z, y
Z, S , y Analyzer/
Evaluator/ Evaluator/
Analyzer Analyzer
Evaluator Inter- Discipline
Consistent Disciplinary
Consistent Solution at Consistent
Solution Optimality Solutio
n MDF IDF
All-at-Once Extreme In-Between
Extreme
1 2 3 4 5
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BI-LEVEL FORMULATIONS

• Industry design environment
• Distributed approach
• Disciplines retain control over their
  respective design
• variables
• Coordination through Project Office

Bi-level formulations attempt to incorporate such
features in the Mathematical definition of the
Problem statement
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BI-LEVEL PROBLEM DECOMPOSITION
DESIGN VECTOR
SINGLE LEVEL BI-LEVEL (CO)
Z ZL ? ZS System level
Zaug Z ? Y Zaug ZS ? ZC
ZS ? zSi , ZC
? zci
zci zcIi ? zcOi
Discipline level
X ? xi
xi xLi ? xsi ? xcIi ?
xcOi
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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 , xcO1
gn , xcOn
Analysis 1
Analysis N
zSi shared variables zcIi zcOi coupling
variables xsi , xcIi xcOi copies of system
targets at discipline level
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COLLABORATIVE OPTIMIZATION System level
Optimization Problem Find Z aug which Minimize
F (ZS) s.t. r (Zaug) 0 F
objective function Zaug design variable
vector(targets issued to sub-spaces) r
non-linear constraint vector, whose elements are
discrepancy functions returned from
solution of the sub space
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 -
zcIi ?? ?? ycOi - zcOi ?? s.t gi
(xi ) ? 0 hi (xi ) 0
ri ri xi
xi The norm in the objective function
ri (xi ) is generally, calculated as L2 norm.

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CONCURRENT SUB-SPACE OPTIMIZATION
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CONCURRENT SUB-SPACE OPTIMIZATION

• Step 1 System Analysis at initial system
  design vector,
• local sensitivities
• 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. Local
• states obtained from disciplinary analysis
• Each subspace return different optima

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CONCURRENT SUB-SPACE OPTIMIZATION

• Step 4 Design database updated during subspace
  optimizations
• 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.

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Bi-Level Integrated System Synthesis - BLISS
X X0 DXOPT Z Z0 DZOPT
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Bi-Level Integrated System Synthesis - BLISS

• Step 1 System Analysis Sensitivity (GSE)
• Step 2 Subsystem objective Fsdf/dXT DXs
• Subsystem optimization

Linear approximation for the coupling variables
for evaluating constraints
Shared variables (system var.) Y held
constant during subsystem optimization
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Bi-Level Integrated System Synthesis - BLISS

• Step 3 Obtain sensitivity of X and F
  (optimal) wrt
• ZS and Y
• These sensitivities link the system and subsystem
  level optimizations (Optimal Design
  Sensitivities)
• At system level use shared variables to further
  improve system objective
• Step 4 System level optimization problem

F(ZS) is obtained as a linear extrapolation
based on the optimum design sensitivity
obtained in each subsystem
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Thank You Visithttp//www.casde.iitb.ac.in/MDO/
4th Meeting of SIG-MDO in March 2004