jan. 28, 2004mdo algorithms - 1 colloquium on mdo, vssc thiruvananthapuram part ii: mdo...
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Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 1Colloquium on MDO, VSSC Thiruvananthapuram
Part II: MDO Architechtures
Prof. P.M. Mujumdar, Prof. K. Sudhakar
Dept. of Aerospace Engineering, IIT Bombay
&
Umakant Joysula, DRDL, Hyderabad
System Design – New Paradigms
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 2Colloquium on MDO, VSSC Thiruvananthapuram
• Coupled System
• Engg. Design Optimization Problem Statement
• Analyzer & Evaluator
• Classification of MDO Architectures
• Single level Architectures / formulations
• Bi-level Architectures / formulations
OUTLINE
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 3Colloquium on MDO, VSSC Thiruvananthapuram
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
DISCIPLINE 1 S1Z
DISCIPLINE 2 S2Z
MULTI DISCIPLINARY
ANALYSIS (MDA)
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 4Colloquium on MDO, VSSC Thiruvananthapuram
• 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
MDO ARCHITECTURE / FORMULATION
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 5Colloquium on MDO, VSSC Thiruvananthapuram
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
OPTIMIZER
Interface
Z
ANALYSIS
Z S
f, g, h
Nested ANalysis and Design (NAND)
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 6Colloquium on MDO, VSSC Thiruvananthapuram
ANALYSIS AND EVALUATOR
Analysis
S
Converged
Z
S
rEvaluatorr = A(Z, S)
so
Iteratorupdate S
Closed AnalysisNested ANalysis and Design (NAND)
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 7Colloquium on MDO, VSSC Thiruvananthapuram
ALTERNATE STATEMENT
Evaluator
rZ, S
Interface
f, h, r, gZ, S
Optimizer • 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)
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 8Colloquium on MDO, VSSC Thiruvananthapuram
p
Analysis v/s Evaluators
*Solving pushed to optimization level
Conventional approach:
INTERFACE
Solve
z
z p
hgf ,,
design variables
pressure load
objective function
nequality constraints
equality constraints
z
p
f
g
h
OPTIMIZER
0p AIC 2. Calculates
1AIC
3. Calculates
1p AIC
Evaluator:Does not solve Evaluates residues for given Computationally inexpensive
, z pOPTIMIZER
INTERFACE
, z p rhgf ,,,
EVALUATOR
, z p r
, design variables
residue
objective function
equality constraints
, equality constraints
z p
r
f
g
h r
A different approach*:
r p AIC
Analysis:Conservation laws of systemNonlinear, iterativeMultidisciplinaryTime intensive
1. Generates AIC
z p
2. Calculates r p AIC
r
0r
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 9Colloquium on MDO, VSSC Thiruvananthapuram
SYSTEM AND DISCIPLINE LEVEL
SYSTEM LEVEL(DISCIPLINE COORDINATOR)
ZS
DISCIPLINE 1ZL1
DISCIPLINE ‘2’ZL2
Z = ( ZLi) (ZSi )
ZL : Local to discipline (Disciplinary Variables)
ZS : Shared by more than one discipline (System
Variables) Y : Coupling functions
ZS1
Y
ZS2
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 10Colloquium on MDO, VSSC Thiruvananthapuram
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
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 11Colloquium on MDO, VSSC Thiruvananthapuram
• 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)
CLASSIFICATION OF MDO ARCHITECTURES
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 12Colloquium on MDO, VSSC Thiruvananthapuram
NAND-NAND FORMULATION (MDF)
SystemOptimizer
z1
z2
z3
f, G
Analyzer 1
g1
y12, y13
Analyzer 3
Analyzer 2
y21, y31
y12, y32
y13, y23
g2
y21, y23
y31, y32
g3
f, g0
Z
SystemCoordinator
Iterator
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 13Colloquium on MDO, VSSC Thiruvananthapuram
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)
NAND-NAND FORMULATION
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 14Colloquium on MDO, VSSC Thiruvananthapuram
SAND-NAND FORMULATION (IDF)
Zaug = { design variables Z, coupling variables Y*} ; y*13 -y13 = 0
SystemOptimizer
Analyzer 1f, g0
z1
g1
y12, y13
z2
g2
y21, y23
Analyzer 3
Analyzer 2
z3
y31, y32
g3
Z, Y*
f, G
System Coordinator
y13, y23 * *
y21, y31 * *
y12, y32 * *
ICC
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 15Colloquium on MDO, VSSC Thiruvananthapuram
SAND-NAND FORMULATION (IDF)
Augmented Design Variable Vector Zaug = ( Z , y*12 , y*13, y*21, y*23, y*31, y*32 )
Design Constraints (DC): g0 0 ( system design constraints)
g1 0 ; g2 0 ; g3 0 (disciplinary design constraints)
Auxiliary Constraints: ( Inter disciplinary Consistency Constraints) y21 - y*21 = 0; y31 - y*31 = 0 y12 - y*12 = 0; y32 - y*32 = 0 ( ICC) y13 - y*13 = 0; y23 - y*23 = 0
Min f (Zaug ) ; subject to constraints ‘DC’ and ‘ICC’
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 16Colloquium on MDO, VSSC Thiruvananthapuram
SAND-SAND FORMULATION
Zaug = { design variables Z, coupling variables Y*, state variables S}
SystemOptimizer
Evaluator 1f, g0
z1, s1 r1
g1
y12, y13
z2, s2
g2
r2
y21, y23
Evaluator 3
Evaluator 2
z3, s3 r3
y31, y32
g3
Z, S, Y*
f, G, R
System Coordinator
y13, y23
y21, y31 *
y12, y32 *
*
*
* *
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 17Colloquium on MDO, VSSC Thiruvananthapuram
SAND-SAND FORMULATION (AAO)
Augmented Design Variable Vector Zaug = ( Z , S, y*12 , y*13, y*21, y*23, y*31, y*32 )
Design Constraints (DC): g0 0 ( system design constraints) g1 0 ; g2 0 ; g3 0 (disciplinary design constraints)
Auxiliary Constraints: y21 - y*21 = 0; y31 - y*31 = 0 y12 - y*12 = 0; y32 - y*32 = 0 ( ICC) y13 - y*13 = 0; y23 - y*23 = 0
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 18Colloquium on MDO, VSSC Thiruvananthapuram
SAND-SAND FORMULATION
Auxiliary Constraints:(Disciplinary Analysis Constraints)
r1 = s1 – E1( z1, y*21 ,y*31) = 0
r2 = s2 – E2( z2, y*12 ,y*32) = 0 (DAC) r3 = s3 – E3( z3, y*13 ,y*23) = 0
Optimization problem statement:
Find Zaug whichMinimize f (Zaug )Subject to‘DC’ , ‘ICC’ and ‘DAC’ as stated
above.
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 19Colloquium on MDO, VSSC Thiruvananthapuram
Single Level MDO Architectures
Analysis 1
Iterations till convergence
Analysis 2
Iterations till convergence
Multi-Disciplinary Analysis (MDA)
Interface
Optimizer
12y
21y
z hgf ,,
1 2z z 1 2s s
Analysis 1
Iterations till convergence
Analysis 2
Iterations till convergence
Disciplinary Analysis
Interface
Optimizeryz , yhgf ,,,
121, yz 212 , yz 211, ys 122 , ys
Evaluator 1
No iterations
Evaluator 2
No iterations
Disciplinary Evaluation
Interface
Optimizerysz ,, ryhgf ,,,,
111 ,, ysz 222 ,, ysz1r 2r
Individual Discipline Feasible (IDF)
All At Once (AAO)
1. Minimum load on optimizer2. 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
1. Optimizer load increases tremendously
2. No useful results are generated till the end of optimization
3. Parallel evaluation4. Evaluation cost relatively
trivial
Iterative; coupled
)0( r)0( r
Multi-Disciplinary Feasible (MDF)
Uncoupled Non-iterative; Uncoupled
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 20Colloquium on MDO, VSSC Thiruvananthapuram
COMPARISON OF SINGLE LEVEL FORMULATIONS
NAND - NAND SAND-NAND SAND-SANDZ Z, y* Z, S , y*
Analyzer/ Evaluator/ Evaluator/ Analyzer Analyzer Evaluator
Inter- Discipline Consistent Disciplinary Consistent Solution atConsistent Solution OptimalitySolution
MDF IDF All-at-Once
Extreme In-Between Extreme
1
2
3
4
5
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 21Colloquium on MDO, VSSC Thiruvananthapuram
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
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 22Colloquium on MDO, VSSC Thiruvananthapuram
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
BI-LEVEL PROBLEM DECOMPOSITION
DESIGN VECTOR
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 23Colloquium on MDO, VSSC Thiruvananthapuram
zn
x1 g1 , xcO1
System level Optimizer
Min f(Z)s.t. rj (Z) = 0 ; j = 1, N*
Analysis 1 Analysis N
z1r1* rn
*
xn
Subspace Optimizer 1
Min r1(x1) = xs1-zs1 + xcI1-zcI1
+ xcO1 – zcO1 s.t. g1(x1) 0
Subspace optimizer N
Min rn(xn) = xsn-zsn + xcIn-zcIn + xcOn– zcOn
s.t. gn(xn) 0
COLLABORATIVE OPTIMIZATION FORMULATION
zSi shared variables ; zcIi & zcOi coupling variables xsi , xcIi & xcOi copies of system targets at discipline level
gn , xcOn
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 24Colloquium on MDO, VSSC Thiruvananthapuram
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**
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 25Colloquium on MDO, VSSC Thiruvananthapuram
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 = r*i ; xi = x*i
The norm in the objective function ri (xi ) is generally, calculated as L2 norm.
COLLABORATIVE OPTIMIZATION
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 26Colloquium on MDO, VSSC Thiruvananthapuram
?System LevelCoordination
ApproximationModel
Process flow Information flow
Convergence
SS01 SS02 SS03
A1
A2
A3
A1
A2
A3
CONCURRENT SUB-SPACE OPTIMIZATION
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 27Colloquium on MDO, VSSC Thiruvananthapuram
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
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 28Colloquium on MDO, VSSC Thiruvananthapuram
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.
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 29Colloquium on MDO, VSSC Thiruvananthapuram
System Analysis and
Sensitivity Analysis
Update Variables
Discipline 1 Optimizationand
Optm. Sensitivity Analysis
initialize X & Z
X = X0 + XOPT
Z = Z0 + ZOPT
Opportunity for Concurrent Processing
Discipline 2j Optimizationand
Optm. Sensitivity Analysis
Discipline k Optimizationand
Optm. Sensitivity Analysis
System Optimization
HumanIntervention BLISS CYCLE
Bi-Level Integrated System Synthesis - BLISS
X = X0 + XOPT
Z = Z0 +ZOPT
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 30Colloquium on MDO, VSSC Thiruvananthapuram
Bi-Level Integrated System Synthesis - BLISS
Step – 1 System Analysis + Sensitivity (GSE) Step – 2 Subsystem objective Fs={df/dX}T Xs
Subsystem optimizationS
T
Given Z , and Y*
Find X local
To Minimize F {df/dX} X
Subject to {g} 0
{X } {X} {X }L U
* * *{ / }TapproxY Y dY dX X
Linear approximation for the coupling variables for evaluating constraints
Shared variables (system var.) & Y* held constant during subsystem optimization
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 31Colloquium on MDO, VSSC Thiruvananthapuram
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
Bi-Level Integrated System Synthesis - BLISS
S
S
S S S
Find Z
To Minimize F(Z )
{ } { } { }L UZ Z Z
F(ZS) is obtained as a linear extrapolation based on the optimum design sensitivity obtained in each subsystem
Jan. 28, 2004 MDO AlgorithmsMDO Algorithms - 32Colloquium on MDO, VSSC Thiruvananthapuram
Thank You Visit
http://www.casde.iitb.ac.in/MDO/
4th Meeting of SIG-MDO in March 2004