Jan. 28, 2004MDO Algorithms - 1 Colloquium on MDO, VSSC Thiruvananthapuram Part II: MDO...

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


• Coupled System

• Engg. Design Optimization Problem Statement

• Analyzer & Evaluator

• Classification of MDO Architectures

• Single level Architectures / formulations

• Bi-level Architectures / formulations

OUTLINE


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)


• 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


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)


ANALYSIS AND EVALUATOR

Analysis

S

Converged

Z

S

rEvaluatorr = A(Z, S)

so

Iteratorupdate S

Closed AnalysisNested ANalysis and Design (NAND)


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)


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


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


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


• 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


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


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


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


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’


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 *

*

*

* *


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


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.


Single Level MDO Architectures

Analysis 1

Iterations till convergence

Analysis 2


Multi-Disciplinary Analysis (MDA)

Interface

Optimizer

12y

21y

z hgf ,,

1 2z z 1 2s s

Analysis 1


Analysis 2


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


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


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


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


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


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**


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


?System LevelCoordination

ApproximationModel

Process flow Information flow

Convergence

SS01 SS02 SS03

A1

A2

A3

A1

A2

A3

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



• 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 coordination process is used as the next design iterate in Step 1.


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


Discipline k Optimizationand


System Optimization

HumanIntervention BLISS CYCLE

Bi-Level Integrated System Synthesis - BLISS

X = X0 + XOPT

Z = Z0 +ZOPT



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


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


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


Thank You Visit

http://www.casde.iitb.ac.in/MDO/

4th Meeting of SIG-MDO in March 2004

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Jan. 28, 2004MDO Algorithms - 1 Colloquium on MDO, VSSC Thiruvananthapuram Part II: MDO...

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