Optimal Multilevel SystemDesign under Uncertainty

NSF Workshop on Reliable Engineering Computing

Savannah, Georgia, 16 September 2004

M. Kokkolaras and P.Y Papalambros

University of Michigan

Z. Mourelatos

Oakland University

Outline

• Design by Decomposition• Hierarchical Multilevel Systems• Analytical Target Cascading

– Deterministic Formulation– Nondeterministic Formulations

• Propagation of Uncertainty• Practical Issues• Example

Optimal System Design

0pxh

0pxg

pxx

),(

),( s.t.

),( min f

Design Target Problem

0pxh

0pxg

TpxRx

),(

),( s.t.

),( min2

2

Design by Decomposition

• When dealing with large and complex engineering systems, an “all-at-once” formulation of the optimal design problem is often impossible to solve

• Original problem is decomposed into a set of linked subproblems

• Typically, the partitioning reflects the hierarchical structure of the organization (different design teams are assigned with different subproblems according to expertise)

VEHICLE

ELECTRONICS CLIMATE CONTROLBODY CHASSISPOWERTRAIN

ENGINE DRIVELINETRANSMISSION

CYLINDER BLOCK…

Decomposition Example

VALVETRAIN

Multilevel System Design

• Multilevel hierarchy of single-level (sub)problems

• Responses of higher-level elements are depend on responses of lower-level elements in the hierarchy

system

subsystem 1 subsystem 2

component 1 component 2 component m

subsystem n

Challenges

• Need to assign design targets for the subproblems to the design teams

• Design teams may focus on own goals without taking into consideration interactions with other subproblems; this will compromise design consistency and optimality of the original problem

Analytical Target Cascading

• Operates by formulating and solving deviation minimization problems to coordinate what higher-level elements “want” and what lower-level elements “can”

• Parent responses rp are functions of

– Children response variables rc1, rc2, …, rcn, (required)

– Local design variables xp (optional)

– Shared design variables yp (optional)

• In the following formulations:– Subscript index pairs denote level and element– Superscript indices denote computation “location”

Mathematical Formulation

1 1

2 2

2 2

,( 1) ( 1) ( 1) ( 1)

2

( 1) ( 1) 2

( 1) ( 1)

minimize

with respect to ,..., , , , ,..., ,

subject to

nn cc ijij

ij

U U r yij ij ij ij ij ij

r yij ij ij iji k i k i k i k

L riji k i k

k C

i k i

r r y y

r r x y y y

r r

y y

1

1

1

2

2

( 1) ( 1)

( 1) ( 1)

( 1) ( 1)

( ,..., , , )

( ,..., , , )

where ( ,..., , , )

ij

ncij

ncij

ncij

L yijk

k C

ij ij iji k i k

ij ij iji k i k

ij ij ij iji k i k

g r r x y 0

h r r x y 0

r f r r x y

element optimization problem pij, where rij is provided by the analysis/simulation model

1( 1) ( 1)( ,..., , , ) cij

ij ij i k i k ij ijr f r r x y

1( 1) ( 1),..., cij

l li k i ky y

( 1) 1 ( 1),..., cij

l li k i kr r

uijruijy

response and sharedvariable values cascaded

down from the parent

response and shared variable values passed

up from the children

optimization inputs

response and sharedvariable values cascaded

down to the children

lijrlijy

( 1) 1 ( 1),..., cij

u ui k i kr r

1( 1) ( 1),..., cij

u ui k i ky y

response and shared variable values passed

up to the parent

optimization outputs

Information Exchange

Multilevel System Designunder Uncertainty

• Multilevel hierarchy of single-level (sub)problems

• Outputs of lower-level problems are inputs to higher-level problems: need to obtain statistical properties of responses

system

subsystem 1 subsystem 2

component 1 component 2 component m

subsystem n

Nondeterministic Formulations

• For simplicity, and without loss of generalization, assume uncertainty in all design variables only

• Introduce random variables (and functions of random variables)

• Identify (assume) distributions

• Use means as design variables assuming known variance

Stochastic Formulation

( 1) ( 1) ( 1) ( 1)1 1

( 1) ( 1)

2 2

2 2

,

2

2

minimize [ ]

with respect to ,..., , , , ,..., ,

[ ]

subject to

ij ij ij

ij iji k i k i k i knn cc ijij

i k i k

ij

U U R Yij ij ij

R Yij ij

lij

k C

R Y Y

R R X Y Y Y

R

E R μ μ μ

μ μ μ μ μ μ

μ ER

( 1) ( 1)

1

1

1

2

2

( 1) ( 1)

( 1) ( 1)

( 1) ( 1)

[ ( ,..., , , ) ]

[ ( ,..., , , ) ]

where ( ,..., ,

i k i k

ij

ncij

ncij

ncij

R

l Yij

k C

ij ij iji k i k

ij ij iji k i k

ij ij iji k i k

Y Yμ μ

E g R R X Y 0

E h R R X Y 0

R f R R X , )ijY

Constraints

• “Hard” and “soft” inequalities

• “Hard” and “soft” equalities

• Typically, a target reliability of satisfying constraints is desired

Probabilistic Formulation

( 1) ( 1) ( 1) ( 1)1 1

( 1) ( 1)

2 2

2 2

,

2

2

minimize [ ]

with respect to ,..., , , , ,..., ,

[ ]

subject to

ij ij ij

ij iji k i k i k i knn cc ijij

i k i k

ij

U U R Yij ij ij

R Yij ij

lij

k C

R Y Y

R R X Y Y Y

R

E R μ μ μ

μ μ μ μ μ μ

μ ER

( 1) ( 1)

1

1

2

2

~

( 1) ( 1)

( 1) ( 1)

[ ( ,..., , , ) 0]

where ( ,..., , , )

i k i k

ij

ncij

ncij

R

l Yij

k C

ij ij ij fi k i k

ij ij ij iji k i k

Y Yμ μ

P g R R X Y P

R f R R X Y

Propagation of Uncertainty

1 2 1 2

1 2

1 2

1 21

2

, ,..., , ,..., , ,...,

then E[Z] , ,...,

and , ,...,

N N i

N

N

N

N X X X X X X i Xi i

X X X

Z X X Xi

ZZ X X X Z X

X

Z

Z

X2

1

i

N

Xi

State of the Art (?):Since functions are generally nonlinear, use first-order approximation(Taylor series expansion around the means of the random variables)

Validity of Linearization

Y(X)

XX

consistency constraints in ATC formulation secure validity

Y

Examples

1 1 2 1,2

2 22 1 2 1 2

21 2

3

2 21 2 1 2

4

5

exp( 7) 10 ~ (6,0.8)

~ (10,2), ~ (10,1)

120

( 5) ( 12)1

30 120

1

Z X X X N

Z X X X N X N

X XZ

X X X XZ

Z

1,2

21 2

~ (5,0.3)

80

( 8 5)

X N

X X

Results

Linear. MCS* lin %

E[Z1]

Var[Z1]1/2

3.6321

1.9386

3.4921

0.9327

4.00

107.85

E[Z2]

Var[Z2]1/2

200

44.721

205.04

45.101

-2.45

-0.84

E[Z3]

Var[Z3]1/2

-5.25

0.8385

-5.3114

0.8407

-1.15

-0.26

E[Z4]

Var[Z4]1/2

-1.0333

0.1166

-1.0404

0.1653

-0.68

29.46

E[Z5]

Var[Z5]1/2

-0.1428

0.00627

-0.1448

0.00630

-1.3

-0.47* 1,000,000 samples

Moment Approximation UsingAdvanced Mean Value Method

1. Consider Z=g(X)

2. Discretize “range” (from 4 (Pf = 0.003%) to 4 (Pf = 99.997%))

3. Find MPP for P[g(X)>0]<(-ifor all i

4. Evaluate Z=g(XMPP), i.e., generate CDF of Z

5. Derive PDF of Z by differentiating CDF numerically

6. Integrate PDF numerically to estimate moments

2 2

E[ ] ( )

Var[ ] ( ) ( )

Z Z

Z Z Z

Z zf z dz

Z z f z dz

Results

Linear. MAM MCS* lin % MAM %

E[Z1]

Var[Z1]1/2

3.6321

1.9386

3.6029

0.9013

3.4921

0.9327

4.00

107.85

3.17

-3.36

E[Z2]

Var[Z2]1/2

200

44.721

203.37

45.203

205.04

45.101

-2.45

-0.84

-0.81

0.22

E[Z3]

Var[Z3]1/2

-5.25

0.8385

-5.3495

0.8423

-5.3114

0.8407

-1.15

-0.26

0.71

0.19

E[Z4]

Var[Z4]1/2

-1.0333

0.1166

-1.0380

0.1653

-1.0404

0.1653

-0.68

29.46

-0.23

0

E[Z5]

Var[Z5]1/2

-0.1428

0.00627

-0.1454

0.00631

-0.1448

0.00630

-1.3

-0.47

0.41

0.15* 1,000,000 samples

Example:Piston Ring/Liner Subassembly

GT Power

Brake-specific fuel consumption (BSFC)

Power loss due to friction

RingPak

Ring and liner surface roughness Liner material properties

Oil consumptionBlow-byLiner wear rate

Lower-level Problem Formulation

5

minimize E[power loss]

with respect to , , ,

subject to P oil consumption 15.3 ( )

P blowby 4.25 10 ( )

P l

R Rliner ringS S liner liner

f

f

Y H

g Phr

kg Ps312iner wear rate 2.4 10 ( )

1 m

10 m

80 GPa 340 GPa

150 BHV 240 BHV

with rin

fm Ps

Y

H

g and line surface roughness ~ ( ,1.0 m) and =3 (P 0.13%) fN

Results and Reliability Assessment = 4 m

= 6.15 m

= 80 GPa

240 BHV

R

L

Y

H

Active Pf, % MCS*

Liner Wear Rate No < 0.13 0

Blow-by No < 0.13 0

Oil Consumption Yes 0.13 0.16

0.03% less reliable than assumed* 1,000,000 samples

Statistical Properties of Power Loss

MAM - PDF MCS – PDF (1,000,000 samples)

Linearization MAM MCS Lin. MAM E[pl] 0.3950 0.3922 0.3932 0.45% -0.25%

Var[pl]1/2 0.0481 0.0309 0.0311 54.6% -0.64%

Upper-level Problem FormulationL * * * * E[power loss] ( , , , )

R Rliner ringS S liner linerf Y H

L

L

minimize E[fuel consumption]

with respect to ,

( E[power loss] ) subject to

E[power loss]

P

P

Linearization MAM MCS* Lin. MAM

E[fuel] 0.5341 0.5341 0.5342 -0.01% -0.01%

Var[fuel]1/2 0.00757 0.00760 0.00759 -0.25% 0.13%

* 1,000,000 samples

Probability Distribution of BSFC

MAM MCS with 1,000,000 samples

Practical Issues

• Computational cost

• Noise/accuracy in the model vs. magnitude of uncertainty in inputs

• Convergence of multilevel approach

Concluding Remarks

• Practical yet rational decision-making support– Value of optimization results is in trends not in numbers– Strategies should involve a mix of deterministic

optimization and stochastic “refinement”

• Need for accurate uncertainty quantification (and propagation)

Error Issues

• y=f(x) + model + metamodel + data

+ num + unc. prop.

• Need to keep ALL errors relatively low

Q & A

Partitioned Group #1 Partitioned Group #2

OSLH Samples

Optimum Symmetric Latin Hypercube (OSLH) Sampling

Cross-Validated Moving Least Squares (CVMLS) Method

Polynomial Regression using Moving Least Squares (MLS) Method

In MLS, sample points are weighted so that nearby samples have more influence on the prediction.

T

j 1

( ) ( ) ( ) ( )m

j jf g a b

x x x a b xGlobal Least Squares : a : vector of constants;

T T

1 2 1( ) ( ) ( ) 1( )

m nb b b x x x x xb x

T

j 1

( ) ( ) ( ) ( ) ( ) ( )m

mls j jf g a b

x x x x a x b xMoving Least Squares : xwfxa ;

1

2

, , 1

( , ) expn

ni i

k k kk

w d x x

x xwhere :

Metamodel Errors• Optimal symmetric Latin hypercube sampling

(200 train points and 150 trial points for Ringpak, 45 train points and 40 trial points for GT-power)

• Moving least squares approximations

Relative errors, % Maximum MeanStandard Deviation

Power loss 8.62 0.37 0.77

Wear rate 9.78 0.72 1.32

Blow-by 3.98 0.37 0.63

Oil consumption 41.8 1.74 3.68

BSFC 0.01 0.005 0.004

Accuracy and Efficiency of Monte Carlo Method

1 12

1

ˆ ˆ1 1% 100 1 100 1

2 2

Example: for a 95% confidence, 1 0.95 0.05, and

1 1 % 100 0.975 196

Since the target probability of failure is 0.0013 (0.13%)

p p p

NpNp

a

p p

Np Np

p

0.9987 % 196 , and with =1,000,000

0.0013 % 196(0.001) 5.43% 0.13% 0.007%

NN

p

99% confidence 99.9% confidence 99.99% confidence

0.13% 0.009% p

0.13% 0.012% p

0.13% 0.014% p