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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
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),( min f
Design Target Problem
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0pxg
TpxRx
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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
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