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Probabilistic Re-Analysis Using Monte Carlo Simulation

Efstratios Nikolaidis, Sirine Salem, Farizal, Zissimos Mourelatos

April 2008

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Definition and SignificanceProbabilistic design optimization• Find design variables • To maximize average utility

RBDO • Find design variables• To minimize loss function • s. t. system failure probability does not exceed allowable value

• Often average utility or system failure probability must be calculated by Monte Carlo simulation. – Vibratory response of a dynamic system: failure domain consists of multiple disjoint

regions

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Definition and Significance

Challenge: High computational cost• Optimization requires probabilistic analyses

of many alternative designs• Each probabilistic analysis requires many

deterministic analyses• Expensive to perform deterministic analysis

of a practical model

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Definition and Significance

Vibratory door displacement

Excitation at engine mounts

Reliability analysis

Monte Carlo Simulation (10,000 replications)

Probability of failure

RBDO

Search for optimum (100- 500 Monte Carlo Simulations)

Optimum

Deterministic FEA

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Outline

1. Objectives and Scope2. Probabilistic Re-analysis

– RBDO problem formulation– Method description– Sensitivity analysis

3. Example– Preliminary Design of Internal Combustion

Engine Conclusion

4. Conclusion

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1. Objectives and Scope

• Present probabilistic re-analysis approach (PRA) for RBDO– Estimate reliability of many designs by performing a

single Monte-Carlo simulation– Integrate PRA in a methodology for RBDO– Demonstrate efficacy

• Design variables are random; can control their average values

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2. Probabilistic Re-analysis

RBDO problem formulation:– Find average values of random design

variables – To minimize cost function – So that

Xμ)( Xμl

allfsys pIPp ]1),,([ PXd

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Reducing computational cost by using Probabilistic Re-analysis

1. Select a sampling PDF and perform one Monte Carlo simulation2. Save sample values that caused failure (failure set)3. Estimate failure probability of all alternative designs by using failure

set in step 2

x1

x2

Failure region

Sampling PDF

Alternative designs Failure set

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• Failure probability

• Confidence in failure probability estimate

• Similar equations are available for average value of a function of design variables (for example utility)

• Values xi are calculated from one Monte-Carlo simulation, same values are used to find failure probabilities of all design alternatives

2

1

2)(/

ˆ ˆ)(

)/(

)1(

1sys

n

i iS

iEp pn

f

f

nns

f

sys

x

μx

X

XXX

syspn

sys stp ˆ2

1,1ˆ

)(

)/(1)(ˆ

)(/

1 iS

in

isys

f

f

np

Ef

x

μxμ

X

XX XX

Sampling PDF

PDF when mean values of design variables = µX

Estimation of failure probability

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Sensitivity analysis

)(

1)|(1

1

)(|

iS

n

i X

Xi

jX

sys

f

f

n

p f

j

E

x

μx

X

XX

• Analytical expression• Can be calculated very efficiently because it is easy to

differentiate PDF of a random variable

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RBDO with Probabilistic Re-analysis

• Find X

• To minimize

• s. t.

• Solution requires only n deterministic analyses

)( Xμl

allf

iS

in

isys p

f

f

np

f

)(

)/(1)( /

1 x

μxμ

X

EXX XX

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RBDO with Probabilistic Re-analysis

Iso-cost curves

Feasible Region

Increased Performance

x2

x1

Optimum

Failure subset

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Efficient Probabilistic Re-analysis:Capabilities

• Calculates system failure probabilities of many design alternatives using results of a single Monte-Carlo simulation

• Does not require calculation of the performance function of modified designs – reuses calculated values of performance function from a single simulation. Cost of RBDO cost of a single simulation

• Non intrusive, easy to program• If PDF of design variables is continuous then system

failure probability varies smoothly as function of design variables

• Highly effective when design variables have large variability

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Challenges

• Works only when all design variables are random

• Requires sample that fills the space of design variables

• Cost of single simulation increases with design variables

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3. Example: RBDO of Internal Combustion Engine

• Preliminary design of flat head internal combustion engine from thermodynamic point of view

• Find average bore, inner and outer diameters, compression ratio and RPM

• To maximize specific power • S. t. system failure probability ≤pf

all (0.4% to 0.67%)

• Failure: any violation of nine packaging and functional requirements

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Design variables(all variables normal)

Standard

Deviation Lower Bound

Upper Bound

Cylinder bore, b , mm

0.4 81.7 82.5

Intake valve diameter, Id , mm

0.15 35.7 36

Exhaust valve

diameter, Ed , mm

0.15 30.2 30.5

Compression ratio,

rc

0.05 9.3 9.39

RPM at peak power/1000,

0.25 5.15 5.65

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Sampling PDF

Average Values

Bore 82.13

Intake valve diameter

35.84

Exhaust valve diameter

30.33

Compression ratio

9.34

RPM 5.31

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Effect of average bore on system failure probability (100,000 replications)

81.6 81.8 82 82.2 82.4 82.60.1

0.05

0

0.05

0.1

dPFb b Ecr EdI EdE E

b

.

81.9 81.95 82 82.05 82.1 82.15 82.2 82.250.004

0.006

0.008

0.01

0.012

Monte-CarloEfficient Reanalysis95% Upper Bound95% Lowel Bound

Bore

Fai

lure

Pro

babi

lity

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RBDO Results for the Engine Design Example (maximum failure probability:

0.0067) Design Variables Initial

Design 1

Initial Design

2

Optimum

Bore 82.023 82.5 82.151 Intake valve diameter 35.7 36 35.857

Exhaust valve diameter 30.2 30.5 30.37 Compression ratio 9.3 9.39 9.315

RPM 5.289 5.65 5.373 psys (PRA)

psys(Monte Carlo, one million replications)

0.0067 0.0071

Objective function l (KW/liter) 51.162

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Specific Power and Probability of Failure

3.E-03

4.E-03

5.E-03

6.E-03

7.E-03

8.E-03

50.7 50.8 50.9 51 51.1 51.2

Specific Power

Fa

ilure

Pro

ba

bili

ty

PRA

Monte Carlo

Liang et al.

95% Upper ConfidenceBound PRA95% Lower ConfidenceBound

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Comparison of efficiencies of standard Monte Carlo and PRA (narrower CI means

higher efficiency of the method)

0.E+00

1.E-04

2.E-04

3.E-04

4.E-04

5.E-04

6.E-04

7.E-04

8.E-04

9.E-04

4.45E-03 5.45E-03 6.42E-03 6.92E-03 7.09E-03

Failure Probability

Hal

f-W

idth

95%

CI

MC

PRA

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Observations

• PRA found an optimum design almost identical as RBDO using FORM (Liang 2007).

• PRA converged to same optimum from different initial designs

• PRA underestimated consistently system failure probability by 5% to 11%.

• 95% confidence intervals have half width = 23% to 28% of system failure probability

• Confidence interval from PRA is 50% wider than that of standard Monte Carlo. This means that PRA needs 225,000 replications to yield results with same accuracy as standard Monte Carlo with 100,000 replications.

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4. Conclusion

• Presented efficient methodology for RBDO using Monte Carlo simulation

• Solves RBDO problems using a single Monte Carlo simulation

• Calculates sensitivity derivatives of system failure probability

• Limitation: methodology, in its present form, works only when all design variables are random