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Assessment of Imprecise Reliability Using Efficient Probabilistic Reanalysis

FarizalEfstratios Nikolaidis

SAE 2007 World Congress

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Outline

• Introduction

• Objective

• Approach

• ExampleCalculation of Upper and Lower Reliabilities of System with

Dynamic Vibration Absorber

• Conclusion

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Introduction

Challenges in Reliability Assessment of Engineering Systems:

– Scarce data, poor understanding of physics • Difficult to construct probabilistic models• No consensus about representation of uncertainty in

probabilistic models

– Calculations for reliability analysis are expensive

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Introduction (continued)

• Modeling uncertainty in probabilistic modelsProbability

• Second-Order Probability: Parametric family of probability distributions. Uncertain distribution parameters, , are random variables with PDF fΘ(θ)

• Reliability - random variable

θx]θxθ XΘ ddffRE F ),()[(1)(

0.97 0.98 0.99 10

0.5

1.

R()

CDF

5

0.97 0.98 0.99 10

0.5

1.

CDF

Introduction (continued)

Interval Approach to Model Uncertainty

Given ranges of uncertain parameters find minimum and maximum reliability

– Finding maximum or minimum reliability: Nonlinear Programming, Monte Carlo Simulation, Global Optimization

– Expensive – requires hundreds or thousands reliability analyses

R R

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Objective

• Develop efficient Monte-Carlo simulation approach to find upper and lower bounds of Probability of Failure (or of Reliability) given range of uncertain distribution parameters

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Approach

General formulation of global optimization problem

Max (Min) PF()

Such that: ],[ θθθ

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Solution of optimization problem

• Monte-Carlo simulation – Select a sampling PDF for the parameters θ

and generate sample values of these parameters. Estimate the reliability for each value of the parameters in the sample. Then find the minimum and maximum values of the values of the reliabilities.

– Challenge: This process is too expensive

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Using Efficient Reliability Reanalysis (ERR) to Reduce Cost

• Importance Sampling

),(

),()(

1

1 θx

θxx

X

X

i

ii

n

i g

fI

nPF

Sampling PDF

True PDF

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Efficient Reliability Reanalysis• If we estimate the reliability for one value the uncertain

parameters θ using Monte-Carlo simulation, then we can find the reliability for another value θ’ very efficiently.

• First, calculate the reliability, R(θ), for a set of parameter values, θ. Then calculate the reliability, R(θ’), for another set of values θ’ as follows:

(2) ),(

),()(

11)(

then

)1( ),(

),()(

11)( If

i is

iii

i is

iii

g

fI

nR

g

fI

nR

θx

θxxθ

θx

θxxθ

X

X

X

X

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Efficient Reliability Reanalysis (continued)

• Idea: When calculating R(’), use the same values of the failure indicator function as those used when calculating R ().

• We only have to replace the PDF of the random variables, fX(x,θ), in eq. (1) with fX(x,θ’).

• The computational cost of calculating R(’) is minimal because we do not have to compute the failure indicator function for each realization of the random variables.

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Using Extreme Distributions to Estimate Upper and Lower Reliabilities

Reliability

PDFParent PDF(Reliabilities in a sample follow this PDF)

PDF of smallest reliability in sample

If we generate a sample of N values of the uncertain parameters θ, and estimate the reliability for each value of the sample, then the maximum and the minimum values of the reliability follow extreme type III probability distribution.

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Algorithm for Estimation of Lower and Upper Probability Using Efficient Reliability Reanalysis

Information about Uncertain Distribution

Parameters

Reliability Analysis

Repeated Reliability

Reanalyses

Estimate of Global Min and Max Failure Probabilities

Fit Extreme Distributions To Failure Probability

Values

Estimate of Global Min And Max Failure ProbabilityFrom Extreme Distributions

Path A

Path B

14Path B: Estimation of Lower and Upper Probabilities

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Example: Calculation of Upper and Lower Failure Probabilities of System with Dynamic Vibration

Absorber

m, n2Dynamic absorber

Original systemM, n1

F=cos(et)

Normalized system amplitude y

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Objectives of Example

• Evaluate the accuracy and efficiency of the proposed approach

• Determine the effect of the sampling distributions on the approach

• Assess the benefit of fitting an extreme probability distribution to the failure probabilities obtained from simulation

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Displacement vs. normalized frequencies

β1

Displacement

β2

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Why this example

• Calculation of failure probability is difficult• Failure probability sensitive to mean

values of normalized frequencies• Failure probability does not change

monotonically with mean values of normalized frequencies. Therefore, maximum and minimum values cannot be found by checking the upper and lower bounds of the normalized frequencies.

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Problem Formulation

Max (Min) R()

Such that : 0.9 ≤ i ≤ 1.1, i = 1, 2

0.05 ≤ i ≤ 0.2, i = 1, 2

0 ≤ R() ≤ 1

i: mean values of normalized frequencies

i: standard deviations of normalized frequencies

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PFmax for IS

0.25

0.27

0.29

0.31

0.33

0.35

0.37

36*25 120*25 36*1000 120*1000

N*m

PF

max

2000

5000

10000

Target PFmax

PFmax vs. number of replications per simulation (n), groups of failure probabilities (N), and failure

probabilities per group (m)

2000 replications

5000 replications

10000 replications

True value of PFmax

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Comparison of PFmin and PFmax for n = 10,000True PFmax=0.332

N m

Proposed Method with ERR

MC

PFmin

(PFmin) PFmax

(PFmax) PFmin

(PFmin) PFmax

(PFmax)

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25 0.03069 (0.0021)

0.27554 (0.0144)

0.032 (0.0017)

0.2763 (0.0045)

1000 0.02333 (0.0016)

0.30982 (0.0190)

0.0251 (0.0016)

0.3106 (0.0046)

120

25 0.03069 (0.0021)

0.3008 (0.0170)

0.032 (0.0018)

0.3004 (0.0046)

1000 0.02333 (0.0016)

0.32721 (0.0200)

0.0239 (0.0016)

0.3249 (0.0047)

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Effect of Sampling Distribution on PFmax

PFmax for n = 10000

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

36*25 120*25 36*1000 120*1000

N*m

PF

max

single sampling

bisampling

M C

True Value

Two sampling distributions

Monte Carlo

One sampling distribution

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CPU TimeCPU time for simulation with n= 10000

N m

CPU Time (sec)

Proposed Method with

ERR

MC

36 25 2.70 151

1000 100 6061

120 25 8.61 503

1000 342 20198

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Maximum Case: 120*1000*10K

0.0

0.2

0.4

0.6

0.8

1.0

1.2

PF

CD

F

M cmax

Data _M C

Ismax

Data_ IS

Fitted extreme CDF of maximum failure probability vs. data

N=120, m=1000, n=10000

Fitted, ERRFitted MC

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Conclusion• The proposed approach is accurate and yields

comparable results with a standard Monte Carlo simulation approach.

• At the same time the proposed approach is more efficient; it requires about one fiftieth of the CPU time of a standard Monte Carlo simulation approach.

• Sampling from two probability distributions improves accuracy.

• Extreme type III distribution did not fit minimum and maximum values of failure probability