1 assessment of imprecise reliability using efficient probabilistic reanalysis farizal efstratios...
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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
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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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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
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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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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