Transcript Monte-Carlo Simulation Approach for Estimation of Imprecise

Assessment of Imprecise
Reliability Using Efficient
Probabilistic Reanalysis
Farizal
Efstratios Nikolaidis
SAE 2007 World Congress
1
Outline
•
•
•
•
Introduction
Objective
Approach
Example
Calculation of Upper and Lower Reliabilities of System with
Dynamic Vibration Absorber
• Conclusion
2
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
3
Introduction (continued)
• Modeling uncertainty in probabilistic models
Probability
• Second-Order Probability: Parametric family of probability
distributions. Uncertain distribution parameters, , are
random variables with PDF fΘ(θ)
• Reliability - random variable
.
1
CDF
0.5
0
0.97
0.98
0.99
1
R()
E ( R) 1   fΘ (θ)[F f X (x, θ) dx] dθ
4
Introduction (continued)
Interval Approach to Model Uncertainty
Given ranges of uncertain parameters find minimum and
maximum reliability
R
.
R
1
CDF
0.5
0
0.97
0.98
0.99
1
– Finding maximum or minimum reliability: Nonlinear
Programming, Monte Carlo Simulation, Global Optimization
– Expensive – requires hundreds or thousands reliability analyses
5
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
6
Approach
General formulation of global optimization
problem
Max (Min)  PF()
Such that: θ  [θ, θ]
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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
8
Using Efficient Reliability
Reanalysis (ERR) to Reduce Cost
• Importance Sampling
f ( x , θ)
1 n
PF   I ( x i ) X i
n i 1
g X ( x i , θ)
True PDF
Sampling 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:
f X ( x i , θ)
1
If R(θ)  1   I i (x i )
n i
g s ( x , θ)
X
(1)
i
then
1
R(θ)  1   I i (x i )
n i
f X (x i , θ)
s
gX
( x i , θ)
(2)
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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
PDF
PDF of smallest
reliability in sample
Parent PDF
(Reliabilities
in a sample
follow this
PDF)
Reliability
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
Path A
Estimate of Global Min and Max Failure
Probabilities
Path B
Fit Extreme Distributions
To Failure Probability
Values
Estimate of Global Min
And Max Failure Probability
From Extreme Distributions
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Path 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, n2
Dynamic absorber
Original system
Normalized system
amplitude y
M, n1
F=cos(et)
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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
16
Displacement vs. normalized
frequencies
Displacement
β2
β1
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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 vs. number of replications per simulation (n),
groups of failure probabilities (N), and failure
PFmax for IS
probabilities
per group (m)
0.37
2000 replications
0.35
True value of PFmax
5000 replications
PFmax
0.33
10000 replications
0.31
0.29
2000
5000
0.27
10000
Target PFmax
0.25
36*25
120*25
36*1000
120*1000
N*m
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Comparison of PFmin and PFmax for n = 10,000
True PFmax=0.332
N
m
Proposed Method
with ERR
MC
PFmin
(PFmin)
PFmax
(PFmax)
PFmin
(PFmin)
PFmax
(PFmax)
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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)
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)
36
120
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Effect of Sampling Distribution on PFmax
PFmax for n = 10000
0.34
Two sampling distributions
0.33
One sampling distribution
PFmax
0.32
Monte Carlo
0.31
0.3
0.29
single sampling
bisampling
0.28
MC
True Value
0.27
36*25
120*25
36*1000
120*1000
N*m
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CPU Time
CPU time for simulation with n= 10000
CPU Time (sec)
N
36
120
m
Proposed
Method with
ERR
MC
25
2.70
151
1000
100
6061
25
8.61
503
1000
342
20198
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Fitted extreme CDF of maximum failure probability vs.
data
N=120,
m=1000, n=10000
Maximum Case: 120*1000*10K
1.2
Fitted, ERR
Fitted MC
1.0
0.6
0.4
M cmax
Dat a _M C
0.2
Ismax
Dat a_IS
0.0
0.
26
0.
26
4
0.
26
8
0.
27
2
0.
27
6
0.
28
0.
28
4
0.
28
8
0.
29
2
0.
29
6
0.
3
0.
30
4
0.
30
8
0.
31
2
0.
31
6
CDF
0.8
PF
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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
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