Application of Probabilistic Sensitivities in
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Transcript Application of Probabilistic Sensitivities in
Probabilistic Sensitivity Measures
Wes Osborn
Harry Millwater
Department of Mechanical Engineering
University of Texas at San Antonio
TRMD & DUST Funding
University of Texas at San Antonio
Objectives
Compute the sensitivities of the probability of fracture with
respect to the random variable parameters, e.g., median,
cov
No additional sampling
Currently implemented:
Life scatter (median, cov)
Stress scatter (median, cov)
Exceedance curve (amin, amax)
Expandable to others
University of Texas at San Antonio
Probabilistic Sensitivities
Three sensitivity types computed
Zone
Conditional - based on Monte Carlo samples
SS, PS, EC
Unconditional - based on conditional results
SS, PS, EC
Disk
Stress scatter - one result for all zones
Exceedance curve - one result for all zones using a particular
exceedance curve (currently one)
Life scatter - different for each zone
95% confidence bounds developed for each
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Conditional Probabilistic
Sensitivities
Enhance existing Monte Carlo algorithm to compute probabilistic
sensitivities (assumes a defect is present)
f X j ( x˜ ) 1
PMC
f X˜ ( x˜ )d x˜ BT
I(x )
j ~ j f X j ( x˜ )
f X i ( x˜ ) 1
E I(x )
BT
~
j f X i ( x˜ )
N
f X i ( x˜ k )
1
1
BT
I(x j )
N k1
j f X j ( x˜ k )
~
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Conditional Probabilistic
Sensitivities
BT - Denotes Boundary Term needed if perturbing the
parameter changes the failure domain, e.g., amin, amax
P
f (x)
a
a
x
dx f (amax ) max f (amax ) min
amax a min amax
amax
amax
a max
a max
a min
f x (x)
dx f (amax )
amax
Thus the boundary term is f(amax). This term is an
upper bound to the true BT in N dimensions
University of Texas at San Antonio
Conditional Probabilistic
Sensitivities
Example lognormal distribution
Sensitivity with respect to the Median ( x˜ )
f (x) 1
ln(x) ln(x˜ )
x˜ f (x) x˜ ln1 COV 2
Sensitivity with respect to the Coefficient of
Variation (stdev/mean)
f (x) 1
COV f (x)
COV ln1COV 2 ln(x˜ ) ln(x)
2
1COV ln1COV
2
2 2
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Sensitivity with Respect to
Median, X˜
~
PMC ~ ln(x) ln(X )
~ E I ( x ) ~
2
X
X ln(1 cov )
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Sensitivity with Respect to
Coefficient of Variation, cov
2
2
˜ ) ln(x)
cov
ln
1
cov
ln(
X
PMC
EI( x˜ )
2
2
2
cov
1 cov ln1 cov
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Sensitivities of Exceedance Curve Bounds
Perturb bounds assuming same slope at end points
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Sensitivity with Respect to
a min
PMC
E[I( x˜ )] fA (amin )
amin
PMC fA (amin )
assumes BT is zero
(i ) N (a min ) i
a min
a min
i
f A (a min ) N (a min ) N (a max )
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Sensitivity with Respect to amax
PMC
fA (amax )
1 E[I( x˜ )]
amax
fA (amax )(1 PMC )
Assumes BT is f(amax)
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Zone Sensitivities
Partial derivative of probability of fracture of zone
with respect to parameter j
P
MC i
i
(1 PFi )
PMC i i
j
j
j
i1
PFi
nˆ
nˆ number of zones affected by j
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Disk Sensitivities
Partial derivative of probability of fracture of disk
with respect to parameter j
P
PF
1
Fi
(1 PF )
i
(1 P )
j
j
Fi
i1
nˆ
nˆ number of zones affected by j
University of Texas at San Antonio
Procedure
For every failure sample:
Evaluate conditional sensitivities
N
f X ( x˜ k )
PMC 1
1
BT
I(x j )
j
N k1
j f X ( x˜ k )
~
i
j
Divide by number of samples
Add boundary term to amax sensitivity
Estimate confidence bounds
Results per zone and for disk
University of Texas at San Antonio
DARWIN Implementation
New code contained in sensitivities_module.f90
zone_risk
accumulate_pmc_sensitivities
accrue expected value results
compute_sensitivities_per_pmc
compute_sensitivities_per_zone
write_sensitivities_per_zone
zone_loop
sensitivities_for_disk
write_disk_sensitivities
University of Texas at San Antonio
Application Problem #1
The model for this example consists of the titanium ring
outlined by advisory circular AC-33.14-1 subjected to
centrifugal loading
Limit State:
g N f 20,000 cycles
Pf P[ g 0]
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Loading
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Model
Titanium ring
24-Zones
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Random Variable
Defect Dist.
amin 3.524
amax 111060
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Results
Random Variables
Pf
amin
Sampling Technique
Finite Difference Technique
8.4047E-10
8.3033E-10
6.0010E-12
5.9921E-12
Pf
a max
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Contd…
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Application Problem #2
Consists of same model, loading conditions, and limit
state
In addition to the defect distribution, random variables Life
Scatter and Stress Multiplier have been added
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Random Variable Definitions
Variable
Median
Cov
Life Scatter
1
0.1
Stress Multiplier
0.001
0.1
amin 3.524
amax 111060
Defect Dist.
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Results
Random Variables
Pf
Sampling Technique
Finite Difference Technique
7.802050E-4
7.901650E-4
SM COV
Pf
1.040530E-3
1.056080E-3
LS COV
Pf
4.745940E-5
5.044580E-5
LS median
Pf
-2.556550E-4
-2.224830E-4
1.148740E-9
2.721670E-8
5.988860E-12
3.180280E-10
SM COV
Pf
amin
Pf
a max
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Contd…
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Conclusion
A methodology for computing probabilistic sensitivities
has been developed
The methodology has been shown in an application
problem using DARWIN
Good agreement was found between sampling and
numerical results
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Example - Sensitivities wrt amin
14 zone AC test case
Sensitivities of the
conditional POF wrt
amin
Zone
Numerical
Analytical
1
1.7881E-05
1.7992E-05
2
1.7881E-05
1.5664E-05
3
1.7881E-05
2.1802E-05
4
1.1325E-04
1.2494E-04
5
4.5300E-04
4.5165E-04
6
1.2100E-03
1.2134E-03
7
2.7239E-03
2.6827E-03
8
1.1921E-05
1.3060E-05
9
5.9604E-06
7.9424E-06
10
5.9604E-06
8.1728E-06
11
1.7881E-05
1.4760E-05
12
3.5763E-05
3.6387E-05
13
1.7881E-04
1.8838E-04
14
1.8716E-03
1.8278E-03
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Probabilistic Sensitivities
Sensitivities for these distributions developed
Normal (mean, stdev)
Exponential (lambda, mean)
Weibull (location, shape, scale)
Uniform (bounds, mean, stdev)
Extreme Value – Type I (location, scale, mean, stdev)
Lognormal Distribution (COV, median, mean, stdev)
Gamma Distribution (shape, scale, mean, stdev)
Sensitivities computed without additional sampling
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Exceedance Curve
a min
amax
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Probabilistic Model
Probability of Fracture per Zone
PF , zone P(i anom alies) P( fracture | i anom alies)
i 1
PF, zone 1 exp PMC
Probability of Fracture of Disk
n
n
k 1
k 1
PF 1 P(no failure in zone k) 1 1 PF, zone k
PMC P( fracture| i anomalies)
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