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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
University of Texas at San Antonio
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˜ ) 



f X i ( x˜ k )
1 N 
1 
 BT
 I(x j )
N k1 
 j f X j ( x˜ k ) 
 ~


University of Texas at San Antonio

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
f x (x)
 
dx  f (amax )
a min amax
a max
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˜  ln 1COV 2

Sensitivity with respect to the Coefficient of

Variation (stdev/mean)
f (x) 1


COV f (x)


COV   ln 1 COV 2  ln( x˜ ) ln( x)
2
1 COV  ln 1 COV 
2

2 2
University of Texas at San Antonio
Sensitivity with Respect to
Median, X˜
~
 ~ ln( x)  ln( X ) 
PMC

~  E I ( x )  ~
2 
X
X  ln( 1  cov ) 

University of Texas at San Antonio
Sensitivity with Respect to
Coefficient of Variation, cov

2

2
˜ ) ln( x)
cov

ln
1
cov

ln(
X



PMC

 EI( x˜ )
2
2
2
cov
1 cov  ln 1 cov 




University of Texas at San Antonio






Sensitivities of Exceedance Curve Bounds
 Perturb bounds assuming same slope at end points
University of Texas at San Antonio
Sensitivity with Respect to
amin
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 ) 

University of Texas at San Antonio
Sensitivity with Respect to amax
PMC
 fA (amax ) 1 E[I( x˜ )]

amax
 fA (amax )(1 PMC )
Assumes BT is f(amax)

University of Texas at San Antonio
Zone Sensitivities
Partial derivative of probability of fracture of zone
with respect to parameter  j


P
MCi
i
 (1 PFi )  
 PMCi  i 



 j
 j 
i1  j
PFi
nˆ
nˆ number of zones affected by  j


University of Texas at San Antonio
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 
i1 
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 i ( x˜ k )
PMC 1 
1 
 BT
 I(x j )
 j
N k1 
 j f X j ( x˜ k ) 
 ~

 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]
University of Texas at San Antonio
Loading
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Model
Titanium ring
24-Zones
University of Texas at San Antonio
Random Variable
Defect Dist.
amin  3.524
amax  111060
University of Texas at San Antonio
Results
Random Variables
Pf
a min
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…
University of Texas at San Antonio
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
University of Texas at San Antonio
Random Variable Definitions
Variable
Median
Cov
Life Scatter
1
0.1
Stress Multiplier
0.001
0.1
amin  3.524
amax  111060
Defect Dist.
University of Texas at San Antonio
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
a min
Pf
a max
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Contd…
University of Texas at San Antonio
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
University of Texas at San Antonio
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
University of Texas at San Antonio
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
University of Texas at San Antonio
Exceedance Curve
amin

amax

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Probabilistic Model
Probability of Fracture per Zone

PF , zone   P(i anomalies )  P( fracture | i anomalies )
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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