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![Page 1: University of Texas at San Antonio Probabilistic Sensitivity Measures Wes Osborn Harry Millwater Department of Mechanical Engineering University of Texas.](https://reader036.fdocuments.net/reader036/viewer/2022062714/56649cfa5503460f949cbe6f/html5/thumbnails/1.jpg)
University of Texas at San Antonio
Probabilistic Sensitivity Measures
Wes Osborn Harry Millwater
Department of Mechanical EngineeringUniversity of Texas at San Antonio
TRMD & DUST Funding
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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
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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
DiskStress scatter - one result for all zonesExceedance 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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University of Texas at San Antonio
Conditional Probabilistic Sensitivities
Enhance existing Monte Carlo algorithm to compute probabilistic sensitivities (assumes a defect is present)
€
∂PMC
∂θ j
= I(x~)
∂fX j( ˜ x )
∂θ j
1
fX j( ˜ x )
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟f ˜ X
( ˜ x )d ˜ x −∞
∞
∫ + BT
= E I(x~)
∂fX i( ˜ x )
∂θ j
1
fX i( ˜ x )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥+ BT
≅1
NI(x j
~
)∂fX i
( ˜ x k )
∂θ j
1
fX j( ˜ x k )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥k=1
N
∑ + BT
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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
∂amax
=∂fx (x)
∂amax
dx + f (amax ) ⋅∂amax
∂amaxamin
amax
∫ − f (amax ) ⋅∂amin
∂amax
=∂fx (x)
∂amax
dx + f (amax )amin
amax
∫
Thus the boundary term is f(amax). This term is an upper bound to the true BT in N dimensions
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University of Texas at San Antonio
Conditional Probabilistic Sensitivities
Example lognormal distribution
€
∂f (x)
∂COV⋅
1
f (x)=
COV ⋅ − ln 1+COV 2( ) + ln( ˜ x )− ln(x)( )
2
( )
1+COV 2( ) ⋅ln 1+COV 2
( )2
Sensitivity with respect to the Coefficient of Variation (stdev/mean)€
∂f (x)
∂˜ x ⋅
1
f (x)=
ln(x)− ln( ˜ x )
˜ x ⋅ln 1+COV 2( )
Sensitivity with respect to the Median
€
( ˜ x )
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University of Texas at San Antonio
Sensitivity with Respect to Median,
€
˜ X
⎥⎦
⎤⎢⎣
⎡
+⋅−
⋅=∂∂
)cov1ln(~
)~
ln()ln()~(~ 2X
XxxIE
X
PMC
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University of Texas at San Antonio
Sensitivity with Respect to Coefficient of Variation,
€
cov
€
∂PMC
∂cov= E I( ˜ x )⋅
cov⋅ − ln 1+ cov2( ) + ln( ˜ X )− ln(x)( )
2
( )
1+ cov2( ) ⋅ln 1+ cov2
( )2
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
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University of Texas at San Antonio
Sensitivities of Exceedance Curve Bounds
Perturb bounds assuming same slope at end points
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University of Texas at San Antonio
Sensitivity with Respect to
€
amin
€
∂PMC
∂amin
= E[I( ˜ x )]⋅ fA(amin )
= PMC ⋅ fA(amin )
[ ]ΨΨ⋅−⋅−=
ΨΨ⋅
∂∂
=∂∂
iA
ii
aNaNaf
a
aN
a
)()()(
)()(
maxminmin
min
min
min
λ
assumes BT is zero
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University of Texas at San Antonio
Sensitivity with Respect to
€
amax
€
∂PMC
∂amax
= fA(amax )⋅ 1− E[I( ˜ x )]( )
= fA(amax )⋅(1− PMC )
Assumes BT is f(amax)
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University of Texas at San Antonio
Zone Sensitivities
€
∂PFi
∂θ j
= (1− PFi) ⋅
∂λ i
∂θ j
⋅PMC i+ λ i ⋅
∂PMC i
∂θ j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
i=1
ˆ n
∑
Partial derivative of probability of fracture of zone with respect to parameter jθ
€
ˆ n number of zones affected by
€
θ j
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University of Texas at San Antonio
Disk Sensitivities
€
∂PF
∂θ j
= (1− PF ) ⋅ λ i ⋅∂PFi
∂θ j
•1
(1− PFi)
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
i=1
ˆ n
∑
Partial derivative of probability of fracture of disk with respect to parameter jθ
€
ˆ n number of zones affected by
€
θ j
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University of Texas at San Antonio
Procedure
For every failure sample: Evaluate conditional sensitivities
Divide by number of samplesAdd boundary term to amax sensitivityEstimate confidence bounds
Results per zone and for disk
€
∂PMC
∂θ j
≅1
NI(x j
~
)∂fX i
( ˜ x k )
∂θ j
1
fX j( ˜ x k )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥k=1
N
∑ + BT
![Page 15: University of Texas at San Antonio Probabilistic Sensitivity Measures Wes Osborn Harry Millwater Department of Mechanical Engineering University of Texas.](https://reader036.fdocuments.net/reader036/viewer/2022062714/56649cfa5503460f949cbe6f/html5/thumbnails/15.jpg)
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
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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:
cyclesNg f 000,20−=
]0[ ≤= gPPf
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University of Texas at San Antonio
Loading
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University of Texas at San Antonio
Model
Titanium ring
24-Zones
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University of Texas at San Antonio
Random Variable
Defect Dist. 524.3min =a 111060max =a
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University of Texas at San Antonio
Results
Random Variables Sampling Technique Finite Difference Technique
mina
Pf
∂
∂
8.4047E-10
8.3033E-10
maxa
Pf∂
∂
6.0010E-12
5.9921E-12
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University of Texas at San Antonio
Contd…
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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
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University of Texas at San Antonio
Random Variable Definitions
Variable Median Cov
Life Scatter 1 0.1
Stress Multiplier 0.001 0.1
Defect Dist. 524.3min =a 111060max =a
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University of Texas at San Antonio
Results
Random Variables Sampling Technique Finite Difference Technique
COV
f
SM
P
∂
∂
7.802050E -4
7.901650E -4
COV
f
SM
P
∂
∂
1.040530E -3
1.056080E -3 COV
f
LS
P
∂
∂
4.745940E -5
5.044580E -5
median
f
LS
P
∂
∂
-2.556550E -4
-2.224830E-4
mina
Pf
∂
∂
1.148740E -9
2.721670E-8
maxa
Pf∂
∂
5.988860E -12
3.180280E-10
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University of Texas at San Antonio
Contd…
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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
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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
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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
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University of Texas at San Antonio
Exceedance Curve
€
amax
€
amin
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University of Texas at San Antonio
Probabilistic Model
€
PF, zone = 1− exp −λ ⋅PMC[ ]
€
PF = 1− P(no failure in zone k) = 1− 1− PF, zone k[ ]k =1
n
∏k =1
n
∏
∑∞
=
⋅=1
, )|()(i
zoneF anomaliesifracturePanomaliesiPP
Probability of Fracture of Disk
Probability of Fracture per Zone
)|( anomaliesifracturePPMC =