University of Texas at San Antonio Probabilistic Sensitivity Measures Wes Osborn Harry Millwater...
-
Upload
edward-cannon -
Category
Documents
-
view
214 -
download
1
Transcript of University of Texas at San Antonio Probabilistic Sensitivity Measures Wes Osborn Harry Millwater...
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
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
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
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
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
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 )
University of Texas at San Antonio
Sensitivity with Respect to Median,
€
˜ X
⎥⎦
⎤⎢⎣
⎡
+⋅−
⋅=∂∂
)cov1ln(~
)~
ln()ln()~(~ 2X
XxxIE
X
PMC
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
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
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
∂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
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)
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
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
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
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:
cyclesNg f 000,20−=
]0[ ≤= gPPf
University of Texas at San Antonio
Loading
University of Texas at San Antonio
Model
Titanium ring
24-Zones
University of Texas at San Antonio
Random Variable
Defect Dist. 524.3min =a 111060max =a
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
University of Texas at San Antonio
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
Defect Dist. 524.3min =a 111060max =a
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
University of Texas at San Antonio
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
€
amax
€
amin
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 =