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 =