Recent Probabilistic Computational Efficiency Enhancements in DARWIN TM

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1 Recent Probabilistic Computational Efficiency Enhancements in DARWIN TM Luc Huyse & Michael P. Enright Southwest Research Institute Harry R. Millwater University of Texas at San Antonio Simeon H.K. Fitch Mustard Seed Software 6 th Annual FAA/Air Force/NASA/Navy Workshop on the Applications of Probabilistic Methods to Gas Turbine Engines March 18, Solomon s Island, MD

2 Outline Background DARWIN s probabilistic fracture mechanics Zone based life prediction Modeling of Uncertainties Importance Sampling Methodology Confidence Bounds Disk Variance Reduction Concluding Remarks

Zone-based Probabilistic Fatigue Life Prediction Limit State Random Variables Defect area & location Stress Fatigue life Inspection time ( X) g( X) = K K 0 Probability of

3 Zone-based Probabilistic Fatigue Life Prediction Limit State Random Variables Defect area & location Stress Fatigue life Inspection time ( X) g( X) = K K 0 Probability of detection (POD) Zone Failure Probability p = P[ F] = δ p Disk Failure Probability C f i i i Pf = P Fi F2 F m P m δ p f i i i= 1 Each zone is idealized as a plate with an initial defect

4 Uncertainties in Risk Analysis Focus here is on defect-related uncertainties: Does a defect occur and what is its size? If present, can the defect be detected during inspection and the part removed? Defect distribution derived from existing data. Are results sensitive to the tail area? Which defects are most likely to cause failure? How accurate is the POD curve? 1 5 Probability of Detection Stress and Life scatter factors (PDF) model the uncertainties of the FE stresses and Fatigue Crack Growth Law Is lognormal PDF appropriate? E+0 1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 Defect area (sq. mils)

5 Importance Sampling Methodology Step 1: Numerical integration of probability of failure without inspection: 1a: Build Response Surface 1b: Compute Critical Defect Size 1c: Perform Integration p F = failure ( d) ( s)dd db ds Step 2: To assess effectiveness of inspections, generate conditional failure samples and determine whether the scheduled inspections can find these defects before K(X)>K C. Conditional failure samples result in K(X)>K C for a number of cycles less than the service life. f D f B ( b) f S

6 Step 1a: Response surface The predicted life scales linearly with the life scatter factor, see da m = b C ( K ) dn fa tter sca ess Str r cto Life values can be clipped if substantially larger than the service life Î computational savings Predicted life Compute the predicted life as a function of the defect size and the stress scatter factor cale) e (log s iz s t c fe De 10

7 Step 1b: Critical defect size ac to r Life s c a tte re ss sc at te rf r fact or St Compute the exceedance probability of the critical defect size. This probability is obtained directly from the defect distribution. 1.0 Pr(d > d*) From response surface, compute the critical defect size for each life and stress scatter value; this is the smallest defect which results in a predicted life equal to the specified service life

8 Step 1c: Perform integration Pr(d > d*) fb(b) fs(s) Numerically integrate: Pr(d > d * b, s ) f B (b) f S ( s ) Stress scatter factor ) Pr(d > d*) fb(b) fs(s e-7 1e-6 1e-5 1e-4 1e-3 1e e-1 1e+0 1e+1 Median value Life scatter factor St re ss sc at te rf ac to r Integration bounds: Lif rf atte e sc a ct o 1.4 r Life scatter: [-5, 3 ] Stress scatter: [-3, +5 ]

9 Step 2: Importance Sampling Now that we know what the failure probability and the conditional failure density is, we can generate samples that fall inside the failure domain. Generate defect size, stress and life scatter factors from new, conditional PDF s This guarantees that K > K C at Service Life (20,000 cycles in example). PDF (1,000,000 Importance samples) 9.0E-5 8.0E-5 7.0E-5 6.0E-5 5.0E-5 4.0E-5 3.0E-5 2.0E-5 1.0E-5 0.0E Predicted life (cycles)

10 Joint Importance Density of Defect Size and Stress Scatter Original joint density Conditional failure density Stress scatter factor Stress scatter factor Initial defect size Initial defect size (joint histogram obtained from 1,000,000 samples)

11 Observations The mode (or high density region) of the conditional density of the stress scatter factor is shifted towards higher values. Need to make sure lognormal model fits actual stress scatter well between the 70 th and 99.9 th percentile. Most likely initial defect size to result in failure after 20,000 flight cycles is less than 1000 square mils. For this case the use of truncated distributions (between 4 and 10 5 sq mils) seems justifiable. Conditional density also suggests which defect size must be detectable for inspection to be effective Note: the results apply to this analysis only Curves change dramatically depending on COV Increasing COV shifts important range of initial defects to the left

12 Monte Carlo Confidence Bounds For small p F a large number of samples N is required to obtain a small COV on the p F estimate. COV 1 p F 95% confidence bounds give indications of the accuracy of the estimate. The confidence bounds can be narrowed at will by using more samples. p F N 2.0E-6 1.5E-6 1.0E-6 5.0E-7 0.0E+0 Without inspection With inspection Monte Carlo (10,000 samples)

13 Importance Confidence Bounds No sampling uncertainty associated with p F,tot,wo without inspection at Service Life (20,000 cycles). Method can skip sampling for zones with no failure contribution 2.0E-6 1.5E-6 1.0E-6 Importance Sampling (100 samples) Without inspection With inspection The IS probabilities converge much faster as the number of samples increases than for Monte Carlo: COV For smaller p F,tot,wo, efficiency gain of Importance Sampling is higher than for MC. p F, tot, wo p F p F N 5.0E-7 0.0E % confidence bounds give indications of the accuracy of the estimate. For DARWIN, Importance Sampling typically needs 100 times fewer samples than Monte Carlo for same accuracy.

14 Importance Sampling: Summary Importance Sampling is expedient and accurate for Risk Analysis. Step 1: Numerical Integration Numerical integration is performed accurately over a sufficiently large domain. Much more accurate failure probability estimation than Monte Carlo; typically 100 times faster. Step 2: Conditional Sampling If one is only interested in p F,tot,wo, no sampling is required for IS; computation ends after numerical integration IS skips the sampling in zones if they do not contribute to the total disk failure risk Importance Sampling densities provide some additional information about the most likely failure combinations.

15 Disk-based Variance Reduction Zone failure probability Is binomial for a sample size n i Can be approximated as a normal distribution for large sample size n i Disk failure probability can also be approximated as normal Objective is to set disk upper confidence bound below target risk Probability Density Change in disk mean Target risk Decreasing disk variance Disk Failure Probability P f As disk variance decreases: mean disk failure probability can increase yet maintain target for a given confidence

16 Confidence Bounds and Target Risk Zone variance pˆ (1 ˆ i pi) σ pˆ i = n Disk mean & variance m µ = δ pˆ Pˆ i i f i= 1 m ˆ = P δi σ pˆ f i i= 1 σ Disk confidence bounds lower bound µ σ Pˆ f k α /2 upper bound + µ σ Pˆ f k α /2 i Pˆ Pˆ f f Probability Density sampling error -k α/2 µ p ˆ f (1-α) confidence interval Target risk k α/2 DARWIN TM currently reports disk confidence bounds Provides guidance for selecting number of Monte Carlo samples

17 Disk Variance Reduction: Allocation of Samples to Zones Uniform Sample Allocation ni 1 = N (m = number of zones ) m Allocate Samples according to Risk contribution factors 2 δ ˆ ip k (1 P ) i α /2 f ni = RCFi N RCFi = N = 2 Pˆ γ Pf Optimal Approach (Wu et al. 2000) f n i = δ m i= 1 p (1 p ) i i i δ p (1 p ) i i i N 2 m kα /2 2 2 γ P f i= 1 N = δi pi(1 pi) 2

POST95-3FBH-3FBH defect distribution #3 FBH 1:1 POD Curve Stress

18 Application: Aircraft Rotor Disk (fixed number of zones) Design life: 20,000 cycles Zones: 44 Random variables: AIA POST95-3FBH-3FBH defect distribution #3 FBH 1:1 POD Curve Stress scatter LN(1.0, 0.2) Life scatter LN(1.0, 0.4) Inspection time N (10000, 0.2)

19 Results 2.0 WITHOUT INSPECTION WITH INSPECTION 1.5 PROBABILITY DENSITY 1.0 LOWER (95%) CONFIDENCE BOUND TARGET RISK UPPER (95%) CONFIDENCE BOUND NORMALIZED PROBABILITY Mean failure probability results are below target risk Upper bound failure probability results are above target risk Apply disk variance reduction techniques

20 Disk Variance Reduction Uniform Approach 4.0 PROBABILITY DENSITY TARGET RISK 100 samples per zone , ,000 UPPER 95% CONFIDENCE BOUND MEETS TARGET RISK AT 100,000 SAMPLES PER ZONE NORMALIZED PROBABILITY Increase number of samples in all zones Target risk satisfied with 100,000 samples per zone Over 4 million total samples for disk

21 Comparison of Disk Variance Reduction Techniques NORMALIZED PROBABILITY Uniform - lower bound (95%) Uniform - mean Uniform - upper bound (95%) RCF - lower bound (95%) RCF - mean RCF - upper bound (95%) Optimal - lower bound (95%) Optimal - mean Optimal - upper bound (95%) TARGET RISK E E E E E+07 NUMBER OF SAMPLES RCF and optimal techniques require fewer samples to meet target Optimal technique slightly more efficient than RCF technique

22 Concluding Remarks Potential efficiency gains for probabilistic fatigue life predictions Use response surface (LAF) for deterministic life prediction Use importance sampling instead of Monte Carlo Apply zone refinement to zones with high RCF Optimal choice of number of samples Disk-based variance reduction can be combined with zone discretization to meet target risk with improved efficiency Zone discretization level primary influence on disk risk mean Disk-based variance reduction (number of samples per zone) primary influence on disk risk COV

EFFICIENT STATISTICAL ANALYSIS OF FAILURE RISK IN ENGINE ROTOR DISKS USING IMPORTANCE SAMPLING TECHNIQUES

EFFICIENT STATISTICAL ANALYSIS OF FAILURE RISK IN ENGINE ROTOR DISKS USING IMPORTANCE SAMPLING TECHNIQUES Luc Huyse & Michael P. Enright 2 Southwest Research Institute San Antonio, Texas 78238 Abstract