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, 2003 - Solomon s Island, MD
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 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
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 4 3 2 1 Stress and Life scatter factors (PDF) model the uncertainties of the FE stresses and Fatigue Crack Growth Law 0.75 0.5 Is lognormal PDF appropriate? 0.25 0 1.E+0 1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 Defect area (sq. mils)
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
Step 1a: Response surface The predicted life scales linearly with the life scatter factor, see da m = b C ( K ) dn 50000 40000 30000 20000 10000 0 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 0.0020 fa tter sca ess Str r cto Life values can be clipped if substantially larger than the service life Î computational savings 60000 Predicted life Compute the predicted life as a function of the defect size and the stress scatter factor. 0.0022 10000 1000 100 cale) e (log s iz s t c fe De 10
Step 1b: Critical defect size 0.8 0.6 0.4 0.2 ac to r 0.0022 0.0020 0.0018 0.0016 0.0014 0.0012 0.0010 0.0008 0.0006 2.2 2.0 1.8 1.6 1.4 1.2 Life s c a tte re ss sc at te rf 0.0 1.0 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. 0.8 0.6
Step 1c: Perform integration Pr(d > d*) fb(b) fs(s) Numerically integrate: Pr(d > d * b, s ) f B (b) f S ( s ) 0.0025 Stress scatter factor 0.0020 60 ) Pr(d > d*) fb(b) fs(s 50 40 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 0.0015 1e-1 1e+0 1e+1 Median value 30 0.0010 20 10 0.4 0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Life scatter factor St 0.0025 re ss 0.0020 sc at te 0.0015 rf ac to 0.0010 r Integration bounds: 0.4 0.6 0.8 Lif 1.0 1.2 rf atte e sc a ct o 1.4 r 1.6 1.8 Life scatter: [-5, 3 ] Stress scatter: [-3, +5 ]
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+0 0 10000 20000 Predicted life (cycles)
Joint Importance Density of Defect Size and Stress Scatter Original joint density Conditional failure density 0.0025 1 10 100 1000 10000 0.0025 1 10 100 1000 Stress scatter factor 0.0020 0.0015 Stress scatter factor 0.0020 0.0015 0.0010 0.0010 10 100 1000 10000 Initial defect size 10 100 1000 10000 Initial defect size (joint histogram obtained from 1,000,000 samples)
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
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) 0 5000 10000 15000 20000
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+0 0 5000 10000 15000 20000 95% 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.
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.
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
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 2 2 2 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
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
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)
Results 2.0 WITHOUT INSPECTION WITH INSPECTION 1.5 PROBABILITY DENSITY 1.0 LOWER (95%) CONFIDENCE BOUND TARGET RISK UPPER (95%) CONFIDENCE BOUND 5.0 0.0 0.0 5.0 1.0 1.5 2.0 NORMALIZED PROBABILITY Mean failure probability results are below target risk Upper bound failure probability results are above target risk Apply disk variance reduction techniques
Disk Variance Reduction Uniform Approach 4.0 PROBABILITY DENSITY 3.0 2.0 TARGET RISK 100 samples per zone 1000 10,000 100,000 UPPER 95% CONFIDENCE BOUND MEETS TARGET RISK AT 100,000 SAMPLES PER ZONE 1.0 0.0 0.90 0.95 1.00 1.05 1.10 1.15 1.20 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
Comparison of Disk Variance Reduction Techniques NORMALIZED PROBABILITY 1.6 1.4 1.2 1.0 8.0 6.0 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 4.0 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 NUMBER OF SAMPLES RCF and optimal techniques require fewer samples to meet target Optimal technique slightly more efficient than RCF technique
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