Bayesian Knowledge Fusion in Prognostics and Health Management A Case Study

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Erika Stokes
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University of Maryland-College Park Ali Moahmamd-Djafari Laboratoire

1 Bayesian Knowledge Fusion in Prognostics and Health Management A Case Study Masoud Rabiei Mohammad Modarres Center for Risk and Reliability University of Maryland-College Park Ali Moahmamd-Djafari Laboratoire des Signaux et Systèmes, UMR 8506 CNRS - SUPELEC - Univ Paris Sud 11

2 Motivation and Background 2

fatigue tests with 2X safety factors Objective: Quantify the

3 P3 Aircraft Health Management Fatigue Cracks SAFE-life design assumes very low probability of crack initiation Full-sacle fatigue tests with 2X safety factors Objective: Quantify the risk associated with fleet life extension (damage-tolerance regime) 3

4 Motivation & Background Today s objective in fleet management is to use an airframe to its maximum service life (total life) [1] Stochastic Physics-of-Failure (PoF) or mechanistic-life approach has proved useful for fleet management. Shortcomings of PoF: Limited knowledge about the underlying PoF Scarcity of relevant material-level test data to estimate model parameters In practice, disconnected from the system being modeled (no feedback) Crack Size Critical size Safe Safe Unsafe FH1 FH2 FH 3 Rogue flaw Assumed initial flaw FH3 [1] Hoffman, Paul C. Fleet management issues and technology needs. International Journal of Fatigue 31, no (November): Flight Hours 4

5 Methodology GOAL: Developing a hybrid prognostics methodology for health management consisting of the following modules: Physics-of-Failure (PoF) Model Non-Destructive Inspection (NDI)-based integrity assessment Knowledge Fusion Module Physics of Failure PoF Model-Based Prediction Non-destructive Inspection AE Monitoring of Crack Growth Feature Extraction Calibration Fatigue Crack Growth Parameters ( K, da/dn) Crack Size Distribution from AE Knowledge Fusion Hybrid Prognostic Model Bayesian Framework Knowledge Fusion Probabilistic AE-Based Diagnostic Inspection Field Data 5

6 Hybrid PHM Approach Hybrid Model for fleet management Pr( a > acritical ) Crack size (a) acrit * * * Meta model a = f ( FLE θ ) 0.01" Prediction for FH Day One (FLE 0%) FLE100% FLE(100+j)% Crack Growth PoF Model On-board NDI π (θ Evidence) = FLE i% Index of Life L( Evidence θ ) p (θ ) p (Evidence) Field Inspection 6

7 Acoustic Emission Monitoring 7

8 AE for Fatigue > Background Acoustic emissions are elastic stress waves generated by a rapid release of energy from localized sources within a material under stress [3]. Fig. from [2] mv Passive technique (good for detecting damage as it accumulates) Global monitoring and localization capability Only good for detecting active defects Highly susceptible to noise µsec AE waveform from crack growth [2] Huang, M. et al., Using acoustic emission in fatigue and fracture materials research. JOM, 50(11), 1-14 [3] Mix, P.E., Introduction to nondestructive testing: a training guide, Wiley-Interscience 8

AE for Fatigue > AE analysis Medium (Aluminum) Received AE Signal (Complex Waveform) Sensor 60 40 Deconvolution of the measured voltage signal from the sensor to evaluate the properties of the source

9 AE for Fatigue > AE analysis Medium (Aluminum) Received AE Signal (Complex Waveform) Sensor Deconvolution of the measured voltage signal from the sensor to evaluate the properties of the source event is extremely difficult. 20 mv Signal at Source (Pulse) µsec AE Features Amplitude Energy Rise time Counts (Threshold crossing) Frequency content Waveform shape 9

10 AE for Fatigue > Estimating Crack Growth Rate Objective: To correlate AE parameters with fatigue crack growth parameters Acoustic Emission Amplitude Energy Rise time Counts Frequency Waveform Fatigue Crack Growth Rate correlation K=f(stress,crack size) da/dn da dc log = β1 log + β2 dn dn One can estimate da/dn, given β1, β2 and AE count rate Log-log Scale AE Count Rate [4] Bassim, M.N., St Lawrence, S. & Liu, C.D., Detection of the onset of fatigue crack growth in rail steels using acoustic emission. ENG FRACT MECH, 47(2),

11 Experimental Procedure 11

12 AE for Fatigue > Experimental Procedure AE sensor Fatigue crack Loading Grips CT Specimen (7075-T6) Crack Growth Clip 12

13 AE for Fatigue > Experimental Procedure > Crack Size Measurement 13

Experimental Procedure > Noise Filtration 300 250 Load 200

of crack surfaces Crack closure Grip noise Other active

14 Experimental Procedure > Noise Filtration Load 200 Noise (AE from any source other that crack growth) Rubbing of crack surfaces Crack closure Grip noise Other active defects Cycles x 10 14

15 Experimental Procedure > Noise Filtration (Hit Type) 15

16 Experimental Procedure > Noise Filtration (Peak Freq.) 16

17 Experimental Procedure > Noise Filtration (Amplitude) 17

18 Experimental Procedure> Noise Filtration AE Discovery Tool developed in MATLAB 18

19 AE for Fatigue > Model Calibration > Bayesian Regression Model Y = β1x + β 2 + ε ε~normal(0,σ) Likelihood n L( D θ ) = i=1 1 y i ( β 1x i + β 2 ) 2 σ Updating parameters via MCMC 1 2 e 2π σ α α1 Integrating over parameters Probability density of da/dn For a given value of measured AE count rate and calibrated model parameters. f (Y X,D) = f (Y X,q )p (q D)dq ﾠﾠﾠ q q = {b1,b2,s } 19

20 AE-based Crack Size Prediction Predicted Growth Rate Input AE recording N Calibrated Model AE Signals ( N)i Feature Extraction (β1,β2 ) σ dc dn i Calibrated Model da dn i ( a)i 20

21 AE-based Crack Size Prediction Sources of Uncertainty: Stochastic nature of the model Initial crack distribution 21

22 Knowledge Fusion Physics of Failure PoF Model-Based Prediction Non-destructive Inspection AE Monitoring of Crack Growth Feature Extraction Calibration Fatigue Crack Growth Parameters ( K, da/dn) Crack Size Distribution from AE Knowledge Fusion Hybrid Prognostic Model Bayesian Framework Knowledge Fusion Probabilistic AE-Based Diagnostic Inspection Field Data 22

23 Knowledge Fusion > Multi-stage state updating 23

24 Knowledge Fusion > Bayesian Model Updating Acoustic Emission: Evidence Simulation: ﾠ (k ) DAE = {DAE k = 1K N e } (k ) DAE = {(x k, E (k ) ) E (k ) ~ f E ( k ) (e(k ) d;x k )} DSM = {(x j, y ij ) i = 1K N s, j =1K N x } ﾠ 24

25 Knowledge Fusion > Bayesian Model Updating > Model Structure Hierarchical Bayesian Model ψ φ λ E Hyper parameters Hyperprior (2nd level) Prior (1st level) Evidence Likelihoo d λ = {θ,σ} (First level prior) φ = { Μθ,Σθ } (Second level prior) Ψ = {MΜθ,ΣΜθ,Ω,η,α,β} (Vector of hyperparameters) Μ θ ~ Normal( Μ Μ θ,σ Μ θ ) Σ θ ~ Wishart (Ω,η) MM θ ΣMθ Ω,η Hyper parameters τ ~ Gamma( α, β ) θ ~ Normal( Μ θ,σ θ ) θ,τ Mθ,Σθ Hyperprior (2nd level) α,β Prior (1st level) Y ~ Lognormal( µ,τ ) µ = f ( X ;θ ) Y Likelihoo d Hyper parameters 25

26 Knowledge Fusion > Bayesian Model Updating > Inference The objective is to infer the model parameters from the simulation and the AE data. p( l,f DSM,DAE ) =? (1) (N e ) (1) (N ) p( l,f DSM,DAE ) = p( l,f DSM,DAE,K,DAE ) = p( l,f DSM, E,K, E e ) ﾠﾠﾠﾠf = E (1 ) (1) (1) (2) (N ) (1) (e ) p( l,f DSM,e, E,K, E e ) de e (1 ) = ﾠﾠf E (1) e (1 ) e ( 2) M = ﾠ e (1 ),K,e ( Ne ) ﾠ (1) (2) (1) (1) (2) (3) (N ) ( 2) (1) (e ). f E ( 2 ) E (1) (e e ).p( l,f DSM,e,e, E,K, E e ).de.de Ne f E (1) (e(1) ).ﾠ ( f E ( k ) E ( k- 1) (e(k ) e(k - 1) ).p( l,f DSM,e(1),e(2),K,e(N e ) ).de(k ) ) k =2 The correlation between AE data at subsequent time instances is captured in the conditional PDF terms that appear in the above equations. 26

27 Knowledge Fusion > Bayesian Model Updating > Inference Now using Bayes rule: p( l,f DSM,e(1),e(2),K,e(N e ) ) ﾵ p( DSM,e(1),e(2),K,e(N e ) l,f ) p( l,f ) = p( DSM l,f ).p(e(1) l,f ).K.p(e(N e ) l,f ).p( l,f ) ﾠ implification based on the independence of ﾠﾠ SM (k ) p E 1 E (k 2 ) (k ),?,? = p E 1?,?," k1,k 2 = 1,K,N e and DAE and conditional independence (1) (N e ) = p( DSMof El s).pgiven l model e l.p l f ).p(f ;y ) (e the ).K.p ( parameters, ﾠi.e.) ( (k) does not appear in the likelihood functions and by using the rules of conditional probability At this level, each of the likelihood terms, p(. λ), can be easily calculated as follows: Nx Ns p( DSM l ) = ﾠﾠ f Y ( y ij l ;x j ) j =1 i=1 p(e(k ) l ) = f Yk (e(k ) l ;x k ) 27

28 Knowledge Fusion > Bayesian Model Updating > Solution Approach DSM and DAE are independent so we can do sequential updating: First update using DSM, use the resulting posterior as the prior for updating with DAE For DAE, discretize the distribution of E(k) s and treat each resulting point as regular evidence. Perform the updating but (k ) (k - 1) weigh the resulting posterior using an appropriate f E E weight (e e ) calculated from the conditional distribution Bayesian updating at each step is performed via MCMC simulation ﾠ ( k) ( k- 1) We use WinBUGS software to find the posterior Weight calculation is performed in MATLAB Large computation time for large data sets and lower discretization error. 28

29 Summary A case study of using Bayesian fusion technique for integrating information from multiple sources in a structural health management problem was presented The simulation data was first used to find the model parameters and then, as crack size estimates from AE became available, the model parameters were updated in light of the new evidence. The mathematical formulation of the problem as well as the setup of the Bayesian inference solution was given. The solution includes treatment of uncertain evidence and also takes into account the correlation between AE observations. The resulting equations should be solved numerically. Efforts are still under way to provide an efficient computational solution to this problem 29

30 Thanks you! Questions? Comments? 30

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