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
Motivation and Background
2
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
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. 11-12 (November): 1631-1637.
Flight Hours 4
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
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
Acoustic Emission Monitoring
7
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]
60
40
mV
20
0
-20
-40
-60 -100
0
100
200
300
400
500
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., 1998. Using acoustic emission in fatigue and fracture materials research. JOM, 50(11), 1-14 [3] Mix, P.E., 2005. 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 event is extremely difficult.
20
mV
Signal at Source (Pulse)
0
-20
-40
-60 -100
0
100
200
300
400
500
µSec
AE Features Amplitude Energy Rise time Counts (Threshold crossing) Frequency content Waveform shape
9
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., 1994. Detection of the onset of fatigue crack growth in rail steels using acoustic emission. ENG FRACT MECH, 47(2), 207-214.
10
Experimental Procedure
11
AE for Fatigue > Experimental Procedure
AE sensor Fatigue crack
Loading Grips
CT Specimen (7075-T6) • Crack Growth Clip 12
AE for Fatigue > Experimental Procedure > Crack Size Measurement
13
Experimental Procedure > Noise Filtration
300
250
Load
200
Noise (AE from any source other that crack growth) Rubbing of crack surfaces Crack closure Grip noise Other active defects
150
100
50
2.506
2.508
2.51 Cycles
2.512
2.514
2.516 4
x 10
14
Experimental Procedure > Noise Filtration (Hit Type)
15
Experimental Procedure > Noise Filtration (Peak Freq.)
16
Experimental Procedure > Noise Filtration (Amplitude)
17
Experimental Procedure> Noise Filtration
AE Discovery Tool developed in MATLAB
18
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
9000
8000 1.75 7000
6000
α2
1.7
5000
1.65
4000
3000 1.6 2000
0.017
0.0175
0.018
0.0185
0.019
0.0195
0.02
0.0205
0.021
0.0215
1000
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
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
AE-based Crack Size Prediction
Sources of Uncertainty:
Stochastic nature of the model Initial crack distribution
21
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
Knowledge Fusion > Multi-stage state updating
23
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
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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
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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 )
ﾠ ﾠ ﾠ
ﾠf
=
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.
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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
ﾠ
ﾠ
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
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
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
Thanks you! Questions? Comments? Email:
[email protected]
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