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
Presentation at the MaxEnt2010
July 2010 Chamonix, France
Copyright 2012 by M. Modarres
2
Motivation and Background
3
P3 Aircraft Health Management
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)
Fatigue Cracks
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) approach has proved useful for fleet management. Shortcomings of PoF: 1. Limited knowledge about the underlying physics of failure 2. Scarcity of relevant material-level test data to estimate model
parameters 3. In practice, disconnected from the system being modeled (no
feedback)
∆FH1 ∆FH2 ∆FH3
Crack Size
Flight Hours
Safe Safe
Unsafe
∆FH’
3
Rogue flaw Assumed initial flaw
Critical size
[1] Hoffman, Paul C. Fleet management issues and technology needs. International Journal of Fatigue 31, no. 11-12 (November): 1631-1637.
5
Methodology
AEMonitoring
of CrackGrowth
Fatigue CrackGrowth
Parameters(K, da/dN)
FeatureExtraction
Calibration
Crack SizeDistribution
from AE
InspectionField Data
ProbabilisticAE-BasedDiagnostic
HybridPrognostic
Model
PoFModel-Based
Prediction
KnowledgeFusion
BayesianFramework
Non-destructive Inspection
Physics of Failure
Knowledge Fusion
GOAL: Developing a hybrid prognostics methodology for health management consisting of the following modules: Physics-of-Failure (PoF) Model NDI-based structural integrity assessment Knowledge Fusion Module
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Hybrid PHM Approach
|FLEfa
Crack Growth PoF Model
On-board NDI
Field Inspection
Index of Life FLE100%
0.01"
FLE(100+j)%
acrit
Day One (FLE 0%)
Crack size (a)
FLE i %
Prediction for FH
Pr( a > acritical )
* * *
Hybrid Model for fleet management
Meta model
)Evidence()(|EvidenceLEvidence|
pp
7
Acoustic Emission Monitoring
8
AE for Fatigue > Background
[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
Fig. from [2]
Acoustic emissions are elastic stress waves generated by a rapid release of energy from localized sources within a material under stress [3].
-100 0 100 200 300 400 500-60
-40
-20
0
20
40
60
Sec
mV
AE waveform from crack growth
Passive technique (good for detecting damage as it accumulates) Global monitoring and localization capability Only good for detecting active defects Highly susceptible to noise
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AE for Fatigue > AE analysis
Signal at Source (Pulse)
Received AE Signal (Complex Waveform)
Medium (Aluminum)
Sensor
-100 0 100 200 300 400 500-60
-40
-20
0
20
40
60
Sec
mV
Deconvolution of the measured voltage signal from the sensor to evaluate the properties of the source event is extremely difficult.
AE Features Amplitude Energy Rise time Counts (Threshold crossing)
Frequency content Waveform shape
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AE for Fatigue > Estimating Crack Growth Rate
Objective: To correlate AE parameters with fatigue crack growth parameters
Amplitude Energy Rise time Counts Frequency Waveform
Crack Growth Rate K=f(stress,crack size)
[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.
One can estimate da/dN, given b1, b2 and AE count rate
da/dN
AE Count Rate
Log-log Scale
21 loglog bb
dNdc
dNda
Acoustic Emission
Fatigue
correlation
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Experimental Procedure
12
AE for Fatigue > Experimental Procedure
AE sensor
Fatigue crack
CT Specimen (7075-T6)
Loading Grips
• Crack Growth Clip
13
AE for Fatigue > Experimental Procedure > Crack Size Measurement
14
Experimental Procedure > Noise Filtration
2.506 2.508 2.51 2.512 2.514 2.516
x 104
50
100
150
200
250
300
Cycles
Lo
ad
Noise (AE from any source other that crack growth) Rubbing of crack surfaces Crack closure Grip noise Other active defects
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Experimental Procedure > Noise Filtration (Hit Type)
16
Experimental Procedure > Noise Filtration (Peak Freq.)
17
Experimental Procedure > Noise Filtration (Amplitude)
18
Experimental Procedure> Noise Filtration
AE Discovery Tool developed in MATLAB
19
AE for Fatigue > Model Calibration > Bayesian Regression
Probability density of da/dN For a given value of measured AE count rate and calibrated model parameters.
Likelihood
Updating parameters via MCMC
Integrating over parameters
bb
,
21
,
dD|,X|YfD,X|Yf
n
1i
2)x(y21 2i1i
e21|DL
bb
1
2
0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02 0.0205 0.021 0.0215
1.6
1.65
1.7
1.75
1000
2000
3000
4000
5000
6000
7000
8000
9000
),σ0Normal(ε~εXY 21 bb
Model
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AE-based Crack Size Prediction
Input AE recording
N)i
FeatureExtraction
AESignals
idNdc
CalibratedModel
idNda
a)i
b1,b2
Calibrated Model
Predicted Growth Rate
N
21
AE-based Crack Size Prediction
Sources of Uncertainty: Stochastic nature of the model Initial crack distribution
22
Knowledge Fusion
AEMonitoring
of CrackGrowth
Fatigue CrackGrowth
Parameters(K, da/dN)
FeatureExtraction
Calibration
Crack SizeDistribution
from AE
InspectionField Data
ProbabilisticAE-BasedDiagnostic
HybridPrognostic
Model
PoFModel-Based
Prediction
KnowledgeFusion
BayesianFramework
Non-destructive Inspection
Physics of Failure
Knowledge Fusion
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Knowledge Fusion > Multi-stage state updating
Knowledge Fusion > Bayesian Model Updating
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xsijjSM NjNiyxD 1,1|),(
k
kE
kkk
kAE
ek
AEAE
xefEExD
NkDD
k ;|~|),(
1|
)()()()(
)(
)(
Acoustic Emission:
Simulation:
Evidence
Knowledge Fusion > Bayesian Model Updating > Model Structure
25
MM SM
,t Y
Hyper parameters
Hyperprior (2nd level)
Prior (1st level)
Likelihood
M ,S
W , h
, b Hyper parameters
);(,Lognormal~
t
XfY
bt ,Gamma~ θθ ,ΣΜNormal~
)(Wishart~
)(Normal~
Ω,ηΣ
,ΣΜΜ
θ
ΜΜθ θθ
l = {,} (First level prior)
f = { M,S } (Second level prior)
Y = {MM,SM,W,h,,b} (Vector of hyperparameters)
Hierarchical Bayesian Model
y f l E
Hyper parameters
Hyperprior (2nd level)
Prior (1st level)
Evidence Likelihood
Knowledge Fusion > Bayesian Model Updating > Inference
26
The objective is to infer the model parameters from the simulation and the AE data.
)()1()()1( ,,,|,,,,|,,|, ee NSM
NAEAESMAESM EEDpDDDpDDp flflfl
?,|, AESM DDp fl
)1(
)1()1()()2()1()1( ,,,,|,
e
NSME deEEeDpef efl
)()1(
)1()()1(
)1( )2(
)1()2()1(
,, 2
)()()2()1()1()(|
)1(
)1()2()()3()2()1()1()2(|
)1(
.,,,,|,.|.
..,,,,,|,.|.
eN
ee
kk
e
ee
N
k
kNSM
kkEEE
e e
NSMEEE
deeeeDpeefef
dedeEEeeDpeefef
fl
fl
The correlation between AE data at subsequent time instances is captured in the conditional PDF terms that appear in the above equations.
Knowledge Fusion > Bayesian Model Updating > Inference
Now using Bayes’ rule:
27
flflfl ,,|,,,,,,,,|, )()2()1()()2()1( peeeDpeeeDp ee NSM
NSM
Simplification based on the independence of DSM and DAE and conditional independence of E(k)’s given the model parameters, i.e.
e)(k)(k)(k ,N,,kk,φ,λ|Epφ,λ,|EEp 121
121
yffllll ; .|.|..|.| )()1( ppepepDp eNSM
flflflfl ,.,|..,|.,| )()1( pepepDp eNSM
f does not appear in the likelihood functions and by using the rules of conditional probability
Nx
j
Ns
ijijYSM xyfDp
1 1
;|| ll
kk
Yk xefep
k;|| )()( ll
At this level, each of the likelihood terms, p( .| l), can be easily calculated as follows:
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 weigh the resulting posterior using an appropriate weight calculated from the conditional distribution Bayesian updating at each step is performed via MCMC simulation 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.
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)1()(|
|)1()(
kkEE eef kk
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
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Thanks you! Questions? Comments?
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