Bayesian Knowledge Fusion in Prognostics and Health Management—A Case Study

Masoud RabieiMohammad Modarres

Center for Risk and ReliabilityUniversity of Maryland-College Park

Ali Moahmamd-Djafari Laboratoire des Signaux et Systèmes, UMR

8506 CNRS - SUPELEC - Univ Paris Sud 11

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) or mechanistic-life approach has proved useful for fleet management.Shortcomings of PoF:

Limited knowledge about the underlying PoFScarcity of relevant material-level test data to estimate model parametersIn practice, disconnected from the system being modeled (no feedback)

∆FH1 ∆FH2∆FH3

Crack Size

Flight Hours

Safe Safe

Unsafe

∆FH’3

Rogue flawAssumed 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 Non-Destructive Inspection (NDI)-based integrity assessment Knowledge Fusion Module

6

Hybrid PHM Approach

( )θ|FLEfa =

Crack Growth PoF Model

On-board NDI

Field Inspection

Index of LifeFLE100%

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|

p

p θθθπ =

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

µSec

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

9

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

µSec

mV

Deconvolution of the measured voltage signal from the sensor to evaluate the properties of the source event is extremely difficult.

AE FeaturesAmplitude

Energy

Rise time

Counts (Threshold crossing)

Frequency content

Waveform shape

10

AE for Fatigue > Estimating Crack Growth Rate

Objective: To correlate AE parameters with fatigue crack growth parameters

AmplitudeEnergyRise timeCountsFrequency 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 β1, β2 and AE count rate

da/dN

AE Count Rate

Log-log Scale

21 loglog ββ +

=

dN

dc

dN

da

Acoustic Emission

Fatigue

correlation

11

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

Load

Noise (AE from any source other that crack growth)

Rubbing of crack surfaces

Crack closure

Grip noise

Other active defects

15

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

ﾠ

f Y | X,D( ) = fq

ﾠﾠﾠY | X,q( )p q | D( )dq

q = b1,b2,s

( ) ∏=

+−−

=n

1i

2)x(y

2

1 2i1i

e2

1|DL σ

ββ

σπθ

α1

α 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 ++= ββModel

20

AE-based Crack Size Prediction

Input AE recording

(∆N)i

FeatureExtraction

AESignals

idN

dc

CalibratedModel

idN

da

(∆a)i

(β1,β2)σ

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

23

Knowledge Fusion > Multi-stage state updating

Knowledge Fusion > Bayesian Model Updating

24

ﾠ

DSM = (x j , y ij ) | i =1K Ns, j =1K Nx

ﾠ

DAE = DAE(k ) | k =1K Ne

DAE(k ) = (xk, E (k )) | E (k ) ~ f

E ( k ) e(k ) |d;xk( ) Acoustic Emission:

Simulation:

Evidence

Knowledge Fusion > Bayesian Model Updating > Model Structure

25

MMθ

ΣMθ θ ,τ Y

Hyper parameters

Hyperprior (2nd level)

Prior (1st level)

Likelihood

Mθ ,Σθ

Ω , η

α , βHyper parameters

( ));(

,Lognormal~

θµτµ

Xf

Y

=

( )βατ ,Gamma~( )θθ ,ΣΜNormal~θ

)(Wishart~

)(Normal~

Ω,ηΣ

,ΣΜΜ

θ

ΜΜθ θθ

λ = θ,σ (First level prior)

φ = Μθ,Σθ (Second level prior)

Ψ = MΜθ,ΣΜθ,Ω,η,α,β (Vector of hyperparameters)

Hierarchical Bayesian Model

ψ φ λ E

Hyper parameters

Hyperprior (2nd level)

Prior (1st level)

Evidence Likelihoo

d

Knowledge Fusion > Bayesian Model Updating > Inference

26

The objective is to infer the model parameters from the simulation and the AE data.

ﾠ

p l ,f | DSM ,DAE( ) = p l ,f | DSM ,DAE(1) ,K ,DAE

(Ne )( ) = p l ,f | DSM , E (1),K , E (Ne )

( )

ﾠ

p l ,f | DSM ,DAE( ) = ?

ﾠ

= fE (1) e(1)

( )p l ,f | DSM ,e(1),E (2),K ,E (Ne )( )de(1)

e (1)

ﾠ

ﾠ

= fE (1) e(1)

( ). fE ( 2) |E (1) e(2) | e(1)

( ).p l ,f | DSM ,e(1),e(2), E (3),K , E (Ne )( ).de(2).

e ( 2)

ﾠ de(1)

e (1)

ﾠ

M

= fE (1) e(1)

( ). fE ( k ) |E ( k- 1) e(k ) | e(k - 1)

( ).p l ,f | DSM ,e(1),e(2),K ,e(Ne )( ).de(k )

( )k =2

Ne

ﾠe (1) ,K ,e ( Ne )

ﾠ

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

ﾠ

p l ,f | DSM ,e(1),e(2),K ,e(Ne )( ) ﾵ p DSM ,e(1),e(2),K ,e(Ne ) | l ,f( )p l ,f( )

Simplification based on the independence of

D

SM and DAE and conditional independence

of E(k)’s given the model parameters, i.e.

ﾠ

p E(k1)

|E(k2)

,?,?

= p E

(k1)|?,?

," k1,k2 =1,K ,Ne

ﾠ

= p DSM | l( ).p e(1) | l( ).K .p e(Ne ) | l( ).p l | f( ).p f ;y( )

ﾠ

= p DSM | l ,f( ).p e(1) | l ,f( ).K .p e(Ne ) | l ,f( ).p l ,f( )

φ does not appear in the likelihood functions and by using the rules of conditional probability

ﾠ

p DSM | l( ) = fY y ij | l ;x j( )i=1

Ns

ﾠj =1

Nx

ﾠ

ﾠ

p e(k ) | l( ) = fYke(k ) | l ;xk( )

At this level, each of the likelihood terms, p( .| λ), 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.

28

ﾠ

fE ( k ) |E ( k- 1) e(k ) | e(k - 1)

( )

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?

Email: masrab@umd.edu