Accuracy of flaw localization algorithms: application to ...€¦ · NDT & SHM Materials Methods...

Alain Le Duff

Groupe Signal Image & Instrumentation (GSII), Groupe ESEO, Angers, FranceLaboratoire d’Acoustique de l’Université du Maine (LAUM UMR CNRS 6613), Le Mans, France

Première journée nationale SHM-France

Accuracy of flaw localization algorithms: application to structures monitoring using

ultrasonic guided waves

2

Who am I?

• Alain Le Duff, Ing., PhD, HDR

• Teacher-researcher at ESEO, Angers, France– Head of the Department of Electronics and Control

Engineering

– Member of the « Signal, Image & Instrumentation Group » (GSII)

• Research fellow at LAUM, Le Mans, France– Instrumentation and signal processing for acoustics

– Field of applications : LDV, Musical acoustics, ENDT, SHM

Alain Le Duff - First National Day SHM France - March, 15th, 2018, Saclay, France

NDT & SHM

Materials

Methods

PassiveAcoustic emission

ActiveGuided waves, imaging

Coda Wave Interferometry

(CWI)

Composite

Concrete

Bio.

Aluminum

Electronics

solutions

Signal processing

Instrumentation

3

NDT & SHM: Work context

NDT & SHM

Materials

Methods

PassiveAcoustic emission

ActiveGuided waves, imaging

Coda Wave Interferometry

(CWI)

Composite

Concrete

Bio.

Aluminum

Electronics

solutions

Signal processing

Instrumentation

TRL (Technology Readiness Level) : 2 6 3

NDT & SHM: Work context

- Guided waves (Lamb modes) in "plate" type structures;- Use of active ultrasound method;- Flaw localization in plate structures.

TransmitterReceiver #1

Receiver #2

Alain Le Duff - First National Day SHM France - March, 15th, 2018, Saclay, France 4

SHM Work context

- Guided waves (Lamb modes) in "plate" type structures;- Use of active ultrasound method;- Flaw localization in plate structures.

TransmitterReceiver #1

Receiver #2


SHM Work context

ScientistEngineerPhysicist

«Real world»- Physics- Acquisition- Noise- …

5

Methodological approach #2



Observes


5




Observes


𝜃Parameter(s)

5



Signalprocessing


Observes

Designs


𝜃Parameter(s)

5



Signalprocessing


Observes

Designs


𝜃Parameter(s)

መ𝜃

Experimentalsignals

መ𝜃 = 𝜃

5



Signalprocessing


Observes

«Real world»model

Models

Designs


𝜃Parameter(s)

5



Signalprocessing


Observes

«Real world»model

Models

Designs


𝜃Parameter(s)

SimulatesSimulator

5



Signalprocessing


Observes

«Real world»model

Models

Designs


𝜃Parameter(s)

SimulatesSimulator

𝜃

A priori Parameter(s)

5



Signalprocessing


Observes

«Real world»model

Models

Designs

Simulatedsignals


𝜃Parameter(s)

SimulatesSimulator

𝜃


5



Signalprocessing


Observes

«Real world»model

Models

Designs

Simulatedsignals


𝜃Parameter(s)

SimulatesSimulator

𝜃


መ𝜃 ≠ 𝜃

መ𝜃

5



Signalprocessing

«Real world»model

Simulatedsignals

Simulator

𝜃


መ𝜃 ≠ 𝜃

መ𝜃𝝈𝒃

Several noise realizations

Statistical study of መ𝜃↓

Monte-Carlo simulations

5



Signalprocessing

«Real world»model

Simulatedsignals

Simulator

𝜃


መ𝜃 ≠ 𝜃

መ𝜃𝝈𝒃

Several noise realizations

Statistical study of መ𝜃↓

Monte-Carlo simulations

CRB (Expresses a lower bound on the variance of estimators of a deterministic parameter)

5



Signalprocessing


𝜃Parameter(s)

መ𝜃

Experimentalsignals

5




• 2 examples

1. Damage localization estimation

1. Temperature estimation



• 2 examples




𝑥 ?

𝑦 ?


• 2 examples




𝑥 ?

𝑦 ?

𝜃

Example #1: damage localization in a plate

• Context– Array of piezoceramic sensors and actuators

– TOF measurements

• Localization accuracy depends on– Spatial distribution of the transducers

– Signal to Noise Ratio (SNR) of acquired signals

– Defect location itself

• Proposition:

– Study of the of damage localization → CRB

– Experimental assessement of CRB in an aluminium plate using Guided Waves (GW)








• Proposition:










• Proposition:




8Alain Le Duff - First National Day SHM France - March, 15th, 2018, Saclay, France

A0

S1

S2x

Y

𝐷(𝑥, 𝑦)

Principle of localization estimation #1

Problem: - damage localization- Isotropic plate- guided waves- TOF- Array of 3 transducers

distributed on the struture.

9

A0

S1

S2

t1

t2


Distance between A0 and S1

𝑑1 = 𝑑1𝑎 + 𝑑1𝑠 = 𝑐 ∙ 𝑡1

= (𝑥 − 𝑥0 )2+(𝑦 − 𝑦0 )

2 + (𝑥 − 𝑥1 )2+(𝑦 − 𝑦1 )2


𝑑2 = 𝑑2𝑎 + 𝑑2𝑠 = 𝑐 ∙ 𝑡2

= (𝑥 − 𝑥0 )2+(𝑦 − 𝑦0 )

2 + (𝑥 − 𝑥2 )2+(𝑦 − 𝑦2 )

2

(𝑥1, 𝑦1)

(𝑥0, 𝑦0) (𝑥2, 𝑦2)

𝐷(𝑥, 𝑦)


9

A0

S1

S2

t1

t2



𝑑1 = 𝑑1𝑎 + 𝑑1𝑠 = 𝑐 ∙ 𝑡1

= (𝑥 − 𝑥0 )2+(𝑦 − 𝑦0 )

2 + (𝑥 − 𝑥1 )2+(𝑦 − 𝑦1 )2


𝑑2 = 𝑑2𝑎 + 𝑑2𝑠 = 𝑐 ∙ 𝑡2

= (𝑥 − 𝑥0 )2+(𝑦 − 𝑦0 )

2 + (𝑥 − 𝑥2 )2+(𝑦 − 𝑦2 )

2

(𝑥1, 𝑦1)

(𝑥0, 𝑦0) (𝑥2, 𝑦2)

𝐷(𝑥, 𝑦)


9

A0

S1

S2

t1

t2



𝑑1 = 𝑑1𝑎 + 𝑑1𝑠 = 𝑐 ∙ 𝑡1

= (𝑥 − 𝑥0 )2+(𝑦 − 𝑦0 )

2 + (𝑥 − 𝑥1 )2+(𝑦 − 𝑦1 )2


𝑑2 = 𝑑2𝑎 + 𝑑2𝑠 = 𝑐 ∙ 𝑡2

= (𝑥 − 𝑥0 )2+(𝑦 − 𝑦0 )

2 + (𝑥 − 𝑥2 )2+(𝑦 − 𝑦2 )

2

(𝑥1, 𝑦1)

(𝑥0, 𝑦0) (𝑥2, 𝑦2)

𝐷(𝑥, 𝑦)

Objective of signal processing

Signal processing

t1

t2(𝑥, 𝑦)

(𝑥0, 𝑦0, 𝑥1, 𝑦1, 𝑥2, 𝑦2)

Extract the damage coordinates(x,y) from t1 & t2

𝝈𝒕


10

A0

S1

S2

ZOOM


(𝑥1, 𝑦1)

(𝑥0, 𝑦0) (𝑥2, 𝑦2)

𝐷(𝑥, 𝑦)

Accuracy of localization estimation #1


(𝒙, 𝒚)

Errors in time delay estimation

↓error on damage position

estimation


11

𝒇(𝝈𝒕)


(𝒙, 𝒚)



estimation


11

𝒇(𝝈𝒕)


(𝒙, 𝒚)



estimation

Uncertainty area(extend of error)

↓Accuracy of the time delay

measurement 𝜎𝑡2

(estimation variance)


12

A0

S1

S2



12

A0

S1

S2


Relationship between« extend of error » and𝑥, 𝑦, 𝑥0, 𝑦0, 𝑥1, 𝑦1, 𝑥2, 𝑦2

and 𝜎𝑡?


12

A0

S1

S2

Cramer-Rao Bounds(CRB)


Relationship between« extend of error » and𝑥, 𝑦, 𝑥0, 𝑦0, 𝑥1, 𝑦1, 𝑥2, 𝑦2

and 𝜎𝑡?


13

Cramer-Rao Bounds (CRB)


• Give a lower bound for the covariance matrix of any unbiaisedestimator

13




• Indicate the lower limit of the variance of the estimations

13





• Provide a benchmark against which the performance of estimators can be compared

13





• Provide a benchmark against which the performance of estimators can be compared

• In this work: the CRBs give an indication on the a priori accuracy of the localization procedure

14

Exact analytical CRB #1


Observed data:

Hypothesis : 𝑛1and 𝑛2 are independant White Additive Gaussian Noises (WAGN)

𝐾11 =𝑥 − 𝑥0𝑑1𝑎

+𝑥 − 𝑥1𝑑1𝑠

𝐾12 =𝑥 − 𝑥0𝑑2𝑎

+𝑥 − 𝑥2𝑑2𝑠

𝐾21 =𝑦 − 𝑦0𝑑1𝑎

+𝑦 − 𝑦1𝑑1𝑠

𝐾22 =𝑦 − 𝑦0𝑑2𝑎

+𝑦 − 𝑦2𝑑2𝑠

CRB 𝑥 = 𝑐2𝜎𝑡1𝜎𝑡2 ∙𝐾212 + 𝐾22

2

𝐾11 ∙ 𝐾22 + 𝐾12 ∙ 𝐾212

CRB 𝑦 = 𝑐2𝜎𝑡1𝜎𝑡2 ∙𝐾112 + 𝐾12

2

𝐾11 ∙ 𝐾22 + 𝐾12 ∙ 𝐾212

𝜎𝑡12 ∝ 𝐺1

2(𝑥, 𝑦) ∙ 𝜎𝑣2

𝜎𝑡22 ∝ 𝐺2

2(𝑥, 𝑦) ∙ 𝜎𝑣2𝐺1 𝑥, 𝑦 = 𝑑1𝑎𝑑1𝑠 𝐺2 𝑥, 𝑦 = 𝑑2𝑎𝑑2𝑠

Time delay estimations variances

Additive noise variance

Cylindrical attenuation

with

𝑡1 = 𝜏1 + 𝑛1 𝑡2 = 𝜏2 + 𝑛2


Exact analytical CRB #2

In summary:

CRB(𝑥, 𝑦)= 𝑓(𝑥, 𝑦, 𝑥0, 𝑦0, 𝑥1, 𝑦1, 𝑥2, 𝑦2, 𝜎𝑣2, c)

Unknown damage location

Actuator and sensors location

Additive noise variance

Sound velocity


Exact analytical CRB #3Numerical validation• Statistical performance of the procedure illustrated by MC simulations• Several locations of the 3 sensors

Monte-Carlo simulations Theoretical CRBs


Exact analytical CRB #3Numerical validation• Statistical performance of the procedure illustrated by MC simulations• Several locations of the 3 sensors

Monte-Carlo simulations Theoretical CRBs

67 N

17

Experimental results #1


Principle of TOF measurement

67 N

17




67 N

17




67 N

17




67 N

17




67 N

17




18



• Theoritical STD• Experimental STD

18




18




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• Performance of the estimations follows the theoretical STD• CRBs good indication of the expected accuracy of a particular configuration of a SHM system

• as a function of the geometry of a transducers array;• and SNR of the data acquisition.

19

𝑥0 𝑡 → 𝑥1 𝑡 ≈ 𝑥0 𝛼𝑡Example : 𝜃 changeT0 T1

𝜽 estimation

« Historical » estimators: cross-correlation - Stretching

Example #2: temperature estimation

• Proposition:

– Study of temperaturechange → CRB

– Experimentalassessement of CRB using GW

• Context– Temperature estimation

– Scale factor estimation CWI (Coda WaveInterferometry)


20

𝑥1 𝑡 = 𝑥0(𝛼𝑡)

𝑥𝑙1(𝑡) =1

𝛼𝑥𝑙0(𝑡 + ln𝛼) Time delay

Exponential transform

cro

ss-c

orr

elat

ion

ET

ET

max

imiz

atio

n


Scale transform based estimator

Work done1. Study of 4 scale factor estimators (including 2 original)

Work completed2. Algorithmic complexity study3. Analytical CRB: as a function of the testing signal parameters and SNR4. Validation Monte-Carlo simulation


Algorithmic complexity and CRB

𝛼 = 1 − 𝐾𝑇 𝑇1 − 𝑇0

Contribution to the scientific community• CRB → influence of

parameters on accuracy• Methods comparison→ selection guide

Other works• Application to composite materials• Temperature compensation

- Concrete (IFSTTAR)- Aluminum (GAUS)

Work completed (cont’d)5. Experimental validation (aluminum plate)

Application to aluminum


Experimental validation #1

Glass-Epoxy (FR4) composite plate

Embedded transducer

Experimentalset-up• Material: glass-epoxy FR4

• Estimator: short-time Xcorrelation• Temperature controlled• Embedded transducers

Estimated Delay vs temperature.

Application to composite material



• Study of the behaviour of concrete specimens under axial loading• Temperature compensation using a reference specimen

Application to concrete



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Conclusions


• CRBs give a good indication of the expected accuracy of a particular configuration of a SHM system as a function of- the geometry of a transducers array;- the SNR of the data acquisition.

• The approximated expression for the CRB provides a way to select optimal values for the signal parameters, especially for the sampling period

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Accuracy of flaw localization algorithms: application to ...€¦ · NDT & SHM Materials Methods...

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