Accuracy of flaw localization algorithms: application to structures monitoring using ultrasonic guided waves

Transcription

1 Première journée nationale SHM-France Accuracy of flaw localization algorithms: application to structures monitoring using ultrasonic guided waves Alain Le Duff Groupe Signal Image & Instrumentation (GSII), Groupe ESEO, Angers, France Laboratoire d Acoustique de l Université du Maine (LAUM UMR CNRS 6613), Le Mans, France

2 Alain Le Duff, Ing., PhD, HDR Who am I? 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 2

3 NDT & SHM: Work context Electronics Instrumentation Signal processing solutions NDT & SHM Materials Methods Aluminum Composite Concrete Active Guided waves, imaging Bio. Coda Wave Interferometry (CWI) Passive Acoustic emission 3

4 NDT & SHM: Work context Electronics NDT & SHM Materials Aluminum Instrumentation solutions Methods Composite Active Guided waves, imaging Coda Wave Interferometry (CWI) Signal processing Concrete Bio. Passive Acoustic emission TRL (Technology Readiness Level) : 2 6 3

5 SHM Work context - Guided waves (Lamb modes) in "plate" type structures; - Use of active ultrasound method; - Flaw localization in plate structures. Receiver #2 Transmitter Receiver #1 4

6 SHM Work context - Guided waves (Lamb modes) in "plate" type structures; - Use of active ultrasound method; - Flaw localization in plate structures. Receiver #2 Transmitter Receiver #1 4

7 Methodological approach #2 «Real world» - Physics - Acquisition - Noise - Scientist Engineer Physicist 5

8 Methodological approach #2 «Real world» - Physics - Acquisition - Noise - Scientist Engineer Physicist Observes 5

9 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Observes Scientist Engineer Physicist 5

10 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Observes Scientist Engineer Physicist Designs Signal processing 5

11 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Observes Experimental signals መθ = θ Scientist Engineer Physicist Designs Signal processing መθ 5

12 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Observes Models Scientist Engineer Physicist Designs Signal processing «Real world» model 5

13 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Observes Simulator Simulates Scientist Engineer Physicist Designs Signal processing Models «Real world» model 5

14 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Observes Simulator θ Simulates Scientist Engineer Physicist Designs Signal processing Models A priori Parameter(s) «Real world» model 5

15 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Observes Simulator θ Simulates Scientist Engineer Physicist Designs Signal processing A priori Parameter(s) Models «Real world» model Simulated signals 5

16 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Observes Simulator θ Simulates Scientist Engineer Physicist Designs Signal processing መθ A priori Parameter(s) Models «Real world» model Simulated signals መθ θ 5

17 Methodological approach #2 Simulator θ σ b Several noise realizations Statistical study of መθ Monte-Carlo simulations Signal processing መθ A priori Parameter(s) «Real world» model Simulated signals መθ θ 5

18 Methodological approach #2 Simulator θ σ b Several noise realizations Statistical study of መθ Monte-Carlo simulations Signal processing መθ A priori Parameter(s) «Real world» model Simulated signals መθ θ CRB (Expresses a lower bound on the variance of estimators of a deterministic parameter) 5

19 Methodological approach #2 Parameter(s) θ «Real world» - Physics - Acquisition - Noise - Experimental signals Signal processing መθ 5

20 Methodological approach #3 2 examples 1. Damage localization estimation 1. Temperature estimation 6

21 Methodological approach #3 2 examples 1. Damage localization estimation x? y? 1. Temperature estimation 6

22 Methodological approach #3 2 examples 1. Damage localization estimation x? y? 1. Temperature estimation θ 6

23 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) 7

24 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) 7

25 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) 7

26 Principle of localization estimation #1 Y S1 D(x, y) Problem: - damage localization - Isotropic plate - guided waves - TOF - Array of 3 transducers distributed on the struture. A0 S2 x 8

27 Principle of localization estimation #2 Distance between A0 and S1 t1 S1 (x 1, y 1 ) d 1 = d 1a + d 1s = c t 1 = (x x 0 ) 2 +(y y 0 ) 2 + (x x 1 ) 2 +(y y 1 ) 2 D(x, y) A0 (x 0, y 0 ) S2 (x 2, y 2 ) t2 Distance between A0 and S2 d 2 = d 2a + d 2s = c t 2 = (x x 0 ) 2 +(y y 0 ) 2 + (x x 2 ) 2 +(y y 2 ) 2 9

28 Principle of localization estimation #2 Distance between A0 and S1 t1 S1 (x 1, y 1 ) d 1 = d 1a + d 1s = c t 1 = (x x 0 ) 2 +(y y 0 ) 2 + (x x 1 ) 2 +(y y 1 ) 2 D(x, y) A0 (x 0, y 0 ) S2 (x 2, y 2 ) t2 Distance between A0 and S2 d 2 = d 2a + d 2s = c t 2 = (x x 0 ) 2 +(y y 0 ) 2 + (x x 2 ) 2 +(y y 2 ) 2 9

29 Principle of localization estimation #2 t1 S1 (x 1, y 1 ) Distance between A0 and S1 d 1 = d 1a + d 1s = c t 1 = (x x 0 ) 2 +(y y 0 ) 2 + (x x 1 ) 2 +(y y 1 ) 2 D(x, y) Objective of signal processing Extract the damage coordinates (x,y) from t1 & t2 t1 t2 Signal processing (x, y) σ t (x 0, y 0, x 1, y 1, x 2, y 2 ) A0 (x 0, y 0 ) S2 (x 2, y 2 ) t2 Distance between A0 and S2 d 2 = d 2a + d 2s = c t 2 = (x x 0 ) 2 +(y y 0 ) 2 + (x x 2 ) 2 +(y y 2 ) 2 9

30 Accuracy of localization estimation #1 S1 (x 1, y 1 ) D(x, y) ZOOM A0 (x 0, y 0 ) S2 (x 2, y 2 ) 10

31 Accuracy of localization estimation #2 Errors in time delay estimation error on damage position estimation (x, y) 11

32 Accuracy of localization estimation #2 Errors in time delay estimation error on damage position estimation (x, y) f(σ t ) 11

33 Accuracy of localization estimation #2 Errors in time delay estimation error on damage position estimation Uncertainty area (extend of error) Accuracy of the time delay measurement σ t 2 (estimation variance) (x, y) f(σ t ) 11

34 Accuracy of localization estimation #3 S1 A0 S2 12

35 Accuracy of localization estimation #3 S1 Relationship between «extend of error» and x, y, x 0, y 0, x 1, y 1, x 2, y 2 and σ t? A0 S2 12

36 Accuracy of localization estimation #3 S1 Relationship between «extend of error» and x, y, x 0, y 0, x 1, y 1, x 2, y 2 and σ t? A0 S2 Cramer-Rao Bounds (CRB) 12

37 Cramer-Rao Bounds (CRB) Give a lower bound for the covariance matrix of any unbiaised estimator 13

38 Cramer-Rao Bounds (CRB) Give a lower bound for the covariance matrix of any unbiaised estimator Indicate the lower limit of the variance of the estimations 13

39 Cramer-Rao Bounds (CRB) Give a lower bound for the covariance matrix of any unbiaised estimator Indicate the lower limit of the variance of the estimations Provide a benchmark against which the performance of estimators can be compared 13

40 Cramer-Rao Bounds (CRB) Give a lower bound for the covariance matrix of any unbiaised estimator Indicate the lower limit of the variance of the estimations 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 13

41 Exact analytical CRB #1 Observed data: t 1 = τ 1 + n 1 t 2 = τ 2 + n 2 Hypothesis : n 1 and n 2 are independant White Additive Gaussian Noises (WAGN) Time delay estimations variances G 1 x, y = d 1a d 1s G 2 x, y = d 2a d 2s Cylindrical attenuation σ t1 2 G 1 2 (x, y) σ v 2 σ t2 2 G 2 2 (x, y) σ v 2 Additive noise variance CRB x = c 2 σ t1 σ t2 K K 22 K 11 K 22 + K 12 K 2 21 with K 21 = y y 0 d 1a + y y 1 d 1s K 11 = x x 0 d 1a + x x 1 d 1s CRB y = c 2 σ t1 σ t2 K K 12 K 11 K 22 + K 12 K 2 21 K 22 = y y 0 d 2a + y y 2 d 2s K 12 = x x 0 d 2a + x x 2 d 2s 14

42 Exact analytical CRB #2 In summary: CRB(x, y)= f(x, y, x 0, y 0, x 1, y 1, x 2, y 2, σ v 2, c) Unknown damage location Actuator and sensors location Additive noise variance Sound velocity 15

43 Exact analytical CRB #3 Numerical validation Statistical performance of the procedure illustrated by MC simulations Several locations of the 3 sensors Monte-Carlo simulations Theoretical CRBs 16

44 Exact analytical CRB #3 Numerical validation Statistical performance of the procedure illustrated by MC simulations Several locations of the 3 sensors Monte-Carlo simulations Theoretical CRBs 16

45 Experimental results #1 67 N Principle of TOF measurement 17

46 Experimental results #1 67 N Principle of TOF measurement 17

47 Experimental results #1 67 N Principle of TOF measurement 17

48 Experimental results #1 67 N Principle of TOF measurement 17

49 Experimental results #1 67 N Principle of TOF measurement 17

50 Experimental results #1 67 N Principle of TOF measurement 17

51 Experimental results #2 Theoritical STD Experimental STD 18

52 Experimental results #2 Theoritical STD Experimental STD 18

53 Experimental results #2 Theoritical STD Experimental STD 18

54 Experimental results #2 Theoritical STD Experimental STD 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. 18

55 Context Example #2: temperature estimation Temperature estimation Scale factor estimation CWI (Coda Wave Interferometry) Proposition: Study of temperature change CRB Experimental assessement of CRB using GW Example : θ change x 0 t x 1 t x 0 αt T0 T1 θ estimation «Historical» estimators: cross-correlation - Stretching 19

56 cross-correlation maximization Scale transform based estimator Work done 1. Study of 4 scale factor estimators (including 2 original) x 1 t = x 0 (αt) Exponential transform x l 1 (t) = 1 α x (t + ln α) Time delay l0 ET ET 20

57 Algorithmic complexity and CRB Work completed 2. Algorithmic complexity study 3. Analytical CRB: as a function of the testing signal parameters and SNR 4. Validation Monte-Carlo simulation 21

58 Experimental validation #1 Application to aluminum Work completed (cont d) 5. Experimental validation (aluminum plate) 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) α = 1 K T T 1 T 0 22

59 Experimental validation #2 Application to composite material Material: glass-epoxy FR4 Estimator: short-time Xcorrelation Temperature controlled Embedded transducers Glass-Epoxy (FR4) composite plate Experimental set-up Embedded transducer Estimated Delay vs temperature. 23

60 Experimental validation #3 Study of the behaviour of concrete specimens under axial loading Temperature compensation using a reference specimen Application to concrete 24

61 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 25