Merging of Experimental and Simulated Data Sets with a Bayesian Technique in the Context of POD Curves Determination

Transcription

1 5 th Euroean-American Worksho on Reliability of NDE Lecture 5 Merging of Exerimental and Simulated Data Sets with a Bayesian Technique in the Context of POD Curves Determination Bastien CHAPUIS *, Nicolas DOMINGUEZ *, Frédéric JENSON *, Pierre WILLAUME **, Thierry YALAMAS ** * CEA LIST Centre de Saclay, France ** PHIMECA Engineering SA, Cournon d'auvergne, France Abstract POD curves are usually described using a arametric model which relates the robability to detect a secific defect to one of its geometrical characteristics, usually its size. The arameters of the model are estimated following a statistical rocedure alied to a set of insection results which are obtained thru dedicated exerimental camaigns. Statistical significance requires that 6 to 8 mock-us containing realistic flaws are fabricated and insected by several insectors. This costly and time consuming rocess must be done for each NDT rocedure for which a measure of NDT reliability is required. Consequently, cost and time reduction of POD trials is currently a major issue. One way to achieve cost reduction is to relace some of the required exerimental data with numerical simulation results. This idea follows the concet of Model Assisted POD (MAPOD). POD curves are no longer estimated from a fully emirical dataset but rather from a mix of exerimental and simulated data. Simulations are erformed using hysics-based models, whose redictions are validated for the considered alication case. In order to make the aroach suitable for industrial needs, it is required that uncertainties introduced in the rocess thru the merging of simulation and exerimental data are assessed. In this resentation, a statistical method based on Bayesian udating is roosed, which mixes numerical simulations and information brought by the measurements. Traditionally, POD curves are assessed using Maximum Likelihood Estimation methods using either hit/miss or signal resonse data. This article only deals with hit/miss data. Following Berens article, the POD is modelled by a log-logistics function. Hit/miss data is treated as a Bernoulli s variable and Bayesian udating is erformed on the POD model to assess the osterior distributions of the POD arameters assuming non-informative rior distributions on them. Finally, Monte Carlo simulations are run to assess the confidence band POD. A ractical imlementation of the aroach to a high frequency eddy current insection for fatigue cracks is resented. License: htt://creativecommons.org/licenses/by-nd/3./ 1

2 References [1] F. Jenson, N. Dominguez, P. Willaume and T. Yalamas, A Bayesian aroach for the determination of POD curves from emirical data merged with simulation results, AIP Conf. Proc. 1511, 1741 (213), htt://dx.doi.org/1.163/

3 Merging of Exerimental and Simulated Data Sets with a Bayesian Technique in the Context of POD Curves Determination Seaker: B. Chauis (1) Authors: F. Jenson (1), N. Dominguez (1), P. Willaume (2), T. Yalamas (2) (1) CEA, LIST, France (2) Phimeca, France October, 7th 213 Background Background Insection reliability is one of the key issues in ensuring safety of critical structural comonents Increasing use of robabilistic aroaches based on statistical criteria such as POD curves and PFAs Are currently obtained thru exensive and time consuming exerimental camaigns Aroaches were roosed to relace some of the exerimental data with simulation results See: MAPOD working grou at The PICASSO euroean roject at Technique in the Context of POD Curves Determination 2 3

4 Objectives Objectives of the study Define a Bayesian aroach to POD curve determination (current aroach relies on classical frequentist statistics) Determine POD curves from emirical data merged with numerical simulation data Definitions Emirical ~ exerimental Data ~ insection results Technique in the Context of POD Curves Determination 3 Outline A Bayesian framework for the determination of POD curves Estimation of POD curves using a functionnal form (Berens aroach) Focus on the hit/data format case Descrition of the Bayesian udating scheme Bayesian aroach to data merging First ste: Bayesian udate with simulation data Alication case: HFET of fatigue cracks in Titanium Simulation suorted POD curve determination (revious results) Second ste: Bayesian udate with emirical data Conclusions and future work Technique in the Context of POD Curves Determination 4 4

5 POD curves relate the detectabilityof a flaw to its size (or to another geometrical characteristic) Usual aroach (MIL-HDBK 1823) consist in: Descrition of the Bayesian framework Estimation of the POD curve using a functional form assuming a functional form for the POD curve estimating the arameters of the function from the insection results estimating the associated confidence bound Berens Hit/miss data format: Log-odds function Signal resonse data format: Cumulative log-normal distribution function Cheng Estimated from a finite size data samle samling uncertainty confidence bound Technique in the Context of POD Curves Determination 5 Descrition of the Bayesian framework Estimation of the POD curve for hit/miss data Hit miss data: log odds model Which can also be exressed as: ex( α + β ln a) POD( a) = 1+ ex( α + β ln a) POD ( a) ln = + ln( a) 1 POD ( a) α β Estimation of αand βbased on the rincile of maximizing the likelihood function of the arameters given an observed samle of measurements (a i, a 1/ ) The likelihood function of the arameters given the data is: L( α, β ) = i= 1 j= 1 n cracks k insections Z ij = or 1 for insection j of crack i i is the robability value to detect crack i. The value is given by the log odds model: = POD a ) i ( i n k Zij i (1 1 Z ij i ) It indicates how likely chosen values of the arameters are, in light of the observed outcome Technique in the Context of POD Curves Determination 6 5

6 Descrition of the Bayesian framework Bayesian udating scheme Assumtions Prior on α: to be defined Prior on β: to be defined Bayesian udating Posterior is roortional to rior times likelihood P(A B) P(A)x P(B A) Exected results Posterior law (samle) for α Posterior law (samle) for β α Likelihood function: The variable follows a Bernouilli law α Non informative β Data: (crack size/ outcome of the insection) or 1 for «Hit-miss» values β The objective is to udate the knowledge on the arameters αand β In order to solve the roblem, the Monte-Carlo Marko Chains technique is used The objective is to roduce a samle which is reresentative of the osterior law Technique in the Context of POD Curves Determination 7 Outline A Bayesian framework for the determination of POD curves Estimation of POD curves using a functionnal form (Berens aroach) Focus on the hit/data format case Descrition of the Bayesian udating scheme Bayesian aroach to data merging First ste: Bayesian udate with simulation data Alication case: HFET of fatigue cracks in Titanium Simulation suorted POD curve determination (revious results) Second ste: Bayesian udate with emirical data Conclusions and future work Technique in the Context of POD Curves Determination 8 6

7 Bayesian aroach to data merging First ste: BU with simulation data POD 1 Hit/miss 1 Simulated data on a HFEC alication case Defect size Prior distributions for arameters (α,β) α Bayesian udate Posterior distributions for arameters (α,β) Bayesian udate with simulated data α median α POD curve for a «choice» of arameter values (median of the distributions for instance) Determination of the lower confidence bound from a large samle of POD curves generated from the arameter distributions Determination of a secific value of interest a 9/95 β β median β Technique in the Context of POD Curves Determination 9 Bayesian aroach to data merging Alication case: HFET of fatigue cracks in Titanium Part Material: Titanium(TA6V) NDT Configuration:High Frequency Eddy Currents Testing (HFET) Geometry: Flat areas Defects: Fatigue cracks Probe: Pencil robe (2MHz) Conditions: In-service (manual) Technique in the Context of POD Curves Determination 1 7

8 Bayesian aroach to data merging POD s in CIVA Aim of the aroach: simulate a set of realistic values for the NDT system resonse «realistic values»: values that take into account uncertainty and noise sources and thus reflect fluctuations around the theoretical resonses Signal resonse Theoretical resonse Signal resonse Fluctuations Flaw dimension Flaw dimension Deterministic aroach Aroach based on uncertainty roagation Technique in the Context of POD Curves Determination 11 Bayesian aroach to data merging POD s in CIVA POD s in CIVA Software: a samling aroach based on existing hysical models Inut arameters With uncertainty Outut Values, With variability Probabilistic characterization of insection caability: POD, PFA, ROC, Uncertainty roagation 1. Definition of the insection setu 2. Descrition of uncertainties on a set of inut arameters 3. Proagation of uncertainty thru existing models (Monte Carlo) 4. Extraction of the outut (the variable of interest) 5. Evaluation of robabilistic criteria such as the POD, PFA Technique in the Context of POD Curves Determination 12 8

9 Bayesian aroach to data merging Design of Numerical Exeriments HFET on Titanium Characteristic variable: crack length (mm) Uncertain arameters: Start scan osition Crack height (mm) Angle of the robe ( ) Electricalcontacts in the crack Corresonds to the osition of the robe for icking the maximum amlitude signal Fatigue cracking is subject to many uncertainties Translated into an additional lift-off using geometrical rule Real cracks exhibit electrical contacts Probability density Probability density Probability density Ratio of electrical contacts (%) Y start osition (mm) Uniform in [-.5;5] (scan increment=1mm) Crack height (ratio of length) Gaussian.5*length +N(,1)*.12*length Probe angle ( ) Gaussian( ;1 ) Samling strategy: Monte Carlo Crack length (mm) Technique in the Context of POD Curves Determination 13 Bayesian aroach to data merging Simulated and Exerimental data Exerimental database 69 cracks from.33 mm to 6.66 mm 5 oerators Exerimental data 1 Signal resonse (%FSH) Crack length (mm) Simulations 1 crack lengths from.25 mm to 5. mm 6 samles er crack length Technique in the Context of POD Curves Determination 14 9

10 Bayesian aroach to data merging Bayesian udate on simulated data Posterior distributions for alha and beta Samle generated thru MCMC Pdf Pdf beta alha beta alha Prior distributions for alha and beta (non informative) Posterior distributions are centered and close to Gaussian distributions (slightly skewed) Samle of arameters values exhibit a statistical deendency Technique in the Context of POD Curves Determination 15 Bayesian aroach to data merging POD Maximum Likelihood estimate (Berens) vs Bayesian Udate POD curves are almost identical Technique in the Context of POD Curves Determination 16 1

11 Bayesian aroach to data merging Comutation of the POD curve with confidence Definition: The 95% lowerconfidence POD curveisthe curvewhich leaveson the left95% of the «POD curves» of the beam. In ractice the POD curve with confidence can be built ointwize, looing on POD values: For in [.1;.99] Findthe POD curveleaving95 % of the curveson the leftatordinate Pick-u the corresonding flawsize a End The POD curvewithconfidence isthe set of oints (a, ). Technique in the Context of POD Curves Determination 17 Bayesian aroach to data merging Lower confidence band at 95% Confidence bound close to the «median» POD curve due to the large size of the simulated data set (6 simulated insections) Technique in the Context of POD Curves Determination 18 11

12 Outline A Bayesian framework for the determination of POD curves Estimation of POD curves using a functionnal form (Berens aroach) Focus on the hit/data format case Descirtion of the Bayesian udating scheme Bayesian aroach to data merging First ste: Bayesian udate with simulation data Alication case: HFET of fatigue cracks in Titanium Simulation suorted POD curve determination (revious results) Second ste: Bayesian udate with emirical data Conclusions and future work Technique in the Context of POD Curves Determination 19 Bayesian aroach to data merging Second ste: Bayesian udate using emirical insection results Simulated data on a HFEC alication case Subset of available emirical data for the HFEC alication case Prior distributions for arameters (α,β) Bayesian udate Posterior distributions for arameters (α,β) Bayesian udate with simulated data First ste results give the rior distributions for the second ste of the rocess Bayesian udate Posterior distributions for arameters (α,β) Bayesian udate with emirical data α POD curve for a «choice» of arameter values (median of the distributions for instance) Determination of the lower confidence bound from a large samle of POD curves generated from the arameter distributions a 9/95 α β Technique in the Context of POD Curves Determination 2 β 12

13 Bayesian aroach to data merging Bayesian udate on emirical data Posterior distributions for alha and beta Samle generated thru MCMC Pdf Pdf beta beta alha Prior distributions for alha and beta (obtained thru BU on simulated data) Mean values of the osterior distributions slightly different Variance of the osterior distributions are reduced Technique in the Context of POD Curves Determination 21 Bayesian aroach to data merging POD BU on simulated data vs. BU on emirical data POD curves are very close BU on exerimental data moves the curve to the right Technique in the Context of POD Curves Determination 22 13

14 Outline A Bayesian framework for the determination of POD curves Estimation of POD curves using a functionnal form (Berens aroach) Focus on the hit/data format case Descirtion of the Bayesian udating scheme Bayesian aroach to data merging First ste: Bayesian udate with simulation data Alication case: HFET of fatigue cracks in Titanium Simulation suorted POD curve determination (revious results) Second ste: Bayesian udate with emirical data Conclusions and future work Technique in the Context of POD Curves Determination 23 Conclusions Bayesian aroach to the determination of POD curves Bayesian aroaches are richer than MLE: we obtain distributions on the inut arameters, not just a single value For a given choice of the arameters (e.g. the median values), the resulting POD curve is very close to the Berens(MLE) one Alication of the method to HFET of titanium late Numerical model results are close to exerience: the second udate done with exerimental data doesn t modified significantly the rior that was obtained with the simulation results The arameter variances are reduced Technique in the Context of POD Curves Determination 24 14

15 Persectives Demonstrate that the method allows minimization of the required emirical data set Aly Bayesian techniques to other roblems: Definition of the inut arameters uncertainties in the simulation suorted aroach to POD curve determination Alication to the signal resonse data format Show that less emirical data are required to achieve a certain confidence level than what is demanded by recommendation documents (MIL-HDBK 1823a) Technique in the Context of POD Curves Determination 25 Thank you for your attention! Technique in the Context of POD Curves Determination 26 15

16 Descrition of the Bayesian framework How to define rior distributions for the arameters of the POD model? Interretation of the arameters α and β αand β are not easily interretable in hysical terms How to evaluate rior knowledge? An equivalent form of the logit model is: With: POD( a) = 1 + ex µ=-α/β σ=π/(β.sqrt(3)) π ln a µ 3 σ 1 In this form: µ=ln(a.5 ) where a.5 is the flaw size which is detected with a robability of 5% σ is the steeness of the POD(a) function. The smaller the value of σ, the steeer the POD(a) function 1 POD 1 POD Dimension caractéristique Dimension caractéristique Technique in the Context of POD Curves Determination 27 Bayesian aroach to data merging Simulated vs. Exerimental POD curves 6 simulated data 4 uncertain arameters (start osition, crack height, robe angle, elec. contacts) Probability Of Detection (%) Observed ratio (exerimental) 3 Estimated POD (exerimental) 2 POD 95 % conf. (exerimental) Estimated POD (simulation) 1 POD 95 % conf. (simulation) Emirical POD (simulation) Crack length (mm) a a ex 9 ex 9 / 95 = 1.5mm = 1.8mm Observed ratio (exerimental) Estimated POD (exerimental) POD 95 % conf. (exerimental) Observed Estimated POD 95 % ratio POD conf. (simulation) a a simu, EC 9 simu, EC 9 / 95 = 1.5mm = 1.7mm POD curves very similar Sloe corrected but stillthe effectof the numberof data Confidence band smaller for simulation dataset becausemore data thanin the exerimental dataset Values of interest very similar Technique in the Context of POD Curves Determination 28 16