Bayesian Approach for Fatigue Life Prediction fro Field Inspection Dawn An and Jooho Choi School of Aerospace & Mechanical Engineering, Korea Aerospace University, Goyang, Seoul, Korea Srira Pattabhiraan and Na Ho Ki University of Florida, Gainesville, FL, USA In fatigue life design of echanical coponents, uncertainties arising fro aterials and anufacturing processes should be taken into account for ensuring reliability. A coon practice is to apply a safety factor in conjunction with a physics odel for evaluating the lifecycle, which ost likely relies on the designer's experience. Due to conservative design, predictions are often in disagreeent with field observations, which akes it difficult to schedule aintenance. In this paper, the Bayesian technique, which incorporates the field failure data into prior knowledge, is used to obtain a ore dependable prediction of fatigue life. The effects of prior knowledge, noise in data, and bias in easureents on the distribution of fatigue life are discussed in detail. By assuing a distribution type of fatigue life, its paraeters are identified first, followed by estiating the distribution of fatigue life, which represents the degree of belief of the fatigue life conditional to the observed data. As ore data are provided, the values will be updated to reduce the confidence interval. The results can be used in various needs such as a risk analysis, reliability based design optiization, aintenance scheduling, or validation of reliability analysis codes. In order to obtain the posterior distribution, the Markov Chain Monte Carlo technique is eployed, which is a odern statistical coputational ethod which effectively draws the saples of the given distribution. Field data of turbine coponents are exploited to illustrate our approach, which counts as a regular inspection of the nuber of failed blades in a turbine disk. Keywords: Fatigue life, Prior distribution, Posterior distribution, Bayesian approach, Markov Chain Monte Carlo Technique, Field Inspection, Turbine blade P(x) P(x y) f X (x) f XY (x,y) f X (x y) P f Noenclature = Probability of rando variable X = x = Conditional probability of X = x with a given Y = y = Probability density function of rando variable X = Joint probability density function of rando variables X and Y = Marginal probability density function = Probability of failure, = Paraeters for noral distribution, = Paraeters for Weibull distribution f life (t) = Distribution of fatigue life Graduate Student Professor, AIAA Meber Graduate Research Assistant, Student Meber Associate Professor, Corresponding author, Departent of Mechanical and Aerospace Engineering, PO Box, University of Florida, Gainesville, FL -, AIAA Meber
P I. Introduction ERFORMANCE of echanical coponents undergoes a change by uncertainties such as environental effects, diensional tolerances, loading conditions, aterial properties and aintenance processes. Especially when the design criterion is fatigue life, it is significantly affected by syste uncertainties. Even with today s odern coputing systes, it is infeasible to include all the relevant uncertain variables into the analytical prediction, since any of the potential inputs are not characterized in the design phase. Approxiation ethods, such as the response surface ethod with Monte Carlo siulation (MCS),, were often eployed to overcoe excessive coputational cost in reliability assessent. To account for the unknown variables, coon practices use so called safety factors or statistical iniu properties in conjunction with the analytical prediction when evaluating lifeties. Due to these conservative estiations, analytical predictions are often in disagreeent with field experience, and a gap exists in correlating the field data with the analytical predictions. Thus, there is an increasing need to iprove the analytical predictions using field data, which collectively represents the real status of a particular achine. Field failure data can be helpful in predicting fatigue life that has uncertainties due to the unknown potential inputs. Recently, for ore reliable life prediction, any studies using field data have been undertaken. In nonfatigue life prediction, Orchard et al. used particle filtering and learning strategies to predict the life of a defective coponent. Marahleh et al. predicted the creep life fro test data, using the Larson-Miller paraeter. Park et al. used an energy-based approach to predict constant aplitude ultiaxial fatigue life. Guo et al. perfored reliability analysis for wind turbines using axiu likelihood function, incorporating test data. In this paper, the Bayesian technique is utilized to incorporate field failure data with prior knowledge to obtain the posterior distribution of the unknown paraeters of the fatigue life. The analytical predictions are obtained either fro nuerical odels or laboratory tests. The field data, although noisy, invariably portray environental factors, easureent errors, and loading conditions, or in short, reality. Since the predictions incorporate field experience, as tie progresses and ore data are available, the probabilistic prediction is continuously updated. This results in a continuous increase of confidence and accuracy of the prediction. In this paper, Markov Chain Monte Carlo (MCMC) technique is eployed as an efficient eans to draw saples of given distribution 9. Consequently, the posterior distribution of the unknown paraeters of the fatigue life is obtained in light of the field data collected fro the inspection. Subsequently, fatigue life is predicted a posteriori based on the drawn saples. The resulting distributions can then be used directly in risk analysis, aintenance scheduling, and financial forecasting by both anufacturers and operators of heavy-duty gas turbines. This presents a quantification of the real tie risk for direct coparison with the volatility of the power arket. The paper is organized as follows. In Section, the Bayesian technique is suarized, particularly with estiating the distribution of fatigue life through identifying the distribution of paraeters. Section discusses the effect of noise and bias on the accuracy of posterior distribution. In Section, five different cases are considered with varying priors and likelihoods, followed by conclusions in Section. II. BAYESIAN TECHNIQUE FOR LIFE PREDICTION In this section, Bayesian inference is explained in the view of updating distribution of fatigue life using test data. The Bayesian theore is first presented in a general for, followed by a specific expression for estiating the distribution of fatigue life. A. Bayes' theore Bayesian inference estiates the degree of belief in a hypothesis based on collected evidence. Bayes forulated the degree of belief using the identity in conditional probability: P( x y) P( x y) P( y) P( y x) P( x ) () where P(x y) is the conditional probability of X = x given Y = y. In the case of estiating the probability of fatigue life using test data, the conditional probability of event X (i.e., fatigue life) when the probability of test Y is available can be written as P( x y) P( y x) P( x) Py () ()
where P(x y) is the posterior probability of fatigue life X for given test y, and P(y x) is called the likelihood function or the probability of obtaining test Y for a given fatigue life x. In Bayesian inference, P(x) is called the prior probability, and P(y) is the arginal probability of Y and acts as a noralizing constant. The above equation can be used to iprove the knowledge of P(x) when additional inforation P(y) is available. Bayes' theore in Eq. () can be extended to the continuous probability distribution with probability density function (PDF), which is ore appropriate for the purpose of the present paper. Let f X be a PDF of fatigue life X. If the test easures a fatigue life Y, it is also a rando variable, whose PDF is denoted by f Y. Then, the joint PDF of X and Y can be written in ters of f X and f Y, as f ( x, y) f ( x Y y) f ( y) f ( y X x) f ( x ) () XY X Y Y X When X and Y are independent, the joint PDF can be written as f ( x, y) f ( x) f ( y ) and Bayesian inference XY X Y cannot be used to iprove the probabilistic distribution of f X (x). Using the above identity, the original Bayes' theore can be extended to the PDF as ( ) ( ) f ( x Y y) X f y X x f x Y f () y Y X () Note that it is trivial to show that the integral of f X (x Y = y) is one by using the following property of arginal PDF: f ( y) f ( y X ) f ( )d () Y Y X Thus, the denoinator in Eq. () can be considered as a noralizing constant. By coparing Eq. () with Eq. (), f ( x Y y ) is the posterior PDF of fatigue life X given test Y = y, and f ( y X x ) is the likelihood function X Y or the probability density value of test Y given fatigue life X = x. When the analytical expressions of the likelihood function, f ( y X x ), and the prior PDF, f Y X (x), are available, the posterior PDF in Eq. () can be obtained through siple calculation. In practical applications, however, they ay not be in the standard analytical for. In such a case, the Markov Chain Monte Carlo (MCMC) siulation ethod can be effectively used, which will be addressed in Section. in detail. When ultiple, independent tests are available, Bayesian inference can be applied either iteratively or all at once. When N nuber of tests are available; i.e., y = {y, y,, y N }, the Bayes' theore in Eq. () can be odified to N f ( x Y y ) f ( y X x) f ( x) () K X Y i X i where K is a noralizing constant. In the above expression, it is possible that the likelihood functions of individual tests are ultiplied together to build the total likelihood function, which is then ultiplied by the prior PDF followed by noralization to yield the posterior PDF. On the other hand, the one-by-one update forula for Bayes' theore can be written in the recursive for as f ( x Y y ) f ( y X x) f ( x), i,, N ( i) ( i ) X i Y i X Ki () ( i where K i is a noralizing constant at i-th update and f ) () x is the PDF of X, updated using up to (i )th tests. In X the above update forula, f () () x is the initial prior PDF, and the posterior PDF becoes a prior PDF for the next X update. In the view of Eqs. () and (), it is possible to have two interesting observations. Firstly, the Bayes' theore becoes identical to the axiu likelihood estiate when there is no prior inforation; i.e., f X (x) = constant. Secondly, the prior PDF can be applied either first or last. For exaple, it is possible to update the posterior distribution without prior inforation and then to apply the prior PDF after the last update. An iportant advantage of Bayes' theore over other paraeter identification ethods, such as the least square ethod and axiu likelihood estiate, is its capability to estiate the uncertainty structure of the identified
paraeters. These uncertainty structures depend on that of the prior distribution and likelihood function. Accordingly, the accuracy of posterior distribution is directly related to that of likelihood and prior distribution. Thus, the uncertainty in posterior distribution ust be interpreted in that context. B. Application to fatigue life estiation In deriving Bayes' theore in the previous section, two sets of inforation are required: a prior PDF and a likelihood function. In estiating fatigue life, the prior distribution can be obtained fro nuerical odels and laboratory tests. Since they can be perfored ultiple ties with different input paraeters that represent various uncertainties, it is possible to evaluate the distribution of fatigue life, which can be served as a prior PDF of fatigue life. On the other hand, the field data cannot be obtained in a laboratory environent. In this section, using field data in calculating the likelihood function is presented. When a gas turbine engine is built and installed in the field, the aintenance/repair reports include the history of the nuber of parts that were defective and replaced at specific operating cycles. Although these data are not obtained under a controlled laboratory environent, they represent reality with various effects of uncertainties in environental factors, easureent errors, and loading conditions. Thus, it is desirable to use these data to update the fatigue life of the specific achine using Bayes' theore. The standard approach to applying Bayes' theore is to use the field data to build the likelihood function, which is basically the sae as the PDF for with fatigue life. However, different fro specien-level tests, the field data cannot be repeated ultiple ties to construct a distribution. Only one data point exists for the specific operation cycles. Thus, the original forulation of Bayes' theore needs to be odified. First, instead of updating the PDF of fatigue life, it is assued that the distribution type of fatigue life is known in advance. This can be a big assuption, but it is possible that different types of distributions can be assued and the ost conservative type can be chosen. Once the distribution type is selected, then it is necessary to identify distribution paraeters. For exaple, in the case of noral distribution, the ean ( ) and standard deviation ( ) need to be identified. In this paper, these distribution paraeters are assued to be uncertain and Bayes' theore is used to update their distribution; i.e., the joint PDF of ean and standard deviation will be updated. In this case, the vector of rando variables is defined as X = {, }, and the joint PDF f X is updated using Bayes' theore. Initially, it is assued that the ean and standard deviation are uncorrelated. A field data set consists of the nuber of hours of operation until inspection (N f ), and the nuber of defective blades (r) out of the total nuber of blades (n). Thus, the field data are represented by y = {N f, n, r}, which are given in Table. Then, the likelihood function is the PDF f Y for given X = {, }. Since the field data is given at fixed N f and n, f Y can be represented in ters of r. Unfortunately, the nuber of defective blades cannot be a continuous nuber because it is an integer. Thus, the likelihood function f Y can be represented using the following probability ass function: n! r n r f y X {, } ( P ) ( P ) () Y f f r!( n r)! where P f is the probability of defects at given N f for given X = {, }. Since the distribution of fatigue life is given as a function of X, the probability of defects can be calculated by N f P, f t;, dt (9) f where, f life is the PDF of fatigue life distribution. The probability ass function in Eq. () is a binoial distribution, which odels the probability distribution of having r defects out of n saples with defect probability of P f. Figure illustrates the relation in Eq. (9). The predictive distribution of life can be estiated using the ean and standard deviation obtained fro the updated joint PDF as life f ( t) N (, ) () life In this paper, since the distribution type of the fatigue life is unknown, two different types are assued: noral and Weibull. The strategy is to select the one that can provide a ore conservative estiate. In the case of the Weibull distribution, the scale and shape paraeters (, ) are used in the Bayesian technique.
As entioned above, the predictive distribution of fatigue life depends on that of the prior distribution and likelihood function. The PDF of life f life (t X) and the prior PDF f(x) are therefore assued as the following cases: Case : f t : noral dist. and f, : non-inforative prior () life Case : f t : noral dist. and f, N,. N.,. () life Case : f t : noral dist. and life f, N,. Inv-,. () Case : f t, : Weibull dist. and f, : non-inforative prior () life Case : f t, : Weibull dist. and f, LogN -.9,. LogN.9,. () life For the likelihood function, noral and Weibull distributions are first considered as fatigue life distributions, and then, are used to calculate the fatigue failure probability P f in likelihood calculation as in Eq. (). In the odel, the associated odel paraeters are, in the case of noral and, in the case of Weibull distribution, respectively. These are taken to be unknown and are estiated using the inspected data. In ters of prior distribution, the probability distribution of fatigue life that was obtained nuerically by conducting reliability analysis is exploited, in which the ean and standard deviation of fatigue life are given by noralized fatigue life and. respectively. Based on these values, different kinds of prior distributions for the two paraeters are undertaken for the Cases, and, in which the coefficient of variation (COV) for the ean and standard deviation are assued as. in coon. Figure : Probability of defects calculation fro life distribution. Table : Field data for inspected turbine blades Engine Hours( N f ) Failed( r )/Total( n ) Engine Hours( N f ) Failed( r )/Total( n ). /. /. / 9. /.9 /. /. /. /. /. /. /. /. / C. MCMC siulation Once the expression for the posterior PDF is available as in Eq. (), one can proceed to saple fro the PDF. A priitive way is to copute the PDF values at a grid of points after identifying the effective range, and saple by the inverse CDF ethod. This ethod, however, has several drawbacks such as the difficulty finding correct location and scale of the grid points, the spacing of the grid, and so on. Especially when a ulti-variable joint PDF is required, the coputational cost is proportional to N, where N is the nuber of grids in one-diension and is
the nuber of variables. On the other hand, the MCMC siulation can be an effective solution as it is less sensitive to the nuber of variables 9. The Metropolis-Hastings (M-H) algorith is a typical ethod of MCMC, which is given in the case of two paraeters (, ) by the following procedure: () (). Initialize,.. For i to N Saple u ~ U., * * * * i i Saple, ~ q,,. if u A,,, i i * * () A p in, p * * i i * *, q,, i i * * i i, q,, else i i * *,, i i i i,, () () where, is the initial value of unknown paraeters to estiate, N is the nuber of iterations or saples, U is the unifor distribution, p, is the posterior PDF (target PDF), and q, is an arbitrarily chosen i i * * distribution. A unifor distribution is used in this study for the sake of siplicity. Thus, q,, and * * i i q,, becoe constants, and q, can be ignored. As an exaple of MCMC, the joint posterior PDF of the unknown paraeters, of the fatigue life using only the first data is shown in Figure, which represents the degree of belief on the concerned paraeters in the for of PDF. The joint posterior PDF using the grid ethod as well as MCMC sapling are shown in Figure (a) and (b), respectively. In Table, statistical oents by the two ethods are copared. As shown in the table, the two ethods agree quite closely but MCMC used () saples, whereas gird used () saples. The difference in the nuber of saples will be significantly increased as ore variables are identified. (a) using grid ethod ( grid) (b) using MCMC ( iterations) Figure : Joint posterior PDF of Case in Eq. () with one test data Table : Statistical oents by the two ethods Grid..... MCMC..9... Cov,
. ANALYTICAL EXAMPLE Although the Bayesian approach has been used extensively in literature, it is iportant to investigate its perforance. In particular, it is iportant to characterize how this ethod identifies unknown paraeters when the experiental data have noise and bias. In this section, the properties of Bayes' theore are studied using analytical exaples. In particular, the effects of noise and bias of data on the final distribution are discussed. The data here are siulated to deonstrate the new analytical technique. These data are then perturbed in order to study the effects of noise and bias errors on the algorith. A. Field data governed by a distribution The first study is to test the accuracy of the updated distribution using Bayes' theore. Table shows an exaple of field data that are used in this section. The total nuber of blades is n =. The field data are generated fro a distribution, B ~ N(, ) (B is actually unknown distribution which should estiated based on the field data but is used to generate the field data) and the objective is to test if the updated distribution recovers the distribution B. For the prior distribution, the ean is uniforly distributed in the interval of [, ], while the standard deviation is also uniforly distributed in the interval of [, ]. Using the four sets of field data in Table, the posterior joint PDF are calculated using Bayes' theore. After obtaining the posterior joint PDF, saples obtained fro the MCMC siulation are used to predict the distribution of life. Figure copares the updated distribution of fatigue life with the distribution B, along with the predicted probability of defects fro field data. In this figure, the red curves represent the % lower bound, ean, and 9% upper bound fro the left hand, respectively. It is clear that both distributions atch each other quite well. When the field data are governed by a particular distribution, Bayes' theore can reproduce the distribution well when ore than sets of data are available. Table : Saple field data for inspected turbine blades Engine Operating hours 9 Nuber of defective blades 9.9....... True distribution. Data points Bayesian updated life... Operating hours x Figure : Coparison between the updated life distribution using Bayes' theore and the original distribution B. Effect of bias and noise In practice, the field data are often accopanied by noise and bias. The forer is caused by variability in easureent environent, while the latter represents systeatic departure, such as device error. The difference is that the forer is rando, while the latter is deterinistic, although its value is unknown. In soe cases, a positive bias is consciously applied to reain conservative. This section analyzes the effect of bias and noise on the updated life distribution using the sae saple data shown in Table.
First, the bias is given in ters of the operating hours of blades. For exaple, bias = % eans that the operating hours of blades is % ore than the noinal operating hours. Figure (a) shows the effect of bias on the distribution of fatigue life. Both positive and negative biases are considered. As expected a negative bias leads to a conservative estiate of the updated life distribution. Fro Table, it can be observed that the standard deviation of the life distribution reains about the sae as the true value, while the ean values are shifted by ± due to the biases. Next, the effect of noise in the data is investigated by randoly perturbing the original data by % and %. Figure (b) shows the updated distribution of fatigue life with the two different levels of noise, and ore specific results are shown in Table, which also includes the effect of the nuber of data. Different fro the case of bias, as the nuber of data increases, the eans converge to a noinal ean. However, the standard deviation is relatively insensitive to the nuber of data; it shows a slight increase with points of data. Although the estiated distribution is inaccurate with high levels of noise, it tends to be conservative when the life with a low probability of failure is estiated. In general, bias can be identified with any data by using the least-square ethod or the Bayesian approach with the bias as an unknown syste paraeter..9.... True distribution. Data with bias +%. Updated life with bias= +%. Data with bias -%. Updated life with bias= -%... Operating hours x (a) life distribution results due to ±% bias Figure : Effect of bias and noise.9.... True distribution. Data with noise %. Updated life with noise %. Data with noise %. Updated life with noise %... Operating hours x (b) life distribution results due to % and % noise Table : Mean and standard deviation of unknown paraeter, deterinistic bias noise % noise % +% -% data= data= data= data= data= data= 9 9 9 Error (%).. (-)....... 9 Error (%).9.9. 9...9... 9 9 9 9. IDENTIFICATION OF MODEL PARAMETER AND PREDICTION OF LIFE DISTRIBUTION A. Posterior distribution of odel paraeter The results of Case where the likelihood is a noral distribution and non-inforative prior are shown in Figure. In Figure (a), the contours of prior distribution, likelihood function and joint posterior PDF of the unknown paraeters are plotted. In these figures, the updated prior and the likelihood are obtained fro the posterior distribution previously obtained and the inspection data, respectively. The posterior distribution is obtained by ultiplying the prior and likelihood, and is used in the next updating step as the prior distribution. Since a non-
using only thest data using st~th data using st~th data using st~th data posterior updated prior μ (noralized posterior μ (noralized μ (noralized updated prior posterior fro field data updated prior fro field data posterior using st~th data updated prior fro field data σ (noralized inforative prior is used, the first update is identical to the first likelihood function. Since data fro a single test can be represented by infinite cobinations of eans and standard deviations, the contours of likelihood becoe straight lines. In Figure (b), the posterior PDFs are plotted in the for of contours. Even if the contours of individual likelihood functions are straight, the uncertainty in the posterior distribution is reduced and show correlation between the ean and standard deviation. It is shown that as ore data are added, the location and range of, oves and narrows down to converge to a certain point. The results indicate our knowledge on the unknown paraeters based on the field inspection. fro field data μ (noralized μ (noralized using only thest data using st~th data using st~th data using st~th data using st~th data σ (noralized (a) likelihood function and updated prior and posterior distribution μ (noralized μ (noralized μ (noralized μ (noralized μ (noralized (b) contour of joint posterior PDF Figure : Updated posterior PDF of Case. The results of Cases are shown in Figure ~Figure 9. The results of Case where the likelihood is still a noral distribution but with norally distributed priors are shown in Figure (a) and (b). Different fro the noninforative prior case in Figure, the level of uncertainties reains alost constant, but their locations continuously change for different field data. As illustrated in Figure (a), the posterior distribution is already narrowed due to the prior distribution. However, since the prior distribution is quite different fro the field data, the centers of posterior distributions gradually ove as ore field data are used. In addition, the final posterior distribution oves toward to the prior distribution copared to that of Figure. Figure and Figure clearly provide the effect of prior distribution. When there is a strong inclination to the prior distribution, then its use can yield the posterior distribution closer to it than the non-inforative prior. Then due to field data that predict a longer fatigue life, the expected life will gradually increase. The results of Case where the likelihood is noral and the prior for the siga is changed to a chi-square distribution with degree of freedo n( = ) and scale paraeter =., which are ore reasonable assuptions due to non-negativity, are shown in Figure (a) and (b). In this case, the standard deviation of the posterior distribution increases with ore field data because the prior has a sall standard deviation than that of field data. If the likelihoods are considered first with a non-inforative prior, as is found in the st~th data and st~th data of Figure (c), and the posterior is obtained by applying the prior at the last stage, which is st~th data in the figure, one gets the standard deviation decreased as ore field data are added, with the final posterior being updated by the prior. The results of Case where the likelihood is the Weibull distribution with scale ( ) and shape () paraeter and a non-inforative prior are shown in Figure, and the results of Case where the likelihood is still the Weibull distribution but with the prior being lognorally distributed are shown in Figure 9. Both cases show the convergent behavior, but the range of, of Case is narrower than that of Case due to the prior. The case of a noninforative prior shows a strong correlation between the ean and standard deviation, which is significantly reduced with the lognoral prior. These cases are the ost reasonable because the Weibull distribution is the best odel for the lifetie. 9
........ using only the st data prior fro field data posterior..... (a) results using the first data Figure : Updated posterior PDF contours of Case........ posterior dist. contour st~th data st~th data only st data st~th data st~th data..... (b) updated posterior PDF contours... using only the st data posterior dist. contour fro field data prior posterior.......... (a) results using only the first data (b) updated posterior PDF Figure : Updated posterior PDF contours of Case... st~th data st~th data st~th data st~th data only st data...... posterior dist. contour st~th data st~th data 9 st~th data (c) updated posterior PDF with prior being applied at the last stage using only the st data using st~th data using st~th data using st~th data using st~th data Figure : Updated posterior PDF contours of Case using only the st data posterior fro field data using st~th data using st~th data using st~th data using st~th data prior Figure 9: Updated posterior PDF contours of Case The final posterior PDFs of all cases are shown in Figure. Figure (a) and (b) are the results of and of the noral distribution and and of the Weibull distribution, respectively. Case in Figure (a) and Case in Figure (b) are the results of non-inforative priors. As was expected, the results are uch wider than the others
due to the non-inforative prior. If specific prior inforation is available, the precision of the posterior distribution is increased...... posterior dist. contour(likeli-n) Case Case Case (a) likelihood is noral distribution Figure : Final posterior PDFs of all cases.... posterior dist. contour(likeli-w) Case Case (b) likelihood is Weibull distribution B. Posterior predictive distribution of fatigue life The posterior PDF obtained in Section. can be used to predict the probability of fatigue life. Recall that the Bayesian inference updates the distribution of ean and standard deviation of the life distribution. Each saple of ean and standard deviation yields a distribution of fatigue life, which can be represented by a cuulative distribution function (CDF). Thus, as a result of Bayesian inference, a distribution of CDF can be obtained. For explanatory purposes, this distribution of CDF can be represented by a confidence interval. It is expected that this interval will be wide when the uncertainty in the ean and standard deviation is large. In general, the uncertainty in ean and standard deviation reduces with greater nubers of field data, and the confidence interval will also be reduced with ore field data. Figure shows the updating process of the predicted fatigue life CDF along with field data for Case. The red stars are the field data at the current update, while the blue stars are the field data up to the previous update. In the sae figure, the dashed red curve is the ean CDF of fatigue life, while the two solid red curves are % and 9% confidence bounds. Copared to the significant noise in the field data, the confidence interval of the predicted CDF is progressively reduced. In order to accoodate safety, it is advised to take a % lower bound of the CDF. In order to show the difference between initial and final distributions of fatigue life, Figure plots the confidence intervals using prior and the final posterior distribution of ean and standard deviation for Cases,, and. It is clear that all initial distributions are overly conservative. This often happens because analytical odels often assue all input rando variables are independent, when they ay be correlated. In addition, aterial properties and design loads are often chosen to be conservative. For the purpose of planning scheduled aintenance, it is often required to choose either % or % life, which is also called B or B life. Table shows the confidence intervals of B and B life, along with lower- and upperbounds of the 9% confidence intervals. In Cases and, which eployed a noral odel, negative values for the life are calculated due to wrong assuptions on the odel; the noral distribution can have negative lives. On the other hand, Cases and are reasonable because they only allow positive values. Since the type of distribution is assued initially, it is advised to choose the ost conservative one, which is Case in this study. Although Case does not have prior inforation copared to Case, its confidence interval has significantly been reduced using enough field data; the confidence intervals at % P f are. and., respectively. However, this ay not always be true. For exaple, Cases and are copared in Figure and Table when only the first data are used in Bayesian inference. Without having the prior distribution, Case shows a significant uncertainty, and the lower-bound of % P f is.9, which is overly conservative and costly due to frequent aintenance. Thus, when the nuber of field data is sall or the field data have excessive noise, the prior inforation plays an iportant role to deterine acceptable aintenance intervals. In suary, the ability to use the prior properly and the ability to choose appropriate statistical odel are the ain advantages of Bayesian inference over the regression ethod.
........ New field data edian 9% C.I. Noralized operating hours (a) using only the st data. Old field data. New field data Median 9% C.I. Noralized operating hours (b) using st ~ th data. Old field data. New field data Median 9% C.I. Noralized operating hours (c) using st ~ th data..... Old field data. New field data Median 9% C.I. Noralized operating hours (d) using st~th data Figure : Updated Process of Case. Old field data. New field data Median 9% C.I. Noralized operating hours (e) using st ~ th data.......... Field data Initial(prior) Final(~th data) Noralized operating hours (a) Case Figure : Final updated distribution of fatigue life.. Field data Initial(prior) Final(~th data) Noralized operating hours (b) Case. Field data Initial(prior) Final(~th data) Noralized operating hours (c) Case Table : Confidence interval of noralized fatigue life at the last stage % P f % P f % lower 9% upper interval % lower 9% upper interval Case -. -..... Case...... Case -.9.9...9. Case.9..... Case.9.9...9.
... Figure : Coparison between Case and Case. Field data Case Case Noralized operating hours Table : Confidence interval of noralized fatigue life at the first stage % P f % P f % lower 9% upper interval % lower 9% upper interval Case.9....9. Case....9... CONCLUSIONS In this paper, a Bayesian updating technique is presented, which incorporates statistical prediction with field data. By using MCMC siulation, saples of odel paraeters (, or, ) are drawn effectively, which are paraeters of the fatigue life distribution. After obtaining saples for a joint posterior PDF of, the fatigue life prediction results are obtained, which have a CDF in confidence intervals due to the uncertainties of the odel paraeters. If there is specific prior inforation of odel paraeters, the precision of the posterior distribution is increased. In case the type of the distribution of the likelihood is not known a priori, it is advised to choose the ost conservative type after exaining several candidates as was found in this study.. ACKNOWLEDGEMENTS This research was supported by Basic Science Research Progra through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (-- and 9-). REFERENCES. Voigt, M., Mücke, R., Vogeler, K., Oeverann, M., Probabilistic Lifetie Analysis of Turbine Blades based on a cobined direct Monte Carlo and Response Surface Approach, ASME Publication, Paper No. GT-9, ASME Turbo Expo, June, Vienna, Austria.. Weiss, T., Voigt, M., Schlus, H., Mücke, R., Becker, K.-H., Vogeler, K., Probabilistic Finite-Eleent-Analysis on turbine blades, Paper No. GT9-9, ASME Turbo Expo, June 9, Orlando, FL.. Marahleh, G., A. R. I. Kheder, and H. F. Haad. Creep-Life Prediction of Service-Exposed Turbine Blades. Materials Science, (),.. Orchard, M., Wu, B., Vachtsevanos, G., A Particle Filtering Fraework for Failure Prognosis, World Tribology Congress III Sep. -,, Washington, D.C., USA. Marahleh, G., Kheder, A.R.I., Haad, H.F., Creep-Life Prediction of Service-Exposed Turbine Blades, Materials Science, Vol., No.,. Park, J., Nelson, D., Evaluation of an energy-based approach and a critical plane approach for predicting constant aplitude ultiaxial fatigue life, International Journal of Fatigue Vol.,, pp. 9
. Guo, H., Watson, S., Tavner, P., Xiang, J., Reliability analysis for wind turbines with incoplete failure data collected fro after the date of initial installation, Reliability Engineering & Syste Safety Vol. 9, Issue, June 9, pp. -. Ki, N. H., S. Pattabhiraan, and L.A. Houck III. Bayesian Approach for Fatigue Life Prediction fro Field Data. ASME Turbo Expo Gas Turbine Technical Congress and Exposition, Glasgow, Scotland, UK, June -,. 9. Andrieu, C., N. de Freitas, A. Doucet, and M. Jordan. An introduction to MCMC for Machine Learning. Machine Learning, :-,.. Bayes, Thoas (). "An Essay towards solving a Proble in the Doctrine of Chances.". Philosophical Transactions of the Royal Society of London :.. Athanasios Papoulis (9), Probability, Rando Variables, and Stochastic Processes, second edition. New York: McGraw-Hill.. Gelan, A., J. B. Carli, H. S. Stern, and D. B. Rubin. Bayesian Data Analysis (second ed.). New York, Chapan & Hall/CRC,.. Coppe, A., Haftka, R. T., Ki., N. H. and Yuan, F.G. Reducing Uncertainty in Daage Growth Properties by Structural Health Monitoring, Annual Conference of the Prognostics and Health Manageent Society, 9. Haldar, A., and Mahadevan, S.,, Probability, Reliability, and Statistical Methods in Engineering Design, John Wiley & Sons Inc., New York.