Crack Growth Prediction Based on Uncertain Parameters Using Ensemble Kalman Filter

Hana, Norihiko; Hatanaka, Rinya; Amaya, Kenji

doi:10.1115/1.4066474

Abstract

We are developing a method that will enable the estimation of crack shapes in such structures as power equipment and social infrastructure with greater precision as well as the prediction of crack growth life under conditions of uncertainty regarding crack perimeter structure and applied loads. Ascertaining the dimensions of cracks is complicated by the influences exerted by external loads on crack propagation as well as the geometry of crack perimeters. The prediction of crack propagation based on uncertain information is an overly conservative approach due to the lack of accuracy. This paper presents a Bayesian estimation of actual crack geometry based on predictions from a physical model of crack growth and measured crack geometry. The uncertainty in the load and the geometry of the crack perimeter are reflected in the crack propagation model. The range over which the uncertain parameters are estimated is updated simultaneously with estimations of the crack shape. Furthermore, we describe how optimal measurement intervals can be identified from the one-period-ahead prediction of crack growth based on a physical model. The application of properly spaced measurements and sequential Bayesian estimation can effectively mitigate the impact of measurement error and parameter uncertainty, thereby enhancing the precision of crack growth prediction. Sequential Bayesian estimation is an Ensemble Kalman Filter, and our physical model of crack propagation is a Paris measure based on fracture mechanics. The efficacy of the methodology presented in this paper is validated by the outcomes of the simulated observed data of a CT specimen.

1 Introduction

Periodic inspections are essential for ensuring the safe use of such large structures as power plants and bridges that are used for extended periods. In power plants, inspections focus on cracks in power equipment and high-pressure vessels. In bridges, inspections concentrate on cracks in steel bridges and metal materials. Note that cracks in electrical equipment and bridges can occur not only in areas that can be visually inspected but also in areas that resist such inspections [1–3].

To examine the rotor of a turbine generator for power equipment, the generator must be disassembled [4]. This process increases the period during which the equipment cannot be used. As a result, robotic inspection is increasingly being used to avoid disassembling such equipment. Given that a robot must enter a confined space that is inaccessible to humans [4–7], the inspection equipment (mounted on it) must be as thin and small as possible. For instance, a robot that is deployed between a turbine generator’s stator and rotor is 20–30 mm thick [8,9]. Similarly, the device that is utilized to inspect a stator by percussion is less than 20 mm thick [9]. In the field of bridge inspection, a comprehensive examination of a large area can be conducted using images [3,10]. Nondestructive methods for crack inspection include percussion [11], ultrasonic [12], infrared [13], microwave [14], X-ray [15], electrical resistance change sensors [16], piezo-electric sensors [17], and image sensors [3,10,18,19]. One method for measuring the crack-induced changes in waves or the current input to test objects is to inspect the changes in the waves or the current output of a smaller device. The waves or currents that can be generated and input by the small device are small, reducing the accuracy of the measurement. The smaller the imaging device is, the narrower the inspection area is and with fewer visible details. If the imaging device is moved to expand the inspection range, obtaining a clear image is difficult. Measuring the crack size by a small sensor or image is subject to error. The size of the crack, including the error, is used to predict its propagation life and to determine when to repair the crack or replace the damaged area. The prediction of crack propagation requires information on the applied load, the crack’s surrounding structure, and its size itself [20–22]. The actual applied loads are neither precise nor uncertain. The dimensions of the crack propagation zone are also uncertain due to dimensional tolerances and assembly errors.

Periodic inspections allow for the proper timing of crack repairs and the replacement of cracked areas [3,23,24]. It is challenging to monitor the condition of the numerous components of large structures in power plants and social infrastructure. The frequency of inspections is set at an appropriate level, and their cost is kept as low as possible [25]. To ensure the safety of more large structures in power plants and social infrastructure, the frequency of inspections must be reduced while maintaining structural safety requirements. To reduce the frequency of inspections, we must improve the accuracy of predicting crack propagation based on uncertain crack size of the applied load information as well as the structure surrounding the crack.

Previous studies have attempted to predict the lifetime of cracks from measurement errors and uncertain parameters, including those by Yiwei [20], Chen [21], and Wang [22]. Yiwei et al. [20] employed an extended Kalman filter to forecast the extent of crack propagation in a hollow cylinder due to the discrepancy between internal and external pressure. This prediction assumes the existence of uncertainty in the pressure differential and the characteristics of crack growth in addition to the crack’s measured length. Chen et al. [21] utilized the particle filtering of ultrasonically measured cracks in real-time to predict crack propagation. Such prediction assumes uncertainty in crack growth properties other than the measured crack length. Wang et al. [22] measured visual crack lengths and predicted the growth lifetime with a modified particle filter. In addition to the measured crack length, uncertainties in the maximum load and crack growth characteristics are assumed. Previous studies did not consider the uncertainty in the shape of the propagated objects.

Previous studies, which predict crack growth lifetimes and set appropriate inspection frequencies, include Vatn [25], Dong [26], and Kim [27]. However, Dong [26] and Kim [27] assume that uncertainty in the parameters affects the physical models of crack propagation without updating the measurement results to narrow the range of uncertainty. Some predictions do not utilize physical models of crack propagation [25]. The prediction accuracy of the crack growth can be enhanced by updating and narrowing the estimated range of the uncertain parameters based on the measured crack length. Prior studies have not yet determined the optimal measurement intervals by updating not only the crack length but also the uncertain parameters at each measurement to predict crack propagation.

To enhance the precision of crack propagation forecasts, it is essential to concurrently refine the accuracy of the time-updated state variable, crack length, and the uncertain parameters that influence crack propagation. Data assimilation [28] is a method of simultaneously estimating state quantities and uncertain parameters based on physical model simulations and observations.

Data assimilation can be conducted in three principal ways: the Kalman filter [29], in which the simulation model is linear and the noise follows a normal distribution; the extended Kalman filter [30], in which a nonlinear simulation model is solved to a linear approximation; and the ensemble Kalman filter (EnKF) [29], in which a nonlinear simulation model is approximated by multiple particles. Additionally, there are particle filters [29] that approximate the model of a nonlinear simulation with multiple particles and do not assume a normal distribution for the noise.

Crack growth models based on fracture mechanics include the Paris model [31], which models the region of stable crack growth, and the Walker model [32], which is an extension of the Paris model. Furthermore, the Collipriest model [32] provides a comprehensive framework for analyzing fatigue cracks, from their initial growth to the transition to unstable crack growth, ultimately leading to ultimate failure. Crack growth simulations based on crack growth models are nonlinear. Chen et al. [21] utilized the particle filtering of ultrasonically measured cracks to predict crack propagation. The fatigue testing of attach lag structures serves to validate the prediction by particle filtering. In the event that both the observed and simulated system noise are a normal distribution, the ensemble Kalman filter is an effective method for calculating the nonlinearity of crack propagation without linearizing it.

In this paper, we used a physical model of crack growth and measurement results to simultaneously estimate uncertain parameters other than crack length with an [29] EnKF) to improve the prediction accuracy of the crack growth lifetime. Furthermore, we demonstrate how to set an appropriate measurement interval based on the results of the ensemble’s one-step-ahead forecast. First, the Paris law [31] is introduced as a physical model for predicting crack growth. The uncertain parameters predicted by the EnKF include load, geometry, and material properties related to crack propagation. In addition to load and geometry, material properties can be predicted simultaneously. It should be noted that the estimated results can be used for purposes other than crack propagation prediction. In this paper, the geometry of the crack propagation target and the load applied to it are uncertain parameters for prediction. The measured crack length and uncertain parameters are defined as random variables. Until the next measurement interval, a one-step-ahead forecast is made based on the uncertain parameters and the crack length estimated by the Paris law in addition to the system noise. The EnKF estimates the ground truth crack length from the one-step-ahead forecast results and the measured crack length. The uncertain parameters are estimated simultaneously. The elapsed time for a crack length to reach a state that requires repair or replacement is predicted from the one-step-ahead forecast. A probability distribution is assumed for the predicted arrival time for each ensemble. Then we derive the measurement interval from the probability distribution of the arrival time and the predetermined reliability. This paper validates our proposed method using the time-series simulated observed data of the values measured on a CT specimen with the incorporated errors. The data obtained from the BeCu material [33] measured using the direct current (DC) current potential method were employed in this validation. BeCu is used in the rotor wedge of turbine generators [34].

2 Prediction by Kalman Filter and Derivation of Measurement Intervals

This section presents a framework for estimating the state of crack propagation using a Kalman filter. An EnKF is employed to estimate nonlinear crack growth. A method is also presented for deriving measurement intervals from the one-step-ahead forecast of the EnKF.

2.1 State Estimation Problem for Crack Propagation.

The framework of the problem of estimating the state of crack propagation is presented. The observed crack length is denoted by a, and the uncertain parameters used to predict the crack propagation’s state are load range $Δ P$ and geometry SS. It predicts the state of the crack propagation from the state equation, which forecasts the growth with uncertain parameters, and the observed equation, which relates the observed crack length to the predicted value.

First, the state equation for predicting crack growth is presented. The parameters that determine crack propagation in metals include stress at the crack tip [35], the crack aperture [35], the stress intensity factor [36], J-integral [37], the plastic strain-based parameters [38], and the inelastic strain energy density [39,40]. All the parameters and crack propagation criteria were determined experimentally for each material. To illustrate, a database of material properties pertinent to crack growth has been constructed with the stress intensity factor as a parameter [41]. A number of physical models of growth have been proposed [31,32,42,43], each of which is based on a material-specific database of crack growth regions where cracks are stable. This paper introduces into the state equation the Paris law [31], a physical model of crack propagation.

Figure 1 shows a schematic diagram of the physical model. Equation is the Paris law

\frac{d a}{d n} = C {(Δ K)}^{m}

(1)

Fig. 1

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Physics model of crack propagation: Stress intensity factor is obtained from initial crack length, shape, and load using finite element method or theoretical equations. Physical model predicts crack growth using stress intensity factor and the Paris law and adds system noise to the predicted crack length to obtain the crack length one-step ahead.

Here,

a (m)

is the fatigue crack length, n is the number of load or displacement repetitions, and

\frac{d a}{d n}

is the crack growth rate. C and m are material constants, and

Δ K [MPa \sqrt{m}]

is the stress intensity factor range. Equation is the stress intensity factor range

Δ K = K_{\max} - K_{\min}

(2)

The crack propagation process is expressed by Eq. from discrete-time-step t at which the load or displacement is repeated and Eq.

a_{t} = a_{t - 1} + {\frac{d a}{d n} |}_{t - 1} \cdot Δ n (t, t - 1)

(3)

Equation is the crack length after the t − 1 crack has propagated to discrete-time-step t. Here,

{\frac{d a}{d n} |}_{t - 1}

is the growth rate calculated for the

a_{t - 1}

crack shape.

n (t - 1)

is the number of load iterations up to discrete-time-step t − 1.

n (t)

is the number of load iterations up to discrete-time-step t. The number of load repetitions from t − 1 to t

Δ n (t, t - 1)

is shown by the following equation:

Δ n (t, t - 1) = n (t) - n (t - 1)

(4)

This study assumes uncertainty in dimensions and loading, which affects the stress intensity factor range. The dimensions and loads are taken as state variables. The state variable vector is given by Eq. . Subscript t denotes discrete time t

x_{t} = {[a_{t}, Δ P_{t}, S S_{t}]}^{T}

(5)

The state equation is shown in Eq. to demonstrate state estimation by the EnKF

x_{t} = f (x_{t - 1}, v_{t})

(6)

Here,

v_{t}

is a system noise term. System noise is uncertainty in the prediction equation for crack growth, load, and structure around the crack. The state equation for each parameter is shown in the following equation:

{\begin{matrix} a_{t} = a_{t - 1} + {\frac{d a}{d n} |}_{t - 1} \cdot Δ n (t, t - 1) + v_{a} \\ Δ P_{t} = Δ P_{t - 1} + v_{Δ P} \\ S S_{t} = S S_{t - 1} + v_{SS} \end{matrix}

(7)

Here, v_a is the system noise term of the prediction equation for crack growth. $v_{Δ P}$ is the system noise term of the load. $v_{SS}$ is the system noise vector of the structure around the crack. Assuming that the time evolution of the crack length, the load, and the structure around the crack occurs randomly, the system noise is assumed to be normal distribution. The mean of the normal distribution is set to zero, and the standard deviation is determined empirically in order to avoid divergence in time evolution.

The observed equation is shown in Eq. with the crack length measurements as observed variable vector

y_{t}

y_{t} = H_{t} x_{t} + w_{t}

(8)

Here,

H_{t}

is the observation matrix that shows the relationship between the observed variable vector and the state variable vector.

y_{t}

is the crack length shown by the following equation:

y_{t} = a_{t}

(9)

where

H_{t}

is the following equation:

H_{t} = [1 0 \dots 0]

(10)

The size of $H_{t}$ is determined by the crack length as a, load range $Δ P$ ⁠, and the number of shape SS parameters. w_t is the observed error of the fatigue crack length. It is assumed that observed data follow the normal distribution, taking measurement error into account. The mean of the normal distribution is set to zero, and the standard deviation is determined based on the measurement error of the method being measured. From Eq. (8), the observed equation is linear, and from Eq. (6), the state equation is a nonlinear combination.

2.2 State Estimation by Ensemble Kalman Filter.

The problem of predicting crack propagation is estimated with the EnKF [29]. Figure 2 shows the flow of the state estimation with it. The state equation is given by Eq. (7). The observed equation is given by Eq. (10).

Fig. 2

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Flow of the Ensemble Kalman Filter: Flow diagram of data assimilation with EnKF to obtain crack length: State quantities are entered into state equations to make a one-step-ahead forecast. New state quantities are estimated from the one-step-ahead forecast and observed results.

The EnKF uses an ensemble approximation of the probability distribution of the state variable vectors [29]. The probability distribution is obtained of a random state variable vector created from the state and observed equations shown in the following equation:

X_{t - 1 | t - 1} = [x_{t - 1 | t - 1}^{(1)}, x_{t - 1 | t - 1}^{(2)}, \dots, x_{t - 1 | t - 1}^{(M)}]

(11)

Here, M of

x^{(M)}

is the total number of ensembles. The symbol t denotes the discretized time-step, whereas

X_{t - 1 | t - 1}

denotes the value of X at time-step t − 1, predicted based on the information available at time-step t − 1. The probability density function based on

Y^{t - 1}

of the state variable vector in Eq. is then given by Eq. in the ensemble approximation

p (x_{t - 1} | Y^{t - 1}) ≐ \frac{1}{M} \sum_{i = 1}^{M} δ (x_{t - 1} - x_{t - 1 | t - 1}^{(i)})

(12)

The EnKF is described using the ensemble approximation in Eq. (12).

In this paper, the probability distribution of initial ensemble values is considered as follows: It is assumed that initial values of crack length follow the normal distribution, taking measurement error into account. The mean of the normal distribution is set to zero, and the standard deviation is determined based on the measurement error of the method being measured. The initial value of the load is assumed to follow a normal distribution. The mean of the normal distribution is entered as is, given the uncertain nature of the information. The standard deviation is then determined empirically from the uncertainty of the information. In other words, the load-bearing capacity of the structure should be reasonably set based on the variation of loads that the structure is subjected to in the actual environment. It is assumed that the initial values of the structure around the crack are uniformly distributed, with the source of uncertainty being dimensional tolerance. In the absence of further information, the range of the uniform distribution is assumed to be the range of the dimensional tolerance. The initial ensemble of state variables is constructed based on the assumed probability distribution.

The flow of the EnKF is shown according to Fig. 2. The initial ensemble is then system updated with the state equations shown in Fig. 2. First, ensemble member

x_{t - 1 | t - 1}^{(i)}

is randomly generated in accordance with the specified probability distribution. The number of ensemble members to be created is represented by the variable M. Prior research [28] has indicated that increasing the number of ensembles may potentially contribute to a reduction in estimation error. The number of ensembles is typically determined through an empirical process, with the aim of balancing the computational cost and accuracy for each problem. Ensemble noise

{(v_{t}^{(i)})}_{i = 1}^{M}

is randomly generated in accordance with the specified probability distribution.

x_{t | t - 1}^{(i)}

denotes the value of

x^{(i)}

at time-step t, predicted based on the information available at time-step t − 1. Update each ensemble member with state Eq. to obtain the ensemble members of the predictive distribution

x_{t | t - 1}^{(i)} = f (x_{t - 1 | t - 1}^{(i)}, v_{t}^{(i)})

(13)

The state variable vector in Eq. is created from the obtained ensemble members.

x_{t | t - 1}^{M}

is the mean in the one-step-ahead forecast of

x_{t}

⁠.

x_{t | t - 1}^{M}

is obtained in the following equation:

x_{t | t - 1}^{M} = \frac{1}{M} \sum_{i = 1}^{M} x_{t | t - 1}^{(i)}

(14)

\tilde{x_{t | t - 1}^{(i)}}

is the error in the one-step-ahead forecast for an ensemble member.

\tilde{x_{t | t - 1}^{(i)}}

is obtained by the following equation:

\tilde{x_{t | t - 1}^{(i)}} = x_{t | t - 1}^{(i)} - x_{t | t - 1}^{M}, i = 1, \dots, M

(15)

\tilde{X_{t | t - 1}}

is the ensemble matrix of the one-step-ahead forecast errors.

\tilde{X_{t | t - 1}}

is expressed by the following equation:

\tilde{X_{t | t - 1}} = [\tilde{x_{t | t - 1}^{(1)}}, \tilde{x_{t | t - 1}^{(2)}}, \dots, \tilde{x_{t | t - 1}^{(M)}}]

(16)

\tilde{X c_{t | t - 1}}

represents the covariance matrix of the forecast errors obtained through the ensemble approximation illustrated in Eq. .

\tilde{X c_{t | t - 1}}

is expressed by the following equation:

\tilde{X c_{t | t - 1}} = \frac{1}{M - 1} \sum_{i = 1}^{M} \tilde{x_{t | t - 1}^{(i)}} {\tilde{x_{t | t - 1}^{(i)}}}^{T}

(17)

The next step is to measurement update shown in Fig. 2. In this study, the Kalman gain is calculated since the observed equation in Eq. is linear. The ensemble of observed errors

{(w_{t}^{(i)})}_{i = 1}^{M}

is created following a normal distribution.

w_{t | t - 1}^{M}

is the mean of the observed errors.

w_{t | t - 1}^{M}

is obtained by the following equation:

w_{t | t - 1}^{M} = \frac{1}{M} \sum_{i = 1}^{M} w_{t | t - 1}^{(i)}

(18)

\tilde{w_{t | t - 1}^{(i)}}

is the error from the mean of the observed errors.

\tilde{w_{t | t - 1}^{(i)}}

is obtained by the following equation:

\tilde{w_{t | t - 1}^{(i)}} = w_{t | t - 1}^{(i)} - w_{t | t - 1}^{M}, i = 1, \dots, M

(19)

\tilde{W_{t | t - 1}}

is the ensemble matrix of

\tilde{w_{t | t - 1}^{(i)}}

⁠.

\tilde{W_{t | t - 1}}

is shown by the following equation:

\tilde{W_{t | t - 1}} = [\tilde{w_{t | t - 1}^{(1)}}, \tilde{w_{t | t - 1}^{(2)}}, \dots, \tilde{w_{t | t - 1}^{(M)}}]

(20)

\tilde{W c_{t | t - 1}}

is the covariance matrix of

\tilde{w_{t | t - 1}^{(i)}}

⁠.

\tilde{W c_{t | t - 1}}

is shown by the following equation:

\tilde{W c_{t | t - 1}} = \frac{1}{M - 1} \sum_{i = 1}^{M} \tilde{w_{t | t - 1}^{(i)}} {\tilde{w_{t | t - 1}^{(i)}}}^{T}

(21)

The Kalman gain is derived from the system update and the measurement update. From the above, the Kalman gain is obtained from

H_{t}, \tilde{X c_{t | t - 1}}

⁠, and

\tilde{W c_{t | t - 1}}

⁠.

\tilde{K_{t}}

is Kalman gain.

\tilde{K_{t}}

is shown by the following equation:

\tilde{K_{t}} = \tilde{X c_{t | t - 1}} {H_{t}}^{T} {(H_{t} \tilde{X c_{t | t - 1}} {H_{t}}^{T} + \tilde{W c_{t | t - 1}})}^{- 1}

(22)

The Kalman filter is updated from the obtained Kalman gains as follows.

x_{t | t}^{(i)}

is the value of

x^{(i)}

at time-step t, predicted based on the information available at time-step t. The posterior estimate of the updated ensemble becomes the following equation:

x_{t | t}^{(i)} = x_{t | t - 1}^{(i)} + \tilde{K_{t}} (y_{t} + w_{t}^{(i)} - H_{t} x_{t | t - 1}^{(i)})

(23)

$x_{t | t}^{(i)}$ represents the updated crack length and uncertain parameters. $x_{t | t}^{(i)}$ is the output from a single EnKF shown in Fig. 2.

X_{t | t}

is the posterior estimate matrix for

x_{t | t}^{(i)}

⁠.

X_{t | t}

is shown by the following equation:

X_{t | t} = [x_{t | t}^{(1)}, x_{t | t}^{(2)}, \dots, x_{t | t}^{(M)}]

(24)

From the above,

x_{t | t}^{M}

denotes the mean of the posterior estimates.

x_{t | t}^{M}

is shown by the following equation:

x_{t | t}^{M} = \frac{1}{M} \sum_{i = 1}^{M} x_{t | t}^{(i)}

(25)

\tilde{x_{t | t}^{(i)}}

is the error from

x_{t | t}^{M}

⁠.

\tilde{x_{t | t}^{(i)}}

is shown by the following equation:

\tilde{x_{t | t}^{(i)}} = x_{t | t}^{(i)} - x_{t | t}^{M}

(26)

\tilde{X_{t | t}}

is the ensemble matrix of

\tilde{x_{t | t}^{(i)}}

⁠.

\tilde{X_{t | t}}

is shown by the following equation:

\tilde{X_{t | t}} = [\tilde{x_{t | t}^{(1)}}, \tilde{x_{t | t}^{(2)}}, \dots, \tilde{x_{t | t}^{(M)}}]

(27)

\tilde{X c_{t | t}}

is the covariance matrix of

\tilde{x_{t | t}^{(i)}}

⁠.

\tilde{X c_{t | t}}

is shown by the following equation:

\tilde{X c_{t | t}} = \frac{1}{M - 1} \sum_{i = 1}^{M} \tilde{x_{t | t}^{(i)}} {\tilde{x_{t | t}^{(i)}}}^{T}

(28)

The EnKF is repeated with the ensemble posterior estimate as the new initial value.

2.3 Derivation of Measurement Intervals by One-Step-Ahead Forecast.

Next, we predict the time to reach the critical crack length from the results of the EnKF one-step-ahead forecast. The measurement timing is obtained by setting the required reliability for the predicted time to reach the critical crack length. The estimation flow is shown in Fig. 3. A schematic diagram of the estimation is shown in Fig. 4.

Fig. 3

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Flow to determine measurement timing based on a one-step-ahead forecast: Add process to determine measurement timing in one-step-ahead forecast to flow in Fig. 2. Approximate crack growth lifetime of one-step-ahead forecast with a probability distribution. Determine measurement timing based on predicted probability distribution and reliability.

Fig. 4

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Methods flow to determine measurement timing based on a one-step-ahead forecast: We predict time to reach critical crack length from results of EnKF one-step-ahead forecast in Fig. 4(a). Vertical axis of Fig. 4(a) is crack length, and horizontal axis is number of iterations expressed as logarithm of 10 bases. Vertical axis of Fig. 4(b) shows distribution of number of cycles to reach critical crack length predicted for each ensemble from final observed results. Horizontal axis of Fig. 4(b) is number of iterations expressed as logarithm of 10 bases. Number of cycles to be reached is approximated by a three-parent Weibull probability distribution in Fig. 4(b). Maximum number of cycles that satisfy predetermined reliability is obtained from an approximate probability distribution. At obtained cycles, crack length is measured. Timing of next measurement for each one is estimated in Fig. 4(c). Vertical and horizontal axis is same as Fig. 4(a). (a) One-period-ahead forecast. (b)Histogram of cycles reaching crack length. (c) One-period-ahead forecast with number of cycles satisfying reliability.

As shown in Fig. 3, the process of determining the measurement timing is added with a one-step-ahead forecast to the flow shown in Fig. 2. The schematic diagram is shown in Fig. 4. Its upper left shows a schematic diagram of the one-step-ahead forecast to the critical crack length. The initial value or the ensemble after filtering by the observed data predicts the number of cycles to be reached, up to the critical crack length, one period in advance. A schematic diagram of the probability distribution approximating the number of cycles to reach the critical crack length is shown in the upper right-hand panel of Fig. 4. The number of cycles in which the one-step-ahead forecast reaches the critical crack length is approximated by a probability distribution. We seek the maximum number of cycles that satisfies the reliability in the approximated probability distribution. One method for determining the reliability is to utilize the value assumed in the design of the equipment in which the crack has occurred. For instance, for pressure vessels, the safety factor required for design is estimated at the reliability of 75% or higher [44]. We propose the number of cycles as the amount to be measured next time.

The subsequent number of cycles to be measured should not be smaller than the number that started the prediction. To limit the range to be predicted, a probability distribution is assumed in this paper in which the probability of a value smaller than the lower limit is zero. The probability distribution functions for which lower bounds exist are the three-moment Weibull distribution [45], the three-moment lognormal distribution [46,47], and the three-moment gamma distribution [48]. In this paper, the distribution of the number of cycles to reach the crack length is assumed to be a three-parent Weibull distribution [45] as shown in the following equation:

f (z | α, β, γ) = {\begin{matrix} \frac{β}{α} (\frac{z - γ}{α}) exp (- (\frac{z - γ}{α})) & if z > γ \\ 0 & if z \leq γ \end{matrix}

(29)

Here, α is a scale parameter, β is a shape parameter, and γ is a location parameter. The three parameters were estimated as maximum likelihood estimation using in MATLAB [49,50].

3 Problem of Predicting Crack Propagation

In this section, we validate our method for estimating the state of the crack propagation presented in Sec. 2. The problem to be validated, the physical model, and the constraints for computing the EnKF are presented. To create simulated observed data, we augmented with noise the crack growth results from the results of Forman et al. [33]. Two methods were used: one in which the number of measurement cycles is predetermined and another in which the measurement interval is determined by the one-step-ahead forecast of the EnKF.

3.1 Problem to Be Validated.

The method presented in Sec. 2 was validated with the crack growth results obtained from a CT specimen and the simulated observed data with noise assuming a normal distribution. A schematic diagram of the CT specimen is shown in Fig. 5. We used the fatigue crack propagation results for BeCu [33]. The propagation of cracks was measured using the DC current potential method. The details of the crack growth properties used in the validation are given in Appendix A. In this paper, the geometry parameters to be SS are thickness B and width W of Fig. 5.

Fig. 5

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Shape of object for which crack length was estimated: Schematic diagram of CT specimen and dimensions indicated by symbols

3.2 Physical Model of Crack Propagation.

We determined the stress intensity factor used in the physical model of the crack propagation. The stress intensity factor can be obtained by theoretical equations based on experimental results or by numerical analysis. A theoretical equation based on the experimental results of CT specimens is presented in ASTM E 647-23a [51]. The finite element method, which is a numerical analysis approach, can be used to obtain the stress intensity factor [52] by displacement extrapolation, stress extrapolation, the J-integral method, and the interaction integral method. In addition, there are numerical methods to reproduce the crack propagation based on the calculated parameters. Numerical analysis includes the finite element method and the particle method, which is one of the meshless methods. Finite element crack propagation methods include extended finite element method [53], where the crack propagates within the element, and other methods [54,55], where the crack propagating element is deleted and the surrounding mesh is regenerated. When these parameters are obtained using the finite element method, the elements surrounding the crack must be reduced in size, which increases the computational cost. Meshless methods, such as the particle method, do not require recreation of the model due to crack propagation because no elements are created, although its computational cost is rather high [56].

In this report, we used the theoretical equation for the CT specimen to reduce the computational cost of the iterative calculations with the EnKF. The formula for the stress intensity factor of the specimen [51] Eq. is shown in Fig. 5

Δ K = \frac{Δ P}{B \sqrt{W}} F (ξ)

(30)

Here, shape function F in Eq. is shown by the following equation:

F (ξ) = \frac{2 + ξ}{{(1 - ξ)}^{(3 / 2)}} (0.886 + 4.46 ξ - 13.32 ξ^{2} + 14.72 ξ^{3} - 5.6 ξ^{4})

(31)

The ξ in Eq. is shown by the following equation:

ξ = \frac{a}{W}

(32)

The valid range of Eq. to Eq. is the following equation:

\frac{a}{W} ≧ 0.2

(33)

3.3 Conditions for Ensemble Calculations.

Next, we show the computational conditions for the ensemble one-step-ahead forecast. Figure 6 shows the flow of updating the Kalman gain from the one-step-ahead forecast and the measurements to obtain the posterior estimates. In this paper, the EnKF model is used when the one-step-ahead forecast is beyond the set range, as shown in Fig. 6. Those ensembles that fall outside the set range are updated to the nearest value within the set range. The Ensemble Kalman Filter must be computed on the entire ensemble, including the ensemble that has been replaced. Equation shows the range within which the ensembles depicted in Fig. 6 can exist

{\begin{matrix} 0 ≦ Δ P \\ 0.0250 ≦ W \\ \frac{W}{20} ≦ B ≦ \frac{W}{4} \\ 0.3 W ≦ a ≦ 0.7 W \end{matrix}

(34)

Fig. 6

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Outlier processing of one-step-ahead forecast results shown in probability density schematic: Vertical axis of Fig. 6 is probability density, and horizontal axis is state quantity. Steps of EnKF calculation are illustrated for each horizontal axis. Set state quantity range within which the ensemble to be predicted realistically exists. The initial step is the creation of a state distribution derived from an ensemble of state variables. Subsequently, a distribution of one-period-ahead forecasts is generated, with each ensemble undergoing an update through the application of system equations. The predicted distribution is then formed by updating the one-period-ahead forecasts that fall outside the state quantity range to the nearest value within that range. Finally, the ensemble is updated with the Kalman gain obtained from the predicted distribution and the observed distribution.

4 Prediction of Crack Propagation Based on Pre-Determined Measurement Timing

4.1 Observed Data and Timing to Predict Cracks.

In this paper, simulated observed data are generated from the crack propagation measurements results of in Forman et al. [33]. The crack length was measured by Forman et al. using the DC current potential method. Simulated truth value

a_{tru}

is generated by interpolating the relationship between the propagated crack length and the number of iterations. Simulated truth value

a_{tru}

is a vector with N_inter elements, which is the number of interpolated cycles. Noise is added to the simulated truth values using Eq. to create the simulated observed data

y_{obs} (j) = a_{tru} (j) + N (0, {(a_{tru} (j) \times NSD)}^{2})

(35)

Here, j denotes the interpolated cycles, and the maximum is N_inter. $N (0, {(a_{tru} (j) \times NSD)}^{2})$ denotes a normal distribution with mean 0 and variance ${(a_{tru} (j) \times NSD)}^{2}$ ⁠. The standard deviation of each simulated truth value was multiplied by the NSD ratio. The NSD ratio was noise standard deviation rate. Simulated spectra were created by multiplying the standard deviation of the simulated truth values by the ratios of five levels: 1%, 2%, 5%, 7%, and 10%. Figure 7 shows the simulated observed data with noise, with a standard deviation of 2% of the ratio added to the simulated truth values. The vertical axis of the figure shows the crack length and the horizontal axis shows the number of iterations.

Fig. 7

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Simulated observed data: Simulated observed results were interpolated to produce results other than the number of cycles measured in previous literature. Measurement error with a mean of 0 and a standard deviation of 2% was assigned to the interpolated results.

Table 1 shows the parameters and the uncertainty values used in the physical model presented so far. The calculated crack growth characteristics are shown in Appendix B. The mean value of initial crack length was employed as the initial crack length of the CT specimens described in the literature [33]. The load at the crack growth test in proximity to the initial number of cycles in this prediction is $1875 kN$ [33]. $1875 (kN)$ is the mean value of the probability distribution of the initial load value. The ground truth of specimen thickness B is $0.00894 m$ and width W is $0.0508 m$ [33].

Table 1

Parameter of ensemble Kalman filter: Parameters of Paris law, a physical model, are shown

Description	Value
Paris’s law parameter: m	8.16
Paris’s law parameter: C	$2.79 \times 10^{- 16}$
Estimated initial crack length	$N (μ_{a 0} = 0.02, {(σ_{a 0} = μ_{a 0} \times 0.01)}^{2})$
Estimated initial load value	$N (μ_{Δ P 0} = 1875, {(σ_{Δ P 0} = μ_{Δ P 0} \times 0.01)}^{2})$
Estimated initial thickness	$U (B_{0} = 0.0089 \pm 0.001)$
Estimated initial width	$U (W_{0} = 0.0508 \pm 0.001)$
System noise of crack length	$N (μ_{asys} = 0, {(σ_{asys} = μ_{a 0} \times 5 \times 10^{- 4})}^{2})$
System noise of load value	$N (μ_{Δ P} = 0, {(σ_{Δ P} = μ_{Δ P 0} \times 5 \times 10^{- 5})}^{2})$
System noise of thickness	$N (μ_{B} = 0, {(σ_{B} = B_{0} \times 1 \times 10^{- 5})}^{2})$
System noise of width	$N (μ_{W} = 0, {(σ_{W} = W_{0} \times 1 \times 10^{- 5})}^{2})$
Measurement noise of crack length	$N (μ_{ameasure} = 0, {(σ_{ameasure} = μ_{a 0} \times 0.02)}^{2})$
Number of ensembles member	$1000$

Description	Value
Paris’s law parameter: m	8.16
Paris’s law parameter: C	$2.79 \times 10^{- 16}$
Estimated initial crack length	$N (μ_{a 0} = 0.02, {(σ_{a 0} = μ_{a 0} \times 0.01)}^{2})$
Estimated initial load value	$N (μ_{Δ P 0} = 1875, {(σ_{Δ P 0} = μ_{Δ P 0} \times 0.01)}^{2})$
Estimated initial thickness	$U (B_{0} = 0.0089 \pm 0.001)$
Estimated initial width	$U (W_{0} = 0.0508 \pm 0.001)$
System noise of crack length	$N (μ_{asys} = 0, {(σ_{asys} = μ_{a 0} \times 5 \times 10^{- 4})}^{2})$
System noise of load value	$N (μ_{Δ P} = 0, {(σ_{Δ P} = μ_{Δ P 0} \times 5 \times 10^{- 5})}^{2})$
System noise of thickness	$N (μ_{B} = 0, {(σ_{B} = B_{0} \times 1 \times 10^{- 5})}^{2})$
System noise of width	$N (μ_{W} = 0, {(σ_{W} = W_{0} \times 1 \times 10^{- 5})}^{2})$
Measurement noise of crack length	$N (μ_{ameasure} = 0, {(σ_{ameasure} = μ_{a 0} \times 0.02)}^{2})$
Number of ensembles member	$1000$

Mean and variance of initial values, system noise, and observed noise for the EnKF are shown. N is normal distribution. mu is mean of normal distribution, σ is standard deviation of normal distribution. Variance of normal distribution is square of standard deviation. Initial values of shapes B and W are uniform distributions U. Number of ensemble members is calculated in this section.

To obtain the amount of a crack’s propagation from Eq. (30), its propagation is integrated over n iterations, as shown in Eq. (7). There is concern that the amount of propagation may deviate from the truth if the range of the number of iterations to be integrated is large. In this paper, the amount of progress of the one-period-ahead forecast is calculated by dividing the next number of measurement cycles into 100 segments. System noise is added to 100 iterations because Eq. (7) is computed for each of the divided calculations.

The result in Fig. 7 is the simulated survey shown in Table 1. We assume that the initial values of the crack and the load as well as the system noise are distributed according to a normal distribution. We further assume that the initial values of shapes B and W have uniform distribution, and that the system noise for shapes B and W is distributed based on a normal distribution. The values used as simulated observed data are shown in Fig. 7. The start of the measurement is set to $2.7 \times 10^{6}$ times. Five observed data are simulated: $2.8 \times 10^{6}, 3 \times 10^{6}, 3.2 \times 10^{6}, 3.4 \times 10^{6}$ ⁠, and $3.5 \times 10^{6}$ cycles.

4.2 Prediction Results.

Figure 8 shows the crack growth predicted by the EnKF. Here the crack length to compare the predicted lifetime is set to $0.032 m$ ⁠. Figure 8(a) shows the predicted results of the crack propagation, where the vertical axis represents the crack length and the horizontal axis represents the number of iterations. Figure 8(b) shows the distribution of the number of cycles in which the one-step-ahead forecast results after the last observed reached a crack length of $0.032 m$ ⁠. The vertical axis indicates the number of ensembles reached, and the horizontal axis indicates the number of iterations. The sample mean, the sample variance, and the sample standard deviation are shown in Fig. 8(b). The sample mean is $3.86 \times 10^{4}$ cycles, which is 0.3% longer than the ground truth lifetime. The sample mean is in good agreement with the ground truth lifetime in Fig. 8(b). The results of the crack growth of the CT specimens validated the proposed method, which predicted the crack growth lifetime with high accuracy. Furthermore, the reliability of the sample mean predicted from the sample variance was validated.

Fig. 8

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Estimated results with noise standard deviation 2%: Vertical axis of Fig. 8(a) is crack length, and horizontal axis is number of iterations expressed as a logarithm of 10 bases. Vertical axis of Fig. 8(b) shows distribution of number of cycles to reach critical crack length of $0.032 m$ predicted for each ensemble from final measurement results. Horizontal axis of Fig. 8(b) is number of iterations expressed as a logarithm of 10 bases. Sample mean, sample variance, and sample standard deviation of predicted number of cycles are shown in Fig.8(b). (a) Estimated result noise standard deviation 2% and (b)distribution of number of cycles to reach a critical crack length of $0.032 m$ ⁠.

Next Fig. 9 shows the uncertain parameter transitions in the progress forecast. Figure 9(a) shows the predicted results for the load range. The vertical axis is the load range; the horizontal axis is the number of repetitions. Figure 9(b) is the thickness of the CT specimen; Fig. 9(c) is its width. The vertical axis is the dimension of the thickness or the width of the CT specimen, and the horizontal axis is the number of iterations. Figure 9(d) is the stress intensity factor range obtained from Eqs. (30) to (31). The vertical axis is the stress intensity factor range, and the horizontal axis is the number of iterations.

Fig. 9

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Variation of parameters and stress intensity factors estimated simultaneously with crack length: Vertical axes in Fig. 9(a) indicate loading range, vertical axes in Fig. 9(b) indicate thickness, vertical axes in Fig. 9(c) indicate width, and vertical axes in Fig. 9(d) indicate stress intensity factor. Horizontal axis of Fig. 9 is number of iterations expressed as a logarithm of 10 bases. (a) Estimated load, (b) estimated thickness, (c) estimated width, and (d) estimated $Δ K$ ⁠.

Such uncertain parameters as load range, width, and thickness do not agree with the ground truth. The stress intensity factor range calculated from the uncertain parameters approaches the ground truth because the former was directly related to the crack length in the physical model. As the load range, width, and thickness are predicted simultaneously to achieve a stress intensity factor range that approaches with the ground truth, there may be discrepancies between the predicted and the ground truth for each of these parameters. Observe that the uncertain parameters of the physical model of the crack propagation vary in relation to one another, thereby bringing the stress intensity factor range closer to the ground truth.

4.3 Impact of Observed Error on Prediction Accuracy.

The accuracy of the prediction of the crack growth life was compared by varying the error of the simulated observed data. The measurement interval is the number of cycles described in Sec. 4.1. The proposed method predicts the crack growth life as simulated observed data, where the NSD in Eq. (35) was varied in five steps: 1%, 2%, 5%, 7%, and 10%. In predicting with EnkF from each simulated observed data, the observed error w is determined from the initial crack length and NSD.

The EnkF forecast employs the observed error indicated by σ_ameasure in Table 1. The value of $μ_{a 0}$ multiplied by the ratio is σ_ameasure. In this paper, the ratio to be multiplied by $μ_{a 0}$ is identical to the NSD ratio.

The number of cycles required to reach the critical crack length of $0.032 m$ was predicted for all the simulated observed data. The sample mean and sample standard deviation of the predicted number of cycles are presented in Fig. 10. The left-hand vertical axis in Fig. 10 is the sample mean, the right-hand vertical axis is the sample standard deviation, and the horizontal axis is the σ_ameasure used in the EnKF. If the value of σ_ameasure is greater, the error associated with the simulated observed data will also be greater.

Fig. 10

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Relation between measurement error and estimated results: Left vertical axis is sample mean of number of cycles to reach critical crack length, and right vertical axis is sample standard deviation of predicted result. Horizontal axis shows standard deviation of measurement error.

From Fig. 10, the difference between the sample mean and the ground truth lifetime is small up to $0.0014 m$ ⁠, where the standard deviation of the observed error corresponds to 7% of the initial crack length. At $0.002 m$ ⁠, where the standard deviation of the observed error corresponds to 10% of the initial crack length, the sample mean predicts a life 20% longer than the ground truth life. The impact of observed error intensified, and even after five measurements, the variation in crack length in EnKF remained unabated, and the number of cycles required to reach the critical crack length exhibited variability, as illustrated in the exemplar depicted in Fig. 4(a). The dataset comprised a greater number of ensembles with diminished estimates of the crack growth rate and a greater number of results predicting a longer number of cycles to reach the critical crack length. The accuracy of the sample mean predictions from the simulated observed data is high, since they are within $\pm 50 %$ of the ground truth lifetime, as shown in Fig. 10.

The sample standard deviations up to a standard deviation of $0.0014 m$ of the observed error are also less than 10% in terms of the coefficient of variation. The predicted sample standard deviations also increase as the simulated observed data error increases, and the sample standard deviations increase significantly in comparison to the sample mean.

The verification results from varying the error of the simulated observed data demonstrated that the sample mean lifetime predicted by the five measurements and the proposed method exhibited a high degree of agreement with the ground truth. The error of the simulated observed data significantly impacted the variation in the number of predicted cycles.

5 Prediction of Crack Propagation Based on Measurement Timing Determined by One-Step-Ahead Forecast

5.1 Observed Data to Predict Cracks.

Our method, which determined the measurement timing from the one-step-ahead forecast was validated. Figure 11 shows the simulated observed data used to predict the crack growth. The vertical axis is the crack length, and the horizontal axis is the number of iterations. The results of Forman et al. [33] are shown in Appendix A. Observed data other than number of cycles measured in prior literature [33] were linearly interpolated. Noise was added to the linearly interpolated results by a formula (35), as described in Sec. 4.1. The simulated observed data are created here with NSD that was set at 2%.

Fig. 11

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Simulated observed data: Simulated observed data were generated by incorporating errors into results obtained from prior literature [33]. The vertical axis is the crack length, and the horizontal axis is the number of iterations. Observed data other than number of cycles measured in prior literature [33] were linearly interpolated. Noise was added to linearly interpolated results by a formula (35), as described in Sec. 4.1. Simulated observed data are created here with NSD that was set at 2%.

For each measurement cycle determined from the one-period-ahead forecast, the simulated observed data shown in Fig. 11 were input to the EnKF. Five observations were conducted with the simulated observed data. Table 2 shows the parameters and the uncertainty values used in the physical model presented so far. The mean of the initial values was changed from Table 1 so that the simulated truth values were included in the variation of the initial values of the load and the crack length. The mean value of initial crack length was employed as the initial crack length of the CT specimens described in the literature [33]. The load at the crack growth test in proximity to the initial number of cycles in this prediction is $1750 kN$ [33]. $1750 kN$ is the mean value of the probability distribution of the initial load value. The initial values for Table 1 are $1.0 times 10^{6}$ cycles. The limiting crack length for the condition that determines the measurement interval is $0.032 m$ ⁠. The reliability that the critical crack length is not reached is set to 99%.

Table 2

Parameter of ensemble Kalman filter: Parameters of Paris law (a physical model) are shown

Description	Value
Paris’s Law parameter: m	8.16
Paris’s Law parameter: C	$2.79 \times 10^{- 16}$
Estimated initial crack length	$N (μ_{a 0} = 0.02, {(σ_{a 0} = μ_{a 0} \times 0.01)}^{2})$
Estimated initial load value	$N (μ_{Δ P 0} = 1750, {(σ_{Δ P 0} = μ_{Δ P 0} \times 0.01)}^{2})$
Estimated initial thickness	$U (B_{0} = 0.0089 \pm 0.001)$
Estimated initial width	$U (W_{0} = 0.0508 \pm 0.001)$
System noise of crack length	$N (μ_{asys} = 0, {(σ_{asys} = μ_{a 0} \times 5 \times 10^{- 4})}^{2})$
System noise of load value	$N (μ_{Δ P} = 0, {(σ_{Δ P} = μ_{Δ P 0} \times 5 \times 10^{- 5})}^{2})$
System noise of thickness	$N (μ_{B} = 0, {(σ_{B} = B_{0} \times 1 \times 10^{- 5})}^{2})$
System noise of width	$N (μ_{W} = 0, {(σ_{W} = W_{0} \times 1 \times 10^{- 5})}^{2})$
Measurement noise of crack length	$N (μ_{ameasure} = 0, {(σ_{ameasure} = μ_{a 0} \times 0.02)}^{2})$
Number of ensemble members	$1000$

Description	Value
Paris’s Law parameter: m	8.16
Paris’s Law parameter: C	$2.79 \times 10^{- 16}$
Estimated initial crack length	$N (μ_{a 0} = 0.02, {(σ_{a 0} = μ_{a 0} \times 0.01)}^{2})$
Estimated initial load value	$N (μ_{Δ P 0} = 1750, {(σ_{Δ P 0} = μ_{Δ P 0} \times 0.01)}^{2})$
Estimated initial thickness	$U (B_{0} = 0.0089 \pm 0.001)$
Estimated initial width	$U (W_{0} = 0.0508 \pm 0.001)$
System noise of crack length	$N (μ_{asys} = 0, {(σ_{asys} = μ_{a 0} \times 5 \times 10^{- 4})}^{2})$
System noise of load value	$N (μ_{Δ P} = 0, {(σ_{Δ P} = μ_{Δ P 0} \times 5 \times 10^{- 5})}^{2})$
System noise of thickness	$N (μ_{B} = 0, {(σ_{B} = B_{0} \times 1 \times 10^{- 5})}^{2})$
System noise of width	$N (μ_{W} = 0, {(σ_{W} = W_{0} \times 1 \times 10^{- 5})}^{2})$
Measurement noise of crack length	$N (μ_{ameasure} = 0, {(σ_{ameasure} = μ_{a 0} \times 0.02)}^{2})$
Number of ensemble members	$1000$

Mean and variance of initial values, system noise, and observed noise for the EnKF are shown. N is Normal distribution. mu is mean of normal distribution, σ is standard deviation of normal distribution. Variance of normal distribution is square of standard deviation. Initial values of B and W are uniform distribution U. The number of ensemble members is calculated in this section.

5.2 Prediction Results.

The results of predicting crack propagation while determining the number of measurement cycles are shown in Fig. 12, which shows the results of the crack length estimation. The vertical axis is the crack length; the horizontal axis is the number of cycles. The average number of cycles resulting in the same crack length in the predicted results following the final EnKF is shown in Fig. 12(a). The average number of cycles resulting in the same crack length is close to the ground truth.

Fig. 12

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Forecasting results by determining timing of next measurement with a one-step-ahead forecast: Vertical axis of Fig. 12(a) is crack length; horizontal axis is number of iterations expressed as a logarithm of 10 bases. Limiting crack length for condition that determines measurement interval is $0.032 m$ ⁠. Reliability that critical crack length is not reached is set to 99%. Vertical axis of Fig. 12(b) shows distribution of number of cycles to reach critical crack length of $0.032 m$ predicted for each ensemble from final measurement results. Horizontal axis of Fig. 12(b) is number of iterations expressed as a logarithm of 10 bases. Sample mean, sample variance, and sample standard deviation of predicted number of cycles are shown in Fig. 12(b). (a) Result of determining and predicting number of measurement cycles with reliability 99%. (b) Distribution of number of cycles to reach a critical crack length of $0.032 m$ ⁠.

The distribution of the number of cycles that reached the critical crack length of $0.032 m$ in the predicted progression after the last EnKF is shown in Fig. 12(b). The vertical axis is the frequency of the occurrence and the horizontal axis is the number of cycles. The sample mean, the sample variance, and the sample standard deviation of the predicted number of cycles are shown in Fig. 12(b). The predicted lifetime, which is 4% longer than the ground truth, is as accurate as that shown in Fig. 10.

The intervals between the measurement cycles determined from the one-step-ahead forecast are shown in Fig. 13. The vertical axis shows the intervals between cycles per measurement cycle; the horizontal axis shows the measurement cycle.

Fig. 13

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Measurement interval determined by one-step-ahead forecast: Vertical axis is number of cycles in measurement interval; horizontal axis is measurement timing. For example, “1–2” indicates number of cycles between first and second measurement timings. Relationship between reliability in determining number of measurement cycles and bar is shown in figure.

From Fig. 13, observe that the first measurement interval is the second longest of the five measurements. This is due to the slow crack growth rate predicted from the initial conditions. The second measurement interval is the longest of the five. Given that the second simulated observed data was smaller than the initial value, we predicted that the crack growth rate would be even slower than the initial condition. As the crack grew, the measurement interval shortened. This is because the crack growth rate in the one-step-ahead forecast increased, while the variation in the number of forecast cycles decreased to reach the critical crack length.

The EnKF method, which determined the measurement interval based on a one-step-ahead forecast and reliability indicating that the critical crack length would not be reached, predicted the crack propagation life within 4% errors. The EnKF method was employed to resolve a time-evolving nonlinear model of crack growth, with the objective of estimating uncertain parameters from both previously measured and newly measured crack lengths. The estimated parameters were then utilized to enhance the precision of crack growth predictions. The ability to reflect changes in the crack length and the growth rate allows for planning for appropriate measurements and repairing frequencies.

6 Conclusions

This paper proposed a method to predict crack growth lifetimes from the ground truth of crack lengths measured by an Ensemble Kalman Filter that incorporated a physical model of crack growth. Our proposed method was validated by applying it to simulated observed data of crack growth in CT specimens. The following are our results:

The Ensemble Kalman Filter improved the accuracy of predicting crack growth lifetimes by updating the measured crack length and the uncertain parameters affecting crack growth. Integrating a physical model of crack growth into the Ensemble Kalman Filter enhanced the accuracy of estimating the crack length and uncertain parameters for each measurement.
A method was proposed to determine the crack measurement interval from the one-step-ahead forecast of the Ensemble Kalman Filter. The number of cycles to reach the criterion crack length is determined from the one-step-ahead forecast of the ensemble. The distribution of the number of cycles is approximated by a probability distribution, and the measurement interval is determined based on the reliability that the reference crack length is not reached.
The proposed method accurately predicted the crack growth life of a CT specimen with noisy simulated observed data of the crack growth results. These data with a standard deviation of 10% of the maximum crack growth result were used to validate the method, and the predicted growth lifetime was accurate after five measurements. The variation of the predicted lifetime increased with the magnitude of the noise.
The Paris law was used as a crack growth model, and the uncertain parameters were the size of the crack growth target and the cyclic loading range. This crack growth model determines the crack length based on the range of the stress intensity factors. The stress intensity factor range is derived from uncertain parameters, which were varied in concert with one another for each crack measurement, thereby approximating the ground truth of the stress intensity factor range.
The measurement interval was determined based on a one-step-ahead forecast, which means that the measurement interval is wider under slow crack growth conditions and shorter when the crack growth is faster. The measurement interval can be set based on the reliability, allowing for efficient prediction of the crack growth lifetime.

In this paper, we assumed that the uncertainty in the physical properties of crack propagation is negligible. We will add uncertainty in the physical properties of crack propagation to improve the accuracy of the predictions for social implementation. The proposed method will be further validated by modifying the target of crack growth prediction and the method of crack measurement. Nondestructive crack measurement methods such as digital image correlation and ultrasonic testing are planned.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

a =: crack length
B =: CT specimen thickness
BeCu =: beryllium copper
C =: fatigue crack growth rate constant of Paris law
$\frac{d a}{d n}$ =: crack growth rate
EnKF =: ensemble Kalman filter
$F (ξ)$ =: shape function of stress intensity factor
$H_{t}$ =: observation matrix
$\tilde{K_{t}}$ =: Kalman gain
M =: total number of ensembles
m =: stress intensity exponent of Paris law
$N$ =: normal distribution
$NSD$ =: noise standard deviation rate
$SS$ =: dimension vector of the structure around the crack
$U$ =: uniform distribution
va =: system noise term of prediction equation for crack growth
$v_{t}$ =: vector of system noise
$v_{SS}$ =: system noise vector of structure around crack
W =: CT specimen width
wt =: vector of observed noise
$\tilde{W_{t | t - 1}}$ =: ensemble matrix of observed errors
$X_{t | t - 1}$ =: ensemble matrix of state variables
$\tilde{X c_{t | t - 1}}$ =: covariance matrix of forecast errors
$y_{t}$ =: vector of observed data
α =: shape parameter of 3-parameter Weibull distribution
β =: scale parameter of 3-parameter Weibull distribution
γ =: location parameter of 3-parameter Weibull distribution
$Δ K$ =: stress intensity factor range
$Δ n$ =: increase in number of repetitions
$Δ P$ =: load range
$ξ$ =: nondimensional shape parameters in shape functions
$v_{Δ P}$ =: system noise term of prediction equation for crack growth

Appendix A: Crack Propagation Results for CT Specimen

The data on crack growth utilized in Secs. 4 and 5 are presented here. Fig. 14 shows the results for BC-R4 and BC-R8 described in previous literature [33]. The vertical and horizontal axes indicate the crack length and the number of iterations.

Fig. 14

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Test results of crack propagation used for prediction: Vertical axis shows crack length; horizontal axis shows number of repetitions in logarithms of 10 bases

Appendix B: Crack Propagation Parameter

The crack propagation characteristics utilized in Tables 1 and 2 are shown in Fig. 15. The vertical axis in the figure shows the crack growth rate, and the horizontal axis shows the stress intensity factor range. The stress ratios of BC-R4 and BC-R8 are both 0.4. The approximate result of the Paris law is shown in Fig. 15. The approximate range is 15 $MPa \sqrt{m}$ from the minimum stress intensity factor range. The values of C and m displayed in the equations in Fig. 15 were utilized in Tables 1 and 2.

Fig. 15

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Approximation of crack growth characteristics by Paris law: Vertical axis is crack growth rate in a logarithm of 10 bases; horizontal axis is stress intensity factor range in a logarithm of 10 bases. Stress ratio is 0.4. Approximate range is 15 $MPa \sqrt{m}$ from minimum stress intensity factor range.

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Crack Growth Prediction Based on Uncertain Parameters Using Ensemble Kalman Filter

Abstract

1 Introduction

2 Prediction by Kalman Filter and Derivation of Measurement Intervals

2.1 State Estimation Problem for Crack Propagation.

2.2 State Estimation by Ensemble Kalman Filter.

2.3 Derivation of Measurement Intervals by One-Step-Ahead Forecast.

3 Problem of Predicting Crack Propagation

3.1 Problem to Be Validated.

3.2 Physical Model of Crack Propagation.

3.3 Conditions for Ensemble Calculations.

4 Prediction of Crack Propagation Based on Pre-Determined Measurement Timing

4.1 Observed Data and Timing to Predict Cracks.

4.2 Prediction Results.

4.3 Impact of Observed Error on Prediction Accuracy.

5 Prediction of Crack Propagation Based on Measurement Timing Determined by One-Step-Ahead Forecast

5.1 Observed Data to Predict Cracks.

5.2 Prediction Results.

6 Conclusions

Data Availability Statement

Nomenclature

Appendix A: Crack Propagation Results for CT Specimen

Appendix B: Crack Propagation Parameter

References

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Crack Growth Prediction Based on Uncertain Parameters Using Ensemble Kalman Filter

Abstract

1 Introduction

2 Prediction by Kalman Filter and Derivation of Measurement Intervals

2.1 State Estimation Problem for Crack Propagation.

2.2 State Estimation by Ensemble Kalman Filter.

2.3 Derivation of Measurement Intervals by One-Step-Ahead Forecast.

3 Problem of Predicting Crack Propagation

3.1 Problem to Be Validated.

3.2 Physical Model of Crack Propagation.

3.3 Conditions for Ensemble Calculations.

4 Prediction of Crack Propagation Based on Pre-Determined Measurement Timing

4.1 Observed Data and Timing to Predict Cracks.

4.2 Prediction Results.

4.3 Impact of Observed Error on Prediction Accuracy.

5 Prediction of Crack Propagation Based on Measurement Timing Determined by One-Step-Ahead Forecast

5.1 Observed Data to Predict Cracks.

5.2 Prediction Results.

6 Conclusions

Data Availability Statement

Nomenclature

Appendix A: Crack Propagation Results for CT Specimen

Appendix B: Crack Propagation Parameter

References

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