Method for calculating the probability of crack detection by eddy current non-destructive testing system

Abstract

The application discloses a kind of vortex nondestructive testing system crack detection probability calculation method, this method first establishes the training dataset needed for polynomial chaos expansion element model. Based on training dataset, polynomial chaos expansion coefficient is calculated, and element model construction is completed. Then using root mean square error and normalized root mean square error verifies the accuracy of element model, if the accuracy requirement is not met, the number of training dataset needs to be increased to reconstruct element model. The element model is used instead of the time-consuming physical model to calculate the large amount of system response data required for crack detection probability, which can significantly improve the efficiency of eddy current nondestructive testing detection probability research.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of eddy current nondestructive testing, in particular to a method for calculating the crack detection probability of an eddy current nondestructive testing system. BACKGROUND

[0002] Nondestructive testing, as one of the popular research fields today, aims to detect the crack defects and material discontinuity in the detection target without damaging it, and is a typical engineering application technology with low investment and high output. The study of detection probability is very important for quantifying the detection capability of the nondestructive testing system for defects and the reliability of the system. Nondestructive testing systems have certain detection errors and uncertainties, which will affect the detection probability of defects. Therefore, it is necessary to study the detection probability of defects considering the detection errors (such as errors introduced by detection engineers operating equipment, or errors caused by objective factors such as experimental environment). Considering that the high uncertainty in the detection system leads to a large number of random variables, it is difficult to achieve the efficiency required by the model-assisted detection probability study only by relying on efficient physical simulation models. Metamodel is a kind of efficient calculation model, which can be used to replace the accurate but time-consuming physical simulation model, so it has important practical significance for accelerating the solution of nondestructive testing detection probability problem. Generally, there are two methods to construct metamodel, one is the intrusive method, and the other is the non-intrusive method. The intrusive method needs to modify the existing model, which makes it difficult to apply to complex problems. SUMMARY

[0003] The present application relates to the field of eddy current nondestructive testing, in particular to a method for calculating the crack detection probability of an eddy current nondestructive testing system.

[0004] Technical scheme: In order to achieve the purpose of the present application, the present application proposes a method for calculating the crack detection probability of an eddy current nondestructive testing system, which comprises the following steps:

[0005] Step 1: Sampling the uncertainty variables in the process of detecting material cracks by the eddy current nondestructive testing system, obtaining the system response at the sampling points corresponding to different lengths of cracks through experiments, taking the normalized sampling points and the corresponding system responses as the required data set, the sampling method is Latin hypercube sampling, and the data set obtained by Monte Carlo sampling is used as the training data set and the validation data set;

[0006] Step 2: Calculate the polynomial chaos expansion coefficients based on the training data set, so as to obtain the polynomial expression of the system response, and complete the construction of the metamodel;

[0007] Step 3: Test the accuracy of the meta-model based on the validation sample set. If the accuracy requirement is not met, increase the amount of training dataset and rebuild the meta-model until the accuracy requirement is met.

[0008] Step 4: Use the constructed meta-model to calculate the system response, and calculate the detection probability of the crack based on the response data.

[0009] Furthermore, the specific steps of step 1 are as follows:

[0010] (1) Uncertain variables x1, x2, ..., x in the process of detecting material cracks using an eddy current nondestructive testing system. m Using Latin hypercube sampling, n sets of sample points are selected. After normalizing these n sets of sample points, the following results are obtained: x ti j Represents the uncertainty variable x in the training dataset i The j-th normalized sample point, i = 1, 2, ..., m; j = 1, 2, ..., n; obtained experimentally with lengths l1, l2, ..., l o The n-dimensional response vectors of the o cracks at the sample points are as follows: o≥2, where, i=1,2,…,o, j=1,2,…,n, Indicates a length of l i The response value of the crack at the j-th sample point will be x t and The training dataset for cracks of length l1 Similarly, the above method is used to obtain lengths l2,…,l o The training dataset corresponding to the crack;

[0011] (2) For the uncertain variables x1, x2, ..., x m Using Monte Carlo sampling, b groups of sample points were selected. After normalizing these b groups of sample points, the following results were obtained:

[0012] x vi j This indicates the uncertainty variable x in the validation dataset. i The j-th normalized sample point, i = 1, 2, ..., m; j = 1, 2, ..., b, has a length of l1, l2, ..., l obtained experimentally. o The b-dimensional response vectors of the o cracks at the sample points are as follows: in, Indicates a length of l i The response value of the crack at the j-th sample point will be xv and Validation data set corresponding to the crack with length l1 Similarly, validation data sets corresponding to the cracks with lengths l2,…,l o are obtained using the above method.

[0013] Further, the specific steps of step 2 are as follows: the system response y is expressed by a polynomial chaos expansion as follows:

[0014]

[0015] where Φ i (X) is an orthogonal polynomial basis function selected according to the type of probability distribution of the uncertainty variable, a i is the expansion coefficient, and P is the number of terms of the polynomial, and the calculation formula is where p is the order of the polynomial chaos expansion model selected according to actual needs, and m is the number of uncertainty parameters.

[0016] Training data set based on the crack with length l1: is the output response vector, is the input x t corresponding to the orthogonal polynomial matrix, and the coefficient a i is calculated as follows:

[0017] a) Set all coefficients to 0, and the residual is the response vector Find a column Φ j in the matrix Φ that is related to the residual;

[0018] b) Increase the coefficient a j in the direction of Φ j , and the residual is until a column Φ k in the matrix Φ that has not been selected is found, so that the residual has the same correlation with Φ j and Φ k , and the projection of the residual is located on the angle bisector of Φ k and Φ j ;

[0019] c) Increase the coefficients {a j , a k} in the direction of the angle bisector of {Φ j , Φ k}, and the residual is until a column Φ h in the matrix Φ that has not been selected is found, so that the residual has the same correlation with Φ j , Φ k , and Φ h , and the projection of the residual is located on the angle bisector of Φ j, Φ k , Φ h ;

[0020] d) the coefficients {a j , a k , a h} are increased along the spatial angular bisector of {Φ j , Φ k , Φ h}, and the residual is until a column Φ z of the matrix Φ is not selected, such that the residual has the same correlation with Φ j , Φ k , Φ h , Φ z as the residual, and the projection of the residual is located on the spatial angular bisector of Φ j , Φ k , Φ h , Φ z ;

[0021] e) repeat the above steps until the error meets the requirements or all columns of the matrix Φ are traversed;

[0022] From the above process, the coefficients are obtained, and the polynomial chaos expansion expression is obtained. The meta-model corresponding to the crack with a length of l1 is constructed, and the above steps are repeated to construct the meta-models corresponding to the cracks with lengths of l2, …, l o .

[0023] Further, the specific steps of step 3 are as follows:

[0024] Input the x v in the validation data set into the meta-model corresponding to the crack with a length of l1 to obtain a b-dimensional response vector and the in the validation data set.The root mean square error RMSE and the normalized root mean square error NRMSE are calculated by the following formula:

[0025]

[0026]

[0027] wherein, is the jth value in the , is the jth value in the , and max and min are the maximum and minimum values in the , respectively. If both the RMSE and the NRMSE are less than the set threshold, the accuracy of the meta-model corresponding to the crack with a length of l1 meets the requirements, and the lengths of l2, …, l​o If the meta-models corresponding to the o cracks with different lengths meet the accuracy requirements, go to step 4; otherwise, increase the sample points of the training data set and return to step 1 to recalculate.

[0028] Further, the specific steps of step 4 are as follows:

[0029] (1) For the uncertain variables x1, x2, …, x m , use the Latin hypercube sampling method to select N groups of sample points, and input the normalized length l1, l2, …, l o corresponding to the crack into the meta-model to obtain o N-dimensional response vectors Wherein,

[0030] (2) Perform logarithmic linear regression on the o*N crack responses:

[0031]

[0032] Wherein, β0 and β1 are the intercept and slope of the linear regression line, respectively, and are calculated by the maximum likelihood method, is an o*N-dimensional response vector L is an o*N-dimensional crack length vector, ε is a random error obeying normal distribution, whose mean value is 0 and standard deviation is σ ε ;

[0033] (3) Calculate the detection probability PoD(l i ) of the crack with length l i by the following formula:

[0034]

[0035] Wherein, is the selected system response threshold, and Ψ is the standard normal distribution function, whose mean value μ and standard deviation σ are:

[0036]

[0037] Beneficial effects: Compared with the prior art, the technical scheme of the present application has the following beneficial technical effects:

[0038] The meta-model in the present application can replace the accurate but time-consuming physical model in the research on crack detection probability by non-destructive testing, greatly improving the calculation efficiency while maintaining the same accuracy, and this non-invasive method solves the shortcomings of the invasive method, i.e. the need to modify the existing model and the difficulty in applying to complex problems. BRIEF DESCRIPTION OF DRAWINGS

[0039] Figure 1It is a flow chart of a calculation method of a crack detection probability of an eddy current nondestructive testing system based on a polynomial chaos expansion metamodel used in an embodiment of the present application.

[0040] Figure 2 It is a logarithmic linear regression analysis graph calculated by a metamodel.

[0041] Figure 3 It is a detection probability curve graph calculated by a metamodel. DETAILED DESCRIPTION

[0042] The technical solutions of the present application are further described below in combination with the drawings and embodiments.

[0043] In this embodiment, an eddy current nondestructive testing model for detecting surface cracks of a metal plate by a finite section coil to obtain impedance changes is established in COMSOL simulation software. The inner diameter of the detection coil is 9.34 mm, the outer diameter is 18.4 mm, the number of turns of the coil is 408, the working frequency is 7000 Hz, the thickness of the metal plate is 12.22 mm, the electrical conductivity is 30.6 MS / m, three cracks with a depth of 5 mm, a width of 0.28 mm and a length of 0.3 mm, 0.4 mm and 0.5 mm are selected. The selected uncertainty variables are the three coordinate positions of the detection coil, and the probability distribution is shown in Table 1.

[0044] Table 1 Empirical distribution of uncertainty variables Unit: mm

[0045]

[0046]

[0047] The specific process of step 1 is as follows:

[0048] (1) x, y, z axis positions are taken as uncertainty variables x1, x2, x3, and 80 sample points are selected for x1, x2, x3 using Latin hypercube sampling method The impedance of the three different length cracks corresponding to the sample points is calculated by COMSOL simulation software, and three 80-dimensional impedance vectors y are obtained t0.3 = [0.001949, 0.001835, …, 0.001464], y t0.4 = [0.003141, 0.002944, …, 0.002367], y t0.5 = [0.004589, 0.004296, …, 0.003464]. The sample points of the variable x1 subject to normal distribution are linearly mapped to standard normal distribution; the variables x2, x3 subject to uniform distribution are linearly mapped to the interval [-1, 1], and the normalized sample points x t are obtained. The xt and y t0.3 The training data set of the crack with the length of 0.3 mm is obtained as follows:

[0049] Similarly, the training data sets of the cracks with the lengths of 0.4 mm and 0.5 mm are obtained

[0050] (2) 1000 sample points are selected for x1, x2 and x3 by using the Monte Carlo sampling method, and the remaining process is the same as (1), to obtain x v and y v0.3 , y v0.4 , y v0.5 , and the verification data set

[0051] The specific process of step 2 is as follows:

[0052] The order of the polynomial chaos expansion model is selected to be 3, and the impedance is expressed by the polynomial chaos expansion as follows: The optimal basis function corresponding to x1 is the Hermite polynomial, and the optimal basis functions corresponding to x2 and x3 are the Legendre polynomials. Based on The coefficients a i are calculated, and the specific expression of the polynomial chaos expansion is obtained, and the construction of the meta-model of the crack with the length of 0.3 mm is completed. Similarly, based on and the construction of the meta-model of the crack with the length of 0.4 mm and 0.5 mm is completed.

[0053] The specific process of step 3 is as follows:

[0054] The x v in the verification data set is input into the meta-model corresponding to the crack with the length of 0.3 mm, and 1000-dimensional impedance vectors The precision threshold required by the meta-model is selected to be: the root mean square error is less than 3% of the standard deviation of the 1000 impedance values in the verification data set, and the normalized root mean square error is less than 1%. The root mean square error (RMSE) and the normalized root mean square error (NRMSE) are calculated according to the following formula

[0055]

[0056]

[0057] wherein, is the jth value in y , y v0.3 j is the jth value in y v0.3 , y v0.3max and y v0.3minThe maximum and minimum values in y v0.3 .

[0058] The RMSE is 1.7e-5 and the NRMSE is 0.46%, both of which are less than the set thresholds 1.75e-5 and 1%. For the crack with length 0.4mm, the RMSE is 2.3e-5 and the NRMSE is 0.98%, both of which are less than the set thresholds 2.7e-5 and 1%; for the crack with length 0.5mm, the RMSE is 3.1e-5 and the NRMSE is 0.36%, both of which are less than the set thresholds 3.9e-5 and 1%. The accuracy of the metamodel for the three cracks meets the requirements.

[0059] The specific process of step 4 is as follows:

[0060] (1) 300 sample points are selected for x1, x2, and x3 using Latin hypercube sampling method, and after normalization, the 300-dimensional impedance vectors corresponding to the metamodels of the cracks with lengths of 0.3mm, 0.4mm, and 0.5mm are obtained

[0061]

[0062] (2) Log-linear regression is performed on the 3*300 impedance values, and the results are shown in Figure 2 .

[0063] (3) According to the regression parameters and the selected system response threshold , several indicators of detection probability are calculated: mean μ, standard deviation σ, crack length a when the detection probability is 50% and 90% 50 and a 90 . Compared with the detection probability indicators of the 300 sample points calculated by COMSOL, the results are shown in Table 2. It can be seen that the relative error of the indicators calculated by the metamodel and COMSOL is less than 1%. The detection probability curve calculated by the metamodel is shown in Figure 3 .

[0064] Table 2 Detection probability indicators

[0065] Indicator COMSOL Metamodel μ -0.87265 -0.87273 σ 0.13680 0.13588 a 50 ]]> 0.41784 0.41780 a 90 ]]> 0.49790 0.49728

Claims

1. A method for calculating the probability of detection of a crack by an eddy current non-destructive testing system, characterized in that, The method comprises the following steps: Step 1: Sampling the uncertain variables in the process of detecting material cracks by the eddy current nondestructive testing system, obtaining the system response at the sampling points corresponding to different lengths of cracks through experiments, taking the normalized sampling points and the corresponding system responses as the required data set, taking the data set obtained by Latin hypercube sampling as the training data set, and taking the data set obtained by Monte Carlo sampling as the verification data set; Step 2: Calculating the polynomial chaos expansion coefficients based on the training data set, thereby obtaining the polynomial expression of the system response, and completing the construction of the meta-model; Step 3: Testing the accuracy of the meta-model based on the verification sample set, if the accuracy requirement is not met, increasing the number of training data sets to re-construct the meta-model until the accuracy requirement is met; Step 4: Calculating the system response using the constructed meta-model, and calculating the detection probability of the crack according to the response data; The specific steps of step 1 are as follows: (1) The uncertainty variables x1, x2, …, x m , select n groups of sample points using Latin hypercube sampling method, and obtain the following after normalizing the n groups of sample points: x ti j , the jth normalized sample point of the uncertainty variable x i in the training data set, i = 1, 2, …, m; j = 1, 2, …, n; through experiments, o cracks with lengths l1, l2, …, l o have n-dimensional response vectors at the sample points respectively: wherein, , the response value of the crack with length l i at the jth group of sample points, and x t and are the training data set of the crack with length l1 Similarly, the training data sets corresponding to the cracks with lengths l2, …, l o are obtained using the above method; (2) For the uncertain variables x1, x2, …, x m , b groups of sample points are selected using the Monte Carlo sampling method, and after normalization, the following is obtained: x vi j x i , i = 1, 2, …, m; j = 1, 2, …, b, are the jth normalized sample points of the uncertainty variable x o in the validation dataset, with lengths l x i , i = 1, 2, …, m; j = 1, 2, …, b, are the jth normalized sample points of the uncertainty variable x v and Similarly, the validation datasets corresponding to the cracks with lengths l o are obtained using the above method.​​ The specific steps of step 2 are as follows: the system response y is expressed by polynomial chaos expansion as follows: wherein, Φ i (X) is an orthogonal polynomial basis function selected according to the probability distribution type of the uncertainty variable, a i is an expansion coefficient, P is the number of terms of the polynomial, and the calculation formula is wherein, p is the order of the polynomial chaos expansion model selected according to actual needs, and m is the number of uncertainty parameters. Based on the training dataset of cracks of length l1: for the output response vector, for the input x t the corresponding orthogonal polynomial matrix, the coefficients a i are calculated as follows: a) Set all coefficients to 0, at which point the residual is the response vector Find the column of the matrix Φ that is most correlated with the residual j ; b) coefficient a j along Φ j increases, at which point the residual is until a column of the matrix Φ k is selected that makes the residual have the same correlation with Φ j and Φ k , at which point the projection of the residual lies on the angle bisector of Φ k and Φ j ; c) the coefficient {a j , a k} increases in the direction of the angle bisector of {Φ j , Φ k}, at which time the residual is until a column of the matrix Φ h is not selected, such that the residual has the same correlation with Φ j , Φ k , Φ h as the selected columns, at which time the projection of the residual lies on the spatial angle bisector of Φ j , Φ k , Φ h ; d) coefficients {a j ,a k ,a h} increase in the direction of the spatial angle bisector of {Φ j ,Φ k ,Φ h}, at which the residual is until a column Φ z of the matrix Φ is not selected, such that the residual has the same correlation with Φ j ,Φ k ,Φ h ,Φ z , at which the projection of the residual is located on the spatial angle bisector of Φ j ,Φ k ,Φ h ,Φ z ; e) Repeat the above steps until the error meets the requirements or all columns in the matrix Φ are traversed; The polynomial chaos expansion expression is obtained from the above process by obtaining the coefficients, and the meta-model of the crack with a length of l1 is constructed. The above steps are repeated to construct the meta-model corresponding to the crack with a length of l2, …, l o n, respectively.

2. The method of claim 1, wherein the probability of detection of a flaw by the eddy current non-destructive testing system is calculated by: The specific steps of step 3 are as follows: x v Input the meta-model corresponding to the crack with length l1 to get the b-dimensional response vector With the validation data set The root mean square error RMSE and the normalized root mean square error NRMSE are calculated by the following formula: wherein, is the jth value in is the jth value in and are the maximum and minimum values in respectively, if both RMSE and NRMSE are less than a set threshold, the crack corresponding to the meta-model with length li is verified to meet the accuracy requirement, and the meta-models corresponding to the cracks with lengths l2,...,lnare verified in turn according to the above steps, if the meta-models corresponding to the cracks with o different lengths all meet the accuracy requirement, step 4 is entered; otherwise, the sample points of the training data set are increased, and step 1 is returned to recalculate. o ​ 3. The method for calculating the crack detection probability of an eddy current nondestructive testing system according to claim 1, characterized in that, The specific steps of step 4 are as follows: (1) For uncertain variables x1, x2, …, x m , N groups of sample points are selected using Latin hypercube sampling method, and after normalization, they are input into the meta-model corresponding to the crack with the length l1, l2, …, l o , and o N-dimensional response vectors are obtained (2) Logarithmic linear regression is performed on the o*N crack responses: where β0and β1are the intercept and slope of the linear regression line, respectively, calculated by the method of maximum likelihood, is an o * N-dimensional response vector L is an o * N-dimensional fracture length vector, ε is a random error following a normal distribution with mean 0 and standard deviation σ ε ; (3) The probability of detection PoD(l) for a crack of length / is calculated by the following formula: i PoD(l) = 1 - exp(-2l2 / (πσ2σ2)) (3) i ) : wherein is a selected system response threshold, and Ψ is a standard normal distribution function with mean μ and standard deviation σ:

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