A method for predicting defects of heavy nodular cast iron based on phase space reconstruction

Abstract

The application discloses a kind of based on phase space reconstruction's thick big nodular cast iron defect prediction method, it is related to nondestructive testing technique, obtain the measured data with thick big nodular cast iron defect, and measured data is divided into training set and test set;By phase space reconstruction, the measured data in training set is converted from low-dimensional data into high-dimensional data, then by prediction algorithm, high-dimensional data is mapped to one-dimensional real number field to obtain the prediction output of next time point, to obtain prediction value;According to the prediction value of prediction algorithm, the structure of BP neural network is established, and the best initial value of the optimal BP neural network weight and threshold value is sought by improved HGAPSO algorithm;Real-time data of thick big nodular cast iron to be tested are input into the BP neural network structure to obtain the defect output result of thick big nodular cast iron.The application can predict the next thick big nodular cast iron defect, reduce nodular cast iron defect and the emergence of anomaly, help to comprehensively predict nodular cast iron defect.

Description

TECHNICAL FIELD

[0001] The present application relates to nondestructive testing technology, more particularly, it relates to a thick nodular cast iron defect prediction method based on phase space reconstruction. BACKGROUND

[0002] Nodular cast iron has high strength, high toughness, good wear resistance, processing performance, and thermal and electrical conductivity due to its unique spheroidal graphite structure. The material composition mainly includes iron, carbon, silicon, manganese and other elements, and nodularizing agent and other alloying elements are added to improve the mechanical properties. Before nodular cast iron is formed, the temperature of the molten iron is as high as 1350-1450℃, which causes chemical elements to react with refractory materials to form brittle compounds. At the same time, due to inaccurate control of carbon equivalent, phosphorus elements, rare earth elements and temperature, defects such as shrinkage cavity, slag inclusion, subsurface porosity, graphite floating and poor nodularity are prone to occur. Traditionally, macroscopic control of the formation process of nodular cast iron relies heavily on the operating experience of on-site personnel, and automatic control cannot be achieved. With the progress of science and technology, researchers have established various mechanism mathematical models to reveal the internal phenomena of the molten iron and reflect the formation mechanism of nodular cast iron. However, these models have the disadvantages of low accuracy and long calculation time. SUMMARY

[0003] The technical problem to be solved by the present application is to provide a thick nodular cast iron defect prediction method based on phase space reconstruction to solve the problems of the prior art.

[0004] The thick nodular cast iron defect prediction method based on phase space reconstruction comprises the following steps:

[0005] S1, obtaining measurement data of thick nodular cast iron defects, and dividing the measurement data into a training set and a test set;

[0006] S2, converting the measurement data in the training set from low-dimensional data to high-dimensional data through phase space reconstruction, and then performing prediction processing on the high-dimensional data through a prediction algorithm to obtain a predicted value;

[0007] S3, establishing a BP neural network structure according to the predicted value of the prediction algorithm, and seeking the best initial value of the optimal BP neural network weight and threshold value through an improved HGAPSO algorithm;

[0008] S4, training the training set using the best initial value of the weight and threshold value of the BP neural network structure, and stopping the training when the mean square error of the BP neural network structure reaches the minimum mean square error; testing the test data of the test set using the trained BP neural network structure, and comparing the test data with the actual value of the thick nodular cast iron defects corresponding to the test data of the test set, if they are consistent, it means that the construction of the BP neural network structure meets the requirements;

[0009] S5, real-time data of the thick nodular cast iron to be tested is collected and input into the BP neural network structure in step S4 to obtain a defect output result of the thick nodular cast iron.

[0010] Preferably, the prediction algorithm is specifically:

[0011] S21, setting k = 1 ~ N, ξ as a domain radius, r * is a prediction accuracy, and l is a prediction sample length; N is a total number of test data in the training set;

[0012] S22, searching for high-dimensional data H k in a domain point H k1 in a domain radius ξ k2 ,..., H kn(ξ) , and making it satisfy:

[0013] ;

[0014] where ‖.‖ represents an infinite norm;

[0015] S23, determining next iteration points H k1+1 ,H k2+1 ,..., H kn(ξ)+1 of n(ξ) domain points, and obtaining first components k1+1 ,H k2+1 ,..., H kn(ξ)+1 of the next iteration points H ;

[0016] S24, fitting parameters , α T by using a least square method, and making it satisfy:

[0017] ;

[0018] where min S is a minimum value of the least square fitting parameters of the above-mentioned class; and is a predicted output of a first component of a state vector ; j is an index parameter of input data, and is a number of input data in the test data;

[0019] S25, substituting , α T obtained in S24 into a local linear model, and predicting by using the local linear model to obtain a prediction value ;

[0020] S26, update k=k+1, repeat S22-S25 until k=N+1, thereby obtaining the predicted values ; combine all the predicted values to obtain the combined predicted value .

[0021] S27, calculate the prediction accuracy:

[0022] ;

[0023] wherein, is the estimated output of the i-th predicted sample after the first data from N, is the actual output of the i-th predicted sample after the first data from N;

[0024] compare the prediction accuracy r with a preset accuracy threshold r * , if r≤r * , stop the calculation; if r>r *, modify ξ, k=N, and repeat S22-S26.

[0025] Preferably, the local linear model is:

[0026] ;

[0027] wherein, is a noise signal subject to normal distribution.

[0028] Preferably, the modification method of the domain radius ξ is:

[0029] ;

[0030] wherein Δξ is a positive adjustment amount, and ξ k is the k-th domain radius ξ.

[0031] Preferably, the improved HGAPSO algorithm is specifically:

[0032] S31, initialize the parameters in the HGAPSO algorithm;

[0033] S32, calculate the position of the next generation particle according to the velocity vector of the particle and a random disturbance function:

[0034] ;

[0035] wherein, is the particle position of the i-th particle after iteration k+1 times; is the velocity vector of the k+1-th generation particle; k t is a random disturbance function;

[0036] S33, calculate a global fitness value of the particle, and determine whether the particle is accepted according to the global fitness value, if the particle is accepted, proceed to S34; otherwise, proceed to S35;

[0037] S34, perform boundary constraint on the particle:

[0038] If , then:

[0039] ;

[0040] If , then:

[0041] ;

[0042] wherein lb is the lower boundary of the particle swarm size, and ub is the upper boundary of the particle swarm size;

[0043] S35, repeat steps S32-S34 until the maximum iteration number, form a new particle swarm, extract the weight and threshold data from the particle dimension D containing the weight and threshold information in the particle swarm, and take the weight and threshold data as the optimal initial values of the weight and threshold of the BP neural network structure training.

[0044] Preferably, in S33, the global fitness value is calculated by the following formula:

[0045] ;

[0046] wherein NEW(k) is the global fitness value after dynamic adjustment in the kth generation of particles, β is a hyperparameter for controlling the decay speed; MSE(k (i) ) refers to the mean square error of the real output defect percentage and the estimated output defect percentage of the neural network corresponding to the ith particle of the particle swarm in the kth iteration.

[0047] Preferably, the optimal global fitness value PRENEW(k) is sought by the following formula:

[0048] .

[0049] Preferably, whether the particle is accepted is determined by the following formula:

[0050]

[0051] wherein if NEW(k) is the optimal global fitness value, PRENEW(k)=NEW(k), and PRENEW(k+1)=NEW(k+1); if MSE(k (i) ​) as the best global fitness value, PRENEW(k) = MSE(k (i) ) and PRENEW(k+1) = MSE(k+1 (i) ).

[0052] Preferably, the mean square error MSE(k (i) ) is calculated by the following formula:

[0053] ;

[0054] wherein, refers to the particle position of the particle containing the estimated output value of the neural network after forward feedback under the D-dimensional weight value and threshold value data; refers to the particle position of the particle containing the actual output value under the D-dimensional weight value and threshold value data. Beneficial effects

[0055] Advantages of the present application

[0056] The present application has the advantages that:

[0057] 1. By phase space reconstruction, prediction algorithm and BP neural network structure, the chaotic characteristics and mutual influence of chemical elements in the formation process of nodular cast iron can be captured, and the next batch of thick nodular cast iron defects can be predicted, so as to reduce the occurrence of nodular cast iron defects and abnormalities, and help to comprehensively predict the defects of nodular cast iron.

[0058] 2. Compared with the traditional HGAPSO algorithm, the improved HGAPSO algorithm is used to optimize the weight and threshold value of the BP neural network in the present application, which can enhance the global exploration ability, improve the convergence speed, improve the algorithm speed and ensure the stability of the solution. BRIEF DESCRIPTION OF DRAWINGS

[0059] Figure 1 is a thick nodular cast iron defect prediction method flowchart of the present application;

[0060] Figure 2 is a BP neural network structure dimension diagram of the present application. DETAILED DESCRIPTION

[0061] The present application will be further described below in conjunction with the embodiments, but does not constitute any limitation on the present application, and any limited number of modifications made by anyone within the scope of the claims of the present application is still within the scope of the claims of the present application.

[0062] Referring to Figure 1 , a thick nodular cast iron defect prediction method based on phase space reconstruction of the present application has the following specific steps:

[0063] ​S1, obtaining measurement data. The factors affecting the defects of the ball mill cast iron mainly come from the contents of 8 chemical elements, the cooling rate and the wall thickness. The specific collection method is: the contents of chemical elements are collected by using a direct-reading spectrometer, a high-frequency infrared carbon-sulfur analyzer and a thermal analyzer, and the cooling rate is measured by using a temperature sensor. The contents of chemical elements of 5220 furnaces are collected in a period of 2 hours, of which the data of the first 5000 furnaces are used for training, and the data of the last 220 furnaces are used for testing.

[0064] The cooling rate can be expressed by the following formula:

[0065] .

[0066] The wall thickness can be expressed by the following formula:

[0067] .

[0068] In the formula, v is the propagation speed of the sound wave in the material, t r is the time required for the sound wave to go back and forth.

[0069] S2, the collected contents of 8 chemical elements, the cooling rate and the wall thickness are normalized. The measurement data in the processed training set are converted from low-dimensional data to high-dimensional data through phase space reconstruction, and then the high-dimensional data are mapped to one-dimensional real number field through a prediction algorithm to obtain the prediction output of the next time point.

[0070] S3, a BP neural network structure is constructed according to the output of the prediction algorithm, and the best initial values of the weights and thresholds in the BP neural network structure are obtained through the improved GAPSO algorithm, so that the BP neural network structure has more efficient training ability.

[0071] S4, the BP neural network structure uses the best initial values of the weights and thresholds to train the training set (i.e. 5000 furnace data), and the network stops training when the minimum mean square error is reached. Finally, the trained BP neural network structure is used to test the data of the test group (i.e. 220 furnace data), and the actual value corresponding to the data of the test group is compared. If they are consistent, it means that the construction of the BP neural network structure meets the requirements. The real-time data of the collected thick nodular cast iron to be tested are input into the BP neural network structure in step S4, and the expected defect output is obtained through the prediction of the BP neural network structure.

[0072] S5, the real-time data of the thick nodular cast iron to be tested are input into the BP neural network structure in step S4 to obtain the defect output of the thick nodular cast iron, so as to realize the prediction of the defects of the thick nodular cast iron.

[0073] In the above S2, the phase space reconstruction is implemented as follows:

[0074] Let y j : R n → R be a second-order differentiable measurement function, where R n → R is a process that maps an n-dimensional vector to the real number field, y j is a mapping function; R n is an n-dimensional Euclidean space, the input vector is n-dimensional; R is a one-dimensional real number field. The measurement sequence can be obtained as follows: where is the output variable; x i is the input measurement data; j represents different chemical elements and input elements such as cooling rate and wall thickness, j = 10 in this embodiment (i.e. 8 chemical elements + 1 cooling rate + 1 wall thickness); N = 5000, i.e. 5000 furnace test data in the training set.

[0075] Embed the above measurement sequence into a d j -dimensional phase space, and the state vector can be expressed as:

[0076] .

[0077] In the formula, represents the data output under the kth furnace and the jth index. If k = 2000 and j = 10, it represents the wall thickness data under the 2000th furnace test data. and d j are the embedding time delay and embedding dimension of the measurement sequence respectively. In the above state vector expression, it needs to satisfy , and d is the total dimension of the space.

[0078] Regarding the embedding time delay and the embedding dimension d j , the chemical element composition data, the cooling rate and the wall thickness of 5000 furnaces are normalized to have a mean of 0 and an amplitude of 1. After normalization, the length of the time sequence is defined as n >> 10d, and then the embedding time delay and the embedding dimension d j are determined as follows:

[0079] 1) Determination of the embedding time delay :

[0080] Based on the autocorrelation function C j (τ), τ is discretized to obtain a τ-C j (τ) chart, and 10 time delay data τ1, τ2,..., τ 10 are obtained. The autocorrelation function formula is as follows:

[0081] .

[0082] where, represents the average value of all i discrete time summation in the jth data of the measurement sequence.

[0083] Observe the time delay curve, take C j (τ) first drop to e -1 When the τ is the optimal time delay, the τ at this time is marked as the embedding time delay .

[0084] 2) The determination of the embedding dimension d j :

[0085] Step 1: Set the initial value: d j =1, the counter M j (d)=0.

[0086] Step 2: In the d=d1+d2+...+d 10 dimensional space, calculate the Euclidean distance between and its nearest neighborhood point :

[0087] .

[0088] Step 3: Update d=d+1, at this time the d+1 dimensional distance is:

[0089] .

[0090] If s j / s≥φ, it is considered that when the dimension d→d+1, a pair of pseudo-neighborhood points appears, accordingly, the counter M j (d)=M j (d)+1. Wherein, φ is a threshold parameter to determine whether the nearest vector point in the phase space is a pseudo-neighborhood point.

[0091] Step 4: Change the value of k, repeat steps 2 and 3 until k goes through all points in the phase space orbit, if M j (d)>δ, then d j =d j +1. Wherein, δ is the upper limit of the percentage of pseudo-neighborhood points, and the reconstruction process is stopped when the upper limit is exceeded.

[0092] Step 5: Update d=d+p, where p is the number of counters in M j (d) greater than δ. Repeat steps 2-4 until all M j (d)<δ, then the optimal embedding dimension d j is obtained.

[0093] The state of the phase space reconstruction evolves according to the following function relationship:

[0094] .

[0095] Based on the output of the phase space reconstruction, in the prediction algorithm, for j = 10 input data, there is the following function mapping:

[0096] .

[0097] Taking prediction as an example, it is predicted by a local linear model: .

[0098] In the formula, is the prediction value of , is a noise signal subject to normal distribution, is a bias value of the local linear model, is a to-be-determined coefficient, which is obtained by least square method. Thus, the prediction algorithm specifically includes:

[0099] S21: Set ξ as the field radius, r * is the prediction accuracy, and l is the prediction sample length.

[0100] S22: Search for the ξ field points H k , H k1 ,..., H k2 of H kn(ξ) , that is, satisfy:

[0101] .

[0102] Among them, ‖.‖ represents the infinite norm.

[0103] S23: Determine the next iteration points H k1+1 , H k2+1 ,..., H kn(ξ)+1 of the n(ξ) field points and the first components .

[0104] S24: Fit the parameters , α T by using the least square method and satisfy the following formula:

[0105] .

[0106] In the formula, min S is the minimum value of the above least square fitting parameters, and then the partial derivatives of and α T are equal to 0, respectively, to obtain , αT data.

[0107] S25: Substitute the data of S24 into the partial linear model, and use the partial linear model to predict the data of S24, to obtain , α T into the partial linear model, and use the partial linear model to predict the data of S24, to obtain . .

[0108] S26: Update k=k+1, repeat S22-S25 until k=N+l.

[0109] Step 7: Calculate the prediction accuracy:

[0110] .

[0111] wherein, is the estimated output of the i-th prediction sample after the first data from N, is the actual output of the i-th prediction sample after the first data from N.

[0112] Compare the size of r and r * , if r≤r * , stop calculation; if r>r*, correct ξ, k=N, repeat steps 2-6.

[0113] Here, the linear correction method is adopted:

[0114] ;

[0115] wherein, Δξ is a positive adjustment amount.

[0116] The final combination of the prediction value is . According to the basis of j=10 and N=5000 furnace data, the prediction value is a 10*5000-dimensional data.

[0117] According to the prediction value of the prediction algorithm, the BP neural network structure is established. According to the principle that the embedding dimension is equal to the input node of the neural network, the neural network structure is established, as shown in Figure 2 , the BP neural network structure is as follows: the number of neural network nodes is 10, the number of hidden layer nodes is 2*10+1 according to the empirical formula, and the number of output layer nodes is the percentage of 5 kinds of unique defect types.

[0118] After the BP neural network structure is established, the improved HGAPSO algorithm is used to seek the optimal initial values of the BP neural network weight and threshold. For the traditional HGAPSO algorithm, the iteration of the particle swarm needs to be constantly updated, which increases the calculation amount of the algorithm. Here, a dynamic adjustment factor and a Gaussian mutation operator are introduced to improve the calculation speed of the HGAPSO and reduce the iteration number. The improved HGAPSO algorithm mainly includes the following steps:

[0119] S31, algorithm initialization:

[0120] S311, population initialization:

[0121] The population size is set to N, the maximum iteration number is set to T, the particle dimension is set to D, and the particle swarm is an NxD matrix. The particle dimension D includes the weight and threshold information between different nodes, for example: the first 10x21 dimensions of the particle are the weights between the input layer and the hidden layer, the 10x21+1th to 10x21+21x5 dimensions of the particle are the weights between the hidden layer and the output layer, and the 10x21+21x5+1th to 10x21+21x5+21 dimensions of the particle are the threshold values of the hidden layer nodes. Then the particle position is randomly generated where i=1, 2, …, N.

[0122] S312, parameter initialization:

[0123] The parameters c1, c2, r1, r2, and the decay factor λ in the particle swarm algorithm formula are initialized.

[0124] S32, particle position update:

[0125] The speed vector formula of the next generation of particles is:

[0126] ;

[0127] is the speed vector of the k+1th generation of particles; w is the inertia weight; c1 and c2 are learning factors; r1 and r2 are random numbers distributed in [0, 1]; is the individual optimal value in the kth generation of particles; is the global optimal value in the kth generation of particles; σ is the Gaussian mutation parameter, i.e. the standard deviation of the noise. In the speed vector formula represents the independent evolution mode of the individual in each generation, and represents the evolution mode of the group in each generation.

[0128] The random disturbance function is:

[0129] where k0 is the initial adjustment factor.

[0130] The next generation particle position is calculated according to the velocity vector of the particle and a random disturbance function:

[0131] ;

[0132] wherein, is the particle position of the i-th particle after the iteration k+1.

[0133] S33, the fitness of the particle whose particle position is calculated as is calculated, and whether the particle is accepted is determined according to the fitness:

[0134] S331, the mean square error between each particle after iteration by the optimization algorithm and the particle before iteration is calculated:

[0135] ;

[0136] wherein, MSE(k (i) ) refers to the mean square error of the actual output defect percentage and the estimated output defect percentage of the neural network corresponding to the i-th particle of the particle swarm in the k-th iteration; refers to the estimated output value of the neural network after forward feedback under the condition that the particle whose particle position is contains D-dimensional weight values and threshold data; refers to the actual output value of the particle whose particle position is contains D-dimensional weight values and threshold data.

[0137] S332, the fitness value is dynamically adjusted:

[0138] .

[0139] wherein, NEW(k) is the global fitness value after dynamic adjustment of the k-th generation of particles, β is a hyperparameter for controlling the decay speed, t is the current iteration number, and T is 1-T.

[0140] After the fitness values of the N particles are obtained, the best global fitness value PRENEW(K) is sought, specifically,

[0141] .

[0142] S333, the particle swarm position is updated:

[0143] .

[0144] S334, boundary constraint:

[0145] If , then:

[0146] .

[0147] If, then:

[0148] .

[0149] S35, repeating steps S32-S34 until the maximum number of iterations, forming a new particle swarm, from the particle swarm containing weight and threshold information in the particle dimension D to extract the weight and threshold data, and the weight and threshold data as the best initial value of the weight and threshold of the BP neural network structure training.

[0150] The above only describes the preferred embodiments of the present application, it should be noted that for those skilled in the art, without departing from the structure of the present application, can also make several variations and improvements, these will not affect the effect of the present application and the practicality of the patent.

Claims

1. A method for predicting defects in heavy nodular cast iron based on phase space reconstruction, characterized by, The method comprises the following steps: S1, acquiring measurement data of thick nodular cast iron defects, and dividing the measurement data into a training set and a test set; S2, normalizing the collected content of 8 chemical elements, cooling rate and wall thickness, converting the measurement data in the processed training set from low-dimensional data to high-dimensional data through phase space reconstruction, and mapping the high-dimensional data to a one-dimensional real number field based on a local linear model through a prediction algorithm to obtain a prediction output of the next time point, so as to obtain a prediction value; S3, establishing a BP neural network structure according to the prediction value, and seeking optimal initial values of BP neural network weights and thresholds through an improved HGAPSO algorithm with a dynamic adjustment factor and a Gaussian mutation operator; S4, training the training set by using the optimal initial values of the BP neural network weights and thresholds, and stopping training when the mean square error of the BP neural network structure reaches the minimum mean square error; testing the test data of the test set by using the trained BP neural network structure, and comparing the test data with the actual value of the thick nodular cast iron defects corresponding to the test data, if they are consistent, it means that the construction of the BP neural network structure meets the requirements; S5, inputting real-time data of the thick nodular cast iron to be tested into the BP neural network structure constructed in step S4 to obtain the defect output result of the thick nodular cast iron.

2. The method for predicting defects of thick ductile cast iron based on phase space reconstruction according to claim 1, characterized in that, The improved HGAPSO algorithm is as follows: S31, initializing parameters in the HGAPSO algorithm; S32, the velocity vector formula of the next generation of particles is: ； is the velocity vector of the k+1th generation particle; w is the inertia weight; is the velocity vector of the kth generation particle; c1 and c2 are both learning factors; r1 and r2 are both random numbers distributed in [0, 1]; is the individual optimal value in the kth generation particle; is the global optimal value in the kth generation particle; , σ is the Gaussian variation parameter; represents the independent evolution mode of the individual in each generation, represents the evolution mode of the group in each generation; The position of the next generation of particles is calculated according to the velocity vector of the particles and a random disturbance function: ； wherein, is the position of the i-th particle after iteration k+1; is the velocity vector of the k+1th generation of particles; k t is a random perturbation function; S33, calculate the particle position as The global fitness value of the particle is determined, and whether the particle is accepted is determined according to the global fitness value. If the particle is accepted, proceed to S34; otherwise, proceed to S35; S34, boundary constraint is performed on the particles: If then: ； If then: ； Wherein, lb is the lower boundary of the particle swarm size, and ub is the upper boundary of the particle swarm size; S35, repeating steps S32-S34 until the maximum iteration number, forming a new particle swarm, extracting weight and threshold data from the particle dimension D containing weight and threshold information in the new particle swarm, and taking the weight and threshold data as the optimal initial values of the weights and thresholds of the BP neural network structure training.

3. The method according to claim 2, wherein, In S33, the global fitness value is calculated by: ； wherein NEW(k) is the global fitness value of the kth generation of particles after dynamic adjustment, β is a hyperparameter controlling the decay rate; MSE(k (i) ) refers to the mean square error of the actual output defect percentage and the estimated output defect percentage of the neural network corresponding to the ith particle of the kth iteration of the particle swarm; t is the current iteration number.

4. The method according to claim 3, wherein, The optimal global fitness value PRENEW(k) is sought by: 。

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