Real-time prediction method of carbon concentration in carburized steel based on a BP neural network

ABSTRACT

Gears are key components of mechanical transmission systems. They need to be carburized and quenched to meet the requirements of internal toughness and external hardness, to obtain higher hardness and wear resistance. Heat treatment engineers can calculate carbon concentration distribution through the finite element analysis method, facing the challenge of high computational cost and high memory requirements. To solve this problem, a real-time prediction method of 1D and 2D carburizing concentration based on a Back Propagation (BP) neural network (BPNN) is proposed. First, carburizing experiments were conducted to verify the accuracy of the carburizing numerical model. Then, by establishing an accurate carburizing model, 37,800 and 177,147 1D and 2D carburizing training samples were generated using the finite element model (FEM) method, respectively. Finally, for 1D carburizing, the average relative error of the training model on the test set is 0.2911%. For 2D carburizing, the average relative error of the reconstructed BPNN model on the test set was 0.4381%, and a real-time performance evaluation was performed. The carbon concentration prediction time for each set of process parameters is only 0.12 s, nearly 175 times faster than FEM calculation, which meets the accuracy and real-time requirements for real-time prediction of carburizing carbon concentration.

KEYWORDS:

• Carburizing
• carbon concentration gradient
• BPNN
• 20Cr2Ni4A

Introduction

In the real world, gears are generally affected by various stresses such as variable load impact, contact stress, pulsating bending stress and friction [1–3] under working conditions. Therefore, gear steel is required to have high toughness, fatigue strength and wear resistance [4]. In order to improve the performance of gears, carburizing and quenching has become the dominant heat treatment process for high-performance hard tooth surfaces today [5]. After carburizing, it is quenched and tempered at low temperature, so that the material obtains high-carbon and low-carbon tempered martensite structures from the surface to the core. It is able to reach high surface hardness and contact fatigue strength, while still maintains good impact toughness in the core [6]. During the atmosphere carburizing process, the carburizing medium decomposes into activated carbon atoms at a specific temperature. These activated carbon atoms are absorbed by the surface of the part, thereby forming a carbon concentration gradient between the surface and the core of the part [7].

In current carburizing production processes, empirical analysis or traditional trial and error methods are usually used to formulate process plans [8]. That is to find the relationship between carburizing process parameters and carbon content gradient changes through experiments or simulations. However, the disadvantages of this method include expensive experimental costs, long research cycles and abundant demands on labor, resources and financial investment. Therefore, a more forward-looking method is to use numerical simulation technology [9] to calculate the carbon concentration distribution under given carburizing process parameters. The optimal conditions for the carburizing process can be effectively determined, thereby reducing experimental costs and research cycles. However, in actual production, the use of numerical simulation methods requires a certain theoretical foundation, and the knowledge of production technology operators is insufficient. In addition, numerical simulation also faces some challenges, such as requiring a large amount of computing resources, including computing time and running memory [10]. Therefore, it is necessary to construct a model for predicting the carbon concentration gradient after atmosphere carburizing to find the optimal atmosphere carburizing process parameters, which have practical application value.

BPNN is an artificial neural network widely used in the fields of data classification, prediction and identification [11–13]. It has a strong learning ability and high nonlinear mapping ability. In materials science, BPNN has been successfully used to study the relationship between material processing parameters and material properties [14], and has achieved significant applications. In the field of heat treatment, it can be used to predict temperatures [15], material thermal performance prediction [16] and heat treatment deformation [17]. This can improve the control and effect during the heat treatment process and provide technical support for the manufacturing of metal parts.

Some scholars use the BPNN model to establish the nonlinear relationship between the surface carbon concentration, the effective carburizing layer depth and the process parameters after vacuum carburizing to find the optimal vacuum carburizing process conditions. The percentage prediction error of the model is 5% [8]. However, the depth of the carburized layer and the surface carbon concentration cannot accurately reflect the carbon concentration gradient changes of the part. This paper established a numerical model of the carburizing process of 20Cr2Ni4A cylindrical specimens through finite element, and designed the furnace sample to be carburized to verify the accuracy of the model. 1800 and 243 sets of different carburizing process parameters were designed using orthogonal experimental methods for 1D and 2D, respectively, according to the actual carburizing process flow, and the carbon concentration distribution results under the corresponding process parameters were calculated. Finally, a prediction model based on BPNN is established with the carburizing process parameters and distance from the surface (1D carburizing) or coordinates (2D carburizing) as inputs, and the carbon concentration value as output. It aims to predict the relationship between carbon concentration distribution and process parameters after atmosphere carburizing heat treatment.

The average relative error of the BPNN model established in this article for the 1D carburizing on the test set is 0.2911%, while that for 2D carburizing is 0.4381%. Moreover, the 2D carbon concentration prediction time for each set of process parameters is only 0.12 s, which is nearly 175 times faster than the finite element model (FEM) calculation. Therefore, the established BPNN model has important application value in optimizing carburizing process parameters, reducing carbon emissions and improving the environment.

Modeling and experimental methods

Numerical model

When carbon atoms diffuse inside the metal, the carbon concentration in the infiltration layer gradually decreases from the surface to the center. The existing carbon concentration difference becomes the diffusion driving force. Under this driving force, the rate at which carbon atoms pass through the interface of a certain layer can be described by carbon flux, namely Fick’s second law [18]: (1) ${\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }\mathrm{t}}={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(D\cdot {\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }x}}\right)}}$(1) where C is the carbon concentration, t is the carburizing time (s) and D is the carbon diffusion coefficient. Considering the effect of alloying elements on the diffusion coefficient, the diffusion coefficient with respect to temperature and carbon concentration is derived as below: (2) $D=0.0047\exp \left(-1.6C\right)\exp \left[{\displaystyle \frac{-\left(37000-6600C\right)}{\mathrm{RT}}}\right]\times S$(2) where R is the gas constant, T is the temperature (Kelvin temperature) and S in the equation is expressed as: (3) $\begin{array}{rl}S=& 1+\left(0.15+0.033\left[\text{\%}S\mathrm{i}\right]\right)\left[\text{\%}\mathrm{Si}\right]-0.0365\left[\text{\%}\mathrm{Mn}\right]\\ & -\left(0.13-0.0055\left[\text{\%}\mathrm{Cr}\right]\right)\left[\text{\%}\mathrm{Cr}\right]\\ & +\left(0.03-0.03365\left[\text{\%}\mathrm{Ni}\right]\right)\left[\text{\%}\mathrm{Ni}\right]\\ & -\left(0.025-0.01\left[\text{\%}\mathrm{Mo}\right]\right)\left[\text{\%}\mathrm{Mo}\right]\\ & -\left(0.03-0.02\left[\text{\%}\mathrm{Al}\right]\right)\left[\text{\%}\mathrm{Al}\right]\\ & -\left(0.016+0.0014\left[\text{\%}\mathrm{Cu}\right]\right)\left[\text{\%}\mathrm{Cu}\right]\\ & -\left(0.22-0.01\left[\text{\%}\mathrm{V}\right]\right)\left[\text{\%}\mathrm{V}\right]\end{array}$(3) The carbon flux on the metal surface represents the mass of carbon atoms entering the metal per unit surface area from the atmosphere per unit time. The carbon flux is expressed as: (4) $-D{\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }x}=\text{β}\cdot \left(C_{\mathrm{g}}-C_s\right)}$(4) where $\text{β}$ is the transfer coefficient, C_g is the carbon concentration in the atmosphere and C₀ is the carbon concentration on the metal surface.

Finite element modeling

The research object of this article is the carburization process of 20Cr2Ni4A cylindrical sample, and to establish a FEM with Abaqus. As shown in Figure 1, in order to perform simplified calculations of the two-dimensional diffusion process, half of the cylindrical sample is taken as the simulation object. A geometric model is established with a height of 48 mm, a diameter of 40 mm and AB as the axis of symmetry, and boundary conditions of temperature and carburizing carbon potential are imposed on the remaining three sides.

Figure 1. Two-dimensional mesh division model of cylindrical sample.

Experiment methods

The experimental material and heat treatment process

In order to verify the accuracy of the simulation results, carburizing experiments were conducted in the device as shown in Figure 2 wx-1000, an atmosphere carburizing multifunctional furnace of Jiangsu Yixin Gear Manufacturing Co., Ltd. The material used in this study is alloy steel 20Cr2Ni4A, prepared in the form of a cylindrical sample of $\mathrm{\varnothing }$ 20 mm $\times$ 48 mm. Table 1 shows the chemical composition of 20Cr2Ni4A steel.

Figure 2. Carburizing furnace.

Table 1. Chemical composition (wt.%) of the base materials.

Compositions	C	Si	Mn	Cr	Ni	P	S	Cu	Fe
Content	0.2	0.23	0.34	1.17	3.41	0.012	0.0076	0.077	Bal.

The carburizing process mainly goes through two stages: carburizing and diffusing, followed by cooling to the quenching temperature. The carburizing process mainly goes through two stages: carburizing and diffusing, followed by cooling to the quenching temperature, the process curve is shown in Figure 3.

Figure 3. Carburizing and quenching process curve.

According to the determined carburizing process of spiral bevel gears, the cylindrical samples were carburized in the furnace to verify the accuracy of the simulation model. As shown in Figure 4, processes 1, 2 and 3 are processes with different carburizing, diffusion and quenching parameters, including temperature, carbon potential and duration. The carburizing diffusion temperature of process 1 and process 2 is 910°C, the carburizing diffusion temperature of process 3 is 920°C, and the quenching temperature of the three processes is 820°C. The carburizing carbon potential of the three processes is 1.2%, the diffusion carbon potential of process 1 and process 3 is 0.8%, and the diffusion carbon potential of process 2 is 0.9%. The specific process parameters are shown in Table 2.

Figure 4. Pictures of furnace samples before carburizing by three different processes. (a) Process 1; (b) process 2 and (c) process 3.

Table 2. Carburizing and quenching process.

Process no.	Carburizing T1/t1 (h)/c1%	Diffusion T2/t2 (h)/c2%	Quenching T3/t3 (h)
1	910/7/1.2	910/2.5/0.8	820/0.7
2	910/3.5/1.2	910/1/0.9	820/0.7
3	920/14.5/1.2	920/6/0.8	820/2

After carburizing, the samples with three different process parameters are released from the furnace as shown in Figure 5. The oxide layer was then removed from the surface of the cylindrical specimen for carbon concentration gradient testing.

Figure 5. Pictures of furnace samples after carburizing by three different processes. (a) Process 1; (b) process 2 and (c) process 3.

Carbon concentration test

After carburizing, the BURKER direct-reading spectrometer is used to test the carbon concentration value as shown in Figure 6. The carbon concentration gradient test uses the analysis principle of atomic emission spectroscopy. The sample is placed directly on the excitation platform, the samples generate atomic emission spectrums excited by the discharging of light source through electrodes. The atomic characteristic analysis spectral lines of each element are obtained through processes such as lighting, spectroscopy and detection, and the accurate content of each element in the material is directly read out after computer data processing.

Figure 6. BRUEKR direct-reading spectrometer.

The test positions of the cylindrical samples of the three different processes are shown in Figure 7(a). First, the surface carbon concentration is measured, using a grinder to grind away the spectrometer test traces, and then the component at the same position is measured until basic. The test area of the test sample should cover the hole at the electrode position (the area pointed by the red arrow in Figure 7(b) is the hole area, and the area pointed by the black arrow is the electrode position area). Therefore, the test position must be a certain distance from the boundary, and there must be a certain distance distribution between the three selected test positions. Except for the edge and corner positions, the carbon concentration gradient distribution is theoretically the same across the entire end surface. The test results of test positions 1–3 should be close. Due to the fluctuation of material composition and test error, there are small errors in the test results of the three positions.

Figure 7. Three different process carbon concentration test positions. (a) Test positions and (b) the schematic diagram of the testing area.

By calculating the carbon concentration results under three sets of different process parameters in the simulation software, a total of 31 points were extracted from the top of each sample inward, with 1 point extracted every 0.13 mm, as shown in Figure 8(a). Comparing the simulation results with the experimental results, it can be seen from Figure 8(b–d) that at a distance of 0.5–4 mm from the surface, the experimental values of the three processes are in good accordance with the simulation values. In the distance of 0–0.3 mm away from the surface, the maximum percentage error is 10%, which is caused by decarburization [19]. During metal heat treatment, the carbon content of the surface layer of steel is lower than the carbon content inside the metal, which is called decarburization. The essence of decarburization is the diffusion of carbon atoms. Because the surface carbon atoms increase when heated, the affinity with oxygen atoms is greater than the affinity with carbon atoms, resulting in the diffusion of surface carbon atoms.

Figure 8. Comparison of experimental values and simulation values. (a) Simulation result point positions; (b) process 1; (c) process 2 and (d) process 3.

The surface carbon concentration of process 2 is higher than that of the other two groups of processes. This is due to the shorter diffusion time of process 2. The rate at which carbon atoms are absorbed by the surface of the steel is much greater than the rate at which carbon atoms diffuse into the interior of the steel. Carbon atoms accumulate on the surface, resulting in a higher surface carbon concentration and a thinner carburized layer during the carburization stage.

Microhardness and microstructure

After the heat treatment process, the microstructures were observed by an optical microscope. Figure 9(b) is a schematic diagram of the microstructure observation of the surface layer (marked as ‘A’), sub-surface layer (marked as ‘B’) and base material part (marked as ‘C’) of the test block. After atmosphere carburizing in process 1, high-hardness martensite was observed in the deep layer within 100 $\mu \mathrm{m}$. At approximately 1200 $\mu \mathrm{m}$ from the surface, the martensite content in the transition layer decreased and was accompanied by lamellar bainite. The structure of the core base material that had not been carburized was mainly bainite [20]. Since the results of processes 2 and 3 were similar to the microstructure of process 1, processes 2 and 3 can refer to the results of process 1.

Figure 9. Microstructure after atmosphere carburizing. (a) Cylindrical sample; (b) sketch map; (c) surface; (d) sub-surface and (e) center.

After carburizing and quenching, the surface microhardness of the sample was tested by a HVS-1000A hardness tester. The Vickers hardness test was carried out in accordance with the national standard GB/T 230.1-2018. The test load was 500 g and the loading time was 10 s. The test position for the hardness gradient is selected at a cross-section 8 mm away from the end face, and the hardness of the sample is measured along the arc normal direction every 0.2 mm. The hardness test position is shown in Figure 10(a). The effective hardened layer depths of the three processes are 0.63, 1.05 and 1.36 mm, respectively. The hardness gradient is shown in Figure 10(b).

Figure 10. Microhardness profile of the carburized layer. (a) Indentation diagram and (b) three different process hardness gradients.

1D carbon concentration prediction

BPNN prediction model

A neural network is a multi-layer feedforward neural network containing three or more layers of neurons. Each layer is composed of multiple neurons, and each neuron between adjacent layers is fully connected, while there is no connection between the neurons in the same layer. BPNN adopts a supervised learning method for training. In forward propagation, the input data are weighted and activated through the activation function of each layer to obtain the output result. In backpropagation, the error of the hidden layer is calculated forward layer-by-layer through the chain rule, and the weight of each neuron is updated based on the gradient descent principle, so that the error gradually decreases [21]. This process continues to iterate until the network output result is close to the real label and reaches the training goal.

Dataset construction

By establishing an accurate numerical model of the 20Cr2Ni4A carburizing process, an orthogonal process combination is designed for large-scale calculations. As shown in Table 3, the carburizing temperature and diffusion temperature are matched and fixed into six groups. The carburizing and diffusion times are set to the same. The carburizing and diffusion carbon potential are set into three and four groups, respectively, with a total of $6\times 3\times 5\times 4\times 5=1800$ different groups process parameters.

Table 3. Simulation generates a training set scheme list.

Temperature	Carburizing		Diffusion
Carburizing–diffusion (T °C)	Carbon potential (c%)	Time t (h)	Carbon potential (c%)	Time t (h)
900–900	1	3	0.8	3
910–900	1.5	6	0.85	6
910–910	2	9	0.9	9
920–900		12	0.95	12
920–910		15		15
920–920

For each set of process parameters, one point is extracted every 0.02 mm from the end of the sample to the inside, and a total of 21 points are extracted to obtain the carbon concentration at the corresponding position as shown in Figure 11.

Figure 11. One-dimensional carburizing carbon concentration point location.

BPNN structure

The structure of a neural network has a great impact on its performance, and how to choose an appropriate network structure is very critical. However, there is currently no complete and unified theoretical method to determine the structure of BPNN in applications, and most of the time it is based on experience. If the number of network layers and neurons is too large, the amount of computation will increase and the training speed will decrease, resulting in a decrease in fault tolerance. However, if the number of network layers and neurons is too small, the network may not be able to achieve the set convergence accuracy. After many attempts, the number of neurons in the three hidden layers is 10, and the topology model is 7 × 10 × 10 × 10 × 1, as shown in Figure 12. The simulation software generates 1800 sets of simulation data of different process parameters. Each set of data contains carbon concentration values at 21 different positions from surface to interior. A total of 21$\times 1800$ = 37,800 samples, of which 26,460 sets of samples are used to train the model. The remaining 11,340 sets of samples are used to test the performance of the model.

Figure 12. Schematic diagram of the BPNN structure.

BPNN training function and training parameters

Choice of transfer function

The functions of the first and second hidden layers of the atmosphere carburizing carbon concentration prediction model constructed in this article are set to the S-shaped tangent function tansig as shown in formula (5): (5) $\mathrm{f}\left(x\right)={\displaystyle \frac{2}{1+e^{-2x}}-1}$(5) The third layer hidden layer function is set to the linear function purelin, as formula (6): (6) $\mathrm{f}\left(x\right)=w\cdot x+b$(6) While, w and b are the weights and thresholds, respectively.

Selection of training parameters

The amount of change during weight adjustment is affected by the learning rate (value range is 0.01∼0.8), and the stability of the system may be affected by a too large learning rate. A learning rate that is too small may cause the training process to converge slowly, resulting in a long training time. When selecting the expected error, the accuracy and structure of the neural network model interact with each other. After many attempts, the expected error was set to 1 × 10⁻⁶, the learning rate was set to 0.01, the momentum factor was set to 0.001, and other parameters were set to default values.

Choice of the training function

BPNN has many advantages and is widely used, but there are also some problems. For example, the algorithm may fall into local extrema and the weights converge to local minimum points, and the learning efficiency is low and the convergence speed is slow, etc. In order to make BPNN more widely used, improvements to the traditional BPNN algorithm mainly include trainscg, traingdm, traingda, trainbfg, trainlm and trainrp [22]. The learning curve when training BPNN based on six algorithms are shown in Figure 13. It can be seen that when using traingda, trainscg and trainbfg training algorithms, the learning curve shows that the iteration stops before the maximum number of iterations is reached. This is because a minimum gradient threshold of 1$\times$10⁻⁷ is set for all algorithms during the training process. Once the gradient of the network weight drops below this threshold, the algorithm will automatically terminate the iteration, assuming that it has approached or reached a local minimum [23]. This early stopping behavior indicates that the network parameters are close enough to the optimal solution and further iteration may have limited improvement in performance. However, the ‘trainlm’ algorithm achieves the expected set goals in the learning curve and the error decreases quickly. This is because trainlm uses the Levenberg-Marquardt algorithm, which is a nonlinear optimization method between Newton's method and gradient descent method. It is not sensitive to over-parameterization problems and can effectively handle redundant parameter problems. The chance that the loss function falls into a local minimum is greatly reduced. Therefore, for carburizing carbon concentration prediction, the trainlm algorithm should be selected.

Figure 13. Learning curve chart of each training algorithm.

Model prediction verification

Before applying the BPNN model, the trained model needs to be verified to evaluate its generalization performance. This can be tested using untrained set of 11,340 test samples. The calculation formula for the relative error of prediction is: (7) $\frac{ϵ={\displaystyle y-\hat{y}}}{\hat{y}\times 100\text{\%}}$(7) While $ϵ$ represents the relative error of prediction, y represents the predicted value of the output parameter and y^{^} represents the actual value of the output parameter.

The minimum relative error of prediction is 2.238 × 10⁻⁵%, the maximum value is 2.8738%, and the average relative error is 0.2911%. This illustrates that the BPNN is feasible in predicting carburizing carbon concentration and can accurately reflect the nonlinear mapping relationship between the atmosphere carburizing process parameters and the gradient change of carbon content after carburizing.

In order to better observe the gradient change of carburizing carbon content, the corresponding input data of six sets of process parameters (as shown in Table 4) were randomly selected from the test set and input into the trained neural network. The test results are, as shown in Figure 14, six different carbon concentration gradient curves are displayed. By comparing the neural network model prediction results with the simulation results, it can be observed that the prediction error percentage is less than 1%, indicating that the model has strong generalization performance.

Figure 14. Comparison of predicted and simulated values for six different processes.

Table 4. Six sets of carburizing process parameters.

Process no.	Carburizing T1/t1 (h)/c1%	Diffusion T2/t2 (h)/c2%	Quenching T3/t3 (h)/c3%
1	900/3/1	900/9/0.85	820/0.7/0.85
2	910/15/1	900/3/0.85	820/0.7/0.85
3	910/6/1	910/12/0.85	820/0.7/0.85
4	920/15/1	900/12/0.85	820/0.7/0.85
5	920/9/1	910/3/0.85	820/0.7/0.85
6	920/3/2	920/15/0.8	820/0.7/0.8

2D carbon concentration prediction

Dataset construction

In the actual production process, it is necessary to obtain the carbon concentration distribution of the cross-section of the workpiece. One-dimensional carburizing cannot obtain the carbon concentration distribution in the corner areas of the two-dimensional carburizing cloud image. Therefore, this article selects a common square shape and uses finite element software to generate carbon concentration distributions under 243 sets of different process parameters. The grid is divided into a total of 729 grid points as shown in Figure 15. The combination of process parameters is shown in Table 5. The temperature, time, carbon potential, x coordinate and y coordinate corresponding to the carburizing and diffusing stages are used as input, and the carbon concentration is the output. There are a total of 729$\times$243 = 177,147 samples, of which 124,002 sets of samples are used to train the network, and the remaining samples are used to test the network performance. Since the number of two-dimensional training samples is much larger than the number of one-dimensional samples, the number of neurons and hidden layers of the original network model is not enough to learn such complex features. Therefore, the BPNN topology model was rebuilt as 8$\times$20$\times$20$\times$20$\times$20$\times$1, the learning rate was set to 1 × 10⁻⁵ and trainlm was selected as the training algorithm.

Figure 15. Square meshing.

Table 5. Simulation generates a training set scheme list.

Temperature	Carburizing		Diffusion
Carburizing–diffusion (T °C)	Carbon potential (c%)	Time t (h)	Carbon potential (c%)	Time t (h)
910–910	1	6	0.8	6
920–910	1.5	9	0.85	9
920–920	2	12	0.9	12

Model prediction verification

In the training performance curve of the BPNN model shown in Figure 16, after 62 rounds of learning and training iterations, the error has been successfully reduced to the preset target error level. This shows that the network has a very fast training speed and can achieve good training results. It can be seen from the training performance curve that in the first 20 rounds or so, the mean square error dropped sharply, and the network convergence speed was significantly rapid. After 50 rounds, the training performance curve is smooth and has no obvious fluctuations, which can better reflect the one-to-one correspondence between the eight input layer parameters and the output. On the test set, the root mean square error is 2.84$\times$10⁻⁵, the average percentage error is 0.4381%, the maximum percentage error is 4.12% and the minimum percentage error is 2.950$\times$10⁻⁶%.

Figure 16. Training performance curve of the BPNN model.

However, the average relative error cannot fully represent the accuracy of the neural network prediction results, because when averaging the prediction errors of each sample, some large errors may be balanced. A more in-depth approach is to compare FEM simulation results with model predictions directly in the 2D view. Therefore, the input data corresponding to three different process parameters (as shown in Table 6) are randomly selected from the test set and input into the trained BPNN. As shown in Figure 17, from left to right are the simulation cloud chart, prediction result cloud chart and percentage error cloud chart. It can be seen from the percentage error cloud charts under three different process parameters that the maximum percentage errors of the three different process parameters are: 2.29%, 1.83% and 3.79%, respectively, indicating that the model has strong generalization. There is no obvious pattern in the distribution of areas with large percentage errors.

Figure 17. From left to right: (1) ground truths, (2) forecast results and (3) percent error.

Table 6. Three sets of carburizing process parameters.

Process no.	Carburizing T1/t1 (h)/ c1%	Diffusion T2/t2 (h)/c2%	Quenching T3/t3(h)/c2%
1	910/6/1	910/12/0.85	820/0.7/0.85
2	910/9/1	910/12/0.8	820/0.7/0.8
3	920/12/2	920/12/0.9	820/0.7/0.9

Real-time performance evaluation

Since the research goal of this article is to achieve real-time prediction of carburizing carbon concentration, requirements are placed on the real-time performance of the mapping model. We measured the time took by a neural network to predict the carbon concentration in the two-dimensional carburizing of a workpiece, as shown in Table 7, and the corresponding process parameters are shown in Table 6. The average prediction time of BPNN was 0.12 s, which met the need for real-time prediction of the carburizing field in the actual carburizing production process, while the average calculation time of FEM software was 21 s.

Table 7. Comparison of prediction time and simulation calculation time.

Process no.	Simulation software time (s)	BPNN time (s)
1	19	0.208909
2	19	0.136368
3	25	0.017733

Conclusions

This paper first establishes a carburizing model, which can obtain the carbon concentration distribution results of the sample under different carburizing process parameters. Through three groups of actual carburizing experiments, samples with effective carburizing depths of 1.44, 1.05 and 1.35 mm were obtained. The test results of the carbon concentration gradient of these three groups of samples were in good agreement with the results of the corresponding calculation model, verifying the accuracy of the model.

An accurate FEM was used to calculate 1800 sets of carburizing results with different 1D carburizing process parameters and 243 sets of carburizing results with different 2D carburizing process parameters. These calculation results serve as BPNN model training samples. Through training with a large amount of historical data, the model can learn the complex nonlinear relationship of process parameters and carbon concentration in atmosphere carburizing. By testing six different training algorithms, it was determined that the Levenberg-Marquardt (LM) algorithm was used to train faster and with smaller training errors. For 1D carburizing, the average relative error of the trained model on the test set is 0.2911%, and the correlation coefficient between the training value and the simulation value reaches 0.99998. For 2D carburizing, the average relative error of the reconstructed BPNN model on the test set is 0.4381%, and a real-time performance evaluation was performed. The carbon concentration prediction time for each set of process parameters is only 0.12 s, which is nearly 175 times faster than the FEM calculation.

Future research can predict the carburizing carbon concentration of an arbitrary 2D shape workpiece, thereby further improving the application scope and effect of the model. In summary, the carbon concentration prediction model for atmosphere carburizing based on BPNN has superiority in both prediction performance and practical application.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was financially supported by National Natural Science Foundation of China [grant number 51905148].