4.1. Case 1
The Okaloosa Armature Tester (OAT) and MCA 103 armature published by the Florida laboratory were taken as the research objects [
25,
26]. The geometric dimensions and material property parameters of the model are shown in
Table 4.
The extrapolation prediction performance of the model was tested using the launch data of SLK 018. The armature has a mass of 249.72 g and flies out after accelerating in the bore for 1.0 ms. The waveform diagram of excitation current is shown in
Figure 10.
Establishing the finite element simulation model of the electromagnetic railgun. Based on the armature conductivity (1.86 × 10
7 S/m) in
Table 3, the numerical simulation results of the armature electromagnetic force under different armature conductivities are obtained, as shown in
Figure 11. Because of the short acceleration time and low flight velocity of the armature in this case, there is no “pseudo-oscillation” in the numerical solution of the armature electromagnetic force under each conductivity. Under the condition of low armature conductivity, the current distribution of the conductor is more uniform, and the armature electromagnetic force is larger.
Through the above numerical simulation, 4000 groups of required sample data are obtained. In order to eliminate the influence of the magnitude and dimension of data on the prediction accuracy and convergence speed, the sample data are normalized and divided into a training set and a test set. Among them, the training set is divided into two kinds: the sample data obtained under the standard armature conductivity and the sample data obtained under different armature conductivity. The test set is the sample data in 0.87–1 ms time under the standard armature conductivity.
The above two training sets are used to train four extrapolated prediction models: Support Vector Regression (SVR), Random Forest (RF), DNN, and DBN-DNN. Then, the test set is used to test the extrapolated prediction model after training. The Mean Absolute Percentage Error (MAPE) is selected as the standard to measure the evaluation performance of the extrapolated prediction model. The value range of MAPE is [0, +∞). The smaller the value is, the higher the accuracy of the prediction model is. The calculation method is
In the formula,
N is the number of samples, and
and
yi are the predicted and simulated value of armature electromagnetic force of the sample, respectively. Considering the randomness of the performance evaluation of each extrapolation prediction model, the four extrapolation prediction models are trained and tested five times, and then the average value of MAPE is taken, as shown in
Table 5.
Compared with the training set under the standard armature conductivity, the sample data in the training set under a different armature conductivity are more sufficient, which enables the extrapolated prediction model to better grasp the overall characteristics of the data. It is more helpful to learn the mapping relationship between the excitation current, time, velocity, armature conductivity variables, and armature electromagnetic force, and the prediction effect is better.
As traditional machine learning models, SVR and RF have a limited ability to deal with input features, a restricted generalization ability, and a low prediction accuracy when solving complex problems. As a deep learning model, DNN can use deep networks and a large number of sample data to learn the multi-level abstract features and the hidden structural representations of data, and it has a high prediction accuracy. In the DBN-DNN model, DBN completes feature extraction by pre-training the coupling relationship of input features, and it provides reasonable initial parameters for DNN training, so DBN-DNN has a greater nonlinear fitting ability and generalization ability than DNN model, and it has the highest prediction accuracy.
Through the comparison of the MAPE average of the four extrapolated prediction models under the two training sets, we know that the training set under different armature conductivities is used to train the DBN-DNN model, and then the test set is used to test the highest accuracy, and the average MAPE is the smallest, which is 0.52%. Thirty-five groups of samples are selected from its test set at equal intervals, and the calculated values of armature electromagnetic force are compared with the predicted values, as shown in
Figure 12.
In
Figure 11, the numerical simulation results of the armature electromagnetic force of 0–0.87 ms under the standard armature conductivity are obtained. The DBN-DNN model is trained with the sample data obtained under different armature conductivity, and then the extrapolated prediction of the armature electromagnetic force of 0.87–1 ms under the standard armature conductivity is obtained by using the test set sample extrapolation prediction. The calculated value is superimposed with the predicted values to obtain the comprehensive armature electromagnetic force of the whole launch process (0–1 ms), as shown in
Figure 13.
According to the armature electromagnetic force under the standard armature conductivity in
Figure 11, the calculated value of the armature velocity is obtained; according to the integrated armature electromagnetic force under the standard armature conductivity in
Figure 13, the combined value of calculation and prediction of the armature velocity is obtained. The two values are similar, and they are compared with the experimental measurement value of the armature movement velocity, as shown in
Figure 14. The experimental measurement value of the armature exit velocity is 247.0 m/s; the calculated value is 245.7 m/s, which is 0.53% smaller than the experimental measurement value; and the combined value of calculation and prediction is 244.9 m/s, which is 0.85% smaller than the experimental measurement value.
4.2. Case 2
In case 1, the armature velocity is low. In order to verify the applicability of the extrapolation prediction method under high-speed armature movement, the 40 mm × 50 mm medium-caliber railgun developed by the Agency for Defense Development (ADD) is taken as the research object [
27]. Using the launch data of test number #26, where the armature mass is 300 g and it flies out after accelerating in the bore for 4.0 ms. The waveform diagram of the excitation current is shown in
Figure 15.
Establishing the finite element simulation model of the electromagnetic railgun. The numerical simulation results of the armature electromagnetic force under different armature conductivities are obtained when 2.50 × 10
7 S/m is used as the standard armature conductivity in this case model, as shown in
Figure 16. Under the standard conductivity, the numerical solution of the armature electromagnetic force appears as “pseudo-oscillation” after 1.41 ms, and the lower the armature conductivity, the later the “pseudo-oscillation” occurs, and the stable numerical simulation of the whole launch process can be realized under 40% standard armature conductivity.
Through the above numerical simulation, 9500 groups of sample data under different armature conductivities of the stable stage are obtained. These sample data are normalized and divided into a training set and a test set. The training set contains the 0–1.13 ms sample data under the standard armature conductivity and all the sample data under low conductivity; the test set is the 1.13–1.41 ms sample data under the standard armature conductivity. Using the training set to train the DBN-DNN model, and then using the test set to test the DBN-DNN model after training. The average MAPE of five training tests is 0.56%.
The DBN-DNN model, which is closest to the average value of MAPE in five training tests, is used to predict the armature electromagnetic force of the pseudo-oscillation stage (1.41–4.0 ms) under the standard conductivity, and the extrapolated prediction value of the armature electromagnetic force is obtained. Then, superimposed with the calculated value of the stable stage under the standard conductivity (0–1.41 ms) in
Figure 16, the integrated armature electromagnetic force of the whole launch process (0–4.0 ms) is obtained, as shown in
Figure 17.
According to the armature kinetic equation and the integrated armature electromagnetic force under the standard armature conductivity obtained in
Figure 17, the armature exit velocity is calculated as 2029.2 m/s. It is 1.03% smaller than the experimental measurement value 2050.3 m/s, which can meet the needs of practical engineering calculation and tests the performance of the armature electromagnetic force extrapolation prediction of the DBN-DNN model.
4.3. Training Strategy
The above two cases use the original training strategy: based on the stable numerical simulation sample data under different conductivities, they train the model together, and then they extrapolate the armature electromagnetic force under the standard conductivity. In order to improve the convergence speed and prediction performance of the model, this paper further proposes an improved training strategy for the transfer of DBN-DNN parameters from the armature electromagnetic force to the standard conductivity under a low conductivity. The details are as follows: First, the DBN-DNN is trained based on the numerical simulation data of 40% standard conductivity, and the current network parameters are saved after the training is completed. They are then used as the initial value of the network parameters under 60% standard conductivity. Similarly, the network parameters after training under 60% standard conductivity are taken as the initial values of network parameters under 80% standard armature conductivity. Based on this strategy, the extrapolation prediction of the armature electromagnetic force under standard armature conductivity is realized.
Using the two cases in this paper, the results of the DBN-DNN model under the two training strategies are compared, as shown in
Table 6. Compared with the original training strategy, the improved training strategy reduces the MAPE value by about 20%, and the prediction effect is better; the improved training strategy improved the training speed of the model by 46.05% and 63.86%, respectively. As a whole, the training speed can be greatly accelerated by improving the training strategy, while the accuracy of the prediction model is guaranteed.
In the DBN-DNN model with an improved training strategy, the solution under different conductivities can be regarded as multiple tasks with similar control equations. Although the initial training error of the network is higher at 40% armature conductivity, the initial training error of the network at 60%, 80%, and a standard armature conductivity will gradually decrease. Additionally, the number of iterations decreases with the training process, and the training speed can become faster and faster so that the solution of the network can converge quickly under the standard armature conductivity, thus accelerating the optimization process of network parameters.