Keywords
1. Introduction
With the potential application of superconductors in flywheel energy storage systems and magnetic bearing [1], [2], [3], high temperature superconductors have gained an increasing interest of superconducting community. However,JcB is a fundamental property of superconductors that limits the maximum current they can carry. In addition, the superconductors are also subjected to a higher electromagnetic force induced by shielding current and magnetic field under the external field [4], [5], [6]. $J_{c}(B)$ of superconductors is affected by many factors, including magnetic field, temperature and strain [7], [8], [9], [10]. For the relationship between $J_{c}(B)$ and the magnetic field, many scholars have proposed different models in the past years. For Bean model [11], [12], the critical current density $J_c$ is independent of the magnetic field, i.e. $ J_{c}=\text { constant } $. In the Kim model [13], [14], $ J_{c}=\alpha /\left(|B|+B_{0}\right) $, which considers the pinning force independent of the magnetic field, $ \alpha=\text { constant } $. In the exponential model [15], [16], $ J_{c}=J_{c 0} \exp \left(-|B| / B_{0}\right) $.
$J_cB$ is of interest due to the limitation of magnetic field and strain in the application of superconductors [7]. Ekin [17] described $J_{c}(B)$ measurement commonly used in experiments, which mainly includes several measurement methods. $J_{c}(B)$ measured by transport method is more accurate, and some researchers [18] have shown that the current sweep rate has an impact on the critical current. On the other hand, indirect measurement methods include magnetization measurements and eddy-current testing. Since the transport method is more suitable for tapes and wires rather than the bulk, so the indirect method is generally adopted in bulk superconductor [19], [20].
The magnetization measurement is one of the contactless methods, where $J_{c}(B)$ of the bulk superconductor can $B_e$ obtained by the increasing and decreasing processes of magnetic field [21], [22]. Firstly, the hysteresis loop of the bulk is measured by Hall sensor or Vibrating Sample Magnetometer (VSM) in experiments [23], [24]. $J_{c}(B)$ of a superconductor can $B_e$ determined by measuring the difference between the hysteresis loops obtained during increasing and decreasing magnetic field sweeps [25]. However, this analytical formula of magnetization measurement method is usually based on the Bean model [21]. Compared with Kim model, an error can $B_e$ observed in a low field [25]. Moreover, the applicability of the magnetization measurement method is restricted in the reverse region of magnetic field. Therefore, it is necessary to determine $J_{c}(B)$ with an efficient prediction method.
Recently, neural networks are popular in classification, clustering, recognition, and prediction [26], which have also been widely used in the field of superconductivity. Neural networks are powerful for manufacture and test of superconducting materials, fault detection, structure design and optimization [27], [28], [29], [30], [31], [32]. Sadeghi et al predicted the quench probability by ANN for six different high-temperature superconducting (HTS) tapes [33]. Ke et al carried out the prediction of magnetic levitation force of superconductor with deep learning [34]. Tomassetti et al optimized the structure of the tokamak device based on the neural networks [35]. The neural networks have also been applied in predicting $J_{c}(B)$ of superconductors. Mohammad et al estimated $J_{c}(B)$ and stress of high temperature superconducting tapes by neural networks [36], [37]. Liu et al trained the neural network with the temperature, magnetic field and angle to estimate $J_{c}(B)$ of tapes, and Zhu et al also gave the n-value of superconductors additionally by the same method [38], [39], [40], [41], [42]. During the prediction of $J_{c}(B)$ of superconducting thin films, three different ANNs were compared in terms of prediction efficiency and computation time [43]. Suresh et al proposed a method based on fuzzy logic to predict the critical current density of HTS tapes and compared with ANN [44].
In above works, $J_{c}(B)$ is studied in the superconducting tapes with ANN. However, there are few works on the prediction of $J_{c}(B)$ in bulk superconductors. Ikuta et al found the obvious magnetostriction as superconducting materials are in applied magnetic field. The maximum relative deformation of the sample measured in the experiment is more than 10−4 [45]. They also pointed out that magnetostriction measurement might $B_e$ a novel approach in the study of superconductivity. After that, the magnetostrictions in the superconductors were studied widely by many researchers [46], [47], [48], [49], [50], [51]. In this work, the ANN is used to solve $J_{c}(B)$ of bulk superconductor with the Kim model. The hysteresis loop and the magnetostriction loop are fed into ANN as the training data, respectively. With the trained ANN, the model parameters α and $B_0$ can $B_e$ obtained, which can characterize the $J_{c}(B)$ curve based on the formula of Kim model. In the Section 2, we derive the hysteresis loop formula and the magnetostriction formula of the cylinder with the Kim model, and calculation of $J_{c}(B)$ through the width of the hysteresis loop are also described. Then, we introduce the ANN, including the partition and normalization of the data set, the activation function and the loss function in the Section 3. Afterwards, the results and the validation of ANN will $B_e$ shown and discussed in the Section 4. Finally, in the Section 5, the main conclusions are summarized.
2. The hysteresis loop and magnetostriction loop
In this section, consider an infinitely long cylinder as shown in the Fig. 1, which is under an external magnetic field $B_e$ (only along z axis). At the same time, the eddy current loss is neglected. The magnetization of a superconducting cylinder is calculated based on the critical state Kim model.
Fig. 1. Infinitely long cylindrical superconductor and an externally applied magnetic field $B_e$ along its axis. |
As shown in Fig. 1, the shielding current can $B_e$ induced during the magnetization. Starting from the zero-field cooling condition, applied magnetic field $B_e$ rises to the maximum value $B_M$, then gradually decreases to 0, later reverses load to $-B_M$, finally decreases the applied magnetic field to 0 again. Fig. 2 shows the whole magnetic field scanning process. The similar distribution of magnetic field was also given in Ref. [52]. In such an excitation process, there are three types of field loops depending on the relationship between the fully penetrating field $B_p$, the maximum external field $B_M$ and $B^*$, where $B^*$ is the external field when the flux density gradient becomes negative during the decreasing branch. Here, we only focus on the field loop when the applied magnetic field is sufficiently large, i.e.$ \bar{B}_{M} \geqslant B_{p}$,$B^{*} \geqslant 0$.
Fig. 2. Schematic diagram of magnetic field distribution under Kim model. (a) The first increasing field branches. (b) The decreasing field branches [52]. |
2.1. Hysteresis loop
The magnetization $M\left(B_{e}\right)$ of a cylinder is calculated as [53]
$M=\frac{2 \pi}{\mu_{0} S} \int_{0}^{R} B(r) \cdot r \cdot d r-\frac{B_{8}}{\mu_{0}}.$
From the Eq. (1), the magnetization can be obtained by the magnetic flux density distribution B(r) inside the cylinder. The Maxwell equation in the cylindrical coordinate is
$-\frac{d}{d r} B(r)=\mu_{0} J_{c}(r)$
The Kim-Anderson model assumes that $J_{c}(r)=\alpha /\left(|B(r)|+B_{0}\right)$ [14]. Substituting Kim model formula into Eq. (2), we get
$\frac{d B}{d r}=-\mu_{0} \frac{\alpha}{B+B_{0}}$
It can be seen from the Ref. [52] that the magnetization process can be divided into five stages: Ⅰ-V. The corresponding boundary condition of each stage is substituted into the Eq. (3). Then $B(r)$ in the cylinder at each stage can be obtained after integration.
The five stages of the magnetization process are shown in Fig. 2. There are two stages during the increasing field branch in the Fig. 2(a). The center of the sample is not penetrated in stage Ⅰ ($0 \leqslant B_{e} \leqslant B_{p}$). The applied magnetic field continues to increase to $B_M$, i.e. $B_{p} \leqslant B_{e} \leqslant B_{M}$ in stage Ⅱ. Fig. 2(b) shows the decreasing branch of the magnetic field. The range of stage Ⅲ is $B_{M} \geqslant B_{e} \geqslant B^{*}$, where $B(r)$ inside the sample gradually shifts direction as $B_e$ decreases. The magnetic field range of stage Ⅳ is $B^{*} \geqslant B_{e} \geqslant 0$. In the stage V, $B_e$ is applied in the opposite direction, $0 \geqslant B_{e} \geqslant-B_{p}$. Through the four stages of Ⅱ to Ⅴ, the loop without the initial stage can $B_e$ presented [54].
The magnetization in stage Ⅰ is as follow [53]:
$M\left(B_{e}\right)=\frac{1}{\mu_{0}}\left[\frac{4(n-m)}{3 n^{2}}\left(m^{\frac{3}{2}}-B_{0}^{3}\right)+\frac{4}{5 n^{2}}\left(m^{\frac{5}{2}}-B_{0}^{5}\right)-B_{0}\left(1-\rho_{1}^{2}\right)-B_{e}\right],$
Where
$m=\left(B_{e}+B_{0}\right)^{2}, \quad n=2 \mu_{0} \alpha R, \quad \rho_{1}=1-\frac{m-B_{0}^{2}}{n}.$
The Br of stage Ⅱ is the same as that of Stage Ⅰ. Magnetization in stage Ⅱ is
$M\left(B_{e}\right)=\frac{1}{\mu_{0}}\left[\frac{4}{3 n^{2}} m^{\frac{3}{2}}+\frac{8}{15 n^{2}}\left((m-n)^{\frac{5}{2}}-m^{\frac{5}{2}}\right)-B_{0}-B_{e}\right].$
The magnetization of stage Ⅲ is [55]
$\begin{aligned} M\left(B_{e}\right)=\frac{1}{\mu_{0}} & {\left[\frac{4\left(m-m_{1}+2 n\right)}{3 n^{2}}\left(\frac{m+m_{1}}{2}\right)^{\frac{3}{2}}+\frac{8}{15 n^{2}}\left(m_{1}-n\right)^{\frac{5}{2}}\right], } \\ & +\frac{1}{\mu_{0}}\left[\frac{-4(2 m+5 n)}{15 n^{2}} m^{\frac{3}{2}}-B_{0}-B_{e}\right] \end{aligned}$
Where
$m_{1}=\left(B_{M}+B_{0}\right)^{2}, \quad \rho_{2}=1-\frac{m_{1}-m}{2 n}$
The magnetizations of stages Ⅳ and V are
$M\left(B_{e}\right)=\frac{1}{\mu_{0}}\left[\frac{8}{15 n^{2}}\left((m+n)^{\frac{5}{2}}-m^{\frac{5}{2}}\right)-\frac{4}{3 n^{2}} m^{\frac{3}{2}}-B_{0}-B_{e}\right],$
$\begin{array}{c} M\left(B_{e}\right)=\frac{1}{\mu_{0}}\left[\frac{4}{5 n^{2}} B_{0}^{5}-\frac{4\left(m_{2}+n\right)}{3 n^{2}} B_{0}^{5}+\frac{8}{15 n^{2}}\left(m_{2}+n\right)^{\frac{5}{2}}+2 B_{0} \rho_{3}^{2}\right] \\ \quad+\frac{1}{\mu_{0}}\left[\frac{4\left(m_{3}-n\right)}{3 n^{2}}\left(m_{3}^{\frac{3}{2}}-B_{0}^{3}\right)-\frac{4}{5 m^{2}}\left(m_{3}^{\frac{5}{2}}-B_{0}^{5}\right)+B_{0}-B_{e}\right], \end{array}$
Where
$m_{2}=B_{0}^{2}-B_{e}^{2}+2 B_{e} B_{0}, \quad m_{3}=\left(B_{e}-B_{0}\right)^{2}, \quad \rho_{3}=1-\frac{m_{3}-B_{0}^{2}}{n}.$
For the Kim model, there are two parameters α and $B_0$, which are represented by a dimensionless parameter $p=\sqrt{2 \mu_{0} \alpha R} / B_{0}$. Then the full penetration field of the cylinder is expressed as [52]
$B_{p}=\sqrt{B_{0}^{2}+2 \mu_{0} \alpha R}-B_{0}=B_{0}\left(\sqrt{1+p^{2}}-1\right).$
For each stage, the magnetization M and applied magnetic field $B_e$ are normalized by $B_p$, which are $\mu M\left(B_{e}\right) / B_{p}$ and $B_{e} / B_{p}$. Finally, the hysteresis loop of superconducting cylinder is plotted in Fig. 3(a).
Fig. 3. (a) Normalized hysteresis loops and (b) magnetostriction loops without initial stage with p=0.3,1,3,10. |
2.2. Magnetostriction loop
$u(r)=\frac{1-\nu}{2 E \mu_{0}} r\left[\frac{1-\nu-2 \nu^{2}}{(1-\nu)^{2} r^{2}} \int_{0}^{r} r^{\prime} B^{2} d r^{\prime}+\frac{1-3 \nu+4 \nu^{2}}{(1-\nu)^{2} R^{2}} \int_{0}^{R} r^{\prime} B^{2} d r^{\prime}-B_{e}^{2}\right].$
Therefore, the magnetostriction of the cylinder can be expressed as the radial displacement of cylinder
$\Delta R=u(R)$
$\begin{array}{c} \frac{\Delta R}{R}=\frac{1-\nu-2 \nu^{2}}{2 E \mu_{0}}\left\{\left(m-n+B_{0}^{2}\right)\left(1-\rho_{4}^{2}\right)+\frac{2 n}{3}\left(1-\rho_{4}^{3}\right)\right. \\ \left.+\frac{8 B_{0}}{15 n^{2}}\left[\left(2 n-2 m+3 n \rho_{4}\right)\left(m-n\left(1-\rho_{4}\right)\right)^{\frac{3}{2}}+(2 m-5 n) m^{\frac{3}{2}}\right]\right\} ; \end{array}$
$\begin{aligned} \frac{\Delta R}{R}=\frac{1-\nu-2 \nu^{2}}{2 E \mu_{0}}\left[m-\frac{n}{3}+\right. & \frac{8 B_{0}}{n^{2}}\left(\frac{2}{15} m^{\frac{5}{2}}-\frac{2}{15}(m-n)^{\frac{5}{2}}-\frac{n}{3} m^{\frac{3}{2}}\right) \\ + & \left.B_{0}^{2}-B_{e}^{2}\right] \end{aligned}$
$\begin{array}{c} \frac{\Delta R}{R}=\frac{1-\nu-2 \nu^{2}}{2 E \mu_{0}}\left\{m+\frac{n}{3}+\left(m_{1}-m-2 n\right) \rho_{2}^{2}+\frac{4 n}{3} \rho_{2}^{3}-B_{e}^{2}\right. \\ +\frac{8 B_{0}}{15 n^{2}}\left[-2\left(m_{1}-n\right)^{\frac{5}{2}}+\left(2 m_{1}-2 n-3 n \rho_{2}\right)\left(m_{1}-n\left(1-\rho_{2}\right)\right)^{\frac{3}{2}}\right], \\ \left.+\frac{8 B_{0}}{15 n^{2}}\left[(2 m+5 n) m^{\frac{3}{2}}-\left(2 m+2 n+3 n \rho_{2}\right)\left(m+n\left(1-\rho_{2}\right)\right)^{\frac{3}{2}}\right]\right\} \end{array}$
$\begin{array}{c} \frac{\Delta R}{R}=\frac{1-\nu-2 \nu^{2}}{2 E \mu_{0}}\left\{m+\frac{n}{3}+B_{0}^{2}+B_{e}^{2}\right. \\ \left.+\frac{8 B_{0}}{15 n^{2}}\left[-2(m+n)^{\frac{5}{2}}+2 m^{\frac{5}{2}}+5 n m^{\frac{3}{2}}\right]\right\} \end{array}$
$\begin{array}{c} \frac{\Delta R}{R}=\frac{1-\nu-2 \nu^{2}}{2 E \mu_{0}}\left\{m_{3}-\frac{n}{3}+\left(m_{2}-m_{3}+2 n\right) \rho_{3}^{2}-\frac{4 n}{3} \rho_{3}^{3}+B_{0}^{2}\right. \\ +\frac{8 B_{0}}{15 n^{2}}\left[-2\left(m_{2}+n\right)^{\frac{5}{2}}+\left(2 m_{2}+2 n+3 n \rho_{3}\right)\left(m_{2}+n\left(1-\rho_{3}\right)\right)^{\frac{3}{2}}\right] \\ \left.+\frac{8 B_{0}}{15 n^{2}}\left[-5 n m_{3}^{\frac{3}{2}}+2 m_{3}^{\frac{5}{2}}+\left(2 n-2 m_{3}+3 n \rho_{3}\right)\left(m_{3}-n\left(1-\rho_{3}\right)\right)^{\frac{3}{3}}\right]\right\} \end{array}$
For each stage of the magnetostriction formula, $B_e$ is normalized by $B_p$. And the $\Delta R / R$ is normalized by $\Delta R_{p} / R$, which is the absolute value of the magnetostriction at the applied field $B_{e}=B_{p}$ of the ascending branch. The normalized magnetostriction loops are shown as Fig. 3(b).
2.3. Critical current obtained by magnetization method
The specific steps for obtaining $J_{c}(B)$ of the bulk superconductor with the magnetization measurement method are as follows. Firstly, the hysteresis loop of the bulk superconductor is measured. Then the difference $\Delta M(H)$ between decreasing and increasing magnetic fields is calculated in the hysteresis loop (see Fig. 4). Finally, $J_{c}(B)$ of the cylindrical superconductor of radius R [25] can $B_e$ obtained by
$J_{c}\left(B=\mu_{0} H\right)=\frac{3 \Delta M(H)}{2 R}$
Fig. 4. The magnetization difference between decreasing and increasing magnetic fields in the hysteresis loop. (H and M are normalized by $H_P$). |
However, the above Eq. (20) is derived based on the Bean model. When $J_{c}(B)$ is obtained by using the above equation and hysteresis loop, the results are given in Fig. 5. One can see that there are errors between $J_{c}(B)$ determined by Eq. (20) and Kim model in low field and reverse region. The similar trend was also discussed in Ref. [25], where the overlap is mainly distributed in middle region. Only for a smaller p (p=0.3), the Kim model can $B_e$ reduced to the Bean model, and the error in low field is less as shown in Fig. 5(a). Therefore, the error of $J_{c}(B)$ may appear with the magnetization measurement method directly. In Fig. 5, the curve in the reverse region is directly reduced to 0, and Eq. (20) is not suitable for the reverse region due to the existence of positive and negative currents in the superconductor. Thus, it is meaningful to reduce the error with ANN.
Fig. 5. Comparison of $J_{c}(H)$ between Kim model (red lines) and magnetization measurement (blue points) at p=0.3,1,3,10. (Hare normalized byHP. $J_{c}(H)$ are normalized by $J_{c}\left(H=H_{p}\right)$). |
3. The introduction of ANN
ANN is a simplified model of the human brain processing information, which can realize artificial intelligence [31], [57]. ANN is composed of a number of neurons that can adjust weights and biases. Neurons are the basic units of neural network, which are interconnected between layers [31]. For complex nonlinear relationships between input data and output data, ANN can achieve classification, feature extraction and curve fitting by adjusting the weights and biases of neurons [58], [59]. For a simplest ANN, there are generally three parts [60]. The first part is the input layer, which receives input data into the network. The second part is the hidden layer, which are composed of one or multiple layers. The main role of this part is calculating the characteristic of the input data through the weight, bias and activation function of each neuron [61]. The third part is the output layer, which offers the results calculated by the neural network. As shown in Fig. 6, the output of each neuron can $B_e$ expressed as following function [62]
$\text { output }=\mathrm{f}\left(\sum_{i} w_{i} x_{i}+b\right) \text {, }$
$w_i$ is the weight and b is the bias. The neural network adjusts these two parameters of each neuron, so that the final prediction of the neural network is gradually close to the output target. f is the activation function of the neuron, which can introduce the nonlinearity into the model [63].
Fig. 6. The structure of ANN (lower) and a neuron model (upper). |
(1)Pure linear:
$F(x)=x$
(2)Log-sigmoid:
$F(x)=\frac{1}{1+\exp (-x)}$
(3)Hyperbolic tangent sigmoid:
$F(x)=\tanh (x)$
As can $B_e$ seen in Fig. 7, the curves of tanh activation function and sigmoid activation function are similar. However, sigmoid function is smoother than tanh function and has a smaller gradient, which is not conducive to weight updating. As can $B_e$ seen in the tanh function plot, it is symmetric about zero, while sigmoid function is asymmetric about zero [63]. Then, it is more suitable to choose the tanh function as the activation function of hidden layer.
Fig. 7. Three common activation functions [37]. |
The loss function is used to assess the error between the predicted value ŷ of the model and the true value y. The root mean square error (RMSE) is usually used as the loss function of ANN, and the accuracy of the model is improved by minimizing the loss function of neural network. When the neural network has a single output, the loss function is as follows:
$E(W, b)=\sqrt{\frac{1}{n} \sum_{i=1}^{n}\left(\hat{y}_{i}-y_{i}\right)^{2}} \text {. }$
When the neural network is multi-output, the loss function of k outputs is
$E(W, b)=\sqrt{\frac{1}{n} \sum_{i=1}^{n} \frac{1}{k} \sum_{j=1}^{k}\left(\hat{y}_{i j}-y_{i j}\right)^{2}}.$
Usually, the data set of the neural network model is divided into three parts: training set, test set and validation set. The training set is fed into the network during training, and the network parameters are adjusted based on its errors. The validation set is used to measure the generalization capability of the neural network, and training stops when generalization is not improving. The test set has no effect on the training of the network, so it is possible to provide independent measures of network performance. It can $B_e$ seen from the above that the proportion of training data has an important influence on the final precision and training speed of neural network model. If the ratio of training set is too high, there is no enough data for model validation and testing, and if the ratio is too low, then the amount of data used for training may not $B_e$ sufficient, which can reduce the accuracy of model.
The data is normalized before training the neural network, because the difference of the magnitude of the factors in the dataset may lead to the reduction of convergence of the ANN [38]. If X is a set of finite real values and not all the elements of X are equal, then x (x∈X) can be normalized to $\left[y_{\min }, y_{\max }\right]$ b
$y=\left(y_{\max }-y_{\min }\right) \times \frac{x-x_{\min }}{x_{\max }-x_{\min }}+y_{\min }.$
In general, the data will be normalized to -1,1.
Here, two approaches based ANN are proposed to estimate $J_{c}(B)$ of bulk superconductor with the Kim model. The inputs are the hysteresis loop width and magnetostriction loop width of the bulk superconductor, respectively. And the outputs are the parameters α and $B_0$ of the Kim model. ANNs are trained using the Levenberg-Marquardt algorithm [65], which is commonly used in ANN. This algorithm usually occupies more memory, but takes less computation time. In addition, the training is automatically stopped when generalization stops improving. The calculation of ANN model is based on a PC with Inter(R) Core(TM) i7-9700 K CPU @ 3.60 GHz.
4. Numerical results
The flowchart of training and prediction is shown in the Fig. 8. The ANN is trained with the hysteresis loop and the magnetostriction loop, respectively. After the parameters α and $B_0$ are estimated by ANN, $J_c$ of Kim model can be obtained.
Fig. 8. The flowchart of ANN trained by (a) hysteresis loop, (b) magnetostriction loop. |
4.1. Prediction of α and $B_0$ based the hysteresis loop
In this section, ANN is trained by hysteresis loop. The training time and accuracy of the ANN can $B_e$ affected by the structure of the ANN, so it is necessary to obtain a suitable structure of the neural network. We set aside 10% of the total data to evaluate the ANN, which is not involved in the training validation and testing of the network. In order to reduce the impact of random fluctuations and provide a valid comparison, the RMSE and training time of ANNs are the average of 500 training runs.
Firstly, we consider the output of two parameters simultaneously or separately with ANN. For the double-output ANN, two parameters α and $B_0$ are output at the same time. For the single-output ANN, the parameters α and $B_0$ are output separately, which needs two training networks. From Fig. 9 and Fig. 10, it can be seen that the RMSE of parameter α shows a decreasing trend or slight change as the number of neurons in the input and hidden layers increases. However, for parameter $B_0$, the RMSE decreases first and then increases with the augment of the hidden layer neurons. It means that only augmenting the number of neurons does not always improve the accuracy for different ANN models. Augmenting the number of neurons means that the ANN can be more accurate in the training set. However, it may lead to the increasing of the RMSE of the validation and overfitting. Compared to the double-output ANN, the RMSE of parameter α in the single-output ANN is an order of magnitude smaller. As shown in Fig. 11, for parameter α and $B_0$, the error distribution of single-output ANN compared with double-output is smaller and concentrated within 1%. In addition, the computation time increases gradually with the augment of the number of neurons in the ANN, as shown in Fig. 12. Therefore, in order to reduce the computation time, the number of neurons should be small while maintaining accuracy. The computation time of both networks is almost equal when the ANN is single output. The sum of the training time for two networks in single-output ANNs is slightly more than the double-output ANN. By taking into account the computational accuracy and the training time, the single-output ANNs are finally selected for training. The structures of ANNs are chosen in Fig. 13.
Fig. 9. The RMSE of double-output ANN with different inputs and hidden layer neurons for (a) the parameter α, (b) the parameter $B_0$. |
Fig. 10. The RMSE of single-output ANN with different inputs and hidden layer neurons for (a) the parameter α, (b) the parameter $B_0$. |
Fig. 11. The error distribution of single-output and double-output ANN (a) the parameter α, (b) the parameter $B_0$. |
Fig. 12. The comparison of training time between single-output (dashes) and double-output (solid lines) ANN. |
Fig. 13. (a) The structure of ANN for parameter α. (b) The structure of ANN for parameter $B_0$. |
For the ratio of training data in total data, the values are chosen as: 50%, 70%, 90%. The ANN structures shown in Fig. 13(a) and (b) are used for the parameter α and $B_0$ respectively. As can be seen from the Table 1, the RMSE is smallest for both parameters as the ratio is 70%. Thus, 70% of the data are used to train ANN in the following part.
Table 1. RMSE of parameter α and $B_0$ for different ratio of training data. |
| Ratio of training data | RMSE of parameter α | RMSE of parameter $B_0$ | ||||
|---|---|---|---|---|---|---|
| train | validation | test | train | validation | test | |
| 50% | 4.84 × 105 | 4.85 × 105 | 4.86 × 105 | 0.004996 | 0.005011 | 0.005009 |
| 70% | 4.82 × 105 | 4.83 × 105 | 4.83 × 105 | 0.004833 | 0.004834 | 0.004848 |
| 90% | 4.93 × 105 | 4.93 × 105 | 4.95 × 105 | 0.004884 | 0.004874 | 0.004911 |
As shown in Fig. 14, the $J_{c}(B)$ estimated by the trained ANN is compared with the theoretical curve. One can see that the predictions are in agreement with the theoretical curves.
Fig. 14. Comparison of ANN predicted $J_c$ (dots) and theoretical curves (lines) at $B_0$=0.1,0.2,0.3,0.4. |
Douine et al proposed a method to obtain $J_{c}(B)$ based on Kim model by using the initial magnetization curve [53], which is called as fitting method later. In order to test the ANN, the proposed ANN is used to obtain the $J_{c}(B)$ with hysteresis loop in the literature. The hysteresis loop data is given in Ref. [66], and $J_{c}(B)$ curves obtained by the three methods are shown in Fig. 15. When $B_e$ is close to 0, $J_c$ of fitting method is one order of magnitude larger than that of ANN method. It can be seen that there is a difference between the fitting method and the magnetization measurement method. In addition, the ANN method is more close to the magnetization measurement method than the fitting method in the middle region.
Fig. 15. Comparison of $J_c$ curves with the Kim model obtained by the fitting method, the ANN method and magnetization measurement method. |
4.2. Prediction of α and $B_0$ based the magnetostriction loop
In section 2.2, the relationship between the magnetostriction and the applied field $B_e$ is obtained. Then the ANN is trained in the same way to obtain $J_{c}(B)$. It is necessary to change training data from hysteresis loop to magnetostriction loop, which means $J_{c}(B)$ can also be obtained by measuring the deformation of the bulk superconductor.
Similarly, we first discuss the network structure of ANN using magnetostriction as the training data. Fig. 16 show the RMSE of double-output ANN with different numbers of input and different numbers of hidden layer neurons. Fig. 17 shows the RMSE of single-output ANN for different numbers of network neurons. At the same time, the total training time required for two single-output ANNs is approximately equivalent to that of a double-output ANN. By considering the computational precision and training time, the ANN network structure is determined as single output. The ANN structure of parameter α is 10-15-1, and the structure of parameter $B_0$ is 10-10-1.
Fig. 16. The RMSE of double-output ANN with different numbers of input and different numbers of hidden layer neurons for (a) parameter α, (b) parameter $B_0$. |
Fig. 17. The RMSE of single-output ANN with different numbers of input and different numbers of hidden layer neurons for (a) parameter α, (b) parameter $B_0$. |
Finally, we compare the error distributions of ANNs trained by hysteresis loop and magnetostriction loop. Fig. 18(a) and (b) is the comparison of the error distribution of parameter α and $B_0$. It can be found that the error of ANN trained by magnetostriction loop is smaller than that of ANN trained by hysteresis loop for parameter α. For parameter $B_0$, the error of ANN trained by hysteresis loop is smaller than that of ANN trained by magnetostriction loop. In the experiment, $J_{c}(B)$ can be obtained by simultaneously measuring hysteresis loop and magnetostriction loop of the bulk superconductor.
Fig. 18. Comparison of the error distribution of ANN based on hysteresis loop and magnetostriction loop for (a) parameter α, (b) parameter $B_0$. |
5. Conclusions and discussion
In this paper, the ANNs are trained by hysteresis loop and magnetostriction loop respectively, and ANNs are used to estimate the parameters of $J_{c}(B)$ in superconductor with the Kim model. Firstly, the hysteresis loops and magnetostriction loops of bulk superconductor with the Kim model are derived. Then, the ANNs are trained by the hysteresis loop and magnetostriction loop. The proposed ANNs can obtain $J_{c}(B)$ of the bulk superconductor through the hysteresis and magnetostriction loops. Compared with fitting method, $J_{c}(B)$ obtained by ANNs is more reasonable. The numerical results given by two ANNs are discussed and compared, and both hysteresis and magnetostriction loops are effective in predicting $J_{c}(B)$ of bulk. The approach presented in this paper can be applied to estimate $J_{c}(B)$ of superconductors for other models as well. For different sample shapes, the approach can also be used to calculate the critical current density. Only a large number of magnetization curves of samples with corresponding shapes need to be obtained as the dataset to train ANN.
In the derivation of the equations for the magnetization and magnetostriction in this paper, it is assumed that the cylinder is long enough. In order to obtain the magnetization and magnetostriction of the bulk superconductor with finite length, numerical simulation on the electromagnetic field can $B_e$ carried out. A large number of magnetization curves can $B_e$ simulated through FEM, and the dataset can improve the ANN model. However, the time required to generate the training set using FEM will also greatly increase compared to the analytical solutions.
In addition, $J_{c}(B)$ of superconducting tape is not only related to magnetic field, but also affected by temperature and angle of applied magnetic field. For the more complex relation, we will predict nonlinear relationships with ANN in the future work.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We acknowledge the support from the National Natural Science Foundation of China (Grant Nos. U2241267, 12172155 and 11872195).
