© H. Zhou et al., Published by EDP Sciences 2024

1 Introduction

Small module tooth-shaped parts (SMTSPs), the module is smaller than 1, made of difficult-to-deformed metals, as one of the most precision transmission components, are widely used in intelligent robots and high-end CNC machine tools [1,2], which are traditionally manufactured by tooth hobbing or tooth shaping, leads to the problems of low material utilization and weak toughness. Splitting spinning refers to a specific spin-forming method by which the rotating circular blank being separated into “Y” shape gradually through the feeding of the splitting roller (SR) with sharp roundness radius [3]. The difficult-to-deformed metals exhibit the high deformation resistance and poor plasticity, for example, yield strength of 30CrMnSiA alloy reaches 835 MPa and the elongation is only 10%. The plasticity of the material can be improved effectively and the deformation resistance can also be reduced greatly under the effect of the pulse current, which is called the electroplastic effect. Therefore, a novel current-assisted splitting spinning (CASS) method was proposed to realize the complete near net forming of the SMTSPs made of difficult-to-deformed metal [4].

In previous research [5], the effects of pulsed current parameters, including current density and duty ratio, on gear tooth saturation were primarily investigated. However, the underlying mechanism of pulsed current in tooth filling remains unclear. Research by Xia et al [6] demonstrated that the forming quality of SMTSP was affected by numerous factors during CASS, including current density, duty ratio, and the radial feeding velocity. These parameters exhibit complex interactions, compounding the difficulties in controlling the forming quality of the spun part. Nevertheless, the CASS process is highly nonlinear, especially when considering multi-objective optimization making the relationships between influencing factors and forming quality remarkably intricate. In previous studies on quality optimization in spinning forming, various researchers employed different methods. Li et al. [7] developed a multi-objective optimization model based on orthogonal experimental design and a response surface model to improve the forming quality of 2219 aluminum alloy thin-walled complex components, achieving high quality with minimal roundness deviation and reduced part-mold contact gaps. Zhang. et al. [8] employed grey relational analysis to investigate the effects of process parameters on wall thickness deviation and warping deformation during the spinning of magnesium alloy hubs. They found that the axial offset of the spinning roller significantly impacted the accuracy of hub spinning. Li et al. [9] established the relationship between process parameters and maximum stress and wall thickness increment of an accumulator shell using a response surface model, obtaining a Pareto-optimal solution set through multi-objective optimization. Zhang et al. [10] used extreme difference analysis and the grey relational analysis to determine suitable rotational speed ratios, feed rates, and friction coefficients for optimizing the outer diameter ovality, generatrix precision, and maximum forming force of shovel spinning pulley components. Khodadadi et al. [11] applied simulated annealing and genetic algorithms to optimize the pile-up edge in internal gear spinning. They observed that the pile-up edge increased with higher feed rates and thinning ratios, with the optimized minimum pile-up edge measured at 0.445 mm. The orthogonal experimental design has proven effective for studying multi-factor, multi-level problems in spinning forming [12,13]. By combining orthogonal experimental design with the NSGA-II algorithm and BP neural network-based multi-objective optimization, the interactive effects between various factors can be thoroughly analyzed. This comprehensive approach helps in determining optimal process parameter combinations, which holds significant practical value for manufacturing high-quality small module gear-shaped components.

In this study, a coupled electric-thermal-mechanical FEA model for the CASSF of SMTSP was developed based on the ABAQUS software, the deformation characteristics of the small module tooth and the mechanism of pulsed current parameters on tooth filling were investigated. Finally, based on orthogonal experimental data, the optimal parameter combination was determined using a BP neural network integrated with the NSGA-II genetic algorithm, and the reliability and accuracy of the optimization results were validated through experiments.

2 Analysis of CASS process of SMTSP

2.1 Principle of CASS of SMTSPs

As shown in Figure 1, the CASS forming system mainly contains mandrel, tailstock, splitting roller, pulse power, brush and synchronization system. The pulse current is supplied by a pulse power, and flows from the positive brush into the brush clamping area of the cup-shaped blank, then flows through the thick-walled region under the splitting roller (SR) into the thin-walled region, and finally exiting via the negative brush at the tailstock.

The cup-shaped blank with differential wall thickness used in CASS is obtained through three-roller stagger spinning forming [14] (Fig. 2a). The SMTSP, made of 30CrMnSiA alloy structural steel, has a diameter-to-module ratio of 160 (Fig. 2b), and its tooth structure is depicted in Figure 2c. The cup-shaped blank is divided into three regions from the cup mouth to the bottom: the thick-walled region, the transitional region, and the thin-walled region. The small modulus teeth are formed in the thick-walled region of the cup-shaped part. Combining with the result of previous work [5,6], the specific process parameters were shown in Table 1.

Fig. 1 Schematic diagram of principle of CASS of SMTSP.

Fig. 2 Schematic diagram of (a) cup-shaped blank and (b) typical SMTSP (c) target tooth.

2.2 Experimental conditions

The CASS experiment was carried out on the HGQX-LSM vertical spinning machine (Fig. 3). The current-assisted system primarily consisted of a high-frequency pulse power supply, electrical brushes, and upper and lower brush support seats. However, the electric sparks and the tooth surface erosion in the forming zone occurs easily when relative motion between the blank and other components existed. Therefore, a “tailstock-blank-pushing ring” current application method was designed.

A temperature control system was designed to prevent excessive temperature rise caused by the Joule heating effect during CASS forming, which monitored the temperature in the tooth-forming zone of the workpiece using an infrared thermometer and adjusted the air compressor's wind speed to control the temperature. Since the wall thickness in the tooth-forming area was only 0.766 mm and the material had a high thermal conductivity of approximately 52 W • (m • K)⁻¹ [15], the internal and external temperatures of the material in the tooth-forming area was uniform. Therefore, only the sample surface temperature was real-time monitored.

Fig. 3 HGQX-LSM vertical spinning machine.

3 Establishment of FEA Model and optimisation approach

3.1 FE modelling of CASS of SMTSP

The mechanical properties of 30CrMnSiA alloy steel were determined through mechanical property testing experiments, and the true stress-strain constitutive relationship is as follows [16]:

$$\sigma =\left\{467.7+429.358[1-\exp (-3.414\epsilon )]\right\}•[1-0.003\ln ({\displaystyle \dot{\epsilon }}/0.01)]•\exp (-0.00962J_P-0.0996d)$$(1)

where, σ refers to the true stress, ε is the true plastic strain, ${\displaystyle \dot{\epsilon }}$ is the strain rate, J_p is the current density, and d is the duty ratio.

Figure 4 shows the FEA model for CASS of SMTSP established by Abaqus/Explicit software. The cup-shaped blank was treated as a deformable body, while the SR and mandrel were defined as rigid bodies. In the simulation, the blank and mandrel remained fixed, while the SR rotated around its own axis and around the center of the blank. The SMTSP studied has 160 teeth along the circumferential direction and only the materials at the transitional and thick-walled regions undergo deformation during CASS process, therefore, 1/72 transitional and thick-walled regions along the circumferential direction of the cup-shaped blank was selected in the FEA model due to the characteristic of rotational symmetry of each tooth, and only one tooth deformation of SMTSP during CASS was simulated to improve the simulation efficiency. To restrict circumferential and axial material flow, a baffle was introduced. The circumferential material flow was constrained by the mutual interaction of the material along the circumference, while the axial flow was limited by the positive brush located below the mouth of the cup-shaped blank. The penalty function contact algorithm and Coulomb friction model (with a friction coefficient of µ = 0.10) were applied to define the interaction between the blank and the SR. The baffle was assumed to be frictionless with respect to the blank.

The supplied method of the pulse current from the mouth of the cup-shaped blank to the tailstock was equivalent to that from the lower conductive block to the mouth, then through the axial interface of the transitional region, and finally to the upper conductive block. Additionally, to account for the thermal and electrical conduction between the upper/lower conductive blocks and the cup-shaped blank, the upper/lower conductive blocks were defined as deformable bodies. A convective heat dissipation coefficients of 25 W.m⁻¹.K⁻¹ was applied to the surface of all components to simulate the effect of convective heat dissipation due to compressed air cooling [17].

Fig. 4 Illustration of FEA model for CASS of SMTSP.

3.2 Validation of FE Model

The CASS experiment was carried out under n₁ = 150 r/mm, v_f = 0.6 mm/min, Δ = 0.2175 mm, J_p = 20 A/mm² and d = 50% to verify the reliability of the FEA model, and the tooth thickness and the dedendum circle diameter of the SMTSP were selected as forming quality indicators [4]. 8 measurement points with an equal intervals of 1.5 mm along axial direction and 45° along the circumferential direction were selected (Fig. 5). The maximum relative errors of tooth thickness and dedendum circle diameter were 4.3% and 6.72%, respectively (Fig. 6), which indicates that the established CASS FEA model of SMTSPs is reasonable.

Fig. 5 Tooth thickness and dedendum circle measuring points along the axial direction of spun workpiece.

Fig. 6 Relative error of the tooth thickness and the dedendum circle obtained by simulation and experiment.

3.3 Training model and optimization method

The training method of BPNN is a widely used machine learning approach in the field of artificial intelligence. The pre-feedback neural network is consisted of an input layer, a hidden layer and an output layer. In this research, it contains three neurons in the input layer, the current density (J_p), duty ratio (d), and radial feeding velocity (v_f). The output layer is trained by Levenberg-Marquardt back propagation algorithm to obtain three output neurons: the tooth profile deviation (F_α), the maximum individual pitch deviation (f_pt), and the tooth saturation (γ). The performance of the trained neural network is evaluated using the Mean Squared Error (MSE) and the correlation coefficient (R), as defined in equation (1) [18] and equation (2) [19]. Minimizing the root mean square error is utilized as a limiting condition to avoid falling into local optimum and overfitting during the training process, and to minimize the error between the ideal output results and the predicted output results.

$$MSE=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{(f_i-f_i^{\text{'}})}^2$$(2)

$$R=\frac{1}{n-1}{\displaystyle \sum _{i=1}^n}(\frac{f_i-{{\displaystyle \bar{f}}}_i}{{\sigma }_i})(\frac{f_i-{{\displaystyle \bar{f}}}_i}{{\sigma }^{\text{'}}})$$(3)

where, n denotes the amount of data, $f_i^{\text{'}}$ denotes the predicted value, ${\displaystyle \overline{f_i^{\text{'}}}}$ and ${\displaystyle \overline{f_i}}$ denotes the average of the desired output and the actual value, σ_i and ${\sigma }_{\text{i}}^{\text{'}}$ denotes the error value between the actual output and the ideal output.

The training data for the BPNN was obtained from the orthogonal experiment. In the orthogonal experiment, the current density, duty ratio, and radial feeding velocity were selected as factors, each with five levels, as detailed in Table 2. The test objectives included tooth profile deviation, maximum individual pitch deviation, and tooth saturation. Numerical simulations were conducted using ABAQUS to evaluate the results for the various combination schemes of the orthogonal test. The designed orthogonal experimental is shown in Table 3.

The sample points obtained from the orthogonal experiment are used to establish the initial data set for the BPNN prediction model, of which 80% of the data are utilized for training, and 20% for validation. Before training, normalization was used uniformly to reduce the effect of error due to magnitude. The Levenberg-Marquardt algorithm is selected as the training algorithm for the developed BPNN model. The data is processed using the dividerand function to randomly assign the training data, the Trainlm function is used for the training of the BPNN and the Purelin function is used for the output layer.

The optimization process in this study is shown in Figure 7. The input sample data is first fed into the input layer of the BPNN, where it is transmitted through the network to the hidden and output layers via connection weights and activation functions. Then the output value of the BPNN is compared with the real value, and the MSE is calculated. This process begins at the output layer and propagates backward through the network, updating the weights layer by layer to minimize the error. Finally, BPNN can gradually learn the mapping relationship between input and output by iterating and updating the weights, so as to achieve the prediction of unknown data. The use of a trained BPNN as a surrogate model to replace traditional mathematical expressions offers significant advantages by reducing computational demands and improving optimization efficiency. The NSGA-II algorithm further enhances the process by employing selection, crossover, and mutation operations to iteratively evolve the population. During this evolutionary process, elite individuals are prioritized, enabling the identification of optimal Pareto front solutions that meet specified conditions. This approach effectively optimizes the CASS of SMTSP.

Fig. 7 Flow diagram of the BP neural network integrated with NSGA-II multi-objective optimization process.

4 Results and discussion

4.1 Evaluation metrics for the forming quality of SMTSPs

To accurately describe the forming quality of SMTSPs, it is essential to detect tooth profile errors. The tooth profile deviation F_α is one of the key indicators used to assess gear geometric accuracy, representing the deviation between the actual tooth profile and the designed tooth profile (as shown in Fig. 8).

The individual pitch deviation refers to the difference between the actual pitch and the design pitch of the gear, and it can be calculated by equation (3).

$$f_{pi}=p_i-p.$$(4)

Thereby the maximum individual pitch deviation can be expressed as follows:

$$f_{pt}=\max \left|f_{pi}\right|$$(5)

where, f_pi is the individual pitch deviation of any gear tooth, p_i is the actual pitch, p is the theoretical pitch, f_pt is the maximum individual pitch deviation (as shown in Fig. 9).

In addition, tooth saturation γ is an important concept in the spinning forming of tooth-shaped parts [5], and it can be calculated by equation (5).

$$\gamma =\frac{S_{\text{a}}}{S_{\text{t}}}$$(6)

where, S_a and S_t are the actual and theoretical cross-section areas of small module tooth, respectively (as shown in Fig. 10).

Fig. 8 Schematic diagram of tooth profile deviation.

Fig. 9 Schematic diagram of individual pitch deviation.

Fig. 10 Tooth profile of spun workpiece.

4.2 Mechanism of tooth filling under CASS

Figure 11 shows the distribution of three-directional stresses of the spun part under n₁ = 150 r/mm, v_f = 0.6 mm/min, Δ = 0.2175 mm, J_p = 20 A/mm² and d = 50%. It shows that the material at the tooth groove (region C) and the center of the gear tooth (region B) is in the state of three-dimensional compressive stresses, which makes the material at region C on both sides flow towards region B. However, the material at the tooth tip (region A) is in the state of tensile stress along the radial direction (RD) and compressive stresses along the tangential direction (TD) and axial direction (AD). This stress state at region A is beneficial for the radial flow of material and filling the tooth.

Figure 12 shows the distribution of three-directional strains of the spun part n₁ = 150 r/mm, v_f = 0.6 mm/min, Δ = 0.2175 mm, J_p = 20 A/mm² and d = 50%. It shows that all of the material at the deformation zone is in a plane strain state and the strain along AD is nearly 0 during CASS. Furthermore, the material at region C is in a state of radial compression and tangential elongation. However, the material is in a state of radial elongation and tangential compression at regions A and B.

Figure 13 shows the distribution of equivalent plastic strain (PEEQ) of the spun workpiece under without and with pulse current. It shows that the stress concentration and deformation resistance of the material decrease and the depth of the deformation zone increases at region C under the condition of pulse current, which indicates that the plastic deformation capacity of the material is improved significantly under the action of pulse current. The material of the deformation zone at region C increases, which makes more materials flow to the gear tooth during the CASS. Therefore, the tooth saturation with the pulse current is much greater than that of without the pulse current.

Fig. 11 Distribution of three-directional stress at (a) the tooth tip, (b) the gear tooth, and (c) the tooth groove under n1 = 150 r/mm, vf = 0.6 mm/min, Δ = 0.2175 mm, Jp = 20 A/mm2 and d = 50%. (σr, σθ and σa denote the radial, tangential and axial stress, respectively; RD, TD and AD denote the radial, tangential and axial direction, respectively).

Fig. 12 Distribution of three-directional strain at (a) the tooth tip, (b) the gear tooth, and (c) the tooth groove under n1 = 150 r/mm, vf = 0.6 mm/min, Δ = 0.2175 mm, Jp = 20 A/mm2 and d = 50%. (LEr, LEθ and LEa denote the radial, tangential and axial true strain, respectively).

Fig. 13 Distribution of PEEQ of the spun workpiece under (a) without and (b) with pulse current.

4.3 Optimization of process parameters for SMTSPs

4.3.1 Predictive analytics of BPNN

The number of hidden layer nodes is crucial for the BP neural network. Too few nodes result in insufficient problem-solving information, while too many lead to longer training times and overfitting, reducing generalization ability. The empirical formula (Eq. (6)) is used to estimate the optimal number of nodes.

$$P=\sqrt{m+n}+\alpha$$(7)

where P is the number of neurons in the hidden layer, m is the number of neurons in the input layer, n is the number of neurons in the output layer, and α is a constant ranging from 1 to 10.

The fitting accuracy of the BP network under different numbers of hidden layer neurons is shown in Table 4. The results indicate that when the number of hidden layer neurons is set to 8, the MSE reaches its minimum value, demonstrating a higher fitting accuracy. Therefore, the number of hidden layer neurons was finalized as 8.

Figures 14 and 15 shows the regression plot of the training and validation sets, where the correlation coefficient R is a statistical indicator of the degree of correlation between the response output values. Usually, after normalization, the correlation coefficient R ranges from [–1, 1]. The R is closer to ±1, the greater the degree of correlation. Where the dashed line represents the desired output curve built using the BPNN and the hollow points represent training date. As shown in the figure, the correlation coefficients of the training and validation sets are 0.9768, 0.9554, 0.9823, 0.9927, 0.9687 and 0.9869 respectively. This indicates that the target output values of the training and validation sets are consistent with the expected output values, which are able to respond to the nonlinear mapping relationship between multiple objectives.

Fig. 14 Regression plot of correlation coefficients for training data set: (a) Fα, (b) fptmin, and (c) γ.

Fig. 15 Regression plot of correlation coefficients for validation data set: (a) Fα, (b) fptmin, and (c) γ.

4.3.2 Multi-objective optimization based on NSGA-II algorithm

For the prediction model established before, the NSGA-II algorithm is used to solve the multi-objective problem, which is transformed into the mathematical model of equation (7). The current density is X₁, duty ratio is X₂ and radial feeding velocity is X₃, they are the input parameters of the CASS. F_αmin, f_ptmin, and −γ_min are three output variables, which are taken as the objective functions. According to the results of the single-factor analysis, the output variables are taken as the constraint functions, so as to find the minimum of the tooth profile deviation, the maximum individual pitch deviation and maximum of the tooth saturation of the SMTSP.

$$\{\begin{array}{l}{F}_{\alpha \min }=f_1(X_1,X_2,X_3)\\ f_{pt\min }=f_2(X_1,X_2,X_3)\\ -{\gamma }_{\min }=f_3(X_1,X_2,X_3)\end{array}Parameters=\{\begin{array}{l}3.5\le X_1\le 17.5\\ 16\le X_2\le 80\\ 0.2\le X_3\le 1\end{array}$$(8)

In the operation process of NSGA-II algorithm, the gamultiobj function is adopted for the minimum value of the optimization of the F_αmin, f_ptmin, and −γ_min, after 300 generations of iterative operations to obtain the relative optimal solution of the Pareto front, as shown in Table 5. As observed from the table, the 12 optimal solution sets require the selection of a satisfactory solution to determine the best process parameters for the CASS and the corresponding optimal target values.

In this study, the method of a satisfaction function was used to select the satisfactory solution. The higher the value of the satisfaction function, the better the target values. The satisfaction function is defined as:

$$S=\frac{1}{\frac{f_1-f_{1\min }}{f_{1\max }-f_{1\min }}+\frac{f_2-f_{2\min }}{f_{2\max }-f_{2\min }}+\frac{f_3-f_{3\min }}{f_{3\max }-f_{3\min }}}$$

where, f_1max, f_1min, f_2max, f_2min, f_3max, and f_3min represent the maximum and minimum values of f₁, f₂, and f₃ within the Pareto optimal solution set, respectively.

By calculating the satisfaction of the Pareto optimal solution for each group, the values of the satisfaction are shown in Table 6.

As shown in the table, the satisfaction function value of the 5th group in the optimal solution set is the highest, indicating that the forming quality of the SMTSP is superior under its corresponding process parameters. Therefore, this group is selected as the satisfactory solution. The corresponding process parameters are J_p = 16.6 A/mm², d = 65 %, and v_f= 0.9 mm/min.

To validate the accuracy of the optimization results and assess the effectiveness of combining the BP network with the multi-objective genetic optimization algorithm, FEM simulations of CASS of SMTSP were conducted using ABAQUS under the optimal solution. A comparison of the gear tooth filling effects before and after optimization is shown in Figure 16. The results demonstrate significant improvements after optimization, with the gear tooth saturation increasing from 88.5% to 97.8%, a tooth profile deviation of 0.01723 mm, and a maximum individual pitch deviation of 0.0144 mm.

Figure 17 shows the spun part under the optimized parameter combination. The figure illustrates that the SMTSP formed by CASS exhibit clear tooth profiles without defects such as misalignment or fractures, indicating high forming quality. Furthermore, the experimental results closely match the predicted values, with an maximum error of only 2.9%, confirming the accuracy of the predictive model.

Fig. 16 Comparison of spun parts before and after optimization.

Fig. 17 SMTSP manufactured under the optimal process parameter.

5 Conclusions

In this study, a coupled electrical-thermal-mechanical FEA model was established using ABAQUS to investigate the mechanism of tooth filling under CASS. Based on the simulation results, an orthogonal experimental dataset was generated to train a surrogate model using a backpropagation neural network (BPNN), which was developed to predict key forming quality metrics, including the tooth profile deviation, the maximum individual pitch deviation, and the tooth saturation. This predictive model was further integrated with the NSGA-II algorithm for multi-objective optimization of process parameters. The major conclusions are listed as follows:

• The application of pulse current significantly enhances the plastic deformation capacity of the material in the tooth groove, promoting better material flow toward the gear tooth. Furthermore, pulse current reduces stress concentration and deformation resistance, deepening the deformation zone and ultimately improving tooth filling.

• The BP neural network combined with the NSGA-II multi-objective genetic optimization algorithm, effectively optimizes the process parameters of CASS. The tooth filling performance is significantly improved, resulting in enhanced forming quality of small-module tooth-shaped parts, with optimal values of J_p = 16.6 A/mm², d = 65%, and v_f = 0.9 mm/min.

• Experiment is conducted based on the optimal process parameters, with the maximum relative error between the optimization results and experimental measurements being only 2.9%, confirming both the feasibility of the process and the accuracy of the predictive model.

Acknowledgments

This work was financially funded by the National Natural Science Foundation of China (No. 52175316), Guangdong Natural Science Foundation Project of Basic and Applied Basic Research (No. 2021A1515011074), and Guangzhou Science and Technology Planning Project (201804010135).

Conflicts of interest

The authors report no declarations of interest.

Data availability statement

The data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

Author contribution statement

Haoyang Zhou: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing − Original Draft, Visualization, Funding acquisition. Qinxiang Xia: Supervision, Conceptualization, Methodology, Validation, Investigation, Resources, Data Curation, Writing − Review & Editing, Visualization, Project administration, Funding acquisition. Gangfeng Xiao: Supervision, Validation, Resources, Writing − review & editing, Conceptualization, Methodology. Can Chen: Validation, Data Curation.

References

All Tables

All Figures

	Fig. 1 Schematic diagram of principle of CASS of SMTSP.
In the text

	Fig. 2 Schematic diagram of (a) cup-shaped blank and (b) typical SMTSP (c) target tooth.
In the text

	Fig. 3 HGQX-LSM vertical spinning machine.
In the text

	Fig. 4 Illustration of FEA model for CASS of SMTSP.
In the text

	Fig. 5 Tooth thickness and dedendum circle measuring points along the axial direction of spun workpiece.
In the text

	Fig. 6 Relative error of the tooth thickness and the dedendum circle obtained by simulation and experiment.
In the text

	Fig. 7 Flow diagram of the BP neural network integrated with NSGA-II multi-objective optimization process.
In the text

	Fig. 8 Schematic diagram of tooth profile deviation.
In the text

	Fig. 9 Schematic diagram of individual pitch deviation.
In the text

	Fig. 10 Tooth profile of spun workpiece.
In the text

	Fig. 11 Distribution of three-directional stress at (a) the tooth tip, (b) the gear tooth, and (c) the tooth groove under n1 = 150 r/mm, vf = 0.6 mm/min, Δ = 0.2175 mm, Jp = 20 A/mm2 and d = 50%. (σr, σθ and σa denote the radial, tangential and axial stress, respectively; RD, TD and AD denote the radial, tangential and axial direction, respectively).
In the text

	Fig. 12 Distribution of three-directional strain at (a) the tooth tip, (b) the gear tooth, and (c) the tooth groove under n1 = 150 r/mm, vf = 0.6 mm/min, Δ = 0.2175 mm, Jp = 20 A/mm2 and d = 50%. (LEr, LEθ and LEa denote the radial, tangential and axial true strain, respectively).
In the text

	Fig. 13 Distribution of PEEQ of the spun workpiece under (a) without and (b) with pulse current.
In the text

	Fig. 14 Regression plot of correlation coefficients for training data set: (a) Fα, (b) fptmin, and (c) γ.
In the text

	Fig. 15 Regression plot of correlation coefficients for validation data set: (a) Fα, (b) fptmin, and (c) γ.
In the text

	Fig. 16 Comparison of spun parts before and after optimization.
In the text

	Fig. 17 SMTSP manufactured under the optimal process parameter.
In the text