*5.3. Process Parameters Optimization*

On the basis of the above results, a second-order regression model is used to obtain the response surface for each objective function. Let *y*<sup>1</sup> be the energy consumption of hot stamping, *y*<sup>2</sup> the thinning rate of the stamping parts, and *y*<sup>3</sup> the thickening rate of the stamping parts. The developed model can then be described as

$$\begin{array}{rcl} y\_1 &=& 261054 + 76.8495 \mathbf{x}\_1 + 2643.08480 \mathbf{x}\_2 - 2455.30317 \mathbf{x}\_3 + 0.26751 \mathbf{x}\_1 \mathbf{x}\_2 \\ &- 0.17018 \mathbf{x}\_1 \mathbf{x}\_2 - 6.13451 \mathbf{x}\_2 \mathbf{x}\_3 + 0.14412 \mathbf{x}\_1^2 - 31.95144 \mathbf{x}\_2^2 + 6.31017 \mathbf{x}\_3^2 \end{array} \tag{14}$$

$$\begin{array}{l} \mathbf{y}\_{2} &= -185.68990 - 0.037136 \mathbf{x}\_{1} + 0.58991 \mathbf{x}\_{2} + 1.68914 \mathbf{x}\_{3} - 0.16617 \mathbf{x}\_{1} \mathbf{x}\_{2} + 3.71659 \times 10^{-3} \mathbf{x}\_{1} \mathbf{x}\_{3} \\ &+ 4.68506 \times 10^{-3} \mathbf{x}\_{2} \mathbf{x}\_{3} + 0.06927 \mathbf{1} \mathbf{x}\_{1}^{2} - 0.055531 \mathbf{x}\_{2}^{2} - 3.83894 \times 10^{-3} \mathbf{x}\_{3}^{2} \end{array} \tag{15}$$

$$\begin{array}{llll} \mathbf{y}\_3 &= 0.78922 - 0.39606 \mathbf{x}\_1 + 0.29588 \mathbf{x}\_2 + 0.059743 \mathbf{x}\_3 + 0.036733 \mathbf{x}\_1 \mathbf{x}\_2 - 2.74212 \times 10^{-3} \mathbf{x}\_1 \mathbf{x}\_3 \\ &- 3.66213 \times 10^{-3} \mathbf{x}\_2 \mathbf{x}\_3 + 0.050898 \mathbf{x}\_1^2 + 0.041834 \mathbf{x}\_2^2 - 6.34584 \times 10^{-5} \mathbf{x}\_3^2 \end{array}, \tag{16}$$

where *x*<sup>1</sup> is the blank holder force, *x*<sup>2</sup> is the stamping speed, and *x*<sup>3</sup> is the forming temperature.

Hot stamping process optimization aims to obtain a set of process parameters that will produce stamping parts with reduced thickness variations and low energy consumption. Therefore, the objective function and constraint conditions in the optimization process can be expressed as

$$\begin{cases} F = \min\left(y\_1, y\_2, y\_3\right) \\ \text{s.t.} \begin{cases} 2 \le x\_1 \le 8 \\ 3 \le x\_2 \le 10 \\ 200 \le x\_3 \le 250 \end{cases} \end{cases} \tag{17}$$

where *y*1, *y*2, and *y*<sup>3</sup> are the objective functions. The goal is to minimize the stamping energy consumption and thickness variations in hot stamping.

NSGA-II is used to solve the multiobjective optimization problem in Equation (17). A series of considered efficient solutions constituting the Pareto frontier is obtained, as shown in Figure 10. The results show that the formability and energy consumption of sheet metals are contradictory. For ZK60 magnesium alloy hot stamping, formability improves with the increase in forming temperature, but the energy consumption of hot stamping increases significantly with the rise of heating temperature. The thinning and thickening rates are also contradictory. With increasing blank holder force, the thinning rate of stamping parts increases gradually, whereas the wrinkling trend of the sheet metal decreases. On the basis of the forming requirements (the thinning and thickening rates of stamping must be less than 10%, and the energy consumption of stamping should be relatively small), the following two groups of compromise solutions are selected, and Table 4 presents the corresponding index values.

**Figure 10.** Pareto optimal solutions of ZK60 magnesium alloy hot stamping.

**Table 4.** Compromise solutions and their corresponding indices.


Figure 11 shows the simulation results of the thickness variation distribution under different compromise solutions. The forming quality of stamping parts under the two groups of process parameters is good and can meet the usage requirements. The thicknesses of the stamping parts vary greatly in terms of punch and die radii. From the straight wall section to the flange area, from the bottom to the top, the thickness variation distribution of the parts presents a gradual increase trend at the parameters of the two compromise solutions, and the thickening phenomenon is shown in the flange area. A comparison of the color distribution of the thickness variation of the stamping parts in the simulation results indicates that the thickness variation distribution of the stamping parts obtained at the parameters of compromise solution 1 is slightly more uniform than that at the parameters of compromise solution 2. The energy consumption at the parameters of compromise solution 2 can be reduced by 17.6% in comparison with those in compromise solution 1. Therefore, considering all energy-economizing indices of hot stamping, the indices of energy consumption and thinning in compromise solution 2 are better than those in compromise solution 1.

**Figure 11.** Thickness variation distribution demonstrated by the numerical simulation under different compromise solutions: (**a**) compromise solution 1, (**b**) compromise solution 2.

## *5.4. Stamping Experiments Verification*

The corresponding experiment is conducted with experimental equipment to verify the feasibility of the optimization results for the hot stamping process, as shown in Figure 12. In accordance with the obtained compromise solutions, the experiments are performed as follows:

**Figure 12.** Equipment and die structures for hot stamping process of tube-shaped parts: (**a**) 100 T partitioned VBHF hydraulic machine, (**b**) die structure, (**c**) physical die.

The temperature in the die was raised to 225 ◦C by internal heating, as shown in Figure 12b,c. Then, the prepared ZK60 magnesium alloy sheet material is placed into the die, and the mold is closed. The heat preservation time is set as 10 min to heat the blank fully. The process parameters of the forming press are set as follows: 8 kN blank holder force and 3 mm/s stamping speed (Table 4). The stamping experiment is then conducted. Three groups of experiments are conducted in accordance with the above experimental steps. Then, to reduce the temperature of the die to 200 ◦C and repeat above experimental steps, the process parameters are set as follows: 4.7 kN blank holder force and 3.3 mm/s stamping speed (Table 4).

Figure 13 exhibits the obtained stamping parts at the parameters of the two compromise solutions. The forming quality of the obtained stamping parts is good, and no evident defects are observed. However, slight wrinkling is identified in the flange area of the stamping parts under the process parameters of compromise solution 2 because of its large thickening rate. Nevertheless, the slight wrinkling will not affect the final use of the product because the flange area of the obtained parts is cut off in actual production. Therefore, compromise solution 2 is the optimal combination of process parameters from a comprehensive perspective.

**Figure 13.** Hot stamping parts of experimentally stamped ZK60 magnesium alloy: (**a**) Obtained part at the parameters of compromise solutions 1; (**b**) Obtained part at the parameters of compromise solutions 2.

The thickness variation of a stamping part is an important index of the forming quality. Therefore, the thickness variation of the stamping parts obtained by the hot drawing experiment under the process parameters of comprise solution 2 is compared with that of the simulation results, as shown in Figure 14. The thickness variation distribution of each shell element in the FE model can be read in the postprocessing of the simulation. The thickness variation distribution of the stamping parts is measured along the symmetrical section of the parts by a micrometer. The results show that the thickness variation distribution obtained by the FE method is consistent with that obtained by the experiment, which further verifies the validity and rationality of the FE process simulation of hot stamping.

**Figure 14.** Thickness variation along the symmetrical section of the experimentally obtained parts and simulation result.

#### **6. Conclusions**

Hot stamping is developed and widely applied in vehicle production according to the lightweight demands of automobiles. But hot stamping is energy intensive due to the high-temperature forming conditions of blanks. To reduce the energy consumption and improve the energy efficiency of hot stamping, an energy-economizing optimization method for hot stamping is proposed. In this method, the process parameters are optimized to reduce the energy consumption of hot stamping while maintaining the required forming quality.

In this study, the mathematical modelling, simulation, and optimization of ZK60 magnesium alloy sheets hot stamping process were addressed in the aim of improving the energy efficiency and forming quality of the stamping parts. For this purpose, a new multiobjective optimization method is proposed and tested in a real industrial size case study experiment. The model is developed and solved using a multiobjective genetic algorithm (NSGA-II), and offers feasible optimized solutions. The comparison between the numerically-predicted technical parameters of the stamps and the real experimental results demonstrates the applicability of the method. Under the optimized conditions in the case study, a substantial energy consumption reduction, i.e., 17.6%, is shown. This method may serve as a reference for finding proper process parameters to solve the energy inefficiency and high energy consumption problems associated with the metal forming fields.

**Author Contributions:** Conceptualization, M.G. and L.L.; methodology, M.G.; software, L.L.; validation, M.G., Q.W. and L.L.; formal analysis, Z.M.; investigation, Z.M.; resources, Q.W.; data curation, Z.M.; writing—original draft preparation, M.G.; writing—review and editing, Q.W.; visualization, L.L.; supervision, L.L.; project administration, Q.W.; funding acquisition, M.G. and Q.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by [key projects of natural science research in colleges and universities of Anhui province China] grant number [KJ2018A0451], [Anhui Major Science and Technology Project] grant number [18030901023], [Suzhou College Scientific Research Foundation Project] grant number [2017JB03], [Suzhou Engineering Research Center for Collaborative Innovation of Mechanical Equipment] grant number [SZ2017ZX07], [Suzhou College Teacher Application Ability Development Workstation] grant number [2018XJYY01], [Opening Project of Suzhou University Research Platform] grant number [2019kyf21, 2019ykf26, 2019ykf27], and [Suzhou College Teacher Application Ability Development Workstation] grant number [2018XJYY01].

**Conflicts of Interest:** We declare that we have no financial or personal relationships with other people or organizations that can inappropriately influence our work. We confirm that none of the material in the paper, in whole or in part, has been published or is under consideration for publication elsewhere. All the authors listed have approved the manuscript.
