Optimization of electromagnetic forming process parameters for curved specimens based on CCD-RSM

Abstract

Electromagnetic forming is an efficient, non-contact high-speed forming technology widely used in aerospace, automotive manufacturing, and other industries. This paper introduces a novel curved workpiece based on an elliptical coil, with characteristics similar to rocket propellant tanks. To optimize key parameters for the electromagnetic forming of curved workpieces, the effects of discharge voltage, coil inner diameter, and coil width on the forming results are studied. Finite element simulations using COMSOL, combined with existing AA 2219-O aluminum alloy high-speed forming experimental data, validate the model’s accuracy. Sensitivity analysis of key parameters for electromagnetic forming is performed, with discharge voltage, coil inner diameter, and coil width selected as factors for the central composite design (CCD). The influence of various process parameters on the forming characteristics of curved workpieces is explored. Using a response surface optimization method with a forming diameter of 100 mm as the goal, a single-objective optimization model is built to determine the optimal parameter combination. Experimental results show that when the discharge voltage is 4403.27 V, the coil inner diameter is 67.7 mm, and the coil width is 3.39 mm, the electromagnetic forming diameter of the workpiece is closest to 100 mm.

Introduction

With the rapid development of the aerospace and transportation industries, there is an increasing demand for enhanced performance in large launch vehicles, primarily reflected in greater payload capacity, lower energy consumption, and extended service life^1,2. The propellant tank of a launch vehicle is one of the major structural components of the rocket body. As a pressure vessel for storing liquid propellants, it not only ensures tank sealing but also serves as a critical platform interface for the fuel pipeline system, electrical control system, and propulsion system, while providing the foundation and space for the installation of other system equipment. Consequently, the tank bottom experiences complex loading conditions, making manufacturing quality crucial³. As a key component affecting the overall reliability of the launch vehicle, the propellant tank demands stringent design and fabrication standards. To facilitate the connection between the propellant tank and the pipeline system, flanged holes of a certain height must be machined into the tank bottom to enable the transport and discharge of liquid media within the tank. However, forming flanged holes on the lobed structure of the tank bottom presents significant challenges due to the complexity of the forming process and the influence of multiple parameters^4,5. Conventional flanging processes require a forming tooling system with a punch and die set, which involves complex procedures and poor adaptability. Hot forming processes, on the other hand, require expensive heating equipment and have high time costs. Additionally, the heating process may alter the alloy’s properties and microstructure, potentially leading to grain coarsening and the growth of secondary phases^6,7.

Conventional flanging processes typically require a forming tool system consisting of both punch and die, which results in complex procedures and limited adaptability. Hot forming processes, on the other hand, involve expensive heating equipment and high time costs. Moreover, the heating process can alter the alloy’s properties and microstructure, potentially leading to issues such as grain coarsening and second-phase growth^8,9. Currently, the most popular high-speed forming methods are electromagnetic forming, and the research on explosive forming and water forming methods is decreasing. Moreover, the specimens with pre-installed holes in this paper are not applicable to the explosive forming and water forming methods.

As a contactless and rapid forming process based on electromagnetic force, electromagnetic forming (EMF) offers advantages such as high deformation speed, superior forming quality, and the absence of mold wear, making it a promising solution for manufacturing large and complex aluminum alloy components. Zhang Wenzhong et al.¹⁰conducted a study on the electromagnetic flanging of circular holes in 5A06 aluminum alloy plates using planar spiral coils. Compared to conventional stamping flanging, the material’s elongation increased from 14% to 31.7%. The process achieved crack-free forming of the workpiece without the need for pre-treatment steps such as grinding and polishing, and the thinning was more uniform. Yu Haiping et al.¹¹ conducted a comparative study on the deformation mechanisms of electromagnetic flanging and stamping flanging for QCr0.8 copper alloy. The inertial effects in electromagnetic forming, by influencing the radial tensile stress on the workpiece, expanded the range of biaxial tensile strain. Under the same flanging coefficient, the electromagnetic flanging height of the workpiece was increased by 28.83% compared to conventional stamping flanging. Chen Mengcai et al.¹² conducted a flanging study on 1060 aluminum alloy circular plate workpieces using an axial-radial dual-coil configuration. The results indicated that, compared to conventional single axial coil loading, the material flow in the flange area of the workpiece and the flanging height were significantly increased.

Su et al.¹³ used numerical simulation to study the flanging forming process of 1.5 mm thick 2219 aluminum alloy. The study found that the plastic strain before collision was determined by the geometry, rather than being dependent on the discharge energy. The collision between the flanged part and the mold was beneficial for forming, and the failure strain in electromagnetic forming was 77.8% higher than that in quasi-static forming. In another study, Su et al.¹⁴ compared the forming results of annealed 2219 aluminum alloy (AA2219-O) sheets under quasi-static (QS), electromagnetic dynamic (EM), and mechanical dynamic (MD) tensile loads. The results showed that the forming limit of AA2219-O sheets under EM tensile load was significantly higher than under QS tensile load (45.4%) and slightly higher than under MD tensile load (3.7%–4.3%). Compared to QS tensile loading, the combined deformation conditions of EM and MD tensile loading significantly improved the forming performance. Tang Tianyu et al.¹⁵ analyzed the effects of different temperatures and strain rates on the flow stress of 2219 aluminum alloy, and made corrections and optimizations to the strain rate term of the traditional Johnson–Cook (J-C) constitutive model. Compared to the conventional J-C constitutive model, the optimized model showed a better agreement between the simulation results of electromagnetic forming and the experimental values. This confirmed that the optimized constitutive model could accurately describe the high-strain-rate deformation behavior of 2219 aluminum alloy.

Paesea et al.¹⁶ proposed a novel electromagnetic coupling calculation model for flat spiral coils, which can accurately predict the forming shape of sheet materials. Experimental validation of discharge forming was conducted under different energy levels. Lai et al.¹⁷ introduced an innovation in their approach by designing a dual-coil system. By changing the polarity of the discharge current, they were able to influence the Lorentz force generated by the coils, which ultimately affected the deformation of the workpiece. Zhang Jun et al.¹⁸ designed a concentric circular stepped coil, with the number of copper wire turns increasing gradually from the inside to the outside. They found that the energy efficiency of this stepped coil was much higher than that of conventional planar spiral coils. Additionally, the drawn parts exhibited a smaller taper and higher mold-conforming accuracy. Lai et al.¹⁹ used coil efficiency F as the optimization objective for uniform pressure coils. They investigated the effects of three parameters: the gap g between the coil and the sheet, the conductor thickness , and the external channel height b. The study found that as the gap g and conductor thickness increased, the coil efficiency decreased. By taking the first derivative of the objective function with respect to b, the critical point was determined, leading to an expression for the optimal channel height. Cui et al.²⁰ designed a novel planar structure uniform pressure coil based on traditional coils. Compared to the structure and magnetic force of conventional uniform pressure coils, the planar uniform pressure coil can generate a more uniform magnetic force distribution on the sheet material. Additionally, it reduces manufacturing difficulty and processing costs, while extending the service life of the coil.

Although electromagnetic forming technology has been extensively studied both domestically and internationally in recent years, research specifically focusing on the electromagnetic forming of AA2219 aluminum alloy remains relatively limited. As a high-strength, weldable aluminum-copper alloy, 2219 aluminum alloy holds significant application value in the aerospace industry. However, due to its high yield strength and low ductility, compared to common aluminum alloys (such as AA 1100, AA 6061, etc.), the deformation mechanism, forming process, and numerical simulation of AA2219 aluminum alloy during electromagnetic forming have not yet been systematically investigated. Furthermore, most electromagnetic forming research has primarily focused on flat plates or simple geometries, research on the deformation characteristics and optimization methods involving complex geometrical components remains insufficient. Against this background, this study investigates the deformation mechanism of AA2219 aluminum alloy during the electromagnetic forming process using curved surface specimens. By integrating Central Composite Design (CCD) experiments with the Response Surface Methodology (RSM), the coil parameters and discharge parameters are optimized to improve forming quality, thereby providing theoretical and data support for the practical application of electromagnetic forming in rocket tank bottom manufacturing.

Numerical simulation

Electromagnetic forming principle

Electromagnetic forming (EMF) utilizes pulsed currents to generate a strong magnetic field in the forming coil, which induces Lorentz forces on the metal workpiece, enabling high-speed plastic deformation, as shown in Fig. 1. According to Faraday’s law of induction, the current induces a magnetic flux in the workpiece, generating eddy currents. The interaction between these eddy currents and the magnetic field produces forces that drive the metal beyond its elastic limit, leading to deformation.

Fig. 1.

Schematic diagram of electromagnetic forming principle.

Pulsed electromagnetic systems are widely used in welding, metal tube compression and expansion, and sheet metal forming. The performance and effectiveness of EMF depend significantly on the geometry of the forming coil, as well as the size and shape of the workpiece. By optimizing the electromagnetic system parameters, efficient and precise metal forming can be achieved.

During the electromagnetic forming (EMF) process, the interaction between the electric and magnetic fields is the core mechanism driving the current to generate electromagnetic force. When a pulsed current flows through the coil, the electromagnetic field governing equations of the forming system, based on Maxwell’s equations, are given as (1)–(6)²¹.

1

2

3

4

5

6

where is the magnetic field intensity, is the induced current density, is the magnetic flux density, is the electric field intensity, is the electrical conductivity of the material, is the magnetic vector potential, and is the electromagnetic force density.

During the electromagnetic forming process, the dynamic response of the workpiece is driven by the electromagnetic force. To describe the deformation behavior of the workpiece, the dynamic equation shown in (7) is adopted.

7

where is the density of the workpiece, is the displacement vector of the workpiece, and is the stress tensor.

Material properties

The material used for the tests is fully annealed AA 2219 aluminum alloy (AA 2219–0). AA 2219-O is a wrought aluminum-copper alloy that can be strengthened through solution heat treatment and aging. Due to its excellent weldability and good mechanical properties at low temperatures, it has found widespread application in the aerospace industry. Table 1 presents the nominal chemical composition of the material¹⁴.

Table 1.

Chemical Composition of AA2219-O.

Al	Cu	Mg	Mn	Fe	Si	Zn	Zr	Ti
Bal	6.5	0.01	0.36	0.21	0.05	0.02	0.18	0.06

Due to the dynamic changes in the electromagnetic field and the high deformation rates, real-time measurement of the deformation information of the specimen (such as stress–strain distribution and electromagnetic force distribution) remains a significant challenge. A finite element model of the curved surface specimen was established using COMSOL 6.2 software. The selected specimen is based on actual data from the bottom of the XX rocket propellant tank, where the outer surface profile of the tank bottom conforms to an elliptical shape. The corresponding equation is shown in Eq. (8), where a and b represent the major and minor axes of the ellipse, respectively. In this study, the major axis was set to 1675 mm and the minor axis to 1646.8 mm.

8

To save computation time, this study selects the specimen sample from the position directly beneath the bottom of the propellant tank. This location features a top diameter of 200 mm, a pre-drilled hole diameter of 80 mm, and a thickness of 2 mm. The test sample is shown in Fig. 2.

Fig. 2.

Specimen sampling location diagram.

Finite element model establishment

The 2D finite element model of the rocket tank bottom specimen, based on the elliptical equation, and the coil model were established using the commercial software COMSOL 6.2, as shown in Fig. 3a, the finite element simulation model is presented, and Fig. 3b illustrates the meshing model.

Fig. 3.

Finite element model. (a) Finite element model. (b) Mesh division of the model.

The structural parameters of the model are listed in Table 2. The model consists of an eight-turn coil, a specimen, and two air domains. Quadrilateral meshing was applied to the coil and the specimen, while triangular meshing was used for the air domains, resulting in a total of 7,019 elements. The definitions of each component in the model are illustrated in Fig. 4. Rigid constraints were applied to the upper and lower boundary conditions to simulate the actual steel mold, with a friction coefficient set at 0.05. A fixed constraint was applied to the right boundary to simulate the distal end of the workpiece.

Table 2.

Structural parameters of the coil and specimen.

Geometric parameters	Values(mm)
Coil width	3
Coil height	5
Inner diameter of coil	64
Number of windings	8
Coil gap	1
Sheet thickness	2
Die radius	54

Fig. 4.

Diagram of element definitions.

In the numerical simulation, a simplified Johnson–Cook (J-C) model is used to describe the deformation behavior of the specimen, considering strain rate hardening¹⁴. Due to the short forming time, the effect of temperature is neglected in the model. The simplified J-C constitutive equation is shown as (9).

9

where is the material’s flow stress, , , C and n are material constants, is the strain rate, and is the reference strain rate.

The material parameters selected for AA 2219-O are shown in Table 3.

Table 3.

Material property parameters of AA 2219-O.

Materials	Density(kg/)	Young’s modulus (GPa)	Poisson’s ratio	A (MPa)	B (MPa)	C	n
2219-O	2820	73.1	0.33	69.5	258.4	0.01616	0.52

The circuit simulation was based on the simplified equivalent discharge circuit of the electromagnetic forming test platform at Huazhong University of Science and Technology, as shown in Fig. 5. In this system, the capacitor () stores electrical energy. Once fully charged, the stored energy is released through the resistor () and inductor (), generating current flow. The inductor () is connected in series with the capacitor to regulate current flow and generate the corresponding magnetic field. The diode () serves to restrict the current direction, preventing reverse flow and protecting circuit components. () and () represent the additional inductance and resistance generated by the workpiece. When current flows through the coil, it induces a varying magnetic field that interacts with the workpiece, generating electromagnetic forces that ultimately facilitate the forming process.

Fig. 5.

Equivalent circuit diagram.

The circuit parameters are shown in Table 4.

Table 4.

Discharge circuit parameters.

Name	U0	C0	R1	L1	Rd
Value	4250	320	25	5	0.1
Unit	V	uF	mΩ	uH	Ω
Describe	Capacitance initial voltage	capacitance	Discharge circuit resistance	Inductance of discharge circuit	Current loop resistance

Using COMSOL’s differential equation module, the circuit component is modeled with differential equations, as shown in Eq. (10).

10

Verification of numerical method accuracy

To verify the accuracy of the numerical method, the electromagnetic forming experiment of the AA 2219-O specimen was simulated. The rated voltage was set to 30 kV, with a peak current of 150 kA and a capacitance of 426 µF¹⁴. A racetrack-shaped coil with a height of 10 mm was used, as shown in Fig. 6b. The specimen dimensions are shown in Fig. 6c, with a thickness of 1.5 mm. In the simulation software, two Φ18 circular holes were constrained to serve as bolt constraints during the forming process. The gap between the specimen and the coil was set to 2 mm, as illustrated in Fig. 6a. Ultimately, the deformation profile and equivalent strain distribution map of the workpiece were obtained and compared with the experimental data, as shown in Fig. 7a. The strain distribution of the specimen is presented in Fig. 7b.

Fig. 6.

Accuracy verification model. (a) Assembly model. (b) Discharge coil model. (c) Verification specimen model.

Fig. 7.

Comparison of experiment and simulation. (a) Workpiece deformation diagram and equivalent strain distribution diagram. (b) Strain distribution diagram.

The error bar chart of plastic deformation is shown in Fig. 8a. As illustrated, the numerical simulation results closely match the experimental results, with a maximum deformation error of 3.65% and a minimum error of 0.32%. The residuals between the experimental and simulated plastic deformation are presented in Fig. 8b, indicating that within the strain range of 0 to 0.4, the simulation results are generally consistent with the experimental data, with the absolute value of the maximum residual remaining below 0.01. These results confirm the accuracy of the numerical simulation, thereby validating the correctness of both the constitutive model for AA2219 aluminum alloy and the differential equations used in the discharge system.

Fig. 8.

Simulation error verification graph. (a) Error bar. (b) Residuals.

According to the design parameters of the electromagnetic forming system, a numerical simulation analysis of the electromagnetic forming process was conducted using the established numerical method. Figure 9 presents contour plots illustrating the variation of stress and electromagnetic force on the workpiece over time, with a time interval of 20 μs. The forming process concludes at 180 μs after discharge. In each time step, the left-side image depicts the stress distribution, while the right-side image shows the electromagnetic force acting on the workpiece. At the end of the forming process, the maximum stress occurs at the fillet region of the workpiece.

Fig. 9.

Shape formation process diagram.

The force characteristics and deformation velocity distribution of the workpiece during the electromagnetic forming process were obtained through numerical simulation, and their corresponding 3D surface plots were generated. As shown in Fig. 10a, the z-axis represents the magnitude of the force experienced by the workpiece throughout the forming process, with the horizontal axis representing time and the vertical axis indicating the axial length of the workpiece, at approximately 40 μs, the electromagnetic force rapidly reaches its peak value and then gradually decreases thereafter. Figure 10b illustrates the deformation velocity of the workpiece during the forming process, using the same coordinate axis settings as the force distribution diagram. The deformation velocity increases rapidly in the initial phase, reaching its peak at 90 μs, and then gradually decreases to a stable state.

Fig. 10.

Workpiece force characteristics and deformation velocity. (a) Workpiece force distribution. (b) Workpiece deformation velocity distribution.

As shown in Fig. 11, the black curve represents the axial stress distribution of the workpiece, while the red curve represents the strain distribution, with the horizontal axis indicating the arc length. The stress and strain exhibit significant non-uniform distribution across different regions of the workpiece, the maximum stress and strain are primarily concentrated at the fillet regions of the deformation zone and around the hole edges. The embedded deformation profile in the figure shows the final forming radius of the specimen, measured at 49.07 mm. These results indicate that the workpiece undergoes intense plastic deformation under the action of electromagnetic force, with the most vulnerable regions for cracking being the fillet area and the edges of the pre-existing hole.

Fig. 11.

Stress–strain distribution diagram.

Objective optimization

Relative sensitivity analysis

Given the numerous structural parameters of the coil, a sensitivity analysis of the coil and process parameters was conducted to reduce finite element analysis time and improve computational efficiency. By varying different parameters, those with significant influence on the optimization objectives were identified. Subsequently, coil parameters were accurately and efficiently optimized using Central Composite Design (CCD) combined with Response Surface Methodology (RSM).

Figure 12 presents the results of the relative sensitivity analysis of electromagnetic forming process parameters, based on the local sensitivity analysis method. The horizontal axis represents variations in coil width, coil height, coil gap, coil turns, coil inner diameter, and discharge voltage, while the vertical axis corresponds to the formed specimen radius. Based on the analysis, the three most sensitive parameters—discharge voltage, coil inner diameter, and coil width—were selected as the experimental factors for the central composite design (CCD).

Fig. 12.

Sensitivity analysis results.

Central composite experimental design

In this study, a central composite design (CCD) consisting of 20 experimental points, including 6 axial points and 8 factorial points, was employed. For the three most sensitive parameters—discharge voltage, coil inner diameter, and coil width—five distinct levels were defined. These parameter levels are classified as − 2, − 1, 0, 1, and 2, as shown in Table 5.

Table 5.

Parameter levels.

Factors	Levels
	− 2	− 1	0	1	2
discharge voltage	3931.82	4000	4100	4200	4268.18
Coil diameter	58.6364	60	62	64	65.3636
coil width	2.15910	2.5	3	3.5	3.84090

Table 6 presents the number of simulations to be performed in the DOE and their execution sequence. The forming diameter for each parameter combination is obtained using COMSOL simulation analysis. The goodness of fit of the regression model and the relationship between parameters are evaluated using analysis of variance (ANOVA) based on the correlation coefficient and the p-value. When the coefficient approaches 1.0, it indicates a more accurate model fit, while a lower p-value signifies greater importance of the corresponding factor.

Table 6.

Number of Simulations and Simulation Sequence.

Run	Discharge voltage	Coil diameter	Coil width	Form diameter
1	3829.55	64.00	3.00	93.46
2	4000.00	68.00	3.50	95.98
3	4500.00	60.00	2.50	97.98
4	4500.00	68.00	3.50	101.588
5	4250.00	64.00	3.00	97.812
6	4250.00	64.00	3.00	97.812
7	4000.00	60.00	2.50	93.18
8	4250.00	64.00	2.15	99.204
9	4000.00	68.00	2.50	98.186
10	4250.00	64.00	3.00	97.772
11	4250.00	64.00	3.84	95.848
12	4250.00	64.00	3.00	97.772
13	4250.00	57.26	3.00	92.124
14	4250.00	70.72	3.00	100.594
15	4250.00	64.00	3.00	97.772
16	4670.45	64.00	3.00	102.33
17	4500.00	68.00	2.50	103.714
18	4250.00	64.00	3.00	97.772
19	4000.00	60.00	3.50	97.718
20	4500.00	60.00	3.50	96.048

Table 7 presents the results of a statistical analysis of the effects of electromagnetic forming process parameters based on the Analysis of Variance (ANOVA) method. Discharge voltage and coil inner diameter are identified as the most critical factors influencing the forming diameter., with P-values far below 0.05, indicating their highly significant impact on the results. Among the interaction terms, the interactions between discharge voltage and coil inner diameter, as well as between coil inner diameter and coil width, are relatively significant.

Table 7.

Analysis of variance (ANOVA) results.

Source	DOF	Adj SS	Adj MS	F	P
Model	9	37.1822	4.1314	12.16	0.000
Linear	3	31.7546	10.5849	31.16	0.000
Discharge voltage	1	15.5906	15.5906	45.90	0.000
Coil diameter	1	15.1697	15.1697	44.66	0.000
Coil width	1	0.9943	0.9943	2.93	0.118
Square	3	0.6443	0.2148	0.63	0.611
Discharge voltage * Discharge voltage	1	0.1620	0.1620	0.48	0.506
Coil diameter * Coil diameter	1	0.3948	0.3948	1.16	0.306
Coil width * Coil width	1	0.0240	0.0240	0.07	0.796
Two-factor interaction	3	4.7832	1.5944	4.69	0.027
Discharge voltage * Coil diameter	1	2.0030	2.0030	5.90	0.036
Discharge voltage * Coil width	1	1.2760	1.2760	3.76	0.081
Coil diameter * Coil width	1	1.5042	1.5042	4.43	0.062
Error	10	3.3965	0.3396
Lack-of-fit	5	3.3959	0.6792
Pure error	5	0.0000	0.0001
Total	19	40.5787

DOF, degree of freedom; Adj SS, adjusted variation due to factors, accounting for errors; Adj MS, average variation per factor, adjusted for degrees of freedom; F, ratio of factor variation to error variation, tests significance; P, probability of observing data assuming no factor effect.

The regression Eq. (11) is a second-order polynomial equation fitted from the data. In this equation, by substituting different values for discharge voltage (A), coil radius (B), and coil width (C), the forming diameter of the workpiece can be calculated, as shown in Table 8. Based on the ANOVA results (Table 7), the adjusted coefficient of determination R² (adj) is 85.86%, and the predicted coefficient of determination R² (pred) is 87.26%, indicating that the regression model exhibits good fitting performance and strong predictive capability within the studied parameter range.

11

Table 8.

Molding diameter prediction results.

Run	Discharge voltage	Coil diameter	Coil width	Form diameter	Predicted diameter	Error (%)
1	3829.55	32.00	3.00	93.46	94.764	1.441
2	4000.00	34.00	3.50	95.98	96.083	0.709
3	4500.00	30.00	2.50	97.98	97.220	1.285
4	4500.00	34.00	3.50	101.588	100.761	1.326
5	4250.00	32.00	3.00	97.812	97.758	0.014
6	4250.00	32.00	3.00	97.812	97.758	0.014
7	4000.00	30.00	2.50	93.18	93.351	1.360
8	4250.00	32.00	2.15	99.204	98.897	1.389
9	4000.00	34.00	2.50	98.186	97.299	1.553
10	4250.00	32.00	3.00	97.772	97.758	0.055
11	4250.00	32.00	3.84	95.848	97.081	2.063
12	4250.00	32.00	3.00	97.772	97.758	0.055
13	4250.00	28.63	3.00	92.124	93.277	1.292
14	4250.00	35.36	3.00	100.594	100.367	0.182
15	4250.00	32.00	3.00	97.772	97.758	0.055
16	4670.45	32.00	3.00	102.33	101.952	0.327
17	4500.00	34.00	2.50	103.714	105.172	0.750
18	4250.00	32.00	3.00	97.772	97.758	0.055
19	4000.00	30.00	3.50	97.718	95.603	3.346
20	4500.00	30.00	3.50	96.048	96.278	0.699

As shown in Fig. 13, the forming diameter increases with the discharge voltage. Similarly, as the coil inner diameter increases, the forming diameter also increases. However, when the coil inner diameter reaches 68 mm, the slope K of the curve decreases, indicating that the influence of the coil inner diameter on the forming diameter is reduced beyond this point. Additionally, an increase in coil width results in a decrease in the forming diameter.

Fig. 13.

Effect of forming diameter.

As shown in Fig. 14, the effects of different parameter combinations on the forming radius are visually presented using three-dimensional surface plots. In Fig. 14a, the forming radius increases with both discharge voltage and coil inner diameter, with coil inner diameter exhibiting a more significant effect. In particular, when the inner diameter increases from 60 to 68 mm, the forming radius shows a substantial growth. The combined influence of discharge voltage and coil inner diameter demonstrates a clear interaction effect, especially at higher discharge voltages and larger inner diameters. In Fig. 14b, discharge voltage exerts a pronounced positive effect on the forming radius, while the influence of coil width is relatively minor. However, when the coil width approaches 4 mm, a slight increase in forming radius can be observed. The interaction effect between discharge voltage and coil width is weak, with the forming radius mainly influenced directly by the discharge voltage. As illustrated in Fig. 14c, the forming radius significantly increases with the enlargement of the coil inner diameter, whereas changes in coil width have little impact. Overall, coil inner diameter and discharge voltage are identified as the most influential factors on forming radius, exhibiting notable interaction effects, while the role of coil width is comparatively secondary.

Fig. 14.

Surface plots of the effects of different parameters on the radius. (a) Surface diagram of form diameter and coil diameter, discharge voltage. (b) Surface diagram of form diameter and coil width, discharge voltage. (c) Form diameter and coil width, coil diameter surface diagram.

Response optimization

Using MINITAB software, an optimization study was conducted to predict the optimal combination of single or multiple response variables. In MINITAB, the target forming diameter was set to 100 mm. As shown in Fig. 15, the optimized results yielded a discharge voltage of 4403.27 V, a coil inner diameter of 67.7 mm, and a coil width of 3.39 mm, is approximately 100 mm. As illustrated in Fig. 16, the displacement contour plot of the formed specimen is presented. The final forming diameter obtained through simulation was 100.1 mm, further verifying the accuracy of the optimization.

Fig. 15.

Optimization result plot.

Fig. 16.

Variation of forming radius over time.

Conclusions

This study conducted a numerical simulation of the workpiece deformation behavior during the electromagnetic forming process, revealing that electromagnetic force plays a dominant role in plastic deformation during the initial forming stage. The simulation results indicate that the electromagnetic force rapidly reaches its peak at approximately 40 μs and then gradually attenuates. The deformation velocity of the workpiece peaks around 90 μs before gradually decreasing. Through sensitivity analysis, this study identified discharge voltage, coil inner diameter, and coil width as the key process parameters influencing the forming outcome. The response surface methodology (RSM) combined with central composite design (CCD) was used to establish the influence patterns of different parameters on the forming diameter. Analysis of variance (ANOVA) results indicate that discharge voltage and coil inner diameter are the most significant factors affecting the forming diameter, with a notable interaction effect between them. In comparison, the influence of coil width on forming diameter is relatively moderate, and its interaction effects are comparatively minor.

Through the regression equation and optimization method, this study successfully achieved an accurate prediction of the forming diameter of a local specimen based on the elliptical line equation for the tank bottom. When the discharge voltage is 4403.27 V, the coil inner diameter is 67.7 mm, and the coil width is 3.39 mm, is approximately 100 mm. The accuracy of the regression equation was verified, ultimately yielding a flanging forming diameter of 100.1 mm.

The results of this study confirm the feasibility of using COMSOL for electromagnetic forming simulation analysis and enhance the reliability of the simulation by comparison with existing experimental data, providing guidance for future electromagnetic forming simulations. The sensitivity analysis of process parameters offers practical guidance for engineering applications. Additionally, the specific data and parameters obtained for electromagnetic flanging forming of rocket tank bottoms, based on the elliptical profile equation, hold significant practical value and provide important reference guidance for actual engineering implementation.

Author contributions

LM wrote the main manuscript text and prepared all the pictures. TJ and WT reviewed the manuscript.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Competing interests

The authors declare no competing interests.