Large ellipsoid parts manufacture using electromagnetic incremental forming with variable blankholder structure

Large ellipsoid parts are prone to wrinkle when forming by using traditional stamping. Aiming to solve the wrinkling problem in large parts, the EMIF method with a variable blank holder is proposed in this paper. The numerical simulation has shown that the sheet material near the blank holder is, as a consequence of stamping, subjected to circumferential compressive stress. When the drawing height was 100 mm, the sheet metal was notably wrinkled. In the electromagnetic forming (EMF) process, the sheet region facing the coil becomes thinner. However, the sheet metal thickness corresponding to the coil edge increases with the increase in forming height. If the EMF forming height is 150 mm, the sheet, which is in contact with the smooth mold, is deformed without a wrinkle. Compared to traditional stamping, the EMF can significantly reduce the sheet metal wrinkling, improving the deformation height of the sheet metal smooth area.


Introduction
With the increasing demand for high-speed trains, large aircraft, and large carrier rockets requiring integral, high-precision, and lightweight structural parts, the sheet parts are developing rapidly. There is a need for producing parts of larger size, thinner walls, deeper cavities, and more complex surfaces, all while using difficult-to-deform materials. The use of high-performance lightweight alloys for manufacturing large integral components is the primary technical direction for increasing the part bearing capacity limit in both aviation and aerospace. For example, the aluminum alloy has several advantages, including the low density, high strength, and corrosion resistance; as such, it has been widely used in the aerospace field.
Large-scale ellipsoid parts are the key components of a rocket tank; the desired shape is mainly achieved by applying mechanical force and liquid pressure to drive part deformation. Currently, there are two main methods for manufacturing the large rocket tank: (1) decomposing the part into smaller parts, which are then welded together after forming, or (2) using heavy equipment to achieve the integral manufacturing of large parts.
For the former (1), there are several examples. Feng et al. [1] designed a melon petal dies through numerical simulation and experimental research; the results were then applied to successfully manufacture the smooth surface and wrinklefree melon flap parts (the deep drawing was used). However, the drawing forming needs multiple mold tests, increasing the mold manufacturing cost. Yang et al. [2] found that pre-deformation introduced during the creep age forming can both reduce the springback and improve the mechanical properties. Xu et al. [3] observed that it is possible to improve the creep deformation and reduce the springback in the nonisothermal creep aging process compared to their isothermal counterparts. Yang et al. [4] established the finite element model of creep aging forming to manufacture the vehicle fuel tank melon flap. By analyzing the creep strain, equivalent stress, and yield strength, the melon flap parts of a carrier rocket fuel tank were successfully manufactured. However, the creep aging forming has several limitations, including the long production cycle and high cost-both the forming molds and hot pressing tanks are expensive. Additionally, accurate springback prediction is another problem encountered when using creep aging [5,6]. For this reason, the traditional rocket tank manufacturing process by using melon petal forming and welding generally has low manufacturing efficiency and poor performance.
Regarding the latter, the use of heavy equipment to produce the large parts (2), China's largest vertical spinning machine with (1000 kN capacity) has the following dimensions: it is 30 m long, 18 m wide, and stands 13 m tall [7]. However, it can process parts with a maximum diameter of up to 2 m. Similarly, Yuan et al. [8] have built the double-acting sheet metal hydraulic forming equipment with the world's largest tonnage-150 MN. The machine is 19.5 m high and has a 4.5m × 4.5-m working table. The huge structure and high price inherent to large equipment pose a great challenge to the equipment manufacturing industry. Therefore, enabling the use of smaller equipment to precisely manufacture the large, thin-walled, and curved aluminum alloy parts is among the critical scientific and technological problems.
However, aluminum alloys are difficult to form due to their poor plasticity (at room temperature). Thus, evaluation and improvement of the formability of such materials is an important research topic. The forming limit diagram (FLD) or forming limit curve (FLC) is often used tools to estimate sheet metal necking limits. Paul [9] compared the common independent strain path to strain/stress-based limiting criteria. Furthermore, the calculation error can be minimized by improving experimental techniques and establishing detailed material data for various strain paths. Paul [10] analyzed the effect of punch geometry, microstructure, pre-straining path, strain rate, and temperature on the FLC. It was concluded that it can be improved either by reducing the non-coherent particles, through the uniform distribution of fine hard phases, or by the introduction of twin and transformation induced plasticity.
The electromagnetic forming (EMF) is a type of special processing method with both a high energy rate and high speed. Compared with the traditional quasi-static forming method, the EMF technology has a higher speed and the advantage of being a non-contact method. This allowed it to greatly improve the material forming limit, simplify mold manufacturing, reduce residual stress, enhance forming accuracy, and easily control the energy, improving the production automation. For example, Balanethiram et al. [11] improved the forming limit in the electromagnetic ring expansion experiment of AA6061-T4 aluminum alloy; such behavior was defined as "hyperplasticity." Oliveira et al. [12] discovered that the principal strain of AA5754 aluminum alloy in EMF increased by approximately 50% compared to quasi-static conditions. Similarly, Thomas et al. [13] reported that the forming limit of AA6063-T6 aluminum tubes formed using the electromagnetic free bulging is two to three times higher compared to quasi-static conditions. Cui et al. [14,15] simulated the electromagnetically assisted forming of V-shaped and U-shaped parts. Cui et al. [16] proposed a novel reversebending method using EMF. The plastic strain increases and stress decreases were found in the sheet-bending region following the electromagnetic forming.
Currently, there are two main strategies for manufacturing large-size parts using the EMF: 1) Using high-energy equipment and a large coil structure, Lai et al. [17] established an electromagnetic forming device without the assistance of traditional stamping. Combined with the electromagnetic blanking and inertial constraints, the 5083-O aluminum alloy hemisphere part with approx. 1000 mm in diameter and 225 mm deep was successfully developed. However, two sets of electromagnetic forming devices over 800-kJ discharge energy and the coil diameters over 800 mm were required. 2) Low-energy equipment and a small coil structureadopted to achieve the integral forming by moving the coil discharge. Cui et al. [18] proposed electromagnetic incremental forming (EMIF). In this method, the working coil moves step-by-step along the mold profile, with each step being discharged twice in each position. The first discharge reduces the distance between the sheet and the mold profile, while the second one makes the sheet and mold fit completely.
Tan et al. [19] used EMIF to successfully develop a doublecurvature panel. The influence of technological parameters such as discharge voltage, capacitance, coil height, discharge path, and coil and mold fit overlap ratio was studied experimentally. Furthermore, Li et al. [20] analyzed the forming mechanism and defect law in large-sized curved surfaces made of aluminum alloy using the EMIF process. They used numerical simulation and experimental verification, finding that the wrinkling was primarily caused by the circumferential compressive stress, which was a consequence of stress wave propagation caused by electric discharge. Finally, Cui et al. [21] proposed the electromagnetic partitioning forming to achieve precise manufacturing and control the springback when producing curved parts.
Cui et al. [22] combined the traditional drawing process and the EMIF aiming to manufacture large curved thinwalled parts made of aluminum alloy. After 36 discharges, curved surface parts with a 580-mm diameter and a height of nearly 75 mm were obtained. When the blank holder radius was set to 390 mm, the deformed sheet surface was smooth, and there were no wrinkles, as shown in Fig. 1a. Inside the dotted red line, the sheet regions deformed using die are shown. Furthermore, to obtain a larger ellipsoid part, the blank holder radius was enlarged to 475 mm. When the forming height was 60 mm, notable wrinkles appear on the sheet surface (see Fig. 1b); therefore, the wrinkling problem must be solved when forming large parts.
In this paper, the authors applied the electromagnetic incremental forming with a variable blank holder structure aiming to manufacture larger parts. The sheet metal thickness-todiameter ratio was approximately 0.1%, causing severe sheet wrinkling in traditional drawing processes. The numerical simulation and the experiment were carried out to analyze the forming process and wrinkling in large parts, both in traditional drawing and EMIF processes.
2 Forming principle Figure 2 shows the EMIF process using variable blank holder structure. The forming system includes the forming die, pressing plate, support plate, working coil, sheet, and hydro-cylinder. The pressing plate and supporting plate are used to press the sheet metal, which is then deformed in deep drawing direction under the action of hydro-cylinder and magnetic force.
The whole forming principle is carried out in two steps: 1) For the blank holder radius R 1 , the part was achieved in steps, with the final forming depth being the H 1 .
2) The blank holder radius size was changed to R 2 , and the part was formed in steps. The final forming depth is set to H 2 .

Wrinkle simulation
The quasi-static forming process is shown in Fig. 3. The sheet is made of 3003-O aluminum alloy and its diameter is 1150 mm with 1 mm thickness. The material yield strength and elastic modulus are 50 MPa and 68 GPa, respectively. The stress-strain relationship of the material properties was determined by carrying out a uniaxial tensile experiment, as shown in Eq. (1). To consider how does the high strain rate affects the EMF process, the material behavior was modeled using the Cowper-Symonds power law, as shown in Eq. (2). The data used in Eq. 1 and Eq. 2 are taken from the paper published by [22].
where σ is the dynamic flow stress, σ qs is the quasi-static flow stress,ε pl is the strain rate, and C m = 6500 s −1 , and n = 4 are specific aluminum alloy parameters. Figure 3a shows the forming system; the blank holder radius R is set to 475 mm, and the distance between the pressing plate and supporting plate is 2 mm. In this paper, the sheet metal thickness-to-diameter ratio is approximately 0.1%. The obvious sheet metal wrinkles will occur if the flange is not clamped by the pressing and supporting plates. The ABAQUS/Explicit procedure was used to analyze the stamping process. During the simulation, the pressing plate, support plate, and die were considered rigid bodies. Figure 3 c and d show the deformation results for the drawing depth of 100 mm. In Fig. 3c, the C3D8 elements were used for sheet metal (a solid unit). On the other hand, in Fig. 3d, the shell element type was used. During the drawing process, the pressing plate, supporting plate, and punch were set as rigid bodies. The friction coefficient used within the study was 0.2. It was evident that the wrinkling will not occur on sheet metal when using solid elements. However, severe wrinkling appeared on sheet metal when using shell elements, mainly since the large solid element stiffness is difficult to distort, yielding the incorrect wrinkling analysis simulation results. In order to Fig. 1 Deformation results using different blank holders sizes: a 390-mm-radius blank holder. b 475-mm-radius blank holder predict the wrinkling of sheet metal by numerical simulation, many scholars use shell element type. Kawka et al. [23] analyzed the wrinkling of conical cups using finite element method (FEM) and experimentally. The shell element was used in FEM. Du et al. [24] established a critical principal strain analytic model for plate wrinkling based on the energy method theory and plastic flow law of material. To reduce shear locking and membrane locking, a 4-node shell element was used in numerical procedure. Du et al. [25] designed a stretching wrinkling test of wedge plate with the boundary constraint conditions of mold. The wedge plate was meshed with shell element in FEM. For these reasons, shell elements were used for the following considerations.
3.2 Single blank holder forming system (R = 475 mm) Figure 4 shows the deformation results when using a single blank holder forming system. The radius of the blank holder is 475 mm. The distance between the pressing plate and supporting plate is set to 1 mm, with the pressing force of 50000 N. The remaining simulation parameters are consistent with Fig. 3d. Additionally, Fig. 4a shows that obvious wrinkles will appear on the sheet when the drawing depth is set to 100 mm.
To further analyze the wrinkling height, path 1 was defined; Fig. 4b shows the displacement forming along the path 1 direction. It is evident that the sheet metal deflection has noticeable waves. The maximum displacement was Fig. 2 The forming principle with a variable blank holder structure: a blank holder radius R 1 and a forming depth H 1 and b blank holder radius R 2 and a forming depth H 2 1) The radial (σ r ) and circumferential stress (σ θ ) of the element 3605 are 89.7 MPa and 83.5 MPa, respectively, meaning that the element is subjected to both radial and circumferential tensile stresses. Furthermore, its radial strain (ε r ), circumferential strain (ε θ ), and thickness strain (ε t ) are 0.0213, 0.0122, and −0.0335, respectively. Based on the presented values, it can be concluded that element 3605 appears to be thinning (reducing in thickness).
2) The radial and circumferential element 5296 stresses are 17 MPa and −71.7 MPa, respectively. Its radial, circumferential, and thickness strains are 0.013, −0.021, and 0.008, respectively. Finally, the element 5296 thickness increases, causing wrinkles in the region near the blank holder.
Node 9597 on element 5296 and node 7765 on element 3605 were extracted, and the stress and strain variation trends are shown in Fig. 5. Node 7765 was subjected to both radial and circumferential tensile stresses and strains; the tensile stress and strain increase with the increase in forming height, decreasing the stress and strain thickness as the drawing height increases. When the forming height was between 0 and 17 mm, node 9597 was acted upon by a low radial tensile stress, while the circumferential stress was practically 0. When the forming height was between 17 and 100 mm, the circumferential compressive stress measured in node 9597 gradually increased. Furthermore, the circumferential compressive stress is much greater than the radial tensile stress, causing an increase in the thickness strain as the drawing height increases.
3.3 Two blank holder forming systems (R 1 = 390 mm, R 2 = 475 mm) Aiming to reduce the sheet metal wrinkling, the forming method using a variable blank holder was proposed. Two blank holder radii were adopted: R 1 = 390 mm and R 2 = 475 Fig. 4 The deformation results for: a displacement, b path 1 data, c stress, and d strain mm, with a total drawing height of 100 mm. Finally, three forming schemes were created by adjusting the drawing height, as shown in Table 1. Figure 6 shows the stress distribution for various blank holder combinations. Figure 6 a, c, and e illustrate the stress distribution for drawing depths of 50 mm, 60 mm, and 70 mm while using the first blank holder (390-mm radius). In that case, element 3897 (near the first blank holder) is subjected to radial tension stress and circumferential compressive stress. With the increase in the forming height, the radial tensile stress decreases while the circumferential compressive stress increases. Figure 6 b, d, and f show the stress distribution when the total forming height was 100 mm for all three schemes. Element 5296 (near the second blank holder) was subjected to greater circumferential compressive stress compared to element 3897. As a result, all three forming schemes with the variable blanking structure caused the noticeable wrinkling on the sheet metal (for the forming height of 100 mm).
Aiming to further analyze the cause behind the sheet metal wrinkling, scheme 2 simulation results were selected. Figure 7 shows the changes in node 9597 stress and strain as time progresses. When the first block holder was used (R 1 = 390 mm), node 9597 was subjected to radial tensile stress and circumferential compressive stress. That corresponds to the radial tensile strain, circumferential compressive strains, and thickness tensile strains. The thickness strain increases sparsely and has a value of 0.057%. Furthermore, when the second blank holder (R 2 = 475 mm) was used, the node 9597 circumferential compressive strain increased significantly, causing the final thickness strain to increase sharply. Its value is 0.58%, given that the forming depth is 100 mm. Figure 8 shows the final sheet deformation result and the change in the path 1 displacement when using scheme 2. The apparent wrinkling can be observed on the sheet, along with the displacement on path 1, which also has obvious height variations. Such observations show that sheet material thickness strain has only increased by 0.58%; however, the sheet material has also displayed the obvious wrinkling behavior. Figure 9a shows the electromagnetic field model established using ANSYS/EMAG software. The electromagnetic field model includes far-field air, near-field air, coils, and sheets. Solid97 element type was used to model the near-field air, coil, and sheet metal, while the Inf111 element type was used for far-field air. Two spiral waist-shaped coils were symmetrically distributed on the sheet metal. For example, the deformed sheet and the coil position were shown in the 19th discharge (see Fig. 9b). Following the coil discharge, Fig. 9c shows the electromagnetic force distribution across the sheet metal.

Finite element model
During the electromagnetic field analysis, the magnetic force acting upon the solid element was taken as the magnetic  force acting on the shell element. The aim was to predict the sheet forming process; the simulation strategy effectiveness were demonstrated by Long et al. [26] and Cui et al. [16]. Figure 9d shows the sheet forming system established using  To solve the wrinkling sheet metal phenomenon in the large blank holders, as shown in Fig. 1b, the EMIF process with variable blank holder (Fig. 2) was adopted. The schematic diagram of the ensuing forming process is displayed in Fig.  10a. In each layer, the pressing plate moves downwards for h n = 15 mm. The coil discharges N times in this layer to ensure that the sheet and the die are in contact. The coil is required to rotate twice and discharge in each layer, primarily to ensure the sheet metal deformation uniformity. Furthermore, it is assumed that the distance between the inner coil wall and the sheet metal center is L (in the nth layer). Figure 10b shows the discharge positions of the first turn; the angle between the coil discharge position 2 and the horizontal sheet metal centerline  is marked as α. During the second turn, the discharge positions were rotated for 0.5α in the counter-clockwise direction, as shown in Fig. 10c.
In this paper, the forming process is carried out as follows: (1) the first blank holder radius R 1 is set to 390 mm and the forming height H 1 is 75 mm; (2) the second blank holder radius R 2 is set to 475 mm and the forming height H 1 is 150 mm. If both blank holders are used, the coil discharges in ten layers to ensure that the sheet is deformed in a step-by-step manner. The drawing height of the pressing plate moves downwards for H, distance L, and rotation angle α, and the number of discharges N are shown in Table 2 and Table 3 for each layer.

4.2
The first blank holder forming system (R 1 = 390 mm, H 1 = 75 mm) According to Table 2, forming data and the process presented in Fig. 10, the final sheet metal deformation shape in the 2nd to 5th layers is shown in Fig. 11. Furthermore, the 10th, 18h, 26th, and 36th coil discharges correspond to the last coil discharge in the 2nd, 3rd, 4th, and 5th layers, respectively. Nodes 6370 and 8960 are located near the first blank holder edge; the displacement difference between said nodes is below 1 mm between the 2nd and 5th layers. Thus, the overall sheet metal deformation is uniform-the sheet surface is smooth, and there are no noticeable wrinkles.
Aiming to find why the sheet did not wrinkle when using the first blank holder, it is necessary to analyze the stress-strain and sheet thickness variation during the drawing and discharging processes. Figure 12a shows the deformation profiles after the 26th discharge and 5th drawing. Moreover, Fig.  12b shows the sheet metal stress distribution after the 5th drawing. Element 3901, located near the blank holder, is subjected to both radial and circumferential compressive stresses, which potentially cause an increase in the sheet metal thickness strain. Figure 12c shows the increments of three principal  strains at node 7778 during the 5th drawing process. Since the radial compressive stress is greater than its circumferential counterpart, the circumferential compressive sheet metal strain increments are smaller. Lastly, Fig. 12d shows the thickness of node 7778 increments in the 5th drawing process. After the 5th drawing is carried out, the deformation results after the 27th coil discharge are shown in Fig. 13. Deformations of sheet metal parts facing the coil are evident. Nodes 6369 and 8970, corresponding to the coil middle and edge, are selected. Moreover, during the 27th coil discharge, the node 6369 reduction in thickness becomes apparent, while the node 8970 thickness slightly increases. Figure 14 shows the changes in stress and strain distribution at nodes 6369 and 8970 depending on the time. The node 6369 was subjected to radial and circumferential tensile stresses at 150 μs, after which the stress oscillated and, finally, decreased. The plastic strain increment at node 6369 during the 27th discharge was a radial tensile strain, circumferential tensile strain, and thickness compressive strain.
Node 8970 was subjected to bi-directional compressive stress at 150 μs. The final strain increment is divided into radial and circumferential compressive strain and thickness tensile strain. This results in a slight increase in the sheet metal thickness corresponding to the coil edge (as shown in Fig. 13). However, the sheet metal does not wrinkle during the 5th drawing process due to a very small increase in the circumferential compressive strain (see Figs. 13 and 14).

4.3
The second blank holder forming system (R 2 = 475 mm, H 2 = 150 mm) Using the forming data provided in Table 3, the final deformed shapes for layers 6 to 9 are shown in Fig. 15. The 46th, 56th, 66th, and 76th coil discharges correspond to the final coil discharges in the 6th, 7th, 8th, and 9th layers. It can be seen that the deformed sheet surface is smooth, with no discernable wrinkles.
Nodes 9953 and 9957 located near the edge of the second blank holder (R 2 = 475 mm) are selected next. The displacement difference between the nodes is approximately 3 mm in the 9th layer. Furthermore, the sheet region deformations near the second blank holder become inhomogeneous as the forming depth increases; therefore, a minor wrinkle could take place at the sheet region near the blank holder. Assuming that the second blank holder (R 2 = 475 mm) is used, variations in stress, strain, and thickness with time caused by the drawing and discharging processes are analyzed. The deformed sheet profiles following the 76th discharge and 10th drawing are shown in Fig. 16a. Moreover, Fig. 16b shows the stress distribution on the sheet after the 10th drawing; element 5296 is subjected to bidirectional compressive stress. Fig. 12 Deformation results after the 5th drawing: a deformation profiles, b the 3D stress distribution, c node 7778 plastic strain increment during the 5th drawing, and d node 7778 thickness changes during the 5th drawing Fig. 13 Results after the 27th coil discharge: a the 3D deformation shape and b changes in thickness (in time) After comparing Fig. 12 and Fig. 16, it is evident that, during the drawing process, the sheet region near the blank holder is subjected to bi-directional compressive stress. When the blank holder diameter increases, the circumferential compressive sheet metal strain increases. Figure 16c shows the changes in stress with time at node 9597 (during the 10th Deformed sheet shapes for different layers: a after the 46th discharge, b after the 56th discharge, c after the 66th discharge, and d after the 76th discharge drawing). In the radial and circumferential directions, sheet metal is subjected to compressive stresses. Additionally, the thickness of sheet metal strain should be increased. Figure 16d shows the radial and circumferential compressive strains which have caused an increase in thickness strain. Furthermore, since the circumferential compressive strain has a significant role in the thickness increase, the wrinkling trend will also increase.
Following the 10th drawing, Fig. 17 shows the sheet deformation results after the 77th coil discharge. Nodes 9145 and 9954 were selected, corresponding to the coil middle and coil edge, respectively. Moreover, during the 77th EMF process, node 9145 thickness decreased significantly, while node 9954 thickness increased. After comparing Figs. 13 and 17, it was clear that, with the increase in the blank holder diameter, the thickness of the sheet region facing the coil edge increases significantly following the coil discharge. Figure 18 shows the changes in stress and strain of nodes 9145 and 9954 as time progresses. At 120 μs, node 9145 is subjected to the bi-directional tensile stress, while the stress varies in the later stages. The node 9145 strain increments during the discharge are as follows: radial and circumferential tensile strain and thickness compressive strain. It can be seen that thickness of node 9145 is reducing; therefore, the coil is in the position opposite to the plate during the discharge process, thus preventing wrinkling.
Node 9954 was subjected to bi-directional compressive stress at 150 μs. The final strain increment ratio of sheet metal is circumferential compressive strain and thickness tensile strain. Such strain arrangement eventually results in a slight increase in the sheet thickness located near the coil edge. Thus, based on the analysis above, it can be stated that: 1) The thickness deformation reduction mainly occurs in the position where the coil is facing the sheet; in that position, there will be no wrinkling; 2) The thickness of the coil edge area increases due to the circumferential compressive stress, which may cause wrinkling. Moreover, with the increase in the blank holder diameter, the coil edge area thickness increment increases.
The final sheet deformation following the ten drawings and 86 coil discharges is shown in Fig. 19. In Fig. 19a, the displacement in the forming direction (along path 1) exceeds 150 mm, but its distribution is not uniform. Since path 1 is close to Fig. 16 Deformation results after the 10th drawing: a the 2D deformed profiles, b the 3D stress distribution, c node 9597 stress during the 10th drawing, and d node 9597 strain variations during the 10th drawing the sheet material blankholder, it indicates that the sheet material close to the blankholder will still be wrinkled after discharge. The wrinkling amplitude was close to 3 mm. In Fig.8, the sheet metal wrinkle reached 5.8 mm when the drawing height was 100 mm. Moreover, the sheet metal (inside the white dotted line) is uniformly deformed and has a smooth surface in Fig. 19b. Local pits and bulges appeared in the area near the blankholder. Compared to the drawing results in Fig.  8, the method adopted in this paper can significantly reduce the wrinkling and avoid it, ensuring that the deformed sheet metal area will be smooth. Finally, the sheet metal thickness distribution is shown in Fig. 19c.  blank holder with a radius of 390 mm; in that case, the sheet metal surface is smooth and without wrinkles (assuming that the drawing height is 60 mm). Based on Fig.  20a, Fig. 20b shows the sheet deformed using the 475-mmradius blank holder. The associated drawing height is 100 mm, and there is apparent wrinkling appearing on the sheet. In other words, the wrinkles appear on the previously smooth surface. Figure 20c shows the process using a 390-mm-radius blank holder. The resulting sheet metal surface is smooth and without wrinkling for the forming height of 75 mm. Finally, Fig. 20d shows the deformation sheet metal when R 2 = 475 mm. The deformed sheet area located inside the red dotted line is smooth and without wrinkling, while the area surrounding the pressing edge displays a lower degree of wrinkling. Such Fig. 19 Final deformation results: a path 1 data, b 3D deformation shape, and c thickness Fig. 20 Experimental results: a quasi-static drawing with 390mm blank holder and H 1 = 60 mm, b quasi-static drawing with 475-mm blank holder and H 2 = 100 mm, c electromagnetic forming with 390-mm blank holder and H 1 = 75 mm, and d electromagnetic forming with 475-mm blank holder and H 2 = 150 mm behavior is mainly caused by the sheet metal corresponding to the coil edge, which will be thicker due to the action of circumferential compressive stress when discharging. Finally, it should also be added that the circumferential compressive strain increases with the increase in the blank holder size.

The experiment
In this paper, the total number of discharges is 86, while the drawing times are 10 for obtaining the final shape (shown in Fig. 20d). The current EMF machine can discharge five times in a minute, meaning that the total discharge time in the forming process is below 20 min. Finally, Fig. 21 shows the distribution of profile deformation and thickness in the experiment and the simulation. It is evident that both the sheet metal deformation profile and thickness distribution are in agreement with the experimental results. In the traditional single point incremental forming (IF), the material flow is problematic. Thus, the sheet deformation occurs by thinning, and the sheet thickness is found through the sine theorem. The minimum thicknesses in the IF process, the numerical simulation, and the experimental data of EMF (from 0 to 350 mm) were 0.691 mm, 0.91 mm, and 0.89 mm, respectively. Thus, the method presented in this paper can inhibit thickness thinning.

Conclusions
1) The electromagnetic forming method supplemented with the use of variable blank holder was proposed to reduce the wrinkling during the manufacturing of large ellipsoid parts. The proposed simulation method was found to accurately predict the sheet forming process in both the quasi-static drawing and electromagnetic forming. 2) In the traditional drawing process, the sheet material near the blank holder was subjected to radial tensile stress and circumferential compressive stress. With the increase in forming height or when the blank holder is increased, the circumferential compressive strain near the blank holder rises. Finally, when the forming height was 100 mm, the sheet material wrinkling became apparent. 3) During the coil discharge, the sheet region facing the coil was subjected to both radial and circumferential tension strain. For this reason, electromagnetic forming is helpful to reduce wrinkling during the forming of large parts. However, with the increase in the blank holder diameter, the sheet metal region corresponding to the coil edge has a minuscule increase in thickness, causing an increase in wrinkling. Finally, when the forming height was 150 mm, the sheet material near the blank holder became wrinkled; however, the sheet region in contact with the dye had no wrinkling.
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Competing interests
The authors declare no competing interests.