Finite Element Analysis of Electromagnetic Forming Process and Optimization of Process Parameters Using RSM

Abstract

Aluminium can benefit from the high-speed forming technique known as electromagnetic forming (EMF). EMF is increasingly used in automotive applications as a result of this capability. This technology depends on Lorentz force (Magnetic force) in the practical forming application which relies on different process parameters like forming a coil. A finite element model for the EMF process is built and studied in this work using the finite element analysis software ANSYS 2022 R1. The affecting process parameters are investigated using the Design of Experiments (DOE) approach. Response Surface Methodology (RSM) of the DOE approach is used by taking process parameters such as coil size, gap, and current density into account. The number of experiments is reduced by using Central Composite Design (CCD), an RSM model. To determine the optimal level of parameters, a magnetic force optimization study is carried out. The parameters of the EMF process (e.g., magnetic force) are investigated through a developed 2D finite element model and validated with available literature.

1. Introduction

A technique called electromagnetic forming (EMF) uses the repellent force produced by opposing magnetic fields within neighboring conductors to shape metals. By applying an electric current through a coil, a magnetic field is produced in this technique. This magnetic field causes the generation of eddy currents which travel in the opposite direction to the coil’s current within the metallic component. The eddy current causes a repulsive force between the workpiece and the forming coil, leading to the deformation of the metal. The resulting repulsive force pushes the workpiece above its yield limit, causing permanent deformation at high strain rates. To develop a proficient EMF system and the evaluation of its effectiveness, it is essential to employ suitable numerical techniques that are not only cost-effective but also well-suited for industrial applications.

In the EMF process, the Lorentz forces (also termed magnetic forces) are generated and utilized to reshape metal sheets at high velocities. As illustrated in Figure 1, the setup for the EMF process experiment consists of significant capacitance with a low-inductance electrical circuit. This circuit is responsible for supplying electrical current to a tool coil used for the forming process. Faraday’s law of induction states that when the current flows through the coil, it causes the induction of magnetic flux in the adjacent conductor (or workpiece), resulting in the generation of eddy currents. As a result of these induced eddy currents, magnetic forces are generated causing deformation of the metal sheet above its elastic limit.

The fundamental equations representing the electromagnetic fields are governed by Maxwell and given in Equations (1)–(5).

$$\mathsf{\Delta }\ast \stackrel{⃛}{E}=-\frac{d\stackrel{⃛}{B}}{dt}$$

(1)

$$\mathsf{\Delta }\ast \stackrel{⃛}{H}=\stackrel{⃛}{J}$$

(2)

$$\mathsf{\Delta }\ast \stackrel{⃛}{B}=0$$

(3)

$$\stackrel{⃛}{B}=\mu \ast \stackrel{⃛}{H}$$

(4)

$$\mathsf{\Delta }\ast \stackrel{⃛}{J}=0$$

(5)

where $\stackrel{⃛}{E}$—Electric Field, $\stackrel{⃛}{H}$—Magnetic Field Intensity, $\stackrel{⃛}{B}$—Magnetic Field Density, μ—Permeability, $\stackrel{⃛}{F}$—Lorentz Force, $\stackrel{⃛}{J}$—Current Density

The current density depends on the electrical conductivity of the workpiece material (σ) which can be represented as per Equation (6):

$$\stackrel{⃛}{J}=\sigma \ast \stackrel{⃛}{E}$$

(6)

As Lorentz stated in 1895, the force density $\stackrel{⃛}{F}$ acting on the workpiece depends upon the magnetic flux density $\stackrel{⃛}{B}$ generated due to the supplied current density $\stackrel{⃛}{J}$ given as:

$$\stackrel{⃛}{F}=\stackrel{⃛}{J}*\stackrel{⃛}{B}$$

(7)

Commercial FEA software packages like ANSYS Emag, COMSOL Multiphysics, NASTRAN, ABAQUS/Explicit, LS-DYNA, etc., are getting popular in industrial applications for sheet metal forming simulation.

In electromagnetic forming, Furth and Waniek [2] introduced the concept that the workpiece is pushed away from the tool during the process. In their proposal, the authors recommended the use of two distinct coils to generate the forces, which create the bulges on large sheets or hollow workpieces. Meriched et al. [3] calculated the deformation of a circular thin sheet using numerical simulation by formulating coupling of electric circuit analysis and electromagnetic force for a flat spiral coil.

El-Azab et al. [4] focused on two dimensional modelling of EMF process. Zhang et al. [5] employed the COMSOL Multiphysics software package to simulate a two-dimensional axisymmetric model for sheet metal workpieces to analyze its dynamic behavior. The accuracy of this numerical model was confirmed by comparing it with experimental findings from Takatsu et al. [6]. The numerical simulation took into account the effects of workpiece deformation and motion on the distribution of magnetic forces and discharge current. Kleiner et al. [7] investigated the deformation of tubular workpieces and flat sheets by studying the influence of process parameters like magnetic pressure and strain rate on it. To simulate the EMF process for the deformation of a thin aluminium sheet, Fenton et al. [8] utilized an Arbitrary Lagrangian–Eulerian (ALE) computer code. They assessed the findings with experimental data which showed excellent consistency. Further, Mamalis et al. [9] simulated a loosely coupled numerical approach to model a two-dimensional axisymmetric aluminum alloy sheet by using LS-DYNA software and ANSYS Finite Element code. The authors used an equivalent circuit method to validate their numerical model with experimental data. Like previous researchers, Luca [10] also developed a two-dimensional finite element model using FLUX2D software, while ALGOR software was employed to calculate the strains and stresses in an alloy sheet (i.e., AlMn_0.5Mg_0.5). These numerical results showed a good level of agreement with experimental results.

Reese et al. [11] emphasized the use of a coarse mesh through the reduction of Gauss points as a means of improving the accuracy of numerical solutions while minimizing computational effort. The authors improved the coupling between mechanical and electromagnetic fields by using the reduced integration/hourglass stabilization method. A finite element model having thermal, magnetic, and mechanical aspects was created by Unger et al. [12] for the EMF process. The authors investigated the multi-field coupling formulation for an Aluminium alloy (AA 6005) plate. For the simulation of the EMF process, Siddiqui et al. [13] formulated two distinct problems: the electromagnetic problem and the mechanical problem. They have used FEMM4.0 finite element code to predict magnetic forces; it is further used as input in finite element software ABAQUS/Explicit.

Imbert et al. [14] used FEA software LS-DYNA for the simulation of the EMF process for conical and V-shaped AA 5754 sheets. After comparing experimental and numerical simulation results, the authors concluded that the formability of the sheets is improved by reducing tool/sheet interaction. Deng et al. [15] employed ANSYS software to simulate a 2D axisymmetric FE model for the electromagnetic attractive force forming process. In this simulation, the magnetic flux was distributed along the flat coil and the experimental findings show that the workpiece was quickly attracted towards the forming coil.

The electromagnetic forming process applied to aluminium tubes was experimentally and numerically analyzed by Khandelwal et al. [16]. The authors studied the deformation of the workpiece by varying the process parameters such as workpiece thickness, standoff distance (i.e., the gap between coil and workpiece), and discharge energy using the ANOVA technique. Siddiqui et al. [17] utilized a finite element code named FORTRAN and FEA software FEMM to conduct a numerical simulation for the EMF process on an Al 1050 tube. The authors compared the numerical results with earlier experimental results reported in the literature. Subsequently, the validated numerical results were utilized to forecast the electromagnetic tube expansion process using FEA software ABAQUS/Explicit. Parez et al. [18] employed various numerical software tools such as Pam-Stamp2G, Sysmagna^®, and Maxwell 3D to simulate the EMF process using sequential coupling and loose coupling. The simulation results were verified by comparing them to experimental data obtained from an Al 1050 sheet. To focus the magnetic field at the precise location of the workpiece, Bahmani et al. [19] employed a field shaper approach in the EMF process. The authors revealed that in comparison with the two-dimensional FE model, a 15% greater magnetic flux density was generated for the three-dimensional FE model.

Jaleel and Narayanan [20] discussed the formability of sheet metal, particularly the perforated stainless-steel sheets like SS 304L. Various studies explored the impact of different parameters such as lubricants, tool diameters, spindle speed and forming techniques like Single Point Incremental Forming (SPIF) on the formability of these sheets. Xie et al. [21] investigated the deformation behavior and formability of aluminum alloys, specifically 2195-SSS and Al–Li alloys, under electromagnetic forming (EMF). The studies showed that EMF enhances the materials’ mechanical properties, improves formability by promoting uniform plastic deformation, and leads to the formation of complex sub-structures and uniformly distributed dislocations. The microstructure evolution and deformation mechanisms in EMF are explored, highlighting the role of multiphysics fields in influencing the materials’ properties. Cao et al. [22] implemented a dual-coil electromagnetic tube forming method to enhance deformation uniformity in comparison to single-coil systems. Experimental and numerical results demonstrated significant enhancements in deformation uniformity and efficiency with the dual-coil system, showcasing its potential for optimizing electromagnetic tube forming processes. Kiarasi et al. [23] investigated the impact of hygrothermal effects on the natural frequencies of functionally graded annular plates with piezo-magneto-electro-elastic layers on an elastic foundation

Many researchers used FEA software like ANSYS, COMSOL, LS-DYNA, ABAQUS, etc., for the numerical simulation of the EMF process due to its user-friendly nature. Based on the literature, it can be inferred that very few researchers investigated the impact of coil size, current density, and air gap (i.e., the distance between the sheet and the coil) on the generated magnetic force. In this paper, the EMF process is simulated to calculate magnetic force. To investigate the influence of coil size, current density, and air gap, the Design of Experiments (DOE) approach is adopted. The use of Response Surface Methodology (RSM) enables the determination of the appropriate number of finite element simulations that need to be conducted.

To find the intensity of current density, air gap, and coil size on magnetic force, an Analysis of Variance (ANOVA) is performed. Also, by RSM, the optimized level of parameters that gives the maximum magnetic force is found.

2. Modelling and Numerical Analysis

To carry out the numerical analysis of the EMF process and compute the Lorentz force (magnetic force) generated, ANSYS 22R1 Emag software is used. As shown in Figure 2, a two-dimensional (2D) axisymmetric finite element model, is created with the geometric parameters listed in

Table 1. Geometric Parameters used for numerical simulation.

Geometric Parameters	Values
Size of the forming coil	(5 × 5) mm2
Number of Coils	10
Thickness of the Sheet	1.5 mm
Length of the Sheet	140 mm

, based on literature results from Asati [24]. The simulation region comprises the tool coil, workpiece (sheet), and air region. It is meshed using 2D quadrilateral four-noded axisymmetric elements (PLANE13).

Table 2. Material properties used for numerical simulation.

Material	Relative Permeability (µ)
Copper (For Forming Coil)	0.99
Aluminum (For metal sheet)	1.003
Air	1

provides the material properties used in the electromagnetic forming process simulation. The 2D finite element model is discretized into 12,480 elements; there were 12,717 nodes in total.

The EMF process simulation has a square-shaped forming coil made of copper. The forming coil is modelled with 10 turns. The current density of 8000 A/m² is given as input to the coil which generates a magnetic field. Since flux parallel boundary conditions are used, magnetic flux is percolated in the nearby region. Figure 3 illustrates the meshed 2D model. Figure 4, Figure 5 and Figure 6 represent the magnetic force vector sum, 2D flux line, and magnetic force vector plot, respectively. Figure 6 shows the magnetic force distribution along the sheet’s length. To check the accuracy of the FE model, it was validated with the findings of Asati et al. [24], giving an error of 0.19% (

Table 3. Validation of present finite element model with Asati et al. [24].

Process Variables	Input Values	Magnetic Force		Percentage (%)
Current density (A/m2)	8000	Asati et al. [24]	Present Model	0.192
Size of square coil (Length × Height) (mm)	5	19,923.9	19,885.6
Gap between coil and sheet (mm)	2

).

3. Results

3.1. Design of Experiments Approach

To simulate the EMF process, the current density, gap, and coil size are considered as the influencing process parameters. A Design of Experiments (DOE) method is used to find different combinations of current density, gap and coil size for simulation purposes. Five different levels are considered for each of these process parameters. The Central Composite Design (as shown in Figure 7) of Response Surface Methodology (RSM) is used for the simulation.

Response Surface Methodology (RSM) uses the Central Composite Design (CCD) extensively to analyze complicated systems with various affecting factors. CCD is valued for its ability to reveal variable interactions and quadratic effects with few experimental runs. This design typically has 20 runs with strategically arranged 6 axial points and 8 corner points, distributed across five levels (−2, −1, 0, 1, 2) for each process parameter. The CCD provides extensive response surface exploration, enabling process parameter optimization. These parameters are adjusted consistently to span the experimental zone, ensuring trustworthy and actionable results. The 20 sample runs, created by using Central Composite Design (CCD), are shown in Figure 7.

Table 4. Parameters levels.

Symbol	Process Parameters of EMF	−2	−1	0	1	2
A	Current density (A/m2)	6000	7000	8000	9000	10,000
B	Size of square coil	3	4	5	6	7
C	(Length × Height) (mm)	1	1.5	2	2.5	3

represents the values of different parameters corresponding to their levels. The magnetic force, to be calculated, is considered a response parameter.

Table 5. DOE table presents various combinations and their corresponding magnetic forces.

Run Order	Current Density (A/m2)	Size of Coil (mm)	Gap (mm)	Magnetic Force (N)
1	9000	4	2.5	15,330.6
2	8000	3	2	6970.78
3	7000	4	1.5	9980.07
4	9000	6	2.5	33,973
5	8000	5	2	19,885.6
6	8000	7	2	39,507.2
7	8000	5	2	19,885.6
8	10,000	5	2	30,012.9
9	9000	6	1.5	36,666.5
10	8000	5	2	19,885.6
11	7000	6	2.5	20,551.6
12	7000	4	2.5	9274.09
13	8000	5	1	20,819.5
14	9000	4	1.5	16,497.7
15	7000	6	1.5	22,181
16	8000	5	2	19,885.6
17	8000	5	2	19,885.6
18	8000	5	2	19,885.6
19	8000	5	3	17,904.4
20	6000	5	2	10,804.6

presents the DOE table containing the required number of simulations and corresponding models. ANSYS 22R1 Emag is utilized to calculate the magnetic force for each combination of process parameters. The impact of three input process factors—current density, air gap, and coil size—on the response variable, i.e., a magnetic force, is determined by applying the Analysis of Variance (ANOVA) method.

The ANOVA results are presented in

Table 6. ANOVA table.

Source	DF	Adj SS	Adj MS	F-Value	p-Value
Model	9	1,469,079,237	163,231,026	2029.37	0.000
Linear	3	1,411,916,745	470,638,915	5851.23	0.000
Current Density (A/m2)—A	1	389,052,350	389,052,350	4836.90	0.000
Gap (mm)—B	1	9,039,313	9,039,313	112.38	0.000
Size of the coil (mm)—C	1	1,013,825,082	1,013,825,082	12,604.41	0.000
Square	3	26,734,812	8,911,604	110.79	0.000
A2	1	2,268,154	2,268,154	28.20	0.000
B2	1	37,561	37,561	0.47	0.510
C2	1	25,542,236	25,542,236	317.55	0.000
2-Way Interaction	3	30,427,680	10,142,560	126.10	0.000
AB	1	290,787	290,787	3.62	0.086
AC	1	29,386,691	29,386,691	365.35	0.000
BC	1	750,202	750,202	9.33	0.012
Error	10	804,342	80,434
Lack-of-Fit	5	804,342	160,868	*	*
Error	5	0	0
Total	19	1,469,883,579

. By conducting the ANOVA using the MINITAB software, a regression equation can be derived that depicts the relationship of the response parameter and the input parameters. This equation also illustrates the interrelationships between the model variables.

The p-value serves as a metric for gauging the importance of the results. Also, whether to “accept or reject” the hypothesis is determined using F-value. Where Adj SS—Adjusted Some of Squares, DF—Degrees of Freedom, Adj MS—Adjusted Mean Squares.

3.2. Regression Equation

By using MINITAB 19 software, a second-order polynomial equation (Equation (8)) is obtained which gives the relationship between the magnetic force and the corresponding values of current density, gap, and coil size for different samples.

Table 7. The DOE table presents magnetic forces generated for various combinations with error percentage.

Run Order	Current Density (A/m2)	Size of Coil (mm)	Gap (mm)	Magnetic Force (N)	Magnetic Force (N)	Error (%)
1	9000	4	2.5	15,330.6	14,833.08	3.24
2	8000	3	2	6970.78	7355.1	5.52
3	7000	4	1.5	9980.07	9870.1	1.1
4	9000	6	2.5	33,973	34,167.15	0.57
5	8000	5	2	19,885.6	19,251.5	0.03
6	8000	7	2	39,507.2	39,211.1	0.74
7	8000	5	2	19,885.6	19,251.5	0.03
8	10,000	5	2	30,012.9	30,317.5	1.01
9	9000	6	1.5	36,666.5	36,657.1	0.025
10	8000	5	2	19,885.6	19,251.5	0.03
11	7000	6	2.5	20,551.6	20,848.15	1.44
12	7000	4	2.5	9274.09	9366.15	0.99
13	8000	5	1	20,819.5	20,903.5	0.40
14	9000	4	1.5	16,497.7	16,283.15	1.29
15	7000	6	1.5	22,181	22,576.15	1.78
16	8000	5	2	19,885.6	19,251.5	0.03
17	8000	5	2	19,885.6	19,251.5	0.03
18	8000	5	2	19,885.6	19,251.5	0.03
19	8000	5	3	17,904.4	17,909.5	0.02
20	6000	5	2	10,804.6	10,585.5	2.02

shows the various combinations generated by the design of experiments and the corresponding magnetic force calculated. It also includes the error percentage between the magnetic force calculated from the regression equation and the Finite Element simulation (ANSYS E_mag).

$$\begin{array}{l}Magnetic Force=\mathrm{52,443}-8.69 A+3991 B-\mathrm{16,227} C+0.000300 A2+155 B2+1007.9 C2\\ -0.381 AB+1.917 AC-612 BC\end{array}$$

(8)

where, A = current density; B = gap; C = coil size

The numerical analysis results are compared using regression analysis to check the compatibility of a regression model. The error in magnetic force between the regression analysis and numerical analysis is found to be less than 6% which indicates the validity of the regression model.

Figure 8 illustrates how various parameters impact magnetic force. Figure 9 demonstrates the maximum magnetic force generated when the current density ranges from 8000 A/m² to 10,000 A/m² concerning the variation of coil size from 6 to 7 mm. Figure 10 displays contour plot, aids in visualizing the influence of current density, and air gap on magnetic force.

According to Figure 11, the highest magnetic force is produced when the coil size falls within the range of 5 to 7 mm.

3.3. Regression Equation

The study on response optimization is conducted using MINITAB 19 software which assists in selecting the best combination of variables. As shown in Figure 12, it optimizes either a single response or multiple responses. These optimized values are further validated through finite element analysis. The optimized combination of “10,000 A/m² current density, 1 mm air gap and 7 mm size square coil” yields a maximum magnetic force of 61,536 N, as shown in

Table 8. Response optimization results.

Variables	Setting
Current density (A/m2)	1000
Size of square coil (Length × Height) (mm)	1
Gap between coil and sheet (mm)	7

.

The magnetic force is calculated using ANSYS Emag for the optimized combination found by MINITAB, resulting in a value of 56,845 N. The error between the magnetic force obtained by the optimized combination in MINITAB (61,536 N) and the FE simulation (56,845 N) is 7%, demonstrating the validity of the optimized model as given in

Table 9. Validation of present finite element model with Asati et al. [24].

Response Variable	Fit	SE Fit	95% CI	95% PI
Magnetic Force	61,536	829	(59,689, 63,383)	(59,584, 63,488)

.

4. Conclusions

This study focuses on an electromagnetic forming (EMF) process simulation using the finite element method (FEM) to calculate the magnetic force generated. After validating the FEM model with the literature result, the influences of current density, gap, and coil size on magnetic force are investigated using the Response Surface Methodology (RSM) approach. The magnetic force is calculated for 20 different combinations of process parameters by using FEM. A regression equation through ANOVA is developed through these results which accurately predicts the magnetic force with less than 6% error. An optimization study is then performed to determine the optimal combination of process parameters for maximum magnetic force. The optimized model is confirmed with further FEM simulations, demonstrating good agreement (error percentage = 7%). Overall, this study provides insights into the EMF process and can aid in future designs.