Exploring Electromagnetic Tube Expansion Based on Field Concentrator: An Analytical Model and Numerical Study

Abstract

This paper delves into the realm of Electromagnetic forming (EMF), a high-speed forming process leveraging pulsed magnetic fields generated via capacitor bank discharge through a coil. Traditionally, numerical analyses of EMF processes focused on electromagnetic tube expansion and the influence of forming conditions. However, in high-energy electromagnetic setups, the magnetic field concentrator adds complexity, transitioning analyses into a 3D modeling domain. This paper consolidates an analysis and numerical investigation inspired by the Muhlbauer approach to elucidate the impact of the field concentrator on electromagnetically expanded tubes. Employing an enhanced finite element model for magnetic tube expansion simulations. This research sheds light on how this additional component influences the EMF process, offering valuable insights into modeling electromagnetic bulging when incorporating the field concentrator.

Index Terms Electromagnetic Forming; Field Concentrator; Finite Element Modeling; Tube Expansion.

I. INTRODUCTION

Nowadays, several natural phenomena and engineering procedures are comprehensively elucidated through the lens of multi-physical models. One such interesting domain is Electromagnetic Metal Forming (EMF), a high-speed forming process in which magnetic pressure is generated (e.g., [¹]) when a coil adjacent to the workpiece is excited by the discharging current of a capacitor bank. This incorporation of electromagnetic principles and metalworking techniques, as described in [¹], encapsulates the intricate association between electromagnetic forces and industrial applications, which demonstrates a dynamic interplay that shapes the foundation of advanced manufacturing methodologies (e.g., [²]-[⁵]).

Although EMF offers certain advantages over other forming methods such as increased formability, reduced wrinkling, and reduced toolmaking costs. From the active literature (e.g., [⁶]-[¹⁴]), dynamic analyses for axisymmetric electromagnetic forming systems, such as the expansion of tubular conductors, were presented. Coupled finite element models (e.g., [¹⁵]-[¹⁸]) also numerically investigated the effect of sample size and forming conditions on the EMF process. However, in high-energy electromagnetic forming processes (e.g., [¹⁹], [²⁰]), system setup often includes an additional component, which consists of a magnetic field concentrator, which is placed between the coil and the workpiece [²¹]-[²⁴].

In fact, a field concentrator or field shaper is an important tool in the electromagnetic forming process, it is used to concentrate the magnetic field (e.g., [²¹]-[²⁴]). Its design and manufacturing are easier, more effective, and quicker than manufacturing only the coil [²²]-[²⁴]. However, this additional component significantly complicates the analysis and the modeling of the process due to coil-concentrator and concentrator-workpiece double coupling [²²].

Following these considerations, the purpose of this paper is to show the effect of field concentrators on EMF systems such as the electromagnetic bulging of tubular workpieces. It presents the analytical model, inspired by the Muhlbauer one for induction furnace with a cold crucible [²⁵], developed to simulate the introduction of the concentrator.

A. Paper Key Contributions

In this paper, we introduce and extensively delve into the intricate details of our numerical investigation and simulation, focusing on unraveling the impact of the field shaper on the process of tube expansion. Overall, in this paper, we make the following contributions.

• we initiate by presenting a comprehensive overview of the context, background, and motivation that inspired us to conduct this research;

• we also provide a detailed state-of-the-art analysis, which allows us to provide a comprehensive vision of the recent and pertinent work related to our research;

• we illustrate the fundamental principle of electromagnetic tube expansion based on a field concentrator, as well as introduce the mathematical models essential for achieving our numerical investigation;

• we conduct a rigorous numerical evaluation and analysis of the significant effect and impact of the field shaper to ensure the effectiveness and reliability of the proposed methods in handling and analyzing the process of tube expansion.

Moreover, this paper serves as a substantial extension of our earlier work [²⁶]. While the previous short paper provided a brief introduction to tube expansion with a focus on the field concentrator, this current study significantly surpasses those contributions. Over an in-depth numerical investigation and comprehensive modeling approach, we delve into the intricacies of electromagnetic tube expansion phenomena. Specifically, the 2D mathematical and analytical models discussed in this paper are crucial for optimizing the design of magnetic forming systems that leverage field concentrators. These models can enhance the industrial applications of magnetic tube expansion setups, including the specific device examined here. Additionally, the electromagnetic and mechanical transparency of such systems strongly depends on the design of the field concentrator. Key factors such as the slot width and coil length play a significant role in determining the system’s effectiveness.

Through this paper, we aim to provide a forward-looking perspective on the convergence of high-speed methods and Electromagnetic industrial applications, our proposed approach is designed to be both versatile and future-proof, accommodating emerging trends and technologies in the rapidly evolving landscape of Electromagnetic methods.

B. Paper Organizations

The remaining part of this paper is organized as follows. Section II focuses the attention on analyzing relevant related work to our research. In Section III, we present the fundamental principle of electromagnetic tube expansion based on a field concentrator. As well as introducing the mathematical models essential for achieving our numerical investigation. After that, in Section IV, we report our extensive applications and the obtained results. Finally, Section V provides conclusions and future work on our research.

II. RELATED WORK

Recently, the process of electromagnetic tube expansion has garnered significant attention from researchers. This Section aims to highlight some of the most appropriate and relevant works related to our research.

[²⁷] introduces a novel method, named Electromagnetic Tube Expansion with Axial Compression (EMTEAC), to address the challenge of significant wall thickness reduction encountered in conventional electromagnetic tube expansion. In EMTEAC, additional coils positioned at each end of the tube generate axial electromagnetic force, complementing the radial force from the driving coil. Finite element analysis is employed to assess the distribution of magnetic flux density and electromagnetic force from the three coils. Simulation results reveal that the axial electromagnetic force in EMTEAC is approximately seven times greater than that in conventional EMTE, enhancing material flow during tube expansion.

In [²⁸], the authors address the challenging task of achieving precise control over the workpiece profile in electromagnetic forming and propose a new dual-coil electromagnetic tube forming method. The innovation lies in implementing automatic feedback control of Lorentz force distribution to achieve highly uniform deformation of tubes. Through a series of experiments and simulations, the effectiveness of this approach is demonstrated, highlighting the impact of key electromagnetic parameters on the deformation behavior of AA6061-O aluminum alloy tubes. The proposed method dynamically adjusts the radial Lorentz force distribution, resulting in significantly improved tube deformation uniformity, approximately 2.7 times better than conventional single-coil electromagnetic forming.

In [²⁹], a comprehensive analysis was conducted to model a multi-turn coil, a field shaper, and an Al6061-T6 tubular specimen, employing a 3D electromagnetic-mechanical approach. The primary goal was to predict the velocity and displacement of the tubular specimen during the manufacturing process. The model incorporated magnetic pressure applied to the specimen, predicting its deformation based on two charging energies. To assess the accuracy of the numerical predictions, experimental tests were carried out, measuring both velocity and displacement using Photon Doppler Velocimetry (PDV). The comparative analysis between numerical and experimental results revealed a substantial agreement, demonstrating the capability of the numerical simulations to effectively capture the intricacies of this complex manufacturing process.

[³⁰] addresses the challenge of implementing electromagnetic attractive forming, a process with significant potential for expanding applications in electromagnetic forming. The obstacle arises from the prevalent dominance of electromagnetic repulsive forces in conventional RLC circuits during the forming process. To overcome this limitation, the research proposes a new discharge circuit topology capable of generating dual-frequency current, thereby substantially increasing electromagnetic attractive forces. Notably, the proposed method achieves this with the use of only a single coil and a single power supply. Both numerical simulations and experimental results validate the efficacy of the novel circuit, highlighting its superior performance in attractive forming compared to existing methods. The developed forming system is demonstrated to successfully apply attractive forming to AA6061-O aluminum alloy tubes, even in the presence of an inner solid die, where the generated attractive force surpasses the repulsive force.

The purpose of [³¹] is to describe a significant contribution in the field of EMF, focusing on enhancing efficiency by addressing energy losses in the process. The key contribution lies in the exploration of die material impact on the deformation of an aluminum tube during EMF. By utilizing two dies with the same geometry but different materials (nylon and steel), the study reveals a noteworthy 5% increase in deformation with the nylon die compared to the steel die at the same discharge energy. This finding suggests that the die material plays a crucial role in the efficiency of the EMF process. Additionally, the study employs finite element simulations, demonstrating a strong agreement of 94% between simulated and experimentally observed results.

[³²] discusses the development of a concise and accurate theoretical model that serves as a link between electrical and thermal data in the electromagnetic tube-forming process. The proposed model offers a faster estimation of workpiece temperature compared to finite element or experimental models. The research focuses on constructing a one-dimensional theoretical model of a semi-coupled electrothermal field for the analysis of electromagnetic tube forming. This model integrates electromagnetic and thermomechanical sub-models, emphasizing a thorough study of crucial process parameters and their interplay. To assess the effectiveness of the proposed approach, the results are compared with those obtained from the finite element method-ANSYS tool and experimental data.

[³³] explores the effects of inner diameter (ID), outer diameter (OD), effective number of turns, and cable connection on the coil's behavior. The study quantifies coil performance through changes in inductance, resistance, current pulse, and tube deformation. Experimental determination and analysis of variations in coil inductance and resistance concerning alterations in ID, OD, and cable connection are presented. The study establishes correlations between the coil's performance, theoretically and experimentally determined current pulse and its overall impact on system inductance. The findings reveal that slight adjustments in diameter and effective turns can result in approximately 30kA changes in current pulse amplitude. Furthermore, the introduction of an additional parallel discharge cable enhances current and frequency by approximately 26% and 23%, respectively.

In [³⁴], they focus on addressing a common issue in conventional electromagnetic tube expansion, specifically the inhomogeneous tube deformation along the axial direction caused by end effects generated by the traditional helix coil. The work proposes a novel solution to this problem by introducing a new concave coil structure to replace the industry-standard helix coil, aiming to enhance the radial electromagnetic force distribution on the tube. The anticipated outcome is an improvement in the axial uniformity of tube deformation. To assess and quantify this improvement, the study introduces a new criterion for deformation uniformity (R-L criterion). Additionally, an electromagnetic-structural coupling finite element model is developed to explore the relationship between the distribution of electromagnetic force generated by the concave coil and the tube's uniformity under varying voltage levels. The effectiveness of the proposed method is validated through a comprehensive series of experimental and simulation analyses.

[³⁵] provides a comprehensive review of electromagnetic forming, by covering fundamental research on process principles, key parameters, workpiece deformations, and energy transfer. It also explores application-oriented research in forming, joining, cutting, and process combinations, including EMF integration into conventional technologies. The discussion extends to material behavior at high strain rates and the equipment used for electromagnetic forming. Despite EMF’s potential for extending forming limits in comparison to quasistatic processes, the research critically examines the limited adoption in industrial manufacturing and identifies areas for further research.

In [³⁶], they introduce a novel method called electromagnetic incremental forming (EMIF) to overcome the limitations of shaping large parts using conventional electromagnetic forming techniques. Leveraging the principle of single-point incremental forming, EMIF utilizes a small coil and discharge energy to induce local deformations in the workpiece at high speed, with these deformations accumulating to shape larger parts. The research investigates the impact factors of processing parameters, including discharge voltage, vent hole, discharging times in a fixed position, and the number of discharge regions, on the final shape of AA3003 aluminum alloy parts in electromagnetic incremental sheet forming. Experimental and simulation results collectively demonstrate the feasibility and effectiveness of this technology in producing large parts.

III. NUMERICAL MODELING

In this Section, we present the fundamental principle of electromagnetic tube expansion based on field concentrator. As well as introducing the mathematical models essential for achieving our numerical investigation.

In general, a larger electromagnetic (EM) pressure can be generated when the working surface of the inductor is formed from a monolithic block of a metal of high hardness and high conductivity. So, large pressures can often be created with inducers mono winding, but because of their low inductance, they are often quite inefficient. Field concentrators can be used to develop strong pressure while being able to expand and adjust the inductance of the coil.

A typical inductor multi-turn is still used to create the magnetic field. It is coupled to an inductor called secondary field shaping. As described schematically in Fig. 1(a), if properly designed, the entire stream flow created by the primary current can be transferred through the slot shaper. When the electrical power available in an EMF setup is not sufficient to develop electromagnetic forces capable of causing wanted deformations, it is often of interest to use a field shaper between the forming coil and the workpiece. The field concentrator (see Fig. 1(b)) allows, besides increasing the concentration of magnetic field by the concentration of coil current, a particular distribution of electromagnetic forces, the use of a single coil to deform parts of different diameters, and the reduction of EM forces on the body of the coil.

Fig. 1
Field Concentrator (a) Top View and Side View (b) Construction of Field Shaper and Tube Compression Setup (c) Eddy Currents in a Field Shaper for Tube Expansion [³⁵].

For a tube expansion, as shown in Fig. 1(c), only the inner surface of the concentrator is driven by the eddy currents created by the inductor. Once these currents reach the isolated slot, they come to loop through the outer surface. Therefore, we are dealing with a transformer whose primary is the coil and secondary is the concentrator. Due to the presence of the slot, the forming system with a field concentrator does not present symmetry and the problem becomes 3D. Therefore, it is necessary to develop a simplified model.

To study the influence of the field concentrator on the EMF systems, we have developed an analytical model, inspired by the Muhlbauer one [²⁵] for an induction furnace with a cold crucible. Since magnetic forming systems use high-frequency discharge currents, the penetration depth of the electromagnetic field $\delta =1/\left(2.\pi .f.{\mu }_0.\sigma \right)$ is relatively low. The marked skin effect leads to a concentration of the induced currents at the surface of the shaper, the current density J and the magnetic induction B tend towards zero in the core.

A. Assumptions

The current flowing in the coil is essentially azimuthal and generates an axial magnetic field in the central part of the coil. This magnetic field induces azimuthal currents, in the inner and outer cylindrical walls of the shaper. Because of the skin effect, these currents can be expressed by their azimuthal linear densities, i_c, which are different on the internal and external walls and we designate by i_c, the linear density of the currents circulating on the walls of the slot (see Fig. 1(c)).

Thereby, to calculate the current on the concentrator, based on the work of [²⁵], we suppose the geometrical conditions presented in Equation (1):

(1) $$\frac{C}{R_{cm}}\ll 1\dots \dots \dots \frac{e}{C}\ll 1$$

where C is the concentrator thickness, e is the slot width, and R_cm is the concentrator's average radius.

Considering a section of the concentrator of height z, we can consider two current loops: $i_c^{ext}$ and $i_c^{int}$. These assumptions allow us to introduce an effective superficial linear current density in the concentrator and to neglect the effect of axial currents. So, the problem becomes axisymmetric.

(2) $$i_c^{eff}\left(z\right)=i_c^{ext}\left(z\right)+i_c^{int}\left(z\right)$$

From the Ampere theorem, current and flow conservation laws presented in Fig. 2(a), we obtain the effective current expression presented in Equation (3):

Fig. 2
Elementary Volume of the Concentrator (a) - Representation of an Inductive Turn (b).

(3) $$i_c^{eff}=\frac{K}{{\mu }_0}\frac{{\partial }^{2}A}{\partial z^2}$$

such as K is defined as the concentrator geometry characteristic parameter, proportional to its thickness and inversely proportional to the width of its slot, presented in Equation (4).

(4) $$K=\frac{2\pi R_{c}C}{e}$$

where R_c is the concentrator's external radius.

Thus, we can use a 2D numerical model for the EMF systems with the field concentrator taking into account the new eddy current on the magnetic shaper. This effective current is calculated analytically based on the Biot-Savart law.

B. Analytical Model

From Biot-Savart law, the magnetic vector potential created on a point $P\left(r,z\right)$, by a turn of radius R and height Z (see Fig. 2(b)), through which flows a current I, is given by:

(5) $$\overrightarrow{A}=\frac{{\mu }_{0}I}{4\pi }\int \frac{\overrightarrow{dl}}{\left|\overrightarrow{P}-{\overrightarrow{P}}_0\right|}$$

In axisymmetric, the vector potential has only one azimuthal component [²⁵], which is written in Equation (6):

(6) $$A_{\theta }=\frac{{\mu }_{0}I}{2\pi }\sqrt{\frac{R}{r}}\left[\frac{\left(2-p^2\right)E_1\left(p\right)-2E_2\left(p\right)}{p}\right]$$

(7) $$p^2=\frac{4rR}{{\left(R+r\right)}^2+{\left(Z-z\right)}^2}$$

where 0 < p < 1, r and z are the coordinates of the considered point, R, and Z represent the coordinates of the current turn and E₁, E₂ represent the elliptic Legendre integrals of the first and second kind, respectively and I is the lineic current density.

(9) $$E_1\left(p\right)={\int }_0^{\frac{\pi }{2}}\frac{d\theta }{\sqrt{1-p^2{\sin }^2\theta }}$$

(9) $$E_2\left(p\right)={\int }_0^{\frac{\pi }{2}}\sqrt{1-p^2{\sin }^2\theta }d\theta$$

Considering that p is very low the calculation of these integrals is done using the following developments, which are inspired from [²⁵]:

(10) $$E_1\left(p\right)=\frac{\pi }{2}\left[1+2\left(\frac{p^2}{8}\right)+9{\left(\frac{p^2}{8}\right)}^2+\cdots \cdots \right]$$

(11) $$E_2\left(p\right)=\frac{\pi }{2}\left[1+2\left(\frac{p^2}{8}\right)-3{\left(\frac{p^2}{8}\right)}^2+\cdots \cdots \right]$$

We take as interpolations:

(12) $$E_1\left(p\right)=1.3862944+0.1119723\left(1-p^2\right)+0.0725296{\left(1-p^2\right)}^2+\left[0.5+0.1213478\left(1-p^2\right)+0.0288729{\left(1-p^2\right)}^2\right]Log\frac{1}{\left(1-p^2\right)}$$

(13) $$E_2\left(p\right)=1+0.4630151\left(1-p^2\right)+0.1077812{\left(1-p^2\right)}^2+\left[0.245272\left(1-p^2\right)+0.0412496{\left(1-p^2\right)}^2\right]Log\frac{1}{\left(1-p^2\right)}$$

with:

(14) $$\frac{\partial p}{\partial z}=-\left[\frac{\left(z-Z\right)p^3}{4rR}\right]$$

(15) $$\frac{\partial p}{\partial r}=\frac{p}{2r}-\frac{p^3}{4}\left[\frac{r+R}{rR}\right]$$

(16) $$\frac{\partial E_1\left(p\right)}{\partial p}=\frac{1}{p}\left[\frac{E_2\left(p\right)}{1-p^2}-E_1\left(p\right)\right]$$

(17) $$\frac{\partial E_2\left(p\right)}{\partial p}=\frac{1}{p}\left[E_2\left(p\right)-E_1\left(p\right)\right]$$

As a consequence, by substituting the magnetic vector potential and its derivatives with their corresponding expressions, we obtain the analytical expressions of the eddy current, $i_c^{eff}$, in the field concentrator as follows:

(18) $$i_c^{eff}=\frac{KI}{4\pi r\sqrt{rR}}\left[F_1{F}_2+p\left(z-Z\right)\left[F_3+F_4{E}_2\left(p\right)+F_5{F}_6\right]\right]$$

with:

(19) $$F_1=\left[-\frac{{\left(z-Z\right)}^2{p}^3}{4rR}+p\right]$$

(20) $$F_2=\left[-E_1\left(p\right)+\left(\frac{R^2+r^2+{\left(z-Z\right)}^2}{{\left(R-r\right)}^2+{\left(z-Z\right)}^2}\right)E_2\left(p\right)\right]$$

(21) $$F_3=\frac{\left(z-Z\right)p^2}{4rR}\left[\frac{E_2\left(p\right)}{1-p^2}-E_1\left(p\right)\right]$$

(22) $$F_4=\left(\frac{-4rR\left(z-Z\right)}{{\left[{\left(R-r\right)}^2+{\left(z-Z\right)}^2\right]}^2}\right)$$

(23) $$F_5=-\frac{\left(z-Z\right)p^2}{4rR}\left[E_2\left(p\right)-E_1\left(p\right)\right]$$

(24) $$F_6=\left(\frac{R^2+r^2+{\left(z-Z\right)}^2}{{\left(R-r\right)}^2+{\left(z-Z\right)}^2}\right)$$

C. Numerical Model

In this electromagnetic tube expansion process, a massif tool coil is placed inside of the metal tube, and an intense transient magnetic field, which is achieved by a pulse current generator, is employed to deform electrically conductive material tubes. A typical EMF system consists of a bank of capacitors, a tube as a workpiece, and a coil. The energy stored in the capacitor bank is discharged through the coil in an alternative, damped electric current, given by Equation (25).

(25) $$I\left(t\right)=V_0\sqrt{\frac{C}{L}}{e}^{-\xi \omega t}.sin\left(2\pi ft\right)$$

where, I(t) is capacitor bank discharging current, V₀ is the charging voltage of capacitors, ξ is the damping term given by: $\xi =\frac{1}{2}R\sqrt{\frac{C}{L}}$ and ω represents the natural frequency given by: $\omega =\sqrt{\frac{1}{LC}}$.

Furthermore, Fig. 3 illustrates the external current density, used within our numerical model, derived from discharging circuit, whose parameters are $R=25.5m\text{Ω}$, $L=2\mu H$, $C=40\mu F$ and $V_0=6kV$.

Fig. 3
External Current Density.

Additionally, a rigorous analysis of electromagnetic tube expansion can be represented as a Multiphysics problem coupling electromagnetic and mechanical phenomena, governed by Equation (26) and Equation (27):

(26) $$\sigma \frac{\partial A}{\partial t}+curl\left(\frac{1}{\mu }.curlA\right)-\sigma .\nu .\left(curlA\right)=J_{ex}$$

where A is the magnetic vector potential, J_ex denotes the current density in the coil, μ represents the magnetic permeability, σ is the electrical conductivity, v is the velocity of the tube, and curl A is the vector operator that describes the circulation (i.e., the movement) of magnetic vector potential A.

(27) $$\rho \frac{{\partial }^{2}u}{d{t}^2}-\nabla .\text{Σ}=F$$

where ρ is density, u represents the displacement vector, Σ is the stress tensor, and F is the electromagnetic force density.

These two equations are solved using the finite element method and are strongly coupled via COMSOL Multiphysics software to determine the electromagnetic and mechanical quantities, primarily tube deformation (in our case, tube expansion).

Moreover, it should be noted that due to the presence of a slit at the field concentrator, the numerical modeling of tube expansion in the presence of this field concentrator must be 3D, despite the axial symmetry of the setup. This approach is more computationally intensive and costly in terms of implementation.

To address this, we have developed an analytical mathematical model to calculate the current at the field concentrator, which will be used as the excitation current for the 2D model of tube expansion.

IV. RESULTS AND DISCUSSION

In this Section, we delve deeper into the numerical simulations of electromagnetic tube expansion applications performed by leveraging the robust capabilities of the COMSOL MULTIPHYSICS software-a powerful tool enabling comprehensive coupling of electromagnetic, mechanical, and thermal systems. This integration facilitated a holistic approach, allowing us to analyze and comprehend the intricate interplay among these fundamental aspects. In this endeavor, our primary focus is mainly directed toward pioneering advancements within industrial processes, with a keen emphasis on the realm of electromagnetic tube expansion techniques.

Our goal is to explore, refine, and innovate upon these widely utilized methods, pushing the boundaries of their efficiency, precision, and applicability within various industrial domains. Through our concentrated efforts, we aspire to unlock novel possibilities and elevate the standards of electromagnetic tube expansion, catering to the evolving needs of modern industries while fostering advancements in manufacturing practices.

In addition, our research examined the consequences of integrating a field concentrator into the magnetic forming installation. By employing the proposed numerical model, we meticulously assessed the impact on the tube expansion system. Within this system, an aluminum tube with the same coil length was utilized, coupled with the incorporation of a copper-based field concentrator, as visually depicted in Fig. 4.

Fig. 4
Tube Bulging Setup with Field Concentrator.

This strategic incorporation aimed to investigate the altered dynamics and enhanced efficiency induced by the introduction of the concentrator within this specific setup. Moreover, the basic dimensions of the forming system, which are similar to the experimental setup employed in previous work (e.g., [³⁷]), serve as the basis for our current investigation. Notably, this system is augmented by the inclusion of a field concentrator and maintains consistency with the parameters outlined in Table I. Building upon the established framework from previous experimentation, the addition of the field concentrator stands as a pivotal modification within this pre-existing setup, warranting a comprehensive analysis to discern its influence on the system dynamics and performance.

TABLE I
SYSTEM PARAMETERS

Parameter	Value
Coil Length - Tube Length - Concentrator Length	40 mm
Coil Radius	12.4 mm
Tube Inner Radius	15 mm
Concentrator Depth	1.4 mm
Concentrator Inner Radius	13 mm
Slot Width	0.18 mm

For what concerns the boundary conditions of our numerical model, for the electromagnetic problem, we assume that the magnetic vector potential is zero at the level of the adopted air box i.e., A = 0.

On the other hand, for the mechanical problem, and in order to ensure the plastic expansion of the tube, we consider that the tube is fixed at the upper and lower edges, which can be formulated as follows: u = 0 for: z = 0 and z = 0.04 m.

A. Effect of the Concentrator Characteristic Parameter

The utilization of this quasi-3D model has provided an invaluable opportunity to investigate the impact of geometric parameters on both electromagnetic and mechanical attributes, specifically focusing on tube deformation. As illustrated in Fig. 5(a) and Fig. 5(b), the variations in magnetic vector potential and eddy current density, respectively showcase a significant reversal induced by the field concentrator upon the tube undergoing deformation. As theoretically anticipated, this reversal becomes evident as the concentrator effectively alters the direction and distribution of the magnetic potential and eddy currents within the deforming tube. Such findings align with theoretical expectations, reaffirming the influential role played by the field concentrator in shaping the electromagnetic deformation process.

Fig. 5
Magnetic Potential at the Tube Center (a) - Variations of Eddy Current (b).

On the other hand, the magnetic field (as depicted in Fig. 5(a)), the eddy currents (illustrated in Fig. 5(b)), and the magnetic forces (as presented in Fig. 6) exhibit a marked increase corresponding to the augmentation of parameter K.

Fig. 6
Variations of Magnetic Force.

This increase in these magnetic aspects aligns closely with both theoretical predictions and experimental validations in previous studies (e.g., [²¹]-[²⁴]). The observed correlation between the parameter K and the amplified magnetic field, eddy currents, and magnetic forces further substantiates and aligns with the established theoretical frameworks and empirical evidence from prior research endeavors. In addition, an intriguing observation emerges when K attains a value of $0.253$, the magnetic field, eddy current, and consequently, the magnetic force reaches a null state. This intriguing outcome suggests that at this specific parameter value, the field concentrator assumes a role akin to that of a magnetic screen.

As shown in Fig. 7(a) and Fig. 7(b), the magnitude of tube deformation notably intensifies as the values of K increase (i.e., low slot widths). This observation aligns perfectly with previous research findings (e.g., [²¹]-[²⁴]), affirming that the presence of the concentrator enables the attainment of augmented deformation without a proportional escalation in the energy input within the setup.

Fig. 7
Tube Deformation at t = 100µs (a) - Displacement at the Tube Center (b).

Nevertheless, it is important to note that for lower values of K, corresponding to greater slot widths, the resultant tube bulging appears to be comparatively reduced. This particular trend sheds light on the inherent limitations of this model, delineating a threshold where the efficacy of the concentrator in facilitating substantial deformation encounters constraints. Consequently, this highlights the nuanced interplay between the concentrator parameters and the resulting deformation, underscoring the model limitations that are based on assumptions, previously presented (see Section III-A) and its capacity to accurately predict outcomes across various parameter ranges.

Therefore, we have highlighted the determining influence of the K parameter of the concentrator (slot width of concentrator) on the magnetic field, the induced currents, and the magnetic forces transmitted to the tube and consequently its deformation. We have also revealed the limits of the developed mathematical model.

B. Effect of the Coil Length

In this second application, in order to study the effect of the coil length, we have considered two cases: A coil shorter than the field concentrator with $L=26.1mm$ and another longer with $L=53.94mm$. We can conclude that the effect of the field concentrator is greater in the case when the coil and the concentrator have the same length, particularly for greater values of K. In the opposite case (i.e., low values of K and great slot width), the screen effect is accentuated in the case of the short coil, which is presented in Fig. 8(a).

Fig. 8
Tube Deformation at $t=300\mu s$ for (a) Short Coil - (b) Long Coil.

However, in the case of a coil longer than the field shaper, the tube bulging is accentuated in the area in front of the field concentrator, particularly for greater values of the field geometrical parameter K, lower values of slot width (see Fig. 8(b)).

Hence, the application of the semi-analytical model and the results obtained showed the significant electromagnetic screening effect provided by the concentrator, for low values of the characteristic factor K. This effect decreases for high values of this parameter where the concentrator, in this case, allows to increase the electromagnetic forces acting on the deforming tube and consequently its expansion is accentuated. Furthermore, we have highlighted the effect of the coil length and its influence on the tube deformation in the presence of the field concentrator.

Finally, as a result, derived from our analysis, we can confirm that the introduction of the field concentrator in electromagnetic forming setups (e.g., tube expansion) increases their efficiency.

C. Discussions

In this Section, we provide a detailed comparison between our proposed approach and the model of Muhlbauer [²⁵], which proves the effectiveness of our approach.

Muhlbauer’s work involves the development of a simplified quasi-three-dimensional semi-analytical model that facilitates the study and calculation of electromagnetic characteristics and quantities for installations with metallic cooling systems, such as induction furnaces with cold crucibles. This model describes the action of the metallic system, resulting in a surface current, which reduces the problem size and lowers calculation time and costs. Additionally, Muhlbauer et al. demonstrated the electromagnetic shielding effect of the cooling system, which mainly depends on a characteristic parameter proportional to the insulation width.

On the other hand, our numerical representation of electromagnetic forming processes, particularly tube expansion, occurring in electromechanical installations with field shapers requires complex three-dimensional mathematical models due to unsteady phenomena, system nonlinearity, and complex geometries-especially with purely three-dimensional field concentrator elements that demand intricate and costly numerical analyses.

However, it is possible to develop simplified models that study the most important characteristics, including electromagnetic and mechanical quantities, while reducing the necessary calculations. This is exemplified by the simplified model, which combines semi-analytical and two-dimensional numerical approaches with axial symmetry, developed in this work for magnetic tube expansion installations with field concentrators.

Our research highlights the decisive influence of the field concentrator on the magnetic tube expansion process. This influence is manifested through the effect of its characteristic parameter K (based on slot width) on the electromagnetic field distribution, induced currents in the workpiece, transmitted magnetic pressure, and plastic tube expansion. Furthermore, practical studies using this model address the effects of the characteristic parameter and the length of the inductor. Calculations performed with this simplified model are fast and cost-effective, making it advantageous for designing field concentrator installations and predicting electromagnetic and mechanical quantities in magnetic tube expansion systems, thereby introducing a highly recommended element of importance.

V. CONCLUSIONS AND FUTURE WORK

In this paper, the investigation conducted underscores the pivotal role of field shapers in tube expansion processes through a combined approach of analytical modeling and numerical simulation. Our findings illuminate the pronounced impact of field concentrators particularly notable for lower characteristic factor values. As the factor K increases, the magnetic forces intensify with the concentrator, which amplifies the tube bulging. Moreover, our exploration of coil length has revealed its significant influence on tube deformation, particularly in conjunction with the field concentrator, thereby enhancing the overall efficiency of electromagnetic forming setups. These insights not only contribute to a deeper understanding of the interplay between field shapers and tube expansion but also offer potential pathways for optimizing electromagnetic forming processes in industrial applications.

Future works are mainly oriented through investigating the efficiency of magnetic forming processes with different field shapers, as well as exploring other electromagnetic forming trends (e.g., [³⁸]-[⁴¹]).

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Publication Dates

• Publication in this collection
  09 Dec 2024
• Date of issue
  2024

History

• Received
  04 Feb 2024
• Reviewed
  14 Apr 2024
• Accepted
  10 Nov 2024