1. Introduction
In recent decades, high-temperature superconductors (HTSs) display high potential for applications in high-field superconducting magnets. Owing to the excellent electromagnetic characteristics of (Re)Ba2Cu3O7-x (REBCO) coated conductors (CCs), the HTSs have gained a lot of attention in accelerators, NMR, MRI, and fusion devices [1], [2], [3], [4], [5]. Recently, the ultra-high-field magnets above 30 T with REBCO coils were fabricated and demonstrated successfully [6], [7], [8]. The screening currents were induced in the superconducting magnets under the external field [9], [10], [11]. Meanwhile, a large mechanical stress and strain is generated in superconducting magnets because of electromagnetic force, which has reference to screening current and magnetic field. Nowadays, numerous magnets have developed by leaps and bounds, a huge mechanical deformation is inevitable, which can cause the rotation of REBCO CC and the critical current's decline. Furthermore, the REBCO CCs can even be damaged due to substantial electromagnetic force and thermal stress [12], [13], [14], [15]. Therefore, the analysis of stresses and strains in high-temperature superconducting coils is crucial to guarantee their stable operation and service life. By improving an understanding of the mechanical stresses in the coil, the coil structure can be optimized to enhance its reliability and performance.
The superconducting coil is exposed to complicated stresses including the winding stress, thermal stress and electromagnetic force for the duration of operation. Utilizing analytical and numerical approaches, the stresses owing to different processes in the superconducting coil are discussed systematically [16], [17], [18], [19], [20]. A larger electromagnetic force in the superconducting coil can generate significant mechanical deformation in REBCO tape. It was reported that the maximum stress can reach several hundreds of MPa owing to the screening current. The mechanical deformation obtained by the contact model is higher in the coils compared to the bulk model [12]. Due to a lower delamination strength, the interfacial failures have been observed and studied in the REBCO tape [21], [22], which restrict its application in the high-field strength cases. Excessive stresses and strains can cause the degradation of critical current in the superconducting coil [23]. In addition, the delamination or damage of superconducting layer may also appear in the REBCO tape for a larger mechanical stress [24], which can affect the safety and reliability of superconducting magnet. In addition, the coupled electromechanical behavior deserves careful consideration in the development of superconducting coil. In the early stages, pioneering researches have investigated the magneto-mechanical coupling effect of low-temperature superconductors [25], [26], [27]. Subsequently, owing to the superior current-carrying capability and high mechanical strength of REBCO tapes, they have received widespread attention and related coupling models have been proposed. For one thing, the rotation of the REBCO tape alters the normal and parallel fields, which leads to the redistribution of tape's current density [28]. For another, the critical current density $J_c$ of tape will degrade with the mechanical strain and become unrecoverable beyond a particular strain limit [29], [30]. In the high-field magnet experiments, the post analysis found a wavy deformation on the side of the conductor owing to the screening current effect, so plastic deformation cannot be ignored in the numerical analysis of mechanical deformation [7], [31]. Based on the impact of strain on the superconductivity, the FLOSSS model was proposed to consider the mechanical behaviours in HTS coils [32]. The results show that the mechanical strain obtained by the coupled model can reduce the error between the simulation and experimental outcomes [33], [34]. In the racetrack coil, the mechanical stress shows the discernible difference between the straight part and circular part [35]. In the recent studies, the impact of turns' separation on the contact resistance was discussed considering a coil without insulation and using the simplified coupled electromechanical model [36]. The electromagnetic response of a pancake coil in the magnetic fields of high strength was studied with equivalent circuit model, where the effect of rotation of REBCO tape is considered [37], [38]. In addition, some researchers have reported the mechanical response in the high-field coil using the connection between the $J_c$ and strain [39], [40].
The mesh movement due to small deformation is neglected during the numerical simulation in the above work. However, the impact of mesh movement will become significant for a larger deformation case owing to the variation of magnetic field distribution arise from movement in geometric configuration. In this paper, a coupled electromagnetic-mechanical model using H-formulation is proposed to investigate the mechanical response of REBCO coil. Particularly, the effects of coil position and overshoot method with plateau are studied. This work includes four main sections. In Section 2, the electromagnetic and mechanical equations of the coupled model and arbitrary Lagrangian-Eulerian (ALE) method are described. In Section 3, the distributions of current density are discussed, and maximum hoop stress and strain are compared. To reduce the mechanical stress induced due to screening current, the overshoot methods with plateau are used. Afterwards, the electromechanical behavior of two-tapes co-winding coil is investigated. And in the last part of the paper, the conclusions of this work are outlined.
2. Electromechanical model with H-formulation
2.1. Kinematics and mechanical equations
As the continuum body undergoes a mechanical deformation, the sample will change from an unaltered configuration to a distorted one. To describe the motion of a continuum medium, the Lagrange description defined under the reference configuration and the Eulerian description defined under the current configuration are adopted. In the reference configuration, the material points are labeled as vector $\boldsymbol{X}$ in the material reference frame. Similarly, in the current configuration, they are labeled as vector $\boldsymbol{x}$ in the space reference frame. The gradient tensor of deformation is $\mathbf{F}=\frac{\partial \mathbf{x}}{\partial \mathbf{X}}$.
For the mechanical behavior of the continuum body, $\sigma$ and $\mathbf{f}_{\mathrm{b}}$ are the Cauchy stress and body force defined in the current configuration, and $\mathbf{S}$ and $\widetilde{\mathbf{f}}_{\mathrm{b}}$ is the second Piola-Kirchoff stress (i.e., the nominal stress) and body force defined in the current configuration, respectively. The transformations of the stress and body force follow,
$ \sigma=J_{\mathrm{d}}^{-1} \mathbf{S} \mathbf{F}^{\mathrm{T}}, \mathbf{f}_{\mathrm{b}}=J_{\mathrm{d}}^{-1} \tilde{\mathbf{f}}_{\mathrm{b}}$
where Jd is the determinant of $\mathbf{F}$.
The strain defined in the reference configuration under the Lagrangian description is $\varepsilon_{\mathrm{G}}$. The strain defined in the current configuration is $\varepsilon_{\mathrm{A}}$ under the Eulerian description.
$ \varepsilon_{\mathrm{G}}=\frac{1}{2}\left(\mathbf{F}^{\mathrm{T}} \mathbf{F}-\mathbf{I}\right), \varepsilon_{\mathrm{A}}=\frac{1}{2}\left(\mathbf{I}-\mathbf{F}^{-\mathrm{T}} \mathbf{F}^{\mathrm{T}}\right)$
where I is the unit tensor.
The components of coil will experience the plastic deformation when a larger strain is applied, in which case, the von-Mises yield criterion is used for obtaining the equivalent stress as,
$ \sigma_{\mathrm{e}}=\sqrt{3 J_{2}}$
2.2. ALE method
The Lagrangian framework and the Eulerian framework are two classical methods used for describing the motion of objects. To combine the advantages of these two frameworks, the ALE description has been proposed, which was originally used to solve fluid dynamics problems [42], [43] and later for analysing large deformation in solids [44], [45], [46]. By introducing the reference coordinate system $O-\varphi_{1} \varphi_{2} \varphi_{3}$, the mesh generation is performed on the reference configuration, where the mesh points can move independently with object and space. Spatial coordinates are dependent on $\mathbf{X}$ and time t, i.e., $x=x(\mathbf{X}, t)$ or mesh coordinates φ and time t, i.e.,$ x=x(\varphi, t)$. The velocity of material point v and velocity of mesh point v¯ is determined by [47]
$ \mathbf{v}=\left.\frac{\partial \mathbf{x}(\mathbf{X}, t)}{\partial t}\right|_{\mathbf{x}}, \overline{\mathbf{v}}=\left.\frac{\partial \mathbf{x}(\varphi, t)}{\partial t}\right|_{\varphi}$
where $(\cdot) \mid \mathbf{x}$ and $\left.(\cdot)\right|_{\varphi}$ represent the material coordinate $\mathbf{X}$ and the mesh coordinate $\varphi$, respectively.
The Eulerian form of the momentum conservation equation is
$ \rho\left(\left.\frac{\partial \mathbf{v}}{\partial t}\right|_{\varphi}+\left(c_{v} \cdot \nabla\right) \mathbf{v}\right)=\nabla \cdot \sigma+\rho \mathbf{b}$
where $\rho$ is the mass density, v is the velocity, and b is the body force, and $\mathbf{C}_{v}$ is convective velocity, which represents the difference between v and $\overline{\mathbf{v}}$.
2.3. Electromagnetic equations
The above equations present the mechanical equations and boundary conditions in different configurations. By considering the mechanical deformation, the electromagnetic parameters can be defined in the reference and current configurations. Based on the conservation properties, the relationships between the corresponding variables are [48],
$ \mathbf{H}=\mathbf{F}^{-\mathrm{T}} \tilde{\mathbf{H}}, \mathbf{B}=J_{\mathrm{d}}^{-1} \mathbf{F} \tilde{\mathbf{B}}$
$$ \mathbf{E}=\mathbf{F}^{-\mathrm{T}} \tilde{\mathbf{E}}-\mathbf{v} \times \mathbf{B}, \mathbf{J}=J_{\mathrm{d}}^{-1} \mathbf{F} \tilde{\mathbf{J}}$
where B is the magnetic flux density and the relationship between B and magnetic field strength H is denoted by $\mathbf{B}=\mu_{0} \mathbf{H}$. Here, μ0 is the permeability of vacuum, E is the electric field and J is the current density. Their components are referred to the variables in the spatial coordinate system. Likewise, $\widetilde{\mathbf{H}}, \widetilde{\mathbf{B}}, \widetilde{\mathbf{E}}$ and $\widetilde{\mathbf{J}}$ are the corresponding variables referred to variables in the material coordinate system. Here, the motion velocity, v during deformation is not neglected. For the H-formulation, the electromagnetic response can be determined by the Faraday's law of electromagnetic induction and Ampere’ law [49], [50]. In the reference and current configurations, the above equations are expressed as,
$ \nabla \times \mathbf{E}=-\mu_{0} \frac{\partial \mathbf{H}}{\partial t}, \tilde{\nabla} \times \tilde{\mathbf{E}}=-\mu_{0} \frac{\partial \tilde{\mathbf{H}}}{\partial t}$
$ \nabla \times \mathbf{H}=\mathbf{J}, \tilde{\nabla} \times \tilde{\mathbf{H}}=\tilde{\mathbf{J}}$
where $\nabla_{x}$ and $\nabla_{X}$ represent the curl operator in the current and reference configurations, respectively. $\mathrm{d}(\cdot) / \mathrm{d} t$ is the material derivative of the B in regard to time.
The current density and electric field follow the E-J power law, which is used to describe the electrical characteristics,
$ \mathbf{E}=\rho \mathbf{J}-\mathbf{u} \times \mathbf{B}, \tilde{\mathbf{E}}=\rho J_{\mathrm{d}}^{-1} \mathbf{C} \tilde{\mathbf{J}}$
$ \rho=\frac{E_{\mathrm{c}}}{J_{\mathrm{c}}(\mathbf{B})} \cdot\left(\frac{|\mathbf{J}|}{J_{\mathrm{c}}(\mathbf{B})}\right)^{n-1}$
$ J_{c}(\mathbf{B})=\frac{J_{c 0}}{\left(1+\frac{\sqrt{k^{2} B_{z}^{2}+B_{r}^{2}}}{B_{0}}\right)^{\alpha}} $
where $E_{c}$ is critical current criterion and equal to 1 µV/cm, and n is the power-index describing the flux creep which is given as 43. $B_{r}$ and $B_{z}$ are the parallel field and the vertical field relative to the tape surface. The parameter values in (12) are $J_{\mathrm{c} 0}=7.24 \times 10^{11} \mathrm{~A} / \mathrm{m}^{2}, B_{0}=0.4674 \mathrm{~T}, k=9.13 \times 10^{-3} \text { and } \alpha=0.7518$[53]. Then, the equation of H-formulation in reference configuration is
$ \tilde{\nabla} \times(\rho \tilde{\nabla} \times \tilde{\mathbf{H}})=-\mu_{0} \frac{\partial \tilde{\mathbf{H}}}{\partial t}$
2.4. Coupling between the strain and electromagnetic response
The interaction between current and magnetic field generates the electromagnetic force as,
$ \mathbf{f}_{\mathrm{b}}=\mathbf{J} \times \mathbf{B}$
As the superconductors experience the mechanical deformation, the deformation in turn will affect the superconductivity of material. Firstly, the tilting or deflection of the REBCO tape can modify the geometric configuration and field distribution surrounding the coil. $\boldsymbol{J}$ in the REBCO tape will redistribute under the same electromagnetic environment by solving the Maxwell's equations. Here, the ALE method is implemented using moving mesh interface in COMSOL Multiphysics. After calculating the radial and vertical displacements obtained in the Solid mechanics interface, the real-time deformation is determined by prescribed deformation node and prescribed mesh displacement node in the moving mesh interface, which is applied to each turn of the coil. Besides, the redistribution of the surrounding mesh during coil deformation is adjusted by free deformation node applied to the surrounding air domain. The similar procedures are also given in [54], [55], in which the ALE method was used for calculating the levitation force between a superconducting block and a permanent magnet guideway. The superconducting block is assumed to be a rigid body and the mechanical deformation is neglected. However, in this work, the ALE method updates the position of the coil in real-time and characterizes the effect of rigid body displacement and mechanical deformation on the electromagnetic field distribution in the tape.
Secondly, anisotropic correlation exists between the $J_{c}$ and B. The fields that are normal and parallel to the tape have different impact on the reduction of $J_{c}$ [56], [57]. It was found that the rotation of tape causes a change in its normal and parallel fields. As a result, the $J_{c}$ will get modified according to the normal and parallel fields. As the mechanical strain occurs in the tape, its titling angle is
$ \theta=\frac{\partial u}{\partial z}$
where $u$ is displacement along the radial direction. Due to the existence of screening current, the uneven electromagnetic force on the tape is mainly concentrated at both ends of the tape. Therefore, the angle θ varies gradually along the tape width in equation (19), and the parallel and perpendicular fields with respect to the tape surface will change accordingly. As illustrated in Fig. 1, the normal and parallel fields considering the titling of the tape are,
$ \left\{\begin{array}{l} B_{\perp}=B_{r} \cos \theta-B_{z} \sin \theta \\ B_{\|}=B_{z} \cos \theta+B_{r} \sin \theta \end{array}\right.$
Fig. 1. Change in the normal and parallel fields after deformation. |
Thirdly, the degradation of the $J_{c}$ with mechanical stress or strain in the REBCO tape has been reported in [58], [59], [60]. In the coupled model, the strain calculated from the mechanical formulation is passed to the electromagnetic formulation. Based on the experimental data given in Ref. [23], the relationship between the critical current and strain is obtained by fitting the data.
$ k(\varepsilon)=\left\{\begin{array}{c} 1-6.28 \times 10^{-8} \times \exp (2147.35 \times \varepsilon), \varepsilon \leqslant 0.67 \% \\ 0.867-582.466 \times(\varepsilon-0.0067), \varepsilon>0.67 \% \end{array}\right.$
The above relationship is also adopted in Ref. [34]. When the strain is less than 0.67%, there is insignificant decline in $J_{c}$ of the tape, thus maintaining a superior current carrying capability. Nevertheless, when the strain is greater than 0.67%, the critical current's dependency on strain increases significantly. Then, the modified relationship is expressed as,
$ J_{\mathrm{c}}(\mathbf{B}, \varepsilon)=\frac{J_{\mathrm{c} 0}}{\left(1+\frac{\sqrt{k^{2} B_{\|}^{2}+B_{\perp}^{2}}}{B_{0}}\right)^{\alpha}} k(\varepsilon)$
3. Mechanical behaviors in high field coil
In the coupled electromechanical model, the mechanical response of the superconducting coil can be determined numerically. Say, the transport current and external field B are applied to a REBCO coil, as shown in Fig. 2. Only the coil's cross-section is taken into account in the 2D axisymmetric model. Here, the superconducting coil without the influence of electromagnetic force is regarded as the reference configuration. The low-temperature superconducting (LTS) coil generates the external magnetic field. To simplify the calculations, the LTS coil is assumed to be Copper coil. A single pancake coil is constructed with 50 turns and is co-wound using the insulated tape. Here, utilizing COMSOL Multiphysics, the electromechanical coupling problem is resolved. It is to be noted that the mesh is updated using the moving mesh interface in a step-by-step manner while considering the deformed configuration.
Fig. 2. Schematic of REBCO coil in a background field. |
The above discussed calculations are represented in a flow chart form in Fig. 3 and are summarized below.
Fig. 3. The flow chart of calculation. |
(1) Firstly, the model for superconducting coil is constructed. The mesh grid is initialized, and the parameters and boundary condition are set.
(2) The working current in the LTS magnet progressively increases until magnetic field reaches 20 T at 500 s. Simultaneously, REBCO coil is charged to 138 A at a rate of 1 A/s and then remained unchanged.
(3) The electromagnetic force is calculated in the magnetic field formulation interface and imported to the solid mechanics interface through general extrusion coupling operator to obtain the displacement of the coil and the tilting angle.
(4) The Jc-B-ε relationship is updated and the mesh point position is adjusted appropriately to make it compatible with the deformation. The electromagnetic behavior for the next time step is also calculated.
(5) Repeat steps 1 to 4 until the last time step.
3.1. Validation of numerical model
The numerical outcomes are contrasted with the experimental data in order to confirm the effectiveness of the coupled model, where the parameters are consistent with those in [28]. $\Delta \varepsilon$ denotes the difference in strain between the top and bottom points of the outermost layer. The outcomes of contrast are shown in Fig. 4, where one can observe that the $\Delta \varepsilon$ determined by coupled model and uncoupled model has good consistency at a low field, while the difference becomes larger as the field increases. Although the predicted results from coupled model are still larger than the experimental results, the trend is more consistent and closer to the experimental data. Similar results are also reported in the previous findings in the literatures [28], [34]. It is worth mentioning that there is a limitation for model validation as the highest magnetic field strength of this experiment is only 5 T. In the future work, we will try to acquire the experimental data of high field and provide the stronger validation.
Fig. 4. As the axial field increases, the strain difference, $\Delta \varepsilon$ from the coupled and uncoupled models is compared with the experimental outcomes [28]. |
3.2. Comparison of numerical results with coupled model using T-A formulation
T-A formulation is widely accepted to deal with the electromagnetic simulation of superconducting coils [61], [62], [63]. The screening current effects were studied considering the coupled electromagnetic-mechanical model with T-A formulation [33], [34]. The T-A formulation is computationally efficient by employing the thin-film approximation for solving electromagnetic field in large scale structure. Due to the thin-film assumption, T-A formulation is mainly used for the electromagnetic calculation of high-temperature superconducting tapes. Compared to T-A formulation, the H formulation is more feasible for the complicated superconducting wires, such as Bi-2212 and Bi-2223 [64]. In Ref. [33], the experimental results and numerical results of superconducting coil are compared in high field, where the external field is 12 T. They pointed out that the coupling model with T-A formulation can characterize the electromagnetic-mechanical response in high field effectively. To verify the validity of the numerical model using H formulation in high field, the numerical results obtained by H formulation and T-A formulation are presented, as shown in Fig. 5. In this numerical study, samples used are REBCO CCs from superpower.inc and Kapton layer of 25 μm thickness for co-winding [65]. The coil's specifications are given in Table 2.
Fig. 5. The maximum values of (a) $\sigma_{\varphi}$ and (b) $\varepsilon_{\varphi}$ for various central magnetic fields, where the external field is the value at the origin in Fig. 2. |
Table 2. Parameters of the coil system. |
| Parameters | Value |
|---|---|
| Number of turns | 50 |
| REBCO tape width; thickness | 4 mm; 0.095 mm |
| REBCO coil inner/outer radius | 25 mm; 30.975 mm |
| Over-banding tape width; thickness | 4 mm; 0.5 mm |
| Bobbin tape width; thickness | 4 mm; 0.5 mm |
| LTS coil inner/outer radius | 150 mm; 210 mm |
| LTS coil height | 550 mm |
| REBCO coil operating current | 138A |
| REBCO coil height | 112 mm |
The maximum values of hoop stress $\sigma_{\varphi}$ and hoop strain $\varepsilon_{\varphi}$ of REBCO coil during the magnetization are shown in Fig. 5. It can be found that as the external field is increased from 0 to 20 T, the mechanical stress and strain determined by the H formulation agree with the those of T-A formulation. Additionally, the hoop current density $J_{\varphi}$ calculated through coupled models with H-formulation and T-A formulation are shown in Fig. 6. The numerical outcomes are also close for both models, and only the maximum values have small differences. The above comparisons prove that the coupled model with H-formulation is effective in predicting the electromagnetic and mechanical responses and this method is still applicable for a large deformation. For the electromechanical coupling model based on the H formulation, we just need to modify the electromagnetic constitutive relation and introduce ALE method.
Fig. 6. The $J_{\varphi}$ distribution obtained through coupled model with H-formulation and T-A formulation, where the external field is 20 T. |
3.3. Influence of coil position on coupling effect
To investigate the impact of coil position on the coupling effect with the coupled model. In this case, REBCO coils are placed at different positions to compare the mechanical stresses between the uncoupled and coupled models under identical operating conditions. Here, the radial coordinate of REBCO coil remains unchanged, and the coordinate of z-axis is changed from 112 mm to 275 mm. The schematic diagram is displayed in the Fig. 7, whereas the numerical results of position A are given in Fig. 5. The mechanical response of REBCO coil at the position B is shown in the Fig. 8. To differentiate between the uncoupled and coupled models, a variable $D$ is defined as
$ D=\frac{\sigma_{\text {uncoupled }}-\sigma_{\text {coupled }}}{\sigma_{\text {uncoupled }}}$
where σuncoupled and σcoupled are the hoop stresses in the uncoupled and coupled models at 20 T, respectively. The difference between strain can be stated in the same way. The results show that the differences between the $\sigma_{\varphi}$ and the $\varepsilon_{\varphi}$ obtained through uncoupled model and the coupled models at 20 T are 11.23% and 13.61%, respectively. However, as the coil reaches position A, the differences in $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ reach 33.56% and 39.28% at 20 T (see Fig. 6), respectively. This indicates that the position of the coil affects the electromagnetic coupling effect, and predicting the mechanical stress with empirical formula may lead to inaccurate estimations.
Fig. 7. Diagram of REBCO coil at different positions, (a) position A and (b) position B. |
Fig. 8. The maximum values of $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ at various central magnetic fields at position B. |
It is to be noted that the differences in mechanical responses between the uncoupled and coupled models do not always change monotonically with the change of position. Here, the maximum values of $\sigma_{\varphi}$ is presented in Fig. 9 as the vertical location changes from 50 mm to 260 mm. When the coil is moved along the z-axis, the $\sigma_{\varphi}$ of the coil gradually increases before eventually decreasing. This is owing to the fact that the radial electromagnetic force on the coil is primarily influenced by the J and B. As the position ascends, the component parallel to the tape surface (i.e., axial field) decreases while the component perpendicular to the tape surface (i.e., radial field) increases, resulting in a greater screening current and electromagnetic force. However, the field perpendicular to the tape surface plays a dominant role in the reduction of $J_c$. When the position of coil continues to ascend, the decrease in $J_c$ plays a dominant role in the reduction of electromagnetic force.
Fig. 9. (a) The $\sigma_{\varphi}$ in the uncoupled and coupled models at 20 T, (b) relationship between variable $D$ and position along z-axis. |
In addition, as the coil position ascends, the variable D also increases at first, and then decreases. This trend is also owing to the variance in $J_c$ between the coupled and uncoupled models. As the variance in $J_c$ becomes smaller, the variance between the hoop stresses for the two models are negligible.
3.4. Mechanical response with CSR method
In the following discussion, the mechanical response as the coil is located at the position B is focused. As discussed in Section 3.3, the REBCO coil experiences significant inhomogeneous stress and strain in high fields due to the screening current effect (SCE) [66], [67], [68], [69]. The screening effect can be reduced to decrease the $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ caused by the screening current. Several approaches are proposed to remove the screening current, such as degaussing method [70] and current sweep reversal (CSR) method. Because of its simplicity and efficiency, the CSR method is commonly employed [71], [72], [73], [74]. To explore the influence of CSR approach on the mitigation of $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$, two overshoot methods based on the discussed coupled model are investigated [75]. The current waveforms of two overshoot methods are illustrated in Fig. 10. In the traditional overshoot method, the maximum exciting current is larger than the operating current. Then, the current in the coil will reduce to the operating current directly. In the overshoot method with plateau, the current reaches the maximum value where it will experience a period of stabilization before reaching the operating current. The overshoot rate is defined as the percentage of the portion exceeding the operating current to the actual operating current, which is typically less than 20%. It is to be noted that the excessive maximum current can lead to a large electromagnetic force, which may damage the coil. For an operating current of 400 A and a central field of 20 T, the overshoot rate is set at 115% of the operating current and the stabilization time is set as 300 s. From Fig. 11, the traditional overshoot method can reduce the hoop stress and strain to approximately 9.94% and 8.17% compared to the no-CSR method, respectively. The overshoot method with plateau can reduce the hoop stress and strain up to 16.76% and 13.85%, respectively. To discuss the influence of the stabilization time on the mechanical response in the overshoot method with plateau, the maximum values of $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ at different moments are analyzed.
Fig. 10. Charging diagram of two CSR methods, (a) traditional overshoot method and (b) overshoot method with plateau. |
Fig. 11. Simulation results of the (a) $\sigma_{\varphi}$ and (b) $\varepsilon_{\varphi}$ in the upper side of the coil under various current operating conditions at 20 T. |
The relationship between maximum values of $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ with various stabilization times is discussed in Fig. 12. Six cases are presented where the stabilization times are 0 s (i.e., traditional overshoot method), 50 s, 100 s, 300 s, 800 s, 2000 s, respectively. It can be found that the $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ decrease rapidly at the beginning, and then the variation becomes smaller with the increase of stabilization time at moment M. A similar tendency is observed in the final stress and strain at moment N. To explain this phenomenon, the $J_{\varphi}$ distribution for different methods is displayed in Fig. 13, which suggests that the maximum values of $J_{\varphi}$ for the overshoot method are smaller than that of the no-CSR method. In the overshoot method with a plateau, the maximum value of screening current progressively decreases as the stabilization time increases. Besides, a reverse screening current is observed to penetrate at both ends of the coil (see Fig. 13 (b)–(d)). The above-mentioned factors collectively result in a reduction in hoop stress and strain. Therefore, the overshoot method with a plateau is more effective in reducing the screening current effect, despite of its requirements for longer operating time.
Fig. 12. The relationships between stabilization time with (a) $\sigma_{\varphi}$, and (b) $\varepsilon_{\varphi}$ in overshoot method with a plateau at different moments. |
Fig. 13. The hoop current density distribution $ J_{\varphi}\left(\mathrm{kA} / \mathrm{mm}^{2}\right) $ at 20 T, (a) no CSR, (b) traditional overshoot method, (c) overshoot method with a plateau at the stabilization time of 300 s, and (d) overshoot method with a plateau at the stabilization time of 2000 s. |
3.5. REBCO coil with two-tapes co-winding
Co-winding technology refers to the simultaneous winding of two or more tapes during the coil preparation process, as shown in Fig. 14. The advantage of co-winding technology is that it increases the current density by decreasing the amount of insulating turns [76]. To avoid any hotspot generation owing to local degradation, the current can redistribute between two tapes using two-tapes co-winding technique, which can improve the reliability of coil [77]. In addition, the co-winding coil can also sustain higher stress and strain [78], [79]. The electromagnetic and mechanical responses of two-tapes co-winding coil are discussed based on electromagnetic-mechanical model. The maximum current of single winding coil reaches 138 A. To compare with the coil wound using single tape, the maximum current of two-tapes co-winding coil is designed to reach 276 A (= 2×138 A). To reduce the computation complexity, the co-wound two-tapes are considered as a single element and the contact between the two tapes are neglected.
Fig. 14. Schematic diagram of single winding coil and co-winding coil. |
The $J_{\varphi}$ distributions of single winding coil and two-tapes co-winding coil at the outermost turn are illustrated in Fig. 15, where the external field is increased from 10 T to 20 T. It is found that the maximum value of $J_{\varphi}$ for the two-tapes co-winding coil is lower than that of the single winding coil. Additionally, $J_{\varphi}$ is more uniformly distributed along the tape width direction because the current can be shared between the co-wound two tapes. Consequently, the $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ of the two-tapes co-winding coil would be lower. The maps of $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ in the single winding coil and co-winding coil are given in Fig. 16. It's not hard to see that the maximum values of $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ are reduced in the two-tapes co-winding coil. Overall, co-winding technology can enhance the performance and stability of superconducting coils, increasing their reliability and efficiency in practical applications.
Fig. 15. $J_{\varphi}$ distribution obtained from single winding coil and co-winding coil at the outermost turn, where the field is increased from 10 T to 20 T. |
Fig. 16. The hoop stress and strain at 20 T in single winding coil and co-winding coil. |
4. Conclusion
In this paper, a coupled electromechanical model is developed with H-formulation and ALE method. A moving mesh interface is included to update the grid redistribution after coil deformation, and the influence of deformation of tape on the $J_c$ is analyzed. By comparing with the experimental results and other electromagnetic–mechanical models, it is shown that the electromechanical model with H-formulation can simulate the electromagnetic and stress/strain distribution of HTS magnets effectively.
Numerical results show that when the magnetic field is low, the error of the hoop stress between the two models is very low. A similar trend is observed for strain case as well. The differences between the mechanical behaviors of the two models increase as the field becomes larger. The coupling effect of electromagnetic field and mechanical deformation is dependent on the position of the REBCO coil. Since the screening current effect leads to uneven stress and strain for HTS coils, the CSR method is employed to reduce the $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$. In the overshoot method with a plateau, the final maximum $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ progressively decrease with increasing stabilization time. For the stabilization time of 300 s, the maximum $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ are reduced by up to 16.76% and 13.85%, respectively, compared to the no-CSR method. Moreover, the final maximum hoop stress and strain will gradually decrease with an increase in the stabilization time. Employing a two-tapes co-winding technology results in lower maximum current density, and lower $\sigma_{\varphi}$ and $\varepsilon_{\varphi}$ compared to the single winding coil. These findings will be helpful in better understanding the electromagnetic and mechanical characteristics of superconducting magnets.
CRediT authorship contribution statement
Huadong Yong: Writing - review & editing, Validation, Conceptualization, Funding acquisition, Supervision, Writing - original draft. Dong Wei: Writing - original draft, Conceptualization, Software, Visualization, Formal analysis, Data curation. Yunkai Tang: Methodology, Validation, Visualization. Donghui Liu: Funding acquisition, Supervision, Writing - review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors acknowledge the support from the National Natural Science Foundation of China (Nos. U2241267, 12172155 and 12302278), National Key Research and Development Program of China (No. 2023YFA1607304), Major Scientific and Technological Special Project of Gansu Province (23ZDKA0009) and the Fundamental Research Funds for the Central Universities (No. lzujbky-2022-48).
