Numerical Analysis of Screening-Current Induced Strain in a 16 T REBCO Insert Within a 20 T Background Field

REBa2Cu3O7-x (REBCO) coated conductor has emerged as a promising material for the development of ultra-high-field (UHF) magnets. However, the presence of screening-current induced strain within this tape-shaped conductor impacts the operational stability of REBCO coils. We focused on the design of an all-superconducting 36 T / 40 mm UHF magnet, which included a 16 T insert magnet consisting of two nested REBCO coils. In this study, the screening-current induced strain in the 16 T magnet was numerically studied using the coupled electromagnetic-mechanical model, which considered the tilting angle of REBCO tapes and the strain dependency of the critical current. We calculated the screening-current induced strain of each pancake individually and compared these results with those calculated by the sequential model. According to the coupled model, the maximum hoop strain was 0.52%, relatively lower than that calculated by sequential model. The most dangerous pancake was estimated to be the outermost pancake of each coils. Additionally, we varied the critical current value of the REBCO tapes and studied the relationship between the critical current value and screening-current induced strain. This work provides a feasible way to calculate the screening-current effect in large-scale HTS magnets.


I. INTRODUCTION
High-Temperature Superconducting (HTS) materials are capable for generating ultra-high magnetic fields inside external magnets.REBCO, the representative HTS conductor, is increasingly chosen for winding UHF inserts to achieve center fields exceeding 30 T. One remarkable work was the LBC3 operated by NHMFL, which achieved a world-The associate editor coordinating the review of this manuscript and approving it for publication was Zhe Zhang .record 45.5 T direct-current field [1].Additionally, a 32 T all-superconducting user magnet has been tested and opened for experiments [2], [3], [4], [5], [6], [7].MIT focused on the 1.3 GHz NMR project, built an 18.8 T HTS magnet (H800) and tested it in 2018 [8], [9], [10], [11].After H800 quenched, they proposed a single-coil insert magnet design named H800N [12].The entire 1.3 GHz NMR magnet was designed to generate a center field of 30.5 T [13].IEECAS achieved a world-record all-superconducting center field of 32.35 T, with a no-insulation (NI) REBCO insert contributing more than 17 T inside a 15 T LTS external magnet [14].CEA wound a 14.5 T HTS insert using the metal-as-insulation (MI) method, reaching a 32.5 T center field inside a resistive external magnet [15].RIKEN focused on the layer-intra noinsulation (LNI) method and generated a 31.4T field by inserting a LNI-REBCO coil inside a 17 T Low-Temperature Superconducting (LTS) external magnet [16].
However, during the excitation test of REBCO insert magnet, the conductor had a higher risk of quench than predicted.Based on recent works, the screening current induced by the perpendicular field was verified to have serious impact on the performance of REBCO coils.Due to the screening-current effect, the transport current inside the REBCO layer tended to concentrate at the edges rather than distributing uniformly [17].This concentrated current induced additional local strain on the REBCO conductor [18], known as the screening-current induced strain (SCS).This mechanism increased the risk of degradation in REBCO tapes, especially when ramping up UHF REBCO inserts [19], [20].Several reinforcing methods were proposed to reduce the SCS, including the striated multifilament tapes [21], over-banding reinforcement [22], [23] and edge-bonding structure [23].
Numerical calculation methods are being improved to estimate SCS with better precision.The T -A formula was developed [24], [25] and verified for simulating REBCO coils effectively [26].Using classic T -A method, the magnetic field generated by REBCO coils was calculated to be relatively lower than the uniform-current model, and the experimental results well-matched the T -A model [27], [28], [29].When combining the Lorenz force induced by screening current into the mechanical model, the SCS was calculated to be significantly larger than the uniform-current model [30], [31], [32].However, it has been observed that SCS was overestimated when running the T -A and mechanical models sequentially [33].An improved called coupled T -A model has been developed, considering the tilting angle of REBCO tapes and the strain dependence of critical current [23], [34].This improved model has been verified to fit the practical results better [23].Considering the increasing focus on the stability study of UHF magnets, a more precised calculation of SCS in UHF magnets was needed.However, running the coupled T -A method directly on large-scale REBCO magnets can be extremely time-consuming, and in some cases, the model may fail to converge to a certain result.
In this work, a 36 T UHF magnet with a 40 mm electromagnetic bore was conceptually designed and numerical analyzed.The electromagnetic design of the 16 T REBCO insert was presented, a subregional-coupled model was developed to estimate the distribution of SCS in this large-scale REBCO magnet.This model calculated the SCS of each REBCO pancake individually, coupling the tiling-angle effect and critical current variation by local hoop strain.By changing the target pancake in individual models, the SCS distribution of the entire 16 T HTS insert could be summarized.A comprehensive comparison was presented, evaluating the hoop strain, local field, and current density distribution between the subregional-coupled model and the sequential model.Furthermore, we changed the critical current level in the inner coil and found out its influence on the SCS distribution.This work presents a practical method for adapting the coupled T -A model to large-scale REBCO magnets.

II. 36 T MAGNET CONFIGURATION
The object of this study is to design and evaluating a 16 T / 40 mm REBCO magnet inside a 20 T all-superconducting external magnet.As shown in Figure 1(a), the inner diameter of 20 T external is 350 mm designed by ASIPP, this 20 T component will not be constructed using single-tape REBCO pancake stacks.Considering fabrication tolerances, the outermost boundary of the 16 T insert is set to 330 mm, generating a 10 mm gap between the insert and external magnets.The magnetic inner diameter is 40 mm, which is suitable for operating as a user magnet.To lower the concentration of magnetic stress, the insert magnet is arranged in two nested coils in series.Figure 1(b) showed the detailed distribution of the axial magnetic field in the center of the 36 T magnet when all the coils are fully charged.The major REBCO pancakes will experience fields larger than 30 T.
Key parameters of the 16 T insert were listed in Table 1.The two coils will both be wound by 4.8 mm-wide, 200 µ m-thick REBCO tapes.The thickness of copper layer is 125 µ m in order to enhance the thermal stability.Each REBCO turn will be co-wound with 50 µ m-thick stainless steel (SS) tapes.This metal-as-insulation (MI) technique was verified to generate much larger contact resistance compared to NI REBCO coils while maintaining feasible self-protecting ability [35], [36].Coil 1 is a stack of 20 double-pancakes (DP), designed to contribute a 5 T center field.Each single pancake (SP) in Coil 1 comprises 90 REBCO turns, with a 24.3 mm-thick over-banding reinforcement applied at the outermost turn.In Coil 2, there are 210 turns in each SP, accompanied by a 29.7 mm-thick over-banding reinforcement.The operation current is set to 260 A, generating 36 T in total.
As seen in the conceptual design, Coil 2 is designed to be significantly larger than Coil 1.The primary purpose of this scheme is to obtain a global optimal condition of maximum SCS inside the whole 16 T insert, in which case the individual maximum SCS inside Coil 1 and Coil 2 were calculated to be nearly the same.The detailed results will be presented in Section IV.

III. NUMERICAL MODEL FOR THE 16 T INSERT MAGNET A. T-A FORMULATION AND MECHANICAL MODELING
The screening current was induced by variations in the local perpendicular field on the REBCO conductors [17].In this work, a coupled electromagnetic-mechanical model was adapted to study the screening-current effect in the 16 T magnet.The electromagnetic field was calculated by T -A  formulation.In REBCO region, T -formula was simplified considering the axisymmetric feature and thin-tape approximation [18], the n-index was defined as 31 based on the formal experience of single-tape testing.J c has strong dependence to the magnetic field, the relationship was fit by anisotropic Kim model [37], [38] in this study as where J c0 , B 0 , α, k are 2.227 × 10 9 A/m 2 , 1.503 T, 0.7415, 9.13 × 10 −3 respectively.The mechanical analysis of 16 T REBCO insert was conducted using a 2D axisymmetric model, where each REBCO turns was separately built.Figure 2(a) illustrated the schematic building of this discrete-contact model.Each domain in the REBCO region was thick as 250 µ m, consisting of one REBCO turn and one co-winding turn.The equivalent Young's modulus of the REBCO domains was 150.7 GPa, referencing material properties in [23].The inner holder and over-banding domains were defined as 316 L stainless steel.All the mechanical components were assumed to deform linearly.
Considering the exist of tilting angle β [22], [23], the modified B ∥ and B ⊥ are ( 115394 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The relationship between critical current density and hoop strain ε ϕ can be expressed as [23] In the coupled T -A model, the tilting angle and J c -ε ϕ variation were extracted into the T-A electromagnetic model.Compared to the sequential model which omitted these mechanical influences, this modification was verified to estimate SCS better [23].

B. SUBREGIONAL-COUPLED MODEL
For large-scale magnets with a large number of REBCO turns and contact surfaces, obtaining the SCS distribution of the entire magnet using the coupled T -A model directly can be extremely time-consuming.We developed a subregional-coupled model that enabled calculating SCS of each SP individually to enhance the computing speed.
Figure 2(b) showed the modeling of a specific target SP in Coil 1.The discrete-coupled model was built exclusively for the target SP, enabling the calculation of both electromagnetic and mechanical characteristics of each turn.In contrast, for the remaining REBCO coils and the external 20 T magnet, a homogeneous T-A model was built to calculate the electromagnetic field only, with each SP simplified as a single block.
Figure 3 showed the calculation process of the subregionalcoupled model.The coupled T-A model for the target SP was built as an individual component in COMSOL Multiphysics ® separated from the homogeneous T-A model.During the calculation process, the homogeneous T-A model was first proceeded.After obtaining the magnetic field results of the external coils, the coupled T -A model was subsequently proceeded.While building the thin-tape T -A model for the target SP, the value of B ⊥ in equation ( 3) was modified to incorporate both the perpendicular field generated by the target SP itself and the external radial field.The calculated Lorentz force transmitted to the 2D solid model was also modified to account for the presence of the external field.The excitation path for both the target and external models was controlled to be the same, assuming that the entire REBCO coils were ramped simultaneously.

C. TOLERANCE
We built a full model that incorporated both the target and external coils within a single component to compare with the proposed subregional method.In this full model, the whole 36 T magnet was built in one component.The target SP was built as an array of 1D thin-tape edges in 90 turns.The other REBCO coils were modeled as single blocks.Considering the modeling method of each coil in the 16 T insert was the same as subregional model, the comparison between these two models can reveal the calculation tolerance of the subregional method.In this comparison, only electromagnetic parameters were calculated, without considering the tilting effect.We evaluated the current density at the certain turns of the target SP, summarized the error field of the subregional method at upper, middle, and lower cut-off lines.The profile paths for these evaluations were presented in Figure 4(a).
As seen in Figure 4(b), the current density distributed similarly, particularly at the outer turns of the target SP.At the innermost turn, the subregional method showed a slightly lower calculated current density in the upper region, while the maximum value remained nearly the same.The error field represented the differences in magnetic field results between the subregional model and full model.In Figure 4(c), the difference between the axial field B z of the two methods did not exceed 0.05 T. The radial field B r exhibited a maximum difference of 0.12 T at mid plane, but the difference was less than 0.02 T along the other two profile paths.
Overall, the subregional method demonstrated negligible deviations in the magnetic field and current distribution compared to the full model.The larger error B r at the mid-plane may have been caused by differences in the mesh grid between the two models.However, this difference should not significantly affect the hoop strain results, as the current density in this region was quite small, and B z remained nearly the same.Therefore, the subregional process of the 16 T magnet model did not cause noticeable differences in the calculation results.

IV. RESULTS AND DISCUSSIONS A. RAMPING SEQUENCE
The excitation path of the numerical model was set as a two-stage process in the FEM model.As shown in Figure 5, the external magnet was ramped up to 20 T while the insert remained uncharged for the first 2600 s.Afterward, the insert magnet was charged to 260 A in the latter 2600 s at a constant speed of 0.1 A/s, generating a total center field of 36 T finally.
Table 2 presented the computing time statistics for one target SP.We didn't compare these results with the full model because the iteration failed to converge when attempting to calculate SCS in the 16 T insert using the full model.For the external coils, only electromagnetic properties were calculated, so it took less than 1 hour for most simulations.The only exception was Coil 2 SP 18, where the external results took over one and a half hours, possibly due to slower convergence in certain mesh grids during iterations.The coupled model for the target SP required a significantly longer time to obtain the mechanical results.For Coil 1, the average computing time for one SP was approximately 4 hours.In the case of Coil 2, which had a greater number of REBCO turns, the average computing time was nearly 19 hours, much longer than that of Coil 1.The calculation for Coil 2 SP 1 took more than 33 hours, probably due to the slower convergence of the coupled model iterations caused by the larger tilting effect.

B. CURRENT DENSITY DISTRIBUTION AND SCS
After proceeding the subregional-coupled model multiple times for different target SPs, we summarized the results as shown in Figure 6.We skipped several SPs as they did not significantly impact our understanding of the electromagnetic and mechanical characteristics of the 16 T insert.In Figure 6(a), the distribution of the normalized current density J ϕ /J c at 260 A @ 36 T was illustrated.The current density within the SPs at the edges of Coil 1 and Coil 2 tended to concentrate in the edges of the REBCO layers.However, the concentration effect was limited for the SPs in the middle region.
Figure 6(b) showed the hoop strain results, with the displayed deformation being 10 times larger than the calculating results.The maximum hoop strain for each SP was primarily located at the upper edge of the innermost turn.More closer the SP was to the edge of the magnet, 115396 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.more local concentrated the hoop strain was in the SP.In the middle SPs of the 16 T insert, the hoop strain results remained relatively consistent along the z-position within a single turn.In Figure 6(c), the maximum hoop strain of each calculated SP were summarized.For Coil 1, the maximum strain ranged from 0.519% in SP 1 to 0.168% in SP 18. Coil 2 had a similar maximum strain of 0.504% in SP 1, with minimal reduction observed in other SPs.SP 28 still had a relatively high maximum value of 0.445%.
Due to the large perpendicular field generated by the 16 T magnet itself, the screening-current effect had a significant impact on the current distribution, particularly for the SPs at the edges of the magnet.This mechanism caused the edge-side SPs to endure more concentrated magnetic stress compared to the middle SPs.Additionally, all the SPs in Coil 2 revealed a strong tendency to separate from the inner holder during operation, especially the middle SPs where the innermost turn experienced uniform and substantial strain.Developing effective methods to reinforce the supporting structure and prevent the separation of REBCO turns will be crucial for maintaining the stability of this user magnet.Despite the strong separating tendency, the distribution patterns of SCS and current density in Coil 2 were similar to Coil 1.Therefore, the following detailed analysis will focus on Coil 1 as a representation.
Figure 7 showed the detailed distribution of hoop strain and current density in the outermost (SP 1) and innermost (SP 18) pancakes of Coil 1.In Figure 7(a), At 0 A, the hoop strain in Coil 1 was close to zero for all turns.When the transport current was gradually ramped up, SP 1 showed a rapid increase in hoop strain at the upper region of each turn, while a reversed strain revealed at the lower region.The maximum strain value decreased from Turn #1 to #90.In SP 18, the hoop strain increased uniformly with the ramping of the transport current, with a relatively high value of 0.17% observed in Turn #1 at 260 A @ 36 T. Regarding the current density, Figure 7(b) showed that significant screening current was induced at the edges of all turns in SP 1 at 0 A @ 20 T. In SP 18, the induced current density was much lower than that in SP 1.When the insert magnet got charged, the concentrated region in SP 1 expanded, the location of the maximum current density also shifted towards the mid plane.In contrast, the concentrated region in SP 18 did not significantly expand, and the current density increased uniformly in Turn #1.In Turn #90, the current density increased significantly in the upper region while remained nearly unchanged in the lower region.
The local concentration of current density significantly impact the strain condition in SPs at the edges of the magnet.The concentrated current induced a significant tension strain at the upper edge of the REBCO turns, while the reversed current made the REBCO tapes tend to shrink at the lower edge.During the rise of transport current, the local perpendicular field also increased, resulting in the expansion of the current-concentrated region.At the same time, the critical current density decreased at the edges due to the increasing field, causing a decrease of current density at that regions.

C. TILTING EFFECT AND COMPARISON WITH SEQUENTIAL MODEL
During the operation of the 36 T magnet, it is hard to prevent the REBCO turns from tilting towards the direction of the normal field, as all the pancakes were dry-wound.In this part, we built a sequential model in which the tilting angle was set as zero throughout the excitation process [29].The 115398 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.subregional method was also adapted in the sequential model.We compared the calculation results of this model with the coupled model to study the effect of tilting angle on SCS results.
Figure 8 showed the tilting angle results of Coil 1 SP 1 and the variation in local field after coupling the tilting effect.In Figure 8(a), it can be seen that the inner turns of Coil 1 SP 1 experienced a maximum rotation of approximately 3 degrees.The detailed distribution pattern was shown in Figure 8(b).In the inner turns of Coil 1 SP 1, the tilting angle was slightly greater at the upper edge than that at the mid plane.In the outer turns, this comparison was reversed.The tilting angle was less evident in the lower regions in Coil 1 SP 1.
The tilting effect had a significant impact on the local field across the REBCO turns.As shown in Figure 8(c), the modified perpendicular field B ⊥ in the upper and middle regions was significantly lower than that calculated by the sequential model.In the mid plane of Coil 1 SP 1, the B ⊥ was calibrated to be nearly zero.At the lower edge, the results from the coupled and sequential models were almost the same, due to the very small tilting angle.Figure 8(d) showed the results of the parallel field B ∥ , which did not vary significantly after considering the tilting effect.
The difference in hoop strain results of Coil 1 between the two models was then studied.As shown in Figure 9, the maximum tilting angle and hoop strain consistently decreased from SP 1 to SP 18 in the coupled model.The sequential model revealed larger maximum hoop strain values in all SPs compared to the coupled model, and the largest strain was calculated to locate in SP 5, reaching 0.62%.Although SP 1 showed the largest tilting angle, the difference in maximum hoop strain of SP 1 between these two models was relatively small.Additionally, for the SPs in the middle region of Coil 1, the difference in maximum hoop strain between the two models decreased as the maximum tilting angle decreased.
The developed subregional-coupled model revealed that the tilting effect significantly varied the local perpendicular field, this modification significantly influenced the distribution of current density within the REBCO layers, leading Operation Factor was defined to be maximum J op /J c , J op was the operation current density as listed in Table 1.The load line represented the relationship between the transport current and the maximum local perpendicular field in Coil 1 SP 1.
to a substantial change in the hoop strain distribution of the REBCO coils.In the 16 T REBCO insert, not only was the maximum SCS value estimated to be lower than before, but the distribution pattern also changed.In the subregional-coupled model, Coil 1 SP 1 experienced the highest hoop strain.Consequently, it is necessary to study effective methods to control the SCS level in Coil 1 SP 1.

D. CRITICAL CURRENT CONTROLLING
SCS was considered to be closely related to the critical current level [33].We built a series of models for Coil 1 SP 1, varying the critical current level of the REBCO conductors in Coil 1 to study the influencing pattern on SCS distribution by controlling the critical current.The J c − B ⊥ curves for each model were showed in Figure 10, along with a load line representing the relationship between the transport current and the maximum local perpendicular field in SP 1 as a reference.The original curve used in the previous subregional-coupled model was represented by the black line, with a maximum operation factor of 0.265.The critical current level was relatively high, considering that Coil 1 endured a smaller perpendicular field than Coil 2. To lower the critical current level, we added a multiplying factor in the Kim model and generated five new J c − B ⊥ curves.The operation factors for these curves were 0.363, 0.451, 0.563, 0.751, and 0.901 respectively.In Jc5, the operation current almost reached the critical value.The J c − B ⊥ curves for all SPs in Coil 1 were varied, while Coil 2 remained unchanged.
The results of hoop strain and normalized current density using these six J c − B ⊥ curves were showed in Figure 11.In Figure 11(a), we included the result assuming a uniform distribution of transport current for comparison.As the critical current decreased, the maximum hoop strain in SP 1 significantly decreased, the magnetic stress no longer concentrated in the upper region of the inner turns.The location of the maximum hoop strain shifted towards the middle turns of SP 1.When the critical current level approached the operation current value, the maximum hoop strain in SP 1 was close to the result calculated by the uniform-current model.The distribution pattern of normalized current density in SP 1 was also influenced by the change in critical current level.As shown in Figure 11(b), the decrease of critical current led to a significant expanding of the current-concentrated region in SP 1.At Jc3, the concentrated region covered all turns of SP 1 completely.Subsequently, at lower critical current levels, the reverse-concentrated region was more compressed.When the critical current level approached the operation current value, the reverse current almost disappeared, the entire SP 1 reached a near-critical condition.
Figure 12 showed the variation of hoop strain at the upper, middle and lower nodes in Turn #1 of Coil 1 SP 1, comparing the Jc0, Jc5 and uniform-current models.Before 2600 s, the results of Jc0 and Jc5 were nearly the same.As the transport current increased, the calculated hoop strain at the three nodes diverged in the Jc0 and Jc5 models.When the transport current was relatively low, the hoop strain at the three nodes showed similar patterns in both Jc0 and Jc5 models.However, as the transport current got larger, the hoop strain at the upper node in Jc5 model did not rise but gradually decrease.Conversely, the reverse strain at the lower node did not decrease but rose to a positive value.When the transport current reached 260 A, the hoop strain at all three nodes in Jc5 model eventually approached the results calculated by the uniform-current model.
The calculation results demonstrated that the critical current value significantly impacted the hoop strain level in large-scale magnets.The most extreme scenario was to lower the critical current level to approach the operation current level, in which case the estimated SCS will be controlled closely to the uniform-current result.However, under such condition, the REBCO coils would have a much higher risk of quench due to the limited critical current.An optimization method could be developed during the electromagnetic design of UHF magnets to determine a specific critical current level that ensured both SCS and the operation factor to be maintained in acceptable ranges.

V. CONCLUSION
In this study, we presented a conceptual design of a 16 T / 40 mm REBCO insert for a 20 T external magnet.The 16 T insert consisted of two nested MI coils, being wound by 200 µ m-thick REBCO tapes and 50 µ m-thick co-winding tapes.To estimate the screening-current induced strain in the 16 T insert, we adapted the coupled T -A model to a subregional-coupled model in order to make the coupled method capable for calculating the SCS distribution in largescale magnets.
The maximum hoop strain reached approximately 0.52% in the inner coil (Coil 1) and 0.51% in the outer coil (Coil 2).The presence of tilting angle in dry-wound REBCO turns significantly altered the local perpendicular field, thereby influenced the hoop strain distribution.By coupling the tilting angle results, we calculated that all the pancakes in Coil 1 would endure a lower hoop strain.Furthermore, the location of the maximum value shifted from SP 5 to the outermost pancake, SP 1.
We also studied the effect of varying the critical current.By limiting the critical current value, the maximum hoop strain could be efficiently reduced.In the extreme condition where the critical current approached the operation current value, the maximum hoop strain approximately matched the result calculated by uniform-current model.Determining a suitable critical current value is a future focus of the 36 T magnet project, as it can effectively control both the maximum hoop strain and the operation factor at an acceptable level.

FIGURE 1 .
FIGURE 1. Schematic view of the 36 T magnet: (a) the whole assembly; (b) the axial field distribution within the 16 T HTS insert assuming a uniform transport current.

FIGURE 2 .
FIGURE 2. (a) Schematic building of the mechanical discrete-contact model.(b) Modeling of the subregional-coupled model.
J c0 was determined by the measured lifting factor from 77 K to 4.2 K by the manufacturer and 77 K self-field test result (approximately 170 A for 4 mm-wide tapes).The entire 16 T insert was assumed to be operated steadily in 4.2 K without potential local heating.The value of conductor thickness in the boundary condition of T -formula was 250 µ m (200 µ m for REBCO tape and 50 µ m for co-winding tape) in homogeneous model and 1 µ m in discrete thin-tape model [27].

FIGURE 3 .
FIGURE 3. Calculation process of the subregional-coupled model.

FIGURE 4 .
FIGURE 4. Comparison between the full model and subregional model (Coil 1 SP 1): (a) profile paths of the results; (b) comparison of current density between the two methods; (c) error field of the subregional method in comparison with the full model.B r represented the radial field along r-axis.B z represented the axial field along z-axis.

FIGURE 5 .
FIGURE 5. Excitation process of the subregional-coupled model.

FIGURE 6 .
FIGURE 6. Calculation results of the 16 T insert by the subregional-coupled model: (a) distribution of normalized current density J ϕ /J c at 260 A; (b) distribution of hoop strain; (c) maximum hoop strain in each calculated SP.

FIGURE 7 .
FIGURE 7. Detailed results of Coil 1 SP 1 and 18 at 0 A, 150 A and 260 A: (a) hoop strain distribution in the innermost (#1) turn and outermost (#90) turn along the z-position; (b) current density distribution in these turns.

FIGURE 8 .
FIGURE 8. Results of the tilting angle and local field calculated by the coupled model and sequential model, with the same profile paths as Figure 4: (a) general distribution of the tilting angle in Coil 1 SP 1.The displayed deformation was 10 times larger than the calculation; (b) distribution of tilting angle along the upper, lower edges and mid-plane of SP 1; results of the local perpendicular (c) and parallel (d) field, respectively.

FIGURE 9 .
FIGURE 9. Maximum hoop strain and tilting angle of certain SPs in Coil 1 calculated by the coupled model and sequential model.

FIGURE 10 .
FIGURE 10.J c − B ⊥ curves of the calculated models for Coil 1 SP 1.Operation Factor was defined to be maximum J op /J c , J op was the operation current density as listed in Table1.The load line represented the relationship between the transport current and the maximum local perpendicular field in Coil 1 SP 1.

FIGURE 11 .
FIGURE 11.Calculation results of Coil 1 SP 1 using different J c − B ⊥ curves: (a) hoop strain distribution, where the 'Uniform' result assumed that the transport current flew uniformly in SP 1; (b) distribution of normalized current density.

FIGURE 12 .
FIGURE 12. Variation of hoop strain at the upper, middle and lower nodes of Turn #1 in Coil 1 SP 1 during the excitation process.Results calculated by the Jc0, Jc5 and uniform-current models were shown.

TABLE 2 .
Required computing time for calculating one target SP using the subregional-coupled model.