## 5 Thermal Analysis with the Finite Element Method (FEM)  
A thermal analysis of the 3 T REBCO magnet was carried out to verify the thermal characteristics and temperature distributions with COMSOL, a finite element method (FEM) program. Figure 11 presents the heat transfer FEM model and the corresponding simplified thermal resistance circuit model. There are 12 thermal contacts in the model, and the heat is transferred from a thermal shield, of which the temperature is maintained at 82.58 K, to the second stage of the cryo-cooler (9.8 K) through thermal contacts. As the thermal contact resistance (TCR) depends on the surface material, surface roughness and contact pressure, it is very difficult to predict the thermal contact resistance accurately [51, 52, 56, 57]. Because the goal of the simulation is to verify the tendency of the temperature distributions, we roughly estimated the thermal contact resistances using values from an earlier study (reference [57]). As the TCR1, TCR4, TCR 6, TCR 8, and TCR 9 represent thermal contacts between aluminum and copper, we averaged thermal contact resistance of Al-Al 490 N and  
Figure 10: (**a**) Center magnetic field and current, and (**b**) voltages and current of eight DPC during the sudden discharge of the 3 T system.  
Cu-Cu 490 N at 10 K to calculate these values. Since indium sheets were inserted between the contact 7 and 12, TCR7 and TCR12 were calculated by thermal contact resistance of In-Cu 490 N at 10 K. On the other hand, TCR 10 was calculated by only Cu-Cu 50 490 N due to the absence of an indium sheet. We assumed that thermal contact 5 is very poor and the TCR5 value was set to 50. Since it was very difficult to find thermal contact resistance between GFRP materials, we also assumed that the thermal contact resistances at 2, 3, and 11 are same with that of the Al-Al 490 N at 10 K. The roughly calculated thermal contact resistance values considering the temperature (10 K) are shown in Table 3.  
Considering boundary conditions such as the thermal contact, heat flux from radiation and the temperature, we carried out FEM simulations and compared these outcomes with the result from measured data using Cernox temperature sensors. The FEM simulation results are shown in Figure 12. Although there are slight errors between the simulation and experimental results due to the thermal contact resistance values, the tendency of the temperature distributions in the simulation result is similar to that of the measured data (T3 > T2 > T4 > T1).  
Figure 11: (**a**) The heat transfer FEM model, and (**b**) simplified thermal contact resistance circuit.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **Thermal Contact Resistance [K \(\times\) m\({}^{2}\)/W]** & **Thermal Contact Materials** \\ \hline TCR1 & \(1.3\times 10^{-3}\) & Copper/Aluminum \\ TCR2 & \(1.11\times 10^{-4}\) & Aluminum/Aluminum \\ TCR3 & \(1.11\times 10^{-4}\) & Aluminum/Aluminum \\ TCR4 & \(1.3\times 10^{-3}\) & Copper/Aluminum \\ TCR5 & 50 & - \\ TCR6 & \(1.3\times 10^{-3}\) & Copper/Aluminum \\ TCR7 & \(1.7\times 10^{-4}\) & Indium/Copper \\ TCR8 & \(1.3\times 10^{-3}\) & Copper/Aluminum \\ TCR9 & \(1.3\times 10^{-3}\) & Copper/Aluminum \\ TCR10 & \(3.33\times 10^{-3}\) & Copper/Copper \\ TCR11 & \(1.11\times 10^{-4}\) & Aluminum/Aluminum \\ TCR12 & \(1.7\times 10^{-4}\) & Indium/Copper \\ \hline \hline \end{tabular}
\end{table}
Table 3: Estimated Thermal Contact Resistance for the FEM Simulation.