## 2 Calculation of the New Thermal Constitutive Matrix \([M_{\tau}]\)
### Numerical Simulation with \([M_{\lambda}]\)  
The model proposed for the validation consisted of a tube that is similar to the stator or the rotor of an electric machine, especially that of an asynchronous electric machine. It consisted of a tube of 2.00 m in height and 2.00 m in diameter. The wall has a thickness of 0.50 m. This wall is composed of two concentric tubes, with a thickness of 0.25 m for each one. The thermal conductivities of the inner and outer tubes were \(\lambda_{1}\) = 50.0 Wm\({}^{-1}\)K\({}^{-1}\) and \(\lambda_{2}\) = 193.0 Wm\({}^{-1}\)K\({}^{-1}\), respectively. The wall domain, \(\Omega_{w}\), is defined as the volume between the outer surface and the inner surface of the tube. The enveloping domain, \(\Omega_{e}\), is defined as the external volume that surrounds the tube. The boundary conditions were the following: \(\Omega_{w}\) is a thermal conductor, \(\Omega_{e}\) is a thermal insulator, and there are no internal heat sources. The boundary conditions were the following: the external temperature, \(T_{0\_ext}\) = 30 \({}^{\circ}\)C and the internal temperature, \(T_{0\_int}\) = 80 \({}^{\circ}\)C. It was considered that the heat advanced in a radial direction, that the upper and lower covers of the tube were perfect insulators and that there were no heat sources inside the tube. In addition, it was considered a pure heat transmission without convection or radiation. Therefore, it is a three-dimensional problem with axial symmetry. To measure the temperatures, both in the 2D and 3D distributions, a parametric cut was made on a segment, in a radial direction, and there, the temperatures were measured, as can be seen in Figure 6.  
Figure 5: Errors histogram for FEMM vs. analytical solution, mesh with 1,013,144 nodes. The inner tube had a thermal conductivity of 50.0 Wm\({}^{-1}\)K\({}^{-1}\), typical of steels, with a value between 46.6 and 51.9 Wm\({}^{-1}\)K\({}^{-1}\). The outer tube had a thermal conductivity of 193.0 Wm\({}^{-1}\)K\({}^{-1}\), typical of aluminum alloys, with a value between 95.3 and 222.0 Wm\({}^{-1}\)K\({}^{-1}\). In Figure 7, we analyzed a two-dimensional temperature distribution, which corresponds to the generatrix rectangle of revolution of the tube wall. This temperature distribution has been generated with FEMM using FEM. The temperature that was used as a reference for the validation of the results was that obtained by a horizontal parametric cut at half the height of the bi-dimensional distribution. Then, in this section, we calculated, through \([M_{\lambda}]\), the heat conduction in the tube of Figure 6. The matrix \([M_{\lambda}]\) was proposed as an alternative to the matrices used in [18; 19]. These results will be compared with those obtained with the new matrix \([M_{\tau}]\) in the Section 3.2. Figure 6: Parametric cut of temperatures in a 3D model with the Cell Method (CM). Figure 7: Temperatures of reference with FEMM and \(\lambda_{1}\) = 50.0 Wm\({}^{-1}\)K\({}^{-1}\) and \(\lambda_{2}\) = 193.0 Wm\({}^{-1}\)K\({}^{-1}\). On the one hand, a 2D FEMM calculation of 11,042 nodes was carried out, and on the other hand, five calculations were made with CM for an increasing number of nodes, ranging from 108 to 13,136 nodes, using \([M_{\lambda}]\). From 4461 nodes, the results were practically coincident between CM 3D and FEMM 2D, which we considered to be the solution to the problem, because being axisymmetric, its 3D equivalent mesh had many more nodes. The results obtained are shown in Figure 8.  
The distribution of errors when comparing CM with the temperatures obtained in FEM is shown in Figure 9. There is a bias between the actual distribution of errors (bar chart in blue) and the theoretical distribution (Gauss bell type, red color chart). This bias is approximately 0.1%. The errors, as expected, decreased with the increase in mesh density, as shown in Table 3.  
Figure 8: Temperatures in the tube with two thermal conductivities using \([M_{\lambda}]\).