## 4 Conclusions  
In this paper, a new constitutive matrix \([M_{\tau}]\) for thermal conduction for tetrahedral meshes, in a steady state thermal regime, has been developed through a new algebraic methodology, using the Cell Method. The results have been compared with those obtained for the same problem by means of the constitutive matrix \([M_{\lambda}]\), developed previously in other works. Taking a 2D axisymmetric model as reference, calculated with the finite element method, the errors obtained with the new matrix \([M_{\tau}]\) are of the order of 0.0025%, much lower than those obtained with \([M_{\lambda}]\). On the other hand, the finite formulation and its associated numerical method, the Cell Method, allows for great flexibility when mathematically modelling a physical phenomenon such as conduction heat transfer. The main advantage of the method proposed in this article is its simplicity. The constitutive matrices developed by previous methods presented complex calculations, while the new constitutive matrix, proposed in this work, depends exclusively on the coordinates of the vertices of the tetrahedra, which constitutes the mesh. In addition, the errors are much smaller with the new matrix, and this allows for meshes of smaller number of elements, obtaining the same precision with a lower temporal cost. As it has been underlined before, the simplicity of the method and its greater precision mean that the new methodology can be applied to more complex problems, such as the calculation of the thermal heating of the rotor and the stator of an electrical machine of much more complex geometry, and with more complex physical properties and boundary conditions, even convective types. This is critical in the problems in transitory regime. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mesh [nodes]** & **102** & **108** & **2328** & **6428** & **12,895** & **References** \\ \hline R\({}^{2}\) [0, +1]. Optimum: +1 & 0.9910 & 0.9932 & 1.0000 & 1.0000 & 1.0000 & [26] \\ RMSPE [–1, +1]. Optimum: 0 & 0.0576 & 0.0508 & 0.0026 & 0.0027 & 0.0008 & [27] \\ MAEP [–1, +1]. Optimum: 0 & 0.0522 & 0.0462 & 0.0023 & 0.0024 & 0.0007 & [27] \\ PBIAS [–1, +1]. Optimum: 0 & 0.0496 & 0.0442 & 0.0014 & 0.0022 & 0.0004 & [28] \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Metrics for thermal simulation using \([M_{\tau}]\). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **FEMM ref: \({}^{\circ}\)C** & **Obtained temp. \({}^{\circ}\)C** & **Error \%** \\ \hline \([M_{\lambda}]\) 13,136 nodes & 64.0653 & 63.6975 & \(-\)0.5741 \\ \([M_{\tau}]\) 12,895 nodes & 63.6959 & 63.6975 & 0.0025 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Temperature errors using \([M_{\lambda}]\) and \([M_{\tau}]\). **Author Contributions:** [PERSON] supported the theory background, developed simulations, analyzed the data, and wrote the paper. [PERSON] and [PERSON] supported the theory background, analyzed the data, and wrote the paper. All of the authors read and approved the final manuscript. **Funding:** This research received no external funding. **Acknowledgments:** We want to acknowledge the administrative and technical support given by the Department of Electronic Engineering and Automatics (DIEA) of the University of Las Palmas de Gran Canaria (ULPGC), Spain. **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**