## 2 Calculation of the New Thermal Constitutive Matrix \([M_{\tau}]\)
\right]\ =\ -\lambda\ [grad(\tau)]_{3\times 1}, \tag{23}\]  
where \(\lambda\) is the thermal conductivity coefficient, and, hence, the heat flow \(\left[Q^{\mu}\right]\) is  
\[\left[Q^{\mu}\right]_{6\times 1}\ =\ \left[\overline{S}\right]_{6\times 3} \left[\overline{q}\right]_{3\times 1}. \tag{24}\]  
Then, what is written in Equation (23) is equal to Equation (18), and therefore \(\left[Q^{\mu}\right]\ =\ [Q^{\mu}]\)  
\[-\lambda\left[\overline{S}\right]_{6\times 3}[grad(\tau)]_{3\times 1}\ =\ [M_{\tau}]_{6\times 6 }[G]_{6\times 4}[\tau]_{4\times 1}. \tag{25}\]  
Replacing Equation (18) and Equation (21) in the second member of Equation (25), we obtain  
\[-\lambda\left[\overline{S}\right]_{6\times 3}[grad(\tau)]_{3\times 1}\ =\ \left[ \overline{S}\right]_{6\times 3}[A_{\tau}]_{3\times 6}[X]_{6\times 1}. \tag{26}\]  
Simplifying the dual faces matrix \(\left[\overline{S}\right]\)  
\[-\lambda\ [grad(\tau)]_{3\times 1}\ =\ [A_{\tau}]_{3\times 6}[X]_{6\times 1}. \tag{27}\]  
Replacing \([X]\) with the values of \([G]\) and \([\tau]\), we obtain  
\[\left[X\right]_{6\times 1}\ =\ [G]_{6\times 4}[\tau]_{4\times 1}\ =\ \left[ \begin{array}{cccc}-1&1&0&0\\ -1&0&1&0\\ -1&0&0&1\\ 0&-1&1&0\\ 0&0&-1&1\\ 0&-1&0&1\end{array} \right]_{6\times 4}\ \ \left[ \begin{array}{c}\tau_{0} \\ \tau_{1}\\ \tau_{2}\\ \tau_{3}\end{array} \right]_{4\times 1}\ =\ \left[ \begin{array}{c}-\tau_{0}+\tau_{1} \\ -\tau_{0}+\tau_{2}\\ -\tau_{0}+\tau_{3}\\ -\tau_{1}+\tau_{2}\\ -\tau_{2}+\tau_{3}\\ -\tau_{1}+\tau_{3}\end{array} \right]_{6\times 1} \tag{28}\]and, therefore,  
\[\left[ \begin{array}{c}x_{0}\\ x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\end{array} \right]_{6\times 1}=\left[ \begin{array}{c}-\tau_{0}+\tau_{1}\\ -\tau_{0}+\tau_{2}\\ -\tau_{0}+\tau_{3}\\ -\tau_{1}+\tau_{2}\\ -\tau_{2}+\tau_{3}\\ -\tau_{1}+\tau_{3}\end{array} \right]_{6\times 1}. \tag{29}\]  
Then, observing Equation (27), replacing the value of \(\left[grad(\tau)\right]_{3\times 1}\), then,  
\[-\lambda\left[ \begin{array}{c}g_{x}\\ g_{y}\\ g_{z}\end{array} \right]_{3\times 1}=\left[A_{\tau}\right]_{3\times 6}\left[ \begin{array}{c}x_{0}\\ x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\end{array} \right]_{6\times 1}, \tag{30}\]  
that is to say  
\[-\lambda\left[ \begin{array}{c}g_{x}\\ g_{y}\\ g_{z}\end{array} \right]_{3\times 1}=\left[ \begin{array}{cccccccc}A_{00}&A_{01}&A_{02}&A_{03 }&A_{04}&A_{05}\\ A_{10}&A_{11}&A_{12}&A_{13}&A_{14}&A_{15}\\ A_{20}&A_{21}&A_{22}&A_{23}&A_{24}&A_{25}\end{array} \right]_{3\times 6}\left[ \begin{array}{c}x_{0}\\ x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\end{array} \right]_{6\times 1}, \tag{31}\]  
and, therefore, developing Equation (31), we obtain the following expression  
\[-\lambda\left[ \begin{array}{c}g_{x}\\ g_{y}\\ g_{z}\end{array} \right]=\left[ \begin{array}{c}A_{00}\ x_{0}+A_{01}\ x_{1}+A_{02}\ x_{2}+A_{03}\ x_{3}+A_{04}\ x_{4}+A_{05}\ x_{5}\\ A_{10}\ x_{0}+A_{11}\ x_{1}+A_{12}\ x_{2}+A_{13}\ x_{3}+A_{14}\ x_{4}+A_{15}\ x_{5}\\ A_{20}\ x_{0}+A_{21}\ x_{1}+A_{22}\ x_{2}+A_{23}\ x_{3}+A_{24}\ x_{4}+A_{25}\ x_{5} \end{array} \right]. \tag{32}\]  
The value of \(g_{x}\) is included in the following equation  
\[-\lambda\ g_{x} =A_{00}\ x_{0}+A_{01}\ x_{1}+A_{02}\ x_{2}+A_{03}\ x_{3}+A_{04}\ x_ {4}+A_{05}\ x_{5}. \tag{33}\]  
Replacing the values \(x_{i}\) developed in Equation (29), we obtain  
\[-\lambda\ g_{x} =A_{00}\ (-\tau_{0}+\tau_{1})+A_{01}\ (-\tau_{0}+\tau_{2})+A_{02}\ (-\tau_{0}+\tau_{3})+ \tag{34}\] \[+A_{03}\ (-\tau_{1}+\tau_{2})+A_{04}\ (-\tau_{2}+\tau_{3})+A_{05}\ (-\tau_{1}+\tau_{3}).\]  
If we group the terms affected by the same temperature value, then  
\[-\lambda\ g_{x} = \tau_{0}(-A_{00}-A_{01}-A_{02})+ \tag{35}\] \[+\tau_{1}(A_{00}-A_{03}-A_{05})+\] \[+\tau_{2}(A_{01}+A_{03}-A_{04})+\] \[+\tau_{3}(A_{02}+A_{04}+A_{05})\,\]  
Following the same procedure, we obtain the terms of \(g_{y}\)  
\[