## 2 Mathematical and Numerical Modeling
### Description of the Cavity Inserted in the Solid Domain Problem
\begin{array}{c}\widetilde{x},\widetilde{y},\ \widetilde{H},\ \widetilde{H}_{0},\ \widetilde{H}_{1},\ \widetilde{H}_{2},\ \widetilde{L},\ \widetilde{L}_{0},\ \widetilde{L}_{1},\ \widetilde{L}_{2},\ \widetilde{S}_{1}=\frac{x,\ y,\ H,\ H_{0},\ H_{1},\ H_{2},\ L,\ L _{0},\ L_{1},\ L_{2},\ S_{1}}{A^{1/2}}\end{array}  \tag{10}\]  
For the sake of brevity, the boundary conditions equations of null flux in the outer surfaces are not addressed here. More details can be seen in the work of Ref. [16] which studied a similar problem but with a T-shaped cavity configuration. The boundary condition of minimal temperature (\(\theta=\theta_{min}\)) is imposed at all cavity surfaces and described in the following equations:  
\[ \begin{array}{c}\theta=\theta_{min}\ in\ 0\ \leq\ \widetilde{y}\ \leq\ \widetilde{S}_{1}-\frac{\widetilde{H}_{1}}{2 sin(90^{\circ}-\alpha)}\ and\ \widetilde{x}=-\frac{\widetilde{L}_{0}}{2}\ or\ \widetilde{x}=\frac{ \widetilde{L}_{0}}{2}\end{array}  \tag{11}\]  
\[ \begin{array}{c}\theta=\theta_{min}\ in\ \widetilde{S}_{1}-\frac{ \widetilde{H}_{1}}{2 sin(90^{\circ}-\alpha)}\ \leq\ \widetilde{y}\ \leq\ \widetilde{L}_{1}sin\alpha+\widetilde{S}_{1}-\frac{ \widetilde{H}_{1}}{2 sin(90^{\circ}-\alpha)}\\ and\ -\frac{\widetilde{L}_{0}}{2}-\widetilde{L}_{1}cos\alpha\ \leq\ \widetilde{x}\ \leq-\frac{ \widetilde{L}_{0}}{2}\ or\ \frac{\widetilde{L}_{0}}{2}\ \leq\ \widetilde{x}\ \leq\ \frac{L_{0}}{2}+\widetilde{L}_{1}cos\alpha\end{array}  \tag{12}\]  
\[ \begin{array}{c}\theta=\theta_{min}\ in\ \widetilde{x}=-\frac{\widetilde{L}_{0}}{2}- \widetilde{L}_{1}cos\alpha\ or\ \widetilde{x}=\frac{\widetilde{L}_{0}}{2}+\widetilde{L}_{1}cos\alpha\\ and\ \widetilde{L}_{1}sin\alpha+\widetilde{S}_{1}-\frac{\widetilde{H}_{1}}{2 sin(90 ^{\circ}-\alpha)}\ \leq\ \widetilde{y}\ \leq\ \widetilde{L}_{1}sin\alpha+\widetilde{S}_{1}+\frac{ \widetilde{H}_{1}}{2 sin(90^{\circ}-\alpha)}\end{array}  \tag{13}\]  
\[ \begin{array}{c}\theta=\theta_{min}\ in\ \widetilde{S}_{1}+\frac{ \widetilde{H}_{1}}{2 sin(90^{\circ}-\alpha)}\ \leq\ \widetilde{y}\ \leq\ \widetilde{H}_{0}-\frac{ \widetilde{H}_{2}}{sin(90^{\circ}-\beta)}\\ and\ \widetilde{x}=-\frac{\widetilde{L}_{0}}{2}\ or\ \widetilde{x}=\frac{ \widetilde{L}_{0}}{2}\end{array}  \tag{14}\]  
\[ \begin{array}{c}\theta=\theta_{min}\ in-\frac{\widetilde{L}_{0}}{2}- \widetilde{L}_{2}cos\beta\ \leq\ \widetilde{x}\ \leq-\frac{\widetilde{L}_{0}}{2}\ or\ \frac{ \widetilde{L}_{0}}{2}\ \leq\ \widetilde{x}\ \leq\ \frac{\widetilde{L}_{0}}{2}+\widetilde{L}_{2}cos\beta\\ and\ \widetilde{H}_{0}-\frac{\widetilde{H}_{2}}{sin(90^{\circ}-\beta)}\ \leq\ \widetilde{y}\ \leq\ \widetilde{H}_{0}-\frac{ \widetilde{H}_{2}}{sin(90^{\circ}-\beta)}+\widetilde{L}_{2}sin\beta\end{array}  \tag{15}\]  
\[ \begin{array}{c}\theta=\theta_{min}\ in\ \widetilde{x}=-\frac{ \widetilde{L}_{0}}{2}-\widetilde{L}_{2}cos\beta\ or\ \widetilde{x}=\frac{ \widetilde{L}_{0}}{2}+\widetilde{L}_{2}cos\beta\\ and\ \widetilde{H}_{0}-\frac{\widetilde{H}_{2}}{sin(90^{\circ}-\beta)}+ \widetilde{L}_{2}sin\beta\ \leq\ \widetilde{y}\ \leq\ \widetilde{H}_{0}+\widetilde{L}_{2}sin\beta\end{array}  \tag{16}\]  
\[ \begin{array}{c}\theta=\theta_{min}\ in\ \widetilde{H}_{0}\ \leq\ \widetilde{y}\ \leq\ \widetilde{H}_{0}+\widetilde{L}_{2}sin\beta\\ and\ -\frac{\widetilde{L}_{0}}{2}-\widetilde{L}_{2}cos\beta\ \leq\ \widetilde{x}\ \leq-\frac{ \widetilde{L}_{0}}{2}\ or\ \frac{\widetilde{L}_{0}}{2}\ \leq\ \widetilde{x}\ \leq\ \frac{ \widetilde{L}_{0}}{2}+\widetilde{L}_{2}cos\beta\end{array}  \tag{17}\]  
\[ \begin{array}{c}\theta=\theta_{min}\ in\ \widetilde{y}=\widetilde{H}_{0}\ and\ -\frac{ \widetilde{L}_{0}}{2}\leq\widetilde{x}\ \leq\ \frac{\widetilde{L}_{0}}{2}\end{array}  \tag{18}\]  
The objective is to minimize the dimensionless maximum excess of temperature, given by:  
\[ \begin{array}{c}\theta_{max}=\frac{T_{max}-T_{mix}}{q^{\prime\prime\cdot} \frac{A}{k}}\end{array}