## 2 Mathematical and Numerical Modeling  
### Description of the Cavity Inserted in the Solid Domain Problem  
Figure 1 shows the double Y-shaped cavity in a two-dimensional domain. The third dimension (\(W\)) is perpendicular to the plane of Figure 1. The gray region in Figure 1 represents the solid domain which has a constant and uniform heat generation at the volumetric rate given by \(q^{{}^{\prime\prime\prime}}\) (\(W\cdot\mathrm{m}^{-3}\)). Due to the symmetry of the domain about the \(y\)-axis, only half of the computational domain of the solid is solved. The thermal conductivity \(k\) is constant in the solid domain, and its outer boundary surfaces are perfectly insulated (adiabatic conditions). The double Y-shaped cavity surfaces are kept at a minimum temperature (\(T_{min}\)). In this case, the cavity acts by removing the heat generated by the solid domain. The lowest imposed temperature simulates a condition where a phase change convection heat transfer flows through the cavity. As a simplifying hypothesis, the heat transfer coefficient on the cavity wall is assumed high enough in such way the convective resistance can be neglected compared to the solid conduction resistance. The problem studied here is subjected to the simplification hypothesis listed below:  
1 The problem is considered at the steady-state;  
2 The heat generation per unit volume (\(q^{{}^{\prime\prime\prime}}\)) is constant in the solid domain;  
3 The thermal conductivity is also constant;  
4 The domain is two-dimensional. Therefore, the problem is modeled by the heat diffusion equation, which after the imposition of the simplification hypothesis is given by [23]:  
\[\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+ \frac{q^{{}^{\prime\prime\prime}}}{k}=0 \tag{1}\]  
where \(T\) is the temperature (\(K\)), \(x\) and \(y\) are the spatial coordinates (m), \(q^{{}^{\prime\prime\prime}}\) is the heat generation per unit volume (\(W\cdot\mathrm{m}^{-3}\)) and \(k\) is the thermal conductivity (\(W\cdot\mathrm{m}^{-1}\cdot K^{-1}\)). The objective of the present study is to determine the optimal geometry (\(H/L\), \(H_{0}/L_{0}\), \(H_{1}/L_{1}\), \(H_{2}/L_{2}\), \(S_{1}/H_{0}\), \(\alpha\), and \(\beta\)) that minimizes the maximum excess of temperature (\(T_{max}-T_{min}\))/(\(q^{{}^{\prime\prime}}A\)). In the present work, four parameters are investigated (\(\alpha\), \(\beta\), \(S_{1}/H_{0}\), and \(H_{0}/L_{0}\)) while the others are kept constant. According to the CD method, the geometrical investigation is subjected to two constraints, the total area of the domain and cavity area, which are given, respectively, by:  
\[A=HL \tag{2}\]  
\[A_{c}=H_{0}L_{0}+2H_{1}L_{1}+2H_{2}L_{2} \tag{3}\]  
The dimensionless form of the Equations (2) and (3) are represented by the following equations:  
\[\widetilde{H}\widetilde{L}=1 \tag{4}\]  
\[\varphi_{c}=\frac{A_{c}}{A}=\widetilde{H}_{0}\widetilde{L}_{0}+2\varphi_{1}+ 2\varphi_{2} \tag{5}\]  
where:  
\[\varphi_{1}=\widetilde{H}_{1}\widetilde{L}_{1} \tag{6}\]  
\[\varphi_{2}=\widetilde{H}_{2}\widetilde{L}_{2} \tag{7}\]  
The temperature fields in the solid domain were determined by solving the heat diffusion equation given in dimensionless form by:  
\[\frac{\partial^{2}\theta}{\partial\widetilde{x}^{2}}+\frac{\partial^{2}\theta }{\partial\widetilde{y}^{2}}+1=0 \tag{8}\]  
In the dimensionless process of the heat diffusion equation, the term \(q^{{}^{\prime\prime\prime}}/k\) is evidenced in all terms, given rise to the unity in the third term of Equation (8). The dimensionless variables can be written as:  
\[\theta=\frac{T-T_{min}}{q^{{}^{\prime\prime\prime}}\cdot\frac{A}{k}} \tag{9}\]  
\[\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+ \frac{q^{{}^{\prime\prime\prime}}}{k}=0 \tag{1}\]0 Computational domain of the double Y-shaped cavity. \[