## 3 Numeric Experiments and Results Discussion  
We intend to examine some examples of challenges of identifying an arbitrary number and planning inclusions, their geometry and their location in a plate, under the influence of temperature and thermal flows, using topological optimization. Let us consider an isotropic flat body in the form of a thin plate occupying the area \(\Omega=\{x\in[0;a],\ y\in[0;b]\}\), inside which are inclusions occupying the area \(\Omega_{i},\ i=\overline{1,m}\), with the thermal conductivity coefficients \(k_{m}\) (Figure 2). We write the classical differential heat equation [40] for the temperature field \(T\) inside the area \(\Omega\)  
\[k\bigg{[}\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial x^{2 }}\bigg{]}=0,\text{ in }\Omega(x,y), \tag{7}\]  
and boundary conditions  
\[T\big{|}_{B_{T}}=T_{0}\,, \tag{8}\]  
\[k\frac{dT}{dn}\bigg{|}_{B_{q}}=q_{0}, \tag{9}\]  
where \(T_{0}\) is the given temperature at \(\Gamma_{T}\); \(q\) is the heat flux, installed on the \(\Gamma_{q}\) and \(n\) is the vector of the normal to the boundary of the area. The equations were reduced to their dimensionless forms in the standard way [46]. Next, problems (7)-(9) were solved by the Finite Element Method using the COMSOL Multiphysics software. Problem 1: Investigating the Convergence of a Solution by the FEM Using the COMSOL Multiphysics Software  
To obtain reliable results, the convergence of the solution by the finite element method was studied using the COMSOL Multiphysics software. The basic material of the plate in the \(\Omega_{1}\) area was steel, with the coefficient of thermal conductivity (CTC) \(k_{1}=44.5\frac{\text{w}}{\text{m-K}}\), and \(\Omega_{2}\)--stood for inclusions of aluminum with the CTC \(k_{2}=238\frac{\text{w}}{\text{m-K}}\). The reliability of the obtained results was considered achieved if the error did not exceed 0.5% of the previous solution. The study was carried out for three types of inclusions related to geometry (rectangle, rhombus, square), but the total number of inclusions was equal to 18. The obtained results are reported in Table 1.  
Let us examine the partition into FE of 18 inclusions in the form of rectangles, which are located: vertically (Table 2), horizontally (Table 3) and at an angle of 45 degrees (Table 4). Let us also study the influence of location and geometry on the identification result. The tables are reported: the width of the rectangle; rectangle length; the number of FEs when splitting into a small grid and the number of FEs when splitting into a large grid. Each table presents data on the partition into finite elements for three similar cases, while gradually decreasing the width of the plate. The partitions are presented for two values of the size inclusion: (a) \(a=2\cdot 10^{-2}\text{ m},\ b=1\cdot 10^{-1}\text{ m}\); (b) \(a=1\cdot 10^{-2}\text{ m},\ b=1\cdot 10^{-1}\text{ m}\). Figure 2: Plate design with marked areas \(\Omega_{1},\Omega_{2},\dots\ \Omega_{m}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Coordinates of Location of Centers** \\
**of Rectangular Inclusions** \\ \hline \hline \(\Omega_{n}=\left\{a=1\cdot 1^{-1},b=2\cdot 10^{-2}\right\},\ n=1,2,\ldots,18\): \\
1. (0.2, 0.9), 2. (0.5, 0.9), 3. (0.8, 0.9), 4. (0.2, 0.75), 5. (0.5, 0.75), 6. (0.8, 0.75), 7. (0.2, 0.6), 8. (0.5, 0.6), 9. (0.8, 0.6), 10. (0.2, 0.45), 11. (0.5, 0.45), 12. (0.8, 0.45), 13. (0.2, 0.3), 14. (0.5, 0.3), 15. (0.8, 0.3), 16. (0.2, 0.15), 17. (0.5, 0.15), 18. (0.8, 0.15) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The area \(\Omega\), divided into 18 rectangular inclusions, with corresponding coordinates of the inclusion center—problem 1.