## 3 Numeric Experiments and Results Discussion
\begin{table}
\begin{tabular}{c c 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. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Coometry and Dimensions Inclusions** & \(\Omega_{n}=\left\{a=1\cdot 10^{-2},\ n=1,2,\ldots,18\right\}\): \\ \begin{table}
\begin{tabular}{c c c c} \hline
**Type of Boundary Conditions** & **Temperature** & **Results of the** \\
**(10)–(12)** & **Distribution** & **Topological Optimization** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Distribution of the thermal field and identification results for rectangular-shaped inclusions (18 elements) rotated by 90 degrees—Problem 3; \(\Omega_{n}=\big{\{}a=1\cdot 1^{-1},b=2\cdot 10^{-2}\big{\}}\). \begin{table}
\begin{tabular}{c c c c} \hline
**Type of Boundary Conditions (10)–(12)** & **Temperature Distribution** & **Results of the** \\  & & **Topological Optimization** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._From the data presented in Table 2, several conclusions can be drawn: (1) The location of rectangular inclusions significantly effects the partition of the grid (\(N_{1}\), \(N_{2}\)); (2) size width is decreased--a, for rectangular inclusions, leads to an increase in the number of FE, and for \(N_{2}\), the increase is by more than 2 times. Problem 2: Investigation of the Influence of the Boundary Conditions of the Thermal Problem on the Identification of a Large Number of Inclusions in the Form of Rectangles  
In this example, the influence of three types of boundary conditions of the thermal problem on the identification of a large number of inclusions in the form of rectangles was considered. The following boundary conditions for the thermal problem (7) were used:  
\[k\frac{dT}{dn}|_{l_{1}}=q=60,T|_{l_{2}}=T|_{l_{3}}=T|_{l_{4}}=0 \tag{10}\]  
\[k\frac{dT}{dn}|_{l_{1}}=k\frac{dT}{dn}|_{l_{2}}=q=60,T|_{l_{3}}=T|_{l_{4}}=0 \tag{11}\]  
\[k\frac{dT}{dn}|_{l_{1}}=k\frac{dT}{dn}|_{l_{2}}=k\frac{dT}{dn}|_{l_{4}}=q=60, T|_{l_{3}}=0 \tag{12}\]  
The results of the distribution of the thermal field and identification of inclusions for the different boundary conditions of the thermal Equation (7) are shown in Table 3. \begin{table}
\begin{tabular}{c c c} \hline
**Type of Boundary Conditions** & **Temperature** & **Results of the** \\
**(10)–(12)** & **Distribution** & **Topological Optimization** \\ \hline \end{tabular}
\end{table} 
Table 4: _Cont._Upon impact of heat flux only to the left boundary of the plate (boundary condition (10) (Table 3a)), the left column of inclusions was identified. Inclusions number 2 and 17 in the center column are incompletely identified (Table 3d). In the case of an impact of heat flux on the left and upper boundary of the plate (boundary condition (11), (Table 3b)), inclusions 17 and 18 were not completely identified (Table 3e). In the case when the heat fluxes impacted on the left, upper and lower boundaries of the plate (boundary condition (12), (Table 3c)), then all inclusions were identified (Table 3f). Thus, the use of the boundary condition (12) allows you to fully identify all inclusions. It can be concluded that these boundary conditions significantly affect the inclusion's identification (Table 3f) and allow them to identify completely.