# Smart Topology Optimization Using Adaptive Neighborhood Simulated Annealing
## 5 Results
### Heat Conduction Problem  
The results for the TO of the cantilever and half-MBB beam shows that the proposed method is as efficient as the gradient-based TO methods. It should be noted that the main significance of the proposed method is the inclusion of sensitivity information without using any gradient information. Since this method works based on a probabilistic search, the final shapes are not as smooth as the other methods. However, the use of finer mesh or the application of some post processing can improve the final results.  
In order to show the usefulness of this method in practical problems, a 3D heat conduction problem was also studied. The schematic of the design domain is shown in Figure 9. In the heat transfer problem, the heat sink is at a different temperature and the heat flows throw the design domain. The formulation of the heat transfer in the FE format is similar to (3). The difference is that in the heat transfer problem \(U\) is the finite global nodal temperature vector, \(F\) is the global thermal load vector, and \(K\) is the global thermal conductivity. The thermal conductivity matrix or stiffness matrix for the heat conduction problem can be obtained by the assembly of element thermal conductivity matrices. The SIMP method gives the thermal conductivity matrix of each element by interpolating the density of each element. The design domain in Figure 9 is in 3D and all the nodes and matrices should be in 3D as well. For isotropic material, the thermal properties are equal in all directions. The objective function of this heat conduction problem to be minimized is the heat transfer compliance.  
According to the literature [8], a \(40\times 40\) element in \(x\) and \(y\) directions can define the problem efficiently. To ease the results comparison, just one element is considered in the \(z\) direction. To determine the proper number of iterations, the optimization process has been performed with different values of \(m\) and the results are shown in Figure 10. From Figure 10, the converged objective function approximated the optimum solution for values of \(m\) greater than 300. The higher the \(m\), the slightly better the results will be, and since the computational costs are low, the value of \(m=5000\) was chosen. The final optimum solution values of heat transfer compliance from the literature [8] and the proposed method are presented in Table 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Volume Fraction** & **Compliance from [6]** & **Compliance from the Method Proposed Herein** \\ \hline
0.5 & 77.42 & 80.87 \\
0.6 & 66.73 & 68.55 \\
0.7 & 59.69 & 60.72 \\
0.8 & 54.55 & 55.60 \\
0.9 & 51.31 & 51.95 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Half-MBB beam compliance for different volume fractions from the literature and the proposed method.  
Figure 9: Schematic of the design domain for the heat conduction problem.  
The final shape of optimized topologies from the literature [8] and the proposed method are presented in Figure 11 for a \(40\times 40\times 1\) element design domain.  
Figure 11: The results for optimized topology of heat conduction in different volume fractions of 0.5, 0.6, 0.7, 0.8, and 0.9.(**a**) Results from the literature. (**b**) Results from the proposed method.  
Figure 10: SA converged objective function value versus value of \(m\) for the heat conduction problem with a 0.5-volume fraction (\(m\) is the number of objective function evaluations to reach thermal equilibrium in a specific temperature).  
As shown in Figure 11 and Table 3, the results from the proposed method and the literature for the heat transfer problem are almost the same. The value of the minimum compliance has also slightly improved.