# Sensitivity Analysis and Filtering of Machinable Parts Using Density-Based Topology Optimization
## 5 Results
### Conclusions  
Through this study, several conclusions can be drawn regarding the parameters to be considered for designers when using TO as the design methodology, as well as how to fix the issues found. The proposed solutions can be interpreted as potential implementations for commercial software packages in the future, as this kind of methodology has not been implemented yet. In the sensitivity analysis, modifications over the penalization and filter radius are analyzed in a concrete 3D case. A total of 92 simulations are showcased, presenting the most complete study of this kind found in the bibliography. The results are discussed in Section 3, but additional notes are included regarding the paper's objective of aiding designers. When designing functional parts, it is important to aim for a low \(\mathrm{M}_{\mathrm{nd}}\) and high machinability. Additionally, the objective function (compliance) should be minimized while meeting the prescribed constraints, the volume fraction in this case study. The main issue detected in the sensitivity analysis is the fact that a low measure of non-discreteness \(\mathrm{M}_{\mathrm{nd}}\) and high machinability cannot be achieved at the same time. This can be seen in Figure 7, but especially in Figure 8. As the volume fraction is the same for all the simulations, the objective is clearly dependent on the penalization, obtaining better results for low penalizations as Figure 9 and Appendix B.1 evidence. However, worse non-discreteness results are obtained for low penalization factors (Figure 7). Machinability trends show that results with a higher filter radius are more manufacturable, as shown in Figure 7.  
Recognizing the limitations in achieving good machinability and non-discreteness, the need for additional tools to achieve the best results in terms of interpretation and manufacturability is acknowledged. For this paper, three additions have been studied and applied (two of them being a novelty): the machining filter, the ellipsoid-shaped density filter, and \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Case (a)** & **Case (b)** & **Case (c)** & **Case (d)** & **Case (e)** & **Case (f)** & **Case (g)** \\ \hline Iterations & 1018 & 1738 & 346 & 2086 & 1087 & 2639 & 790 \\ \(\mathrm{M}_{\mathrm{nd}}\) & 21.5065 & 0.8477 & 28.3512 & 3.901 & 1.1824 & 5.2443 & 6.1459 \\ Machinability & 89.577 & 94.381 & 95.751 & 100 & 95.479 & 100 & 100 \\ Compliance & 38.0111 & 19.5020 & 59.1714 & 136.4004 & 21.1516 & 46.4197 & 34.3937 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparative relevant data of the verification cases. the Heaviside step filter. The machining filter was developed for obtaining machinable parts in a concrete direction. The ellipsoid-shaped density filter has the aim of obtaining results with concrete features in certain selective directions (looking at, for example, the results of the sensitivity analysis). The Heaviside step filter is applied for obtaining low interpretation of the results or, in other words, low measure of non-discreteness. Table 2 and Appendix C evidence the successful addition of the three filters separately and combined. In the cases where the machining filter is applied (cases (d) and (f)), a machinability of 100% is achieved with a higher computation cost (more iterations until convergence and more time per iteration). In the cases where the Heaviside step filter is used (cases (b), (d), (e), and (f)), \(\mathrm{M_{nd}}\) is sensibly lower than \"standard\" cases. Finally, for the cases where the ellipsoid-shaped density filter is applied (cases (c) and (e)), the machinability is a bit better in comparison to those cases without the filter. Considering this, and especially looking at cases (f) and (g), the behavior end effectiveness of the developed filters is verified, as a low M subindex is obtained while having 100% machinability. The incrementation of the iterations until convergence is a normal consequence of adding extra constraints (filters in this case) that increase nonlinearity.