# Sensitivity Analysis and Filtering of Machinable Parts Using Density-Based Topology Optimization
## 3 Sensitivity Analysis  
The results of the TOs are presented in tables, which can be found in the Appendices B.1-B.4. Note that the relative density \(\mathrm{x_{i}}\) of each element is represented with a greyscale factor in Appendices B.2-B.4, where a darker element represents a value closer to 1. Only elements with a relative density \(\mathrm{x_{i}}>0.5\) are shown. Numerical results for the analysis are shown in Table 1, being  
* Sim. ID: the identifier of the specific TO case. * Penalization, p, and Filter Radius, R.
* Iterations: the number of iterations required to achieve the convergence condition. Note that a maximum of 5000 iterations was set, so if the TO reaches this value, convergence is not reached. * It. Time: the average time the TO took in every iteration. * Objective, the compliance of the structure, calculated as in Equation (8): \[\mathrm{c(x)}=\mathbf{U}^{\mathrm{T}}\mathbf{K}\mathbf{U}=\sum_{n=1}^{n} \mathrm{E_{e}(x_{e})u_{e}^{T}k_{0}u_{e}}\] (8) where \(\mathbf{K}\), \(\mathbf{U}\), and \(\mathbf{F}\) are the global stiffness matrix and the global displacement and force vectors, respectively. N is the total number of elements, \(\mathrm{u_{e}}\) is the e element displacement vector, \(\mathrm{k_{0}}\) is the element stiffness matrix, and \(x_{e}\) is the e element density. * Non-Discreteness: the described parameter in Section 2.3.1. * Machinability: the described parameter in Section 2.3.2. Figure 4: Element directly accessible for a tool detection process. In blue, solid phase; in green, the element under analysis in coordinates (_x,y_). Red arrows indicate hypothetical tool trajectory. Figure 5: Element accessible for a tool through a neighbor detection process. In blue, solid phase; in green, the element under analysis in coordinates (_x,y_). Red arrows indicate hypothetical tool trajectory. Next, in Figures 6 and 7, some remarkable results are displayed along with fitted curves of the discrete results data. As the filter radius increases, it is expected that higher filter radius values lead to simpler shapes, as the density filter in Equation (2) affects a greater number of elements. However, when the value of the radius becomes very large, the results start to exhibit discontinuities, and the computational cost increases. Nevertheless, the fitted data depicted in Figure 6 reveal an unexpected trend. It is shown that as the filter radius grows, the TO process requires fewer iterations and less time per iteration for computation. This finding contradicts the initial expectation, as a larger filter radius entails analyzing more elements in each iteration. Figure 7 provides an analysis of the TOs' manufacturability. In Figure 7, higher penalization values lead to slightly lower M\({}_{\text{nd}}\). This is an expected outcome, since higher penalization corresponds to a more aggressive interpolation of the SIMP function defined by Equation (1) and Figure 2.  
Regarding non-discreteness, the results exhibit regular and predictable behavior for filter radii ranging from Sim. ID 19-72. These values increase with the corresponding filter radii. Similarly, machinability also increases with the filter radius. As mentioned previously, a higher radius corresponds to simpler shapes that are more easily machinable. Figure 6: Comparison of the iterations and time per iteration in every TO. (**a**) Sim. ID vs. Iterations graph: dots are the discrete data, and the blue line represents the corrected data. (**b**) Sim. ID vs. Mean Iteration Time: dots are the discrete data, and the blue line represents the corrected data. Figure 7: Comparison of the measure of non-discreteness and machinability in every TO. (**a**) Non-Discreteness vs. Sim. ID: dots are the discrete data, and the blue line represents the corrected data. (**b**) Machinability vs. Sim. ID: dots are the discrete data, and the blue line represents the corrected data. In Figure 7, it is evident that results become more inconsistent and show greater dispersion and poor predictability when large filter radii are used.