## 4 Numerical Results
### Control Arm Optimization
There were slight differences in the final iteration volume fraction between the FEM and RPIM results. \begin{table}
\begin{tabular}{c c c} \hline \hline
**PR** & **Method** & **Solution** \\ \hline  & & Volume  
To facilitate the visual comparison, two solutions from each method were selected from Table 3. Figure 6 shows two solutions of the previously presented optimisation of the initially solid control arm with density reduction restriction in the boundary conditions zone for \(PR=5\%\). The chosen iteration was the very last one, in order to analyse the final result of the used algorithm. \begin{table}
\begin{tabular}{c c c} \hline \hline
**PR** & **Method** & **Solution** \\ \hline \hline  & & Volume fraction \\ \hline
0 & & \\ \hline  & & it. 8 \\ FEM & & it. 15 \\ FPIM & & it. 33 \\  & & it. 8 \\ RPIM & & it. 15 \\  & & it. 8 \\  & & it. 9 \\ FEM & & it. 13 \\ FPIM & & it. 18 \\ RPIM & & it. 9 \\ \hline \end{tabular}
\end{table} 
Table 2: Optimised solutions obtained with discretization D2. In each figure is indicated the mass at the iteration \(i\) (\(m_{i}\)) with respect to the initial mass (\(m_{o}\)). Figure 6: Optimized solutions for discretization D4, with a density reduction restriction in the boundary conditions sections, for \(PR=5\%\), at iteration 30 for (**A**) FEM method and (**B**) RPIM method. \begin{table}
\begin{tabular}{c c c} \hline \hline
**PR** & **Method** & **Solution** \\ \hline \hline \multicolumn{3}{c}{Volume fraction.} \\ \hline
0 & & 1 \\ \hline  & it. 8 & it. 15 & it. 24 & it. 30 \\  & FEM & & \\
5\% & \(\text{m}_{i}=88.11\%\)\(\text{m}_{o}\) & \(\text{m}_{i}=78.53\%\)\(\text{m}_{o}\) & \(\text{m}_{i}=65.57\%\)\(\text{m}_{o}\) & \(\text{m}_{i}=60.49\%\)\(\text{m}_{o}\) \\  & RFIM & & \\  & & \\  & & \\  & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimised solutions obtained with discretization D3. In each figure is indicated the mass at the iteration \(i\) (\(m_{i}\)) with respect to the initial mass (\(m_{o}\)). Since FEM and RPIM formulations are different (from a mathematical point of view), for the same material model they lead to different variable fields (very close, but different). Because the adopted optimisation algorithm is iterative, and the solution of the next iteration is dependent on the solution of the previous iteration, it is not straightforward that FEM and RPIM analyses tend to the same solution. In this work, it was found that FEM and RPIM analyses allow us to obtain similar results, but with observable differences. For instance, observing just the last column of Table 1, it is possible to observe that for a \(PR=5\%\), RPIM allows one to achieve in the upper part of the model a more defined trabecular arrangement (the trabeculae are more noticeable). Additionally, in the centre part of the model, a thin diagonal trabecula appears in both FEM and RPIM. However, a closer look allows us to observe that RPIM is capable of suggesting a capital, connecting the thin diagonal trabecula to the thick centre diagonal trabecula (which will better distribute the stresses). The same phenomenon can be observed in Table 2 (\(PR=5\%\)). In Tables 3 and 4, \begin{table}
\begin{tabular}{c c c} \hline \hline
**PR** & **Method** & **Solution** \\ \hline \hline \multicolumn{3}{c}{Volume fraction} \\ \hline
0 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimised solutions obtained with discretization D4. In each figure is indicated the mass at the iteration \(i\) (\(m_{i}\)) with respect to the initial mass (\(m_{o}\)). comparing the last column results, it is possible to observe that RPIM allows for once again more defined trabeculae arrangement (all the trabeculae predicted by the RPIM are much more defined than the ones predicted by the FEM). Visually, the RPIM solution exhibits a clearer truss-like structure and a wider range of intermediate densities compared with the FEM solution. Both of these characteristics are potentially advantageous for additive manufacturing, as they can influence the component's structural performance.