## 3 Optimization Design
### Evaluation of Results and Discussion  
This step aims to verify the stress, displacement, and natural frequency results for optimal design. The optimal design found in Tables 2 and 5 was reconstructed in a 3D model using INVENTOR 2018 software, and simulated on ANSYS 18.2 Workbench software. Figures 10 and 11 illustrate the simulation results.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Parameters**} & \multicolumn{3}{c}{**DO-NSGA-II**} & \multicolumn{3}{c}{**RBDO-NSGA-II**} \\ \cline{2-6}  & **Point A** & **Point B** & **Point D** & **Point C** & **Point E** \\ \hline \(T_{1}\) (mm) & 0.48 & 1.2 & 1.4 & 1.2 & 1.15 \\ \(T_{2}\) (mm) & 0.84 & 0.46 & 0.5 & 0.61 & 0.61 \\ \(T_{3}\) (mm) & 0.5 & 0.63 & 0.74 & 0.6 & 0.61 \\ Maximum stress (MPa) & 84.76 & 73.56 & 77.33 & 73.7 & 72.33 \\ Input displacement (\(\mu\)m) & 16.25 & 13.81 & 13.63 & 13.5 & 13.5 \\ Input stiffness (\(K_{in}\)) & 48.21 & 53.11 & 53.03 & 59.26 & 59.26 \\ Natural frequency \(f_{1}(\text{Hz})\) & 1500 & 1499 & 1499 & 1500 & 1499 \\ Amplification ratio \(f_{2}\) & 3.405 & 3.13 & 3.105 & 3.104 & 3.101 \\ \hline \multirow{3}{*}{\(\beta^{FORM}\)} & \(\beta_{1}\) & 2.25 (98.77\%) & 4.20 (100\%) & 8.92 (100\%) & 4.11 (99.99\%) & 8.89 (100\%) \\  & \(\beta_{2}\) & 10.88 (100\%) & 12.96 (100\%) & 11.58 (100\%) & 14.20 (100\%) & 9.49 (100\%) \\ \cline{1-1}  & \(\beta_{3}\) & 0.00 (42.91\%) & 0.00 (48.26\%) & 1.77 (96.13\%) & 3.00 (99.87\%) & 3.05 (99.88\%) \\ \cline{1-1}  & \(\beta_{4}\) & 0.00 (43.11\%) & 0.004 (50.16\%) & 1.30 (90.24\%) & 3.00 (99.87\%) & 3.10 (99.90\%) \\ \hline \end{tabular}
\end{table}
Table 4: Comparison of DO and RBDO solutions.  
\begin{table}
\begin{tabular}{l l l l} \hline \multicolumn{1}{l}{**Design Variables**} & \multicolumn{1}{l}{**Values (mm)**} & \multicolumn{1}{l}{**Design Variables**} & \multicolumn{1}{l}{**Values**} \\ \hline \(T_{1}\) & 1.2 & \(f_{2}\) & 3.104 \\ \(T_{2}\) & 0.61 & \(\sigma_{max}\) & 73.7 MPa \\ \(T_{3}\) & 0.6 & \(D_{out}\)\((D_{y})\) & 40.0 \(\mu\)m \\ \(f_{1}\) & 1500 Hz & \(\delta_{x}\) & 0 \(\mu\)m \\ \(R_{0}\) & 99.87\% & \(\beta_{3}\) & 3 \\ \hline \end{tabular}
\end{table}
Table 5: Reliability analysis optimization results (point C).  
As shown in Figure 10, with a given input displacement \(y_{in}=13.5\)\(\upmu\)m provided by the PZT actuator, the output displacement \(y_{out}=39.65\)\(\upmu\)m in the \(y\)-direction is induced, which is equivalent to the amplification ratio \(A_{amp}=2.937\), and the parasitic motion in the \(x\)-direction of the tool fixture is completely eliminated \((\delta_{x}=0\)\(\upmu\)m). The maximum concentrated stress \(\sigma_{max}=73.705\) MPa satisfies the constraint for the allowable strength of the material. The finite element method ANSYS ADPL is used to evaluate the stiffness of the structure, including the input stiffness \((K_{in})\) and output stiffness \((K_{out})\). To calculate \(K_{in}\), an input force \(F_{in}=800\) N is applied to the input port, and the displacement of this  
Figure 10: FEM simulation results: (**a**) total displacement, (**b**) displacement in the \(x\)-direction, and (**c**) equivalent stress.  
point is measured as \(y_{in}=13.5\) um. It is easy to calculate that \(K_{in}=59.26\) N/um. The output stiffness is also calculated in the same way, by applying the output force \(F_{out}=50\) N to the output port, and the obtained displacement is \(y_{out}=39.65\) um. Therefore, the output stiffness \(K_{out}=5.2\) N/um. The relation between the input and output displacement is illustrated in Figure 11, which is stably linear in the operational range from 0 to 13.5 um.  
In the numerical evaluation of the first natural frequency in Figure 12a, it can be seen that \(f_{0}=1500\) Hz. There is a slight discrepancy between the FEM simulation and the optimal results in Table 4, due to the round-up of the design parameters for better fabrication in the following step.  
Figure 11: The simulated results of input displacement vs. output displacement and stress.  
Figure 12: _Cont._