## 5 Application Discussion
### A Capacitive Accelerometer
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Results**} & \multicolumn{2}{c}{**Case 1:2**} & \multicolumn{2}{c}{**Case 2:3**} & \multicolumn{2}{c}{**Case 3:5**} \\  & & **Dimensions** & **Dimensions** & **Dimensions** \\ \hline \multirow{5}{*}{Proposed method} & \(N_{F}\) & 128 & 142 & 198 \\ \cline{2-5}  & Iterations & 3 & 4 & 4 \\ \cline{2-5}  & \multirow{2}{*}{\(\mathbf{X}_{j}^{*}\) (mm)} & (0.200, 0.185, 0.000) & (0.200, 0.185, 0.000) & (0.200, 0.187, 0.000) \\  & & (0.200, 0.186, 0.000) & (0.200, 0.186, 0.000) & (0.200, 0.188, 0.000) \\  & & (0.200, 0.188, 0.000) & (0.200, 0.187, 0.000) & (0.200, 0.190, 0.000) \\ \cline{2-5}  & \multirow{2}{*}{\(v_{j}^{*},Bel_{j}^{*}\)} & (1.35 \(\upmu\)m, 85.1\%) & (1.41 \(\upmu\)m, 85.8\%) & (1.74 \(\upmu\)m, 86.4\%) \\  & & (1.70 \(\upmu\)m, 96.5\%) & (1.55 \(\upmu\)m, 96.4\%) & (2.06 \(\upmu\)m, 96.1\%) \\  & & (2.00 \(\upmu\)m, 100.0\%) & (1.65 \(\upmu\)m, 100.0\%) & (2.62 \(\upmu\)m, 100.0\%) \\ \hline \multirow{2}{*}{Reference solution} & \multirow{2}{*}{\(v_{j}^{r},Bel_{j}^{r}\)} & (1.35 \(\upmu\)m, 85.1\%) & (1.41 \(\upmu\)m, 85.8\%) & (1.71 \(\upmu\)m, 85.8\%) \\  & & (1.69 \(\upmu\)m, 95.5\%) & (1.55 \(\upmu\)m, 96.4\%) & (2.02 \(\upmu\)m, 95.4\%) \\ \cline{1-1}  & & (2.00 \(\upmu\)m, 100.0\%) & (1.65 \(\upmu\)m, 100.0\%) & (2.56 \(\upmu\)m, 99.9\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The optimal set of the micro-force sensor problem. nominal values are \(\overline{\alpha}_{i}=0\) mm, \(i=1,\,2,\,3\). All of them are uncertain variables, respectively caused by machining errors and assembly errors. According to the existing samples, the marginal BPAs are listed in Figure 10.  
After being given a series of \(Bel_{j}^{T}=80\%\), \(85\%\), \(90\%\), \(95\%\), \(99.9\%\), Equation (26) can be rewritten as Equation (27):  
\[ \begin{array}{l}\min v\\ \text{st.}\,Bel\left(f=\frac{A(X_{\mathbf{x}})}{\Delta T}\leq v\right)\geq Bel_{j} ^{T}\end{array} \Bigg{\}},j=1,\,2,\,\ldots,\,6 \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Part** & **Fixed** & **Movable** & **Coil** & **Counter** & **Block 1** & **Block 2** \\ \hline Material & Silicon & Silicon & Copper & Wolfram & Wolfram & Aluminium \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material of accelerometer parts. Figure 8: The sensing structure of a capacitive accelerometer. Figure 10: The marginal BPAs of variables in the accelerometer problem. BPA: basic probability assignment; FD: frame of discernent. Figure 9: The FEM of the capacitive accelerometer. For easy reproduction of results, the response surface of \(A(\mathbf{X},\mathbf{\alpha})\) is constructed as follows:  
\[ \begin{array}{l}A=1.207X_{1}^{2}-0.430X_{1}X_{2}-18.06X_{1}+1.004X_{2}^{2}-13.9 74X_{2}\\ +100.9\alpha_{1}^{2}-9.0\alpha_{1}(\alpha_{1}-1)+89.0\alpha_{2}^{2}-7.2\alpha_ {2}+40.9\alpha_{3}^{2}-6.7\alpha_{3}+144.0\end{array}  \tag{28}\]  
Next, the EBRO is performed by the proposed method and the double-loop method. The initial design point is selected as \(\overset{\text{-}}{\mathbf{X}}^{(0)}=(9.6\text{mm},\,9.6 mm)\), and \(f\overset{\text{-}}{\mathbf{X}}^{(0)}=0.565\upmu\text{m}/^{\circ}\text{C}\). All results are given in Table 5. The proposed method converges the optimal set after four iteration steps. Each element of the optimal set is very close to that of the reference solution. This indicates to some extent the convergence and accuracy of the proposed method. As for efficiency, the performance function evaluations of the proposed method are done 171 times. Compared to the double-loop method (12,842 times), the efficiency of this method has a definite advantage. Given that hundreds of simulations or dozens of hours of computation are acceptable for most engineering applications, it is feasible to directly call the time-consuming simulation model when performing the EBRO in practice. On the other hand, the proposed method provides six design options under different robustness requirements in this example. The higher the robustness, the greater the performance threshold.