## 4 Results and Discussion
The average computation time of the step of the solution of the complex eigenvalue problem in each iteration for the traditional subspace method was 4.3108 s, while the computation time of the same step for the proposed method was 1.5779 s, as shown in Table 1. The average computation time of the linear calculation of the global matrix of the sensitivity number was 6.1492 s, while the computation time of the same step in a parallel method was 0.9175 s. The modal assurance criteria and filter of the sensitivity numbers were simple vector multiplications and cost little computation effort. Thus, the proposed method saved approximately 76.14% of the computation time compared with the traditional linear method for sensitivity analysis of the topology optimization process of a clamped-clamped microbeam resonator. Similarly, a clamped-free microbeam resonator of \(30\times 6\,\upmu\)m was chosen as a numeric example for comparison. The whole design region was discretized into a \(300\times 60\) finite element mesh with a total of 55,083 DOFs, including the displacement and temperature. Figure 5: Topology optimization results for a clamped-clamped microbeam resonator. (**a**) The original structure. (**b**) The optimization result where the traditional subspace method is implemented in the finite element analysis step. (**c**) The optimization result where the proposed method is implemented in the finite element analysis step. It can be observed in the optimized structure for the traditional method (Figure 6b) and the proposed method (Figure 6c) that the void regions near the clamped side and in the middle were quite alike, with some difference at the border. This may be because the truncation error may affect the local optimizing direction of the topology optimization method. Once again, the error of the normalized thermoelectric dissipation rate was acceptable, and the computational efficiency was improved significantly for the proposed method, as shown in Table 2.  
Thus, the proposed method saved significant computation time in comparison with the traditional linear method for the sensitivity analysis of the thermodynamically coupled system, with little loss of accuracy of the optimization result. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Clamped-Clamped Microbeam (400 \(\times\) 40 mesh)**} \\ \cline{2-4}  & **Eigenvalue Solution** & **Global Sensitivity Number** & **Combined Time** \\ \hline Traditional method (s) & 4.3108 & 6.1492 & 10.46 \\ Proposed method (s) & 1.5779 & 0.9175 & 2.4954 \\ Time saved & 63.4\% & 85.08\% & 76.14\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison of the average computational times of each iteration of topology optimization of the clamped-clamped microbeam resonator with the traditional linear method and the proposed method. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Clamped-Free Microbeam (300 \(\times\) 60 mesh)**} \\ \cline{2-4}  & **Eigenvalue Solution** & **Global Sensitivity Number** & **Combined Time** \\ \hline Traditional method (s) & 5.778 & 9.6368 & 15.4148 \\ Proposed method (s) & 1.2508 & 1.7626 & 3.0134 \\ Time saved & 78.35\% & 81.71\% & 80.45\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The comparison of the average computational times of each iteration of topology optimization of the clamped-free microbeam resonator with the traditional linear method and the proposed method. Figure 6: Topology optimization results for a clamped-free microbeam resonator. (**a**) The original structure. (**b**) The optimization result where the traditional subspace method is implemented in the finite element analysis step. (**c**) The optimization result where the proposed method is implemented in the finite element analysis step.