## 4 Results and Discussion  
In this study, the efficient repeated analysis- and parallel-based modal modification analysis method proposed was compared to the traditional method with two examples of thermodynamically damped systems. The efficient reduced-basis preconditioner method and parallel technique of calculation of the sensitivity number developed herein were compared to the traditional subspace method and linear computation of the sensitivity number in an iterative topology optimization process for complex eigenfrequency and sensitivity analysis [32]. The proposed repeated analysis method was implemented in an iterative topology optimization process to validate its accuracy and computational cost, and the examples were constructed in Matlab. As stated before, the convergence of the Neumann series expansion was dependent on the spectral radius of the \(\mathbf{Y}\) term in parentheses in Equation (12). However, \(\Delta\mathbf{A}\) in \(\mathbf{Y}\) was altered each time as the material distribution changed in the topology optimization process. Thus, a posterior validation of the convergence of the Neumann series expansion was conducted in each iteration of the optimization process. The same material parameters and evolutionary optimization parameters were implemented in the topology optimization process for the traditional linear method and the proposed repeated analysis- and parallel-based method. The material properties were a [PERSON]'s modulus of \(157\,\mathrm{G}\mathrm{P}\mathrm{a}\), density of \(2330\,\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}\), [PERSON]'s ratio of \(0.22\), thermal expansion coefficient of \(2.6\,\times\,10^{-6}\,\mathrm{K}^{-1}\), specific heat per unit mass of \(700\,\mathrm{J}\,/\,(\mathrm{k}\mathrm{g}\cdot\mathrm{K})\), and thermal conductivity of \(90\,\mathrm{W}\,/\,(\mathrm{m}\cdot\mathrm{K})\) with no penalty applied to these parameters. The evolutionary optimization parameters were set to a volume limitation of \(0.8\), evolutionary rate for each iteration of 0.01, and mesh-independent filter radius of 4. A normalization process was adopted for the objective function \(\varphi_{1}\) for better comparison. A clamped-clamped microbeam resonator of \(40\times 4\,\upmu\)m was chosen as a numeric example for optimizing the quality factor of the first-order mode shape, where both the proposed method and traditional method were implemented in the finite element analysis process for a complex eigenvalue solution. The whole design region was discretized into a \(400\times 40\) finite element mesh with a total of 49,323 degrees of freedom (DOFs), including the displacement and temperature. The optimization results with the traditional subspace method and the proposed method (\(r=10\)) for finite element analysis are shown in Figure 5.  
It can be observed in the optimized structure for the traditional method (Figure 5b) and the proposed method (Figure 5c) that the positions of the main voids were nearly the same, which indicates that the main optimizing directions of the optimization methods containing two different eigenvalue solution methods were the same. The difference at the border of the voids may be explained by the truncation error, which may slightly affect the local optimizing direction of the topology optimization method. On the other hand, the error of the normalized thermoelastic dissipation rate was also acceptable, and the optimized structure for the proposed method even had a better dissipative rate than the optimized structure conducted by the traditional subspace method.