# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 4 Results and Discussion
### Validation of Efficiency and Accuracy  
The dual-phase lag heat conduction model is more representative of the actual operating conditions of microresonators at the microscale and below. Therefore, this section discusses the optimization results of different solutions considering the non-Fourier dual-phase lag heat conduction model. The clamped-clamped microbeam resonator is often used as the key component of micro devices, and its quality factor will directly affect the performance of these devices [44; 45]. Thus, the clamped-clamped microbeam resonator is chosen as the example in the validation process. Considering a clamped-clamped microbeam with the size of \(8\times 1\,\mathrm{\SIUnitSymbolMicro m}\), the entire design domain is discretized using a 2D finite element method into a grid mesh of \(320\times 40\) 2D isoparametric thermoelastic finite elements, with a total system degree of freedom of \(39\),\(483\). The evolutionary optimization parameters were set as \(l_{v}=0.8\), \(er=0.008\), \(r_{\mathrm{min}}=4\), and the material penalty factors were set as \(p_{E}=2\), \(p_{\rho}=1\), \(p_{\kappa}=3\), \(p_{a}=1\), \(p_{C_{0}}=1\). The temperature phase-lag was set to \(\tau_{T}=0.1\,\mathrm{\SIUnitSymbolMicro s}\), while the phase-lag of heat flux was set to \(\tau_{q}=0.005\,\mathrm{\SIUnitSymbolMicro s}\). As shown in Figure 4, for the case of non-accelerated method, the normalized quality factor of the microbeam after optimization was \(130.07\%\) of the initial structure. The optimization result for the accelerated method with the order of reduced basis \(r=10\) was \(134.39\%\) of the initial structure. The material distribution and contour map of temperature are similar for the optimization results of non-accelerated (Figure 4b) and accelerated (Figure 4c) methods, with some differences at the border of the vacations, this might be caused by the truncation error of mode shape and reduced basis process of the accelerated method. On the other hand, the error of the normalized quality factor between these two methods is also acceptable, this indicates that the accelerated method can solve the optimization problem with accuracy. As shown in Table 1, the average computation time per iteration of the traditional subspace method for solving eigenvalue problems without acceleration is 26.3567 s, while the same step takes 8.1933 s with the accelerated method. As shown in Figure 4d, the non-accelerated method converges after 39 iterations, with a total computation time of 1453.64 s, including other operations such as sensitivity analysis. As shown in Figure 4e, the accelerated method converges after 45 iterations, with a total computation time of 859.32 s. As the posterior evaluation process takes effect in the iterative optimization process, which can be observed in iterations 28 to 32, the accelerated method takes more total iterations compared to the non-accelerated method. Therefore, the acceleration method proposed in this chapter saves 68.91% of the computational time of the eigenvalue solution step. Although in order to maintain the robustness of the optimization process, the accelerated method takes more iterations to achieve an optimized structure than the non-accelerated method, the accelerated method saves approximately 40.88% of the total computation time of the optimization process compared to the non-accelerated method. Note that for traditional state-space solution of finite element analysis problems of the thermoelastic system, the complexity of the problem will increase exponentially with the degrees of freedom. Thus, for larger scale problems, the percentage of the computational time saved will increase with the scale of the problem, while for some problems, the time and storage consumed may be unbearable and exceed the maximum of many computation devices.