# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 4 Results and Discussion
### Optimal Design of MEMS Uniaxial Mirror  
There are many thermoelastic MEMS applications related to microresonators, such as MEMS switches, energy harvesters, and actuators [48]. Among them, the MEMS switch is widely used in medical devices, mobile phones, radio frequency, and other devices [49]. Take the micro-electromechanical single-axis mirror structure in Figure 7 as an example [1], this mirror can be used as a switch-like component for high-speed switching, which requires high accuracy and stability, that is, the thermoelastic damping is small. Its xoy-plane can be expressed as shown in Figure 7a, where the left side is two vibrating driving beams, the vibrating direction extends along the \(z\)-axis, and the right side is the working mirror. Therefore, it can be simplified as a three-dimensional structural optimization problem, as shown in Figure 7b, where the gray part is the non-design domain of the working mirror, the yellow part is the design domain, and the left side is the fixed constraint. The quality factor is considered the objective function and frequency constraints are imposed to reduce the excessive reduction of the structural frequency during the optimization process. The material parameters are the same as in former examples, and considering the degrees of freedom of the structure, only the optimization design in the  
Figure 6: The finite element analysis and optimization results of the grid beam: (**a**) result of the proposed FEM, (**b**) result of the COMSOL software, (**c**) optimization result of the proposed efficient and robust design method. range of Fourier heat conduction is considered. As shown in Figure 7b, the structural non-design domain mirror size is 400 \(\upmu\)m \(\times\) 800 \(\upmu\)m \(\times\) 40 \(\upmu\)m and is discretized into a 80 \(\times\) 160 \(\times\) 8 mesh with 3D isoparametric thermoelastic finite elements. The design domain is two driving beams with fixed support on the left side, and the right side is rigidly connected with the mirror surface, the size of a single beam is 400 \(\upmu\)m \(\times\) 100 \(\upmu\)m \(\times\) 40 \(\upmu\)m and the beam is discretized into a 80 \(\times\) 20 \(\times\) 8 mesh with 3D isoparametric thermoelastic finite elements. The total degrees of freedom of the 3D system are 590,436. For the classic 3D state-space solution, the total degrees of freedom reaches the number of 1,180,872, which will significantly impair the efficiency of the solution. Moreover, considering the impact of intermediate variables of the solution, there is a high probability that it exceeds the storage limit of conventional computers. We validate the solution of the 3D thermoelastic finite element analysis of the proposed method, and compare the structural frequency and mode shape with the finite element thermoelastic analysis module in the COMSOL software. As shown in Figure 8a,b, the original structure frequency obtained by using the accelerated method in this paper is \(3.9225\times 10^{4}\) Hz, and the frequency obtained by commercial software is \(3.9105\times 10^{4}\) Hz, the error is less than 1%. The quality factor of the original structure obtained by the accelerated solution is 14,164, and the quality factor of the COMSOL software solution is 14,196, and the error is acceptable. Therefore, it can be considered that the finite element system equation solving module in the software of this paper is feasible. Similarly, as commercial software lacks a thermoelastic optimization module, the optimized structure is compared with only the original structure obtained by finite element analysis of this paper. The temperature contour map of the optimized result is shown in Figure 8c, with grid lines retained and only half of the structure displayed in the \(z\)-axis direction on the right side to highlight the internal configuration of the optimized structure. The optimized structural quality factor is 30,110, which is improved by 212.58% compared to the original structure.