# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 4 Results and Discussion
### Optimal Design of Grid Beams  
In microelectromechanical systems, periodic grid beam structures are often used as key components of actuators, sensors, and energy-absorbing structures [46; 47]. Among them, grid beams considering in-plane vibrations are widely used, such as the capacitive load sensor and actuator structure. For the in-plane vibrating grid beam sensor, its quality factor affects the measurement accuracy and system stability, and its resonance frequency affects the transmission sensitivity. Therefore, the quality factor is vital to the grid beam system, and improving the quality factor will affect the overall performance of the system. Thus, in this section, an in-plane vibrating grid beam with phase-lag heat conduction law is considered as an example of the design problem of thermoelastic damping. The sketch map of the structure is shown in Figure 5a, it can be observed that the grid structure is symmetric. Therefore, for the convenience of solving, the left side of the symmetrical structure is selected, and an elemental structure of five beams is chosen as an example in this research, as shown in Figure 5b. In the whole structure, the gray part is the non-design region, these parts of the beams have to be unchanged to maintain the connection with the support end, the finite element grid with a size of 5 \(\upmu\)m \(\times\) 132 \(\upmu\)m and a discretization of 20 \(\times\) 528 2D isoparametric thermoelastic finite elements. The yellow part is the design, where each individual beam has a size of 95 \(\upmu\)m \(\times\) 10 \(\upmu\)m and a mesh discretized to 380 \(\times\) 40 2D isoparametric thermoelastic finite elements. For the 2D microgrid beams, the system has a total 267,027 degrees of freedom. For the classic 2D space-state system solution, the total degrees of freedom is 534,054, due to the iterative characteristic of the optimization process and the medium parameters of the solution, the computation cost is inevitably high. In terms of boundary conditions, the displacement boundary condition is considered as the left fixed support constraint, and the thermal boundary condition is considered an insulated thermal boundary. The material parameters of single crystal silicon are used as reference. Furthermore, as the existing commercial software lack the solution function considering non-Fourier phase lag heat conduction, the Fourier heat conduction is considered in the validation step of the structure (\(\tau_{q}=0,\tau_{\mathrm{T}}=0\)). As the energy dissipation rate is crucial for the system, the optimization objective of the thermoelastic is to maximize the system's quality factor, subject to a volume constraint of 0.8. First, the proposed 2D efficient thermoelastic finite element analysis solver is validated by comparing the structure's frequencies and modes with the COMSOL software's finite element thermoelastic analysis module. As shown in Figure 6, the original full structure's frequency obtained using this software was \(1.4207\times 10^{6}\) Hz, while the frequency obtained through the commercial software was \(1.4144\times 10^{6}\) Hz, with an error of less than 1%. The original structure's quality factor obtained using the accelerated method was 7456, and the quality factor obtained using the COMSOL software was 7481, with an error also less than 1%. Therefore, it can be concluded that the finite element system equation solver in this paper is stable and feasible. Figure 5: Numerical example of the grid beam, (**a**) is the sketch map of the grid beam, (**b**) is the design and non-design regions of the grid beam. Due to the lack of thermoelastic damping optimization module in existing commercial software packages, the optimization result is compared with the original structure solved by the proposed finite element method.