# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 5 Conclusions  
With the reduction in structural scales, the influence of thermoelastic damping becomes gradually dominant, determining the upper limit of the performance of high-frequency MEMS microresonators. Thus, enlarging thermoelastic damping based on the topology optimization design plays a critical role in improving the dynamical performance of microresonators. However, the challenges of computational cost and robustness are critical obstacles in the topology optimization design of microresonators. This is due to the iterative process of optimizing large-scale viscoelastic structures.  
This study proposes an efficient and robust topology optimization method for minimizing the thermoelastic damping of large-scale microresonators. An evolutionary structural optimization method is adopted to passively determine the search direction of optimizing large-scale thermoelastic structures. To efficiently reduce the computational cost of the iterative process of optimizing process, a model reduction method is developed based on the projection-based model reduction method whose reduced basis is generated within the Neumann series subspace. However, the projection-based model reduction method may be unstable when topology modifications are made during an iteration optimization process. To ensure robustness, a modal validation technique is first implemented in the iterative process to stabilize the iteration and narrow down the search domain, and a posterior evaluation of the Neumann series expansion is then developed to retain the convergence of the projection-based model reduction method. Furthermore, the efficiency and accuracy of the proposed topology optimization method are validated through numerical examples.  
Three numerical examples are discussed, results show that the proposed method saves significant computation time in comparison with the traditional linear method in the sensitivity analysis of the thermodynamic structures, with little loss of accuracy of the optimization result. Furthermore, the proposed method has the capability of designing complex large-scale structures.  
Figure 8: The finite element analysis and optimization results of the uniaxial mirror: (**a**) result of the proposed FEM, (**b**) result of the COMSOL software, (**c**) optimization result of the proposed efficient and robust design method.  
The feasibility of the efficient and robust method is demonstrated in the gradient-based topology optimization problem for thermoelastic structures. Thus, the research work can be implemented in designing large-scale thermoelastic structures under phase-lag heat conduction law. Furthermore, the topology optimization method proposed in this work combines the matrix reconstruction and parameterized model reduction techniques for thermoelastic systems under the non-Fourier heat conduction law. Thus, it is hopeful for the eigensolution analysis of thermoelastic structures under the phase-lag heat conduction law and expedite the application of thermoelastic structures arising in different subjects, including conceptual design problems, damage analysis, and smart or adaptive structures, and further research in these areas is worth pursuing.  
Conceptualization, Y.F. and L.L.; formal analysis, L.L.; funding acquisition, L.L.; investigation, Y.F., L.L. and Y.H.; methodology, Y.F. and Y.H.; project administration, L.L.; resources, L.L. and Y.H.; software, Y.F.; supervision, L.L. and Y.H.; validation, Y.F.; visualization, Y.F.; writing--original draft, Y.F. and L.L.; writing--review and editing, L.L. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Key Research and Development Program of China (grant no. 2021 YFB174600), the National Natural Science Foundation of China (grant no. 52175095), and the Young Top-Notch Talent Cultivation Program of Hubei Province of China.  
Not applicable.  
The authors declare no conflict of interest.