# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 3 Efficient and Robust Topology Optimization Method
### Modal Validation Process  
According to the definition of the quality factor in the case of small dissipation (Equation (1)), the quality factor of the structure is dependent on the eigenvalues and mode shapes of the structure. Thus, during the iterative optimization process, a different mode shape may result in a different quality factor; the same order of mode shape has to be ensured in the optimization process. For this reason, we use a modal assurance criteria-based (MAC-based) validation process to identify the first-order mode in the iterative optimization process. First, the first-order left and right eigenvectors \(\boldsymbol{\
abla}_{1}^{1}\) and \(\boldsymbol{\Phi}_{1}^{1}\) of a solid original structure without cavities are used as the reference group for the first iteration in the optimization process. Then, the modal assurance criteria are introduced to the process as follows [42]:  
\[\text{MAC}_{1}^{j,j+1}=\frac{\left[\left(\boldsymbol{\
abla}_{1}^{j+1}\right)^ {\text{T}}\boldsymbol{\Phi}_{1}^{j}\right]^{2}}{\left[\left(\boldsymbol{ \
abla}_{1}^{j}\right)^{\text{T}}\boldsymbol{\Phi}_{1}^{j}\right]\left[\left( \boldsymbol{\
abla}_{1}^{j+1}\right)^{\text{T}}\boldsymbol{\Phi}_{1}^{j+1} \right]} \tag{34}\]  
In the above equation, \(\boldsymbol{\
abla}_{1}^{j}\) and \(\boldsymbol{\Phi}_{1}^{j}\) represent the first-order left and right eigenvectors in the \(j\)-th iterative loop, respectively, while \(\boldsymbol{\
abla}_{1}^{j+1}\) and \(\boldsymbol{\Phi}_{1}^{j+1}\) represent the first-order left and right eigenvectors in the \((j+1)\)-th iterative loop. \(\text{MAC}_{1}^{j,j+1}\) represents the similarity between the first-order mode in the \((j+1)\)-th iterative loop and the original structure's first-order mode, which ranges from 0 to 1. A larger MAC value indicates a higher similarity between the two modes, while a smaller value indicates a lower similarity. By searching through all modes in the \((j+1)\)-th iteration, the mode most similar to the reference group's left and right eigenvectors \(\boldsymbol{\
abla}^{j}\) and \(\boldsymbol{\Phi}^{j}\) is selected, and the corresponding eigenfrequency is used to calculate the quality factor of the current iteration step to maintain the stability of the structure modes during the optimization process. Furthermore, as mentioned in the previous section, the solution of the fully-coupled system means heavy computational cost. While in the case of topology optimization problems, the solution of multiple eigenvalues and local modes will significantly affect the search domain and accuracy of the optimization method. To narrow down the search domain and improve the efficiency of the optimization process, the resonance frequency of the working mode shape in the \(j\)-th iterative loop will be used as the reference frequency in the next loop and only the system mode shapes with nearby resonance frequencies will be solved in the \((j+1)\)-th iterative loop. Then, the working mode shape of the structure can be obtained precisely and efficiently during the optimization process. From the literature above, the MAC-based validation process can reduce the search domain and stabilize the process of the iterative optimization method. The computational efficiency of topology optimization methods based on finite element analysis and sensitivity gradients is heavily influenced by the number of degrees of freedom of the system. However, the fully-coupled nature of thermoelastic damping makes the system equations and sensitivity analysis more difficult to solve. Particularly for large and complex thermoelastic structures, the repetitive analysis process in iterative optimization further exacerbates the issue of low computational efficiency. On the other hand, current commercial software mainly focuses on decoupling and analyzing multi-field coupling systems, with optimization design modules only including statics and weak coupling analysis, lacking corresponding design tools for thermoelastic structures.