# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 3 Efficient and Robust Topology Optimization Method
### Posterior Evaluation Technique  
The convergence of the Neumann series expansion depends on the eigenvalues of its non-identity terms [43]. Therefore, for the convergent Neumann series (26), the necessary and sufficient condition can be obtained as follows:  
\[\rho(\xi_{i}\mathbf{B}_{m}^{-1}\Delta\mathbf{A})<1 \tag{35}\]  
where \(\rho(\bullet)\) denotes the spectral radius of the matrix \((\bullet)\), i.e., it requires that the absolute value of all eigenvalues of the coefficient matrix are less than 1. The maximum eigenvalue of the matrix \(\mathbf{B}_{m}^{-1}\Delta\mathbf{A}\) can be obtained by solving the following eigenvalue problem for its minimum eigenvalue:  
\[\mathbf{B}_{m}\mathbf{y}_{i}=s_{i}\Delta\mathbf{A}\mathbf{y}_{i} \tag{36}\]  
As the thermal-elastic damping of the system is small, the known undamped (real part) frequencies can be used to approximate. Combining the minimum eigenvalue \(s_{\text{min}}\) and Equation (24) yields  
\[|s_{\text{min}}|<|\lambda_{i}| \tag{37}\]  
Note that the original eigenvalue \(\lambda_{i}\) is used herein to approximate the modified eigenvalue \(\lambda_{mi}\). As shown in Equations (24) and (35), the convergence of the Neumann series expansion depends on the system eigenvalues \(\lambda_{i}\), system matrix \(\mathbf{B}\), and the system matrix perturbation \(\Delta\mathbf{A}\). However, due to the thermoelastic damping in the system, the perturbation \(\Delta\mathbf{A}\) is uncertain as it may change during the evolutionary optimization process when materials are added or removed from different locations. Therefore, the convergence of the Neumann series expansion cannot be verified by a priori methods. The convergence of the Neumann series expansion directly affects the efficiency and accuracy of the subspace dimensionality reduction method, and thus affects the optimization process. If the series diverges, it may cause oscillations in the iterative optimization process and cannot guarantee the convergence of the optimization algorithm. As the technique of accelerating the solution of eigenproblems serves the optimization algorithm, and the convergence of the Neumann series cannot be verified in an a priori way, we adopt a posterior evaluation technique to ensure the convergence of the optimization algorithm. As shown in Algorithm 2, the parameter \(evalnum\) is the number of iterations the posterior process has taken effect on. In the first iteration of evaluation, the evolution rate is adjusted, while in later iterations, the sensitivities are adjusted to ensure the convergence of the Neumann series expansion. The adjustment of sensitivities is repeated until the Neumann series expansion converges to ensure the robustness of the topology optimization process. ```
Input:\(\lambda_{i},s_{\text{min}},er,\left(\frac{\partial\lambda_{mi}}{\partial p_{e (\text{low})}}\right)^{(k)}\), \(\left(\frac{\partial\lambda_{mi}}{\partial p_{e(\text{high})}}\right)^{(k)}, \mathcal{P}_{er},\mathcal{P}_{e(\text{low})},\mathcal{P}_{e(\text{high})}.evalnum\) Output:\(\lambda_{mi}\)
1 if\(|s_{\text{min}}|>|\lambda_{i}|\)then
2 if\(evalnum=1\)then
3\(er^{(k+1)}=\mathcal{P}_{er}\times er^{(k)}\)
4 else
5\(\left(\frac{\partial\lambda_{mi}}{\partial p_{e(\text{low})}}\right)^{(k+1)}= \mathcal{P}_{e(\text{low})}\Bigg{(}\frac{\partial\lambda_{mi}}{\partial p_{e (\text{low})}}\Bigg{)}^{(k)}\)
6\(\left(\frac{\partial\lambda_{mi}}{\partial p_{e(\text{high})}}\right)^{(k+1)}= \mathcal{P}_{e(\text{high})}\Bigg{(}\frac{\partial\lambda_{mi}}{\partial p_{e (\text{high})}}\Bigg{)}^{(k)}\)
7 end if
8\(evalnum=evalnum+1\)
9 return to last iteration
10 else
11\(evalnum=1\)
12\(\lambda_{mi}=\lambda_{i}\)
13 continue to next step
14 end
```  
**Algorithm 2**Posterior evaluation for Neumann series subspace  
The evolution rate in gradient-based structural optimization, which is the overall volume reduction rate per iteration, directly affects the change of system matrix \(\Delta\mathbf{A}\).