# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 3 Efficient and Robust Topology Optimization Method
### Posterior Evaluation Technique
Therefore, when the Neumann series expansion does not converge, the optimization process first returns to the structure obtained in the previous iteration and adjusts the evolution rate based on it to ensure convergence:  
\[er^{(k+1)}=\mathcal{P}_{er}\times er^{(k)} \tag{38}\]  
where \(er^{(k)}\) denotes the evolution rate in the \(k\)-th iteration, and \(\mathcal{P}_{er}\) is the penalty function for the evolution rate. After adjusting the evolution rate, a new structure and the corresponding system matrix \(\Delta\mathbf{A}\) are obtained through optimization. If the new structure satisfies the convergence of the Neumann series expansion, the optimization process continues. If the new structure obtained after adjusting the evolution rate still does not satisfy the convergence of the Neumann series expansion, the optimization process returns to the structure from the previous iteration. Then, the weights of the sensitivities of each element are adjusted based on the structure retained from the previous optimization. For the inefficient elements that were discarded in the previous optimization (\(e(\text{low})\)) and the efficient elements that were added in the current optimization (\(e(\text{high})\)), their penalty factors are increased to encourage the algorithm to choose other regions for optimization in the next iteration. \[\left(\frac{\partial\lambda_{mi}}{\partial p_{e(\text{low})}}\right)^{(k+1)}= \mathcal{P}_{e(\text{low})}\Bigg{(}\frac{\partial\lambda_{mi}}{\partial p_{e( \text{low})}}\Bigg{)}^{(k)}\quad,\quad\left(\frac{\partial\lambda_{mi}}{ \partial p_{e(\text{high})}}\right)^{(k+1)}=\mathcal{P}_{e(\text{high})}\Bigg{(} \frac{\partial\lambda_{mi}}{\partial p_{e(\text{high})}}\Bigg{)}^{(k)} \tag{39}\]  
where \(\mathcal{P}e(\text{low})\) and \(\mathcal{P}e(\text{high})\) are penalty functions for inefficient and efficient elements, respectively. After adjusting the element sensitivities, the optimization obtains a new structure and system matrix change \(\Delta\mathbf{A}\), and optimizes for a new structure. If the new structure obtained after adjusting the sensitivities satisfies the convergence condition of the Neumann series expansion, the optimization process continues. Otherwise, based on the optimized elements in the current structure, the process returns to the previous structure and adjusts the element sensitivities to obtain a new structure. This process is repeated until the Neumann series expansion of the new structure converges. By combining the traditional ESO convergent checks with the efficient modal analysis and posterior evaluation technique, the efficiency and robustness of the topology optimization method can be assured. The flowchart of the whole process is shown in Figure 3.  
Figure 3: Flowchart of the efficient and robust topology optimization method. Step 1 : Initialize mesh, boundary condition, evolutionary parameters, and penalty factors. Step 2: Construct global system matrices using the reconstruction technique in Section 3.1. Step 3: Efficient modal analysis can be implemented based on Algorithm 1. Step 4: Use posterior evaluation for Neumann series subspace in Algorithm 2 to ensure convergence so that the proposed method is robust. Step 5: Validate the mode shape of the system using Equation (34). With the help of the modal assurance criteria, the same order of mode shape is ensured, and the search domain of the optimization method is reduced. Step 6: Analyze the global sensitivities through Equation (18) and construct a new design according to global sensitivity. Step 7: Repeat Steps 2-6 until the objective function converges.