# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 2 Statement of the Topology Optimization Problem
### Definition of Optimization Problem
As mentioned before, this paper mainly focuses on the evolutionary structural optimization method to improve the quality factor of the structure. The sensitivity of the \(i\)-th order eigenvalue \(\lambda_{i}\) of the system to the design variable \(p_{e}\) can be expressed as:  
\[\frac{\partial\lambda_{i}}{\partial p_{e}} = \boldsymbol{\Psi}_{i}^{\mathrm{T}}\bigg{(}\frac{\partial\boldsymbol {\Lambda}}{\partial p_{e}}-\lambda_{i}\frac{\partial\boldsymbol{\mathrm{B}}}{ \partial p_{e}}\bigg{)}\boldsymbol{\phi}_{i} \tag{17}\]  
As shown in Equation (17), the right-hand side can be obtained by solving the system equation. Therefore, this equation reveals an explicit relationship between system eigenvalue sensitivity and design variables. As mentioned before, we choose an evolutionary structural topology optimization method based on finite element analysis to solve the fully-coupled thermoelastic optimization problem in Equation (16). Since gradient-based evolutionary structural optimization methods require the sensitivity of the objective function with respect to the design variables, and considering that the displacement field, temperature field, and characteristic frequency of the system may be functions of the design variables themselves, according to the chain rule, the sensitivity of the objective function with respect to the design variables can be expressed as follows. Consequently, the sensitivity of the quality factor \(Q\) with respect to the element pseudo-density \(p_{e}\) can be obtained using the chain rule and the relationship between the system's eigenvalues and quality factor:  
\[\frac{\partial Q^{-1}}{\partial p_{e}}=2\frac{\Im\bigg{(}\frac{\partial \lambda i}{\partial pe}\bigg{)}\Re(\lambda_{i})-\Im(\lambda_{i})\Re\bigg{(} \frac{\partial\lambda i}{\partial pe}\bigg{)}}{\left[\Re(\lambda_{i})\right]^{ 2}} \tag{18}\]  
This equation represents the explicit expression of the sensitivity of the objective function \(Q\) with respect to the design variable \(p_{e}\), and it contains terms related to the hysteresis time (such as \(\boldsymbol{\mathrm{S}}_{T}\), \(\boldsymbol{\mathrm{H}}_{q}\), and \(\boldsymbol{\mathrm{F}}_{q}\)) as indicated by the definition of \(\lambda_{i}\). The optimization process of an ESO method can be discretized into several steps, as shown in Figure 2. For a large-scale design problem, the finite element analysis step of the fully-coupled system, as marked in red, requires heavy computational cost, while the repeated reconstruction of the structure further exacerbates the computational burden. According to the definition and standard process of the ESO method, the convergence of the optimization process can be acquired when the structure reaches the design volume fraction, and the objective function changes little betw  
Figure 2: Flowchart of the optimization process: The finite element analysis of the fully-coupled system incurs a significant computational cost, and the repeated reconstruction of the structure further exacerbates the computational burden. As mentioned before, the search direction, search domain, efficiency, and convergence are vital to the design process. For instance, the gradient-based ESO method determines the search direction. The search domain needs to be narrowed down. The efficiency of the method needs to be improved, and the convergence of the optimization process needs to be validated. In the next section, we will propose an efficient and robust topology optimization method to solve this optimization problem with efficiency and accuracy.