# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 2 Statement of the Topology Optimization Problem
### Definition of Optimization Problem  
The objective of the topology optimization process is to minimize the thermoelastic dissipation of the microbeam resonator. There are several methods for material structure topology optimization design, which can be roughly classified into gradient-based and non-gradient methods. The gradient-based method determines the search direction by computing the sensitivity gradient of the design variables. One specific branch of the gradient-based method is evolutionary structural optimization and its derivative algorithms, where the element pseudo-densities are the design variables. In this approach, a value of \"1\" for an element pseudo-density signifies that the element is filled with material, while a value of \"0\" denotes that the element is empty. By iteratively adjusting the element pseudo-densities based on the sensitivity of the design variable and the optimization evolution rate, the material in the structure is either added or removed, thereby altering its dynamical performance. The evolutionary iterative optimization process converges when the optimized structure reaches the target volume fraction within the constraints, and the objective function converges. The advantage of evolutionary structural optimization lies in its clear and concise concept of element sensitivity. Elements with high design variable sensitivity make a greater contribution to the objective function and are retained, while those with low sensitivity are removed [37]. In addition, the structure obtained by its optimization has no intermediate density and is closer to the \"0-1\" topology configuration. Furthermore, since the iterative algorithm considers the relative values of the design variable sensitivity, there is no strict requirement for the positive or negative signs of the sensitivity, making it more convenient in handling multiphysics optimization problems than other optimization methods such as density-based methods. Thus, the evolutionary structural optimization method is chosen as the design method to determine the search direction of the design problem. For the optimization of microstructures with the non-Fourier phase-lag heat conduction law, the thermoelastic optimization model can be expressed in a specific structural topology optimization form by combining topology optimization-related constraints, design variables \(p_{e}\), and objective function \(\mathbb{C}\). Taking the \(i\)-th quality factor \(Q_{i}\) of the microresonator as the objective function, the matrix form of the thermoelastic optimization problem can be obtained as follows:  
Max \[\mathbb{C}=Q_{i}\] _s. t._ \[\mathbf{A}\boldsymbol{\Phi}_{i}=\lambda_{i}\mathbf{B}\boldsymbol{ \Phi}_{i}\] (16a) \[\sum_{e=1}^{N}\Omega_{e}p_{e}-l_{e}\Omega_{0}\leq 0\] (16b) \[l_{f}f^{\rm(origin)}-f\leq 0\] (16c) \[p_{e}=p_{\rm min}\ {\rm or}\ 1,\quad(e=1,2,\ldots N)\] (16d)  
The eigenvalue equation in (16a) is the main constraint of the optimization problem, where \(\mathbf{A}\) and \(\mathbf{B}\) are the system matrices mentioned earlier. Through finite element analysis, both free vibration boundary conditions and adiabatic boundary conditions are included in (16a). In addition, volume constraint (16b) and element volume constraint (16d) are imposed on the optimization process to obtain a lightweight structure and ensure that the optimization process is not affected by the singularity of the overall system matrix. Here, \(p_{e}\) is the pseudo-density of the \(e\)-th element, which takes the value of 1 or a small positive number close to zero \(p_{\rm min}\). \(N\) is the number of finite elements in the entire structure, \(l_{v}\) is the volume fraction of the optimized structure, \(\Omega_{e}\) is the volume of each element, and \(\Omega_{0}\) is the initial volume of the structure. Furthermore, a frequency constraint (16c) is imposed to ensure the stability of the optimization process and prevent excessive reduction of structural frequency.