# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 2 Statement of the Topology Optimization Problem
### State-Space Method for Large-Scale Microresonators with Phase-Lag Heat Conduction Law  
Combining Equations (3) and (6), we obtain  
\[\left\{ \begin{array}{l}\mathbf{M\dot{U}}+\mathbf{KU}-\mathbf{G\Theta}= \mathbf{0}\\ \mathbf{S\Theta}+\mathbf{S}_{T}\mathbf{\dot{\Theta}}+\mathbf{H\dot{\Theta}}+ \mathbf{F\dot{U}}+\mathbf{H}_{q}\mathbf{\ddot{\Theta}}+\mathbf{F}_{q}\mathbf{ \ddot{U}}=\mathbf{0}\end{array} \right. \tag{10}\]  
Equation (10) represents a unified thermoelastic governing equation that takes into account different heat conduction models. The full coupling between the temperature field and displacement field in the equation of motion and the heat conduction equation requires a simultaneous solution of the governing equations of heat conduction and motion. Moreover, the introduction of high-order terms related to the phase-lag time, such as \(\mathbf{S}_{T}\mathbf{\hat{\Theta}}\), \(\mathbf{H}_{q}\mathbf{\hat{\Theta}}\), and \(\mathbf{F}_{q}\mathbf{\hat{U}}\), further increases the difficulty of solving this fully-coupled system equation. There are several ways to solve the combined Equation (10). Direct decoupling of (10) is usually achieved by ignoring the coupling terms in the motion equation or the heat conduction equation, but this destroys the original equation form and neglects the impact of thermoelastic behavior on energy dissipation. Iterative methods such as the [PERSON] iteration method are generally inefficient for solving fully-coupled high-order equations. Therefore, based on the idea of expanded complexity and descended order, this paper constructs a low-order state-space model that retains the thermoelastic influence while preserving the possibility of a subsequent accelerated solution. Unit matrices related to displacement rate and temperature rate are added:  
\[-\mathbf{I}\mathbf{\dot{U}}+\mathbf{I}\mathbf{\dot{U}} =\mathbf{0} \tag{11}\] \[-\mathbf{I}\mathbf{\dot{\Theta}}+\mathbf{I}\mathbf{\dot{\Theta}} =\mathbf{0} \tag{12}\]  
Filling in the zero terms, we obtain the state-space method of large-scale microresonators with the phase-lag heat conduction law as  
\[\left[ \begin{array}{cccc}\mathbf{H}&\mathbf{H}_{q}&\mathbf{F}&\mathbf{F}_{q }\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{M}\\ \mathbf{I}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{I}&\mathbf{0}\end{array} \right]\left[ \begin{array} []{c}\mathbf{\dot{\Theta}}\\ \mathbf{\dot{\Theta}}\\ \mathbf{\dot{U}}\\ \mathbf{\dot{U}}\end{array} \right]+\left[ \begin{array}{cccc}\mathbf{S}& \mathbf{S}_{T}&\mathbf{0}&\mathbf{0}\\ -\mathbf{G}&\mathbf{0}&\mathbf{K}&\mathbf{0}\\ \mathbf{0}&-\mathbf{I}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-\mathbf{I}\end{array} \right]\left[ \begin{array} []{c}\mathbf{\dot{\Theta}}\\ \mathbf{\dot{\Theta}}\\ \mathbf{U}\\ \mathbf{\dot{U}}\end{array} \right]=\mathbf{0} \tag{13}\]  
Equation (13) represents the governing equation of the system taking into account non-Fourier phase-lag heat conduction. In the 2D thermoelastic eigenvalue problem, the number of nodes \(N_{n}\) directly affects the degrees of freedom of the system, where the basic degrees of freedom \(N_{d}\) are given by \(N_{d}=3N_{n}\) (including two displacement degrees of freedom and one temperature degree of freedom per node). However, the state-space method increases the degrees of freedom, and for the non-Fourier dual-phase lag heat conduction combined Equation (20), the degrees of freedom of the state-space solution increase to \(N_{s}=6N_{n}\). Similarly, for the 3D thermoelastic eigenvalue problem, the basic degrees of freedom are \(N_{d}=4N_{n}\), and for the non-Fourier dual-phase lag heat conduction combined Equation (20), the dimension of the state-space solution increases to \(N_{s}=8N_{n}\).