# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 2 Statement of the Topology Optimization Problem
### Governing Equations of Thermolestic Structures  
For a thermoelastic system constructed with linear isotropic thermoelastic material, the equation of motion can be obtained through the conservation law of momentum [36]:  
\[\frac{E}{1+\
u}\epsilon\cdot\mathbf{\
abla}+\frac{E\
u}{(1+\
u)(1-2\
u)}\mathbf{\
abla} [\mathrm{Tr}(\mathbf{\epsilon})]-\frac{E\alpha}{1-2\
u}\mathbf{\
abla}\theta+\tilde{ \mathbf{b}}=\rho\ddot{\mathbf{u}} \tag{2}\]  
where \(\theta\) is the temperature field, \(E\) is the elastic modulus, \(\alpha\) is the coefficient of thermal expansion, \(T_{0}\) is the reference temperature, and \(\
u\) is Poisson's ratio. This paper mainly considers the dynamic characteristics of structures, i.e., free vibration analysis (modal analysis), so the body force \(\tilde{\mathbf{b}}\), external force, and corresponding load terms are zero. Combining the finite element method and classical global matrix assembly method, and considering boundary conditions, the global equation of motion can be expressed as  
\[\mathbf{M}\dot{\mathbf{U}}+\mathbf{K}\mathbf{U}-\mathbf{G}\mathbf{\Theta}=\mathbf{0} \tag{3}\]  
where \(\mathbf{M}\), \(\mathbf{K}\), and \(\mathbf{G}\) are the global mass, stiffness, and thermal stress matrices of the whole structure, \(\mathbf{U}\) and \(\mathbf{\Theta}\) are the global vectors of displacement and temperature, respectively. Note that the thermal stress term (\(\mathbf{G\Theta}\)) is related to the temperature field of the structure. Therefore, the solution of the structural equation of motion requires the use of the structural heat conduction equation. On the other hand, the dual-phase lag heat conduction model takes the following form [20,22]:  
\[\mathbf{q}(\mathbf{r},t)+\tau_{q}\mathbf{\dot{q}}(\mathbf{r},t)=-\kappa\big{[} \boldsymbol{\
abla}T(\mathbf{r},t)+\tau_{T}\boldsymbol{\
abla}T(\mathbf{r},t )\big{]}, \tag{4}\]  
where \(\mathbf{q}\) and \(\boldsymbol{\
abla}T\) are the heat flux and temperature gradient, respectively. \(\mathbf{r}\) and \(t\) are, respectively, the position vector and time. \(\tau_{q}\) and \(\tau_{T}\) are the phase lags on heat flux and temperature, respectively. Overdot denotes the derivative with respect to time. The following equation describes the behavior of heat conduction and is known as the governing equation of heat conduction:  
\[\kappa\Delta\theta=C_{v}\dot{\theta}+\frac{E\kappa T_{0}}{1-2\
u}\text{Tr}( \epsilon) \tag{5}\]  
where \(C_{v}=\rho C_{p}\) represents the specific heat capacity per unit volume, \(\Delta\) represents the Laplacian operator, \(\kappa\) is the thermal diffusivity, \(\dot{\theta}\) is the rate of change of the temperature field, and \(\dot{\epsilon}\) is the strain rate tensor. Similar to the thermal stress term in the governing equation, the heat generation term (\(\mathbf{FU}\)) caused by strain in the thermal conduction equation is related to the structural displacement field, and therefore, the solution of the structural thermal conduction equation requires the use of the motion equation. Combining Equations (4) and (5), the global equation of heat conduction considering the non-Fourier heat conduction dual-phase lag model can be expressed as [36]:  
\[\mathbf{S\Theta}+\mathbf{S}_{T}\mathbf{\dot{\Theta}}+\mathbf{H\dot{\Theta}}+ \mathbf{FU}+\mathbf{H}_{q}\mathbf{\ddot{\Theta}}+\mathbf{F}_{q}\mathbf{\ddot{ U}}=\mathbf{0} \tag{6}\]  
where  
\[\left\{ \begin{array}{l}\mathbf{S}_{T}=\tau_{T}\mathbf{S}\\ \mathbf{H}_{q}=\tau_{q}\mathbf{H}\\ \mathbf{F}_{q}=\tau_{q}\mathbf{F}\end{array} \right. \tag{7}\]  
The governing heat conduction equation can take various forms depending on the chosen phase lag. In single-phase lag non-Fourier heat conduction models, the phase lag is usually applied to the heat flux [21]. Therefore, only the heat flux lag form is considered. When the temperature lag \(\tau_{T}\) is equal to zero, Equation (6) degenerates to a single-phase lag form.