# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 3 Efficient and Robust Topology Optimization Method
### Reconstruction of the System Matrices  
According to the definition of thermoelastic damping, the system matrices become complex when considering the effects of thermoelastic damping. For the governing equation of motion, let \(\mathbf{M}\) be a positive definite mass matrix, \(\mathbf{K}\) be a non-negative definite symmetric stiffness matrix, and \(\mathbf{G}\) be the thermal expansion matrix in the equation of motion. As mentioned above, the asymmetry of the combined system matrices \(\mathbf{A}\) and \(\mathbf{B}\) leads to the complex eigenvalue problem, which is difficult and time-consuming to solve. The key difficulty in solving the eigensolution is the decomposition and inversion step of the combined matrix. To facilitate the decomposition and inversion of the combined terms in the system equations, we construct the lumped mass matrix by making its diagonal elements proportional to the corresponding diagonal elements of the original consistent mass matrix. The lumped mass matrix is a diagonally positive definite matrix obtained from the traditional consistent mass matrix, which is widely used in finite element dynamic analysis. Its accuracy has been verified through numerical examples [38; 39]. It is assumed that the total sum of all elements in both types of mass matrices is equal, that is  
\[\left[\mathbf{M}\right]_{ii}^{\text{humped}}=\frac{\left[\mathbf{M}\right]_{ii }^{\text{consistent}}}{\sum\left[\mathbf{M}\right]_{ii}^{\text{consistent}}} \sum\sum\left[\mathbf{M}\right]_{ij}^{\text{consistent}} \tag{19}\]  
where \(\left[\mathbf{M}\right]_{ij}\) represents the element in the \(i\)-th row and \(j\)-th column of matrix \(\mathbf{M}\), and \(\sum\) is the summation operator for the elements in the matrix. For convenience, we use \(\mathbf{M}\) to denote the lumped mass matrix in the following text. Similarly, the heat generation matrix \(\mathbf{H}\) in the heat conduction control equation is also constructed as a positive definite diagonal matrix. For thermal-elastic coupling systems that consider non-Fourier dual-phase lag heat conduction models, the full coupling between the motion equation and the heat conduction equation requires a simultaneous solution of both equations. In the previous chapters, the order of the thermoelastic system was reduced by applying a state-space transformation to the system equations due to the influence of high-order time derivatives in the motion and heat conduction system equations. However, this increases the dimensionality of the system equations, and the classic direct combination method for control equations shown in Equation (13) results in two asymmetric matrices for the inherent terms of the system equations. This leads to a slow direct solution of the system equations due to the inversion and decomposition process of the global matrix. To avoid the extremely time-consuming decomposition and inversion steps in solving the asymmetric matrix of the system equation, this paper combines the system matrices \(\mathbf{M}\) and \(\mathbf{H}\) into a single matrix to reduce computation time. Moreover, for the thermoelastic system with the non-Fourier dual-phase lag heat conduction model considered in this paper, the second-order term related to time cannot be directly transformed. Therefore, the first term on the left side of the equation is constructed as a triangular matrix to simplify the inversion and decomposition steps:  
\[\left[ \begin{array}{ccc}\mathbf{H}_{q}&\mathbf{0}&\mathbf{F}_{q}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{M}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{I}\end{array} \right]\left[ \begin{array} []{c}\dot{\mathbf{\Theta}}\\ \dot{\mathbf{\Theta}}\\ \dot{\mathbf{U}}\\ \dot{\mathbf{U}}\\ \end{array} \right]+\left[ \begin{array}{ccc}\mathbf{S}_{T}+\mathbf{H}& \mathbf{S}+\mathbf{F}&\mathbf{0}&\mathbf{0}\\ -\mathbf{I}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&-\mathbf{G}&\mathbf{0}&\mathbf{K}\\ \mathbf{0}&\mathbf{0}&-\mathbf{I}&\mathbf{0}\\ \end{array} \right]\left[