# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 3 Efficient and Robust Topology Optimization Method
### Projection-Based Model Reduction Method  
To maintain the robustness of the modified eigenproblem while reducing the degrees of freedom of the system, this section uses the projection-based model reduction method based on the Neumann series subspace \(\mathcal{R}_{r}\). Based on the \(r\)-th order Neumann subspace basis vectors, the modified eigenvector can be expressed as a linear combination of the original high-fidelity subspace in the Neumann series subspace \(\mathcal{R}_{r}\):  
\[\mathbf{\Phi}_{mi}{=}\sum_{k=0}^{r-1}\alpha_{i}^{(k)}\mathbf{q}_{i}^{(k)}=\mathbf{ R}_{i}\mathbf{\alpha}_{i} \tag{30}\]  
where \(\mathbf{R}_{i}\) is the space spanned by the orthonormalization basis \(\mathbf{q}_{i}^{(k)}\) in the Neumann series subspace \(\mathcal{R}_{r}\) and \(\mathbf{\alpha}_{i}\) is the unknown parameters to be solved. Note that the previous equation used the following relation:\[\mathbf{R}_{i}=\left[ \begin{array}{cccc}\mathbf{q}_{i}^{(0)},&\mathbf{q}_{i}^{(1)},&\ldots,&\mathbf{q}_{i}^{(r-1)}\end{array} \right]\text{,}\quad\boldsymbol{ \alpha}_{i}=\left[ \begin{array}{cccc}\alpha_{i}^{(0)},&\alpha_{i}^{(1)},& \ldots,&\alpha_{i}^{(r-1)}\end{array} \right]^{\mathrm{T}} \tag{31}\]  
According to the projection-based model reduction method, the degrees of freedom of the system matrices can be reduced as the following form [25]:  
\[\mathbf{B}_{R}=\mathbf{R}_{i}^{\mathrm{T}}\mathbf{B}_{m}\mathbf{R}_{i}\text{,} \quad\mathbf{A}_{R}=\mathbf{R}_{i}^{\mathrm{T}}\mathbf{A}_{m}\mathbf{R}_{i} \tag{32}\]  
Because \(N_{s}\gg r\), the reduced system matrices are very small. **Remark 2**. : _Because \(\mathbf{R}_{i}\) is a reduced-basis matrix operator and \(\mathbf{A}_{m}\) is not necessarily a self-adjoint matrix, the robustness of the rectangular basis matrix \(\mathbf{R}_{i}\) must be improved by using the Gram-Schmidt orthogonalization algorithm. Thus, \(\mathbf{R}_{i}\) is the space spanned by the orthonormalization basis \(\mathbf{q}_{i}^{(k)}\) and can be generated by using the \(r\)-dimensional recursive sequence already defined in (28) based on the Gram-Schmidt orthogonalization algorithm. That is, the rectangular basis matrix \(\mathbf{R}_{i}\) should be normalized such that each column becomes linearly independent._  
Since the approximate eigenvectors should satisfy the eigenproblem (22), the unknown coefficients \(\boldsymbol{\alpha}_{i}\) can be determined by substituting the approximate eigenvectors into Equation (22). Combining Equations (22) and (32), the standard eigenvalue problem of the system can be replaced by a reduced eigenvalue problem:  
\[(\lambda_{mi}\mathbf{B}_{R}-\mathbf{A}_{R})\boldsymbol{\alpha}_{i}=\mathbf{0} \tag{33}\]  
The dimension of the reduced system matrix is \(r\times r\), which is much smaller than the original system matrix \(N_{s}\times N_{s}\). Therefore, it is only necessary to solve the reduced eigenvalue problem, which reduces the size of the eigenvalue problem from \(N_{s}\times N_{s}\) to \(r\times r\). Thus, the projection-based model reduction method can save the computational cost significantly. After solving for coefficients \(\boldsymbol{\alpha}_{i}\), the modified eigenvector can be obtained using Equation (30). As shown in Algorithm 1, the construction of the \(r\) order subspace of the Neumann series expansion combined with the reduced-basis solution improves the efficiency of the modal analysis. Numerous studies have shown that the computational complexity of the traditional method for solving the combined eigenvalue equation is \(O(N_{s}^{3})\)[40, 41]. Since the calculation scale of solving the basis vectors is relatively large compared to other steps, we mainly focus on the computational complexity of solving the basis vectors.