# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 3 Efficient and Robust Topology Optimization Method
### Reduced Basis in Neumann Series Subspace  
During the iteration of topology optimization, structural configuration changes often occur by changing its material distribution to improve the mechanical system's performance. Therefore, the system matrix change caused by changes in structural design parameters can be represented as follows:  
\[\mathbf{A}_{m}=\mathbf{A}+\Delta\mathbf{A},\quad\mathbf{B}_{m}=\mathbf{B}+ \Delta\mathbf{B} \tag{21}\]  
where \(\mathbf{B}_{m}\) and \(\mathbf{A}_{m}\) are the system matrices after structural configuration changes, and \(\Delta\mathbf{B}\) and \(\Delta\mathbf{A}\) represent the changes of the system matrices. The eigenvalue problem after structural configuration changes can be expressed as  
\[\left(\lambda_{mi}\mathbf{B}_{m}-\mathbf{A}_{m}\right)\mathbf{\Phi}_{mi}= \mathbf{0} \tag{22}\]  
where \(\lambda_{mi}\) and \(\mathbf{\Phi}_{mi}\) are, respectively, the \(i\)th eigenvalue and eigenvector of the modified viscoelastic structures.  
Recalling Equation (21) and using some matrix manipulations, the modified eigenproblem (22) can be rewritten as  
\[\left(\mathbf{I}_{N}-\xi_{i}\mathbf{B}_{m}^{-1}\Delta\mathbf{A}\right)\mathbf{ \Phi}_{mi}=\xi_{i}\mathbf{B}_{m}^{-1}\mathbf{A}\mathbf{\Phi}_{mi} \tag{23}\]  
where  
\[\xi_{i}=\frac{1}{\lambda_{mi}} \tag{24}\]  
Applying Neumann series expansion to implement the inverse matrix of the coefficient matrix, \(\left(\mathbf{I}_{N}-\xi_{i}\mathbf{B}_{m}^{-1}\Delta\mathbf{A}\right)^{-1}\), one obtains  
\[\mathbf{\[PERSON]}_{mi}=\xi_{i}\sum_{k=0}^{\infty}\left(\xi_{i}\mathbf{B}_{m}^{-1} \Delta\mathbf{A}\right)^{k}\mathbf{B}_{m}^{-1}\mathbf{A}\mathbf{\[PERSON]}_{mi} \tag{25}\]Using the first \(r\) expansion terms and the initial eigenvector \(\mathbf{\[PERSON]}_{i}\) to approximate the modified eigenvector \(\mathbf{\[PERSON]}_{mi}\), we can obtain an approximate approach for calculating the modified eigenvector \(\mathbf{\[PERSON]}_{mi}\) as  
\[\mathbf{\[PERSON]}_{mi}=\xi_{i}\sum_{k=0}^{r}\left(\xi_{i}\mathbf{B}_{m}^{-1}\Delta\mathbf{ A}\right)^{k}\mathbf{B}_{m}^{-1}\mathbf{A}\mathbf{\Phi}_{i} \tag{26}\]  
**Definition 1** (**Neumann series subspace**).: _The Neumann series subspace \(\mathcal{R}_{r}\) is formed by an \(r\)-dimensional recursive sequence \(\left\{\mathbf{r}_{i}^{(0)},\mathbf{r}_{i}^{(1)},\mathbf{r}_{i}^{(2)},\cdots,\mathbf{r}_{i}^{( n-1)}\right\}\) generated by original system matrices \(\mathbf{A}\) and \(\mathbf{B}\), the changes in original system matrices \(\Delta\mathbf{A}\) and \(\Delta\mathbf{B}\), as well as the \(i\)th eigenvector \(\mathbf{\Phi}_{i}\) of the original eigenproblem defined in Equation (18). That is,_  
\[\mathcal{R}_{r}(\mathbf{A},\mathbf{B};\mathbf{A}_{m},\mathbf{B}_{m};\mathbf{\Phi} _{i})\doteq\text{span}\left\{\mathbf{r}_{i}^{(0)},\mathbf{r}_{i}^{(1)},\ldots,\mathbf{r}_ {i}^{(r-1)}\right\} \tag{27}\]  
_where_  
\[\begin{array}{l}\mathbf{r}_{i}^{(0)}=\left(\mathbf{B}+\Delta\mathbf{B} \right)^{-1}\mathbf{A}\mathbf{\Phi}_{i}\\ \mathbf{r}_{i}^{(k)}=\left(\mathbf{B}+\Delta\mathbf{B}\right)^{-1}\Delta \mathbf{A}\mathbf{r}_{i}^{(k-1)},\quad\forall k=2,3\ldots,r\end{array} \tag{28}\]  
Using a Neumann series subspace \(\mathcal{R}_{r}\), it is convenient to use known modes to approximate the modified eigenvectors as the following form according to the approximate approach defined in Equation (26).  
\[\mathbf{\Phi}_{mi}\approx\sum_{k=0}^{r}\xi_{i}^{-(k+1)}\mathbf{r}_{i}^{(k)} \tag{29}\]  
where the \(r\)-dimensional recursive sequence has already been defined in (28).  
**Remark 1**.: _In the classic subspace method for solving the characteristic equation, the most computationally intensive part is often the inversion step, i.e., the process of solving \(\mathbf{B}_{m}^{-1}\). However, for the reconstructed system matrix \(\mathbf{B}_{m}\) obtained from the lumped mass matrix \(\mathbf{M}\) and heat generation matrix \(\mathbf{H}\), which were previously discussed, the matrix decomposition and inversion steps have very low computational cost since \(\mathbf{B}_{m}\) is a diagonal matrix._