## 3 Efficient Modal Modification Analysis Method
### Neumann Series Expansion- and Reduced Basis-Based Preconditioner Method  
As stated before, the mass and heat generation matrices are lumped matrices to efficiently solve the combined equation. A Neumann series expansion- and reduced basis-based preconditioner method is proposed herein. Due to the fully coupled characteristic of the system, the coupled terms in governing the equations of motion (Equation (1)) and heat conduction (Equation (3)) have to be solved at the same time. For two-dimensional thermoelastically coupled eigenvalue problems, the degrees of freedom of the system \(N_{d}\) is \(N_{d}=3N_{n}\) (including the displacement \(2N_{n}\) and temperature \(1N_{n}\)), where \(N_{n}\) is the number of nodes. Thus, the traditional solution of the combined equation (Equation (5)) suffers an increase in the dimensions of \(N_{s}=5N_{n}\). The increase in computational cost could be further magnified by the decomposition and inverse process of the solution. The Krylov subspace-based methods are often used to solve the eigenvalue problems for their simplicity and efficiency in generating orthonormalized bases [47; 48]. The standard Krylov method of the standard eigenvalue problem in the form of Equation (7) can be formulated as [49]  
\[\mathcal{K}_{n}(\mathbf{A},\mathbf{B};\boldsymbol{\Phi}_{i})=\text{span} \Big{\{}\boldsymbol{\Phi}_{i},\mathbf{B}^{-1}\mathbf{A}\boldsymbol{\Phi}_{i},...,(\mathbf{B}^{-1}\mathbf{A})^{n-1}\boldsymbol{\Phi}_{i}\Big{\}} \tag{13}\]  
However, considering the increase in the dimensions and repeated process in the optimization procedure, solving the eigenproblem directly with the traditional Krylov method could be time- and computational cost-consuming. Thus, we use a Neumann series expansion based technique to construct a lower-order subspace of the system. As mentioned above, structural modification is encountered frequently in each iteration of the topology's optimal designs to enhance the performance of structural or mechanical systems by altering their substructures. Suppose a change in the design parameters resulting in modified system matrices, which can be expressed as  
\[\mathbf{A}_{m}=\mathbf{A}+\Delta\mathbf{A},\quad\mathbf{B}_{m}=\mathbf{B}+ \Delta\mathbf{B} \tag{14}\]Here, \(\mathbf{B}_{m}\) and \(\mathbf{A}_{m}\) are the modified matrices. \(\Delta\mathbf{B}\) and \(\Delta\mathbf{A}\) denote the change in the system matrices. The modified eigenproblem can then become  
\[(\lambda_{mi}\mathbf{B}_{m}-\mathbf{A}_{m})\mathbf{\Phi}_{mi}=\mathbf{0} \tag{15}\]  
A preconditioner is used here to turn this eigenproblem into a standard Neumann series expansion form. The preconditioners have been widely used in the solution of linear partial differential equations within finite element eigenvalue problems [50; 51], which occur frequently in thermal and dynamical systems. There are many types of preconditioners, including the diagonal preconditioners, the reduced-basis preconditioners, and the differential preconditioners, which can be interpreted as a differential operator [45; 52]. The preconditioners are frequently (or even necessarily) used in the subspace solutions of partial differential equations [53]. Based on different types of preconditioners, preconditioned iterative methods were proposed to solve large-scale eigenproblems with a slow convergence [47]. Among all the types of preconditioners, the diagonal preconditioner is famous for its simplicity and fast convergence. Thus, a diagonal preconditioner \((\lambda_{mi}\mathbf{B}_{m})^{-1}\) is chosen herein and pre-multiplied to the modified eigenproblem in Equation (15) [54]. Note that for a specific \(i\)th mode shape, the system eigenvalue \(\lambda_{mi}\) is scalar, while the system matrix \(\mathbf{B}_{m}\) has been constructed as a diagonal matrix. Thus, the preconditioner adds little to the computational effort of the inverse step in constructing the basis.