## 2 The Topology Optimization Problem
### Material Model  
When using the SIMP model for topology optimization problems involving eigenvalue optimization, such as dynamic topology optimization problems and topology optimization problems with consideration of the stability constraints, artificial modes (also known as local modes) may appear in low-density areas, and these artificial modes will have an adverse impact on the optimization process. Many scholars have studied the problem of false modes and put forward corresponding solutions. To avoid artificial modes, in this research, the method proposed by [PERSON] and [PERSON] 5 was adopted to calculate the element stiffness matrix and geometric stiffness matrix using different material interpolation models. When calculating the element stiffness matrix, the expression for the material interpolation model used is as follows  
\[E(x)=E_{\min}+(E_{0}+E_{\min})x_{e}^{P} \tag{5}\]  
where \(E_{\min}\) is a non-zero lower bound introduced to ensure the positive definition of the total stiffness matrix of the structure. In the optimization problem considering stability constraints, [PERSON] and [PERSON] 5 suggested that this should be \(10^{-6}\). \(E_{0}\) represents the elastic modulus of solid material, \(x_{e}\) represents the density of the element, and \(P\) is the penalty factor, which is taken as three when interpolating the element stiffness.  
When calculating the geometric stiffness matrix, the expression of the material interpolation model is as follows:  
\[E_{G}=E_{0}x_{e}^{P} \tag{6}\]  
where \(P\) is the penalty factor, which is taken as three when interpolating the element stiffness.  
In this research, three density fields were used to achieve the final optimal design. The density filtering method proposed by [PERSON] [39] was used to avoid numerical instability, and the projection filter proposed by [PERSON] et al. [40] was used to obtain a black and white design. In the expression below, \(x_{e}\) is the initial density field, \(\widetilde{x}_{e}\) is the density field filtered by the density filter, and \(\hat{x}_{e}\) is the density field filtered by the projection. The relationship between \(x_{e}\) and \(\widetilde{x}_{e}\) can be expressed as follows:  
\[\widetilde{x}_{e}=\frac{\sum_{i=1}^{m}h_{e,i}x_{i}}{\sum_{i=1}^{m}h_{e,i}} \tag{7}\]  
where \(h_{e,i}=\max(0,r_{min}-dist(\Omega_{i},\ \Omega_{e}))\), \(r_{min}\) is the filter radius, and \(dist(\Omega_{i},\ \Omega_{e})\) is expressed as the distance between the center of the element \(\Omega_{i}\) and the element \(\Omega_{e}\), and \(\hat{x}_{e}\) is obtained with a projection filter of \(\widetilde{x}_{e}\):  
\[\hat{x}_{e}=\frac{\tanh(\beta\eta)+\tanh(\beta(\widetilde{x}_{e}-\eta))}{ \tanh(\beta\eta)+\tanh(\beta(1-\eta))} \tag{8}\]  
where \(\eta\) and \(\beta\) are projection parameters, and \(\eta\in[0,1]\), \(\eta\in[1,\infty]\). Therefore, when calculating the objective function and the sensitivity of each constraint to the design variables, the chain rule needs to be adopted in the following form:  
\[\frac{\partial\mathcal{C}}{\partial x_{e}}=\frac{\partial\mathcal{C}}{ \partial\hat{x}_{e}}\frac{\partial\hat{x}_{e}}{\partial\widetilde{x}_{e}} \frac{\partial\widetilde{x}_{e}}{\partial x_{e}} \tag{9}\]