## 2 Topological Optimization Problem with Stress Constraints under Aperiodic Load
### Topology Optimization Model after Coalescing Stress Replacement  
The design domain is divided into n finite element meshes, and each element corresponds to a design variable. The design variable can be described as:  
\[x_{i}\in\{0,1\}\,i=1,2,\cdots n \tag{11}\]  
The design variable can be understood as the relative density of the element, and more directly, it can be considered as an indicator of the presence or absence of the element material. When \(x_{i}=1\), it indicates that the \(i\)th element is a solid element with material. Conversely, it indicates that the \(i\)th element is an empty element without material. The interpolation model between the design variables and the elastic matrix of the element material is established.  
\[\mathbf{D}_{i}=x_{i}\mathbf{D}_{0} \tag{12}\]  
where \(\mathbf{D}_{i}\) represents the material elastic matrix corresponding to the \(i\)th element.\(\mathbf{D}_{0}\) represents the elastic matrix of the solid materials, which is only related to the material properties and is a constant matrix.  
According to the elastic mechanics theory, the stress at a point \(a\) of the solid structure is as follows:  
\[\sigma_{a}=[\sigma_{xx},\sigma_{ay},\sigma_{az},\tau_{axy},\tau_{ayz},\tau_{ azx}]^{T} \tag{13}\]  
In the finite element analysis, the following formula can be described as:  
\[\sigma_{a}=\mathbf{D}\mathbf{B}_{a}\mathbf{u} \tag{14}\]  
where \(\mathbf{B}_{a}\) is the strain matrix of point \(a\), and \(\mathbf{u}\) is the displacement vector of point \(a\).  
Therefore, the equivalent stress model of the element can be described as:  
\[\sigma_{i}=\frac{\mathbf{D}_{i}\mathbf{B}_{i}\mathbf{u}(x)}{x_{i}}=\mathbf{D }_{0}\mathbf{B}_{i}\mathbf{u}_{i} \tag{15}\]  
where \(\sigma_{i}\) is the stress vector of the \(i\)th element, \(\mathbf{B}_{i}\) is the strain matrix of the centroid of the \(i\)th element, and \(\mathbf{u}_{i}(x)\) is the node displacement matrix of the \(i\)th element. The von Mises stress can be described as:  
\[\sigma_{vm}=(\mathbf{\sigma}^{T}\mathbf{V}\mathbf{\sigma})^{1/2} \tag{16}\]  
where \(\sigma_{vm}\) is the von Mises stress, \(\mathbf{\sigma}\) is the element stress vector, and \(\mathbf{V}\) is the stress coefficient matrix.  
Because the constraint condition in this optimization problem model is the maximum Mises stress, which has the local attribute of the stress constraint, its size cannot be described by a clear mathematical analytical formula, and the stress sensitivity of each element needs to be calculated in the optimization process. In order to reduce the calculation cost caused by the local stress constraint, the aggregation function is usually used to obtain a global stress measure to approximate the maximum Mises stress of the structure. In this article, the P-norm condensation function is adopted, and it can be described specifically as:  
\[\sigma_{PN}=(\sum_{j=1}^{n}\sigma_{vm,j}^{p})^{\frac{1}{p}} \tag{17}\]  
where \(\sigma_{PN}\) is the cohesion stress value of the structure and \(p\) is the stress norm value considered to be set; \(\sigma_{vm,i}\) is the Mises stress value at the center of the \(i\)th element. From the expression form of this expression, when \(p=1\), \(\sigma_{PN}\) is the sum of the Mises stress values of all the elements. When p approaches infinity, \(\sigma_{PN}\) is the exact Mises stress value of the structure.  
After the maximum Mises stress is replaced by the condensed stress, the new topological optimization mathematical model can be described as:  
\[\begin{split}\text{find}:& x_{i}=[x_{1},x_{2}, \cdots,x_{n}]\\ \text{min}:& V=\sum_{i=1}^{n}V_{i}x_{i}\\ \text{s.f.}:&\mathbf{M}\mathbf{\dot{U}}+\mathbf{ \dot{C}}\mathbf{\dot{U}}+\mathbf{K}\mathbf{U}=\mathbf{F}\\ &\sigma_{PN}\leq\sigma_{vm}^{*}\\ & x_{i}=0\text{ or }1\end{split} \tag{18}\]