## 2 Topological Optimization Problem with Stress Constraints under Aperiodic Load  
### Transient Dynamics Analysis  
In the actual engineering problems, the load form borne by the structure is often dynamic. The general motion control equation expressed in terms of time is as follows:  
\[\mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{C}\ddot{\mathbf{u}}(t)+\mathbf{Ku}(t)= \mathbf{F}(t) \tag{1}\]  
where \(\mathbf{M}\) is the structural mass matrix, \(\mathbf{C}\) is the structural damping matrix, \(\mathbf{K}\) is the structural stiffness matrix, \(\mathbf{F}(t)\) is the node load vector varying with time, and \(\ddot{\mathbf{u}}\), \(\dot{\mathbf{u}}\), and \(\mathbf{u}\) are the node acceleration, node velocity, and node displacement vector, respectively. There are two methods to solve the dynamic finite element equations, namely the direct integration method and the modal superposition method. The basic idea of the direct integration method is to divide the time domain into a series of time points, which divides the time domain into several time intervals and then solves the solution that can satisfy Formula (1) at the discrete time points. The modal superposition method is to superimpose the responses of the system under various modes, but the proportion of the weight factors of each order is different. In this article, the birth-death element method is used to realize the addition and deletion of elements in the process of topological optimization. This process is a nonlinear analysis; so, it can only use the direct integration method but not the mode superposition method. The common direct integration methods include the [PERSON] method, [PERSON] method, [PERSON] method, central difference method, etc. This section focuses on the Newmark method. The Newmark method assumption:  
\[\ddot{\mathbf{u}}^{t+\Delta t}=\dot{\mathbf{u}}^{t}+[(1-\beta)]\ddot{\mathbf{ u}}^{t}+\beta\ddot{\mathbf{u}}^{t+\Delta t}\Delta t \tag{2}\]  
\[\mathbf{u}^{t+\Delta t}=\mathbf{u}^{t}+\Delta t\dot{\mathbf{u}}^{t}+\left[ \left(\frac{1}{2}-\gamma\right)\ddot{\mathbf{u}}^{t}+\gamma\ddot{\mathbf{u}}^{ t+\Delta t}\right]\Delta t^{2} \tag{3}\]  
where \(\gamma\) and \(\beta\) are the Newmark integral parameters. When \(\beta\geq 0.5\) and \(\gamma\geq 0.25(0.5+\beta)^{2}\), the Newmark method has unconditional stability. \(\ddot{\mathbf{u}}\), \(\dot{\mathbf{u}}\), and \(\mathbf{u}\) represent, respectively, the node displacement vector, velocity vector, and acceleration vector at time \(t\); \(\mathbf{u}^{t+\Delta t}\), \(\dot{\mathbf{u}}^{t+\Delta t}\), and \(\ddot{\mathbf{u}}^{t+\Delta t}\) represent, respectively, the node displacement vector, velocity vector, and acceleration vector at time \(t+\Delta t\). To calculate the structural displacement vector \(\mathbf{u}^{t+\Delta t}\), the equation of motion at time \(t+\Delta t\) can be expressed as:  
\[\mathbf{M}\ddot{\mathbf{u}}^{t+\Delta t}+\mathbf{C}\dot{\mathbf{u}}^{t+\Delta t }+\mathbf{Ku}^{t+\Delta t}=\mathbf{F}^{t+\Delta t} \tag{4}\]  
The effective stiffness matrix and effective load at time \(t_{n+1}\) can be expressed as:  
\[\mathbf{\overline{K}}=\mathbf{K}+\frac{1}{\alpha_{0}\Delta t^{2}}\mathbf{M}+ \frac{\beta}{\alpha_{1}\Delta t^{2}}\mathbf{C} \tag{5}\]\[\overline{\mathbf{F}}^{t+\Delta t} = \mathbf{F}^{t+\Delta t}+\mathbf{M}\Big{(}a_{0}\mathbf{u}^{t}+a_{2} \mathbf{\dot{u}}^{t}+a_{3}\mathbf{\dot{u}}^{t}\Big{)}+ \tag{6}\] \[\mathbf{C}\Big{(}a_{1}\mathbf{u}^{t}+a_{4}\mathbf{\dot{u}}^{t}+a_{ 5}\mathbf{\dot{u}}^{t}\Big{)}\]  
where \(a_{0}=\frac{1}{\gamma\Delta t^{2}}\), \(a_{1}=\frac{\beta}{\gamma\Delta t}\), \(a_{2}=\frac{1}{\gamma\Delta t}\), \(a_{3}=\frac{1}{2\gamma}-1\), \(a_{4}=\frac{\beta}{\gamma}-1\), \(a_{5}=\frac{\Delta t}{2}\Big{(}\frac{\beta}{\gamma}-2\Big{)}\), \(a_{6}=\Delta t(1-\beta)\), \(a_{7}=\beta\Delta t\).