# Fail-Safe Topology Optimization Using Damage Scenario Filtering
## 2 Theory of Fail-Safe Structural Topology Optimization
### Theory of Fail-Safe Topology Optimization  
The formulation for minimizing compliance in classical deterministic topology optimization is as follows:  
\[ \begin{array}{ll}\text{Find}&\mathbf{\rho}=(\rho_{1},\,\rho_{2},\cdots,\rho_{N -1},\rho_{N})^{\mathsf{T}}\\ \underset{\mathbf{\rho}}{\text{min}}&f(\mathbf{\rho})\\ \text{s.t.}&\mathbf{Ku}=\mathbf{F}\\ &V_{(\mathbf{\rho})}-V_{\text{max}}\leq 0\\ &0<\rho_{e}\leq 1\end{array}  \tag{2}\]  
where \(\mathbf{\rho}\) represents the vector of topological design variables for the structure, \(f\) denotes the flexibility of the structure, and \(\mathbf{K}\) signifies the stiffness matrix of the structure. The displacement vector \(\mathbf{u}\) is the result of the applied load vector \(\mathbf{F}\) on the structure. Furthermore, \(V\) and \(V_{\text{max}}\), respectively denote the material volume of the structural design domain and the material volume after structural optimization. To establish the mathematical model for fail-safe topology optimization, it is essential to formulate the element stiffness matrix for the local damage model. By assigning the material stiffness of components within pre-defined damaged regions as \(E_{\text{min}}\) and representing the corresponding set of elements as \(N_{a}\), the remaining set of elements without local damage is denoted as \(N_{e}\backslash N_{a}\). Here, \(N_{e}\) encompasses the collection of all elements in the entire structure. Consequently, the overall stiffness matrix of the structure can be expressed as  
\[\mathbf{K}_{m}^{a}=\sum\limits_{j\in N_{a}}\mathbf{K}_{j}+\sum\limits_{j\in N _{e}\backslash N_{a}}\mathbf{K}_{j} \tag{3}\]  
Integrating the element stiffness matrices, with \(\sum\limits_{j\in N_{a}}\mathbf{K}_{j}\) representing the damaged region and \(\sum\limits_{j\in N_{e}\backslash N_{a}}\mathbf{K}_{j}\) representing the undamaged region, yields the overall stiffness matrix \(\mathbf{K}_{m}^{a}\) for the structure. Here, \(\mathsf{m}\) represents the \(m\)-th localized damage scenario for the structure, and \(\mathsf{j}\) denotes the element number. Subsequently, by establishing the finite equilibrium equations, the structure's flexibility \(f^{(m)}\) under the \(m\)-th damage condition is determined through the following equation:  
\[f^{(m)}(\mathbf{\rho})=(\mathbf{u}^{m})^{\mathsf{T}}\mathbf{K}_{m}^{a}\mathbf{u}^ {m} \tag{4}\]  
Once the design domain and local damage zones are established, distinct local damages will have varying effects at different positions within the design domain. The fail-safetopology optimization method considers all potential impacts of local damage, enabling the simultaneous attainment of an optimally designed structure that is insensitive to local damages. This is defined here as the total number of local damage scenarios to be considered during optimization iterations. For each structural design obtained through optimization iterations, the most critical damage scenario is identified by comparing the structural strain energy under different damage scenarios. The objective function is then set as the minimum structural strain energy under the most severe damage scenario, thereby obtaining the mathematical model for fail-safe structural topology optimization. \[ \begin{split}\mathbf{K}_{m}^{a}\mathbf{u}^{m}=\mathbf{F}\\ f_{r}(\mathbf{\rho})=\log\biggl{(}\sum\limits_{i=1}^{m}\exp \Bigl{(}\gamma f^{(i)}(\mathbf{\rho})\Bigr{)}\biggr{)}/\gamma\end{split}  \tag{5}\]  
where \(f_{r}(\mathbf{\rho})\) represents the adjusted objective function, while \(\mathbf{F}\) and \(\mathbf{u}^{m}\), respectively, denote the nodal displacement vector and nodal load vector of the structure in the global coordinate system. \(V(\mathbf{\rho})\) and \(V_{\text{max}}\) represent the overall volume of the structure and the maximum volume constraint in the optimization problem, respectively. \(\mathbf{K}\) is the overall stiffness matrix of the structure, obtained through the integration of the element stiffness matrix to establish the equilibrium equation and subsequently derive the displacement field.