# Fail-Safe Topology Optimization Using Damage Scenario Filtering
## 2 Theory of Fail-Safe Structural Topology Optimization
### Damage Scenario Filtering  
In general, there are two main factors contributing to the high computational complexity of the fail-safe structural topology optimization method. Firstly, it involves solving the structural finite element equations to compute sensitivity information for each local damage scenario. Secondly, the number of local damage scenarios to be considered during the optimization process is a significant factor since optimization information needs to be obtained for each of these scenarios. Figure 1 illustrates the optimization process for the nth iteration, where m\({}^{n}\) represents the number of damage scenarios to be considered, m\({}^{n}_{\text{decrease}}\) denotes the reduced number of damage scenarios, and m\({}^{n}_{\text{improved}}\) indicates the number of damage scenarios to be considered after adopting the improved algorithm.  
For the first scenario, the combined approximation reanalysis method can be employed to enhance the efficiency of obtaining optimization information. The second scenario constitutes a primary focus of this study. Through the extension of the active set method, this approach can be considered as a means of reducing the computational complexity of fail-safe structural optimization by simplifying the number of local damage scenarios.  
When addressing issues that are present in standard fail-safe topology optimization, the use of density-based filters is employed to calculate the unit densities in the local damage region. This, in turn, allows for the assessment of the impact of local damage on the structure, eliminating situations where there are either no units or very few units in the local damage region. This ensures that even with a loss in stiffness of certain materials in the design domain, the structure's overall integrity remains largely unaffected.  
In the aforementioned context, there is no longer a need to perform sensitivity calculations for local damage scenarios. It suffices to consider the sensitivity information under the corresponding damage scenarios as the sensitivity information for the structure without local damage. Referring to Figure 2, one can comprehend the distinctions among traditional topology optimization methods, fail-safe topology optimization methods, and the improved fail-safe optimization method employed in this study, especially regarding the number of finite element analyses and sensitivity calculations during the optimization process.  
\[A_{0}=\int\limits_{N_{i}\in\Omega}\rho(x_{i})d\Omega \tag{14}\]  
where A\({}_{0}\) is defined as the sum of the unit density in the damaged area by comparing it to the predefined density threshold, A, and regions where A\({}_{0}\leq\) A are considered to contribute lower than the given threshold and are excluded from further optimization, thus disregarding failure. Conversely, regions where A\({}_{0}\geq\) A are identified as failure areas contribute higher than the threshold, and they are simulated for failure in subsequent optimizations. Adjusting the threshold reasonably helps regulate the number of damage scenarios, thereby enhancing the computational efficiency of the fail-safe optimization method. To enhance readers' comprehension of this approach, pseudo-MATLAB code will be presented in Algorithm 1.  
Figure 1: Flow charts of traditional fail-safe topology optimization method and proposed method.