# Fail-Safe Topology Optimization Using Damage Scenario Filtering
## 3 Numerical Benchmarks
### L-Bracket  
The well-known L-shaped beam, as illustrated in Figure 8, is discretized using four-node finite elements with a size of 1 \(\times\) 1 mm\({}^{2}\), resulting in a total of 10,000 elements. In the final topology, only 35% of the material is available. The top of the L-shaped bracket is fixed, and an external load (F = 1) is applied at the bottom right corner, as shown at the top of Figure 8. The gray area in the figure represents the prescribed non-active region. The width of the non-active region is equal to the side length of the local damage model. The dimensions of the local damage model are L = 4, used to optimize the L-shaped bracket. Figure 9 displays the L-shaped bracket structure obtained through the standard fail-safe optimization method, followed by the improved structural design using the fail-safe topology optimization algorithm. The density filter 'A' threshold is adjusted to 0.01, 0.1, 1, 2, and 4, respectively. From the final results, it is challenging to discern significant differences between them. When the side length of the damage model is L = 4, the optimization results from both the traditional fail-safe optimization method and the improved fail-safe optimization algorithm with different thresholds are nearly identical. Figure 10 illustrates the iteration time curves for the density-filtered failure-safe optimization method under the condition of a local damage model with L = 3. The filter thresholds are set to 0.01, 0.1, 1, 2 and 4, and a comparison is made with the standard failure-safe optimization method. In Figure 11, the horizontal axis represents the filter threshold, while the vertical axis depicts the reduction in the average iteration steps during the optimization process and the fail-safe topology optimization, respectively (for varying thresholds). Figure 12 illustrates the average iteration time before and after the improvement of the fail-safe structural optimization method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{L = 5} & A & 0.01 & 0.1 & 1 & 2 & 4 \\ \cline{2-7}  & Rate & 13.7\% & 13.9\% & 22.2\% & 26.7\% & 35.8\% \\ \hline \multirow{2}{*}{L = 14} & A & 10 & 21 & 32 & 43 & 54 \\ \cline{2-7}  & Rate & 0 & 0 & 1.6\% & 16.1\% & 35.6\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reduction rate in the average iteration time during the optimization process of the cantilever beam. Figure 8: Model of L-shaped beam. Figure 11: Comparison of average iteration times for each method when L = 4. Figure 10: The change in iteration time in the optimization process of the optimization method. Figure 9: Comparison of optimization results of L-shaped beam when L = 4. Figure 13 depicts the fail-safe optimization designs for the L-shaped bracket obtained using the fail-safe optimization method with a local damage model size of L = 6 and filter thresholds set to 0.01, 0.1, 1, 2, and 4. After analyzing the optimization results of both the standard fail-safe optimization method and the improved fail-safe optimization method, it can be observed from Figures 13 and 14 that the larger the threshold of the density filter, the greater the differences between the final optimization results of the respective algorithms. Additionally, with the increase in the filter threshold, the time required for each iteration during the optimization process gradually decreases. However, when L = 6, although the entire optimization solution can converge to the optimal structure, stability issues may arise during the optimization process. This is because the size of the local damage model directly affects the variation in sensitivity information during the optimization process. As the scale of the local damage model increases, stability and convergence issues in the optimization process become more pronounced. By observing Table 2, it can be noted that by adjusting the threshold of the filter, the computational effort required for fault-tolerant structural topology optimization can be effectively reduced.