# Fail-Safe Topology Optimization Using Damage Scenario Filtering
## 3 Numerical Benchmarks
### Cantilever Beam  
Taking the cantilever beam, which is frequently discussed in many reference studies, as an example (as shown in Figure 3), the design domain of the cantilever beam is discretized into 120 \(\times\) 40 elements, forming a total of 4800 elements. The entire design domain consists of 4800 elements. A vertical load (F = 1) is applied at the bottom right corner of the cantilever beam, with a fixed constraint applied to the left edge of the cantilever beam. The volume fraction of the solid material to be preserved is limited to 0.4, representing the deformation energy of the structure under local damage, as well as the proportion of the second local failure scenario in all failure scenarios. Subsequently, we can establish a density iteration. Firstly, the cantilever beam is optimized to achieve a patch size of 5 \(\times\) 5. Additionally, local damage near the loading position, indicated by the red arrow, is not considered during the optimization process. The loading conditions remain unchanged, mitigating the impact of stress concentration at the loading point. Figure 4 presents the optimized cantilever structure obtained through the fail-safe optimization algorithm for the damage model (L = 5). This includes designs from both traditional fail-safe optimization and the improved algorithm. The nominal design represents the optimization result from the traditional algorithm, while the others denote fail-safe optimization designs from the improved algorithm. In the designs obtained through the improved algorithm, adjusting the threshold of the density filter to 0.01, 0.1, 1, 2, and 4, respectively, produces corresponding fail-safe optimization designs. Comparing the designs obtained through the improved algorithm with those from the standard fail-safe optimization algorithm reveals that the improved fail-safe optimization algorithm not only achieves the desired fail-safe optimization designs, but also significantly reduces the computational effort required for optimization. As shown in Figures 4 and 5, the larger the threshold of the filter, the smaller the computational effort needed for the optimization process. Figure 6a illustrates the cantilever structure obtained through the nominal fail-safe optimization algorithm when adjusting the size of the local damage model to L = 14. As the area of the local damage region increases, the threshold of the damage scenario filter also needs to increase, set at 10, 21, 32, 43, and 54, respectively. From Figures 6 and 7, it is evident that the enhanced fault-tolerant optimization algorithm can achieve the desired structural design and reduce the computational effort required for optimization. However, as the filter threshold increases, the difference between the design obtained through the improved algorithm and the standard design becomes more significant. Clearly, the optimization process deviates from the correct direction, which is not the expected outcome. The  
Figure 3: Local damage model of cantilever beam. judicious and efficient selection of the filter threshold value warrants further research, which is not detailed here. Figure 4: Optimization of cantilever structure with L = 5. (**a**) Comparison of optimization results. (**b**) The change in time in the optimization process of optimization methods. Figure 5: Comparison of average iteration times for each method when L = 5. Figure 6: Optimization of cantilever structure with L = 14. (**a**) Comparison of optimization results. (**b**) The change in time in the optimization process of optimization methods. Figure 7: Comparison of average iteration times for each method when L = 14. Clearly, by assessing the effectiveness of the local damage scenarios, the topological optimization calculations for fail-safe structures can be significantly reduced. Furthermore, this study seamlessly integrates the preconditioned conjugate gradient method with the fail-safe topology optimization algorithm to enhance the computational efficiency of the fail-safe topology optimization algorithm.