Issue 78

M. Habashneh et alii, Fracture and Structural Integrity, 78 (2026) 409-431; DOI: 10.3221/IGF-ESIS.78.24

and cost. Conversely, a denser network of slender members can increase the exposed structural surface area and the number of connection regions, potentially increasing surface-protection requirements and introducing additional locations that may become relevant to fatigue performance under dynamic loading. Therefore, the selection of the evolutionary rate involves a trade-off between structural stiffness and topology complexity rather than stiffness alone. Manufacturing cost, surface protection requirements, and fatigue-related performance are not explicitly included in the present optimization formulation. These considerations motivate the extension to dynamic loading identified in the future-work discussion, where the fatigue performance of the resulting members and connection regions can be investigated explicitly. U-Shaped Plate: effect of filter radius The effect of the filter radius is investigated using r min = 8, 10, and 12 mm, while keeping the mesh size and evolutionary rate fixed at h=2 mm and ER = 0.01. The resulting topologies are shown in Fig. 16, and the corresponding displacement values are listed in Tab. 6. Similar to the previous benchmarks, the filter radius strongly influences the minimum feature size in the optimized design. This effect is especially noticeable for the U-shaped plate because of its complex geometry and multiple load-transfer paths. Although all cases retain the same overall structural concept, clear differences can be observed in the number and shape of the internal diagonal members. The baseline case ( r min = 12 mm, Fig. 16(c)) produces a balanced topology with well-defined diagonal ribs and a clear bridging member across the notch. Several load paths connect the arms to the left web, resulting in an efficient force-transfer system. Reducing the filter radius to 8 mm (Fig. 16(a)) generates the most detailed topology. More diagonal members appear throughout the structure, and the internal layout becomes more refined. The diagonal ribs are thinner, the triangular regions are smaller, and the distinction between solid and void areas becomes clearer. Additional members also reinforce the region around the notch. This richer network of load paths improves the distribution of forces and leads to the stiffest response among the three cases. For r min = 10 mm (Fig. 16(b)), the topology becomes more consolidated. The number of diagonal members decreases, and the structure relies more on thicker boundary members. The bridging strut across the notch remains present but appears wider and less refined. Overall, the design becomes simpler and more dependent on a limited number of primary load-carrying members. This behavior is similar to that observed for higher evolutionary rates, although the simplification is driven here by stronger length-scale control rather than faster material updates.

Figure 16: Effect of filter radius on the optimized topology of the U-shaped plate: (a) ( r min = 8 mm); (b) ( r min = 10 mm); (c) baseline case ( r min = 12 mm). The displacement results further highlight the effect of the filter radius on structural performance. The three cases show a clear relationship between topology and stiffness. Designs with smaller filter radii develop a more detailed internal bracing system and a larger number of load-transfer paths, leading to improved stiffness. In contrast, increasing the filter radius simplifies the topology by removing finer structural features, which results in higher displacement values and lower stiffness. These observations are consistent with the trends found in the L-shaped beam and pinned plate examples. The filter radius remains the parameter with the strongest influence on the final topology, while the evolutionary rate mainly controls the completeness of the internal bracing network and the overall structural efficiency. Mesh size has a smaller effect, although its influence is more noticeable in the U-shaped plate because of the greater geometric complexity. Overall, the results demonstrate that all three parameters contribute to the quality and performance of the optimized design. The similar behavior observed across the three benchmark problems increases confidence in the robustness and general applicability of the proposed optimization approach.

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