Issue 78

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

U-Shaped Plate: effect of evolutionary rate The effect of the evolutionary rate is investigated using three values: ER = 0.01, 0.02, and 0.03, while keeping the mesh size and filter radius fixed at 2 mm and 12 mm, respectively. The optimized topologies are shown in Fig. 15, and the corresponding displacement values are listed in Tab. 5. All three cases retain the same overall structural concept. The optimized layouts consist of diagonal members connecting the left web to the arm tips, a horizontal strut spanning the notch, and a continuous outer boundary. However, increasing the evolutionary rate gradually reduces the complexity of the internal bracing system. The baseline case ( ER = 0.01, Fig. 15(a)) produces the most detailed topology. Numerous diagonal members are formed inside the upper and lower arms, creating a dense load-transfer network. This indicates that the slower material update process allows the optimization to develop a more refined structural layout and utilize the available material more efficiently. For ER = 0.02 (Fig. 15(b)), the main topology remains unchanged, but the number of internal diagonal members decreases. The structure becomes less detailed, and the primary load-carrying members become more dominant. The horizontal bridging strut across the notch is still clearly preserved. The most aggressive case ( ER = 0.03, Fig. 15(c)) results in the simplest layout. Only a limited number of diagonal members remain, while the structure relies more heavily on the outer boundary members. The region around the notch also becomes less defined. This behavior suggests that larger evolutionary rates reduce the ability of the optimization process to develop finer structural features, leading to a simpler topology dominated by a smaller number of major load-carrying members.

Figure 15: Effect of evolutionary rate on the optimized topology of the U-shaped plate: (a) baseline case ( ER = 0.01); (b) ( ER = 0.02); (c) ( ER = 0.03). The displacement results clearly reflect the effect of the evolutionary rate on structural stiffness. The baseline case ( ER = 0.01) produces a displacement of 0.0375 mm. Increasing the rate to 0.02 raises the displacement to 0.0410 mm, while ER = 0.03 gives the highest value of 0.042 mm. These correspond to increases of 9.3% and 12.0%, respectively, compared with the baseline case. The continuous increase in displacement indicates that higher evolutionary rates reduce the effectiveness of the internal bracing system and lead to lower structural stiffness. This is the largest displacement variation observed among all parametric studies of the U-shaped plate, highlighting the strong influence of the evolutionary rate on both topology and performance. Similar trends were also observed for the L-shaped beam and pinned plate examples. The results suggest that lower evolutionary rates are generally preferable in geometrically nonlinear BESO optimization, as they allow the algorithm to develop more stable and efficient load-transfer paths.

Evolutionary Rate (ER)

Type

U2 (mm)

Variation from Baseline (%)

0.01 0.02 0.03

Baseline

0.0375 0.0410

-

Intermediate Aggressive

+9.3% +12.0%

0.042

Table 5: Effect of evolutionary rate on displacement at the load node.

From a practical engineering perspective, however, the lower evolutionary rate should not be regarded as universally preferable solely because it produces a stiffer optimized configuration. The more detailed topology obtained at lower evolutionary rates contains a larger number of slender internal members, whereas increasing the evolutionary rate produces a simpler layout with fewer secondary struts. Such simplification may be advantageous in terms of manufacturing complexity

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