Issue 78

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

The baseline (ER = 0.02, Fig. 10(b)) gives a well-balanced symmetric X-shaped layout with four clear diagonal struts and a reasonable secondary bracing network. The conservative case (ER = 0.01, Fig. 10(a)) produces a richer secondary member system because the slower update pace gives the sensitivity field more time to settle between iterations. That said a slight asymmetry appears in the bracing pattern suggesting that very low evolutionary rates can let small numerical imbalances accumulate over time. The aggressive case (ER = 0.04, Fig. 10(c)) produces a much simpler layout. The secondary bracing almost disappears and the topology reduces to a partial inverted-V arch connecting only the top midspan node to the two bottom corners. The upper corner supports are no longer engaged which means a significant portion of the available structural capacity goes unused. This happens because rapid large-scale material changes amplify sensitivity oscillations that the algorithm cannot resolve properly under large-displacement conditions.

Figure 10: Effect of evolutionary rate on the optimized topology of the pinned plate: (a) conservative evolutionary rate ( ER = 0.01); (b) baseline case ( ER = 0.02); (c) aggressive evolutionary rate ( ER = 0.04). The displacement values support what the topologies show. The baseline gives U 2 = 0.051 mm and the conservative rate gives U 2 = 0.0512 mm which is practically the same. The aggressive rate gives U 2 = 0.054 mm which is 5.9% higher than the baseline. This increase comes directly from the topological simplification at ER = 0.04. With only two corner supports engaged instead of four the structure has fewer load-carrying members and is measurably more flexible as a result. The key finding here is that the evolutionary rate does more than just affect secondary member detail. At aggressive enough settings it can change the primary load-transfer topology entirely and noticeably reduce the structural efficiency of the final design.

Variation from Baseline (%)

U 2 (mm)

Evolutionary Rate (ER)

Type

0.01 0.02 0.04

Conservative

0.0512

+0.4%

Baseline

0.051 0.054

—

Aggressive

+5.9%

Table 2: Effect of ER on vertical displacement at the load node.

Pinned Plate: effect of filter radius The role of spatial regularization is assessed by comparing optimized topologies obtained with r min = 60 mm (baseline), r min = 30 mm, and r min = 90 mm, while the mesh element size and evolutionary rate are held constant at h = 10 mm and ER = 0.02, respectively. The resulting optimized layouts are presented in Fig. 11, and the corresponding vertical displacements at the load application node are reported in Tab. 3. Among the three parametric dimensions investigated for the pinned plate benchmark, the filter radius produces the most pronounced and structurally significant variation in both topology and global stiffness. While mesh refinement and evolutionary rate primarily affected the refinement and completeness of secondary members, the filter radius directly governs the minimum length scale enforced on the optimized material distribution and thereby exerts a controlling influence on the fundamental character of the load-carrying architecture. The baseline configuration ( r min = 60 mm, Fig. 11(b)) produces the familiar X-shaped topology with four diagonal struts and a moderate secondary bracing system, as previously described. Reducing the filter radius to r min = 30 mm (Fig. 11(a)) yields the most structurally articulated result among all cases examined for the pinned plate. With weaker spatial regularization, the sensitivity field is allowed to vary over shorter length scales, enabling the algorithm to resolve finer

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