Issue 78

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

geometric features and distribute material more intricately across the domain. The result is a fully developed X-frame in which all four diagonal struts are well-defined and robustly connected, with a distinctive triangular void opening emerging at the top-center of the domain between the two upper struts. This topological feature is mechanically meaningful: the triangular void indicates that the material connecting the upper struts has been identified as structurally inefficient by the optimization, and its removal concentrates stiffness into the strut boundaries while reducing unnecessary material in the low-stress interior. The overall layout at r min = 30 mm engages all four corner supports through dedicated and clearly resolved compression paths, producing the most complete and efficient load-transfer network among all filter radius cases. Increasing the filter radius to r min = 90 mm (Fig. 11(c)) has the opposite effect. The sensitivity field is regularized over a substantially wider neighborhood, suppressing fine-scale geometric features and enforcing a coarser minimum member size across the domain. The result is a heavily consolidated topology in which the secondary bracing and upper diagonal struts are largely eliminated, reducing the optimized layout to a dominant inverted-V arch connecting the midspan load node to the two bottom corner supports through thick, smooth, and continuous members. The upper corner supports are effectively disengaged from the primary load path, and the structural solution converges toward the same topological simplification observed under aggressive evolutionary rates, though through a fundamentally different mechanism. Whereas aggressive ER suppresses secondary members through sensitivity oscillations, large r min suppresses them through explicit geometric coarsening that prohibits the formation of slender features below the enforced length scale.

Figure 11: Effect of filter radius on the optimized topology of the pinned plate: (a) small filter radius ( r min = 30 mm); (b) baseline case ( r min = 60 mm); (c) large filter radius ( r min = 90 mm). The vertical displacement results clearly show the effect of the filter radius on structural stiffness. For r min =30 mm, the displacement is 0.049 mm, which is 3.9% lower than the baseline case. This improvement is due to the more complete X shaped load path that connects all four supports and spreads the load more effectively. The baseline case ( r min = 60 mm ) gives a displacement of 0.051 mm, while increasing the filter radius to 90 mm raises the displacement to 0.055 mm, corresponding to a 7.8% increase. This is the largest displacement change observed among all parametric studies of the pinned plate. The results indicate that the filter radius has a strong influence on stiffness because it determines how efficiently the support system is utilized. Smaller filter radii create more detailed load-transfer paths and improve structural performance, whereas larger radii produce simpler layouts but reduce stiffness.

Variation from Baseline (%)

Filter Radius r min (mm)

U 2 (mm)

Type

30 60 90

Small

0.049 0.051 0.055

− 3.9%

Baseline

—

Large

+7.8%

Table 3: Effect of r min on vertical displacement at the load node.

U-Shaped Plate The third example is a U-shaped plate, which is the most geometrically complex case considered in this study. The design domain consists of a 160 mm × 120 mm rectangular plate with an 80 mm × 20 mm rectangular cutout at the center of the right edge, forming the U-shaped geometry shown in Fig. 12. The structure includes two horizontal arms connected by a solid web on the left side. Pin supports are applied along the entire left edge. Two concentrated loads of 950 N are applied

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