Issue 78

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

at the right free-end nodes of the upper and lower arms, as indicated in Fig. 12. The upper load acts vertically upward, whereas the lower load acts vertically downward. This loading arrangement creates a splitting effect that activates both the bending behavior of the arms and the resistance of the connecting web. It also generates high stress concentrations around the inner corners of the notch, making the optimization results sensitive to the selected numerical parameters. The model is discretized using four-node S4 shell elements. The material properties are a Young's modulus of 210 GPa and a Poisson's ratio of 0.3. A constant thickness is assumed throughout the domain. The target volume fraction is set to 0.5. The baseline case uses a mesh size of 2 mm, an evolutionary rate of 0.01, and a filter radius of 12 mm. As in the previous examples, a one-parameter-at-a-time study is performed. Mesh sensitivity is investigated using element sizes of 2, 3, and 4 mm. The influence of the evolutionary rate is examined for values of 0.01, 0.02, and 0.03. The effect of the filter radius is evaluated using values of 8, 10, and 12 mm.

Figure 12: U-shaped plate design domain.

U-Shaped Plate: baseline response The baseline optimization result for the U-shaped plate is shown in Fig. 13. The final optimized topology is presented in Fig. 13(a), while Fig. 13(b) shows the evolution of mean compliance and volume fraction during the optimization process over 186 iterations. The optimized design exhibits a complex load-carrying system that reflects both the geometry of the structure and the opposing loading condition. Several diagonal members extend from the left web toward the free ends of the upper and lower arms, creating an efficient internal bracing network. These members transfer the applied loads to the supported boundary through direct load paths. A horizontal member is also retained across the notch opening, connecting the upper and lower arms. This member plays an important role in maintaining the structural integrity of the system and preventing excessive separation between the two arms. Material in low-contribution regions is removed, resulting in a lightweight topology while preserving the main load-transfer paths. The final layout resembles a truss-like structure with a highly efficient force-flow mechanism. The convergence history is more complex than that observed for the L-shaped beam and pinned plate examples. The volume fraction decreases steadily from 1.0 and reaches the target value of 0.5 at around iteration 80. The mean compliance gradually increases as material is removed. A noticeable peak appears near iteration 70, which is associated with material removal around the notch corners where stress concentrations are highest. After this temporary increase, the structure adapts and develops alternative load paths through the surrounding diagonal members. The compliance then stabilizes at approximately 0.386 J, indicating convergence. The optimization requires 186 iterations, considerably more than the pinned plate case, which converged in 61 iterations. This longer evolution process reflects the greater geometric and topological The influence of mesh discretization is examined using element sizes of 2 mm, 3 mm, and 4 mm, while keeping the evolutionary rate and filter radius fixed at 0.01 and 12 mm, respectively. The optimized topologies are shown in Fig. 14, and the corresponding displacement values are listed in Tab. 4. All three mesh sizes produce the same overall structural concept. The optimized layouts consist of diagonal members extending from the left web toward the arm tips, together with a horizontal strut spanning the notch opening. This consistency indicates that the main load-transfer mechanism is controlled by the geometry and loading conditions rather than the mesh density. The horizontal bridging member appears in all cases, highlighting its importance in maintaining stiffness and load continuity. The main differences are observed in the level of detail of the secondary members. The fine mesh case (h=2 mm, Fig. 14(a)) generates the most detailed topology, with several complexity of the U-shaped plate. U-Shaped Plate: effect of mesh size

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