Issue 78

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

case. The problem is modeled as a two-dimensional continuum under quasi-static conditions and geometric nonlinearity is included in both the equilibrium solution and the sensitivity calculation.

Figure 7: Pinned plate design domain.

The domain is meshed using four-node S4 shell finite elements. The material has a Young's modulus of 210 GPa and a Poisson's ratio of 0.3 with uniform thickness throughout. The target volume fraction is set to V f = 0.4. The baseline uses h = 10 mm, ER = 0.02 and r min = 60 mm. Following the same one-at-a-time approach used for the L-shaped beam, mesh size is varied across h = 15 mm, 10 mm and 7.5 mm, evolutionary rate is tested at ER = 0.01, 0.02 and 0.04, and filter radius is evaluated at r min = 30 mm, 60 mm and 90 mm. Pinned Plate: baseline response The baseline result for the pinned plate is shown in Fig. 8. The final topology appears in Fig. 8(a) and the convergence histories are shown in Fig. 8(b). The optimized layout takes an X-shaped form with four diagonal struts running from the loaded top-edge node toward each of the four pinned corners. This makes physical sense given the symmetric support arrangement and central load which together create a biaxial force field. The most direct way to carry the load is to connect it straight to each corner support and that is exactly what the optimizer finds. The interior of the domain between the struts is mostly void confirming that this region does little useful work under the given loading. The convergence histories in Fig. 8(b) behave as expected. The volume fraction drops smoothly from 1.0 and levels off near iteration 46 when the V f = 0.4 target is reached. The mean compliance rises steadily as material is removed and then flattens out in the final iterations. The stable terminal values of both curves confirm that the algorithm has converged and that the solution is numerically sound under the baseline parameter set.

Figure 8: Baseline optimization results for the pinned plate: (a) final optimized topology (black: solid elements; red: void elements); (b) evolution histories of mean compliance and volume fraction.

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