Issue 78

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

pinned plate exhibits almost identical optimized layouts under the two formulations, with both solutions developing the same dominant X-shaped load-transfer mechanism. This topological agreement is consistent with the negligible difference in the corresponding displacement values. In contrast, the U-shaped benchmark exhibits a considerably stronger influence of geometric nonlinearity on the optimized material distribution. Although both solutions retain the main horizontal load-transfer paths and diagonal connections around the opening, the geometrically nonlinear formulation develops a substantially richer internal bracing network, with several additional diagonal members and smaller void regions in both the upper and lower arms. The linear solution is comparatively simpler and is dominated by a smaller number of major load-carrying members. This difference, together with the approximately 11.9% variation in displacement, demonstrates that geometric nonlinearity can significantly influence the topology evolution when the structural geometry and deformation pattern are more complex. Overall, the comparison shows that the influence of geometric nonlinearity is problem-dependent. For the pinned plate, and to a lesser extent the L-shaped benchmark, the principal topology remains largely unchanged, whereas the U-shaped benchmark shows a clear difference in both the detailed material distribution and structural response. Therefore, the additional computational effort associated with geometrically nonlinear BESO becomes particularly relevant for problems in which large-deformation effects substantially modify the evolving load-transfer paths.

Final displacement ଶ (mm) 4.590

Total CPU time (min)

Problem

Analysis

Final compliance

BESO iterations

Linear

276.9

64 55 61 61 99

13 11 10 15 18

L-shaped

Geometrically nonlinear

5.650

275

Linear

0.0504

3.8

Pinned plate

Geometrically nonlinear

0.0510 0.0425

3.7

Linear

0.380 0.388

U-shaped plate

Geometrically nonlinear

0.037 32 Table 7: Quantitative comparison of the linear and geometrically nonlinear baseline BESO solutions. 186

Figure 17: Comparison of the final optimized layouts obtained using geometrically nonlinear and linear BESO formulations for the (a) L-shaped, (b) pinned plate, and (c) U-shaped plate benchmarks. Black and red regions represent solid and void elements, respectively.

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