Issue 78

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

Filter Radius r min (mm)

U2 (mm)

Variation from Baseline (%)

8

0.0390

+4.0% +6.7%

10 12

0.040

0.0375 - Table 6: Effect of r min on displacement at the load node.

The present parametric investigation adopts a one-parameter-at-a-time approach, with mesh size, evolutionary rate, and filter radius varied independently while the remaining parameters are held fixed. This approach enables the individual influence of each parameter to be isolated; however, it does not permit direct quantification of possible coupling effects among the parameters. Such interactions may be relevant because the effective spatial regularization introduced by the filter radius depends on the underlying mesh discretization, while the evolutionary rate controls the magnitude of material changes between successive optimization iterations. Therefore, no conclusion regarding the magnitude or significance of parameter interactions is drawn from the present results. A systematic multi-parameter investigation, for example using a factorial design, is recommended in future work to quantify possible interaction effects under geometrically nonlinear conditions Quantitative and computational assessment Although the optimized topologies provide a direct visual indication of the influence of the investigated parameters, the corresponding displacement variations provide a quantitative measure of their structural significance. For the pinned-plate benchmark, changing the mesh size resulted in displacement variations of up to 3.9%, whereas the maximum variations associated with evolutionary rate and filter radius were 5.9% and 7.8%, respectively. For the geometrically more complex U-shaped benchmark, the influence was more pronounced, with maximum variations of 9.3% for mesh size and 12.0% for evolutionary rate. These results demonstrate that the observed topological differences are not merely graphical; changes in the numerical and algorithmic parameters can lead to measurable changes in the stiffness of the optimized structures. Comparison with linear BESO solutions To provide a quantitative reference for assessing the influence of geometric nonlinearity, additional BESO optimizations were performed using a geometrically linear finite element formulation for the three benchmark problems. The geometry, material properties, boundary conditions, loading, target volume fraction, and baseline BESO parameters were kept identical to those used in the corresponding geometrically nonlinear analyses. The quantitative results are summarized in Tab. 7, while the corresponding optimized layouts are presented separately in Fig. 17. For the L-shaped benchmark, the linear and geometrically nonlinear solutions yield very similar final compliance values of 276.9 and 275, respectively. The linear optimization converges after 64 BESO iterations and requires 13 min, whereas the geometrically nonlinear solution converges after 55 iterations and requires 11 min. Thus, for this particular benchmark, introducing geometric nonlinearity does not increase the overall computational cost and produces only a small difference in the final compliance. For the pinned plate, the difference in global structural response is particularly small. The linear solution gives ଶ ൌ 0.05096 mm and a final compliance of 3.8, whereas the geometrically nonlinear solution gives ଶ ൌ 0.0510 mm and a final compliance of 3.7. The difference in displacement is less than 0.1%, indicating that geometric nonlinearity has only a minor influence on the global stiffness of this benchmark under the adopted loading conditions. Both formulations require 61 BESO iterations. However, the computational time increases from 10 min for the linear formulation to 15 min for the geometrically nonlinear formulation, corresponding to an increase of 50%. A more pronounced difference is observed for the U-shaped benchmark. The linear optimization results in a displacement of 0.04256 mm and a final compliance of 0.380, whereas the geometrically nonlinear solution gives ଶ ൌ 0.0375 mm and a final compliance of 0.388. The displacement of the nonlinear optimized design is approximately 11.9% lower than that of the linear optimized design. Furthermore, the linear formulation converges after 99 BESO iterations and requires 18 min, while the geometrically nonlinear formulation requires 186 iterations and 32 min. This represents an increase of approximately 77.8% in computational time, demonstrating the additional computational effort associated with nonlinear equilibrium calculations for this more complex benchmark. The corresponding optimized layouts are compared in Fig. 17. For the L-shaped benchmark, the linear and geometrically nonlinear formulations recover essentially the same principal load-transfer mechanism, including the vertical member along the left boundary and the diagonal members transferring the load through the re-entrant region into the horizontal arm. Only relatively small differences are observed in the detailed shape and thickness of the secondary members. Similarly, the

427