Issue 78

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

L-Shaped Beam The first structure tested is the L-shaped beam which is a well known benchmark problem in topology optimization research [13]. It is commonly used as a challenging test case because the re-entrant corner introduces geometric complexity that makes the optimization process more demanding. The design domain starts as a 400 mm by 400 mm square plate and a 250 mm by 250 mm square region is cut from one corner to create the L-shape as shown in Fig. 2. The problem is treated as a two-dimensional continuum and the analysis is carried out under quasi-static conditions. The boundary conditions and applied load are set up as illustrated in Fig. 2. The loading was chosen specifically to produce significant large-displacement effects so that geometric nonlinearity plays an active role in both how the structure reaches equilibrium and how the element sensitivity numbers evolve during the optimization.

Figure 2: L-shaped beam benchmark: dimensional layout.

The domain is meshed using four-node S4 shell finite elements. A concentrated load of 12 kN is applied at the designated load point. A homogeneous, isotropic, elastic material representative of aluminum-type materials is considered, with a Young’s modulus of 70 GPa and a Poisson’s ratio of 0.3 and a uniform thickness of 10 mm is used throughout. The BESO algorithm runs with a maximum addition ratio of 1% and a convergence tolerance of 1%. A baseline case is defined first to serve as the reference point for all comparisons. This uses a mesh element size of h = 3 mm, an evolutionary rate of ER = 0.02 and a filter radius of r min = 60 mm. From this starting point each parameter is varied one at a time while the other two stay at their baseline values. For the mesh study element sizes of h = 6 mm, 3 mm and 2 mm are tested with ER and r min kept fixed. For the evolutionary rate study values of ER = 0.01, 0.02 and 0.04 are used while h and r min remain unchanged. For the filter radius study values of r min = 18 mm, 21 mm and 24 mm are tested while h and ER stay at their baseline settings. L-Shaped Beam: baseline response The baseline result is presented first as a reference for all subsequent comparisons. Fig. 3 shows the final topology and the convergence histories of mean compliance and volume fraction. The optimized layout forms a clear load-transfer network with a dominant curved strut near the re-entrant corner redirecting forces from the vertical arm into the horizontal arm and a continuous vertical web along the left boundary carrying the main load. The convergence histories behave as expected. The volume fraction drops steadily and levels off once the target is reached. The mean compliance rises as material is removed and then stabilizes indicating that the algorithm has converged. The smooth and consistent behavior of both curves confirms that the implementation is numerically stable under the baseline settings and the result agrees well with the reference solution in [13]. L-Shaped Beam: effect of mesh size Mesh sensitivity is studied by testing element sizes of h = 6 mm, 3 mm and 2 mm while keeping ER and r min fixed. The resulting topologies are shown in Fig. 4. In all three cases the same primary load-carrying mechanism appears: a vertical web along the left arm and a curved corner strut channeling forces into the horizontal arm. This consistency confirms that the main structural response is driven by the geometry and boundary conditions rather than the mesh. Differences show up mainly in the secondary bracing inside the lower arm. The baseline and fine mesh cases both produce four triangular cells formed by slender diagonal struts. The fine mesh result additionally shows sharper member boundaries and slightly more slender diagonals. The coarse mesh simplifies this to three triangular subdivisions with thicker members that make up for the reduced resolution. Overall mesh refinement does not change the primary topology but it does improve how well the secondary load paths are captured.

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