Issue 78

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

At each iteration, elements are ranked in ascending order of their filtered sensitivity numbers. An element is removed from the current solid domain if its sensitivity falls below the deletion threshold:

th i del   

(18)

Conversely, a void element is reinstated as solid if its sensitivity exceeds the addition threshold:

th i add   

(19)

The two thresholds th th add  are determined at each iteration from the ranked sensitivity list to satisfy the target volume simultaneously for both removal and addition operations, ensuring that the prescribed volume constraint is met at every step of the evolutionary process. Computational framework The full computational procedure is shown in Fig. 1. The implementation runs through a coupled MATLAB and ABAQUS setup where each software handles a different part of the process. MATLAB takes care of the optimization logic. At the start of each iteration the BESO parameters are read in and the element sensitivities are computed and filtered. The target volume for the next iteration is then calculated and the design variables are updated to give a new material distribution. ABAQUS handles the structural side of things. It builds the design domain with the appropriate mesh and boundary conditions and then runs a geometrically nonlinear finite element analysis to get the displacement field and internal forces for the current design. The sensitivity data from this analysis is sent back to MATLAB to drive the next update. It should be noted that the nonlinear equilibrium convergence and the BESO optimization convergence represent two distinct criteria. At each BESO iteration, the nonlinear finite element analysis is first iterated using the Newton–Raphson method until the residual between the external and internal force vectors satisfies the prescribed equilibrium tolerance. The resulting converged displacement field is then used to evaluate the element sensitivities. At the optimization level, the material distribution evolves toward the prescribed target volume fraction; once this target is reached, the volume is held constant and the BESO iterations continue until the optimization convergence criterion is satisfied. N UMERICAL EXAMPLES hree benchmark structures are used in this study to examine how discretization and algorithmic parameters affect the outcome of geometrically nonlinear BESO optimization. The three models are an L-shaped beam, a pinned rectangular plate and a U-shaped plate. The terms ‘beam’ and ‘plate’ are used here to identify the conventional benchmark geometries; all three examples are modeled as two-dimensional continuum domains subjected to in-plane loading. They were chosen because each one presents a structurally different situation. The L-shaped beam involves a re entrant corner and a response that is largely driven by bending. The pinned plate develops a biaxial stress field with competing tension and compression. The U-shaped plate introduces two notch corners and a more complex load transfer pattern where forces must find multiple paths through the domain. Taken together the three geometries create a well rounded testing environment for studying how mesh size, evolutionary rate and filter radius each affect the optimized topology, the compliance convergence history and the regularity of the sensitivity field. Before the parametric study begins the nonlinear BESO implementation is validated using the L-shaped beam by comparing results against the reference solution reported in [13]. Once this is confirmed the same parametric investigation is carried out across all three models using consistent algorithmic conditions so that the results can be meaningfully compared. In the following numerical examples, ଶ denotes the displacement component in the global 2-direction (vertical direction) measured at the load application point and is used to quantify the structural response of the optimized configurations. T del  and

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