Issue 78

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

Structural Optimization method known as BESO has received significant attention from researchers. Part of its appeal comes from the straightforward logic behind it and the fact that it produces clear solid-void designs that are relatively easy to interpret and translate into real structures [3,8]. BESO works by looking at how much each element contributes to the overall structural stiffness. Elements that contribute little are removed and elements in regions that need more material are added back in. This process repeats each iteration based on sensitivity numbers calculated from the compliance objective [1,21]. One practical advantage of this discrete approach is how it handles nonlinear problems. Density based methods often run into trouble under large displacements because intermediate element densities can cause the tangent stiffness matrix to become indefinite or even negative definite which then requires extra regularization just to keep the solution stable [9]. BESO avoids this issue naturally since elements are either fully solid or fully void at all times. That said numerical problems like checkerboard patterns and mesh dependency can still appear in the sensitivity field. To deal with this sensitivity filtering is commonly applied which introduces two additional parameters into the process. These are the filter radius which controls the spatial smoothing range and the evolutionary rate which determines how fast material is added or removed between iterations [16,26]. Most topology optimization work has been done assuming linear elastic material behavior and small displacements. This is a reasonable starting point for many problems but it breaks down when structures experience large displacements or rotations. When geometric nonlinearity comes into play both the equilibrium response and the sensitivity distribution change compared to what a linear analysis would give and the resulting optimized topology can look quite different [7,12]. A number of studies have shown that ignoring these effects can lead to designs that are misleading or even unsafe especially for slender or flexible structures where bending dominates the response. The foundational work in applying BESO to this class of problems was done by Huang and Xie [10]. Their work covered both force loading and displacement loading cases and showed that the method is capable of producing stable converged solutions that do not depend on mesh size as long as appropriate sensitivity filtering and history averaging are used. Subsequent work by Zhu et al. [24] extended these ideas to broader classes of nonlinear structural problems, while Habashneh and Rad [4] proposed an enhanced BESO formulation incorporating elasto-plastic material behavior alongside geometric nonlinearity. More recently, reliability-based geometrically nonlinear BESO formulations have been developed for elasto-plastic materials using total Lagrangian finite element formulations [5]. An improved BESO method introducing an adaptive evolutionary rate strategy for nonlinear structures has also been proposed to accelerate optimization convergence while accounting for both material and geometric nonlinearity simultaneously [19]. The stress-constrained extension of BESO to geometrically nonlinear continuum structures has further demonstrated that the discrete formulation has strong potential to overcome numerical instabilities that arise in density-based methods when large displacements cause tangent stiffness matrices to become indefinite [20]. Furthermore, topology optimization frameworks for the nonlinear dynamic response of three-dimensional beam structures based on BESO have been developed, extending the applicability of the method to time-dependent nonlinear problems [23]. Despite these advances, the inclusion of geometric nonlinearity substantially amplifies the sensitivity of optimization results to numerical discretization and algorithmic settings. Parameters that are routinely treated as secondary choices in linear analyses, namely finite element mesh size, evolutionary rate, and filter radius, can strongly influence convergence behavior, numerical stability, and the physical consistency of the resulting topology in nonlinear problems [22,25]. When the mesh is too coarse secondary load paths can disappear from the solution entirely and localized stress redistribution becomes harder to capture. If the evolutionary rate is set too high the sensitivity field starts to fluctuate in ways that are difficult to control especially when large displacements are involved. The filter radius brings its own challenges too. Set it too small and mesh dependency remains a problem. Set it too large and the optimization starts smoothing out structural features that actually carry physical meaning. A dedicated study on filter radius selection confirmed that getting this parameter right is not trivial since it shapes both the numerical behavior and the level of detail in the final topology [15]. Kamocsai et al. [11] looked at this kind of parameter sensitivity in the context of steel I-section beams optimized under multiple loading conditions using BESO and found that the web topology was quite responsive to changes in the algorithmic settings. More recently the use of coupled ABAQUS-MATLAB setups for running BESO has become more common and this has made parameter selection even more consequential since the way the finite element solver and the evolutionary algorithm interact has a direct impact on how reliable and reproducible the results are [18]. Despite how commonly these parameters appear in published nonlinear BESO work they are rarely studied in a deliberate or isolated way. Most studies either pick values without much justification or tune them to suit a specific example without explaining how those choices would transfer to a different problem. This makes it difficult for other researchers to reproduce results or apply the same methodology to new structures with any confidence. The present study is a direct attempt to fill this gap by treating mesh size evolutionary rate and filter radius as the primary subjects of investigation rather than background settings.

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