Issue 78

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

This paper studies how mesh size, evolutionary rate and filter radius each affect the outcome of topology optimization when geometric nonlinearity is present. The work is built around a large-displacement finite element formulation and three benchmark structures are used to cover a range of different structural behaviors. The first is an L-shaped beam which involves a re-entrant corner and bending-dominated response. The second is a rectangular plate pinned at all four corners and loaded at the midspan of the top edge which produces a biaxial stress field. The third is a U-shaped plate loaded at the arm tips in opposing directions which creates a more complex multi-path load transfer problem. Each parameter is changed one at a time while the rest stay fixed. This makes it possible to see clearly what each parameter does on its own without interference from the others. Nodal displacement at the load point is tracked alongside the topology at each stage to give a measurable indication of how stiff the optimized structure actually is. The results are intended to give researchers and practitioners a clearer basis for choosing these parameters when working with nonlinear BESO. More broadly the study makes the case that parameter values borrowed from linear BESO practice should be reconsidered before being used in problems where large displacements play a significant role. It should be emphasized that the geometrically nonlinear finite element and BESO formulations adopted herein are established methodologies and are not presented as new developments. Likewise, some of the geometries considered in the numerical investigation have previously been employed as benchmark problems. The contribution of the present study instead lies in the systematic isolation and quantitative assessment of mesh size, evolutionary rate, and filter radius under geometrically nonlinear conditions. Unlike previous studies in which these parameters primarily served as prescribed numerical settings within reliability-based or elasto-plastic optimization frameworks, the present work treats them as the principal variables of investigation and evaluates their effects on topology evolution, structural response, convergence behavior, and computational performance across three structurally distinct benchmark problems. Geometrically nonlinear finite element formulation he structural response under large-displacement conditions is governed by a geometrically nonlinear formulation that adopts the Green-Lagrange strain tensor as the kinematic measure. Unlike the infinitesimal strain tensor used in linear analyses, the Green-Lagrange strain captures both linear and nonlinear contributions of the displacement gradients, thereby accounting for the geometric stiffening and internal force redistribution that accompany large rotations and deformations:   , , , , ij i j j i k i k j u u u u     (1) j x . The first two terms on the right-hand side constitute the classical linear strain contribution, while the third term represents the nonlinear geometric coupling that becomes significant under large deformations. Within the finite element discretization, the incremental form of the strain-displacement relationship is expressed through a nonlinear strain-displacement matrix   B U which depends on the current deformation state:   d d   B U U (2) This state-dependent character of is the fundamental source of geometric nonlinearity in the structural response and distinguishes the present formulation from geometrically linear analysis, where remains constant throughout the analysis. The nonlinearity considered in the present study is exclusively geometric, arising from the nonlinear kinematic terms of the Green–Lagrange strain tensor and the deformation-dependent strain–displacement relationship. No material nonlinearity is considered. The material is assumed to be homogeneous, isotropic, and elastic in the reference configuration, with the constitutive response defined using the second Piola–Kirchhoff stress tensor and its work-conjugate Green–Lagrange strain measure.   0 p e ijkl kl C    S (3) T T HEORETICAL FRAMEWORK 1 2 where , i j u denotes the partial derivative of the displacement component i u with respect to coordinate

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