PSI - Issue 79

Marco Piacentini et al. / Procedia Structural Integrity 79 (2026) 394–403

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2. Materials and Methods

2.1. Structure Generation

To generate a set of diverse 2D periodic beam representative volume elements (RVEs) based on Voronoi tessellation, the Python libraries SciPy and Shapely were employed. By varying the number of seeds η , the number of cells within each RVE were systematically controlled. To introduce controlled anisotropy, a customized procedure was developed. After generating the initial isotropic Voronoi tessellation, an a ffi ne compression C is applied to impose a prescribed anisotropic aspect ratio α , followed by a rotation R to impart a global orientation defined by the rotation angle θ . The corresponding deformation tensors are defined as R = cos θ − sin θ sin θ cos θ , C = 1 /α 0 0 1 . (1) As commonly done in the literature to ensure tessellation periodicity Ko¨ll and Hallstro¨m (2014), the Voronoi diagram was generated over a 3 × 3 periodically repeated tiling of the seed points (Fig. 1). After tessellation, only the central tile was retained as the RVE (Fig. 2). In the present method—to obtain a square RVE after applying the prescribed deformation RC to the tessellation—a counter-deformation D = C − 1 R T was applied to the seeds prior to tessellation (Fig. 1). To ensure a Timoshenko-beam-compatible geometry, each strut length l was constrained to l ≥ 8 d by merging vertices closer than 8 d , where d is the global strut diameter. To avoid generating short struts during the trimming of the 3 × 3 lattice to the central RVE, each cell in the lattice was slightly contracted toward its centroid before trimming. These merging and shrinking steps introduced minor morphological perturbations and occasionally caused periodicity violations at high η . The diameter was fixed at d = 0 . 22 mm to approximate a realistic trabecular thickness Bregoli et al. (2024), although beam-model constraints prevented the use of physiological volume fractions ( ϕ ) and geometries. As detailed in Section 3.1, the resulting ϕ values remain below 4%, whereas physiological values exceed 15% even for osteoporotic bone Bregoli et al. (2024). Owing to the low density, the RVE side length was set to 75 mm to ensure structural representativeness. A total of 10 000 diverse tessellations were generated, uniformly sampling the design space illustrated in Fig. 3. For reproducibility, a fixed global random seed was used for all random number generation.

Fig. 1. Generated seeds tiled in a 3 × 3 periodic lattice before (black) and after (red) the counter-deformation D = C − 1 R T .