PSI - Issue 64

Nicola Nisticò et al. / Procedia Structural Integrity 64 (2024) 2238–2245 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Figure 1. Multi-scale approach and definition of length scales. Object representations derived from digital elevation ( Drăgut and Eisank, 2011)

Geometrical model can be generated using strategies based on Voronoi (1908) and Delaunay (1934) methods. Delaunay strategy is focused on discretizing a plane region into triangles (Fig. 2a) while maximizing the smallest angle of these triangles and avoiding the creation of very narrow regions. Each triangle is inscribed in a circumcircle, and none of the circumcircles contain any of the n points. Voronoi strategy aims to discretize the plane into n regions, ensuring that each region contains one seed and the points within each region are closer to the region's seed than to any other seeds (Fig. 2b). On the other hand, the vertices of the n generated triangles correspond to the seeds of the n Voronoi regions, so that the Voronoi tessellation is considered dual to the Delaunay triangulation. In a 3D space, Delaunay regions are composed of tetrahedrons inscribed in spheres, with their vertices serving as the seeds of the Voronoi 3D regions. Examples of lattice elements together with their cross sections are reported in Figure 2c,d,e. A recent overview of lattice models (Zhu et al., 2022), further explains the adopted Delaunay procedure that involves: randomly placing spherical particles within a given volume; using the centers of the particles to generate Delaunay tetrahedra; performing a dual (Voronoi modified) tessellation on the tetrahedra to define facets representing potential failure planes. For each tetrahedron, triangular facets are defined, and the facets associated with each particle form a polyhedral cell system: further example of lattice cells are reported in Figure 3. Both the Delaunay and Voronoi methods: 1) can be combined, in an irregular lattice model used for crack propagation prediction (Bolander and Sukumar, 2005); 2) can be respectively adopted, as demonstrated by Bolander and Saito (1998), for a rigid-body spring model and a beam-spring model. Extended 3D models have been proposed by Fascetti et al. (2016b) to model FRP beams (Figure 4).

Figure 2. 2D and 3D example of lattice models (a) 2D set point generation and Delaunay triangulation (dashed lines) (b) 2D Voronoi tessellation (gray solid lines) (c) lattice element (red line) (d) 2D model: lattice element and relative cross section (solid gray lines) (e) 3D model: example of lattice element (red) and cross section (gray surface) (Gaetani et al., 2019)

Figure 3. Lattice model: (a) adjacent cells (b) lattice struts and contact (c) Lattice cells sharing a projected contact area composed of triangular facets (Fascetti et al., 2018)

Figure 4. FRP beam model. (a) nodes; (b) connections; (c) Delaunay Tetrahedralization. (Fascetti et al., 2016b)