**2. Analytical models of structures**

In material science Voronoi tessellations are extensively used to model grain geometry with the purpose of calculating the properties of polycrystalline aggregates (Cailletaud et al., 2003; Kovaˇc & Cizelj, 2005), modeling short crack initiation and propagation or modeling intergranular stress corrosion cracks (Kamaya & Itakura, 2009; Musienko & Cailletaud, 2009; Simonovski & Cizelj, 2011b). A Voronoi tessellation is a cell structure constructed from randomly positioned points, also referred to as Poisson points. For polycrystalline aggregates we can think of these points as points where the solidification starts and then uniformly extends in all directions. A solidification front then expands until it collides with another one, thus creating a grain boundary. In geometrical terms the grain boundary is obtained by introducing lines perpendicular to lines connecting neighboring Poisson points. The result is a set of convex polygons/polyhedra, see Fig. 1. Examples of 2D Voronoi tessellations can be seen in Fig. 2. Additional details on mathematical background and applications are available in (Aurenhammer, 1991; Okabe et al., 2000).

Three dimensional Voronoi tessellations can be created using the same approach. A number of mathematical and programming packages like Matlab have the option of constructing such tessellations. Fig. 3 shows examples of 3D Voronoi tessellations generated using Qhull algorithm (*Qhull code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection about a Point*, n.d.) and implemented in (Petriˇc, 2010).
