Voronoi-Delaunay tessellation

The way of the best choice to model PS s structure on both molecular and supramolecular levels begins with allocation of primary building units (PBUs), which without gaps and overlaps would fill a 3D space occupied by a PS. An universal method for allocation of such PBUs in both ordered and randomly arranged PSs, formed of packings of convex particles (or pores), is based on the construction of the assembles of Voronoi polyhedra (V-polyhedra) and Delaunay simplexes (or D-poly-hedra), which form Voronoi-Delaunay tessellation [100],... 

Figure 10.3 Illustration of Voronoi/Delaunay tessellation in 2D space (Voronoi polyhedra are represented by dashed lines, and Delaunay simplices by solid lines). For the collection of points with 3D coordinates, such as atoms of the protein-ligand complex, Delaunay simplices are tetrahedra whose vertices correspond to the atoms.

The geometric algorithm of Sastry et al. (1997a) is based on a Voronoi-Delaunay tessellation (Tanemura et al., 1983). It consists of three basic steps. 

Conventional Unstructured Grid Methods In general any grid that is not structured is an unstructured grid. Of particular importance are Voronoi tessellations and their dual the Delaunay tessellation. In three dimensions Voronoi cells are convex polyhedra and Delaunay cells are tetrahedra. In two dimensions Voronoi cells are convex polygons and Delaunay cells are triangles. 

Fig. 2 Illustration of our developing procedure of automatic mesh subdivision on the basis of Voronoi Polygon and Delaunay Tessellation a) Initial geometric model, b) Automatic node generation, c) Intermediate search for new DT s in sequence by edge control, d) Automatic triangulation and e) Transformation to quadrilateral elements.

Figure 9.27 Example of Voronoi (V, dotted line) and Delaunay (D, solid line) tessellations for a 2D case.

The geometry of V-polyhedra and V-tessilations was elaborated by Voronoi and Delaunay [143,144], They have shown that all PBU/Ps have a convex shape, each facet is common for two neighboring PBU/Ps, each edge is formed by no less than d PBU/Ps (d is the dimension of V-tessellation), and no less than d + 1 PBU/Ps intersect at each vertex. We write no less but an exact coincidence usually takes place. 

The Voronoi tessellation divides space into polyhedral regions that are closer to the center of a given particle than to any other. Joining pairs of particle centers whose Voronoi polyhedra share a face yields a dual tessellation of space into Delaunay simplices. 

Corresponding system plastic-strain increments are also obtained at the atomic level from the displacement gradients between the four relevant neighboring corner atoms of Delaunay tetrahedra for each external distortion increment and are allocated subsequently as an atomic site average to each Voronoi polyhedral atom environment by a special procedure of double space tessellation developed by Mott et al. (1992) for this purpose, leading eventually to volume averages of strain-increment tensors of all Voronoi atom environments to attain the system-wide strain-inerement tensor. 

An algorithm of Voronoi tessellation [16] and Delaunay triangulation [17] was used as basis for precise determination of all adjacent particle neighbors. 

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