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.

R. K. Singh, A. Tropsha and I. I. Vaisman, Delaunay tessellation of proteins four body nearest-neighbor propensities of amino acid residues., J. Comput. Biol, 1996, 3, 213-221. 

The Delaunay tessellation is particularly important because it provides a tool to decompose the continuum void space into discrete pores, which is essential for pore-scale modeling. An important drawback to using the Delaunay tessellation for extracting pore structure is that it leads to a fixed pore coordination number of 4, which is a geometric artifact in a real packing, not all voids conform to the tessellation s tetrahedral structure. Experimentally, the average... 

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. 

W. Zheng, S. J. Cho, I. I. Vaisman, A. Tropsha, A new approach to protein fold recognition based on Delaunay tessellation of protein structure, in Pacific Symposium on Biocomputing 97,... 

While this two-scale model allows for a natural connection between the two scales (i.e., the inclusion-boundary behavior and the atomistic cell behavior are connected via the common strain transformation), it also is limited by the required uniformity of strain in the atomistic inclusion (through the periodic continuation conditions on the atomistic cell) and in each of the tetrahedra of the Delaunay tessellation. 

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.

The algorithms above add points as they deem fit in the interior of the domain using Delaunay tessellation, an algorithm that can partition a region into non overlapping triangles for any specified set of nodal positions. For example, from the nodal positions stored in P, the triangular partition can be constructed from... 

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