Conference Paper

# A Novel Delaunay Simplex Technique for Detection of Crystalline Nuclei in Dense Packings of Spheres.

DOI: 10.1007/11424758_84 In proceeding of: Computational Science and Its Applications - ICCSA 2005, International Conference, Singapore, May 9-12, 2005, Proceedings, Part I

Source: DBLP

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**ABSTRACT:**The paper presents an algorithm for calculating the three-dimensional Voronoi-Delaunay tessellation for an ensemble of spheres of different radii (additively-weighted Voronoi diagram). Data structure and output of the algorithm is oriented toward the exploration of the voids between the spheres. The main geometric construct that we develop is the Voronoi S-network (the network of vertices and edges of the Voronoi regions determined in relation to the surfaces of the spheres). General scheme of the algorithm and the key points of its realization are discussed. The principle of the algorithm is that for each determined site of the network we find its neighbor sites. Thus, starting from a known site of the network, we sequentially find the whole network. The starting site of the network is easily determined based on certain considerations. Geometric properties of ensembles of spheres of different radii are discussed, the conditions of applicability and limitations of the algorithm are indicated. The algorithm is capable of working with a wide variety of physical models, which may be represented as sets of spheres, including computer models of complex molecular systems. Emphasis was placed on the issue of increasing the efficiency of algorithm to work with large models (tens of thousands of atoms). It was demonstrated that the experimental CPU time increases linearly with the number of atoms in the system, O(n).Journal of Computational Chemistry 12/2006; 27(14):1676-92. · 3.84 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this chapter we apply a computational geometry technique to investigate the structure of packings of hard spheres. The hard sphere model is the base for understanding the structure of many physical matters: liquids, solids, colloids and granular materials. The structure analysis is based on the concept of the Voronoi Diagram (Voronoi-Delaunay tessellation), which is well known in mathematics and physics. The Delaunay simplexes are used as the main instrument for this work. They define the simplest structural elements in the three-dimensional space. A challenging problem is to relate geometrical characteristics of the simplexes (e.g. their shape) with structural properties of the packing. In this chapter we review our recent results related to this problem. The presented outcome may be of interest to both mathematicians and physicists. The idea of structural analysis of atomic systems, which was first proposed in computational physics, is a subject for further mathematical development. On the other hand, physicists, chemists and material scientists, who are still using traditional methods for structure characterization, have an opportunity to learn more about this new technique and its implementation. We present the analysis of hard sphere packings with different densities. Our method permits to tackle a renowned physical problem: to reveal a geometrical principle of disordered packings. The proposed analysis of Delaunay simplexes can also be applied to structural investigation of other molecular systems.10/2008: pages 13-45; - [Show abstract] [Hide abstract]

**ABSTRACT:**The concept of Procrustes distance is applied to the shape analysis of the Delaunay simplexes. Procrustes distance provides a measure of coincidence of two point sets {x<sub>i</sub>} and {y<sub>i</sub>}, i=1..N. For this purpose the variance of point deviations is calculated at the optimal superposition of the sets. It allows to characterize the shape proximity of a given simplex to shape of a reference one, e.g. to the shape of the regular tetrahedron. This approach differs from the method used in physics, where the variations of edge lengths are calculated in order to characterize the simplex shape. We compare both methods on an example of structure analysis of dense packings of hard spheres. The method of Procrustes distance reproduces known structural results; however, it allows to distinguish more details because it deals with simplex vertices, which define the simplex uniquely, in contrast to simplex edges.3rd International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2006, Banff, Alberta, Canada, July 2-5, 2006; 01/2006

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## Marina L. Gavrilova |