Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework

Measuring kinetic coefficients by molecular dynamics simulation of zone melting

Molecular dynamics simulations are performed to measure the kinetic coefficient at the solid-liquid interface in pure gold. Results are obtained for the (111), (100) and (110) orientations. Both Au(100) and Au(110) are in reasonable agreement with the law proposed for collision-limited growth. For Au(111), stacking fault domains form, as first reported by Burke, Broughton and Gilmer [J. Chem. Phys. {\bf 89}, 1030 (1988)]. The consequence on the kinetics of this interface is dramatic: the measured kinetic coefficient is three times smaller than that predicted by collision-limited growth. Finally, crystallization and melting are found to be always asymmetrical but here again the effect is much more pronounced for the (111) orientation.Comment: 8 pages, 9 figures (for fig. 8 : [email protected]). Accepted for publication in Phys. Rev.

A parallel algorithm for Delaunay triangulation of moving points on the plane

Delaunay Triangulation(DT) is one of the important geometric problems that is used in various branches of knowledge such as computer vision, terrain modeling, spatial clustering and networking. Kinetic data structures have become very important in computational geometry for dealing with moving objects. However, when dealing with moving points, maintaining a dynamically changing Delaunay triangulation can be challenging. So, In this case, we have to update triangulation repeatedly. If points move so far, it is better to rebuild the triangulation. One approach to handle moving points is to use an incremental algorithm. For the case that points move slowly, we can give a faster algorithm than rebuilding it. Furthermore, sequential algorithms can be computationally expensive for large datasets. So, one way to compute as fast as possible is parallelism. In this paper, we propose a parallel algorithm for moving points. we propose an algorithm that divides datasets into equal partitions and give every partition to one block. Each block satisfay the Delaunay constraints after each time step and uses delete and insert algorithms to do this. We show this algorithm works faster than serial algorithms

Topological defect motifs in two-dimensional Coulomb clusters

The most energetically favourable arrangement of low-density electrons in an infinite two-dimensional plane is the ordered triangular Wigner lattice. However, in most instances of contemporary interest one deals instead with finite clusters of strongly interacting particles localized in potential traps, for example, in complex plasmas. In the current contribution we study distribution of topological defects in two-dimensional Coulomb clusters with parabolic lateral confinement. The minima hopping algorithm based on molecular dynamics is used to efficiently locate the ground- and low-energy metastable states, and their structure is analyzed by means of the Delaunay triangulation. The size, structure and distribution of geometry-induced lattice imperfections strongly depends on the system size and the energetic state. Besides isolated disclinations and dislocations, classification of defect motifs includes defect compounds --- grain boundaries, rosette defects, vacancies and interstitial particles. Proliferation of defects in metastable configurations destroys the orientational order of the Wigner lattice.Comment: 14 pages, 8 figures. This is an author-created, un-copyedited version of an article accepted for publication in J. Phys.: Condens. Matter. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version is available online at 10.1088/0953-8984/23/38/38530