414 research outputs found
Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations
One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology
On the computation of zone and double zone diagrams
Classical objects in computational geometry are defined by explicit
relations. Several years ago the pioneering works of T. Asano, J. Matousek and
T. Tokuyama introduced "implicit computational geometry", in which the
geometric objects are defined by implicit relations involving sets. An
important member in this family is called "a zone diagram". The implicit nature
of zone diagrams implies, as already observed in the original works, that their
computation is a challenging task. In a continuous setting this task has been
addressed (briefly) only by these authors in the Euclidean plane with point
sites. We discuss the possibility to compute zone diagrams in a wide class of
spaces and also shed new light on their computation in the original setting.
The class of spaces, which is introduced here, includes, in particular,
Euclidean spheres and finite dimensional strictly convex normed spaces. Sites
of a general form are allowed and it is shown that a generalization of the
iterative method suggested by Asano, Matousek and Tokuyama converges to a
double zone diagram, another implicit geometric object whose existence is known
in general. Occasionally a zone diagram can be obtained from this procedure.
The actual (approximate) computation of the iterations is based on a simple
algorithm which enables the approximate computation of Voronoi diagrams in a
general setting. Our analysis also yields a few byproducts of independent
interest, such as certain topological properties of Voronoi cells (e.g., that
in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI;
Ref [51] points to a freely available computer application which implements
the algorithms; to appear in Discrete & Computational Geometry (available
online
A Randomized Incremental Algorithm for the Hausdorff Voronoi Diagram of Non-crossing Clusters
In the Hausdorff Voronoi diagram of a family of \emph{clusters of points} in
the plane, the distance between a point and a cluster is measured as
the maximum distance between and any point in , and the diagram is
defined in a nearest-neighbor sense for the input clusters. In this paper we
consider %El."non-crossing" \emph{non-crossing} clusters in the plane, for
which the combinatorial complexity of the Hausdorff Voronoi diagram is linear
in the total number of points, , on the convex hulls of all clusters. We
present a randomized incremental construction, based on point location, that
computes this diagram in expected time and expected
space. Our techniques efficiently handle non-standard characteristics of
generalized Voronoi diagrams, such as sites of non-constant complexity, sites
that are not enclosed in their Voronoi regions, and empty Voronoi regions. The
diagram finds direct applications in VLSI computer-aided design.Comment: arXiv admin note: substantial text overlap with arXiv:1306.583
Kinetic Geodesic Voronoi Diagrams in a Simple Polygon
We study the geodesic Voronoi diagram of a set S of n linearly moving sites inside a static simple polygon P with m vertices. We identify all events where the structure of the Voronoi diagram changes, bound the number of such events, and then develop a kinetic data structure (KDS) that maintains the geodesic Voronoi diagram as the sites move. To this end, we first analyze how often a single bisector, defined by two sites, or a single Voronoi center, defined by three sites, can change. For both these structures we prove that the number of such changes is at most O(m³), and that this is tight in the worst case. Moreover, we develop compact, responsive, local, and efficient kinetic data structures for both structures. Our data structures use linear space and process a worst-case optimal number of events. Our bisector KDS handles each event in O(log m) time, and our Voronoi center handles each event in O(log² m) time. Both structures can be extended to efficiently support updating the movement of the sites as well. Using these data structures as building blocks we obtain a compact KDS for maintaining the full geodesic Voronoi diagram
Shortest Path Problems on a Polyhedral Surface
We develop algorithms to compute shortest path edge sequences, Voronoi diagrams, the Fréchet distance, and the diameter for a polyhedral surface
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