Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright (C) 2009 John Wiley & Sons, Ltd.

Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities

Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Voronoi diagrams—a survey of a fundamental geometric data structure

Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

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Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose, this article introduces the formal notion of the family of a-shapes of a finite point set in R3. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter a e R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time 0(n2), worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing are mentioned.

Three-dimensional alpha shapes

Hydrodynamic cosmological simulations at present usually employ either the Lagrangian smoothed particle hydrodynamics (SPH) technique or Eulerian hydrodynamics on a Cartesian mesh with (optional) adaptive mesh refinement (AMR). Both of these methods have disadvantages that negatively impact their accuracy in certain situations, for example the suppression of fluid instabilities in the case of SPH, and the lack of Galilean invariance and the presence of overmixing in the case of AMR. We here propose a novel scheme which largely eliminates these weaknesses. It is based on a moving unstructured mesh defined by the Voronoi tessellation of a set of discrete points. The mesh is used to solve the hyperbolic conservation laws of ideal hydrodynamics with a finite-volume approach, based on a second-order unsplit Godunov scheme with an exact Riemann solver. The mesh-generating points can in principle be moved arbitrarily. If they are chosen to be stationary, the scheme is equivalent to an ordinary Eulerian method with second-order accuracy. If they instead move with the velocity of the local flow, one obtains a Lagrangian formulation of continuum hydrodynamics that does not suffer from the mesh distortion limitations inherent in other mesh-based Lagrangian schemes. In this mode, our new method is fully Galilean invariant, unlike ordinary Eulerian codes, a property that is of significant importance for cosmological simulations where highly supersonic bulk flows are common. In addition, the new scheme can adjust its spatial resolution automatically and continuously, and hence inherits the principal advantage of SPH for simulations of cosmological structure growth. The high accuracy of Eulerian methods in the treatment of shocks is also retained, while the treatment of contact discontinuities improves. We discuss how this approach is implemented in our new code arepo, both in 2D and in 3D, and is parallelized for distributed memory computers. We also discuss techniques for adaptive refinement or de-refinement of the unstructured mesh. We introduce an individual time-step approach for finite-volume hydrodynamics, and present a high-accuracy treatment of self-gravity for the gas that allows the new method to be seamlessly combined with a high-resolution treatment of collisionless dark matter. We use a suite of test problems to examine the performance of the new code and argue that the hydrodynamic moving-mesh scheme proposed here provides an attractive and competitive alternative to current SPH and Eulerian techniques.

E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh

An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its ź(n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn≤216, the range of the experiments. It is conjectured that the average number of edges it creates--a good measure of its efficiency--is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theLp metric for 1

A faster divide-and-conquer algorithm for constructing delaunay triangulations

A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram ofn independent and identically distributed points. The method is applied to derive exact asymptotic bounds on the expected number of vertices of the Voronoi diagram of points chosen from the uniform distribution on the interior of ad-dimensional ball; it is shown that in this case, the complexity of the diagram is ?(n) for fixedd. An algorithm for constructing the Voronoid diagram is presented and analyzed. The algorithm is shown to require only ?(n) time on average for random points from ad-ball assuming a real-RAM model of computation with a constant-time floor function. This algorithm is asymptotically faster than any previously known and optimal in the average-case sense.

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Higher-dimensional voronoi diagrams in linear expected time

This work is the first to validate theoretically the suspicions of many researchers — that the “average” Voronoi diagram is combinatorially quite simple and can be constructed quickly. Specifically, assuming that dimension d is fixed, and that n input points are chosen independently from the uniform distribution on the unit d-ball, it is proved that the expected number of simplices of the dual of the Voronoi diagram is T(n) (exact constants are derived for the high-order term), anda relatively simple algorithm exists for constructing the Voronoi diagram in T(n) time.It is likely that the methods developed in the analysis will be applicable to other related quantities and other probability distributions.

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Higher-dimensional Voronoi diagrams in linear expected time

The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(logdn) for any polytope, the expected number of vertices is Q(logd-' n) for any simple polytope, and the expected number of facets is O(logd' n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time. EXTREME POINTS; GEOMETRIC PROBABILITY; AVERAGE-CASE ANALYSIS OF ALGORITHMS

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On the convex hull of random points in a polytope

The vascular tissue of higher plants consists of specialized cells that differ from all other cells with respect to their shape and size, their organellar composition, their extracellular matrix, the type of their plasmodesmata, and their physiological functions. Intact and pure vascular tissue can be isolated easily and rapidly from leaf blades of common plantain (Plantago major), a plant that has been used repeatedly for molecular studies of phloem transport. Here, we present a transcriptome analysis based on 5,900 expressed sequence tags (ESTs) and 3,247 independent mRNAs from the Plantago vasculature. The vascular specificity of these ESTs was confirmed by the identification of well-known phloem or xylem marker genes. Moreover, reverse transcription-polymerase chain reaction, macroarray, and northern analyses revealed genes and metabolic pathways that had previously not been described to be vascular specific. Moreover, common plantain transformation was established and used to confirm the vascular specificity of a Plantago promoter-β-glucuronidase construct in transgenic Plantago plants. Eventually, the applicability and usefulness of the obtained data were also demonstrated for other plant species. Reporter gene constructs generated with promoters from Arabidopsis (Arabidopsis thaliana) homologs of newly identified Plantago vascular ESTs revealed vascular specificity of these genes in Arabidopsis as well. The presented vascular ESTs and the newly developed transformation system represent an important tool for future studies of functional genomics in the common plantain vasculature.

Rex A. Dwyer

Papers

A faster divide-and-conquer algorithm for constructing delaunay triangulations

Higher-dimensional voronoi diagrams in linear expected time

Higher-dimensional Voronoi diagrams in linear expected time

On the convex hull of random points in a polytope

Common Plantain. A Collection of Expressed Sequence Tags from Vascular Tissue and a Simple and Efficient Transformation Method