837 research outputs found
Robust and Efficient Delaunay Triangulations of Points on Or Close to a Sphere
International audienceWe propose two ways to compute the Delaunay triangulation of points on a sphere, or of rounded points close to a sphere, both based on the classic incremental algorithm initially designed for the plane. We use the so-called space of circles as mathematical background for this work. We present a fully robust implementation built upon existing generic algorithms provided by the Cgal library. The efficiency of the implementation is established by benchmarks
One machine, one minute, three billion tetrahedra
This paper presents a new scalable parallelization scheme to generate the 3D
Delaunay triangulation of a given set of points. Our first contribution is an
efficient serial implementation of the incremental Delaunay insertion
algorithm. A simple dedicated data structure, an efficient sorting of the
points and the optimization of the insertion algorithm have permitted to
accelerate reference implementations by a factor three. Our second contribution
is a multi-threaded version of the Delaunay kernel that is able to concurrently
insert vertices. Moore curve coordinates are used to partition the point set,
avoiding heavy synchronization overheads. Conflicts are managed by modifying
the partitions with a simple rescaling of the space-filling curve. The
performances of our implementation have been measured on three different
processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we
have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds
to a generation rate of over 55 million tetrahedra per second. We finally show
how this very efficient parallel Delaunay triangulation can be integrated in a
Delaunay refinement mesh generator which takes as input the triangulated
surface boundary of the volume to mesh
Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions
We describe the implementation of algorithms to construct and maintain
three-dimensional dynamic Delaunay triangulations with kinetic vertices using a
three-simplex data structure. The code is capable of constructing the geometric
dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points
is triangulated. Time evolution of the triangulation is not only governed by
kinetic vertices but also by a changing number of vertices. We use
three-dimensional simplex flip algorithms, a stochastic visibility walk
algorithm for point location and in addition, we propose a new simple method of
deleting vertices from an existing three-dimensional Delaunay triangulation
while maintaining the Delaunay property. The dual Dirichlet tessellation can be
used to solve differential equations on an irregular grid, to define partitions
in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly
nomenclature), referee suggestions included, typos corrected, bibliography
update
Load-Balancing for Parallel Delaunay Triangulations
Computing the Delaunay triangulation (DT) of a given point set in
is one of the fundamental operations in computational geometry.
Recently, Funke and Sanders (2017) presented a divide-and-conquer DT algorithm
that merges two partial triangulations by re-triangulating a small subset of
their vertices - the border vertices - and combining the three triangulations
efficiently via parallel hash table lookups. The input point division should
therefore yield roughly equal-sized partitions for good load-balancing and also
result in a small number of border vertices for fast merging. In this paper, we
present a novel divide-step based on partitioning the triangulation of a small
sample of the input points. In experiments on synthetic and real-world data
sets, we achieve nearly perfectly balanced partitions and small border
triangulations. This almost cuts running time in half compared to
non-data-sensitive division schemes on inputs exhibiting an exploitable
underlying structure.Comment: Short version submitted to EuroPar 201
A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations
Computing Delaunay triangulations in involves evaluating the
so-called in\_sphere predicate that determines if a point lies inside, on
or outside the sphere circumscribing points . This
predicate reduces to evaluating the sign of a multivariate polynomial of degree
in the coordinates of the points . Despite
much progress on exact geometric computing, the fact that the degree of the
polynomial increases with makes the evaluation of the sign of such a
polynomial problematic except in very low dimensions. In this paper, we propose
a new approach that is based on the witness complex, a weak form of the
Delaunay complex introduced by Carlsson and de Silva. The witness complex
is defined from two sets and in some metric space
: a finite set of points on which the complex is built, and a set of
witnesses that serves as an approximation of . A fundamental result of de
Silva states that if .
In this paper, we give conditions on that ensure that the witness complex
and the Delaunay triangulation coincide when is a finite set, and we
introduce a new perturbation scheme to compute a perturbed set close to
such that . Our perturbation
algorithm is a geometric application of the Moser-Tardos constructive proof of
the Lov\'asz local lemma. The only numerical operations we use are (squared)
distance comparisons (i.e., predicates of degree 2). The time-complexity of the
algorithm is sublinear in . Interestingly, although the algorithm does not
compute any measure of simplex quality, a lower bound on the thickness of the
output simplices can be guaranteed.Comment: 24 page
Introducing Quantum Ricci Curvature
Motivated by the search for geometric observables in nonperturbative quantum
gravity, we define a notion of coarse-grained Ricci curvature. It is based on a
particular way of extracting the local Ricci curvature of a smooth Riemannian
manifold by comparing the distance between pairs of spheres with that of their
centres. The quantum Ricci curvature is designed for use on non-smooth and
discrete metric spaces, and to satisfy the key criteria of scalability and
computability. We test the prescription on a variety of regular and random
piecewise flat spaces, mostly in two dimensions. This enables us to quantify
its behaviour for short lattices distances and compare its large-scale
behaviour with that of constantly curved model spaces. On the triangulated
spaces considered, the quantum Ricci curvature has good averaging properties
and reproduces classical characteristics on scales large compared to the
discretization scale.Comment: 43 pages, 27 figure
JIGSAW-GEO (1.0): locally orthogonal staggered unstructured grid generation for general circulation modelling on the sphere
An algorithm for the generation of non-uniform, locally-orthogonal staggered
unstructured spheroidal grids is described. This technique is designed to
generate very high-quality staggered Voronoi/Delaunay meshes appropriate for
general circulation modelling on the sphere, including applications to
atmospheric simulation, ocean-modelling and numerical weather prediction. Using
a recently developed Frontal-Delaunay refinement technique, a method for the
construction of high-quality unstructured spheroidal Delaunay triangulations is
introduced. A locally-orthogonal polygonal grid, derived from the associated
Voronoi diagram, is computed as the staggered dual. It is shown that use of the
Frontal-Delaunay refinement technique allows for the generation of very
high-quality unstructured triangulations, satisfying a-priori bounds on element
size and shape. Grid-quality is further improved through the application of
hill-climbing type optimisation techniques. Overall, the algorithm is shown to
produce grids with very high element quality and smooth grading
characteristics, while imposing relatively low computational expense. A
selection of uniform and non-uniform spheroidal grids appropriate for
high-resolution, multi-scale general circulation modelling are presented. These
grids are shown to satisfy the geometric constraints associated with
contemporary unstructured C-grid type finite-volume models, including the Model
for Prediction Across Scales (MPAS-O). The use of user-defined mesh-spacing
functions to generate smoothly graded, non-uniform grids for multi-resolution
type studies is discussed in detail.Comment: Final revisions, as per: Engwirda, D.: JIGSAW-GEO (1.0): locally
orthogonal staggered unstructured grid generation for general circulation
modelling on the sphere, Geosci. Model Dev., 10, 2117-2140,
https://doi.org/10.5194/gmd-10-2117-2017, 201
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