5,048 research outputs found
Pomelo, a tool for computing Generic Set Voronoi Diagrams of Aspherical Particles of Arbitrary Shape
We describe the development of a new software tool, called "Pomelo", for the
calculation of Set Voronoi diagrams. Voronoi diagrams are a spatial partition
of the space around the particles into separate Voronoi cells, e.g. applicable
to granular materials. A generalization of the conventional Voronoi diagram for
points or monodisperse spheres is the Set Voronoi diagram, also known as
navigational map or tessellation by zone of influence. In this construction, a
Set Voronoi cell contains the volume that is closer to the surface of one
particle than to the surface of any other particle. This is required for
aspherical or polydisperse systems.
Pomelo is designed to be easy to use and as generic as possible. It directly
supports common particle shapes and offers a generic mode, which allows to deal
with any type of particles that can be described mathematically. Pomelo can
create output in different standard formats, which allows direct visualization
and further processing. Finally, we describe three applications of the Set
Voronoi code in granular and soft matter physics, namely the problem of
packings of ellipsoidal particles with varying degrees of particle-particle
friction, mechanical stable packings of tetrahedra and a model for liquid
crystal systems of particles with shapes reminiscent of pearsComment: 4 pages, 9 figures, Submitted to Powders and Grains 201
Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results
We achieve a detailed understanding of the -sided planar Poisson-Voronoi
cell in the limit of large . Let be the probability for a cell to
have sides. We construct the asymptotic expansion of up to
terms that vanish as . We obtain the statistics of the lengths of
the perimeter segments and of the angles between adjoining segments: to leading
order as , and after appropriate scaling, these become independent
random variables whose laws we determine; and to next order in they have
nontrivial long range correlations whose expressions we provide. The -sided
cell tends towards a circle of radius (n/4\pi\lambda)^{\half}, where
is the cell density; hence Lewis' law for the average area of
the -sided cell behaves as with . For
the cell perimeter, expressed as a function of the polar
angle , satisfies , where is known Gaussian
noise; we deduce from it the probability law for the perimeter's long
wavelength deviations from circularity. Many other quantities related to the
asymptotic cell shape become accessible to calculation.Comment: 54 pages, 3 figure
Calculation of the Voronoi boundary for lens-shaped particles and spherocylinders
We have recently developed a mean-field theory to estimate the packing
fraction of non-spherical particles [A. Baule et al., Nature Commun. (2013)].
The central quantity in this framework is the Voronoi excluded volume, which
generalizes the standard hard-core excluded volume appearing in Onsager's
theory. The Voronoi excluded volume is defined from an exclusion condition for
the Voronoi boundary between two particles, which is usually not tractable
analytically. Here, we show how the technical difficulties in calculating the
Voronoi boundary can be overcome for lens-shaped particles and spherocylinders,
two standard prolate and oblate shapes with rotational symmetry. By decomposing
these shapes into unions and intersections of spheres analytical expressions
can be obtained.Comment: 19 pages, 8 figure
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