27,326 research outputs found
VoroCrust: Voronoi Meshing Without Clipping
Polyhedral meshes are increasingly becoming an attractive option with
particular advantages over traditional meshes for certain applications. What
has been missing is a robust polyhedral meshing algorithm that can handle broad
classes of domains exhibiting arbitrarily curved boundaries and sharp features.
In addition, the power of primal-dual mesh pairs, exemplified by
Voronoi-Delaunay meshes, has been recognized as an important ingredient in
numerous formulations. The VoroCrust algorithm is the first provably-correct
algorithm for conforming polyhedral Voronoi meshing for non-convex and
non-manifold domains with guarantees on the quality of both surface and volume
elements. A robust refinement process estimates a suitable sizing field that
enables the careful placement of Voronoi seeds across the surface circumventing
the need for clipping and avoiding its many drawbacks. The algorithm has the
flexibility of filling the interior by either structured or random samples,
while preserving all sharp features in the output mesh. We demonstrate the
capabilities of the algorithm on a variety of models and compare against
state-of-the-art polyhedral meshing methods based on clipped Voronoi cells
establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed
images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf.
Supplemental materials available on
https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd
Helmholtz Fermi Surface Harmonics: an efficient approach for treating anisotropic problems involving Fermi surface integrals
We present a new efficient numerical approach for representing anisotropic
physical quantities and/or matrix elements defined on the Fermi surface of
metallic materials. The method introduces a set of numerically calculated
generalized orthonormal functions which are the solutions of the Helmholtz
equation defined on the Fermi surface. Noteworthy, many properties of our
proposed basis set are also shared by the Fermi Surface Harmonics (FSH)
introduced by Philip B. Allen [Physical Review B 13, 1416 (1976)], proposed to
be constructed as polynomials of the cartesian components of the electronic
velocity. The main motivation of both approaches is identical, to handle
anisotropic problems efficiently. However, in our approach the basis set is
defined as the eigenfunctions of a differential operator and several desirable
properties are introduced by construction. The method demonstrates very robust
in handling problems with any crystal structure or topology of the Fermi
surface, and the periodicity of the reciprocal space is treated as a boundary
condition for our Helmholtz equation. We illustrate the method by analysing the
free-electron-like Lithium (Li), Sodium (Na), Copper (Cu), Lead(Pb), Tungsten
(W) and Magnesium diboride (MgB2).Comment: Accepted for publication in New Journal of Physics (NJP). 28 pages, 9
figure
Topological Quantum Gate Construction by Iterative Pseudogroup Hashing
We describe the hashing technique to obtain a fast approximation of a target
quantum gate in the unitary group SU(2) represented by a product of the
elements of a universal basis. The hashing exploits the structure of the
icosahedral group [or other finite subgroups of SU(2)] and its pseudogroup
approximations to reduce the search within a small number of elements. One of
the main advantages of the pseudogroup hashing is the possibility to iterate to
obtain more accurate representations of the targets in the spirit of the
renormalization group approach. We describe the iterative pseudogroup hashing
algorithm using the universal basis given by the braidings of Fibonacci anyons.
The analysis of the efficiency of the iterations based on the random matrix
theory indicates that the runtime and the braid length scale
poly-logarithmically with the final error, comparing favorably to the
Solovay-Kitaev algorithm.Comment: 20 pages, 5 figure
Distance-Sensitive Planar Point Location
Let be a connected planar polygonal subdivision with edges
that we want to preprocess for point-location queries, and where we are given
the probability that the query point lies in a polygon of
. We show how to preprocess such that the query time
for a point~ depends on~ and, in addition, on the distance
from to the boundary of~---the further away from the boundary, the
faster the query. More precisely, we show that a point-location query can be
answered in time , where
is the shortest Euclidean distance of the query point~ to the
boundary of . Our structure uses space and
preprocessing time. It is based on a decomposition of the regions of
into convex quadrilaterals and triangles with the following
property: for any point , the quadrilateral or triangle
containing~ has area . For the special case where
is a subdivision of the unit square and
, we present a simpler solution that achieves a
query time of . The latter solution can be extended to
convex subdivisions in three dimensions
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