10,411 research outputs found
Optimal Point Placement for Mesh Smoothing
We study the problem of moving a vertex in an unstructured mesh of
triangular, quadrilateral, or tetrahedral elements to optimize the shapes of
adjacent elements. We show that many such problems can be solved in linear time
using generalized linear programming. We also give efficient algorithms for
some mesh smoothing problems that do not fit into the generalized linear
programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was
presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This
is the final version, and will appear in a special issue of J. Algorithms for
papers from SODA '9
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 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
An advancing front Delaunay triangulation algorithm designed for robustness
A new algorithm is described for generating an unstructured mesh about an arbitrary two-dimensional configuration. Mesh points are generated automatically by the algorithm in a manner which ensures a smooth variation of elements, and the resulting triangulation constitutes the Delaunay triangulation of these points. The algorithm combines the mathematical elegance and efficiency of Delaunay triangulation algorithms with the desirable point placement features, boundary integrity, and robustness traditionally associated with advancing-front-type mesh generation strategies. The method offers increased robustness over previous algorithms in that it cannot fail regardless of the initial boundary point distribution and the prescribed cell size distribution throughout the flow-field
A Bayesian Approach to Manifold Topology Reconstruction
In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated
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