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Sparse Voronoi Refinement
 IN PROCEEDINGS OF THE 15TH INTERNATIONAL MESHING ROUNDTABLE
, 2006
"... ... a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in outputsensitive time O(nlog(L/s) + m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordina ..."
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Cited by 36 (24 self)
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... a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in outputsensitive time O(nlog(L/s) + m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordinates, this matches the optimal time bound of Θ(n log n + m). Our new technique uses interleaving: we maintain a sparse mesh as we mix the recovery of input features with the addition of Steiner vertices for quality improvement.
A practical Delaunay meshing algorithm for a large class of domains
 Proceedings of the 16th International Meshing Roundtable
, 2007
"... Summary. Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes [7]. This class includes polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all nonmanifolds. In contrast to previous approaches, the ..."
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Cited by 25 (7 self)
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Summary. Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes [7]. This class includes polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all nonmanifolds. In contrast to previous approaches, the algorithm does not impose any restriction on the input angles. Although this algorithm has a provable guarantee about topology, certain steps are too expensive to make it practical. In this paper we introduce a novel modification of the algorithm to make it implementable in practice. In particular, we replace four tests of the original algorithm with only a single test that is easy to implement. The algorithm has the following guarantees. The output mesh restricted to each manifold element in the complex is a manifold with proper incidence relations. More importantly, with increasing level of refinement which can be controlled by an input parameter, the output mesh becomes homeomorphic to the input while preserving all input features. Implementation results on a disparate array of input domains are presented to corroborate our claims. 1
Size Complexity of Volume Meshes vs. Surface Meshes
, 2007
"... Typical volume meshes in three dimensions are designed to conform to an underlying twodimensional surface mesh, with volume mesh element size growing larger away from the surface. The surface mesh may be uniformly spaced or highly graded, and may have fine resolution due to extrinsic mesh size conc ..."
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Cited by 13 (12 self)
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Typical volume meshes in three dimensions are designed to conform to an underlying twodimensional surface mesh, with volume mesh element size growing larger away from the surface. The surface mesh may be uniformly spaced or highly graded, and may have fine resolution due to extrinsic mesh size concerns. When we desire that such a mesh have good aspect ratio, we require that some spacefilling scaffold vertices be inserted off the surface. We analyze the number of scaffold vertices in a setting that encompasses many existing volume meshing algorithms. We show that for surfaces of bounded variation, the number of scaffold vertices will be linear in the number of surface vertices.
SVR: Practical engineering of a fast 3D meshing algorithm
 In International Meshing Roundtable
, 2007
"... Summary. The recent Sparse Voronoi Refinement (SVR) Algorithm for mesh generation has the fastest theoretical bounds for runtime and memory usage. We present a robust practical software implementation of the SVR for meshing a piecewise linear complex in 3 dimensions. Our software is competitive in r ..."
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Cited by 10 (8 self)
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Summary. The recent Sparse Voronoi Refinement (SVR) Algorithm for mesh generation has the fastest theoretical bounds for runtime and memory usage. We present a robust practical software implementation of the SVR for meshing a piecewise linear complex in 3 dimensions. Our software is competitive in runtime with state of the art freely available packages on generic inputs, and on pathological worse cases inputs, we show SVR indeed leverages its theoretical guarantees to produce vastly superior runtime and memory usage. The theoretical algorithm description of SVR leaves open several data structure design options, especially with regard to point location strategies. We show that proper strategic choices can greatly effect constant factors involved in runtime. 1
Sparse Parallel Delaunay Mesh Refinement
, 2007
"... The authors recently introduced the technique of sparse mesh refinement to produce the first nearoptimal sequential time bounds of O(n lg L/s+m) for inputs in any fixed dimension with piecewiselinear constraining (PLC) features. This paper extends that work to the parallel case, refining the same i ..."
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Cited by 8 (1 self)
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The authors recently introduced the technique of sparse mesh refinement to produce the first nearoptimal sequential time bounds of O(n lg L/s+m) for inputs in any fixed dimension with piecewiselinear constraining (PLC) features. This paper extends that work to the parallel case, refining the same inputs in time O(lg(L/s) lg m) on an EREW PRAM while maintaining the work bound; in practice, this means we expect linear speedup for any practical number of processors. This is faster than the best previously known parallel Delaunay mesh refinement algorithms in two dimensions. It is the first technique with work bounds equal to the sequential case. In higher dimension, it is the first provably fast parallel technique for any kind of quality mesh refinement with PLC inputs. Furthermore, the algorithm’s implementation is straightforward enough that it is likely to be extremely fast in practice.
Dynamic wellspaced point sets
 In SCG ’10: Proceedings of the 26th Annual Symposium on Computational Geometry
, 2010
"... In a wellspaced point set the Voronoi cells all have bounded aspect ratio, i.e., the distance from the Voronoi site to the farthest point in the Voronoi cell divided by the distance to the nearest neighbor in the set is bounded by a small constant. Wellspaced point sets satisfy some important geom ..."
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Cited by 7 (5 self)
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In a wellspaced point set the Voronoi cells all have bounded aspect ratio, i.e., the distance from the Voronoi site to the farthest point in the Voronoi cell divided by the distance to the nearest neighbor in the set is bounded by a small constant. Wellspaced point sets satisfy some important geometric properties and yield quality Voronoi or simplicial meshes that can be important in scientific computations. In this paper, we consider the dynamic wellspaced pointsets problem, which requires computing the wellspaced superset of a dynamically changing input set, e.g., as points are inserted or deleted. We present a dynamic algorithm that allows inserting/deleting points into/from the input in worstcase O(log ∆) time, where ∆ is the geometric spread, a natural measure that is bounded by O(log n) when input points are represented by logsize words. We show that the runtime of the dynamic update algorithm is optimal in the worst case by showing that there exists inputs and modifications that require Ω(log ∆) Steiner points to be inserted to the output. Our algorithm generates sizeoptimal outputs: the resulting output sets are never more than a constant factor larger than the minimum size necessary. A preliminary implementation indicates that the algorithm is indeed fast in practice. To the best of our knowledge, this is the first time and sizeoptimal dynamic algorithm for wellspaced point sets.
Triangulations with Locally Optimal Steiner Points
, 2007
"... We present two new Delaunay refinement algorithms, second an extension of the first. For a given input domain (a set of points or a planar straight line graph), and a threshold angle α, the Delaunay refinement algorithms compute triangulations that have all angles at least α. Our algorithms have the ..."
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Cited by 7 (0 self)
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We present two new Delaunay refinement algorithms, second an extension of the first. For a given input domain (a set of points or a planar straight line graph), and a threshold angle α, the Delaunay refinement algorithms compute triangulations that have all angles at least α. Our algorithms have the same theoretical guarantees as the previous Delaunay refinement algorithms. The original Delaunay refinement algorithm of Ruppert is proven to terminate with sizeoptimal quality triangulations for α ≤ 20.7◦. In practice, it generally works for α ≤ 34 ◦ and fails to terminate for larger constraint angles. The new variant of the Delaunay refinement algorithm generally terminates for constraint angles up to 42◦. Experiments also indicate that our algorithm computes significantly (almost by a factor of two) smaller triangulations than the output of the previous Delaunay refinement algorithms.
Construction of sparse wellspaced point sets for quality tetrahedralizations
 In IMR
, 2007
"... ..."
Theory of a practical Delaunay meshing algorithm for a large class of domains
 Algorithms, Architecture and Information System Security, World Scientific Review Volume
"... Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes. These domains include polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all nonmanifold spaces. The algorithm is guaranteed to capture the in ..."
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Cited by 3 (1 self)
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Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes. These domains include polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all nonmanifold spaces. The algorithm is guaranteed to capture the input topology at the expense of four tests, some of which are computationally intensive and hard to implement. The goal of this paper is to present the theory that justifies a refinement algorithm with a single disk test in place of four tests of the previous algorithm. The algorithm is supplied with a resolution parameter that controls the level of refinement. We prove that, when the resolution is fine enough (this level is reached very fast in practice), the output mesh becomes homeomorphic to the input while preserving all input features. Moreover, regardless of the refinement level, each kmanifold element in the input complex is meshed with a triangulated kmanifold. Boundary incidences among elements maintain the input structure. Implementation results reported in a companion paper corroborate our claims.