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42
Size Complexity of Volume Meshes vs. Surface Meshes
, 2007
"... Typical volume meshes in three dimensions are designed to conform to an underlying two-dimensional 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 15 (14 self)
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Typical volume meshes in three dimensions are designed to conform to an underlying two-dimensional 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 space-filling 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.
Topological inference via meshing
- IN SOCG: PROCEEDINGS OF THE 26TH ACM SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
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SVR: Practical Engineering of a Fast 3D Meshing Algorithm
, 2007
"... 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 wi ..."
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Cited by 10 (8 self)
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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.
Dynamic wellspaced point sets
- In SCG ’10: Proceedings of the 26th Annual Symposium on Computational Geometry
, 2010
"... In a well-spaced 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. Well-spaced point sets satisfy some important geom ..."
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Cited by 9 (5 self)
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In a well-spaced 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. Well-spaced 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 well-spaced point-sets problem, which requires computing the well-spaced 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 worst-case O(log ∆) time, where ∆ is the geometric spread, a natural measure that is bounded by O(log n) when input points are represented by log-size 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 size-optimal 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 size-optimal dynamic algorithm for well-spaced point sets.
Sparse Parallel Delaunay Mesh Refinement
, 2007
"... The authors recently introduced the technique of sparse mesh refinement to produce the first near-optimal sequential time bounds of O(n lg L/s+m) for inputs in any fixed dimension with piecewise-linear constraining (PLC) features. This paper extends that work to the parallel case, refining the same ..."
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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 near-optimal sequential time bounds of O(n lg L/s+m) for inputs in any fixed dimension with piecewise-linear 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 numberof processors. This is faster than the best previously known parallel Delaunay mesh refinement algorithms in two dimensions. It is thefirst technique with work bounds equal to the sequential case. In higher dimension, it is the first provably fast parallel technique forany kind of quality mesh refinement with PLC inputs. Furthermore, the algorithm's implementation is straightforward enough that it islikely to be extremely fast in practice.
and Alper Üngör. Construction of sparse well-spaced point sets for quality tetrahedralizations
- In Int. Meshing Roundtable
, 2008
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Delaunay mesh generation of three dimensional domains
, 2007
"... Delaunay meshes are used in various applications such as finite element analysis, computer graphics rendering, geometric modeling, and shape analysis. As the applications vary, so do the domains to be meshed. Although meshing of geometric domains with Delaunay simplices have been around for a while, ..."
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Cited by 4 (0 self)
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Delaunay meshes are used in various applications such as finite element analysis, computer graphics rendering, geometric modeling, and shape analysis. As the applications vary, so do the domains to be meshed. Although meshing of geometric domains with Delaunay simplices have been around for a while, provable techniques to mesh various types of three dimensional domains have been developed only recently. We devote this article to presenting these techniques. We survey various related results and detail a few core algorithms that have provable guarantees and are amenable to practical implementation. Delaunay refinement, a paradigm originally developed for guaranteeing shape quality of mesh elements, is a common thread in these algorithms. We finish the article by listing a set of open questions.
Optimal-time dynamic mesh refinement: preliminary results
, 2006
"... We present early results on a dynamic mesh refinement algorithm. Using a variant of the Sparse Voronoi Refinement algorithm and applying the technique of Self-Adjusting Computation, we find that we expect to run in O(polylog n) time per update on points sets in arbitrary dimension. This is based on ..."
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Cited by 4 (3 self)
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We present early results on a dynamic mesh refinement algorithm. Using a variant of the Sparse Voronoi Refinement algorithm and applying the technique of Self-Adjusting Computation, we find that we expect to run in O(polylog n) time per update on points sets in arbitrary dimension. This is based on some theoretical results, along with experimental results from an implementation.
Size Competitive Meshing without Large Angles
"... We present a new meshing algorithm for the plane, Overlay Stitch Meshing (OSM), accepting as input an arbitrary Planar Straight Line Graph and producing a triangulation with all angles smaller than 170 ◦. The output triangulation has competitive size with any optimal size mesh having equally bounde ..."
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Cited by 3 (2 self)
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We present a new meshing algorithm for the plane, Overlay Stitch Meshing (OSM), accepting as input an arbitrary Planar Straight Line Graph and producing a triangulation with all angles smaller than 170 ◦. The output triangulation has competitive size with any optimal size mesh having equally bounded largest angle. The competitive ratio is O(log(L/s)) where L and s are respectively the largest and smallest features in the input. OSM runs in O(nlog(L/s) + m) time/work where n is the input size and m is the output size. The algorithm first uses Sparse Voronoi Refinement to compute a quality overlay mesh of the input points alone. This triangulation is then combined with the input edges to give the final mesh.
