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34
Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery
 In Eleventh International Meshing Roundtable
, 2002
"... In two dimensions, a constrained Delaunay triangulation (CDT) respects a set of segments that constrain the edges of the triangulation, while still maintaining most of the favorable properties of ordinary Delaunay triangulations (such as maximizing the minimum angle). CDTs solve the problem of enfor ..."
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Cited by 44 (1 self)
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In two dimensions, a constrained Delaunay triangulation (CDT) respects a set of segments that constrain the edges of the triangulation, while still maintaining most of the favorable properties of ordinary Delaunay triangulations (such as maximizing the minimum angle). CDTs solve the problem of enforcing boundary conformityensuring that triangulation edges cover the boundaries (both interior and exterior) of the domain being modeled. This paper discusses the threedimensional analogue, constrained Delaunay tetrahedralizations (also called CDTs), and their advantages in mesh generation. CDTs maintain most of the favorable properties of ordinary Delaunay tetrahedralizations, but they are more difficult to work with, because some sets of constraining segments and facets simply do not have CDTs. However, boundary conformity can always be enforced by judicious insertion of additional vertices, combined with CDTs. This approach has three advantages over other methods for boundary recovery: it usually requires fewer additional vertices to be inserted, it yields provably good bounds on edge lengths (i.e. edges are not made unnecessarily short), and it interacts well with provably good Delaunay refinement methods for tetrahedral mesh generation.
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 42 (26 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.
Quality Meshing with Weighted Delaunay Refinement
 SIAM J. Comput
, 2002
"... Delaunay meshes with bounded circumradius to shortest edge length ratio have been proposed in the past for quality meshing. The only poor quality tetrahedra called slivers that can occur in such a mesh can be eliminated by the sliver exudation method. This method has been shown to work for periodic ..."
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Cited by 40 (7 self)
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Delaunay meshes with bounded circumradius to shortest edge length ratio have been proposed in the past for quality meshing. The only poor quality tetrahedra called slivers that can occur in such a mesh can be eliminated by the sliver exudation method. This method has been shown to work for periodic point sets, but not with boundaries. Recently a randomized pointplacement strategy has been proposed to remove slivers while conforming to a given boundary. In this paper we present a deterministic algorithm for generating a weighted Delaunay mesh which respects the input boundary and has no poor quality tetrahedron including slivers. This success is achieved by combining the weight pumping method for sliver exudation and the Delaunay refinement method for boundary conformation. We show that an incremental weight pumping can be mixed seamlessly with vertex insertions in our weighted Delaunay refinement paradigm. 1
Generating WellShaped Delaunay Meshes in 3D
, 2001
"... A triangular mesh in 3D is a decomposition of a given geometric domain into tetrahedra. The mesh is wellshaped if the aspect ratio of every of its tetrahedra is bounded from aboveby a constant. It is Delaunayifthe interior of the circumsphere of each of its tetrahedra does not contain any other mes ..."
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Cited by 31 (0 self)
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A triangular mesh in 3D is a decomposition of a given geometric domain into tetrahedra. The mesh is wellshaped if the aspect ratio of every of its tetrahedra is bounded from aboveby a constant. It is Delaunayifthe interior of the circumsphere of each of its tetrahedra does not contain any other mesh vertices. Generating a wellshaped Delaunay mesh for any 3D domain has been a long term outstanding problem. In this paper, wepresent an efficient 3D Delaunay meshing algorithm that mathematically guarantees the wellshape quality of the mesh, if the domain does not have acute angles. The main ingredient of our algorithm is a novel refinement technique which systematically forbids the formation of slivers, a family of bad elements that none of the previous known algorithms can cleanly remove, especially near the domain boundary  needless to say, that our algorithm ensure that there is no sliver near the boundary of the domain.
Dense Point Sets Have Sparse Delaunay Triangulations
"... Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms ..."
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Cited by 29 (2 self)
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Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since threedimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worstcase running time \Omega (n2). However, this behavior is almost never observed in practice except for highlycontrived inputs. For all practical purposes, threedimensional Delaunay triangulations appear to have linear complexity. This frustrating
Parallel Delaunay Refinement: Algorithms and Analyses
 In Proceedings, 11th International Meshing Roundtable
, 2002
"... In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of "independent" points to insert into the domain, and then update the Delaunay trian ..."
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Cited by 25 (4 self)
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In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of "independent" points to insert into the domain, and then update the Delaunay triangulation. We show that such a set of independent points can be constructed efficiently in parallel and that the number of iterations needed is O(log&sup2;(L/s)), where L is the diameter of the domain, and s is the smallest edge in the output mesh. In addition, we show that the insertion of each independent set of points can be realized sequentially by Ruppert's method in two dimensions and Shewchuk's in three dimensions. Therefore, our parallel Delaunay refinement methods provide the same element quality and mesh size guarantees as the sequential algorithms in both two and three dimensions. For quasiuniform meshes, such as those produced by Chew's method, we show that the number of iterations can be reduced to O(log(L/s)). To the best of our knowledge, these are the first provably polylog(L/s) parallel time Delaunay meshing algorithms that generate wellshaped meshes of size optimal to within a constant.
An Experimental Study of Sliver Exudation
, 2001
"... We present results on a twostep improvement of mesh quality in threedimensional Delaunay triangulations. The rst step re nes the triangulation by inserting sinks and eliminates tetrahedra with large circumradius over shortest edge length ratio. The second step assigns weights to the vertices to ..."
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Cited by 18 (0 self)
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We present results on a twostep improvement of mesh quality in threedimensional Delaunay triangulations. The rst step re nes the triangulation by inserting sinks and eliminates tetrahedra with large circumradius over shortest edge length ratio. The second step assigns weights to the vertices to eliminate slivers. Our experimental ndings provide evidence for the practical eectiveness of sliver exudation.
Generating WellShaped ddimensional Delaunay Meshes
"... A ddimensional simplicial mesh is a Delaunay triangulation if the circumsphere of each of its simplices does not contain any vertices inside. A mesh is wellshaped if the maximum aspect ratio of all its simplices is bounded from above by a constant. It is a longterm open problem to generate wells ..."
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Cited by 17 (0 self)
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A ddimensional simplicial mesh is a Delaunay triangulation if the circumsphere of each of its simplices does not contain any vertices inside. A mesh is wellshaped if the maximum aspect ratio of all its simplices is bounded from above by a constant. It is a longterm open problem to generate wellshaped ddimensional Delaunay meshes for a given polyhedral domain. In this paper, we present a re nementbased method that generates wellshaped ddimensional Delaunay meshes for any PLC domain with no small input angles. Furthermore, we show that the generated wellshaped mesh has O(n) dsimplices, where n is the smallest number of dsimplices of any almostgood meshes for the same domain. Here a mesh is almostgood if each of its simplices has a bounded circumradius to the shortest edge length ratio.
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 15 (14 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.