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Tetrahedral Mesh Generation by Delaunay Refinement
 Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... Given a complex of vertices, constraining segments, and planar straightline constraining facets in E 3 , with no input angle less than 90 ffi , an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradiustoshortest edge ratios are no greater than two ..."
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Cited by 115 (7 self)
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Given a complex of vertices, constraining segments, and planar straightline constraining facets in E 3 , with no input angle less than 90 ffi , an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradiustoshortest edge ratios are no greater than two. The sizes of the tetrahedra can provably grade from small to large over a relatively short distance. An implementation demonstrates that the algorithm generates excellent meshes, generally surpassing the theoretical bounds, and is effective in eliminating tetrahedra with small or large dihedral angles, although they are not all covered by the theoretical guarantee. 1 Introduction Meshes of triangles or tetrahedra have many applications, including interpolation, rendering, and numerical methods such as the finite element method. Most such applications demand more than just a triangulation of the object or domain being rendered or simulated. To ensure accurate results, the triangles or tetr...
WellSpaced Points for Numerical Methods
, 1997
"... mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; ..."
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Cited by 44 (2 self)
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mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; (3) solving the linear system. The approximation error and convergence of the numerical method depend on the geometric quality of the mesh, which in turn depends on the size and shape of its elements. For example, the shape quality of a triangular mesh is measured by its element's aspect ratio. In this work, we shift the focus to the geometric properties of the nodes, rather than the elements, of well shaped meshes. We introduce the concept of wellspaced points and their spacing functions, and show that these enable the development of simple and efficient algorithms for the different stages of the numerical solution of PDEs. We first apply wellspaced point sets and their accompanying technology to mesh coarsening, a crucial step in the multigrid solution of a PDE. A good aspectratio coarsening sequence of an unstructured mesh M0 is a sequence of good aspectratio meshes M1; : : : ; Mk such that Mi is an approximation of Mi\Gamma 1 containing fewer nodes and elements. We present a new approach to coarsening that guarantees the sequence is also of optimal size and width up to a constant factor the first coarsening method that provides these guarantees. We also present experimental results, based on an implementation of our approach, that substantiate the theoretical claims.
A Condition Guaranteeing the Existence of HigherDimensional Constrained Delaunay Triangulations
 Proceedings of the Fourteenth Annual Symposium on Computational Geometry
, 1998
"... Let X be a complex of vertices and piecewise linear constraining facets embedded in E d . Say that a simplex is strongly Delaunay if its vertices are in X and there exists a sphere that passes through its vertices but passes through and encloses no other vertex. Then X has a ddimensional constra ..."
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Cited by 38 (3 self)
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Let X be a complex of vertices and piecewise linear constraining facets embedded in E d . Say that a simplex is strongly Delaunay if its vertices are in X and there exists a sphere that passes through its vertices but passes through and encloses no other vertex. Then X has a ddimensional constrained Delaunay triangulation if each kdimensional constraining facet in X with k d \Gamma 2 is a union of strongly Delaunay ksimplices. This theorem is especially useful in E 3 for forming tetrahedralizations that respect specified planar facets. If the bounding segments of these facets are subdivided so that the subsegments are strongly Delaunay, then a constrained tetrahedralization exists. Hence, fewer vertices are needed than in the most common practice in the literature, wherein additional vertices are inserted in the relative interiors of facets to form a conforming (but unconstrained) Delaunay tetrahedralization. 1 Introduction Many applications can benefit from triangulations...
A TimeOptimal Delaunay Refinement Algorithm in Two Dimensions
 In Symposium on Computational Geometry
, 2005
"... We propose a new refinement algorithm to generate sizeoptimal qualityguaranteed Delaunay triangulations in the plane. The algorithm takes O(n log n + m) time, where n is the input size and m is the output size. This is the first timeoptimal Delaunay refinement algorithm. 1 ..."
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Cited by 25 (2 self)
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We propose a new refinement algorithm to generate sizeoptimal qualityguaranteed Delaunay triangulations in the plane. The algorithm takes O(n log n + m) time, where n is the input size and m is the output size. This is the first timeoptimal Delaunay refinement algorithm. 1
Guaranteedquality triangular mesh generation for domains with curved boundaries
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
, 2002
"... Guaranteedquality unstructured meshing algorithms facilitate the development of automatic meshing tools. However, these algorithms require domains discretized using a set of linear segments, leading to numerical errors in domains with curved boundaries. We introduce an extension of Ruppert’s Delaun ..."
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Cited by 23 (3 self)
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Guaranteedquality unstructured meshing algorithms facilitate the development of automatic meshing tools. However, these algorithms require domains discretized using a set of linear segments, leading to numerical errors in domains with curved boundaries. We introduce an extension of Ruppert’s Delaunay refinement algorithm to twodimensional domains with curved boundaries and prove that the same quality bounds apply with curved boundaries as with straight boundaries. We provide implementation details for twodimensional boundary patches such as lines, circular arcs, cubic parametric curves, and interpolated splines. We present guaranteedquality triangular meshes generated with curved boundaries, and propose solutions to some problems associated with the use of curved boundaries. Copyright c
A PointPlacement Strategy for Conforming Delaunay Tetrahedralization
 Proceedings of the Eleventh Annual Symposium on Discrete Algorithms
, 2000
"... A strategy is presented to find a set of points which yields a Conforming Delaunay tetrahedralization of a threedimensional PiecewiseLinear Complex (PLC). This algorithm is novel because it imposes no angle restrictions on the input PLC. In the process, an algorithm is described that computes a ..."
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Cited by 20 (0 self)
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A strategy is presented to find a set of points which yields a Conforming Delaunay tetrahedralization of a threedimensional PiecewiseLinear Complex (PLC). This algorithm is novel because it imposes no angle restrictions on the input PLC. In the process, an algorithm is described that computes a planar conforming Delaunay triangulation of a Planar StraightLine Graph (PSLG) such that each triangle has a bounded circumradius, which may be of independent interest. 1 Introduction In many two and threedimensional geometric modeling problems, notably the numerical approximation of the solution to a Partial Differential Equation with a FiniteElement type method [SF73], it is very desirable to obtain a triangulation (tetrahedralization) that respects the domain of interest. The task of forming such decompositions, along with ensuring that the elements of the decompositions satisfy applicationspecific quality requirements, is sometimes referred to as unstructured mesh generation. Se...
GuaranteedQuality Parallel Delaunay Refinement for Restricted Polyhedral Domains
, 2004
"... We describe a distributed memory parallel Delaunay refinement algorithm for simple polyhedral domains whose constituent bounding edges and surfaces are separated by angles between 90 o to 270 o inclusive. With these constraints, our algorithm can generate meshes containing tetrahedra with circumradi ..."
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Cited by 19 (8 self)
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We describe a distributed memory parallel Delaunay refinement algorithm for simple polyhedral domains whose constituent bounding edges and surfaces are separated by angles between 90 o to 270 o inclusive. With these constraints, our algorithm can generate meshes containing tetrahedra with circumradius to shortest edge ratio less than 2, and can tolerate more than 80 % of the communication latency caused by unpredictable and variable remote gather operations. Our experiments show that the algorithm is efficient in practice, even for certain domains whose boundaries do not conform to the theoretical limits imposed by the algorithm. The algorithm we describe is the first step in the development of much more sophisticated guaranteed–quality parallel mesh generation algorithms.
A Robust Procedure to Eliminate Degenerate Faces from Triangle Meshes
 VISION, MODELING AND VISUALIZATION (VMV01
, 2001
"... When using triangle meshes in numerical simulations or other sophisticated downstream applications, we have to guarantee that no degenerate faces are present since they have, e.g., no well defined normal vectors. In this paper we present a simple but effective algorithm to remove such artifacts from ..."
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Cited by 19 (2 self)
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When using triangle meshes in numerical simulations or other sophisticated downstream applications, we have to guarantee that no degenerate faces are present since they have, e.g., no well defined normal vectors. In this paper we present a simple but effective algorithm to remove such artifacts from a given triangle mesh. The central problem is to make this algorithm numerically robust because degenerate triangles are usually the source for all kinds of numerical instabilities. Our algorithm is based on a slicing technique that cuts a set of planes through the given polygonal model. The mesh slicing operator only uses numerically stable predicates and therefore is able to split faces in a controlled manner. In combination with a custom tailored mesh decimation scheme we are able to remove the degenerate faces from meshes like those typically generated by tesselation units in CAD systems.
Sweep Algorithms for Constructing HigherDimensional Constrained Delaunay Triangulations
 In Proceedings of the sixteenth annual symposium on Computational geometry, Kowloon, Hong Kong
, 2000
"... I discuss algorithms for constructing constrained Delaunay triangulations (CDTs) in dimensions higher than two. If the CDT of a set of vertices and constraining simplices exists, it can be constructed in O(n v n s ) time, where n v is the number of input vertices and n s is the number of output dsi ..."
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Cited by 19 (0 self)
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I discuss algorithms for constructing constrained Delaunay triangulations (CDTs) in dimensions higher than two. If the CDT of a set of vertices and constraining simplices exists, it can be constructed in O(n v n s ) time, where n v is the number of input vertices and n s is the number of output dsimplices. The CDT of a starshaped polytope can be constructed in O(n s log n v ) time, yielding an efficient way to delete a vertex from a CDT. 1 Introduction Mesh generation and interpolation can benefit from triangulations that have properties similar to Delaunay triangulations, but are constrained to contain specified faces. These constraints may arise because a mesh must conform to the shape of an object, or because of the desire to interpolate a discontinuous function. The constrained Delaunay triangulation (CDT) [5, 1, 11] is a Delaunaylike triangulation that conforms to constraints. In two dimensions, the input is a planar straight line graph (PSLG) X , which is a set of vertices a...
Robust Three Dimensional Delaunay Refinement
 IN THIRTEENTH INTERNATIONAL MESHING ROUNDTABLE
, 2004
"... The Delaunay Refinement Algorithm for quality meshing is extended to three dimensions. The algorithm accepts input with arbitrarily small angles, and outputs a Conforming Delaunay Tetrahedralization where most tetrahedra have radiustoshortestedge ratio smaller than some user chosen µ > 2. Those t ..."
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Cited by 18 (3 self)
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The Delaunay Refinement Algorithm for quality meshing is extended to three dimensions. The algorithm accepts input with arbitrarily small angles, and outputs a Conforming Delaunay Tetrahedralization where most tetrahedra have radiustoshortestedge ratio smaller than some user chosen µ > 2. Those tets with poor quality are in well defined locations: their circumcenters are describably near input segments. Moreover, the output mesh is well graded to the input: short mesh edges only appear around close features of the input. The algorithm has the added advantage of not requiring a priori knowledge of the "local feature size," and only requires searching locally in the mesh.