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Delaunay refinement for piecewise smooth complexes
- Proc. 18th Annu. ACM-SIAM Sympos. Discrete Algorithms
, 2007
"... We present a Delaunay refinement algorithm for meshing a piecewise smooth complex in three dimensions. The algorithm protects edges with weighted points to avoid the difficulty posed by small angles between adjacent input elements. These weights are chosen to mimic the local feature size and to sati ..."
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Cited by 32 (5 self)
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We present a Delaunay refinement algorithm for meshing a piecewise smooth complex in three dimensions. The algorithm protects edges with weighted points to avoid the difficulty posed by small angles between adjacent input elements. These weights are chosen to mimic the local feature size and to satisfy a Lipschitz-like property. A Delaunay refinement algorithm using the weighted Voronoi diagram is shown to terminate with the recovery of the topology of the input. Guaranteed bounds on the aspect ratios, normal variation and dihedral angles are also provided. To this end, we present new concepts and results including a new definition of local feature size and a proof for a generalized topological ball property. 1
Delaunay Meshing of Piecewise Smooth Complexes without Expensive Predicates
"... Recently a Delaunay refinement algorithm has been proposed that can mesh piecewise smooth complexes which include polyhedra, smooth and piecewise smooth surfaces, and nonmanifolds. However, this algorithm employs domain dependent numerical predicates, some of which could be computationally expensive ..."
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Cited by 10 (3 self)
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Recently a Delaunay refinement algorithm has been proposed that can mesh piecewise smooth complexes which include polyhedra, smooth and piecewise smooth surfaces, and nonmanifolds. However, this algorithm employs domain dependent numerical predicates, some of which could be computationally expensive and hard to implement. In this paper we develop a refinement strategy that eliminates these complicated domain dependent predicates. As a result we obtain a meshing algorithm that is practical and implementation-friendly.
Particle Systems for Adaptive, Isotropic Meshing of CAD Models
"... Summary. We present a particle-based approach for generating adaptive triangular surface and tetrahedral volume meshes from CAD models. Input shapes are treated as a collection of smooth, parametric surface patches that can meet non-smoothly on boundaries. Our approach uses a hierarchical sampling s ..."
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Cited by 5 (1 self)
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Summary. We present a particle-based approach for generating adaptive triangular surface and tetrahedral volume meshes from CAD models. Input shapes are treated as a collection of smooth, parametric surface patches that can meet non-smoothly on boundaries. Our approach uses a hierarchical sampling scheme that places particles on features in order of increasing dimensionality. These particles reach a good distribution by minimizing an energy computed in 3D world space, with movements occurring in the parametric space of each surface patch. Rather than using a pre-computed measure of feature size, our system automatically adapts to both curvature as well as a notion of topological separation. It also enforces a measure of smoothness on these constraints to construct a sizing field that acts as a proxy to piecewise-smooth feature size. We evaluate our technique with comparisons against other popular triangular meshing techniques for this domain. Key words: Adaptive meshing, particle systems, tetrahedral meshing, CAD 1
and Alper Üngör. Construction of sparse well-spaced point sets for quality tetrahedralizations
- In Int. Meshing Roundtable
, 2008
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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 non-manifold spaces. The algorithm is guaranteed to capture the in ..."
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Cited by 5 (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 non-manifold 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 k-manifold element in the input complex is meshed with a triangulated k-manifold. Boundary incidences among elements maintain the input structure. Implementation results reported in a companion paper corroborate our claims.
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.
A New Approach to Output-Sensitive Voronoi Diagrams
"... We describe a new algorithm for computing the Voronoi diagram of a set of n points in constant-dimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest ..."
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We describe a new algorithm for computing the Voronoi diagram of a set of n points in constant-dimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures. 1
An Efficient Computation of Handle and Tunnel Loops via Reeb Graphs
"... A special family of non-trivial loops on a surface called handle and tunnel loops associates closely to geometric features of “handles” and “tunnels” respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplificatio ..."
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Cited by 2 (0 self)
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A special family of non-trivial loops on a surface called handle and tunnel loops associates closely to geometric features of “handles” and “tunnels” respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplification / repair, and surface parameterization, to feature and shape recognition. Many of the existing efficient algorithms for computing non-trivial loops cannot be used to compute these special type of loops. The two algorithms known for computing handle and tunnel loops provably have a serious drawback that they both require a tessellation of the interior and exterior spaces bounded by the surface. Computing such a tessellation of three dimensional space around the surface is a non-trivial task and can be quite expensive. Furthermore, such a tessellation may need to refine the surface mesh, thus causing the undesirable side-effect of outputting the loops on an altered surface mesh. In this paper, we present an efficient algorithm to compute a basis for handle and tunnel loops without requiring any 3D tessellation. This saves time considerably for large meshes making the algorithm scalable while computing the loops on the original input mesh and not on some refined version of it. We use the concept of the Reeb graph which together with several key theoretical insights on linking number provide an initial set of loops that provably constitute a handle and a tunnel basis. We further develop a novel strategy to tighten these handle and tunnel basis loops to make them geometrically relevant. We demonstrate the efficiency and effectiveness of our algorithm as well as show its robustness against noise, and other anomalies in the input.
A Fast Algorithm for Well-Spaced Points and Approximate Delaunay Graphs
, 2013
"... We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time O(2 O(d) (n log n + m)), where n is ..."
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Cited by 2 (0 self)
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We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time O(2 O(d) (n log n + m)), where n is the input size, m is the output point set size, and d is the ambient dimension. The constants only depend on the desired element quality bounds. To gain this new efficiency, the algorithm approximately maintains the Voronoi diagram of the current set of points by storing a superset of the Delaunay neighbors of each point. By retaining quality of the Voronoi diagram and avoiding the storage of the full Voronoi diagram, a simple exponential dependence on d is obtained in the running time. Thus, if one only wants the approximate neighbors structure of a refined Delaunay mesh conforming to a set of input points, the algorithm will return a size 2 O(d) m graph in 2 O(d) (n log n + m) expected time. If m is superlinear in n, then we can produce a hierarchically well-spaced superset of size 2 O(d) n in 2 O(d) n log n expected time.
Vertex Deletion for 3D Delaunay Triangulations
"... Abstract. We show how to delete a vertex q from a three-dimensional Delaunay triangulation DT(S) in expected O(C ⊗ (P)) time, where P is the set of vertices neighboring q in DT(S) and C ⊗ (P) is an upper bound on the expected number of tetrahedra whose circumspheres enclose q that are created during ..."
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Cited by 1 (0 self)
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Abstract. We show how to delete a vertex q from a three-dimensional Delaunay triangulation DT(S) in expected O(C ⊗ (P)) time, where P is the set of vertices neighboring q in DT(S) and C ⊗ (P) is an upper bound on the expected number of tetrahedra whose circumspheres enclose q that are created during the randomized incremental construction of DT(P). Experiments show that our approach is significantly faster than existing implementations if q has high degree, and competitive if q has low degree. 1
