Results 1  10
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11
Tetrahedral Grid Refinement
, 1995
"... Zusammenfassung Tetrahedral Grid Refinement. We present a refinement algorithm for unstructured tetrahedral grids, which generates possibly highly nonuniform but nevertheless consistent (closed) and stable triangulations. Therefore we first define some local regular and irregular refinement rules t ..."
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Cited by 51 (1 self)
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Zusammenfassung Tetrahedral Grid Refinement. We present a refinement algorithm for unstructured tetrahedral grids, which generates possibly highly nonuniform but nevertheless consistent (closed) and stable triangulations. Therefore we first define some local regular and irregular refinement rules that are applied to single elements. The global refinement algorithm then describes how these local rules can be combined and rearranged in order to ensure consistency as well as stability. It is given in a rather general form and includes also grid coarsening. 1991 Mathematics Subject Classifications: 65N50, 65N55 Key words: Tetrahedral grid refinement, stable refinements, consistent triangulations, green closure, grid coarsening. Verfeinerung von TetraederGittern. Es wird ein Verfeinerungsalgorithmus fur unstrukturierte TetraederGitter vorgestellt, der moglicherweise stark nichtuniforme aber dennoch konsistente (d.h. geschlossene) und stabile Triangulierungen liefert. Dazu definieren w...
Efficient Algorithms for Petersen's Matching Theorem
, 1999
"... Petersen's theorem is a classic result in matching theory from 1891, stating that every 3regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, ..."
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Cited by 24 (3 self)
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Petersen's theorem is a classic result in matching theory from 1891, stating that every 3regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3regular graphs. We have developed an O(n log^4 n)time algorithm for perfect matching in a 3regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.
Simplicial Grid Refinement: On Freudenthal's Algorithm and the Optimal Number of Congruence Classes
 NUMER. MATH
, 1998
"... In the present paper we investigate Freudenthal's simplex refinement algorithm which can be considered to be the canonical generalization of Bank's well known red refinement strategy for triangles. Freudenthal's algorithm subdivides any given (n)simplex into 2 n subsimplices, in such a way that r ..."
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Cited by 15 (0 self)
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In the present paper we investigate Freudenthal's simplex refinement algorithm which can be considered to be the canonical generalization of Bank's well known red refinement strategy for triangles. Freudenthal's algorithm subdivides any given (n)simplex into 2 n subsimplices, in such a way that recursive application results in a stable hierarchy of consistent triangulations. Our investigations concentrate in particular on the number of congruence classes generated by recursive refinements. After presentation of the method and the basic ideas behind it, we will show that Freudenthal's algorithm produces at most n!=2 congruence classes for any initial (n)simplex, no matter how many subsequent refinements are performed. Moreover, we will show that this number is optimal in the sense that recursive application of any affine invariant refinement strategy with 2 n sons per element results in at least n!=2 congruence classes for almost all (n)simplices.
Spacetime Meshing with Adaptive Refinement and Coarsening
 SCG'04
, 2004
"... We propose a new algorithm for constructing finiteelement meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain# and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain [0, T] i ..."
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Cited by 15 (9 self)
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We propose a new algorithm for constructing finiteelement meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain# and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain [0, T] in constant running time per tetrahedron in IR using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.
Simplex and Diamond Hierarchies: Models and Applications
, 2010
"... Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through simplex bisection. Such decompositions, originally developed for finite elements, are extensively used ..."
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Cited by 9 (4 self)
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Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multiresolution models of scalar fields, such as terrains, and static or timevarying volume data. They have also been used as an alternative to quadtrees and octrees as spatial access structures and in other applications. In this state of the art report, we distinguish between approaches that focus on a specific dimension and those that apply to all dimensions. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplexes, called diamonds, as the modeling primitive. This leads to two classes of data structures and to different query approaches. We present the hierarchical models in a dimension–independent manner, and organize the description of the various applications, primarily interactive terrain rendering and isosurface extraction, according to the dimension of the domain.
Spacetime Meshing for Discontinuous Galerkin Methods
 Department of Computer Science, University of Illinois at UrbanaChampaign
, 2005
"... Important applications in science and engineering, such as modeling traffic flow, seismic waves, electromagnetics, and the simulation of mechanical stresses in materials, require the highfidelity numerical solution of hyperbolic partial differential equations (PDEs) in space and time variables. Man ..."
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Cited by 4 (2 self)
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Important applications in science and engineering, such as modeling traffic flow, seismic waves, electromagnetics, and the simulation of mechanical stresses in materials, require the highfidelity numerical solution of hyperbolic partial differential equations (PDEs) in space and time variables. Many interesting physical problems involve nonlinear and anisotropic behavior, and the PDEs modeling them exhibit discontinuities in their solutions. Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve such PDEs arising from wave propagation phenomena. To support an accurate and efficient solution procedure using SDG methods and to exploit the flexibility of these methods, we give a meshing algorithm to construct an unstructured simplicial spacetime mesh over an arbitrary simplicial space domain. Our algorithm is the first spacetime meshing algorithm suitable for efficient solution of nonlinear phenomena in anisotropic media using novel discontinuous Galerkin finite element methods for implicit solutions directly in spacetime. Given a triangulated ddimensional Euclidean space domain M (a simplicial complex) and initial conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured simplicial mesh of the (d + 1)dimensional spacetime domain M × [0,∞). Our algorithm uses a nearoptimal number of spacetime elements, each with bounded temporal aspect ratio for any finite prefix M × [0,T] of spacetime. Unlike Delaunay meshes, the facets of our mesh satisfy gradient constraints that allow interleaving the construction of the mesh by adding new space
A local adaptive refinement method with multigrid preconditionning illustrated by multiphase flows simulations
, 2008
"... Abstract. The aim of this paper is to describe some numerical aspects linked to incompressible threephase flow simulations, thanks to CahnHilliard type model. The numerical capture of transfer phenomenon in the neighborhood of the interface require a mesh thickness which become crippling in the ca ..."
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Cited by 4 (4 self)
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Abstract. The aim of this paper is to describe some numerical aspects linked to incompressible threephase flow simulations, thanks to CahnHilliard type model. The numerical capture of transfer phenomenon in the neighborhood of the interface require a mesh thickness which become crippling in the case where it is applied to the whole computational domain. This suggests the use of a local refinement method which allows to dynamically focus on problematic areas. The notion of refinement pattern, introduced for Lagrange finite elements, allows to build a conceptual hierarchy of nested conformal approximation spaces which is then used to implement the socalled CHARMS local refinement methods. Properties of these methods are proved ensuring in particular the conformity of approximation spaces at every time of simulations. Furthermore, the multilevel structure obtained by this method, is used to construct multigrid preconditioners. Finally, after a validation on a model problem, the performance of the whole method is illustrated on an example of a liquid lens spreading between two stratified fluids. Résumé. L’objectif de l’article est de décrire certains aspects numériques liés à la simulation d’écoulements incompressibles à trois phases non miscibles, à l’aide de modèles à interfaces diffuses de type CahnHilliard. La capture numérique des phénomènes de transfert au voisinage des interfaces requiert
CompactlyEncoded Optical Flow Fields For MotionCompensated Video Compression And Processing
, 1997
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ADAPTIVE SPACETIME MESHING FOR DISCONTINUOUS GALERKIN METHODS
"... Abstract. Spacetimediscontinuous Galerkin (SDG) finite element methods are used to solve hyperbolic spacetime partial differential equations (PDEs) to accurately model wave propagation phenomena arising in important applications in science and engineering. Tent Pitcher is a specialized algorithm, i ..."
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Cited by 2 (0 self)
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Abstract. Spacetimediscontinuous Galerkin (SDG) finite element methods are used to solve hyperbolic spacetime partial differential equations (PDEs) to accurately model wave propagation phenomena arising in important applications in science and engineering. Tent Pitcher is a specialized algorithm, invented by Üngör and Sheffer [2000], and extended by Erickson et al. [2005], to construct an unstructured simplicial (d + 1)dimensional spacetime mesh over an arbitrary ddimensional space domain. Tent Pitcher supports an accurate, local, and parallelizable solution strategy by interleaving mesh generation with an SDG solver. When solving nonlinear PDEs, previous versions of Tent Pitcher must make conservative worstcase assumptions about the physical parameters which limit the duration of spacetime elements. Thus, these algorithms create a mesh with many more elements than necessary. In this paper, we extend Tent Pitcher to give the first spacetime meshing algorithm suitable for efficient simulation of nonlinear phenomena using SDG methods. Given a triangulated 2dimensional Euclidean space domain M corresponding to time t = 0 and initial conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured tetrahedral mesh in the spacetime domain E 2 × R. For every target time T � 0, our algorithm meshes the spacetime volume M × [0, T] with a bounded number of nondegenerate tetrahedra. A recent extension of Tent Pitcher due to Abedi et al. [2004] adapts the spatial size of spacetime elements in 2D×time to a posteriori estimates of numerical error. Our extension of Tent Pitcher retains the ability to perform adaptive refinement and coarsening of the mesh. We thus obtain the first adaptive nonlinear Tent Pitcher algorithm to build spacetime meshes
Modeling Multiresolution 3D Scalar Fields through Regular Simplex Bisection
"... We review modeling techniques for multiresolution threedimensional scalar fields based on a discretization of the field domain into nested tetrahedral meshes generated through regular simplex bisection. Such meshes are described through hierarchical data structures and their representation is chara ..."
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We review modeling techniques for multiresolution threedimensional scalar fields based on a discretization of the field domain into nested tetrahedral meshes generated through regular simplex bisection. Such meshes are described through hierarchical data structures and their representation is characterized by the modeling primitive used. The primary conceptual distinction among the different approaches proposed in the literature is whether they treat tetrahedra or clusters of tetrahedra, called diamonds, as the modeling primitive. We first focus on representations for the modeling primitive and for nested meshes. Next, we survey the applications of these meshes to modeling multiresolution 3D scalar fields, with an emphasis on interactive visualization. We also consider the relationship of such meshes to octrees. Finally, we discuss directions for further research.