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55
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 187 (15 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
A Pliant Method for Anisotropic Mesh Generation
"... A new algorithm for the generation of anisotropic, unstructured triangular meshes in two dimensions is described. Inputs to the algorithm are the boundary geometry and a metric that specifies the desired element size and shape as a function of position. The algorithm is an example of what we call p ..."
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Cited by 86 (2 self)
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A new algorithm for the generation of anisotropic, unstructured triangular meshes in two dimensions is described. Inputs to the algorithm are the boundary geometry and a metric that specifies the desired element size and shape as a function of position. The algorithm is an example of what we call pliant mesh generation. It first constructs the constrained Delaunay triangulation of the domain, then iteratively smooths, refines, and retriangulates. On each iteration, a node is selected at random, it is repositioned according to attraction/repulsion with its neighbors, the neighborhood is retriangulated, and nodes are inserted or deleted as necessary. All operations are done relative to the metric tensor. This simple method generates high quality meshes whose elements conform well to the requested shape metric. The method appears particularly well suited to surface meshing and viscous flow simulations, where stretched triangles are desirable, and to timedependent remeshing problems.
Quality Mesh Generation in Higher Dimensions
, 1996
"... We consider the problem of triangulating a ddimensional region. Our mesh generation algorithm, called QMG, is a quadtreebased algorithm that can triangulate any polyhedral region including nonconvex regions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation ..."
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Cited by 50 (7 self)
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We consider the problem of triangulating a ddimensional region. Our mesh generation algorithm, called QMG, is a quadtreebased algorithm that can triangulate any polyhedral region including nonconvex regions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles. Finally, our algorithm is guaranteed never to overrefine the domain in the sense that the number of simplices produced by QMG is bounded above by a factor times the number produced by any competing algorithm, where the factor depends on the aspect ratio bound satisfied by the competing algorithm. The QMG algorithm has been implemented in C++ and is used as a mesh generator for the finite element method.
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 49 (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.
Isotropic remeshing of surfaces: A local parameterization approach
 In Proceedings of 12th International Meshing Roundtable
, 2003
"... We present a method for isotropic remeshing of arbitrary genus surfaces. The method is based on a mesh adaptation process, namely, a sequence of local modifications performed on a copy of the original mesh, while referring to the original mesh geometry. The algorithm has three stages. In the first s ..."
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Cited by 45 (4 self)
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We present a method for isotropic remeshing of arbitrary genus surfaces. The method is based on a mesh adaptation process, namely, a sequence of local modifications performed on a copy of the original mesh, while referring to the original mesh geometry. The algorithm has three stages. In the first stage the required number or vertices are generated by iterative simplification or refinement. The second stage performs an initial vertex partition using an areabased relaxation method. The third stage achieves precise isotropic vertex sampling prescribed by a given density function on the mesh. We use a modification of Lloyd’s relaxation method to construct a weighted centroidal Voronoi tessellation of the mesh. We apply these iterations locally on small patches of the mesh that are parameterized into the 2D plane. This allows us to handle arbitrary complex meshes with any genus and any number of boundaries. The efficiency and the accuracy of the remeshing process is achieved using a patchwise parameterization technique.
A global approach to automatic solution of jigsaw puzzles
 In Proc. Conf. Computational Geometry
, 2002
"... Abstract. We present a new algorithm for automatically solving jigsaw puzzles by shape alone. The algorithm can solve more difficult puzzlesthan could be solved before, without the use of backtracking or branchandbound. The algorithm can handle puzzles in which pieces border more than four neighbo ..."
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Cited by 40 (0 self)
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Abstract. We present a new algorithm for automatically solving jigsaw puzzles by shape alone. The algorithm can solve more difficult puzzlesthan could be solved before, without the use of backtracking or branchandbound. The algorithm can handle puzzles in which pieces border more than four neighbors, and puzzles with as many as 200 pieces. The algorithm follows the overall strategy of previous algorithms but applies a number of new techniques, such as robust fiducial points, “highestconfidencefirst”search, and frequent global reoptimization of partial solutions. 1
Isotropic Surface Remeshing
, 2003
"... This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a userspecified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image ..."
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Cited by 35 (3 self)
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This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a userspecified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi tessellation in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly lowpass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.
Interactive modeling of topologically complex geometric detail
 ACM Trans. Graph
, 2004
"... Volume textures aligned with a surface can be used to add topologically complex geometric detail to objects in an efficient way, while retaining an underlying simple surface structure. Adding a volume texture to a surface requires more than a conventional twodimensional parameterization: a part of ..."
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Cited by 29 (1 self)
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Volume textures aligned with a surface can be used to add topologically complex geometric detail to objects in an efficient way, while retaining an underlying simple surface structure. Adding a volume texture to a surface requires more than a conventional twodimensional parameterization: a part of the space surrounding the surface has to be parameterized. Another problem with using volume textures for adding geometric detail is the difficulty in rendering implicitly represented surfaces, especially when they are changed interactively. In this paper we present algorithms for constructing and rendering volumetextured surfaces. We demonstrate a number of interactive operations that these algorithms enable.
Flips in Planar Graphs
, 2006
"... We review a selection of results concerning edge flips in triangulations and planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another. We st ..."
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Cited by 29 (5 self)
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We review a selection of results concerning edge flips in triangulations and planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another. We study the problem both from a combinatorial perspective (where only a combinatorial embedding of the graph is specified) and a geometric perspective (where the graph is embedded in the plane, vertices are points and edges are straightline segments). We highlight both the similarities and differences of the two settings, describe many extensions and generalizations, outline several applications and mention open problems.
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 20 (10 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.