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Isosurface stuffing: Fast tetrahedral meshes with good dihedral angles
 Special issue on Proceedings of SIGGRAPH 2007
, 2007
"... org/10.1145/1239451.1239508. Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the ..."
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Cited by 59 (3 self)
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org/10.1145/1239451.1239508. Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the fi rst page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specifi c permission and/or a fee. Permissions may be
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.
A fast solver for the Stokes equations with distributed forces in complex geometries
 J. Comput. Phys
"... We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded ..."
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Cited by 41 (10 self)
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We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded Boundary Integral method, is based on Anita Mayo’s work for the Poisson’s equation: “The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions”, SIAM Journal on Numerical Analysis, 21 (1984), pp. 285–299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström’s method. The rectangular domain problem is discretized by finite elements for a velocitypressure formulation with equal order interpolation bilinear elements (£¥ ¤£¥ ¤). Stabilization is used to circumvent the ¦¨§�©������� � condition for the pressure space. For the integral equations, fast matrix vector multiplications are achieved via an ���¨���� � algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsify lowrank blocks. The regular grid solver is a Krylov method (Conjugate Residuals) combined with an optimal twolevel Schwartzpreconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates. Key Words: Stokes equations, fast solvers, integral equations, doublelayer potential, fast multipole methods, embedded domain methods, immersed interface methods, fictitious
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
Star splaying: an algorithm for repairing Delaunay triangulations and convex hulls
 SCG ’05: PROCEEDINGS OF THE TWENTYFIRST ANNUAL ACM SYMPOSIUM ON COMPUTATIONAL GEOMETRY, ACM
, 2005
"... Star splaying is a generaldimensional algorithm that takes as input a triangulation or an approximation of a convex hull, and produces the Delaunay triangulation, weighted Delaunay triangulation, or convex hull of the vertices in the input. If the input is “nearly Delaunay” or “nearly convex” in a ..."
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Cited by 26 (0 self)
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Star splaying is a generaldimensional algorithm that takes as input a triangulation or an approximation of a convex hull, and produces the Delaunay triangulation, weighted Delaunay triangulation, or convex hull of the vertices in the input. If the input is “nearly Delaunay” or “nearly convex” in a certain sense quantified herein, and it is sparse (i.e. each input vertex adjoins only a constant number of edges), star splaying runs in time linear in the number of vertices. Thus, star splaying can be a fast first step in repairing a highquality finite element mesh that has lost the Delaunay property after its vertices have moved in response to simulated physical forces. Star splaying is akin to Lawson’s edge flip algorithm for converting a triangulation to a Delaunay triangulation, but it works in any dimensionality.
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. Thos ..."
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Cited by 24 (4 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.
Meshing volumes bounded by smooth surfaces
 Proc. 14th Internat. Meshing Roundtable
, 2005
"... This paper introduces a threedimensional mesh generation algorithm for domains bounded by smooth surfaces. The algorithm combines a Delaunaybased surface mesher with a Ruppertlike volume mesher, to get a greedy algorithm that samples the interior and the boundary of the domain at once. The algori ..."
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Cited by 23 (2 self)
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This paper introduces a threedimensional mesh generation algorithm for domains bounded by smooth surfaces. The algorithm combines a Delaunaybased surface mesher with a Ruppertlike volume mesher, to get a greedy algorithm that samples the interior and the boundary of the domain at once. The algorithm constructs provablygood meshes, it gives control on the size of the mesh elements through a userde ned sizing eld, and it guarantees the accuracy of the approximation of the domain boundary. A noticeable feature is that the domain boundary has to be known only through an oracle that can tell whether a given point lies inside the object and whether a given line segment intersects the boundary. This makes the algorithm generic enough to be applied to a wide variety of objects, ranging from domains de ned by implicit surfaces to domains de ned by levelsets in 3D greyscaled images or by pointset surfaces. 1
Generaldimensional constrained delaunay and constrained regular triangulations i: Combinatorial properties
 Discrete and Computational Geometry
, 2005
"... Twodimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nicely shaped” triangles that optimize several criteria, and they are easy to construct and update. The pr ..."
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Cited by 21 (2 self)
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Twodimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nicely shaped” triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions. The main contributions are rigorous, theorytested definitions of constrained Delaunay triangulations and piecewise linear complexes (geometric domains that incorporate nonconvex faces with “internal ” boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.
Meshing 3D domains bounded by piecewise smooth surfaces
"... This report provides an algorithm to mesh 3D domains bounded by piecewise smooth surfaces. The algorithm may handle as well subdivisions of the domain forming non manifold surfaces. The boundaries and constraints are assumed to be described as a complex formed by a set of vertices, a set of curved ..."
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Cited by 20 (2 self)
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This report provides an algorithm to mesh 3D domains bounded by piecewise smooth surfaces. The algorithm may handle as well subdivisions of the domain forming non manifold surfaces. The boundaries and constraints are assumed to be described as a complex formed by a set of vertices, a set of curved segments and a set of surface patches. Each curve segment is assumed to be a piece of a closed smooth curves and each surface patch is assumed to be included in a smooth surface without boundary. The meshing algorithm is a Delaunay refinement and it uses the notion of restricted Delaunay triangulation to approximate the input curved segments and surfaces patches. The algorithm is shown to end up with a set of vertices whose restricted Delaunay triangulation to any input feature forms an homeomorphic and accurate approximation of this feature. The algorithm also provides guarantees on the size and shape of facets approximating the input surface patches and on the size and shape of the tetrahedra in the domain. In its actual state the algorithm suffers from a severe angular restriction on input constraints. It basically assumes that linear subspaces which are tangent to distinct input features on a common point form angles measuring at least 90 degrees.
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.