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65
Nodal discontinuous Galerkin methods on graphics processors,
 J. Comp. Phys.,
, 2009
"... Abstract Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. ..."
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Cited by 52 (4 self)
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Abstract Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. Lately, another property of DG has been growing in importance: The majority of a DG operator is applied in an elementlocal way, with weak penaltybased elementtoelement coupling. The resulting locality in memory access is one of the factors that enables DG to run on offtheshelf, massively parallel graphics processors (GPUs). In addition, DG's highorder nature lets it require fewer data points per represented wavelength and hence fewer memory accesses, in exchange for higher arithmetic intensity. Both of these factors work significantly in favor of a GPU implementation of DG. Using a single US$400 Nvidia GTX 280 GPU, we accelerate a solver for Maxwell's equations on a general 3D unstructured grid by a factor of 40 to 60 relative to a serial computation on a currentgeneration CPU. In many cases, our algorithms exhibit full use of the device's available memory bandwidth. Example computations achieve and surpass 200 gigaflops/s of net applicationlevel floating point work. In this article, we describe and derive the techniques used to reach this level of performance. In addition, we present comprehensive data on the accuracy and runtime behavior of the method.
Aggressive Tetrahedral Mesh Improvement
 In Proc. of the 16th Int. Meshing Roundtable
, 2007
"... Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh cleanup. ” Our goal is to aggressively optimize the worst t ..."
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Cited by 39 (4 self)
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Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh cleanup. ” Our goal is to aggressively optimize the worst tetrahedra, with speed a secondary consideration. Mesh optimization methods often get stuck in bad local optima (poorquality meshes) because their repertoire of mesh transformations is weak. We employ a broader palette of operations than any previous mesh improvement software. Alongside the best traditional topological and smoothing operations, we introduce a topological transformation that inserts a new vertex (sometimes deleting others at the same time). We describe a schedule for applying and composing these operations that rarely gets stuck in a bad optimum. We demonstrate that all three techniques—smoothing, vertex insertion, and traditional transformations—are substantially more effective than any two alone. Our implementation usually improves meshes so that all dihedral angles are between 31 ◦ and 149 ◦ , or (with a different objective function) between 23 ◦ and 136 ◦. 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.
Virtual Video Camera: ImageBased Viewpoint Navigation Through Space and Time
"... Figure 1: Viewpoint navigation space: time and camera directions span our navigation space. Each cube represents one video frame. The navigation space is partitioned into tetrahedra with video frames as vertices. Tetrahedral edges denote correspondence fields between video frames. Our virtual video ..."
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Cited by 21 (11 self)
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Figure 1: Viewpoint navigation space: time and camera directions span our navigation space. Each cube represents one video frame. The navigation space is partitioned into tetrahedra with video frames as vertices. Tetrahedral edges denote correspondence fields between video frames. Our virtual video camera view is interpolated by warping and compositing the four video frames of the enclosing tetrahedron (blue) in realtime. 2
TETRAHEDRAL MESH GENERATION FROM VOLUMETRIC BINARY AND GRAY SCALE IMAGES
"... We report a general purpose mesh generator for creating finiteelement surface or volumetric mesh from 3D binary or grayscale medical images. This toolbox incorporates a number of existing free mesh processing utilities and enables researchers to perform a range of mesh processing tasks for imageb ..."
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Cited by 19 (0 self)
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We report a general purpose mesh generator for creating finiteelement surface or volumetric mesh from 3D binary or grayscale medical images. This toolbox incorporates a number of existing free mesh processing utilities and enables researchers to perform a range of mesh processing tasks for imagebased mesh generation, including raw image processing, surface mesh extraction, surface resampling, and multiscale/adaptive tetrahedral mesh generation. We also implemented robust algorithms for meshing opensurfaces and subregion labeling. Atomic meshing utilities for each processing step can be accessed with simple interfaces, which can be streamlined or executed independently. The toolbox is compatible with Matlab or GNU Octave. We demonstrate the applications of this toolbox for meshing a range of challenging geometries including complex vessel network, human brain and breast. Index Terms — Mesh generation, Finite element analysis, Medical image
FeaturePreserving Adaptive Mesh Generation for Molecular Shape Modeling and Simulation
, 2007
"... We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a n ..."
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Cited by 18 (7 self)
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We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: featurepreservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvment of meshes in a broad range of applications.
PoissonNernstPlanck Equations for Simulating Biomolecular DiffusionReaction Processes I: Finite Element Solutions
, 2010
"... In this paper we developed accurate finite element methods for solving 3D PoissonNernstPlanck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Ner ..."
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Cited by 16 (2 self)
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In this paper we developed accurate finite element methods for solving 3D PoissonNernstPlanck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the NernstPlanck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, wellposed PNP equations. An inexactNewton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an AdamsBashforthCrankNicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the NernstPlanck equation, and theoretically proved that the transformed formulation is always associated with an illconditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molec
Edge transformations for improving mesh quality of marching cubes
 IEEE TVCG
"... Abstract—Marching Cubes is a popular choice for isosurface extraction from regular grids due to its simplicity, robustness, and efficiency. One of the key shortcomings of this approach is the quality of the resulting meshes, which tend to have many poorly shaped and degenerate triangles. This issue ..."
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Cited by 15 (5 self)
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Abstract—Marching Cubes is a popular choice for isosurface extraction from regular grids due to its simplicity, robustness, and efficiency. One of the key shortcomings of this approach is the quality of the resulting meshes, which tend to have many poorly shaped and degenerate triangles. This issue is often addressed through postprocessing operations such as smoothing. As we demonstrate in experiments with several data sets, while these improve the mesh, they do not remove all degeneracies and incur an increased and unbounded error between the resulting mesh and the original isosurface. Rather than modifying the resulting mesh, we propose a method to modify the grid on which Marching Cubes operates. This modification greatly increases the quality of the extracted mesh. In our experiments, our method did not create a single degenerate triangle, unlike any other method we experimented with. Our method incurs minimal computational overhead, requiring at most twice the execution time of the original Marching Cubes algorithm in our experiments. Most importantly, it can be readily integrated in existing Marching Cubes implementations and is orthogonal to many Marching Cubes enhancements (particularly, performance enhancements such as outofcore and acceleration structures). Index Terms—Meshing, marching cubes. Ç 1
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSONBOLTZMANN EQUATION
"... ABSTRACT. We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear PoissonBoltzmann equation (PBE). We first examine the twoterm regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of t ..."
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Cited by 14 (8 self)
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ABSTRACT. We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear PoissonBoltzmann equation (PBE). We first examine the twoterm regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the PoissonBoltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this twoterm regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L ∞ estimates to establish quasiorthogonality. To provide a highquality geometric model as input to the AFEM algorithm, we also describe a class of featurepreserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.
SOT: Compact representation for tetrahedral meshes
"... The Corner Table (CT) promoted by Rossignac et al. provides a simple and efficient representation of triangle meshes, storing 6 integer references per triangle (3 vertex references in the V table and 3 references to opposite corners in the O table that accelerate access to adjacent triangles). The C ..."
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Cited by 12 (4 self)
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The Corner Table (CT) promoted by Rossignac et al. provides a simple and efficient representation of triangle meshes, storing 6 integer references per triangle (3 vertex references in the V table and 3 references to opposite corners in the O table that accelerate access to adjacent triangles). The Compact Half Face (CHF) proposed by Lage et al. extends CT to tetrahedral meshes, storing 8 references per tetrahedron (4 in the V table and 4 in the O table). We call it the Vertex Opposite Table (VOT) and propose a sorted variation, SVOT, which does not require any additional storage and yet provides, for each vertex, a reference to an incident corner from which an incident tetrahedron may be recovered and the star of the vertex may be traversed at a constant cost per visited element. We use a set of powerful wedgebased operators for querying and traversing the mesh. Finally, inspired by tetrahedral mesh encoding techniques used by Weiler et al. and by Szymczak and Rossignac, we propose our Sorted O Table (SOT) variation, which eliminates the V table completely and hence reduces storage requirements by 50 % to only 4 references and 9 bits per tetrahedron, while preserving the vertextoincidentcorner references and supporting our wedge operators with a linear average cost.