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17
Discrete Laplace operators: No free lunch
, 2007
"... Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set ..."
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Cited by 30 (0 self)
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Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends wellknown and widelyused operators.
Discrete Laplacians on General Polygonal Meshes
"... While the theory and applications of discrete Laplacians on triangulated surfaces are well developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non ..."
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Cited by 7 (2 self)
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While the theory and applications of discrete Laplacians on triangulated surfaces are well developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing nonplanar and nonconvex polygons. Our construction is guided by closely mimicking structural properties of the smooth Laplace–Beltrami operator. Among other features, our construction leads to an extension of the widely employed cotan formula from triangles to polygons. Besides carefully laying out theoretical aspects, we demonstrate the versatility of our approach for a variety of geometry processing applications, embarking on situations that would have been more difficult to achieve based on geometric Laplacians for simplicial meshes or purely combinatorial Laplacians for general meshes. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations, Geometric algorithms, languages, and systems.
Wellcentered Planar Triangulation  An Iterative Approach
, 2007
"... We present an iterative algorithm to transform a given planar triangle mesh into a wellcentered one by moving the interior vertices while keeping the connectivity fixed. A wellcentered planar triangulation is one in which all angles are acute. Our approach is based on minimizing a certain energy ..."
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Cited by 7 (4 self)
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We present an iterative algorithm to transform a given planar triangle mesh into a wellcentered one by moving the interior vertices while keeping the connectivity fixed. A wellcentered planar triangulation is one in which all angles are acute. Our approach is based on minimizing a certain energy that we propose. Wellcentered meshes have the advantage of having nice orthogonal dual meshes (the dual Voronoi diagram). This may be useful in scientific computing, for example, in discrete exterior calculus, in covolume method, and in spacetime meshing. For some connectivities with no wellcentered configurations, we present preprocessing steps that increase the possibility of finding a wellcentered configuration. We show the results of applying our energy minimization approach to small and large meshes, with and without holes and gradations. Results are generally good, but in certain cases the method might result in inverted elements.
Triangulation of simple 3D shapes with wellcentered tetrahedra, in: R.V. Garimella (Ed
 Proceedings of the 17th International Meshing Roundtable
, 2008
"... Abstract. A completely wellcentered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and other fields. We show how to triangulate simple do ..."
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Cited by 6 (3 self)
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Abstract. A completely wellcentered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and other fields. We show how to triangulate simple domains using completely wellcentered tetrahedra. The domains we consider here are space, infinite slab, infinite rectangular prism, cube, and regular tetrahedron. We also demonstrate single tetrahedra with various combinations of the properties of dihedral acuteness, 2wellcenteredness, and 3wellcenteredness. 1.
Numerical method for Darcy flow derived using Discrete Exterior Calculus
"... Abstract. We derive a numerical method for Darcy flow, hence also for Poisson’s equation in first order form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is its discretization on simplicial complexes such as triangle ..."
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Cited by 3 (1 self)
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Abstract. We derive a numerical method for Darcy flow, hence also for Poisson’s equation in first order form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is its discretization on simplicial complexes such as triangle and tetrahedral meshes. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete nonhomogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. Our method requires the use of meshes in which each simplex contains its circumcenter. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solution in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this paper. We also include a discussion of the boundary condition in terms of exterior calculus. 1.
Circumcentric dual cells with negative area
, 2009
"... The circumcentric dual complex associated with a triangulation is becoming a structure of interest in discrete differential geometry. It arises naturally in formulations of discrete exterior calculus and, in the two dimensional case that concerns us here, it provides an elegant interpretation of dis ..."
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Cited by 2 (0 self)
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The circumcentric dual complex associated with a triangulation is becoming a structure of interest in discrete differential geometry. It arises naturally in formulations of discrete exterior calculus and, in the two dimensional case that concerns us here, it provides an elegant interpretation of discrete Laplace operators based on the cotangent formula. If the primal triangulation is Delaunay, then the circumcentric dual complex is the Voronoi diagram of the vertices. On the other hand, if a primal edge is not locally Delaunay, the length of the corresponding dual edge will be negative. In many applications this does not present a problem. However in this note we draw attention to the possibility that the dual cell to a primal vertex may have negative area, and we discuss some of the implications. We review the definition of circumcentric dual cells and provide simple explicit constructions of triangle configurations in which a primal vertex has a circumcentric dual cell with negative area. 1 Circumcentric dual cells and their area
MAPPING HIPPOCAMPAL ATROPHY WITH A MULTISCALE MODEL OF SHAPE
"... We construct a multiscale model of shape of surfaces in 3D space based on the heat kernel associated with the LaplaceBeltrami operator. The model is applied to the analysis of longitudinal neuroimaging data collected by the Alzheimer’s Disease Neuroimaging Initiative. We use measures of shape defo ..."
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We construct a multiscale model of shape of surfaces in 3D space based on the heat kernel associated with the LaplaceBeltrami operator. The model is applied to the analysis of longitudinal neuroimaging data collected by the Alzheimer’s Disease Neuroimaging Initiative. We use measures of shape deformation energy to quantify, compare and create maps of regional changes in hippocampal shape in normal aging, progression of Alzheimer’s disease and mild cognitive impairment over a oneyear period. Index Terms — Shape space, heat kernel, hippocampal atrophy, Alzheimer’s disease, ADNI.
Implicit Euler scheme.
, 908
"... The implicit Euler scheme of time variable and discrete exterior calculus can be united to find an unconditional stable approach, which is called implicit discrete exterior calculus. This technique for solving Maxwell’s equations in time domain is discussed, which provides flexibility in numerical c ..."
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The implicit Euler scheme of time variable and discrete exterior calculus can be united to find an unconditional stable approach, which is called implicit discrete exterior calculus. This technique for solving Maxwell’s equations in time domain is discussed, which provides flexibility in numerical computing on manifold. For some problems, it takes much less computational time to use the implicit method with larger time steps, even taking into account that one needs to solve equations at each step. This algorithm has been implemented on Java development plateform for simulating TE/M waves in vacuum.
Discrete Exterior Calculus and Computation Electromagnetism
, 908
"... Computational electromagnetism is concerned with the numerical study of Maxwell equations. By choosing a discrete Gaussian measure on prism lattice, we use discrete exterior calculus and lattice gauge theory to construct discrete Maxwell equations in vacuum case. We implement this scheme on Java dev ..."
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Computational electromagnetism is concerned with the numerical study of Maxwell equations. By choosing a discrete Gaussian measure on prism lattice, we use discrete exterior calculus and lattice gauge theory to construct discrete Maxwell equations in vacuum case. We implement this scheme on Java development plateform to simulate the behavior of electromagnetic waves.
Computation Electromagnetism and Discrete Exterior Calculus
, 908
"... Figure 1: Simulation of Guassian pluse on happy buddha by DEC Computational electromagnetism is concerned with the numerical study of Maxwell equations. By choosing a discrete Gaussian measure on prism lattice, we use discrete exterior calculus and lattice gauge theory to construct discrete Maxwell ..."
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Figure 1: Simulation of Guassian pluse on happy buddha by DEC Computational electromagnetism is concerned with the numerical study of Maxwell equations. By choosing a discrete Gaussian measure on prism lattice, we use discrete exterior calculus and lattice gauge theory to construct discrete Maxwell equations in vacuum case. We implement this scheme on Java development plateform to simulate the behavior of electromagnetic waves.