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 well-known and widely-used operators.

We present an iterative algorithm to transform a given planar triangle mesh into a well-centered one by moving the interior vertices while keeping the connectivity fixed. A well-centered planar triangulation is one in which all angles are acute. Our approach is based on minimizing a certain energy that we propose. Well-centered 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 space-time meshing. For some connectivities with no well-centered configurations, we present preprocessing steps that increase the possibility of finding a well-centered 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.

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 non-homogeneous 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.

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

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-planar and non-convex 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.

We undertake a study of the local properties of 2-Gabriel meshes: manifold triangle meshes each of whose faces has an open Euclidean diametric ball that contains no mesh vertices. We show that, under mild constraints on the dihedral angles, such meshes are Delaunay meshes: the open geodesic circumdisk of each face contains no mesh vertex. The analysis is done by means of the Delaunay edge flipping algorithm and it reveals the details of the distinction between these two mesh structures. In particular we observe that the obstructions which prohibit the existence of Gabriel meshes as homeomorphic representatives of smooth surfaces do not hinder the construction of Delaunay meshes. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry

We construct a multi-scale model of shape of surfaces in 3D space based on the heat kernel associated with the Laplace-Beltrami 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 one-year period. Index Terms — Shape space, heat kernel, hippocampal atrophy, Alzheimer’s disease, ADNI.

An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, together with some other observations, are used to describe restrictions on the local combinatorial structure of simplicial meshes in which every simplex is well-centered. In particular, it is shown that in a 3-well-centered (2well-centered) tetrahedral mesh there are at least 7 (9) edges incident to each interior vertex, and these bounds are sharp. Moreover, it is shown that, in stark contrast to the 2-dimensional analog, where there are exactly two vertex links that prevent a well-centered triangle mesh in R 2, there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in R 3. 1

Numerical simulation provides a effective tool for studying both the spatial and temporal nature of acoustic field on 3D or 4D timespace. The paper deals with the description of discrete exterior calculus scheme for the wave equation. This method can be directly implemented on manifold, which is the generation of finite difference time domain method from flat space to curved space.

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.