Related papers: Delaunay Hodge Star
In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial…
Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but…
Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant functions on unstructured meshes satisfying an orthogonality…
I present a 3D advancing-front mesh refinement algorithm that generates a constrained Delaunay mesh for any piecewise linear complex (PLC) and extend this algorithm to produce truly Delaunay meshes for any PLC. First, as in my recently…
Suppose $C$ is a compact, $n$-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an…
We compute the convex hull in $\mathbb{C}^2$ of an arbitrary finite subgroup of ${\mathbb{C}^*}^2$. The combinatorics are dictated by continued fractions in a natural way. This reproves a theorem of Smilansky, with a slightly stronger…
We provide a brief overview on the application of the exterior calculus of differential forms to the ab initio formulation of field theories on random simplicial lattices. In this framework, discrete analogues of the exterior derivative and…
Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space $\mathbb{R}^m$ triangulates the convex hull of $P$, provided that $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunay…
A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean…
For a given point set $S$ in a plane, we develop a distributed algorithm to compute the $\alpha-$shape of $S$. $\alpha-$shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for…
We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The…
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…
Analyzing electromagnetic fields in complex, multi-material environments presents substantial computational challenges. To address these, we propose a hybrid numerical method that couples discrete exterior calculus (DEC) with surface…
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star,…
Delaunay protection is a measure of how far a Delaunay triangulation is from being degenerate. In this short paper we study the protection properties and other quality measures of the Delaunay triangulations of a family of lattices that is…
The finite volume Laplacian can be defined in all dimensions and is a natural way to approximate the operator on a simplicial mesh. In the most general setting, its definition with orthogonal duals may require that not all volumes are…
Triangle meshes remain the most popular data representation for surface geometry. This ubiquitous representation is essentially a hybrid one that decouples continuous vertex locations from the discrete topological triangulation.…
Scott Wilson introduced the notion of combinatorial Hodge star operators on a compact oriented triangulated manifold $M$, which act on the singular cohomology ring of $M$. Such an operator depends on both a triangulation $\mathscr K$ of $M$…
Discrete exterior calculus offers a coordinate--free discretization of exterior calculus especially suited for computations on meshes over curved manifolds. The discretization of the wedge product, that would be compatible with discrete…
Maxwell's equations can be solved numerically in space manifold and the time by discrete exterior calculus as a kind of lattice gauge theory.Since the stable conditions of this method is very severe restriction, we combine the implicit…