16 research outputs found
Delaunay Hodge Star
We define signed dual volumes at all dimensions for circumcentric dual
meshes. We show that for pairwise Delaunay triangulations with mild boundary
assumptions these signed dual volumes are positive. This allows the use of such
Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge
star operator can now be correctly defined for such meshes. This operator is
crucial for DEC and is a diagonal matrix with the ratio of primal and dual
volumes along the diagonal. A correct definition requires that all entries be
positive. DEC is a framework for numerically solving differential equations on
meshes and for geometry processing tasks and has had considerable impact in
computer graphics and scientific computing. Our result allows the use of DEC
with a much larger class of meshes than was previously considered possible.Comment: Corrected error in Figure 1 (columns 3 and 4) and Figure 6 and a
formula error in Section 2. All mathematical statements (theorems and lemmas)
are unchanged. The previous arXiv version v3 (minus the Appendix) appeared in
the journal Computer-Aided Desig
Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
We develop exterior calculus approaches for partial differential equations on
radial manifolds. We introduce numerical methods that approximate with spectral
accuracy the exterior derivative , Hodge star , and their
compositions. To achieve discretizations with high precision and symmetry, we
develop hyperinterpolation methods based on spherical harmonics and Lebedev
quadrature. We perform convergence studies of our numerical exterior derivative
operator and Hodge star operator
showing each converge spectrally to and . We show how the
numerical operators can be naturally composed to formulate general numerical
approximations for solving differential equations on manifolds. We present
results for the Laplace-Beltrami equations demonstrating our approach.Comment: 22 pages, 13 figure
A geometric description of Discrete Exterior Calculus for general triangulations
We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML).Peer Reviewe
A geometric description of Discrete Exterior Calculus for general triangulations
We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general
triangulations, relying only on Vector Calculus and Matrix Algebra. We present
DEC numerical solutions of the Poisson equation and compare them against those
found using the Finite Element Method with linear elements (FEML).Comment: We made some corrections, included one more example and added several
figure
Weighted Triangulations for Geometry Processing
In this article we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion
of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primal-dual
structure (through combinations of angles, areas, and lengths) and, in the process, uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing