The Diamond Laplace for Polygonal and Polyhedral Meshes

We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge togeth...

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Bibliographic Details
Published in:Computer graphics forum Vol. 40; no. 5; pp. 217 - 230
Main Authors: Bunge, A., Botsch, M., Alexa, M.
Format: Journal Article
Language:English
Published: Oxford Blackwell Publishing Ltd 01.08.2021
Subjects:
Apexes
CCS Concepts
Computer graphics
Computing methodologies → Mesh models
DDFV
Diamonds
Discrete Differential Geometry
Discrete Laplace Operator
Divergence
Mathematics of computing → Discretization
Operators (mathematics)
Polygons
ISSN:0167-7055, 1467-8659
Online Access:Get full text
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Summary:We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge together with its corresponding primal element — an edge for surface meshes and a face for volumetric meshes. We call the operator resulting from taking the divergence of the gradient Diamond Laplacian. Additional vertices used for the construction are represented as affine combinations of the original vertices, so that the Laplacian operator maps from values at vertices to values at vertices, as is common in geometry processing applications. The construction is local, exactly the same for all types of meshes, and results in a symmetric negative definite operator with linear precision. We show that the accuracy of the Diamond Laplacian is similar or better compared to other discretizations. The greater versatility and generally good behavior come at the expense of an increase in the number of non‐zero coefficients that depends on the degree of the mesh elements.
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ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.14369