Discrete 2-Tensor Fields on Triangulations

Geometry processing has made ample use of discrete representations of tangent vector fields and antisymmetric tensors (i.e., forms) on triangulations. Symmetric 2‐tensors, while crucial in the definition of inner products and elliptic operators, have received only limited attention. They are often d...

Celý popis

Uloženo v:
Podrobná bibliografie
Vydáno v:Computer graphics forum Ročník 33; číslo 5; s. 13 - 24
Hlavní autoři: de Goes, Fernando, Liu, Beibei, Budninskiy, Max, Tong, Yiying, Desbrun, Mathieu
Médium: Journal Article
Jazyk:angličtina
Vydáno: Oxford Blackwell Publishing Ltd 01.08.2014
Témata:
Analysis
Anisotropy
Calculus
Categories and Subject Descriptors (according to ACM CCS)
Computer graphics
Computer Graphics [I.3.5]: Computational Geometry and Object Modeling-Curve and surface representations
Exact solutions
Mathematical analysis
Operators
Representations
Studies
Tensors
Topological manifolds
Triangulation
ISSN:0167-7055, 1467-8659
On-line přístup:Získat plný text
Tagy: Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
Popis
Shrnutí:Geometry processing has made ample use of discrete representations of tangent vector fields and antisymmetric tensors (i.e., forms) on triangulations. Symmetric 2‐tensors, while crucial in the definition of inner products and elliptic operators, have received only limited attention. They are often discretized by first defining a coordinate system per vertex, edge or face, then storing their components in this frame field. In this paper, we introduce a representation of arbitrary 2‐tensor fields on triangle meshes. We leverage a coordinate‐free decomposition of continuous 2‐tensors in the plane to construct a finite‐dimensional encoding of tensor fields through scalar values on oriented simplices of a manifold triangulation. We also provide closed‐form expressions of pairing, inner product, and trace for this discrete representation of tensor fields, and formulate a discrete covariant derivative and a discrete Lie bracket. Our approach extends discrete/finite‐element exterior calculus, recovers familiar operators such as the weighted Laplacian operator, and defines discrete notions of divergence‐free, curl‐free, and traceless tensors–thus offering a numerical framework for discrete tensor calculus on triangulations. We finally demonstrate the robustness and accuracy of our operators on analytical examples, before applying them to the computation of anisotropic geodesic distances on discrete surfaces.
Bibliografie:istex:5CC72AC73270E57D46F3813BF96275B97DB70046
ark:/67375/WNG-QHFB7MCH-9
ArticleID:CGF12427
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ObjectType-Article-1
ObjectType-Feature-2
content type line 23
ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.12427