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...
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| Published in: | Computer graphics forum Vol. 33; no. 5; pp. 13 - 24 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
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Oxford
Blackwell Publishing Ltd
01.08.2014
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | 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. |
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| AbstractList | 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. 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. [PUBLICATION ABSTRACT] |
| Author | Budninskiy, Max Desbrun, Mathieu Liu, Beibei Tong, Yiying de Goes, Fernando |
| Author_xml | – sequence: 1 givenname: Fernando surname: de Goes fullname: de Goes, Fernando organization: Caltech – sequence: 2 givenname: Beibei surname: Liu fullname: Liu, Beibei organization: MSU – sequence: 3 givenname: Max surname: Budninskiy fullname: Budninskiy, Max organization: Caltech – sequence: 4 givenname: Yiying surname: Tong fullname: Tong, Yiying organization: MSU – sequence: 5 givenname: Mathieu surname: Desbrun fullname: Desbrun, Mathieu organization: Caltech |
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| Copyright | 2014 The Author(s) Computer Graphics Forum © 2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2014 The Eurographics Association and John Wiley & Sons Ltd. |
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| References_xml | – reference: Hu H.C.: Some variational principles in elasticity and plasticity. Acta Phys. Sin. 10, 3 (1954), 259-290. 2, 4 – reference: Hildebrandt K., Polthier K., Wardetzky M.: On convergence of metric and geometric properties of polyhedral surfaces. Geometriae Dedicata 123, 1 (2006), 89-112. 9 – reference: Arnold D.N., Winther R.: Nonconforming mixed elements for elasticity. Math. Models Methods Appl. Sci. 13, 3 (2003), 295-307. 2 – reference: Elcott S., Tong Y., Kanso E., Schröder P., Desbrun M.: Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 1 (2007). 2 – reference: Narain R., Samii A., O'Brien J.F.: Adaptive anisotropic remeshing for cloth simulation. ACM Trans. Graph. 31, 6 (2012). 2 – reference: Springborn B., Schröder P., Pinkall U.: Conformal equivalence of triangle meshes. ACM Trans. Graph. 27, 3 (2008). 2 – reference: Ben-Chen M., Butscher A., Solomon J., Guibas L.: On discrete Killing vector fields & patterns on surfaces. Comp. Graph. Forum 29, 5 (2010), 1701-1711. 2, 4, 9, 11 – reference: Kovacs D., Myles A., Zorin D.: Anisotropic quadrangulation. Computer Aided Geometric Design 28, 8 (2011), 449-462. 2, 4 – reference: Laidlaw D., Weickert J.: Visualization and Processing of Tensor Fields: Advances and Perspectives, 1st ed. Springer Publishing Company, Incorporated, 2009. 2 – reference: Fisher M., Schröder P., Desbrun M., Hoppe H.: Design of tangent vector fields. ACM Trans. Graph. 26, 3 (2007). 1 – reference: Meier D.L.: Constrained transport algorithms for numerical relativity. i. development of a finite-difference scheme. The Astrophysical Journal 595 (2003), 980-991. 2 – reference: Pinkall U., Polthier K.: Computing discrete minimal surfaces & their conjugates. Exp. Math. 2, 1 (1993), 15-36. 7, 12 – reference: Arnold D.N., Awanou G., Winther R.: Nonconforming tetrahedral mixed finite elements for elasticity. Math. Models Methods Appl. 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| SubjectTerms | 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 |
| Title | Discrete 2-Tensor Fields on Triangulations |
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