research-article

Subdivision exterior calculus for geometry processing

Authors:

Fernando de Goes,

Mathieu Desbrun,

Tony DeRoseAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 35, Issue 4

Article No.: 133, Pages 1 - 11

https://doi.org/10.1145/2897824.2925880

Published: 11 July 2016 Publication History

Abstract

This paper introduces a new computational method to solve differential equations on subdivision surfaces. Our approach adapts the numerical framework of Discrete Exterior Calculus (DEC) from the polygonal to the subdivision setting by exploiting the refin-ability of subdivision basis functions. The resulting Subdivision Exterior Calculus (SEC) provides significant improvements in accuracy compared to existing polygonal techniques, while offering exact finite-dimensional analogs of continuum structural identities such as Stokes' theorem and Helmholtz-Hodge decomposition. We demonstrate the versatility and efficiency of SEC on common geometry processing tasks including parameterization, geodesic distance computation, and vector field design.

Supplementary Material

ZIP File (a133-degoes-supp.zip)

Supplemental files.

Download
10.53 MB

MP4 File (a133.mp4)

Download
300.09 MB

References

[1]

Alexa, M., and Wardetzky, M. 2011. Discrete Laplacians on general polygonal meshes. ACM Trans. Graph. 30, 4, Art. 102.

Digital Library

[2]

Arnold, D. N., Falk, R. S., and Winther, R. 2006. Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1--155.

[3]

Auchmann, B., and Kurz, S. 2006. A geometrically defined discrete Hodge operator on simplicial cells. IEEE Trans. on Magnetics 42, 4, 643--646.

[4]

Babuska, I., and Suri, M. 1994. The p and h-p versions of the finite element method: Basic principles and properties. SIAM Review 36, 4, 578--632.

Digital Library

[5]

Back, A., and Sonnendr�cker, E. 2014. Finite element Hodge for spline discrete differential forms: Application to the Vlasov-Poisson system. Appl. Numer. Math. 79, 124--136.

Digital Library

[6]

Barendrecht, P. J. 2013. Isogeometric Analysis for Subdivision Surfaces. Master's thesis, Eindhoven University of Technology.

[7]

Bossavit, A., Ed. 1998. Computational Electromagnetism. Academic Press.

[8]

Bossavit, A. 2000. Computational electromagnetism and geometry. (5): The 'Galerkin Hodge'. J. Japan Soc. Appl. Electromagn. & Mech. 8, 2, 203--9.

[9]

Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., and L�vy, B. 2010. Polygon Mesh Processing. AK Peters.

[10]

Buffa, A., Rivas, J., Sangalli, G., and V�zquez, R. 2011. Isogeometric discrete differential forms in three dimensions. SIAM J. Numer. Anal. 49, 2, 818--844.

Digital Library

[11]

Buffa, A., Sangalli, G., and V�zquez, R. 2014. Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations. J. Comput. Phys. 257, 1291--1320.

Digital Library

[12]

Cirak, F., Scott, M. J., Antonsson, E. K., Ortiz, M., and Schr�der, P. 2002. Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision. Comput. Aided Des. 34, 137--148.

[13]

Cottrell, J. A., Hughes, T. J. R., and Bazilevs, Y. 2009. Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley Publishing.

[14]

Crane, K., de Goes, F., Desbrun, M., and Schr�der, P. 2013. Digital geometry processing with discrete exterior calculus. In ACM SIGGRAPH Courses.

Digital Library

[15]

Crane, K., Weischedel, C., and Wardetzky, M. 2013. Geodesics in heat: A new approach to computing distance based on heat flow. ACM Trans. Graph. 32, 5, Art. 152.

Digital Library

[16]

Dahmen, W. 1986. Subdivision algorithms converge quadratically. J. Comput. Appl. Math. 16, 2, 145--158.

Digital Library

[17]

de Goes, F., Memari, P., Mullen, P., and Desbrun, M. 2014. Weighted triangulations for geometry processing. ACM Trans. Graph. 33, 3, Art. 28.

Digital Library

[18]

Desbrun, M., Meyer, M., Schr�der, P., and Barr, A. H. 1999. Implicit fairing of irregular meshes using diffusion and curvature flow. ACM SIGGRAPH, 317--324.

Digital Library

[19]

Desbrun, M., Kanso, E., and Tong, Y. 2008. Discrete differential forms for computational modeling. In Discrete Differential Geometry, A. I. Bobenko et al. (Eds), vol. 38 of Oberwolfach Seminars. 287--324.

[20]

Elcott, S., Tong, Y., Kanso, E., Schr�der, P., and Desbrun, M. 2007. Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26, 1, Art. 4.

Digital Library

[21]

Fisher, M., Schr�der, P., Desbrun, M., and Hoppe, H. 2007. Design of tangent vector fields. ACM Trans. Graph. 26, 3, Art. 56.

Digital Library

[22]

Frankel, T. 2004. The Geometry of Physics: An Introduction. Cambridge University Press.

[23]

Grinspun, E., Krysl, P., and Schr�der, P. 2002. CHARMS: A simple framework for adaptive simulation. ACM Trans. Graph. 21, 3, 281--290.

Digital Library

[24]

He, L., Schaefer, S., and Hormann, K. 2010. Parameterizing subdivision surfaces. ACM Trans. Graph. 29, 4, Art. 120.

Digital Library

[25]

Hirani, A. 2003. Discrete Exterior Calculus. PhD thesis, Caltech.

Digital Library

[26]

Hughes, T., Cottrell, J., and Bazilevs, Y. 2005. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 39-41, 4135--4195.

[27]

Jüttler, B., Mantzaflaris, A., Perl, R., and Rumpf, M. 2016. On numerical integration in isogeometric subdivision methods for PDEs on surfaces. Comput. Methods Appl. Mech. Eng. 302, 131--146.

[28]

Liu, S., Jacobson, A., and Gingold, Y. 2014. Skinning cubic Bézier splines and Catmull-Clark subdivision surfaces. ACM Trans. Graph. 33, 6, Art. 190.

Digital Library

[29]

Liu, B., Mason, G., Hodgson, J., Tong, Y., and Desbrun, M. 2015. Model-reduced variational fluid simulation. ACM Trans. Graph. 34, 6, Art. 244.

Digital Library

[30]

Loop, C., van Gelder, D., Litke, N., El Guerrab, R., Elmieh, B., and Kraemer, M. 2013. OpenSubdiv from research to industry adoption. In ACM SIGGRAPH Courses.

Digital Library

[31]

Lounsbery, M., DeRose, T. D., and Warren, J. 1997. Multiresolution analysis for surfaces of arbitrary topological type. ACM Trans. Graph. 16, 1, 34--73.

Digital Library

[32]

McCormick, S. F. 1984. Multigrid Methods for Variational Problems: Further Results. SIAM J. Numer. Anal. 21, 2, 255--263.

[33]

Mullen, P., Tong, Y., Alliez, P., and Desbrun, M. 2008. Spectral conformal parameterization. Comput. Graph. Forum 27, 5, 1487--1494.

Digital Library

[34]

Mullen, P., Memari, P., de Goes, F., and Desbrun, M. 2011. HOT: Hodge-optimized triangulations. ACM Trans. Graph. 30, 4, Art. 103.

Digital Library

[35]

Munkres, J. R. 1984. Elements of Algebraic Topology. Addison-Wesley.

[36]

Nguyen, T., Karčiauskas, K., and Peters, J. 2014. A comparative study of several classical, discrete differential and isogeometric methods for solving Poissons equation on the disk. Axioms, 3, 280--299.

[37]

Niessner, M., Loop, C., Meyer, M., and Derose, T. 2012. Feature-adaptive GPU rendering of Catmull-Clark subdivision surfaces. ACM Trans. Graph. 31, 1, Art. 6.

Digital Library

[38]

Riffnaller-Schiefer, A., Augsd�rfer, U. H., and Fellner, D. W. 2015. Isogeometric Analysis for Modelling and Design. In Eurographics (short papers).

[39]

Stam, J. 1998. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. ACM SIGGRAPH, 395--404.

Digital Library

[40]

Stam, J. 2003. Flows on surfaces of arbitrary topology. ACM Trans. Graph. 22, 3, 724--731.

Digital Library

[41]

Strang, G., and Fix, G. 1973. An Analysis of the Finite Element Method. Wellesley-Cambridge.

[42]

Thomaszewski, B., Wacker, M., and Strasser, W. 2006. A consistent bending model for cloth simulation with corota-tional subdivision finite elements. In Symp. Comp. Anim., 107--116.

Digital Library

[43]

Wang, K., Weiwei, Tong, Y., Desbrun, M., and Schr�der, P. 2006. Edge subdivision schemes and the construction of smooth vector fields. ACM Trans. Graph. 25, 3, 1041--1048.

Digital Library

[44]

Wang, K. 2008. A subdivision approach to the construction of smooth differential forms. PhD thesis, Caltech.

Digital Library

[45]

Wardetzky, M., Mathur, S., K�lberer, F., and Grinspun, E. 2007. Discrete Laplace operators: No free lunch. In Symp. Geom. Process., 33--37.

Digital Library

[46]

Warren, J., and Weimer, H. 2001. Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann Publishers Inc.

Digital Library

[47]

Whitney, H. 1957. Geometric Integration Theory. Princeton University Press.

[48]

Zhou, K., Huang, X., Xu, W., Guo, B., and Shum, H.-Y. 2007. Direct manipulation of subdivision surfaces on GPUs. ACM Trans. Graph. 26, 3, Art. 91.

Digital Library

[49]

Zorin, D., and Schr�der, P. 2000. Subdivision for modeling and animation. In ACM SIGGRAPH Courses.

Cited By

King NSu HAanjaneya MRuuth SBatty C(2024)A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry ProcessingACM Transactions on Graphics10.1145/367365243:5(1-26)Online publication date: 9-Aug-2024
https://dl.acm.org/doi/10.1145/3673652
Akalin KFinnendahl USorkine‐Hornung OAlexa M(2024)Mesh Parameterization Meets Intrinsic TriangulationsComputer Graphics Forum10.1111/cgf.1513443:5Online publication date: 31-Jul-2024
https://doi.org/10.1111/cgf.15134
Chen MZuo WChen YZhao OCheng BZhao J(2024)Parametric topology optimization design and analysis of additively manufactured joints in spatial grid structuresEngineering Structures10.1016/j.engstruct.2023.117123300(117123)Online publication date: Feb-2024
https://doi.org/10.1016/j.engstruct.2023.117123
Show More Cited By

Index Terms

Subdivision exterior calculus for geometry processing
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations in finite fields
      2. Discretization

Recommendations

Edge subdivision schemes and the construction of smooth vector fields
SIGGRAPH '06: ACM SIGGRAPH 2006 Papers

Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete ...
Discrete exterior calculus for meshes with concyclic polygons
Abstract
Discrete exterior calculus (DEC) is a numerical method for solving partial differential equations on meshes with applications in computer graphics, numerics, and physical simulations. It discretizes PDEs in a way such that important ...
Graphical abstract
Highlights
- Lifting one of the last mesh requirements for using DEC.
- Allows for using ...
Edge subdivision schemes and the construction of smooth vector fields

Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 35, Issue 4

July 2016

1396 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/2897824

Issue’s Table of Contents

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 11 July 2016

Published in�TOG�Volume 35, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

44
Total Citations
View Citations
672
Total Downloads

Downloads (Last 12 months)52
Downloads (Last 6 weeks)11

Reflects downloads up to 19 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

King NSu HAanjaneya MRuuth SBatty C(2024)A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry ProcessingACM Transactions on Graphics10.1145/367365243:5(1-26)Online publication date: 9-Aug-2024
https://dl.acm.org/doi/10.1145/3673652
Akalin KFinnendahl USorkine‐Hornung OAlexa M(2024)Mesh Parameterization Meets Intrinsic TriangulationsComputer Graphics Forum10.1111/cgf.1513443:5Online publication date: 31-Jul-2024
https://doi.org/10.1111/cgf.15134
Chen MZuo WChen YZhao OCheng BZhao J(2024)Parametric topology optimization design and analysis of additively manufactured joints in spatial grid structuresEngineering Structures10.1016/j.engstruct.2023.117123300(117123)Online publication date: Feb-2024
https://doi.org/10.1016/j.engstruct.2023.117123
Weill–Duflos CCoeurjolly DLachaud J(2024)Digital Calculus Frameworks and Comparative Evaluation of Their Laplace-Beltrami OperatorsDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-57793-2_8(93-106)Online publication date: 15-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57793-2_8
Zhang JDumas JFei YJacobson AJames DKaufman D(2023)Progressive Shell Qasistatics for Unstructured MeshesACM Transactions on Graphics10.1145/361838842:6(1-17)Online publication date: 5-Dec-2023
https://dl.acm.org/doi/10.1145/3618388
Bunge AAlexa MBotsch MKim J(2023)Discrete Laplacians for General Polygonal and Polyhedral MeshesSIGGRAPH Asia 2023 Courses10.1145/3610538.3614620(1-49)Online publication date: 6-Dec-2023
https://dl.acm.org/doi/10.1145/3610538.3614620
Liu HGillespie MChislett BSharp NJacobson ACrane K(2023)Surface Simplification using Intrinsic Error MetricsACM Transactions on Graphics10.1145/359240342:4(1-17)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592403
Chen ZKaufman DSkouras MVouga E(2023)Complex Wrinkle Field EvolutionACM Transactions on Graphics10.1145/359239742:4(1-19)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592397
Kuth BOberberger MChajdas MMeyer Q(2023)Edge‐Friend: Fast and Deterministic Catmull‐Clark Subdivision SurfacesComputer Graphics Forum10.1111/cgf.1486342:8Online publication date: 3-Aug-2023
https://doi.org/10.1111/cgf.14863
Mancinelli CNazzaro GPellacini FPuppo E(2023)b/Surf: Interactive Bézier Splines on Surface MeshesIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2022.317117929:7(3419-3435)Online publication date: 1-Jul-2023
https://doi.org/10.1109/TVCG.2022.3171179
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents