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Discrete Willmore Flow

Bobenko, Alexander I. and Schröder, Peter (2005) Discrete Willmore Flow. In: SIGGRAPH '05 ACM SIGGRAPH 2005 Courses. ACM , New York, NY, Art. No. 5. https://resolver.caltech.edu/CaltechAUTHORS:20160725-114846575

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Abstract

The Willmore energy of a surface, ∫(H^2 - K) dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity. As such this energy and its associated gradient flow play an important role in digital geometry processing, geometric modeling, and physical simulation. In this paper we consider a discrete Willmore energy and its flow. In contrast to traditional approaches it is not based on a finite element discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries of the underlying continuous theory in the discrete setting. We derive the relevant gradient expressions including a linearization (approximation of the Hessian), which are required for non-linear numerical solvers. As examples we demonstrate the utility of our approach for surface restoration, n-sided hole filling, and non-shrinking surface smoothing.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1145/1198555.1198664DOIPaper
http://dl.acm.org/citation.cfm?doid=1198555.1198664PublisherPaper
ORCID:
AuthorORCID
Schröder, Peter0000-0002-0323-7674
Additional Information:© 2005 ACM. This work was supported in part by NSF (DMS-0220905, DMS-0138458, ACI-0219979), DFG (Research Center MATHEON “Mathematics for Key Technologies,” Berlin), DOE (W-7405-ENG-48/B341492), nVidia, the Center for Integrated Multiscale Modeling and Simulation, Alias, and Pixar. Special thanks to Kevin Bauer, Oscar Bruno, Mathieu Desbrun, Ilja Friedel, Cici Koenig, Nathan Litke, and Fabio Rossi.
Funders:
Funding AgencyGrant Number
NSFDMS-0220905
NSFDMS-0138458
NSFACI-0219979
Deutsche Forschungsgemeinschaft (DFG)UNSPECIFIED
Department of Energy (DOE)W-7405-ENG-48/B341492
nVidiaUNSPECIFIED
Center for Integrated Multiscale Modeling and SimulationUNSPECIFIED
AliasUNSPECIFIED
PixarUNSPECIFIED
Subject Keywords:Geometric Flow; Discrete Differential Geometry; Willmore Energy; Variational Surface Modeling; Digital Geometry Processing
Classification Code:CR Categories: G.1.8 [Numerical Analysis]: Partial Differential Equations—Elliptic equations; Parabolic equations; Finite difference methods; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid and object representa
Record Number:CaltechAUTHORS:20160725-114846575
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160725-114846575
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69194
Collection:CaltechAUTHORS
Deposited By: Kristin Buxton
Deposited On:25 Jul 2016 20:35
Last Modified:09 Mar 2020 13:19

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