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Stanford CS 468 - Lecture Notes

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Eurographics Symposium on Geometry Processing (2005)M. Desbrun, H. Pottmann (Editors)Discrete Willmore FlowAlexander I. Bobenko1and Peter Schröder21TU Berlin2CaltechAbstractThe Willmore energy of a surface,R(H2− K)dA, as a function of mean and Gaussian curvature, captures thedeviation of a surface from (local) sphericity. As such this energy and its associated gradient flow play an impor-tant role in digital geometry processing, geometric modeling, and physical simulation. In this paper we considera discrete Willmore energy and its flow. In contrast to traditional approaches it is not based on a finite ele-ment discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries ofthe underlying continuous theory in the discrete setting. We derive the relevant gradient expressions including alinearization (approximation of the Hessian), which are required for non-linear numerical solvers. As exampleswe demonstrate the utility of our approach for surface restoration, n-sided hole filling, and non-shrinking surfacesmoothing.Categories and Subject Descriptors (according to ACM CCS): G.1.8 [Numerical Analysis]: Partial Differential Equa-tions; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.6.8 [Simulation and Model-ing]: Types of Simulation.Keywords: Geometric Flow; Discrete Differential Geometry; Willmore Energy; Variational Surface Modeling;Digital Geometry Processing.1. IntroductionThe Willmore energy of a surface S ⊂ R3is given asEW(S) =ZS(H2− K) dA = 1/4ZS(κ1− κ2)2dA,where κ1and κ2denote the principal curvatures, H =1/2(κ1+ κ2) and K = κ1κ2the mean and Gaussian curva-ture respectively, and dA the surface area element. Immer-sions of surfaces which minimize this energy are of greatinterest in several areas:• Theory of surfaces: the Willmore energy of a surfaceis conformally invariant [Bla29] making it an importantfunctional in the study of conformal geometry [Wil00];• Geometric modeling: for compact surfaces with fixedboundary a minimizer of EW(S) is also a minimizerof total curvatureRSκ12+ κ22dA which is a stan-dard functional in variationally optimal surface model-ing [ LP88, WW94, Gre94];• Physical modeling: thin flexible structures are governedby a surface energy of the formE(S) =ZSα + β (H − H0)2− γK dA,the so-called Canham-Helfrich model [Can70, Hel73](H0denotes the “spontaneous” curvaturewhich plays an important role in thin-shells [GKS02, BMF03, GHDS03]). For α = H0= 0,β = γ the Canham-Helfrich model reduces to theWillmore energy.In all of these application areas one typically deals with theassociated geometric flow˙S = −∇E(S),(time derivatives are denoted by an overdot) which drivesthe surface to a minimum of the potential energy given byE(S). In the theory of surfaces as well as in geometric mod-eling one is interested in critical points of E(S). In physicalmodeling the solution shape is characterized by a balance ofexternal and internal forces. In this setting the internal forcesare a function of the Willmore gradient.c The Eurographics Association 2005.A.I. Bobenko & P. Schröder / Discrete Willmore FlowContributions In this paper we explore a novel, discreteWillmore energy [Bob05] and introduce the associated geo-metric flow for piecewise linear, simplicial, 2-manifoldmeshes. In contrast to earlier approaches the discrete flowis not defined through assemblies of lower level discrete op-erators, nor does the numerical treatment employ operatorsplitting approaches. Instead the discrete Willmore energy,defined as a function of the vertices of a triangle mesh, isused directly in a non-linear numerical solver to affect theassociated flow as well as solve the static problem. Since thediscrete formulation has the same symmetries as the continu-ous problem, i.e., it is Möbius invariant, the associated prop-erties, such as invariance under scaling, carry over exactlyto the discrete setting of meshes. To deal effectively withboundaries we introduce appropriate boundary conditions.These include position and tangency constraints as well asa free boundary condition. We demonstrate the method withsome examples from digital geometry processing and geo-metric modeling.1.1. Related WorkWe distinguish here between discrete geometric flows, i.e.,flows based on discrete analogues of continuous differentialgeometry quantities, and those based on discretizations ofcontinuous systems. The guiding principle in the construc-tion of the former is the preservation of symmetries of theoriginal continuous system, while the latter is based on tra-ditional finite element or finite difference approaches whichin general do not preserve the underlying symmetries. Thereis also a broad body of literature which uses linearized ver-sions of the typically non-linear geometric functionals. Suchapproaches are not based on intrinsic geometric properties(e.g., replacing curvatures with second derivatives) but ratherdepend on the particular parameterization chosen. For thisreason we will not further consider them here.Discrete Flows In the context of mesh based geometricmodeling a number of discrete flows have been consid-ered. For example, Desbrun et al. [DMSB99] used meancurvature flow ( α = 1, β = γ = H0= 0) to achieve de-noising of geometry. Pinkall and Polthier [PP93] used arelated approach, area minimizing flow, to construct dis-crete minimal surfaces. Cr itical points of the area functionalalso play an important role in the construction of discreteharmonic functions [DCDS97], their use in parameteriza-tions [EDD∗95, DMA02], and the construction of confor-mal structures for discrete surfaces [Mer01, GY03]. Sincethe underlying “membrane” energy is second order only, itcannot accomodate G1continuity conditions at the bound-ary of the domain. These are important in geometric mod-eling for the construction of tangent plane continuous sur-faces. Fourth order flows on the other hand can accomo-date position and tangency conditions at the boundary. Per-haps the simplest fourth order flow is surface diffusion, i.e.,flow by the Laplace-Beltrami operator of mean curvature,˙S = −∆SH. Such discrete flows were studied by Schneiderand Kobbelt [SK01], Xu et al. [XPB05], and Yoshizawa andBelyaev [YB02]. In each case the approach was based ontaking the square of a discrete Laplace-Beltrami operatorcombined with additional simplifications to ease implemen-tation. Unfortunately surface diffusion flow


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