A level set formulation for willmore flow.pdf

A level set formulation for Willmore ?ow M. Droske, M. Rumpf ? June 2, 2004 Abstract Alevel set formulation of Willmore ?owis derived using the gradient ?owper- spective. Starting from single embedded surfaces and the corresponding gradient ?ow, the metric is generalized to sets of level set surfaces using the identi?cation of normal velocities and variations of the level set function in time via the level set equation. The approach in particular allows to identify the natural dependent quantities of the derived variational formulation. Furthermore, spatial and tempo- ral discretization are discussed and some numerical simulations are presented. AMS Subject Classi?cations:35K55, 53C44, 65M60, 74S05 1 Introduction LetMbe ad-dimensional surface embedded inIR d+1and denote byxthe identity map onM. Consider the energy e[M] := 12 Z M h 2dA wherehis the mean curvature onM, i. e.,his the sum of the principle curvatures onM. The correspondingL 2-gradient ?ow – the Willmore ?ow – is given by the geometric evolution problem [34, 32, 21] ? tx(t) = ? M(t)h(t)n(t) +h(t) (kS(t)k 22 ? 12 h(t) 2)n(t), which de?nes for a given initial surfaceM 0a family of surfacesM(t)fort≥0with M(0) =M 0. HereS(t)denotes the shape operator onM(t),n(t)the normal ?eld onM(t), andk·k2the Frobenius norm on the space of endomorphisms on the tangent bundleT M(t). Nowwe considerM(t)to be given implicitly as a speci?c level set of a corresponding functionφ(t) : ?→IRfor a domain??IR d+1. Thus, the evolution ofM(t)can be described by an evolution ofφ(t). In our case, the level set equation? tφ(t) + ?Numerical Analysis and puting, Lotharstr. 65, University of Duisburg, 47057 Duisburg, Germany. e-mail:{droske,rumpf}***@- 1 k?φ(t)kV= 0(cf. the book of Osher and Paragios [25] for a detailed study), withV being the speed of propagation of the level set ofφ(t), turns into the equation ? tφ+k?φ(t)k μ? Mh+h(t) (kS(t)k 22? 12 h(t) 2) ?= 0, (1) with initial dataφ(0) =φ 0. Hereφ 0implicitly