Computational p-Willmore Flow with Conformal Penalty

Anthony Gruber, Eugenio Aulisa

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The unsigned p-Willmore functional introduced in the work of Mondino [2011] generalizes important geometric functionals, which measure the area and Willmore energy of immersed surfaces. Presently, techniques from the work of Dziuk [2008] are adapted to compute the first variation of this functional as a weak-form system of equations, which are subsequently used to develop a model for the p-Willmore flow of closed surfaces in R3. This model is amenable to constraints on surface area and enclosed volume and is shown to decrease the p-Willmore energy monotonically. In addition, a penalty-based regularization procedure is formulated to prevent artificial mesh degeneration along the flow; inspired by a conformality condition derived in the work of Kamberov et al. [1996], this procedure encourages angle-preservation in a closed and oriented surface immersion as it evolves. Following this, a finite-element discretization of both procedures is discussed, an algorithm for running the flow is given, and an application to mesh editing is presented.

Original languageEnglish
Article number3369387
JournalACM Transactions on Graphics
Issue number5
StatePublished - Sep 2020


  • Willmore flow
  • conformal geometry
  • mesh editing
  • surface fairing
  • surface remeshing

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