P-Willmore flow with quasi-conformal mesh regularization
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The p-Willmore functional introduced by Gruber et al. unifies and extends important geometric functionals which measure the area, total mean curvature, and Willmore energy of immersed surfaces. Techniques of Dziuk and Elliott are adapted to compute the first variation of the p-Willmore functional, which is then used to develop a finite-element model for the p-Willmore flow of closed surfaces in R^3 that is amenable to constraints on surface area and enclosed volume. Further, a minimization-based regularization procedure is formulated to prevent mesh degeneration along the flow. Inspired by a conformality condition due to Kamberov et al., this procedure is generally-applicable to all 2-D meshes and ensures that a given surface mesh remains nearly conformal as it evolves, at the expense of first-order accuracy in the flow. Finally, discretization of the p-Willmore flow is discussed, where some basic results on consistency and numerical stability are demonstrated and a fully-automated algorithm is presented for implementing the closed surface p-Willmore flow with mesh regularization.
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