# Computational p-Willmore Flow with Conformal Penalty

## TL;DR

This paper develops a computational model for the p-Willmore flow of surfaces in 3D, incorporating conformal penalty regularization to preserve mesh quality and applying finite-element methods for practical mesh editing.

## Contribution

It introduces a novel p-Willmore flow model with conformal penalty regularization and finite-element discretization for surface evolution and mesh editing.

## Key findings

- The model decreases p-Willmore energy monotonically.
- Conformal penalty effectively prevents mesh degeneration.
- Application demonstrated in mesh editing tasks.

## Abstract

The unsigned p-Willmore functional introduced in \cite{mondino2011} generalizes important geometric functionals which measure the area and Willmore energy of immersed surfaces. Presently, techniques from \cite{dziuk2008} 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 $\mathbb{R}^3$. This model is amenable to constraints on surface area and enclosed volume, and is shown to decrease the p-Willmore energy monotonically over time. In addition, a penalty-based regularization procedure is formulated to prevent artificial mesh degeneration along the flow; inspired by a conformality condition derived in \cite{kamberov1996}, this procedure encourages angle-preservation in a closed and oriented surface immersion as it evolves. Following this, a finite-element discretization of both systems is discussed, and an application to mesh editing is presented.