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 R 3 . 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.

中文翻译：

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具有保形惩罚的计算 p-Willmore 流

Mondino [2011] 的工作中引入的无符号 p-Willmore 泛函概括了重要的几何泛函，它测量了浸没表面的面积和 Willmore 能量。目前，来自 Dziuk [2008] 工作的技术适用于将该泛函的第一个变体计算为弱形式方程组，随后用于开发 R 中闭合曲面的 p-Willmore 流模型3. 该模型适用于对表面积和封闭体积的约束，并显示出单调降低 p-Willmore 能量。此外，制定了基于惩罚的正则化程序，以防止沿流动的人工网格退化；受 Kamberov 等人的工作中得出的保形条件的启发。[1996]，这个过程鼓励在封闭和定向的表面浸入中保持角度，因为它的发展。在此之后，讨论了这两个过程的有限元离散化，给出了运行流程的算法，并提出了在网格编辑中的应用。