An energy-stable parametric finite element method for the Willmore flow in three dimensions

TL;DR

This work tackles the numerical simulation of Willmore flow and related curvature-dependent geometric gradient flows for closed surfaces in 3D. It introduces an energy-stable parametric finite element framework built on two new geometric identities and a weak evolution equation for the mean curvature, enabling a fully discrete scheme that dissipates the Willmore energy. The method extends to general functionals $W(Γ)=\int_{Γ} f(\mathcal{H}) \, dA$, including a first PFEM for Gauss curvature flow, and incorporates a tangential-motion control mechanism to maintain mesh quality. Numerical experiments confirm robust energy dissipation, convergence near second order, and meaningful morphologies toward known equilibrium shapes, while highlighting challenges in long-time simulations on complex geometries and suggesting future work on Helfrich-type models.

Abstract

This work develops novel energy-stable parametric finite element methods (ES-PFEM) for the Willmore flow and curvature-dependent geometric gradient flows of surfaces in three dimensions. The key to achieving the energy stability lies in the use of two novel geometric identities: (i) a reformulated variational form of the normal velocity field, and (ii) incorporation of the temporal evolution of the mean curvature into the governing equations. These identities enable the derivation of a new variational formulation. By using the parametric finite element method, an implicit fully discrete scheme is subsequently developed, which maintains the energy dissipative property at the fully discrete level. Based on the ES-PFEM, comprehensive insights into the design of ES-PFEM for general curvature-dependent geometric gradient flows and a new understanding of mesh quality improvement in PFEM are provided. In particular, we develop the first PFEM for the Gauss curvature flow of surfaces. Furthermore, a tangential velocity control methodology is applied to improve the mesh quality and enhance the robustness of the proposed numerical method. Extensive numerical experiments confirm that the proposed method preserves energy dissipation properties and maintain good mesh quality in the surface evolution under the Willmore flow.

Figures (7)

• Figure 1: Numerical errors $e^h$ of different initial surfaces at different time $t$: (a) unit sphere, and (b) ellipsoid with $a = 2$ and $b = 1$. Recall the time step size $\tau = \frac{1}{180}h^2$.
• Figure 2: Discretized Willmore energy $W^m$ of different initial surfaces: (a) ellipsoid with $a = 2$ and $b = 1$, $h = 0.075$; (b) torus with $R = \sqrt{2}$ and $r = \sqrt{2}/2$, $h = 0.099$.
• Figure 3: Iteration number of \ref{['iteration']} for different initial surfaces: ellipsoid with $a = 2$ and $b = 1$, $h = 0.15$ and $\tau =0.0005$; (b) torus with $R = \sqrt{2}$ and $r = \sqrt{2}/2$, $h =0.14$ and $\tau = 0.0005$.
• Figure 4: Morphological evolution of an ellipsoid with $a = 3$ and $b = 1$ under the Willmore flow at (a) $t = 0$, (b) $t = 0.2$, (c) $t = 0.5$, and (d) $t = 2$. The mesh size and time step size is chosen as $h = 0.25$ and $\tau = 0.01$ (Figures have been scaled).
• Figure 5: Morphological evolution of a tours with $R = \sqrt{2}$ and $r = \sqrt{2}/2$ under the Willmore flow at (a) $t = 0$, (b) $t = 0.05$, (c) $t = 0.1$, and (d) $t = 0.4$. The mesh size and time step size is chosen as $h = 0.15$ and $\tau = 0.0001$ (Figures have been scaled).

Theorems & Definitions (26)

• lemma 1
• proof
• lemma 2
• proof
• remark 1
• lemma 3: A new transport theorem
• proof
• remark 2
• theorem 1: Energy dissipation
• proof