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A Variational Discretization Method for Mean Curvature Flows by the Onsager Principle

Yihe Liu, Xianmin Xu

TL;DR

This work develops a variational discretization for mean curvature flows using the Onsager principle, deriving continuous and discrete evolutions for standard, volume-preserving, and wetting problems. The semi-discrete schemes preserve energy dissipation and are solved with an efficient improved Euler time step, with mesh-quality stabilized by a penalty term. Numerical experiments show robust performance, including second-order spatial convergence for analytic solutions and correct stationary states in wetting, demonstrating the method’s effectiveness for planar curves and its potential for higher dimensions. Overall, the approach provides a principled, energy-stable framework for geometric flows with practical implications for mesh management and multi-physics interfaces.

Abstract

The mean curvature flow describes the evolution of a surface (a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical method for mean curvature flows by using the Onsager principle as an approximation tool. We first show that the mean curvature flow can be derived naturally from the Onsager variational principle. Then we consider a piecewise linear approximation of the curve and derive a discrete geometric flow. The discrete flow is described by a system of ordinary differential equations for the nodes of the discrete curve. We prove that the discrete system preserve the energy dissipation structure in the framework of the Onsager principle and this implies the energy decreasing property. The ODE system can be solved by the improved Euler scheme and this leads to an efficient fully discrete scheme. We first consider the method for a simple mean curvature flow and then extend it to the volume preserving mean curvature flow and also a wetting problem on substrates. Numerical examples show that the method has optimal convergence rate and works well for all the three problems.

A Variational Discretization Method for Mean Curvature Flows by the Onsager Principle

TL;DR

This work develops a variational discretization for mean curvature flows using the Onsager principle, deriving continuous and discrete evolutions for standard, volume-preserving, and wetting problems. The semi-discrete schemes preserve energy dissipation and are solved with an efficient improved Euler time step, with mesh-quality stabilized by a penalty term. Numerical experiments show robust performance, including second-order spatial convergence for analytic solutions and correct stationary states in wetting, demonstrating the method’s effectiveness for planar curves and its potential for higher dimensions. Overall, the approach provides a principled, energy-stable framework for geometric flows with practical implications for mesh management and multi-physics interfaces.

Abstract

The mean curvature flow describes the evolution of a surface (a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical method for mean curvature flows by using the Onsager principle as an approximation tool. We first show that the mean curvature flow can be derived naturally from the Onsager variational principle. Then we consider a piecewise linear approximation of the curve and derive a discrete geometric flow. The discrete flow is described by a system of ordinary differential equations for the nodes of the discrete curve. We prove that the discrete system preserve the energy dissipation structure in the framework of the Onsager principle and this implies the energy decreasing property. The ODE system can be solved by the improved Euler scheme and this leads to an efficient fully discrete scheme. We first consider the method for a simple mean curvature flow and then extend it to the volume preserving mean curvature flow and also a wetting problem on substrates. Numerical examples show that the method has optimal convergence rate and works well for all the three problems.
Paper Structure (21 sections, 3 theorems, 117 equations, 14 figures, 5 tables)

This paper contains 21 sections, 3 theorems, 117 equations, 14 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The Onsager principle as an approximation tool
  3. Mean curvature flow
  4. Derivation of the continuous equation by the Onsager principle
  5. A discrete mean curvature flow by the Onsager principle
  6. Stabilization for uniform distribution of vertexes
  7. Mean curvature flow in high dimensional spaces
  8. Volume preserving mean curvature flow
  9. Derivation of the dynamic equation by the Onsager principle
  10. Discretization of the dynamic equation by the Onsager principle
  11. Wetting problems
  12. Derivation of the dynamic equation by the Onsager principle
  13. Discretization of the dynamic equation by the Onsager principle
  14. Numerical experiments
  15. Mean curvature flow
  16. ...and 6 more sections

Key Result

Theorem 3.1

Suppose $\dot{X}=(\dot{\vec{x}}_{1}^\top,\dot{\vec{x}}_{2}^\top,...,\dot{\vec{x}}_{n}^\top)^\top$ is the solution of the system AV=G, then we have where the equality holds if and only if $\dot{\vec{x}}_i=0\ (0\leq i \leq n)$.

Figures (14)

  • Figure 1: Selection of interpolation points.
  • Figure 2: Schematic diagram of a liquid-vapor-solid system.
  • Figure 3: A non-closed curve whose two endpoints always lie on the same horizontal line.
  • Figure 4: Selection of interpolation points.
  • Figure 5: Evolution of a unit circle under mean curvature flow (n=40).
  • ...and 9 more figures

Theorems & Definitions (5)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1