On the numerical approximation of hyperbolic mean curvature flows for surfaces

TL;DR

The paper advances the numerical treatment of two hyperbolic mean curvature flows for surfaces in ${\mathbb R}^3$, one with $g(s)=1$ (Gurtin–Podio-Guidugli type) and the other with $g(s)=1+\tfrac{1}{2}s$ (LeFloch–Smoczyk type). It develops a normal-parametrized, parametric formulation, a finite element method for the general case, and a finite difference scheme for axisymmetric surfaces, including careful initialization and stable discretizations that preserve key energy-like quantities. The numerical experiments include convergence tests on spheres and axisymmetric nonspherical geometries, and demonstrations of finite-time singularity formation such as curvature blow-up and torus pinch, validating the methods against analytic solutions where available. The results provide practical tools for simulating these geometric hyperbolic flows and offer insight into their nontrivial singularity behavior, with potential implications for problems in crystal interfaces and related geometric dynamics.

Abstract

The paper addresses the numerical approximation of two variants of hyperbolic mean curvature flow of surfaces in $\mathbb R^3$. For each evolution law we propose both a finite element method, as well as a finite difference scheme in the case of axially symmetric surfaces. We present a number of numerical simulations, including convergence tests as well as simulations suggesting the onset of singularities.

Figures (12)

• Figure 1: Sketch of $\Gamma(t)$ and $\mathcal{S}(t)$, as well as the unit vectors $e_1$, $e_2$ and $e_3$.
• Figure 2: The flow \ref{['eq:Gurtinbeta0']} starting from a unit sphere. Above we show the evolutions of $|\mathcal{S}^m_h|$ (solid) and $A^m$ (dashed) over time for $V_0=0$, $V_0=1$ and $V_0=-1$. The final times for these computations are $T=0.85$, $T=1.7$ and $T=0.5$, respectively. Below we show the corresponding evolutions of $1/{\mathcal{K}}^m_\infty$ over time.
• Figure 3: The flow \ref{['eq:Gurtinbeta0']}, with $V_0=0$, starting from a $2:1:1$ cigar. Above we visualize $\mathcal{S}^m_h$ from \ref{['eq:fea_b']} at times $t=0,0.25,T=0.5$ in two ways: a frontal view and a cut through the $x_1$-$x_2$ plane. Below we show $x^m$ from \ref{['eq:app:fd']} with $g(s)=1$ at times $t=0,0.25,T$, as well as a plot with $|\mathcal{S}^m_h|$ (solid) and $A^m$ (dashed), and a plot of $1/\mathcal{K}^m_\infty$ (right) over time.
• Figure 4: The flow \ref{['eq:Gurtinbeta0']}, with $V_0=1$, starting from a $2:1:1$ cigar. Above we visualize $\mathcal{S}^m_h$ from \ref{['eq:fea_b']} at times $t=0,0.25,0.5,0.75,1,T=1.1$ in two ways: a frontal view and a cut through the $x_1$-$x_2$ plane. Below we show $x^m$ from \ref{['eq:app:fd']} with $g(s)=1$ at times $t=0,0.25,0.5,0.75,1,T$, as well as a plot with $|\mathcal{S}^m_h|$ (solid) and $A^m$ (dashed), and a plot of $1/\mathcal{K}^m_\infty$ (right) over time.
• Figure 5: The flow \ref{['eq:Gurtinbeta0']}, with $V_0=0$, starting from a torus with radii $R=2$ and $r=1$. Above we visualize $\mathcal{S}^m_h$ from \ref{['eq:fea_b']} at times $t=0,1,T=1.3$ in two ways: a frontal view and a cut through the $x_1$-$x_2$ plane. Below we show $x^m$ from \ref{['eq:app:fd']} with $g(s)=1$ at times $t=0,1,T$, as well as a plot with $|\mathcal{S}^m_h|$ (solid) and $A^m$ (dashed), and a plot of $1/\mathcal{K}^m_\infty$ (right) over time.

Theorems & Definitions (4)

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