A minimizing movements approach for crystalline eikonal-curvature flows of spirals

TL;DR

The paper addresses numerical simulation of spiral evolution under crystalline curvature flow with driving force in planar domains. It develops a minimizing movements framework, reinterpreting the evolution via a level-set embedding using a preassigned phase $\theta$ and centers $a_j$, and avoids the need for signed-distance functions by solving a polyhedral total-variation type problem with a Split Bregman solver. Existence and uniqueness of the discrete evolving sequence are established, and the method is augmented with a polyhedral shrinkage operation to handle polygonal anisotropies; stability is ensured through velocity cut-offs to manage fattening. Numerical experiments demonstrate close agreement with front-tracking benchmarks for single spirals, and certify the method’s ability to handle merging, interlaced motions, and illusory patterns across multiple anisotropies, highlighting its robustness and versatility for crystal-growth-like spiral dynamics.

Abstract

We propose an algorithm for evolving spiral curves on a planar domain by normal velocities depending on the so-called crystalline curvatures. The algorithm uses a minimizing movement approach and relies on a special level set method for embedding the spirals. We present numerical simulations and comparisons demonstrating the efficacy of the proposed numerical algorithm.

Figures (17)

• Figure 1: Spiral step function $H_{\Sigma_{\textup{d}}}$.
• Figure 2: Graph of $|H_{\Sigma_{\textup{d}}} - H_{\Sigma_h}|$. In the left panel, the dashed and solid lines indicates $\Sigma_{\textup{d}} (t)$ and $\Sigma_h (t)$, respectively.
• Figure 3: Overlapping snapshots of two spirals at $t = 0, 0.4, 0.6$ and $0.8$, computed using the front-tracking model (dashed curves) and Algorithm \ref{['algo: w/o dist']} (solid curves), with $\Delta x = 1/150$ ($s=3$).
• Figure 4: Graphs of the distances $\mathcal{D} (t)$ and the area difference $\mathcal{A} (t)$ between the spirals computed using the front-tracking model and Algorithm \ref{['algo: w/o dist']}. The lines with $\blacksquare$, $\circ$, $\bullet$, $\triangle$, $\blacktriangle$ correspond the cases of $s=1,2,3,4,5$, respectively.
• Figure 5: The number of outer loops used in the simulations involving three different values of $s$. The horizontal axis reveals the number of time steps performed in the evolution, while the vertical axis shows the number of outer loops.

Theorems & Definitions (15)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Theorem 2.6
• proof
• Theorem 2.7
• Remark 3.1