A finite element approach for minimizing line and surface energies arising in the study of singularities in liquid crystals

TL;DR

This work tackles a Plateau-like energy minimization problem for a two-dimensional surface $T\subset\mathbb{R}^3$ that includes area outside a particle $E$, a surface density on the inclusion boundary, and a boundary-length penalty weighted by $\beta$ via $\mathcal{E}_0(T)=\mathbb{M}(T\Omega)+\int_{\mathcal{M}}|\nu_3|\,d\mu_{T\mathcal{M}}+\beta\mathbb{M}(\partial T+\Gamma)$. Reformulating in the space of currents and using a curl-based representation allows convexification and a total-variation-type minimization framework, which is discretized with finite elements and solved by ADMM. The authors establish Γ-convergence of the discrete scheme to the continuous limit, implement the method in FEniCS with mesh adaptation, and demonstrate rich defect structures across spherical, peanut, donut, and croissant particle geometries, revealing how defects and energies depend on $\beta$ and particle orientation. They introduce a symmetry-breaking shift $d_\Gamma$ to obtain unique minimizers and provide physical interpretations for colloidal particles in nematic liquid crystals, highlighting the framework’s potential for analyzing line and surface energies in complex geometries. Overall, the paper delivers a robust numerical pipeline for coupled surface-boundary energy minimization with obstacle constraints, applicable to defect topology in LC systems and related variational problems. Key contributions include a novel FE-ADMM approach for a convexified limit energy, Γ-convergence guarantees, and detailed numerics across multiple particle geometries that connect geometry, topology, and defect structures.

Abstract

Motivated by a problem originating in the study of defect structures in nematic liquid crystals, we describe and study a numerical algorithm for the resolution of a Plateau-like problem. The energy contains the area of a two-dimensional surface $T$ and the length of its boundary $\partial T$ reduced by a prescribed curve to make our problem non-trivial. We additionally include an obstacle $E$ for $T$ and pose a surface energy on $E$. We present an algorithm based on the Alternating Direction Method of Multipliers that minimizes a discretized version of the energy using finite elements, generalizing existing TV-minimization methods. We study different inclusion shapes demonstrating the rich structure of minimizing configurations and provide physical interpretation of our findings for colloidal particles in nematic liquid crystal.

Figures (20)

• Figure 1: \ref{['fig:num_isoperim_ineq-1layer']} Visualization of choices of $e_n$ and $\tilde{e}_{n,i}$ in the proof of Lemma \ref{['lem:num_isoperim_ineq']}. After finitely many steps an edge in $A_\delta$ is reached. \ref{['fig:num_isoperim_ineq-2layer']} Possible "bubble" in $\mathcal{M}_\delta$ having a thickness of two layers which allows for diverging mass while edges stay localized in one part of $\mathcal{M}_\delta\setminus A_\delta$. Such configurations are not possible in our situation due to the assumption of a one-layer thickness of $\mathcal{M}_\delta$.
• Figure 2: Left: Mesh after removing the interior cells. The colors represent the values of $p_\mathrm{max}$: $10^5$ inside the particle (red), $|\nu_3|$ in the boundary layer (shades of blue), $1$ otherwise (grey). Right: Magnification of a part of the mesh.
• Figure 3: Different isosurfaces of a terminal configuration obtained for $\beta=1$ after $4000$ iterations with $d_\Gamma=0$.
• Figure 4: Observed defect configurations: Saturn ring (a) for small values of $\beta$ and dipole (b) for large $\beta$. The line $S$ is indicated in red and $T$ in blue.
• Figure 5: Energy of minimizers for different values of $\beta$ around a sphere of radius $1$. The mesh consists of around $449\, 000$ cells of size $h=0.03$ around the particle surface.

Theorems & Definitions (13)

• Proposition 2.1: Minimizers have finite mass
• proof
• Remark 2.2: Relation to total variation minimization
• Remark 2.3: Differential forms
• Definition 3.1: Finite element spaces
• Proposition 3.2
• proof
• Remark 3.3: Relation to discrete total variation minimization
• Lemma 3.4
• proof