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

TL;DR

This paper develops a finite element numerical algorithm based on ADMM to minimize complex surface and line energies in liquid crystal defect studies, accounting for obstacles and boundary conditions.

Contribution

It introduces a novel finite element approach with ADMM for minimizing energies involving surfaces, boundaries, and obstacles in liquid crystal models.

Findings

Demonstrates the algorithm's effectiveness on various inclusion shapes.

Reveals complex minimizing configurations with physical relevance.

Provides insights into defect structures in nematic liquid crystals.

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.