Projection-Free Method for the Full Frank-Oseen Model of Liquid Crystals

TL;DR

The paper develops a projection-free gradient flow for the full Frank-Oseen energy of nematic liquid crystals, discretized with finite elements and applied on general shape-regular meshes. It proves Gamma-convergence of the discrete minimizers to the continuous problem under the only requirement that Frank constants are positive, and extends the framework to include a fixed magnetic field. The method maintains energy decrease and controls constraint violations without requiring mesh acuteness, enabling robust modeling of defects. Computational results demonstrate defect arrangements and field-induced transitions (Fréedericksz) as the Frank constants and magnetic field vary, validating both the theory and the practicality of the approach for complex geometries. The work advances numerical analysis and simulation of liquid crystals by enabling accurate, stable computations for the full Frank-Oseen model with defects and external fields.}

Abstract

Liquid crystals are materials that experience an intermediate phase where the material can flow like a liquid, but the molecules maintain an orientation order. The Frank-Oseen model is a continuum model of a liquid crystal. The model represents the liquid crystal orientation as a vector field and posits that the vector field minimizes some elastic energy subject to a pointwise unit length constraint, which is a nonconvex constraint. Previous numerical methods in the literature assumed restrictions on the physical constants or had regularity assumptions that ruled out point defects, which are important physical phenomena to model. We present a finite element discretization of the full Frank-Oseen model and a projection free gradient flow algorithm for the discrete problem in the spirit of Bartels (2016). We prove Gamma-convergence of the discrete to the continuous problem: weak convergence of subsequences of discrete minimizers and convergence of energies. We also prove that the gradient flow algorithm has a desirable energy decrease property. Our analysis only requires that the physical constants are positive, which presents challenges due to the additional nonlinearities from the elastic energy.

Figures (8)

• Figure 1: By searching in tangent directions and damping with parameter $\tau$ yields $|\mathbf{n}^{k}_h(z)+\tau d_t\mathbf{n}^{k+1}_h(z)|=|\mathbf{n}^k_h(z)|^2+\tau^2|d_t\mathbf{n}^{k+1}_h(z)|^2$.
• Figure 2: Plot of discrete unit length constraint errors $\Vert I_h[|\mathbf{n}_h^\infty|^2-1]\Vert_{L^p(\Omega)}$ for $p=1,\infty$. Note that Theorem \ref{['thm:energy_decrease']} and Corollary \ref{['cor:L1-constraint']} imply that $\textsf{err}_1(\mathbf{n}_h^{\infty})\lesssim h$ and $\textsf{err}_\infty(\mathbf{n}_h^{\infty})\lesssim 1$ provided $\tau h^{-1}\leq C_\textrm{stab}$. The computational results corroborate the theory.
• Figure 3: Slice of the projected director field at $\{x=0\}$. Initial configuration (left) and computed minimizer (right) with $k_1=k_3=1$ and $k_2=.1$ and numerical parameters $h=\tau2^{-9/2}, \varepsilon= 10^{-3}/2$ (the stopping parameter of Algorithm \ref{['alg:grad-flow']}). Twist is preferred to splay and bend, in agreement with Helein's condition \ref{['eq:helene-condition']} for $k_2$ sufficiently small relative to $k_1$ and $k_3$.
• Figure 4: Initial and final projected director fields at $\{z=0\}$ from Algorithm \ref{['alg:grad-flow']} with $k_i=1$ for $i=1,2,3$, $h=\tau = 1/16$ and $\varepsilon = 10^{-4}$. This corresponds to the one-constant case $c_0=1, c_1=c_2=c_3=0$ in \ref{['eq:E-tilde2']}. A degree 2 defect for the initial condition splits into two degree 1 defects.
• Figure 5: Final projected director fields at $\{z=0\}$ from Algorithm \ref{['alg:grad-flow']} and $k_1=1,k_2=.75$ and $k_3 = 1,3,5$. The equilibrium configuration changes from two bending degree 1 defects for $k_3=1$ to two degree 1 splay like defects for $k_3=5$.

Theorems & Definitions (38)

• Lemma 2.1: saddle splay
• Proposition 1: modified energy
• Proposition 2: explicit form of $E$
• proof
• Lemma 2.2: properties of $E$
• Remark 1: modified energy $E$
• Remark 2: simplifications of $E$
• Lemma 2.3: discrete unit length constraint
• Lemma 2.4: discrete Sobolev inequality
• Lemma 3.1: recovery sequence