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A Convergent Finite Element Scheme for the Q-Tensor Model of Liquid Crystals Subjected to an Electric Field

Max Hirsch, Franziska Weber

TL;DR

This work addresses the numerical solution of the Landau-de Gennes Q-tensor model for nematic liquid crystals subjected to an electric field. It develops a fully discrete finite element scheme based on convex splitting of the bulk potential and a truncation operator to enforce boundedness of the Q-tensor, proving uniform energy stability and solvability via a Leray-Schauder fixed point argument. The authors establish convergence to a weak solution in the zero-polarization limit ($\varepsilon_3=0$) using Aubin-Lions-Simon compactness and carefully pass to the limit through the coupled elliptic and parabolic equations. Numerical experiments illustrate the method's capability to capture the Fréedericksz transition and demonstrate the essential role of the truncation in maintaining well-posedness and physically meaningful Q-tensor values.

Abstract

We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which we show that it is possible to simulate the Fréedericksz transition with this scheme.

A Convergent Finite Element Scheme for the Q-Tensor Model of Liquid Crystals Subjected to an Electric Field

TL;DR

This work addresses the numerical solution of the Landau-de Gennes Q-tensor model for nematic liquid crystals subjected to an electric field. It develops a fully discrete finite element scheme based on convex splitting of the bulk potential and a truncation operator to enforce boundedness of the Q-tensor, proving uniform energy stability and solvability via a Leray-Schauder fixed point argument. The authors establish convergence to a weak solution in the zero-polarization limit () using Aubin-Lions-Simon compactness and carefully pass to the limit through the coupled elliptic and parabolic equations. Numerical experiments illustrate the method's capability to capture the Fréedericksz transition and demonstrate the essential role of the truncation in maintaining well-posedness and physically meaningful Q-tensor values.

Abstract

We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which we show that it is possible to simulate the Fréedericksz transition with this scheme.
Paper Structure (17 sections, 8 theorems, 164 equations, 9 figures)

This paper contains 17 sections, 8 theorems, 164 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Electrostatic Energy Density Derivation
  3. Related works
  4. Outline
  5. The numerical scheme
  6. Convex Splitting
  7. Definition of the Numerical Scheme
  8. Solvability of the scheme
  9. Convergence
  10. Numerical Results
  11. Constant Initial Director Angle
  12. Electric Field Increasing Magnitude
  13. Truncation
  14. Fréedericksz transition
  15. Convergence
  16. ...and 2 more sections

Key Result

Lemma 2.1

We can write this bulk potential as where If $\beta_1 \ge \max\{|b|,a\}$ and $\beta_2\ge\max\{|b|,c\}$, then $\mathcal{F}_1(Q)$ and $\mathcal{F}_2(Q)$ are convex functions.

Figures (9)

  • Figure 1: Director field imposed on colored contour plot of electric potential $u$ for the constant initial director experiment in Section \ref{['sec:constant-initial-director']}.
  • Figure 2: Maximum magnitude Q-tensor entry and maximum eigenvalue over time for the constant initial director experiment in Section \ref{['sec:constant-initial-director']}.
  • Figure 3: Director field imposed on colored contour plot of electric potential $u$ for the increasing magnitude electric field experiment in Section \ref{['sec:increasing-magnitude']}.
  • Figure 4: Maximum magnitude Q-tensor entry over time for the increasing magnitude electric field experiment in Section \ref{['sec:increasing-magnitude']}.
  • Figure 5: Maximum absolute value of Q-tensor entries (in blue) and maximum eigenvalue of solutions (in orange) over time for the truncation experiment in Section \ref{['sec:truncation-experiment']}. The dashed green line gives the value $R/d$ at which Q-tensor entries are truncated.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Definition 1.1
  • Lemma 2.1: Convex Splitting of $\mathcal{F}_B$
  • proof
  • Theorem 2.2: Trace-Free and Symmetry Preservation Generalized
  • proof
  • Theorem 2.3: Energy Stability
  • proof
  • Remark 2.4: Physical parameter range
  • Corollary 2.5
  • proof
  • ...and 10 more