Core Concepts
Efficient linear numerical schemes for simulating nematic liquid crystals using Q-tensor theory are proposed.
Abstract
The article introduces three linear numerical schemes for modeling nematic liquid crystals using the Landau-de Gennes Q-tensor theory. It discusses the importance of liquid crystals in various applications and the unique properties of nematic liquid crystals. The focus is on the development of efficient numerical schemes to accurately represent the dynamics of these materials. The schemes presented aim to balance accuracy, efficiency, and computational complexity. The content is structured into sections covering the theoretical background, numerical schemes, and computational results to validate the proposed methods.
Introduction
Liquid crystals are versatile materials used in various applications.
Nematic liquid crystals exhibit properties of both solids and liquids.
Landau-de Gennes Q-Tensor Model
Overview of the Q-tensor model for nematic liquid crystals.
Description of the Landau-de Gennes free energy function and elastic energy density.
System Dynamics
Dynamics of the system described by an L2-gradient flow.
Formulation of the problem and boundary conditions.
Numerical Schemes
Challenges in developing numerical schemes due to nonlinearity.
Introduction of linear, unconditionally energy-stable finite element schemes.
Comparison of first-order and second-order schemes for accuracy and efficiency.
First Order Unconditionally Energy Stable Decoupled Scheme (UES1D)
Proposal of a linear, first-order accurate, energy-stable numerical scheme.
Decoupling of unknown variables for computational efficiency.
Second Order Optimal Dissipation Coupled Scheme (OD2C)
Adaptation of a second-order optimal dissipation algorithm for coupled numerical scheme.
Discussion on accuracy and computational complexity.
First Order Optimal Dissipation Decoupled Scheme (OD1D)
Modification of the optimal dissipation algorithm to decouple unknowns.
Sequential computation of unknowns for efficiency.
Stats
The first scheme is based on using a truncation procedure of the energy.
The second scheme uses a modified second-order accurate optimal dissipation algorithm.
The third scheme uses a technique to decouple the unknowns for improved computational efficiency.
Quotes
"We propose three new efficient linear numerical schemes for simulating nematic liquid crystals using a Q-tensor model."