Efficient Moving Mesh Methods for $Q$-Tensor Models of Nematic Liquid Crystals

Abstract

This version is available at https://strathprints.strath.ac.uk/51133/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. EFFICIENT MOVING MESH METHODS FOR Q-TENSOR MODELS OF NEMATIC LIQUID CRYSTALSCRAIG S. MACDONALD * , JOHN A. MACKENZIE * , ALISON RAMAGE * , AND CHRISTOPHER J.P. NEWTON * Abstract. This paper describes a robust and efficient numerical scheme for solving the system of six coupled partial differential equations which arises when using Q-tensor theory to model the behaviour of a nematic liquid crystal cell under the influence of an applied electric field. The key novel feature is the use of a full moving mesh partial differential equation (MMPDE) approach to generate an adaptive mesh which accurately resolves important solution features. This includes the use of a new monitor function based on a local measure of biaxiality. In addition, adaptive time-step control is used to ensure the accurate predicting of the switching time, which is often critical in the design of liquid crystal cells. We illustrate the behaviour of the method on a one-dimensional time-dependent problem in a Pi-cell geometry which admits two topologically different equilibrium states, modelling the order reconstruction which occurs on the application of an electric field. Our numerical results show that, as well as achieving optimal rates of convergence in space and time, we obtain higher levels of solution accuracy and a considerable improvement in computational efficiency compared to other moving mesh methods used previously for liquid crystal problems.

A global 2D well-posedness result on the order tensor liquid crystal theory

“…Some Developments in the order tensor Theory. Although the Q-tensor theory has received much attention in several disciplines as Physics [3], numerical analysis [14], mathematical analysis [16], the solvability study of the related system has not received numerous investigations, yet. We recall here some recent results.…”

A Variational Lagrangian Scheme for a Phase Field Model: A Discrete Energetic Variational Approach

“…our scheme has advantage in capturing the thin diffuse interface in equilibria with a small number of mesh points. Our approach can be extended to general gradient flow system, especially those have equilibria with singularity, sharp interface and free boundary, such as Landau-de Gennes model for liquid crystal [42,63,60] and GinzburgLandau model for superconductivity [23].…”

A BDF2 Energy-Stable Scheme for a General Tensor-Based Model of Liquid Crystals

“…This model is widely used in liquid crystal flows [2,7,12,17,20,22]. The well-posedness of the Q-tensor model has been studied in [4,8,9,13] and numerical approaches are discussed in [11,14]. In the framework of the model (1.2), a variety of numerical codes have been developed, yet there are no systematic efforts to analyse the variational and dissipative nature of the method.…”