As devices using liquid crystals become ever smaller and increasingly complex, there is a commensurate increase in the need for more effective numerical modelling tools in the area. In this thesis, an adaptive finite element method is used to solve a non-linear singularly perturbed boundary value problem which arises from a one dimensional Q-tensor model of liquid crystals.The adaptive non-uniform mesh is generated by equidistribution of a selection of strictly positive monitor functions. By an appropriate selection of the monitor function parameters, it is shown that the computed numerical solution converges at an optimal rate with respect to the mesh density and that the solution accuracy is robust to the size of the singular perturbation parameter.A robust and efficient numerical scheme is then used to solve the system of six coupled partial differential equations which arises from Q-tensor theory. 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. The behaviour of the method is illustrated on a one-dimensional time-dependent problem in a π-cell geometry with an applied electric field. The numerical results show that, as well as achieving optimal rates of convergence in space and time, higher levels of solution accuracy and a considerable improvement in computational efficiency are obtained compared to other moving mesh methods used by previous authors on similar problems.The numerical scheme is then extended to tackle a two-dimensional π-cell problem. It is shown that the adaptive moving mesh method copes well with the presence of moving defects, with the mesh adapting and relaxing to capture the motion, growth and annihilation of the defects.
|Date of Award||1 Apr 2014|
- University Of Strathclyde
|Sponsors||EPSRC (Engineering and Physical Sciences Research Council)|
|Supervisor||John MacKenzie (Supervisor) & Alison Ramage (Supervisor)|