Computational and analytical aspects of energy minimisation problems in cholesteric, ferronematic and smectic liquid crystals

Liquid crystals are widely used in display devices and their indispensable applications have driven more than a century of scientific investigations. They are of great interest in physics, for their striking defect structures, e.g., defect walls and focal conics in smectics; and in mathematics, for the questions arising in their modelling and analysis. Two successful mathematical theories are the Oseen--Frank (vector-based) and Landau--de Gennes (tensor-based) theories for nematics. In the former, the order parameter is simple but a nonlinear constraint must be enforced in the optimisation. The latter theory becomes more appealing in characterising complex defects, as it supports defects (e.g., half charge defects) that Oseen--Frank does not. However, when it comes to the phenomenological modelling of other phases of liquid crystals such as smectics, mathematical theories have not been extensively studied. This thesis takes a step forward in understanding several modelling and implementation issues related to three phases of liquid crystals: cholesterics, ferronematics and smectics.
In the first part of this thesis, we propose an augmented Lagrangian-type preconditioner to construct efficient solvers for Oseen--Frank problems arising in cholesterics. We analyse two advantages of the augmented Lagrangian formulation: (i) it helps in controlling the Schur complement matrix, enabling the development of block preconditioners; (ii) it improves the discrete enforcement of the unit-length constraint of the director. Since the augmentation makes the director block harder to solve, we develop a robust multigrid algorithm for the augmented block. The resulting preconditioner is an efficient and robust approach for solving director-based models of liquid crystals.
The second part is devoted to investigating defect structures (e.g., jumps of the director and magnetisation vector) in ferronematics, through numerical bifurcation analysis. Novel bifurcations of the ferronematic problem of interest are studied to give a more complete picture of solution landscapes as the parameter space varies. The reported numerical results validate the corresponding theoretical analysis of Dalby & Majumdar , and show us the potential of the Landau--de Gennes theory in characterising complicated defects.
Convinced by the successful application of the Landau--de Gennes model in ferronematics, we move to developing effective models of smectic-A liquid crystals in the last part of this thesis. We propose a new continuum model, solving for a real-valued smectic order parameter for the density variation and a tensor-valued nematic order parameter for the director orientation. This expands on an idea mentioned by Ball & Bedford. The model is challenging to discretise due to the high regularity of the density variation; to address this, a continuous interior penalty discretisation is employed. Numerical analysis and experiments are performed to confirm the effectiveness of the proposed model and discretisation. The model numerically captures important defect structures in focal conic domains and oily streaks for the first time.

Advisor: Farrell, P