An adaptive homotopy multi-grid method for molecule orientations of high dimensional liquid crystals

The liquid crystal molecule orientation is arranged by minimizing the so-called Oseen–Frank energy functional. For a better understanding of these complicated orientation singularities, simplified models resulting from specific choices of elastic constants are always of interest. In this paper a pseudo Newton method together with a multi-grid linear system solver or preconditioner is used to compute the orientation of liquid crystal molecules based on a simplified Oseen–Frank energy functional. The penalty method is used to deal with the unit-length constraint of liquid crystal molecules. The Newton and multi-grid methods do not converge when some parameters are small. A homotopy algorithm combined with mesh refinement strategies in order to deal with small parameter cases is studied and is found to be very robust in computing the solution of the model. The method is implemented to compute the orientation of liquid crystal molecules in domains of typical shapes and with various rotational boundary conditions in 2D and 3D. Interesting singularity patterns are observed.