  Defects in liquid crystals are of great practical importance and theoretical
interest. Despite tremendous efforts, predicting the location and transition of
defects under various topological constraint and external field remains to be a
challenge. We investigate defect patterns of nematic liquid crystals confined
in three-dimensional spherical droplet and two-dimensional disk under different
boundary conditions, within the Landau-de Gennes model. We implement a spectral
method that numerically solves the Landau-de Gennes model with high accuracy,
which allows us to study the detailed static structure of defects. We observe
five types of defect structures. Among them the 1/2-disclination lines are the
most stable structure at low temperature. Inspired by numerical results, we
obtain the profile of disclination lines analytically. Moreover, the connection
and difference between defect patterns under the Landau-de Gennes model and the
Oseen-Frank model are discussed. Finally, four conjectures are made to
summarize some important characteristics of defects in the Landau-de Gennes
theory. This work is a continuing effort to deepen our understanding on defect
patterns in nematic liquid crystals.
