  Multiple equilibrium states arise in many physical systems, including various
types of liquid crystal structures. Having the ability to reliably compute such
states enables more accurate physical analysis and understanding of
experimental behavior. This paper adapts and extends a deflation technique for
the computation of multiple distinct solutions arising in the context of
modeling equilibrium configurations of nematic and cholesteric liquid crystals.
The deflation method is applied as part of an overall free-energy variational
approach and is modified to fit the framework of optimization of a functional
with pointwise constraints. It is shown that multigrid methods designed for the
undeflated systems may be applied to efficiently solve the linear systems
arising in the application of deflation. For the numerical algorithm, the
deflation approach is interwoven with nested iteration, creating a dynamic and
efficient method that further enables the discovery of distinct solutions.
Finally, four numerical experiments are performed demonstrating the efficacy
and accuracy of the algorithm in detecting important physical phenomena,
including bifurcation and disclination behaviors. The final numerical
experiment expands the algorithm to model cholesteric liquid crystals and
illustrates the full discovery power of the deflation process.
