  Anisotropic fluids, such as nematic liquid crystals, can form non-spherical
equilibrium shapes known as tactoids. Predicting the shape of these structures
as a function of material parameters is challenging and paradigmatic of a
broader class of problems that combine shape and order. Here, we consider a
discrete shape optimization approach with finite elements to find the
configuration of two-dimensional and three-dimensional tactoids using the
Landau de Gennes framework and a Q-tensor representation. Efficient solution of
the resulting constrained energy minimization problem is achieved using a
quasi-Newton and nested iteration algorithm. Numerical validation is performed
with benchmark solutions and compared against experimental data and earlier
work. We explore physically motivated subproblems, whereby the shape and order
are separately held fixed, respectively, to explore the role of both and
examine material parameter dependence of the convergence. Nested iteration
significantly improves both the computational cost and convergence of numerical
solutions of these highly deformable materials.
