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Anisotropic fluids, such as nematic liquid crystals, can form non-sphericalequilibrium shapes known as tactoids. Predicting the shape of these structuresas a function of material parameters is challenging and paradigmatic of abroader class of problems that combine shape and order. Here, we develop adiscrete shape optimization approach with finite elements to find theconfiguration of a two-dimensional tactoid using the Landau de Gennes frameworkand a Q-tensor representation. Efficient solution of the resulting constrainedenergy minimization problem is achieved using a quasi-Newton and nestediteration algorithm. Numerical validation is performed with benchmark solutionsand compared against experimental data and earlier work. We explore physicallymotivated subproblems, whereby the shape and order are separately held fixed,to explore the role of both and examine material parameter dependence of theconvergence. Nested iteration significantly improves both the computationalcost and convergence of numerical solutions of these highly deformablematerials.

Nonlinear Methods for Shape Optimization Problems in Liquid Crystal Tactoids