As materials possessing mesophases with characteristics of both liquids and organized solids, liquid crystals exhibit an array of interesting physical properties, including dielectric and flexoelectric coupling, inspiring a wide range of applications. In addition to prevalent use in modern display technologies, liquid crystals are applied, for example, to nanoparticle organization, the manufacture of nanoporous solids, and the design of effective actuators, such as light driven motors and artificial muscles. Accurate and efficient numerical simulation of liquid crystal behavior is used to optimize device design, analyze experiments, and suggest the presence of new physical phenomena. Mathematical models of liquid crystals present a number of interesting challenges for the design of theoretically supported computational techniques. Such challenges include highly nonlinear systems, point-wise unit-length constraints, strong coupling with electric and hydrodynamic effects, and stable configurations incorporating discontinuities, among others.
In this talk, we focus on the Frank-Oseen model of liquid crystals, introducing the elastic model for equilibrium configurations, its extension incorporating electric fields, and briefly discuss the addition of hydrodynamic effects. We consider the construction of theoretically supported approaches for such systems and examine a number of methods aimed at addressing different aspects of efficient simulation ranging from well-posed finite-element discretizations to reliable a posteriori error estimators. These methods expand the existing set of computational tools available for effective simulation of liquid crystal behavior. Finally, we highlight some of the interesting open questions and ongoing work in this area. This is joint work with a number of collaborators including James Adler and Tim Atherton (Tufts), Scott MacLachlan (Memorial), Patrick Farrell (Oxford), and Tom Manteuffel (Colorado Boulder).