I start with an overview of nematic liquid crystals (LCs) and their applications, including how they are modeled, such as Oseen-Frank, Landau-de Gennes, and the Ericksen model. For the rest of the talk, I will focus on Landau-de Gennes (LdG) and Ericksen.
Next, I describe some of the analytical difficulties of these models. For example, the Ericksen model exhibits a non-smooth constraint (for the PDE solution), and the LdG model with uniaxiality enforced as a hard constraint is also non-smooth. I will then discuss related numerical analysis issues that arise and how we handle these difficulties with a structure-preserving finite element method (FEM) for computing energy minimizers. We prove stability and consistency of the method without regularization, and \(\Gamma\)-convergence of the discrete energies towards the continuous one as the mesh size goes to zero. Numerical simulations will be presented in two and three dimensions, some of which include non-orientable line fields, using a provably robust minimization scheme. Finally, I will conclude with some current problems and an outlook to future directions.
Time: March 5, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang