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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Wujun Zhang
University of Maryland
Title: A finite element method for nematic liquid crystals with variable degree of orientation

Abstract: We consider the simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field and its degree of orientation, which minimize a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for the director field, which allows for line and plane defects to have finite energy.

We present a structure preserving discretization of the liquid crystal energy without regularization, and show that it is consistent. We prove convergence of the continuous piecewise linear finite solutions as the mesh size goes to zero. We develop a weighted gradient flow scheme for computing discrete equilibrium solutions and prove that it has a strict energy decrease property. We present simulations in two and three dimensions that exhibit both line and plane defects and illustrate key features of the method.

Recording: http://mm.math.cmu.edu/recordings/cna/CNA-Zhang-Nov-11-2014.avi
Recording: http://mm.math.cmu.edu/recordings/cna/CNA-Zhang-Nov-11-2014.mp4
Date: Tuesday, November 11, 2014
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer