Dept. of Mathematics and Computer Science, Kent State University, Kent, OH 44242

Abstract

We will present a scalable parallel finite difference algorithm for computing
the equilibrium configurations, of the order-parameter tensor field for
nematic liquid crystals, in rectangular and ellipsoidal regions, by mimization
of the Landau - de Gennes free energy functional. In this formulation, we
solve for a symmetric traceless 3x3 tensor at each point, thus giving rise to
large non-linear algebraic systems. Due to the complexity of the expressions
symbolic algebra techniques are used in the production of the code. The
target architectures for our implementation are primarily SIMD machines, with
3 dimensional rectangular grid networks, such as the Wavetracer DTC as opposed
to hypercube networks such as the Thinking Machines Corporation CM-2. We will
describe certain problems encountered in implementing multigrid methods on
such architectures and contrast the performance of multigrid with other
iterative solvers. We remark that the more computationally intensive Landau -
de Gennes formulation permits the modelling of phenomena such as line defects
that do not have finite free energy in the more common vector-based
Oseen-Zocher-Frank model. It also allows more general behaviors such as
*non-uniform order* and *biaxiality* to be considered.