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Numerical solution of the dynamics of director fields in nematic liquid crystals

Aursand, Peder
Doctoral thesis
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http://hdl.handle.net/11250/2374944
Utgivelsesdato
2015
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  • Institutt for matematiske fag [1396]
Sammendrag
Since their discovery in the late 1800s, liquid crystals have become an important

part of the technology of the modern world. As a consequence the

study of anisotropic liquids in general, and liquid crystals in particular, has

grown into a large interdisciplinary field involving physics, mathematics,

chemistry and biology to name a few.

In a series of papers we consider numerical solution of the evolution of

the director, a vector valued field giving the local average orientation of the

long axis of molecules in nematic liquid crystals. The flow field is assumed

to be stationary throughout this work. We consider both the free elastic

dynamics of the director as well as the case with applied electric fields on a

finite domain.

We study the dynamics of the 1D Fréedericksz transition, where an

applied electric field forces reorientation in the director field. The director

is assumed strongly anchored and the boundaries. Herein, we study the

role of inertia and dissipation on the time evolution of the director eld

during the reorientation. In particular, we show through simulations that

inertia will introduce standing waves that might e

ect transition time of

the reorientation, but only for very small time scales or extremely high

molecular inertia.

The Fréedericksz transition is also numerically studied with weak boundary

anchoring. For this problem it has been shown analytically that there

exists a hierarchy of meta-stable equilibrium con gurations. This is in sharp

contrast to the strongly anchored case, where the equilibrium is globally well

defined. We derive an implicit numerical scheme for this problem and show

the well-posedness of the discrete equation system. The method can be

used for the fully nonlinear model with coupled electric field. Through simulations

we show that the director can transition into different meta-stable

states given different small perturbations to the initial data.

The numerical solution of variational wave equations describing the elastic

dynamics of nematic liquid crystals is considered in both 1D and 2D.

Using energy respecting Runge{Kutta Discontinuous Galerkin methods we

show that numerical solutions that either conserve or dissipate a discrete

version of the energy can be obtained by efficient time marching. The dissipative

scheme uses a dissipative up-winding at the cell interfaces combined

with a shock-capturing method.

Finally, we consider the application of nonintrusive sampling methods

for uncertainty quantification for the elastic problem with uncertain Frank

constants. The multi-level Monte Carlo (MLMC) method has been successfully

applied to systems of hyperbolic conservation laws, but its applicability

to other nonlinear problems is unclear. We show that MLMC is 5-10 times

more efficient in approximating the mean compared to regular Monte Carlo

sampling, when applied to variational wave equations in both 1D and 2D.
Består av
Paper 1: Aursand, Peder; Ridder, Johanna. The Role of Inertia and Dissipation in the Dynamics of the Director for a Nematic Liquid Crystal Coupled with an Electric Field. Communications in Computational Physics 2015 ;Volum 18.(1) http://dx.doi.org/10.4208/cicp.220414.231214a First published in Communications in Computational Physics / Volume 18 / Issue 01 / July 2015, published by Global Science Press

Paper 2: Peder Aursand, Gaetano Napoli and Johanna Ridder. On the dynamics of the weak Fréeedericksz transition for nematic liquid crystals

Paper 3: Peder Aursand and Ujjwal Koley. Local discontinuous Galerkin schemes for a nonlinear variational wave equation modeling liquid crystals

Paper 4: Peder Aursand and Ujjwal Koley. High-order energy stable numerical schemes for a nonlinear variational wave equation modeling nematic liquid crystals in two dimensions

Paper 5: Peder Aursand and Jonas Sukys. Uncertainty quantification for nonlinear waves in liquid crystals using multi-level Monte Carlo
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NTNU
Serie
Doctoral thesis at NTNU;2015:303

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