@article {25496,
	title = {Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms},
	journal = {Liquid Crystals},
	year = {2017},
	publisher = {Taylor \& Francis Online},
	abstract = {We study the problem of a cholesteric liquid crystal confined to an elliptical channel. The system is geometrically frustrated because the cholesteric prefers to adopt a uniform rate of twist deformation, but the elliptical domain precludes this. The frustration is resolved by deformation of the layers or introduction of defects, leading to a particularly rich family of equilibrium configurations. To identify the solution set, we adapt and apply a new family of algorithms, known as deflation methods, that iteratively modify the free energy extremisation problem by removing previously known solutions. A second algorithm, deflated continuation, is used to track solution branches as a function of the aspect ratio of the ellipse and preferred pitch of the cholesteric.},
	doi = {10.1080/02678292.2017.1365385},
	author = {David B {Emerson} and Patrick Emmet {Farrell} and James Haley {Adler} and Mary C. {MacLachlan} and Timothy J. {Atherton}}
}
@article {25491,
	title = {Combining Deflation and Nested Iteration for Computing Multiple Solutions of Nonlinear Variational Problems},
	journal = {SIAM Journal on Scientific Computing},
	volume = {39},
	number = {1},
	year = {2016},
	pages = {29-52},
	publisher = {SIAM},
	abstract = {This paper compares the performance of penalty and Lagrange multiplier approaches for the necessary unit-length constraint in the computation of liquid crystal equilibrium configurations. Building on previous work in [SIAM J. Sci. Comput., 37 (2015), pp. S157--S176;\&nbsp;SIAM J. Numer. Anal., 53 (2015), pp. 2226--2254], the penalty method is derived and well-posedness of the linearizations within the nonlinear iteration is discussed. In addition, the paper considers the effects of tailored trust-region methods in the context of finite-element discretizations and nested iteration for both formulations. Such methods are aimed at increasing the efficiency and robustness of each algorithm{\textquoteright}s nonlinear iterations. Three representative elastic equilibrium problems are considered to examine each method{\textquoteright}s performance. The first two configurations have analytical expressions for their exact solutions and, therefore, convergence to the true solution is considered. The third problem considers complicated boundary conditions, relevant in ongoing research, simulating surface nano-patterning. Finally, a novel multigrid scheme is introduced and tested for electrically and flexoelectrically coupled models to establish scalability for highly complicated applications. The Lagrange multiplier method is found to outperform the penalty method in a number of measures, the developed trust regions are shown to improve robustness, and nested iteration proves highly effective at reducing computational costs.\&nbsp;},
	doi = {10.1137/16M1058728},
	author = {James Haley {Adler} and David B. {Emerson} and MacLachlan, Scott and Thomas A. {Manteuffel}}
}
