scholarly journals Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

2021 ◽  
Author(s):  
Jingmin Xia ◽  
Patrick E. Farrell ◽  
Florian Wechsung
Keyword(s):  
Liquid Crystals ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Mass Matrix ◽  
Unit Length ◽  
Mesh Size ◽  
Schur Complement ◽  
Cholesteric Liquid Crystals ◽  
Multigrid Algorithm ◽  
The Cost

AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

scholarly journals Synthesis and Characterization of Novel Bio-Chiral Dopants Obtained from Bio-Betulin Produced by a Fermentation Process

Crystals ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 785
Author(s):  
Veridiana G. Guimarães ◽  
Anastasiia Svanidze ◽  
Tianyi Guo ◽  
Pawan Nepal ◽  
Robert J. Twieg ◽  
...  
Keyword(s):  
Liquid Crystals ◽  
Natural Product ◽  
Fermentation Process ◽  
Synthesis And Characterization ◽  
Cholesteric Liquid Crystals ◽  
And Performance ◽  
Chiral Dopants

Cholesteric liquid crystals are frequently produced by the addition of chiral dopants to achiral nematic hosts. We report here the synthesis and performance of chiral dopants obtained from bio-betulin produced by a fermentation process. An important aspect of this work is to point out that the fermentation process used to obtain the starting materials is much easier and cheaper when carried out in large volumes than isolating it from the natural product. The performance of the dopants obtained from bio-betulin is indistinguishable from those obtained from commercially available synthetic betulin.

scholarly journals Pitch tuning induced by optical torque in heliconical cholesteric liquid crystals

2019 ◽  
Vol 1 (3) ◽  
Author(s):  
G. Nava ◽  
F. Ciciulla ◽  
O. S. Iadlovska ◽  
O. D. Lavrentovich ◽  
F. Simoni ◽  
...  
Keyword(s):  
Liquid Crystals ◽  
Cholesteric Liquid Crystals ◽  
Optical Torque

scholarly journals An augmented Lagrangian method for solving total variation (TV)-based image registration model

2020 ◽  
Vol 14 ◽  
pp. 174830262097353
Author(s):  
Noppadol Chumchob ◽  
Ke Chen
Keyword(s):  
Image Registration ◽  
Augmented Lagrangian ◽  
Numerical Solutions ◽  
Augmented Lagrangian Method ◽  
Closed Form Solution ◽  
Numerical Algorithms ◽  
Lagrangian Method ◽  
Form Solution ◽  
The Other ◽  
Other Hand

Variational methods for image registration basically involve a regularizer to ensure that the resulting well-posed problem admits a solution. Different choices of regularizers lead to different deformations. On one hand, the conventional regularizers, such as the elastic, diffusion and curvature regularizers, are able to generate globally smooth deformations and generally useful for many applications. On the other hand, these regularizers become poor in some applications where discontinuities or steep gradients in the deformations are required. As is well-known, the total (TV) variation regularizer is more appropriate to preserve discontinuities of the deformations. However, it is difficult in developing an efficient numerical method to ensure that numerical solutions satisfy this requirement because of the non-differentiability and non-linearity of the TV regularizer. In this work we focus on computational challenges arising in approximately solving TV-based image registration model. Motivated by many efficient numerical algorithms in image restoration, we propose to use augmented Lagrangian method (ALM). At each iteration, the computation of our ALM requires to solve two subproblems. On one hand for the first subproblem, it is impossible to obtain exact solution. On the other hand for the second subproblem, it has a closed-form solution. To this end, we propose an efficient nonlinear multigrid (NMG) method to obtain an approximate solution to the first subproblem. Numerical results on real medical images not only confirm that our proposed ALM is more computationally efficient than some existing methods, but also that the proposed ALM delivers the accurate registration results with the desired property of the constructed deformations in a reasonable number of iterations.

scholarly journals An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Numerical Experiments ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Constrained Optimization Problems ◽  
Strongly Stationary ◽  
Type Constraint

AbstractA reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

scholarly journals Light-driven rotation of gratings formed by electroconvection patterns in cholesteric liquid crystals

2021 ◽  
pp. 116366
Author(s):  
Ying Xiang ◽  
Hongzhen Jing ◽  
Hao Chen ◽  
Jun zhang ◽  
Xiaoyu Ding ◽  
...  
Keyword(s):  
Liquid Crystals ◽  
Cholesteric Liquid Crystals

scholarly journals Optical Response Associated with the Orientation and Structure of Liquid Crystals with Respect to Phosphatidylcholine Concentration

Crystals ◽  
2021 ◽  
Vol 11 (6) ◽  
pp. 678
Author(s):  
Yuqi Han ◽  
Yan Jiang ◽  
Wei Guo ◽  
Bing Li ◽  
Lu Zhang ◽  
...  
Keyword(s):  
Liquid Crystal ◽  
Liquid Crystals ◽  
Reflectance Spectrum ◽  
Optical Response ◽  
The Self ◽  
Cholesteric Liquid Crystals ◽  
Anchoring Effect ◽  
Self Assembling ◽  
Sensitization Effect ◽  
Structure Of Liquid

Based on the anchoring effect due to the self-assembling behavior of the phospholipid molecules at the interface between the liquid crystal and water phases on the orientation of liquid crystals, the optical response associated with the orientation and structure of liquid crystals with respect to the concentration of 1,2-didodecanoyl-sn-glycero-3-phosphocholine solution has been investigated. The optical response owing to changes in the orientation and structure of the mixed cholesteric liquid crystals with respect to the change in the concentration of phosphatidylcholine has been obtained. Moreover, the feasibility of using as-prepared mixed cholesteric liquid crystals to measure the phosphatidylcholine concentration has been verified. A methodology to measure the reflectance spectrum by using mixed cholesteric liquid crystals to sensitize the phosphatidylcholine concentration has been further realized. The sensitization effect of the mixed cholesteric liquid crystals on the measurement of phosphatidylcholine concentration was also verified.

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