Local stability and a renormalized Newton Method for equilibrium liquid crystal director modeling

We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to problems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., submitted]. The characterization of local stability of solutions is complicated by the double saddle-point structure, and here we develop efficiently computable criteria in terms of minimum eigenvalues of certain projected Schur complements. We also propose a modified outer iteration (“Renormalized Newton Method”) in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The Renormalized Newton Method bears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two

The effect of elastic anisotropy on nematic disclination lines

No abstract available

Computational fluid dynamics for nematic liquid crystals

Due to recent advances in fast iterative solvers in the field of computational fluid dynamics, more complex problems which were previously beyond the scope of standard techniques can be tackled. In this paper, we describe one such situation, namely, modelling the interaction of flow and molecular orientation in a complex fluid such as a liquid crystal. Specifically, we consider a nematic liquid crystal in a spatially inhomogeneous flow situation where the orientational order is described by a second rank alignment tensor. The evolution is determined by two coupled equations: a generalised Navier-Stokes equation for flow in which the divergence of the stress tensor also depends on the alignment tensor and its time derivative, and a convection-diffusion type equation with non-linear terms that stem from a Landau-Ginzburg-DeGennes potential for the alignment. In this paper, we use a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress to be cast in the form of an orientation-dependent force. This effectively decouples the flow and orientation, with each appearing only on the right-hand side of the other equation. In this way, difficulties associated with solving the fully coupled problem are circumvented and a stand-alone fast solver, such as the state-of-the-art preconditioned iterative solver implemented here, can be used for the flow equation. A time-discretised strategy for solving the flow-orientation problem is illustrated using the example of Stokes flow in a lid-driven cavity

A preconditioned nullspace method for liquid crystal director modelling

We present a preconditioned nullspace method for the numerical solution of large sparse linear systems that arise from discretizations of continuum models for the orientational properties of liquid crystals. The approach effectively deals with pointwise unit-vector constraints, which are prevalent in such models. The indefinite, saddle-point nature of such problems, which can arise from either or both of two sources (pointwise unit-vector constraints, coupled electric fields), is illustrated. Both analytical and numerical results are given for a model problem

IFISS : a computational laboratory for investigating incompressible flow problems

The IFISS Incompressible Flow & Iterative Solver Software package contains software which can be run with MATLAB or Octave to create a computational laboratory for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions, together with state-of-the-art preconditioned iterative solvers for the resulting discrete linear equation systems. In this paper we give a flavour of the code's main features and illustrate its applicability using several case studies. We aim to show that IFISS can be a valuable tool in both teaching and research

A moving mesh method for modelling defects in nematic liquid crystals

The properties of liquid crystals can be modelled using an order parameter which describes the variability of the local orientation of rod-like molecules. Defects in the director field can arise due to external factors such as applied electric or magnetic fields, or the constraining geometry of the cell containing the liquid crystal material. Understanding the formation and dynamics of defects is important in the design and control of liquid crystal devices, and poses significant challenges for numerical modelling. In this paper we consider the numerical solution of a Q-tensor model of a nematic liquid crystal, where defects arise through rapid changes in the Q-tensor over a very small physical region in relation to the dimensions of the liquid crystal device. The efficient solution of the resulting six coupled partial differential equations is achieved using a finite element based adaptive moving mesh approach, where an unstructured triangular mesh is adapted towards high activity regions, including those around defects. Spatial convergence studies are presented using a stationary defect as a model test case, and the adaptive method is shown to be optimally convergent using quadratic triangular finite elements. The full effectiveness of the method is then demonstrated using a challenging two-dimensional dynamic Pi-cell problem involving the creation, movement, and annihilation of defects

Preconditioning for radial basis function partition of unity methods

Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF--PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner

Adaptive grid methods for Q-tensor theory of liquid crystals : a one-dimensional feasibility study

This paper illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory of liquid crystals. We present the results of an initial study using a simple one-dimensional test problem which illustrates the feasibility of applying adaptive grid techniques in such situations. We describe how the grids are computed using an equidistribution principle, and investigate the comparative accuracy of adaptive and uniform grid strategies, both theoretically and via numerical examples

Substantial metabolic activity of human brown adipose tissue during warm conditions and cold-induced lipolysis of local triglycerides

Current understanding of in vivo human brown adipose tissue (BAT) physiology is limited by a reliance on positron emission tomography (PET)/computed tomography (CT) scanning, which has measured exogenous glucose and fatty acid uptake but not quantified endogenous substrate utilization by BAT. Six lean, healthy men underwent 18fluorodeoxyglucose-PET/CT scanning to localize BAT so microdialysis catheters could be inserted in supraclavicular BAT under CT guidance and in abdominal subcutaneous white adipose tissue (WAT). Arterial and dialysate samples were collected during warm (∼25°C) and cold exposure (∼17°C), and blood flow was measured by 133xenon washout. During warm conditions, there was increased glucose uptake and lactate release and decreased glycerol release by BAT compared with WAT. Cold exposure increased blood flow, glycerol release, and glucose and glutamate uptake only by BAT. This novel use of microdialysis reveals that human BAT is metabolically active during warm conditions. BAT activation substantially increases local lipolysis but also utilization of other substrates such as glutamate

Glucocorticoids Acutely Increase Brown Adipose Tissue Activity in Humans, Revealing Species-Specific Differences in UCP-1 Regulation

SummaryThe discovery of brown adipose tissue (BAT) in adult humans presents a new therapeutic target for metabolic disease; however, little is known about the regulation of human BAT. Chronic glucocorticoid excess causes obesity in humans, and glucocorticoids suppress BAT activation in rodents. We tested whether glucocorticoids regulate BAT activity in humans. In vivo, the glucocorticoid prednisolone acutely increased 18fluorodeoxyglucose uptake by BAT (measured using PET/CT) in lean healthy men during mild cold exposure (16°C–17°C). In addition, prednisolone increased supraclavicular skin temperature (measured using infrared thermography) and energy expenditure during cold, but not warm, exposure in lean subjects. In vitro, glucocorticoids increased isoprenaline-stimulated respiration and UCP-1 in human primary brown adipocytes, but substantially decreased isoprenaline-stimulated respiration and UCP-1 in primary murine brown and beige adipocytes. The highly species-specific regulation of BAT function by glucocorticoids may have important implications for the translation of novel treatments to activate BAT to improve metabolic health