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

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

Hyperbolic Conservation Laws with Relaxation Terms: A Theoretical and Numerical Study

Hyperbolic relaxation systems is an active field of research, with a largenumber of applications in physical modeling. Examples include modelsfor traffic flow, kinetic theory and fluid mechanics. This master s thesis is a numerical and theoretical analysis of such systems, and consists of two main parts: The first is a new scheme for the stable numerical solution of hyperbolic relaxation systems using exponential integrators. First and second-order schemes of this type are derived and some desirable stability and accuracy properties are shown. The scheme is also used to solve a granular-gas model in order to demonstratethe practical use of the method. The second and largest part of this thesis is the analysis of the solutionsto 2 × 2 relaxation systems. In this work, the link between the the sub-characteristic condition and the stability of the solution of the relaxationsystem is discussed. In this context, the sub-characteristic condition andthe dissipativity of the Chapman Enskog approximation are shown to beequivalent in both 1-D and 2-D. Also, the dispersive wave dynamics of hyperbolic relaxation systems isanalyzed in detail. For 2 × 2 systems, the wave-speeds of the individualFourier-components of the solution are shown to fulfill a transitional sub-characteristic condition. Moreover, the transition is monotonic in thevariable ξ = kε, where ε is the relaxation time of the system and k is thewave-number. A basic 2 × 2 model is used both as an example-model in the analyticaldiscussions, and as a model for numerical tests in order to demonstratethe implications of the analytical results

A two-component nonlinear variational wave system

We derive a novel two-component generalization of the nonlinear variational wave equation as a model for the director field of a nematic liquid crystal with a variable order parameter. The equation admits classical solutions locally in time. We prove that a special semilinear case is globally well-posed. We show that a particular long time asymptotic expansion around a constant state in a moving frame satisfies the two-component Hunter–Saxton system

Thermodynamic modeling with equations of state: present challenges with established methods

Equations of state (EoS) are essential in the modeling of a wide range of industrial and natural processes. Desired qualities of EoS are accuracy, consistency, computational speed, robustness and predictive ability outside of the domain where they have been fitted. In this work, we review present challenges associated with established models, and give suggestions on how to overcome them in the future. The most accurate EoS available, multiparameter EoS, have a second artificial Maxwell loop in the two-phase region that gives problems in phase-equilibrium calculations and exclude them from important applications such as treatment of interfacial phenomena with mass based density functional theory. Suggestions are provided on how this can be improved. Cubic EoS are among the most computationally efficient EoS, but they often lack sufficient accuracy. We show that extended corresponding state EoS are capable of providing significantly more accurate single-phase predictions than cubic EoS with only a doubling of the computational time. In comparison, the computational time of multiparameter EoS can be orders of magnitude larger. For mixtures in the two-phase region, however, the accuracy of extended corresponding state EoS has a large potential for improvement. The molecular-based SAFT family of EoS are preferred when predictive ability is important, e.g. for systems with strongly associating fluids or polymers where few experimental data are available. We discuss some of their benefits and present challenges. A discussion is presented on why predictive thermodynamic models for reactive mixtures such as CO2-NH3 and CO2-H2O-H2S must be developed in close combination with phase- and reaction equilibrium theory, regardless of the choice of EoS. After overcoming present challenges, a next-generation thermodynamic modeling framework holds the potential to improve the accuracy and predictive ability in a wide range of applications such as process optimization, computational fluid dynamics, treatment of interfacial phenomena and processes with reactive mixtures.publishedVersionCopyright © 2017 American Chemical Society. This is an open access article published under an ACS AuthorChoice License, which permits copying and redistribution of the article or any adaptations for non-commercial purposes

An exponential time-differencing method for monotonic relaxation systems

We present first and second-order accurate exponential time differencing methods for a special class of stiff ODEs, denoted as monotonic relaxation ODEs. Some desirable accuracy and robustness properties of our methods are established. In particular, we prove a strong form of stability denoted as monotonic asymptotic stability, guaranteeing that no overshoots of the equilibrium value are possible. This is motivated by the desire to avoid spurious unphysical values that could crash a large simulation. We present a simple numerical example, demonstrating the potential for increased accuracy and robustness compared to established Runge-Kutta and exponential methods. Through operator splitting, an application to granular-gas flow is provided.acceptedVersio

A combined fluid-dynamic and thermodynamic model to predict the onset of rapid phase transitions in LNG spills

Transport of liquefied natural gas (LNG) by ship occurs globally on a massive scale. The large temperature difference between LNG and water means LNG will boil violently if spilled onto water. This may cause a physical explosion known as rapid phase transition (RPT). Since RPT results from a complex interplay between physical phenomena on several scales, the risk of its occurrence is difficult to estimate. In this work, we present a combined fluid-dynamic and thermodynamic model to predict the onset of delayed RPT. On the basis of the full coupled model, we derive analytical solutions for the location and time of delayed RPT in an axisymmetric steady-state spill of LNG onto water. These equations are shown to be accurate when compared to simulation results for a range of relevant parameters. The relative discrepancy between the analytic solutions and predictions from the full coupled model is within 2% for the RPT position and within 8% for the time of RPT. This provides a simple procedure to quantify the risk of occurrence for delayed RPT for LNG on water. Due to its modular formulation, the full coupled model can straightforwardly be extended to study RPT in other systems.Comment: 22 pages, 11 figure

Evidence-Based Self-Management for Spondyloarthritis Patients

The file attached to this record is the author's final peer reviewed version. open access JournalWe present a concept including a set of tools for self-management for patients suffering from axial spondyloarthritis (SpA). This concept involves patient-recorded outcome measures, both subjective assessment and clinical measurements, that are used to present recommendations. We report from experiences made while implementing a proof of this concept and analyse it from several perspectives. Our work resulted in proposing a self-management tool for the patient, improving the methodology for clinical measurements of rotation exercises, and proof the viability for using on-board sensors in smart phones. Further, since sensors collect data in a medical setting, we present ethical considerations

On the dispersive wave-dynamics of 2 x 2 relaxation systems

We consider hyperbolic conservation laws with relaxation terms. By studying the dispersion relation of the solution of general linearized 2 × 2 hyperbolic relaxation systems, we investigate in detail the transition between the wave dynamics of the homogeneous relaxation system and that of the local equilibrium approximation. We establish that the wave velocities of the Fourier components of the solution to the relaxation system will be monotonic functions of a stiffness parameter φ = εξ, where ε is the relaxation time and ξ is the wave number. This allows us to extend in a natural way the classical concept of the sub-characteristic condition into a more general transitional sub-characteristic condition. We further identify two parameters β and γ that characterize the behavior of such general 2 × 2 linear relaxation systems. In particular, these parameters define a natural transition point, representing a value of φ where the dynamics will change abruptly from being equilibrium-like to behaving more like the homogeneous relaxation system. Herein, the parameter γ determines the location of the transition point, whereas β measures the degree of smoothness of this transition. Copyright© 2013 World Scientific Publishing Co. All rights reserved.acceptedVersio

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

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