18 research outputs found
Recent Analytic Development of the Dynamic \u3cb\u3eQ\u3c/b\u3e-Tensor Theory for Nematic Liquid Crystals
Liquid crystals are a typical type of soft matter that are intermediate between conventional crystalline solids and isotropic fluids. The nematic phase is the simplest liquid crystal phase, and has been studied the most in the mathematical community. There are various continuum models to describe liquid crystals of nematic type, and Q-tensor theory is one among them. The aim of this paper is to give a brief review of recent PDE results regarding the Q-tensor theory in dynamic configurations
Convergence analysis of a fully discrete energy-stable numerical scheme for the Q-tensor flow of liquid crystals
We present a fully discrete convergent finite difference scheme for the
Q-tensor flow of liquid crystals based on the energy-stable semi-discrete
scheme by Zhao, Yang, Gong, and Wang (Comput. Methods Appl. Mech. Engrg. 2017).
We prove stability properties of the scheme and show convergence to weak
solutions of the Q-tensor flow equations. We demonstrate the performance of the
scheme in numerical simulations
A Convergent Finite Element Scheme for the Q-Tensor Model of Liquid Crystals Subjected to an Electric Field
We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to
an electric field and develop a fully discrete numerical scheme for its
solution. The scheme uses a convex splitting of the bulk potential, and we
introduce a truncation operator for the Q-tensors to ensure well-posedness of
the problem. We prove the stability and well-posedness of the scheme. Finally,
making a restriction on the admissible parameters of the scheme, we show that
up to a subsequence, solutions to the fully discrete scheme converge to weak
solutions of the Q-tensor model as the time step and mesh are refined. We then
present numerical results computed by the numerical scheme, among which, we
show that it is possible to simulate the Fr\'eedericksz transition with this
scheme
Energy stable and maximum bound principle preserving schemes for the Q-tensor flow of liquid crystals
In this paper, we propose two efficient fully-discrete schemes for Q-tensor
flow of liquid crystals by using the first- and second-order stabilized
exponential scalar auxiliary variable (sESAV) approach in time and the finite
difference method for spatial discretization. The modified discrete energy
dissipation laws are unconditionally satisfied for both two constructed
schemes. A particular feature is that, for two-dimensional (2D) and a kind of
three-dimensional (3D) Q-tensor flows, the unconditional
maximum-bound-principle (MBP) preservation of the constructed first-order
scheme is successfully established, and the proposed second-order scheme
preserves the discrete MBP property with a mild restriction on the time-step
sizes. Furthermore, we rigorously derive the corresponding error estimates for
the fully-discrete second-order schemes by using the built-in stability
results. Finally, various numerical examples validating the theoretical
results, such as the orientation of liquid crystal in 2D and 3D, are presented
for the constructed schemes
On the foundations of thermodynamics
On the basis of new, concise foundations, this paper establishes the four
laws of thermodynamics, the Maxwell relations, and the stability requirements
for response functions, in a form applicable to global (homogeneous), local
(hydrodynamic) and microlocal (kinetic) equilibrium.
The present, self-contained treatment needs very little formal machinery and
stays very close to the formulas as they are applied by the practicing
physicist, chemist, or engineer. From a few basic assumptions, the full
structure of phenomenological thermodynamics and of classical and quantum
statistical mechanics is recovered.
Care has been taken to keep the foundations free of subjective aspects (which
traditionally creep in through information or probability). One might describe
the paper as a uniform treatment of the nondynamical part of classical and
quantum statistical mechanics ``without statistics'' (i.e., suitable for the
definite descriptions of single objects) and ``without mechanics'' (i.e.,
independent of microscopic assumptions). When enriched by the traditional
examples and applications, this paper may serve as the basis for a course on
thermal physics.Comment: 78 page
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal