arXiv subjects

Jingmin Xia

Publications and source records attributed to Jingmin Xia.

11 recordsLinked to original sources

arXiv2026

Augmented Lagrangian preconditioning for a simplified Ericksen--Leslie model of nematic liquid crystals

The numerical solution of the simplified Ericksen--Leslie model for nematic liquid crystals is challenging because the flow and director equations are strongly coupled and because incompressibility and the unit-length condition must be enforced simultaneously. A Lagrange multiplier formulation avoids a small Ginzburg--Landau parameter, but the Newton systems have a double saddle-point structure. We develop an augmented Lagrangian block preconditioner in which both constraints are augmented while their discrete enforcement remains multiplier based. After finite element discretization and backward Euler time integration, the Newton increments are grouped into velocity--director and pressure-multiplier variables. A block-diagonal approximation of the coupled velocity-director block then leads to separate, physically scaled approximations of the pressure and director-multiplier Schur complements. Manufactured-solution tests show the expected spatial accuracy and first-order temporal convergence for the primary variables; the multiplier error reaches a spatial-error floor on the fixed mesh used in the temporal study. In the reported parameter ranges, the outer FGMRES iteration counts are nearly mesh independent, remain stable under time-step and viscosity variation, and improve as the augmentation parameters increase. A smooth benchmark also exhibits monotone decay of the computed total energy.

math.NA↗

arXiv2026

Penalty-scaling effects in nonsymmetric interior-penalty DG discretizations of viscous rotating shallow-water equations

We investigate how the scaling of the interior-penalty parameter affects nonsymmetric interior-penalty Galerkin (NIPG) discretizations of the viscous rotating shallow-water equations in geopotential variables. The hyperbolic terms are approximated by a local Lax--Friedrichs flux, while viscosity acts on the momentum variables through a penalty law $\mu_e=\sigma h_e^{-\beta}$. The standard choice $\beta=1$ and the super-penalized choice $\beta=3$ are compared with a symmetric interior-penalty Galerkin reference. For the diffusion form, we establish consistency, continuity for $\beta\ge 1$, and an exact coercivity identity in the momentum DG seminorm. Manufactured-solution tests show that super-penalization can recover the expected momentum $L^2$ accuracy, whereas the coupled geopotential variable need not exhibit the same improvement. Rotating and topography-aware tests further show that the standard scaling generally gives the better accuracy-cost compromise for the explicit implementation considered here.

math.NA↗

arXiv2026

A Landau-de Gennes Type Theory for Cholesteric-Helical Smectic-Smectic C* Liquid Crystal Phase Transitions

We present a rigorous mathematical analysis of a modified Landau-de Gennes (LdG) theory modeling temperature-driven phase transitions between cholesteric, helical smectic, and smectic C* phases. This model couples a tensor-valued order parameter (nematic orientational order) with a real-valued order parameter (smectic layer modulation). We establish the existence of energy minimizers of the modified LdG energy in three dimensions, subject to Dirichlet conditions, and rigorously analyze the energy minimizers in two asymptotic limits. First, in the Oseen--Frank limit, we show that the global minimizer strongly converges to a minimizer of the Landau-de Gennes bulk energy. Second, in the limit of dominant elastic constants, we prove that the global minimizers converge to a classical helical director profile. Finally, through stability analysis and bifurcation theory, we derive the complete sequence of symmetry-breaking transitions with decreasing temperature-from the cholesteric phase (with in-plane twist and no layering) to an intermediate helical smectic phase (with in-plane twist and layering), and ultimately to the smectic C* phase (with out-of-plane twist and layering). These theoretical results are supported by numerical simulations.

math.AP↗

arXiv2025

Solving time-fractional diffusion equations with Robin boundary conditions via fractional Hamiltonian boundary value methods

In this paper, we propose a novel numerical scheme for solving time-fractional reaction-diffusion problems with Robin boundary conditions, where the time derivative is in the Caputo sense of order $\alpha\in(0,1)$. The existence and uniqueness of the solution is proved. Our proposed method is based on the spectral collocation method in space and Fractional Hamiltonian boundary value methods in time. For the considered spectral collocation method, the basis functions used are not the standard polynomial basis functions, but rather adapt to Robin boundary conditions, and the exponential convergence property is provided. The proposed procedure achieves spectral accuracy in space and is also capable of getting spectral accuracy in time. Some numerical examples are provided to support the theoretical results.

math.NA↗

arXiv2025

Tensorial Model of Chiral Smectic C Liquid Crystals

We propose a continuum tensorial model for chiral smectic C (SmC$^*$) liquid crystals using a tensor-valued order parameter $\mathbf{Q}$ to describe orientational order and a real-valued order parameter $\delta\rho$ to capture layer modulation. This model accounts for the coupled effects of nematic alignment, smectic layering, chirality and spontaneous polarisation inherent in SmC$^*$ systems. The model is validated through a series of numerical experiments -- helix, bistable switching, bookshelf and chevron, and defect in SmC$^*$. Our results highlight the model's capability to describe complex phenomena in SmC$^*$ liquid crystals, providing a robust foundation for further theoretical and applied studies on phase transitions.

cond-mat.soft↗

arXiv2024

A Simple Tensorial Theory of Smectic C Liquid Crystals

The smectic C (smC) phase represents a unique class of liquid crystal phases characterised by the layered arrangement of molecules with tilted orientations with respect to layer normals. Building upon the real-valued tensorial smectic A (smA) model in [Xia et al., PRL, 126, 177801 (2021)], we propose a new continuum mathematical model for smC (and smA) by introducing a novel coupling term between the real tensor containing orientational information and density variation, to control the tilt angle between directors and the layer normal (the tilt angle is zero for smA and nonzero for smC). To validate our proposed model, we conduct a series of two- and three-dimensional numerical experiments that account for typical structures in smectics: chevron patterns, defects, dislocations and toroidal focal conic domains (TFCDs). These results also reveal the phenomenological differences between smA and smC configurations.

cond-mat.soft↗

arXiv2023

Colloidal smectics in button-like confinements: experiment and theory

Liquid crystals can self-organize into a layered smectic phase. While the smectic layers are typically straight forming a lamellar pattern in bulk, external confinement may drastically distort the layers due to the boundary conditions imposed on the orientational director field. Resolving this distortion leads to complex structures with topological defects. Here, we explore the configurations adopted by two-dimensional colloidal smectics made from nearly hard rod-like particles in complex confinements, characterized by a button-like structure with two internal boundaries (inclusions): a two-holed disk and a double annulus. The topology of the confinement generates new structures which we classify in reference to previous work as generalized laminar and generalized Shubnikov states. To explore these configurations, we combine particle-resolved experiments on colloidal rods with three complementary theoretical approaches: Monte-Carlo simulation, first-principles density functional theory and phenomenological $\mathbf{Q}$-tensor modeling. This yields a consistent and comprehensive description of the structural details. In particular, we characterize a nontrivial tilt angle between the direction of the layers and symmetry axes of the confinement.

cond-mat.soft↗

arXiv2022

Variational and numerical analysis of a $\mathbf{Q}$-tensor model for smectic-A liquid crystals

We analyse an energy minimisation problem recently proposed for modelling smectic-A liquid crystals. The optimality conditions give a coupled nonlinear system of partial differential equations, with a second-order equation for the tensor-valued nematic order parameter $\mathbf{Q}$ and a fourth-order equation for the scalar-valued smectic density variation $u$. Our two main results are a proof of the existence of solutions to the minimisation problem, and the derivation of a priori error estimates for its discretisation of the decoupled case (i.e., $q=0$) using the $\mathcal{C}^0$ interior penalty method. More specifically, optimal rates in the $H^1$ and $L^2$ norms are obtained for $\mathbf{Q}$, while optimal rates in a mesh-dependent norm and $L^2$ norm are obtained for $u$. Numerical experiments confirm the rates of convergence.

math.NA↗

arXiv2021

One-dimensional ferronematics in a channel: order reconstruction, bifurcations and multistability

We study a model system with nematic and magnetic orders, within a channel geometry modelled by an interval, $[-D, D]$. The system is characterised by a tensor-valued nematic order parameter $\mathbf{Q}$ and a vector-valued magnetisation $\mathbf{M}$, and the observable states are modelled as stable critical points of an appropriately defined free energy. In particular, the full energy includes a nemato-magnetic coupling term characterised by a parameter $c$. We (i) derive $L^\infty$ bounds for $\mathbf{Q}$ and $\mathbf{M}$; (ii) prove a uniqueness result in parameter regimes defined by $c$, $D$ and material- and temperature-dependent correlation lengths; (iii) analyse order reconstruction solutions, possessing domain walls, and their stabilities as a function of $D$ and $c$ and (iv) perform numerical studies that elucidate the interplay of $c$ and $D$ for multistability.

math.AP↗

arXiv2021

Structural Landscapes in Geometrically Frustrated Smectics

A phenomenological free energy model is proposed to describe the behavior of smectic liquid crystals, an intermediate phase that exhibits orientational order and layering at the molecular scale. Advantageous properties render the functional amenable to numerical simulation. The model is applied to a number of scenarios involving geometric frustration, leading to emergent structures such as focal conic domains and oily streaks and enabling detailed elucidation of the very rich energy landscapes that arise in these problems.

cond-mat.soft↗

arXiv2020

Augmented Lagrangian preconditioners for the Oseen-Frank model of nematic and cholesteric liquid crystals

We propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen-Frank model arising in 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.

math.NA↗