On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach

Chun, Liu; Wang, Yiwei

doi:10.1016/j.jcp.2020.109566

Cited by 28 publications

(36 citation statements)

References 65 publications

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“…where M(x) is a positive-definite matrix that determines the dynamics approaching the equilibrium, Q ∈ R 5 is the vectorized order parameter Q. By a discrete energetic variational approach [81], the gradient flow equation of Q corresponds to energy-dissipation law (B6) can be approximated by the gradient flow equations of the coefficients q lnm , given by…”

Section: Appendix B: Numerical Methodsmentioning

confidence: 99%

Dynamic tuning of the director field in liquid crystal shells using block copolymers

Noh

Wang

Liang

et al. 2020

Phys. Rev. Research

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When an orientationally ordered system, like a nematic liquid crystal (LC), is confined on a self-closing spherical shell, topological constraints arise with intriguing consequences that depend critically on how the LC is aligned in the shell. We demonstrate reversible dynamic tuning of the alignment, and thereby the topology, of nematic LC shells stabilized by the nonionic amphiphilic block copolymer Pluronic F127. Deep in the nematic phase, the director (the average molecule orientation) is tangential to the interface, but upon approaching the temperature T NI of the nematic-isotropic transition, the director realigns to normal. We link this to a delicate interplay between an interfacial tension that is nearly independent of director orientation, and the configuration-dependent elastic deformation energy of an LC confined in a shell. The process is primarily triggered by the heating-induced reduction of the nematic order parameter, hence realignment temperatures differ by several tens of degrees between LCs with high and low T NI , respectively. The temperature of realignment is always lower on the positive-curved shell outside than at the negative-curved inside, yielding a complex topological reconfiguration on heating. Complementing experimental investigations with mathematical modeling and computer simulations, we identify and investigate three different trajectories, distinguished by their configurations of topological defects in the initial tangential-aligned shell. Our results uncover a new aspect of the complex response of LCs to curved confinement, demonstrating that the order of the LC itself can influence the alignment and thereby the topology of the system. They also reveal the potential of amphiphilic block copolymer stabilizers for enabling continuous tunability of LC shell configuration, opening doors for in-depth studies of topological dynamics as well as novel applications in, e.g., sensing and programed soft actuators.

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Section: Appendix B: Numerical Methodsmentioning

confidence: 99%

Dynamic tuning of the director field in liquid crystal shells using block copolymers

Noh

Wang

Liang

et al. 2020

Phys. Rev. Research

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“…Hence the stationary points of the discrete free energy F h (Ξ) are exactly the numerical solutions of the Euler-Lagrange equation (δF)/(δϕ) = 0 obtained by a Galerkin method. The idea of discretization first can be extended to dynamics cases with variational structure; we refer the interested reader to Doi, Zhou, Di and Xu (2019) and Liu and Wang (2020a) for some recent developments. In particular, by using the strategy of 'discretize-then-variation', Liu and Wang (2020b) proposed a variational Lagrangian scheme for a phase field model, which has an advantage in capturing the thin diffuse interface with a small number of mesh points.…”

Section: Energy-minimization-based Approachmentioning

confidence: 99%

Modelling and computation of liquid crystals

Wang¹,

Zhang

2021

Acta Numerica

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Liquid crystals are a type of soft matter that is intermediate between crystalline solids and isotropic fluids. The study of liquid crystals has made tremendous progress over the past four decades, which is of great importance for fundamental scientific research and has widespread applications in industry. In this paper we review the mathematical models and their connections to liquid crystals, and survey the developments of numerical methods for finding rich configurations of liquid crystals.

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“…26,48,70,71 From a numerical perspective, the EnVarA formulation also provides a guide line to develop structure-preserving numerical schemes for different systems. [49][50][51] For an isothermal and closed system, an energy-dissipation law is often given by…”

Section: Energetic Variational Approach To the Micro-macro Modelmentioning

confidence: 99%

“…69, applying the kernel regularization to the equation may not preserve variational structure at the particle level. To preserve the variational structure, we apply the particle approximation at the energy-dissipation law level and employ a discrete energetic variational approach 50,69 to derive a coarse-grained micro-macro model with a particle approximation first. A structure-preserving particle-FEM discretization is developed for the coarse-grained model.…”

Section: Introductionmentioning

confidence: 99%

On a deterministic particle-FEM discretization to micro-macro models of dilute polymeric fluids

Bao¹,

Li²,

Wang³

2021

Preprint

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In this paper, we propose a deterministic particle-FEM discretization to micro-macro models of dilute polymeric fluids, which combines a finite element discretization to the macroscopic fluid dynamic equation with a variational particle scheme to the microscopic Fokker-Planck equation. The discretization is constructed by a discrete energetic variational approach, and preserves the microscopic variational structure in the semi-discrete level. Numerical examples demonstrate the accuracy and robustness of the proposed numerical scheme for some special external flows with a wide range of flow rates.

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On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach

Cited by 28 publications

References 65 publications

Dynamic tuning of the director field in liquid crystal shells using block copolymers

Dynamic tuning of the director field in liquid crystal shells using block copolymers

Modelling and computation of liquid crystals

On a deterministic particle-FEM discretization to micro-macro models of dilute polymeric fluids

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