An energy law preserving C0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics

Lin, Ping; Liu, Chun; Zhang, Hui

doi:10.1016/j.jcp.2007.09.005

Cited by 76 publications

(110 citation statements)

References 18 publications

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“…Equations (7)(8)(9)(10) form the governing equations for our two-phase system. For discretization using second-order finite elements, the fourth-order Cahn-Hilliard equation is decomposed into two second-order equations [20,35].…”

Section: A Diffuse Interface Modelmentioning

confidence: 99%

“…Because of its energy-based formalism and the physical picture of the diffuse-interface model, it has some unique features among interface-capturing methods [8]: (i) The evolution of the interface is self-consistent and requires no ad hoc intervention such as the re-initialization in level set methods. (ii) The theory has an energy law that ensures well-posedness in numerical computation [9,10].…”

Section: Introductionmentioning

confidence: 99%

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3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

Zhou

Yue

Feng

et al. 2010

Journal of Computational Physics

117

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-This work presents a three-dimensional finite-element algorithm, based on the phase-field model, for computing interfacial flows of Newtonian and complex fluids. A 3D adaptive meshing scheme produces fine grid covering the interface and coarse mesh in the bulk. It is key to accurate resolution of the interface at manageable computational costs.The coupled Navier-Stokes and Cahn-Hilliard equations, plus the constitutive equation for non-Newtonian fluids, are solved using second-order implicit time stepping. Within each time step, Newton iteration is used to handle the nonlinearity, and the linear algebraic system is solved by preconditioned Krylov methods. The phase-field model, with a physically diffuse interface, affords the method several advantages in computing interfacial dynamics.One is the ease in simulating topological changes such as interfacial rupture and coalescence.Another is the capability of computing contact line motion without invoking ad hoc slip conditions. As validation of the 3D numerical scheme, we have computed drop deformation in an elongational flow, relaxation of a deformed drop to the spherical shape, and drop spreading on a partially wetting substrate. The results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate. Excellent agreement is achieved provided that the 3D interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid. Since our model involves several coupled partial differential equations and we use a fully implicit scheme, the matrix inversion requires a large memory. This puts a limit on the scale of problems that can be simulated in 3D, especially for viscoelastic fluids.

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Section: A Diffuse Interface Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

Zhou

Yue

Feng

et al. 2010

Journal of Computational Physics

117

View full text Add to dashboard Cite

show abstract

“…Subsequently there have been many developments and refinements. Various mixed methods have been considered in [2,3,13]; spectral schemes have been developed in [6]; methods with estimates independent of the penalty parameter were developed in [3,27]; and "semi-implicit" schemes which only require the solution of a linear system at each time step are developed in [13]. All of these schemes for this simplified model utilize the first order implicit Euler stepping scheme to evolve the solution in time.…”

Section: Related Resultsmentioning

confidence: 99%

“…The numerical approximation of solutions to the Lin-Liu equations has been studied extensively [2,6,13,[27][28][29]. These papers consider approximations of the weak statement…”

Section: Lin-liu Equationmentioning

confidence: 99%

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Walkington¹

2010

ESAIM: M2AN

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Abstract. Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.Mathematics Subject Classification. 76A15, 65M12, 65M60, 76M10.

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“…With regard to CPU cost, the methods proposed herein end up having eight degrees of freedom per node, only beated by the methods in [23,24] (for the penalized problem) and [4] (for the limit problem); the method in [25] does not introduce any auxiliary unknown but it requires C 1 finite element approximations, dramatically increasing the CPU cost. Furthermore, the methods proposed herein are both unconditionally stable, as the ones in [24,4] (for the penalized problem).…”

Section: Discussionmentioning

confidence: 99%

Finite element approximation of nematic liquid crystal flows using a saddle-point structure

Badia

Guillén-González

Gutiérrez-Santacreu

2011

Journal of Computational Physics

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Abstract. In this work, we propose finite element schemes for the numerical approximation of nematic liquid crystal flows, based on a saddle-point formulation of the director vector sub-problem. It introduces a Lagrange multiplier that allows to enforce the sphere condition. In this setting, we can consider the limit problem (without penalty) and the penalized problem (using a Ginzburg-Landau penalty function) in a unified way. Further, the resulting schemes have an stable behavior with respect to the value of the penalty parameter, a key difference with respect to the existing schemes. Two different methods have been considered for the time integration. First, we have considered an implicit algorithm that is unconditionally stable and energy preserving. The linearization of the problem at every time step value can be performed using a quasi-Newton method that allows to decouple fluid velocity and director vector computations for every tangent problem. Then, we have designed a linear semi-implicit algorithm (i.e. it does not involve nonlinear iterations) and proved that it is unconditionally stable, verifying a discrete energy inequality. Finally, some numerical simulations are provided.

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An energy law preserving C0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics

Cited by 76 publications

References 18 publications

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Finite element approximation of nematic liquid crystal flows using a saddle-point structure

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