Decoupled energy stable schemes for phase-field vesicle membrane model

doi:10.1016/j.jcp.2015.09.025

Journal of Computational Physics

Volume 302, 1 December 2015, Pages 509-523

https://doi.org/10.1016/j.jcp.2015.09.025 Get rights and content

Abstract

We consider the numerical approximations of the classical phase-field vesicle membrane models proposed a decade ago in Du et al. (2004) [6]. We first reformulate the model derived from an energetic variational formulation into a form which is suitable for numerical approximation, and establish the energy dissipation law. Then, we develop a stabilized, decoupled, time discretization scheme for the coupled nonlinear system. The scheme is unconditionally energy stable and leads to linear and decoupled elliptic equations to be solved at each time step. Stability analysis and ample numerical simulations are presented thereafter.

Introduction

In cell biology, a vesicle is a small organelle within a cell, consisting of fluid enclosed by a lipid bilayer membrane. There have been many experimental and analytic studies on the configurations and deformations of elastic vesicle bio-membranes [2], [6], [12], [13], [14], [17]. In the last decade, using the energetic, variational diffuse interface approach, Du et al. proposed a phase-field model to simulate the deformations of simple vesicles coupled with incompressible flow fields [6], [7], [8], [10], in which, the Helfrich bending elastic energy of the surface is replaced by a phase field functional. The evolution equations are then resulted from the variations of the action functional of the free energy.

The diffuse-interface/phase-field models, whose origin can be traced back to [9], [33], have been proved efficient with much success. A particular advantage of the phase-field approach is that they can often be derived from an energy-based variational formalism, leading to well-posed nonlinear coupled systems that satisfy thermodynamics-consistent energy dissipation laws. Thus it is especially desirable to design numerical schemes that preserve the energy dissipation law at the discrete level. Due to the rapid changes near the interface, the non-compliance of energy dissipation laws of the numerical scheme may lead to spurious numerical solutions if the grid and time step sizes are not carefully controlled [11], [21]. Another main advantage of energy stable schemes is that they can be easily combined with an adaptive time stepping strategy [22], [23], [24], [26], [27], [28], [35].

To construct the numerical schemes for the typical phase-field models coupled with the hydrodynamics, in particular, the Allen–Cahn or Cahn–Hilliard equations, the main difficulties include (i) the coupling between the velocity and phase function through the convection term in the phase equation and nonlinear stress in the momentum equation; (ii) the coupling of the velocity and pressure through the incompressibility constraint; (iii) the stiffness of the phase equation associated with the interfacial width. For the phase-field vesicle membrane model [6], [7], [8], [10], things are about to get even worse due to some extra nonlinear terms with second order derivatives. To the best of the authors' knowledge, there does not exist any easy-to-implement and energy stable scheme for this model so far.

Thus, for the phase-field membrane vesicle model, the main purpose of this paper is to construct a time discretization scheme which (a) satisfies a discrete energy law; and (b) leads to decoupled elliptic equations to solve at each time step. This is by no means an easy task due to many highly nonlinear terms and the couplings among the velocity, pressure and phase function.

The rest of the paper is organized as follows. In Section 2, we introduce the phase-field vesicle membrane model and derive the energy dissipation law. In Section 3, we reformulate the PDE to an equivalent form, construct a decoupled, energy stable numerical scheme, and give the stability analysis. In Section 4, we present the spatial discretization using the finite element method. In Section 5, we present some numerical results to illustrate the accuracy and efficiency of the proposed scheme and summarize our contributions. Some concluding remarks are given in Section 6.

Section snippets

Models

The equilibrium shape of a vesicle membrane is determined by minimizing the elastic bending energy [3], [4], $E = \int_{Γ} (a_{1} + a_{2} {(H - c_{0})}^{2} + a_{3} K) d s,$ where $H = \frac{k_{1} + k_{2}}{2}$ represents mean curvature of the membrane surface; $K = k_{1} k_{2}$ is Gaussian curvature; $k_{1}$ , $k_{2}$ are two principle curvatures; $a_{1}$ is the surface tension; $a_{2}$ , $a_{3}$ are bending rigidities; $c_{0}$ represents spontaneous curvature; Γ is a smooth compact surface in the domain $Ω \in R^{3}$ .

If we consider the model to be isotropic, i.e., the spontaneous curvature $c_{0} = 0$ and

Alternative formulation and its decoupled energy stable scheme

The coupled nonlinear system (2.7), (2.8), (2.9) actually presents formidable challenges for algorithm design, implementation as well as numerical analysis. Although many numerical schemes perform well in practice, the question of their stability remains open [8]. The emphasis of our algorithm development is placed on designing numerical schemes that are not only easy-to-implement, but also satisfy a discrete energy dissipation law. We will design schemes that in particular can overcome the

Spatial discretization

In this paper, we take the finite element method for the spatial discretization to test the numerical scheme (3.8), (3.9), (3.10), (3.11), (3.12).

Numerical simulations

In this section, we present some numerical experiments using the schemes constructed in Sections 3 and 4. We use the inf–sup stable $Iso - P 2 / P 1$ element [32] for the velocity and pressure, and linear element for the phase function ϕ.

Conclusions and remarks

In this paper we construct a stabilized, decoupled, time discretization numerical scheme for the phase-field vesicle membrane model. Even though the model was proposed over a decade and applied in many literatures [6], [7], [8], [10], however, there are still not any energy stable schemes available.

Our scheme is unconditionally energy stable, and is quite easy to implement. At each time step, one only needs to solve Poisson type equations due to the decoupling and linearity. All nonlinear terms

Acknowledgments

G. Ji is partially supported by NSFC-11271048 and Fundamental Research Funds for the Center University. H. Zhang is partially supported by NSFC-RGC-11261160486, NSFC-11471046 and the Ministry of Education Program for New Century Excellent Talents Project NCET-12-0053. X. Yang is partially supported by National Science Foundation DMS-1418898, National Science Foundation DMS-1200487, AFOSR FA9550-12-1-0178, SC Epscor Gear Fund and NSFC-11471312.

References (35)

S. Dong et al.
A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios
J. Comput. Phys.
(2012)
Q. Du et al.
A phase field approach in the numerical study of the elastic bending energy for vesicle membranes
J. Comput. Phys.
(2004)
J. Hua et al.
Energy law preserving $C^{0}$ finite element schemes for phase field models in two-phase flow computations
J. Comput. Phys.
(2011)
H. Johnston et al.
Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term
J. Comput. Phys.
(2004)
C. Liu et al.
A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method
Physica D
(2003)
J.M. Lopez et al.
An efficient spectral-projection method for the Navier–Stokes equations in cylindrical geometries
J. Comput. Phys.
(1998)
J. Pyo et al.
Gauge–Uzawa methods for incompressible flows with variable density
J. Comput. Phys.
(2007)
J. Shen et al.
An efficient moving mesh spectral method for the phase-field model of two-phase flows
J. Comput. Phys.
(2009)
Z. Zhang et al.
An adaptive time-stepping strategy for solving the phase field crystal
J. Comput. Phys.
(2013)
F. Boyer et al.
Numerical schemes for a three component Cahn–Hilliard model
ESAIM Math. Model. Numer. Anal.
(2011)

J.W. Cahn et al.

A microscopic theory for domain wall motion and its experimental verification in Fe–Al alloy domain growth kinetics

J. Phys., Colloq.

(1977)

P.G. Ciarlet

Introduction to Linear Shell Theory

(1998)

P.G. Ciarlet

Mathematical Elasticity, vol. III: Theory of Shells

(2000)

Q. Du et al.

Retrieving topological information for phase field models

SIAM J. Appl. Math.

(2005)

Q. Du et al.

Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation

Commun. Pure Appl. Anal.

(2005)

P.C. Hohenberg et al.

Theory of dynamic critical phenomena

Rev. Mod. Phys.

(1977)

Q. Du et al.

A phase field formulation of the Willmore problem

Nonlinearity

(2005)

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View full text

Decoupled energy stable schemes for phase-field vesicle membrane model

Abstract

Introduction

Section snippets

Models

Alternative formulation and its decoupled energy stable scheme

Spatial discretization

Numerical simulations

Conclusions and remarks

Acknowledgments

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Physica D

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Numerical schemes for a three component Cahn–Hilliard model

ESAIM Math. Model. Numer. Anal.

A microscopic theory for domain wall motion and its experimental verification in Fe–Al alloy domain growth kinetics

J. Phys., Colloq.

Introduction to Linear Shell Theory

Mathematical Elasticity, vol. III: Theory of Shells

Retrieving topological information for phase field models

SIAM J. Appl. Math.

Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation

Commun. Pure Appl. Anal.

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Nonlinearity