Finite element approximation for the dynamics of fluidic two-phase biomembranes

Barrett, John W.; Garcke, Harald; Nürnberg, Robert

doi:10.1051/m2an/2017037

Cited by 29 publications

(19 citation statements)

References 41 publications

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“…In addition, we mention that even for closed surfaces, Gaussian curvature contributions play a role in the context of two-phase models, where the Gauss curvature moduli depend on the phase. See Jülicher and Lipowsky (1996); Elliott and Stinner (2013); Barrett et al (2016a) for numerical approximations of such models. It is the goal of this paper to derive and analyze a finite element approximation of L 2 -gradient flows for curvature functionals of Willmore and Helfrich type that allow also for Gaussian curvature and (semi-)free boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature

Barrett

Garcke

Nürnberg

2017

IMA Journal of Numerical Analysis

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We study numerical approximations for geometric evolution equations arising as gradient flows for energy functionals that are quadratic in the principal curvatures of a two-dimensional surface. Beside the well-known Willmore and Helfrich flows we will also consider flows involving the Gaussian curvature of the surface. Boundary conditions for these flows are highly nonlinear, and we use a variational approach to derive weak formulations, which naturally can be discretized with the help of a mixed finite element method. Our approach uses a parametric finite element method, which can be shown to lead to good mesh properties. We prove stability estimates for a semidiscrete (discrete in space, continuous in time) version of the method and show existence and uniqueness results in the fully discrete case. Finally, several numerical results are presented involving convergence tests as well as the first computations with Gaussian curvature and/or free or semi-free boundary conditions.

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Section: Introductionmentioning

confidence: 99%

Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature

Barrett

Garcke

Nürnberg

2017

IMA Journal of Numerical Analysis

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“…Simpler systems of synthetic multicomponent vesicles, whose membranes can be composed of different lipid species, have been used to study the rich patterns and accompanying morphologies which emerge from elastic heterogeneity (Baumgart, Hess & Webb 2003; Veatch & Keller 2003). These findings have been corroborated and expanded upon using both numerical and analytical techniques (Elliott & Stinner 2013; Barrett, Garcke & Nürnberg 2017), which in turn are of use when attempting to infer membrane properties experimentally (Engelhardt, Duwe & Sackmann 1985; Baumgart et al. 2005; Tian et al.…”

Section: Introductionmentioning

confidence: 86%

Swinging and tumbling of multicomponent vesicles in flow

2022

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Biological membranes are host to proteins and molecules which may form domain-like structures resulting in spatially varying material properties. Vesicles with such heterogeneous membranes can exhibit intricate shapes at equilibrium and rich dynamics when placed into a flow. Under the assumption of small deformations and a two-dimensional system, we develop a reduced-order model to describe the fluid-structure interaction between a viscous background shear flow and an inextensible membrane with spatially varying bending stiffness and spontaneous curvature. Material property variations of a critical magnitude, relative to the flow rate and internal/external viscosity contrast, can set off a qualitative change in the vesicle dynamics. A membrane of nearly constant bending stiffness or spontaneous curvature undergoes a small amplitude swinging motion (which includes tangential tank-treading), while for large enough material variations the dynamics pass through a regime featuring tumbling and periodic phase-lagging of the membrane material, and ultimately for very large material variation to a rigid-body tumbling behaviour. Distinct differences are found for even and odd spatial modes of domain distribution. Full numerical simulations are used to probe the theoretical predictions, which appear valid even when studying substantially deformed membranes.

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“…The diffuse interface formulation has been introduced in [155,157]. Other variants have also been studied [10,58,69,40,84,154,274,184], together withs analytical studies [70,331,53] and numerical approximations [118,93,162,161,438] for the deformation and dynamics of vesicles. The effective phase field modeling of Gaussian curvature energy also leads to an interesting development of the diffuse interface Euler-Poincaré characteristics [156,53] that can be used to detect topological change of the implicitly defined interface within the phase field framework beyond the biophysical applications.…”

Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

The phase field method for geometric moving interfaces and their numerical approximations

Feng

2020

Handbook of Numerical Analysis

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This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science. Key words and phrases. Phase field method, geometric law, curvature-driven flow, geometric nonlinear PDEs, finite difference methods, finite element methods, spectral methods, discontinuous Galerkin methods, adaptivity, coarse and fine error estimates, convergence of numerical interfaces, nonlocal and stochastic phase field models, microstructure evolution, biology and image science applications.

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Finite element approximation for the dynamics of fluidic two-phase biomembranes

Cited by 29 publications

References 41 publications

Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature

Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature

Swinging and tumbling of multicomponent vesicles in flow

The phase field method for geometric moving interfaces and their numerical approximations

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