A finite element method to compute three-dimensional equilibrium configurations of fluid membranes: Optimal parameterization, variational formulation and applications

Rangarajan, Ramsharan; Gao, Huajian

doi:10.1016/j.jcp.2015.05.001

Cited by 21 publications

(27 citation statements)

References 60 publications

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“…A simple solution to this issue is to then update the reference configuration Γ t0 . This kind of parametrization, proposed by Rangarajan & Gao (2015), generalizes the classical Monge parametrization, which is recovered by setting Γ t0 to a plane, M to its constant normal and h to the height of the surface Γ t with respect to the plane (do Carmo 2016). We finally note that for this kind of surface parametrization, we have…”

Section: An Ale Parametrization Based On An Offsetmentioning

confidence: 99%

Modelling fluid deformable surfaces with an emphasis on biological interfaces

2019

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Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex, or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry, and geometry, and govern important biological processes such as cellular traffic, division, migration, or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid-fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of Arbitrarily Lagrangian-Eulerian (ALE) formulations, wellestablished for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimize a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager's formalism and our ALE formulation, to deal with the resulting stiff system of higher-order of partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models for the first time in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.

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Section: An Ale Parametrization Based On An Offsetmentioning

confidence: 99%

Modelling fluid deformable surfaces with an emphasis on biological interfaces

2019

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“…This reference shows that scheme 'a' is highly accurate and performs much better than scheme 'A'. It is also shown that applying the stabilization stresses (47) and (48) only to the in-plane part is much more accurate than applying it throughout the system (i.e. in both Eqs.…”

Section: In-plane Shear and Bulk Stabilizationmentioning

confidence: 99%

“…Under quasi-static conditions, liquid membranes and shells therefore do not provide any resistance to in-plane shear deformations and thus need to be stabilized. Various stabilization methods have been proposed in the past, considering artificial viscosity [40,52], artificial stiffness [28] and normal offsets -either as a projection of the solution (with intermediate mesh update steps) [52], or as a restriction of the surface variation [48]. The instability problem is absent, if shear stiffness is present, e.g.…”

Section: Introductionmentioning

confidence: 99%

A stabilized finite element formulation for liquid shells and its application to lipid bilayers

Sauer

Duong

Mandadapu

et al. 2017

Journal of Computational Physics

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This paper presents a new finite element (FE) formulation for liquid shells that is based on an explicit, 3D surface discretization using C 1 -continuous finite elements constructed from NURBS interpolation. Both displacement-based and mixed displacement/pressure FE formulations are proposed. The latter is needed for area-incompressible material behavior, where penalty-type regularizations can lead to misleading results. In order to obtain quasi-static solutions for liquid shells devoid of shear stiffness, several numerical stabilization schemes are proposed based on adding stiffness, adding viscosity or using projection. Several numerical examples are considered in order to illustrate the accuracy and the capabilities of the proposed formulation, and to compare the different stabilization schemes. The presented formulation is capable of simulating non-trivial surface shapes associated with tube formation and protein-induced budding of lipid bilayers. In the latter case, the presented formulation yields non-axisymmetric solutions, which have not been observed in previous simulations. It is shown that those non-axisymmetric shapes are preferred over axisymmetric ones.

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“…Indeed, bilayers are highly dynamical, but due to the complexity of the chemical and hydrodynamical effects involved, theory and experiments have focused on equilibrium. For instance, the classical bending model of Helfrich (Helfrich, 1973;Lipowsky, 1991;Jülicher and Lipowsky, 1993;Staykova et al, 2013) has been very successful in understanding equilibrium conformations (Steigmann, 1999;Capovilla and Guven, 2002;Tu and Ou-Yang, 2004;Feng and Klug, 2006;Rangarajan and Gao, 2015;Sauer et al, 2017), but is insufficient to understand the reconfigurations of membranes when subjected to transient stimuli. To address this challenge, models and simulations coupling membrane hydrodynamics and elasticity (Arroyo and DeSimone, 2009;Arroyo et al, 2010;Rahimi and Arroyo, 2012;Rangamani et al, 2013;Rodrigues et al, 2013;Barrett et al, 2016) or elasticity and the phaseseparation of chemical species (Embar et al, 2013;Elliott and Stinner, 2013) are emerging in recent years, but only provide initial steps towards a general dynamical framework.…”

Section: Introductionmentioning

confidence: 99%