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

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.

• Publication year

  2016

• Venue

  Journal of Computational Physics

• Publication date

  2016-01-15

• Fields of study

  Physics, Materials Science, Computer Science, Engineering

• Identifiers
  DOI 10.1016/j.jcp.2016.11.004arXiv 1601.03907
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• On the theoretical foundations of thin solid and liquid shells

  2017cited by this paper
• A finite element method to compute three-dimensional equilibrium configurations of fluid membranes: Optimal parameterization, variational formulation and applications

  2015cited by this paper
• Membrane tension and peripheral protein density mediate membrane shape transitions

  2015cited by this paper
• An unbiased computational contact formulation for 3D friction

  2015cited by this paper
• Endocytic proteins drive vesicle growth via instability in high membrane tension environment

  2015cited by this paper
• Mechanisms shaping cell membranes.

  2014cited by this paper
• Mechanical collapse of confined fluid membrane vesicles

  2014cited by this paper
• Stabilized finite element formulations for liquid membranes and their application to droplet contact

  2014influential reference
• Protein-induced membrane curvature alters local membrane tension.

  2014influential reference
• A model for the generation and interconversion of ER morphologies

  2014cited by this paper
• Simulating vesicle-substrate adhesion using two phase field functions

  2014cited by this paper
• ON TWO-FIELD NURBS-BASED ISOGEOMETRIC FORMULATION FOR INCOMPRESSIBLE MEDIA PROBLEMS

  2013cited by this paper
• Continuum Modeling of Cell Membranes

  2013cited by this paper
• A finite element method for viscous membranes

  2013cited by this paper
• An adaptive meshfree method for phase-field models of biomembranes. Part I: Approximation with maximum-entropy basis functions

  2013cited by this paper
• Spontaneous tubulation of membranes and vesicles reveals membrane tension generated by spontaneous curvature.

  2013cited by this paper
• Ultrastructural Imaging of Endocytic Sites in Saccharomyces cerevisiae by Transmission Electron Microscopy and Immunolabeling

  2013cited by this paper
• Two-component coarse-grained molecular-dynamics model for the human erythrocyte membrane.

  2012cited by this paper
• Internal forces, tension and energy density in tethered cellular membranes.

  2012cited by this paper
• Fluid membrane vesicles in confinement

  2012cited by this paper
• Shape dynamics, lipid hydrodynamics, and the complex viscoelasticity of bilayer membranes [corrected].

  2012cited by this paper
• Interaction between surface shape and intra-surface viscous flow on lipid membranes

  2012cited by this paper
• A computational formulation for constrained solid and liquid membranes considering isogeometric finite elements

  2012influential reference
• Plasma membrane reshaping during endocytosis is revealed by time-resolved electron tomography.

  2012influential reference
• A level set projection model of lipid vesicles in general flows

  2011cited by this paper
• A novel two-layer, coupled finite element approach for modeling the nonlinear elastic and viscoelastic behavior of human erythrocytes

  2011cited by this paper
• Determinants of endocytic membrane geometry, stability, and scission

  2011cited by this paper
• Eisosome proteins assemble into a membrane scaffold

  2011cited by this paper
• Cell motility: the integrating role of the plasma membrane

  2011influential reference
• Modeling and computation of two phase geometric biomembranes using surface finite elements

  2010cited by this paper
• Isogeometric finite element data structures based on Bézier extraction of T‐splines

  2010cited by this paper
• Multiscale simulation of erythrocyte membranes.

  2010cited by this paper
• Monte Carlo simulations of fluid vesicles with in-plane orientational ordering.

  2010cited by this paper
• A class of dynamin-like GTPases involved in the generation of the tubular ER network.

  2009cited by this paper
• Formation of Protocell-like Vesicles in a Thermal Diffusion Column

  2009cited by this paper
• Modeling protein-mediated morphology in biomembranes

  2009influential reference
• Mechanisms shaping the membranes of cellular organelles.

  2009cited by this paper
• Relaxation dynamics of fluid membranes.

  2009cited by this paper
• Isogeometric Analysis: Toward Integration of CAD and FEA

  2009influential reference
• Nonlinear Finite Element Methods

  2008cited by this paper
• Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics

  2007cited by this paper
• How proteins produce cellular membrane curvature

  2006cited by this paper
• Eisosomes mark static sites of endocytosis

  2006cited by this paper
• Finite element modeling of lipid bilayer membranes

  2006cited by this paper
• Rough sheets and smooth tubules.

  2006cited by this paper
• Convergence of numerical approximations to a phase field bending elasticity model of membrane deformations

  2006cited by this paper
• Membrane curvature and mechanisms of dynamic cell membrane remodelling

  2005cited by this paper
• Dynamin and the actin cytoskeleton cooperatively regulate plasma membrane invagination by BAR and F-BAR proteins.

  2005cited by this paper
• The biogenesis of multivesicular endosomes

  2004influential reference
• BAR Domains as Sensors of Membrane Curvature: The Amphiphysin BAR Structure

  2004influential reference
• Mechanics of the human red blood cell deformed by optical tweezers

  2003cited by this paper
• Membrane tube formation from giant vesicles by dynamic association of motor proteins

  2003cited by this paper
• On the variational theory of cell-membrane equilibria

  2003cited by this paper
• Formation and interaction of membrane tubes.

  2002cited by this paper
• Computational Contact Mechanics

  2002cited by this paper
• The Many Shapes of Mitochondrial Membranes

  2001cited by this paper
• Fully C1‐conforming subdivision elements for finite deformation thin‐shell analysis

  2001cited by this paper
• Fluid Films with Curvature Elasticity

  1999influential reference
• Endoplasmic reticulum membrane tubules are distributed by microtubules in living cells using three distinct mechanisms.

  1998cited by this paper
• NUMERICAL OBSERVATION OF NONAXISYMMETRIC VESICLES IN FLUID MEMBRANES

  1998cited by this paper
• The Cell

  1997cited by this paper
• Vesicular instabilities: The prolate-to-oblate transition and other shape instabilities of fluid bilayer membranes.

  1995cited by this paper
• Finite Element Procedures

  1995influential reference
• Microtubules and the endoplasmic reticulum are highly interdependent structures

  1986cited by this paper
• Elastic Properties of Lipid Bilayers: Theory and Possible Experiments

  1973cited by this paper
• The finite element method with Lagrangian multipliers

  1973influential reference
• The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell.

  1970cited by this paper
• Vectors, Tensors and the Basic Equations of Fluid Mechanics

  1962cited by this paper

• A computational model for short-range van der Waals interactions between beams and shells

  2026cites this paper
• (2+δ)-dimensional theory of the electromechanics of lipid membranes. III. Constitutive models.

  2025cites this paper
• A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds - Stabilized finite element formulation

  2025cites this paper
• A curvilinear surface ALE formulation for self-evolving Navier–Stokes manifolds – general theory and analytical solutions

  2025influential citation
• Shear stresses in fluid and solid membranes with bending elasticity.

  2024cites this paper
• Mesoscopic modeling of membranes at cellular scale

  2024cites this paper
• A finite strain model for fiber angle plasticity of textile fabrics based on isogeometric shell finite elements

  2024cites this paper
• Irreversible thermodynamics of curved lipid membranes. II. Permeability and osmosis.

  2024cites this paper
• Arbitrary Lagrangian–Eulerian finite element method for lipid membranes

  2024cites this paper
• A computational model of red blood cells using an isogeometric formulation with T-splines and a lattice Boltzmann method

  2024cites this paper
• Osmotically driven beading instability in axons: Continuum theory, perturbation analysis and finite element implementation

  2024cites this paper
• Computational Modeling of Coupled Interactions of Fluid Membranes with Embedded Filaments

  2023cites this paper
• A Review of Continuum Mechanics for Mechanical Deformation of Lipid Membranes

  2023cites this paper
• A simple and efficient hybrid discretization approach to alleviate membrane locking in isogeometric thin shells

  2023cites this paper
• Biomimetic torene shells

  2023cites this paper
• A novel constitutive model for surface elasticity at finite strains suitable across compressibility spectrum

  2023cites this paper
• A NURBS-based inverse analysis of swelling induced morphing of thin stimuli-responsive polymer gels

  2022cites this paper
• An isogeometric finite element formulation for boundary and shell viscoelasticity based on a multiplicative surface deformation split

  2022cites this paper
• Modeling curvature-resisting material surfaces with isogeometric analysis

  2022cites this paper
• A new anisotropic bending model for nonlinear shells: Comparison with existing models and isogeometric finite element implementation

  2022influential citation
• Mem3DG: Modeling membrane mechanochemical dynamics in 3D using discrete differential geometry

  2021influential citation
• A Finite Element-Based Coarse-Grained Model for Cell–Nanomaterial Interactions by Combining Absolute Nodal Coordinate Formula and Brownian Dynamics

  2021cites this paper
• A general theory for anisotropic Kirchhoff-Love shells with embedded fibers and in-plane bending

  2021cites this paper
• Numerical simulation of the viral entry into a cell driven by receptor diffusion

  2021cites this paper
• Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff–Love kinematicsand revealed by a three-dimensional computational framework

  2021influential citation
• A gauge-fixing procedure for spherical fluid membranes and application to computations

  2021cites this paper
• Numerical shape optimization of the Canham-Helfrich-Evans bending energy

  2021cites this paper
• A general isogeometric finite element formulation for rotation-free shells with embedded fibers and in-plane bending

  2021cites this paper
• A viscous active shell theory of the cell cortex

  2021cites this paper
• A general isogeometric finite element formulation for rotation‐free shells with in‐plane bending of embedded fibers

  2021cites this paper
• The Mechanics and Thermodynamics of Tubule Formation in Biological Membranes

  2020cites this paper
• Isogeometric continuity constraints for multi-patch shells governed by fourth-order deformation and phase field models

  2020cites this paper
• Transport Phenomena in Fluid Films with Curvature Elasticity

  2020cites this paper
• A general theory for anisotropic Kirchhoff–Love shells with in-plane bending of embedded fibers

  2020cites this paper
• Efficient isogeometric finite element formulations for coupled systems on deforming surfaces

  2019cites this paper
• Computational modeling of active deformable membranes embedded in three-dimensional flows.

  2019cites this paper
• Interactions between model inclusions on closed lipid bilayer membranes

  2019cites this paper
• Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations

  2019cites this paper
• A NURBS-based inverse analysis of thermal expansion induced morphing of thin shells

  2019cites this paper
• On the use of local maximum entropy approximants for Cahn–Hilliard phase-field models in 2D domains and on surfaces

  2019cites this paper
• A mechanical model reveals that non-axisymmetric buckling lowers the energy barrier associated with membrane neck constriction

  2019influential citation
• An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

  2019influential citation
• Non-axisymmetric shapes of biological membranes from locally induced curvature

  2019cites this paper
• Out-of-equilibrium mechanochemistry and self-organization of fluid membranes interacting with curved proteins

  2019cites this paper
• Thin shell formulations: Axisymmetric and 3D models

  2019cites this paper
• Modelling fluid deformable surfaces with an emphasis on biological interfaces

  2018influential citation
• A finite membrane element formulation for surfactants

  2018cites this paper
• Modeling Membrane-Protein Interactions

  2018cites this paper
• Modeling Membrane Curvature Generation due to Membrane–Protein Interactions

  2018cites this paper
• The multiplicative deformation split for shells with application to growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity

  2018cites this paper
• Non-body-fitted fluid-structure interaction: Divergence-conforming B-splines, fully-implicit dynamics, and variational formulation

  2018cites this paper
• Hydrodynamic flows on curved surfaces: Spectral numerical methods for radial manifold shapes

  2018cites this paper
• On the Computational Modeling of Lipid Bilayers Using Thin-Shell Theory

  2018influential citation
• The linear analysis of thin shell problems using the numerical manifold method

  2018cites this paper
• A NURBS-based Inverse Analysis for Reconstruction of Nonlinear Deformations of Thin Shell Structures

  2018cites this paper
• Onsager’s Variational Principle in Soft Matter: Introduction and Application to the Dynamics of Adsorption of Proteins onto Fluid Membranes

  2018cites this paper
• Arbitrary Lagrangian-Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces

  2018influential citation
• Irreversible thermodynamics of curved lipid membranes.

  2017influential citation
• Theory and algorithms to compute Helfrich bending forces: a review

  2017influential citation
• Review and perspective on soft matter modeling in cellular mechanobiology: cell contact, adhesion, mechanosensing, and motility

  2017cites this paper
• Review and perspective on soft matter modeling in cellular mechanobiology: cell contact, adhesion, mechanosensing, and motility

  2017cites this paper
• A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries

  2017influential citation
• On the theoretical foundations of thin solid and liquid shells

  2017cites this paper
• An isogeometric finite element formulation for phase transitions on deforming surfaces

  2017cites this paper
• A monolithic fluid-structure interaction formulation for solid and liquid membranes including free-surface contact

  2017cites this paper
• A frictional sliding algorithm for liquid droplets

  2016influential citation
• Review and Perspective in Mechanics

  2015cites this paper
• General neck condition for the limit shape of budding vesicles.

  2015cites this paper
• ICES REPORT 19-12 July 2019 An adaptive space-time phase ■ eld formulation for dynamic fracture of brittle shells based on LR NURBS

  year unknowncites this paper