Arbitrary Lagrangian--Eulerian finite element method for curved and deforming surfaces. I. General theory and application to fluid interfaces

Abstract

An arbitrary Lagrangian--Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf--sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.

arXiv comment59 pages, 16 figures

References in corpus (24)

• Formation and Interaction of Membrane Tubes
• Lecture Notes on General Relativity
• Dynamics of filaments and membranes in a viscous fluid
• Stresses in lipid membranes
• Relaxation time of the topological T1 process in a two-dimensional foam
• Protein-induced membrane curvature changes membrane tension
• Endocytic proteins drive vesicle growth via instability in high membrane tension environment
• Membrane-Protein Interactions in a Generic Coarse-Grained Model for Lipid Bilayers
• Modelling fluid deformable surfaces with an emphasis on biological interfaces
• Incompressible fluid problems on embedded surfaces: Modeling and variational formulations
• Membrane geometry with auxiliary variables and quadratic constraints
• Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics
• Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds
• Solving the incompressible surface Navier-Stokes equation by surface finite elements
• A stabilized finite element formulation for liquid shells and its application to lipid bilayers
• Activating membranes
• The irreversible thermodynamics of curved lipid membranes
• Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes
• An isogeometric finite element formulation for phase transitions on deforming surfaces
• Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranes
• Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
• Curvature-coupling dependence of membrane protein diffusion coefficients
• Shear-Driven Circulation Patterns in Lipid Membrane Vesicles
• A monolithic fluid-structure interaction formulation for solid and liquid membranes including free-surface contact

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• Geometry and dynamics of lipid membranes: The Scriven--Love number
• A shallow physics-informed neural network for solving partial differential equations on surfaces
• A numerical approach for fluid deformable surfaces with conserved enclosed volume
• Transport Phenomena in Fluid Films with Curvature Elasticity
• On equilibrium states of fluid membranes
• Absolute vs Convective Instabilities and Front Propagation in Lipid Membrane Tubes
• Force Dipole Interactions in Tubular Fluid Membranes
• Efficient reduction of vertex clustering using front tracking with surface normal propagation restriction
• Computational Modeling of Coupled Interactions of Fluid Membranes with Embedded Filaments
• Stokes flow of an evolving fluid film with arbitrary shape and topology
• Shear stresses in fluid and solid membranes with bending elasticity
• A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds - Stabilized finite element formulation
• Vortex Dynamics in Tubular Fluid Membranes
• The irreversible thermodynamics of curved lipid membranes II: Permeability and osmosis
• A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds -- General theory and analytical solutions
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