Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and Newton methods

International audienceThis framework is concerned with the numerical modeling of the dynamics of individual biomembranes and capillary interfaces in a surrounding Newtonian fluid. A level set approach helps to follow the interface motion. Our method features the use of high order fully implicit time integration schemes that enable to overcome stability issues related to the explicit discretization of the highly non-linear bending force or capillary force. At each time step, the tangent systems are derived and the resulting nonlinear problems are solved by a Newton–Raphson method. Based on the signed distance assumption, several inexact Newton strategies are employed to solve the capillary and vesicle problems and guarantee the second-order convergence behavior. We address in detail the main features of the proposed method, and we report several experiments in the two-dimensional case with the aim of illustrating its accuracy and efficiency. Comparative investigations with respect to the fully explicit scheme depict the stabilizing effect of the new method, which allows to use significantly larger time step sizes

Modelling fluid deformable surfaces with an emphasis on biological interfaces

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, well-established 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 minimizes 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 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 in their full three-dimensional generality, accounting for finite curvatures and finite shape changes. This article has been published in a revised form in Journal of fluid mechanics

Modelling fluid deformable surfaces with an emphasis on biological interfaces

This article has been published in a revised form in Journal of fluid mechanics, http://dx.doi.org/10.1017/jfm.2019.341. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2019Fluid 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, well-established 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 minimizes 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 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 in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.Peer ReviewedPostprint (author's final draft