Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30855
An Implicit Methodology for the Numerical Modeling of Locally Inextensible Membranes

Authors: Aymen Laadhari


We present in this paper a fully implicit finite element method tailored for the numerical modeling of inextensible fluidic membranes in a surrounding Newtonian fluid. We consider a highly simplified version of the Canham-Helfrich model for phospholipid membranes, in which the bending force and spontaneous curvature are disregarded. The coupled problem is formulated in a fully Eulerian framework and the membrane motion is tracked using the level set method. The resulting nonlinear problem is solved by a Newton-Raphson strategy, featuring a quadratic convergence behavior. A monolithic solver is implemented, and we report several numerical experiments aimed at model validation and illustrating the accuracy of the proposed method. We show that stability is maintained for significantly larger time steps with respect to an explicit decoupling method.

Keywords: Finite Element Method, navier-stokes, Newton method, level set, inextensible membrane, liquid drop

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 843


[1] P.R. Amestoy and I.S. Duff and J. Koster and J.-Y. L’Excellent, A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling, SIAM J. Matrix Anal. Appl., 2001, 23(1):15-41.
[2] J.W. Barrett, H. Garcke and R. N¨urnberg, Finite element approximation for the dynamics of asymmetric fluidic biomembranes, preprint No. 03/2015, University Regensburg, Germany (2015).
[3] J.W. Barrett, H. Garcke and R. N¨urnberg, Stable finite element approximations of two-phase flow with soluble surfactant, J. Comput. Phys., 297 (2015), pp. 530–564
[4] T. Biben, K. Kassner and C. Misbah, Phase-field approach to three-dimensional vesicle dynamics, Phys. Rev. E. 2005;72:049121.
[5] F. BREZZI AND M. FORTIN, Mixed and hybrid finite element methods, Springer New York, 15 (1991).
[6] P.B. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theor. Biol., 26 (1970), pp. 61–81
[7] H. Deuling and W. Helfrich, The curvature elasticity of fluid membranes: a catalogue of vesicle shapes, J. Phys. 1976;37:1335–45.
[8] C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Meth. Engng., 2009, 79: 1309-1331.
[9] W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforsch, 1973;28c:693–703.
[10] S. Hysing, A new implicit surface tension implementation for interfacial flows, Int. J. Numer. Methods Fluids, 51 (6) (2006), pp. 659–672
[11] M. Kraus, W. Wintz, U. Seifert and R. Lipowsky, Fluid vesicles in shear flow, Phys. Rev. Lett. 77 (1996) 3685–3688.
[12] Y. Kim and M.-C. Lai, Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method, J. Comput. Phys., 229 (12) (2010), pp. 4840–4853
[13] A. Laadhari, C. Misbah and P. Saramito, On the equilibrium equation for a generalized biological membrane energy by using a shape optimization approach, Physica D: Nonlinear Phenomena 239 (2010) 1567–1572.
[14] A. Laadhari, R. Ruiz-Baier and A. Quarteroni, Fully Eulerian finite element approximation of a fluid-structure interaction problem in cardiac cells, Int. J. Numer. Meth. Engng. 96 (2013) 712–738.
[15] A. Laadhari, P. Saramito and C. Misbah, An adaptive finite element method for the modeling of the equilibrium of red blood cells, Int. J. Numer. Meth. Fluids 80 (2016) 397–428.
[16] A. Laadhari, P. Saramito and C. Misbah, Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods, J. Comput. Phys. 263 (2014) 328–352.
[17] J. Lowengrub, J-J. Xu and A. Voigt, Surface phase separation and flow in a simple model of multicomponent drops and vesicles, Fluid Dyn. Mater. Proc. 2007;3(1):1–19.
[18] J. Katsaras and T. Gutberlet T, Lipid bilayers: structure and interactions, Springer-Verlag, Berlin, 2001.
[19] M.P.I. Forum, MPI: A Message-Passing Interface Standard, http://www. (Accessed: 28.11.2016).
[20] MUMPS: MUltifrontal Massively Parallel Solver, http://mumps. (Accessed: 28.11.2016).
[21] S. Osher and J.A. Sethian Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1) (1988), pp. 12–49.
[22] Paraview: Parallel visualization application, (Accessed: 28.11.2016).
[23] P. Saramito, Efficient C++ finite element computing with Rheolef, CNRS-CCSD ed., 2013. Saramito/rheolef/rheolef-refman.pdf (Accessed: 22.09.16).
[24] D. Salac and M. Miksis, A level set projection model of lipid vesicles in general flows, J. Comput. Phys., 230 (2011), pp. 8192–8215
[25] D. Salac and M. Miksis , Reynolds number effects on lipid vesicles, J. Fluid Mech. 711 (2012) 122–146.
[26] U. Seifert, Configurations of fluid membranes and vesicles, Adv. Phys. 46 (1997) 13–137.
[27] T. Williams and C. Kelley, Gnuplot: An Interactive Plotting Program (Accessed: 28.11.2016).
[28] H. Zhao and E. S. G. Shaqfeh , The dynamics of a vesicle in simple shear flow, J. Fluid Mech. 674 (2011) 578–604.