**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**32019

##### An Implicit Methodology for the Numerical Modeling of Locally Inextensible Membranes

**Authors:**
Aymen Laadhari

**Abstract:**

**Keywords:**
Finite element method,
Newton method,
level set,
Navier-Stokes,
inextensible membrane,
liquid drop.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1339686

**References:**

[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. mpi-forum.org (Accessed: 28.11.2016).

[20] MUMPS: MUltifrontal Massively Parallel Solver, http://mumps. enseeiht.fr/index.php (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, http://paraview.org (Accessed: 28.11.2016).

[23] P. Saramito, Efficient C++ finite element computing with Rheolef, CNRS-CCSD ed., 2013. http://www-ljk.imag.fr/membres/Pierre. 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 http://www.gnuplot.info (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.