An Implicit Methodology for the Numerical Modeling of Locally Inextensible Membranes
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:74641", author = "Aymen Laadhari", title = "An Implicit Methodology for the Numerical Modeling of Locally Inextensible Membranes", abstract = "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, Newton method, level set,
Navier-Stokes, inextensible membrane, liquid drop.", volume = "10", number = "12", pages = "680-8", }
