A finite element scheme for the evolution of orientational order in fluid membranes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 1, p. 1-31

We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as well as the coupling of the local order of the constituent molecules of the membrane to its curvature. We propose an alternative to the model in [J.B. Fournier and P. Galatoa, J. Phys. II 7 (1997) 1509-1520; N. Uchida, Phys. Rev. E 66 (2002) 040902] which replaces a Ginzburg-Landau penalization for the length of the order parameter by a rigid constraint. We introduce a fully discrete scheme, consisting of piecewise linear finite elements, show that it is unconditionally stable for a large range of the elastic moduli in the model, and prove its convergence (up to subsequences) thereby proving the existence of a weak solution to the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.

DOI : https://doi.org/10.1051/m2an/2009040
Classification:  35K55,  74K15
Keywords: biomembrane, orientational order, curvature
@article{M2AN_2010__44_1_1_0,
author = {Bartels, S\"oren and Dolzmann, Georg and Nochetto, Ricardo H.},
title = {A finite element scheme for the evolution of orientational order in fluid membranes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {1},
year = {2010},
pages = {1-31},
doi = {10.1051/m2an/2009040},
zbl = {pre05693764},
mrnumber = {2647752},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_1_1_0}
}

Bartels, Sören; Dolzmann, Georg; Nochetto, Ricardo H. A finite element scheme for the evolution of orientational order in fluid membranes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 1, pp. 1-31. doi : 10.1051/m2an/2009040. http://www.numdam.org/item/M2AN_2010__44_1_1_0/

[1] F. Alouges, A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708-1726. | Zbl 0886.35010

[2] F. Alouges, A new finite element scheme for Landau-Lifchitz equations. Discrete Contin. Dyn. Syst. Ser. S 1 (2008) 187-196. | Zbl 1152.35304

[3] J.W. Barrett, S. Bartels, X. Feng and A. Prohl, A convergent and constraint-preserving finite element method for the $p$-harmonic flow into spheres. SIAM J. Numer. Anal. 45 (2007) 905-927. | Zbl 1155.35055

[4] J.W. Barrett, H. Garcke and R. Nürnberg, On the parametric finite element approximation of evolving hypersurfaces in ${ℝ}^{3}$. J. Comput. Phys. 227 (2008) 4281-4307. | Zbl 1145.65068

[5] S. Bartels, Stability and convergence of finite-element approximation schemes for harmonic maps. SIAM J. Numer. Anal. 43 (2005) 220-238 (electronic). | Zbl 1090.35014

[6] T. Baumgart, S.T. Hess and W.W. Webb, Imaging co-existing domains in biomembrane models coupling curvature and tension. Nature 425 (2003) 832-824.

[7] T. Biben and C. Misbah, An advected-field model for deformable entities under flow. Eur. Phys. J. B 29 (2002) 311-316.

[8] P. Biscari and E.M. Terentjev, Nematic membranes: Shape instabilities of closed achiral vesicles. Phys. Rev. E 73 (2006) 051706.

[9] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York, USA (1991). | Zbl 0788.73002

[10] P.B. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theort. Biol. 26 (1970) 61-81.

[11] Y.M. Chen, The weak solutions to the evolution problems of harmonic maps. Math. Z. 201 (1989) 69-74. | Zbl 0685.58015

[12] C.H.A. Cheng, D. Coutand and S. Shkoller, Navier-Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 39 (2007) 742-800 (electronic). | Zbl 1138.74022

[13] P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. | Zbl 0999.65129

[14] K. Deckelnick, G. Dziuk and C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139-232. | Zbl 1113.65097

[15] Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198 (2004) 450-468. | Zbl 1116.74384

[16] Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212 (2006) 757-777. | Zbl 1086.74024

[17] G. Dziuk, Computational parametric Willmore flow. Numer. Math. 111 (2008) 55-80. | Zbl 1158.65073

[18] E. Evans, Bending resistance and chemically induced moments in membrane bilayers. Biophys. J. 14 (1974) 923-931.

[19] J.B. Fournier and P. Galatoa, Sponges, tubules and modulated phases of para-antinematic membranes. J. Phys. II 7 (1997) 1509-1520.

[20] A. Freire, S. Müller and M. Struwe, Weak compactness of wave maps and harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 725-754. | Numdam | Zbl 0924.58011

[21] M. Giaquinta and S. Hildebrandt, Calculus of variations I: The Lagrangian formalism, Grundlehren der Mathematischen Wissenschaften 310, [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, Germany (1996). | Zbl 0853.49002

[22] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston, USA (1985). | Zbl 0695.35060

[23] C. Grossmann and H.-G. Roos, Numerical treatment of partial differential equations. Universitext, Springer, Berlin, Germany (2007). Translated and revised from the 3rd (2005) German edition by Martin Stynes. | Zbl 1180.65147

[24] W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C 28 (1973) 693-703.

[25] W. Helfrich and J. Prost, Intrinsic bending force in anisotropic membranes made of chiral molecules. Phys. Rev. A 38 (1988) 3065-3068.

[26] J.T. Jenkins, The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math. 32 (1977) 755-764. | Zbl 0358.73074

[27] M.A. Johnson and R.S. Decca, Dynamics of topological defects in the ${l}_{{\beta }^{\text{'}}}$ phase of 1,2-dipalmitoyl phosphatidylcholine bilayers. Opt. Commun. 281 (2008) 1870-1875.

[28] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and quasilinear elliptic equations. Academic Press, New York, USA (1968). Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. | Zbl 0164.13002

[29] T. C. Lubensky and F.C. Mackintosh, Theory of “ripple” phases of bilayers. Phys. Rev. Lett. 71 (1993) 1565-1568.

[30] F.C. Mackintosh and T.C. Lubensky, Orientational order, topology, and vesicle shapes. Phys. Rev. Lett. 67 (1991) 1169-1172. | Zbl 0990.92502

[31] S.J. Marrink, J. Risselada and A.E. Mark, Simulation of gel phase formation and melting in lipid bilayers using a coarse grained model. Chem. Phys. Lipids 135 (2005) 223-244.

[32] S.T.-N.J.F. Nagle, Structure of lipid bilayers. Biochim. Biophys. Acta 1469 (2000) 159-195.

[33] P. Nelson and T. Powers, Rigid chiral membranes. Phys. Rev. Lett. 69 (1992) 3409-3412.

[34] R. Oda, I. Huc, M. Schmutz and S.J. Candau, Tuning bilayer twist using chiral counterions. Nature 399 (1999) 566-569.

[35] M.S. Pauletti, Parametric AFEM for geometric evolution equations coupled fluid-membrane interaction. Ph.D. Thesis, University of Maryland, USA (2008).

[36] R.E. Rusu, An algorithm for the elastic flow of surfaces. Interfaces Free Bound. 7 (2005) 229-239. | Zbl pre02215421

[37] U. Seifert, Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1997) 13-137.

[38] J.V. Selinger and J.M. Schnur, Theory of chiral lipid tubules. Phys. Rev. Lett. 71 (1993) 4091-4094.

[39] D. Steigmann, Fluid films with curvature elasticity. Arch. Ration. Mech. Anal. 150 (1999) 127-152. | Zbl 0982.76012

[40] M. Struwe, Geometric evolution problems, in Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), IAS/Park City Math. Ser. 2, Amer. Math. Soc., Providence, USA (1996) 257-339. | Zbl 0847.58012

[41] N. Uchida, Dynamics of orientational ordering in fluid membranes. Phys. Rev. E 66 (2002) 040902.

[42] E.G. Virga, Variational theories for liquid crystals, Appl. Math. Math. Comput. 8. Chapman & Hall, London, UK (1994). | Zbl 0814.49002

[43] T.J. Willmore, Riemannian geometry, Oxford Science Publications. The Clarendon Press Oxford University Press, New York, USA (1993). | Zbl 0797.53002