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A finite element scheme for the evolutionof orientational order in fluid membranes

Published online by Cambridge University Press:  09 October 2009

Sören Bartels ,
Georg Dolzmann  and
Ricardo H. Nochetto
Sören Bartels
Affiliation:
Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 6, 53115 Bonn, Germany.
Georg Dolzmann
Affiliation:
NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany.
Ricardo H. Nochetto
Affiliation:
Mathematics Department and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA.
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Abstract

We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as wellas 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. II7 (1997) 1509–1520; N. Uchida, Phys. Rev. E66 (2002) 040902] which replacesa Ginzburg-Landau penalization for the length of theorder parameter by a rigid constraint.We introduce a fully discrete scheme, consisting of piecewise linearfinite 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 solutionto the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.

Keywords

Biomembraneorientational ordercurvature
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 1 , January 2010 , pp. 1 - 31
Copyright
© EDP Sciences, SMAI, 2009

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References

Alouges, F., A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708–1726. CrossRef
Alouges, F., A new finite element scheme for Landau-Lifchitz equations. Discrete Contin. Dyn. Syst. Ser. S 1 (2008) 187–196. CrossRef
Barrett, J.W., Bartels, S., Feng, X. and Prohl, A., A convergent and constraint-preserving finite element method for the $p$ -harmonic flow into spheres. SIAM J. Numer. Anal. 45 (2007) 905–927. CrossRef
Barrett, J.W., Garcke, H. and Nürnberg, R., On the parametric finite element approximation of evolving hypersurfaces in ${\Bbb R} ^3$ . J. Comput. Phys. 227 (2008) 4281–4307. CrossRef
Bartels, S., Stability and convergence of finite-element approximation schemes for harmonic maps. SIAM J. Numer. Anal. 43 (2005) 220–238 (electronic). CrossRef
Baumgart, T., Hess, S.T. and Webb, W.W., Imaging co-existing domains in biomembrane models coupling curvature and tension. Nature 425 (2003) 832–824. CrossRef
Biben, T. and Misbah, C., An advected-field model for deformable entities under flow. Eur. Phys. J. B 29 (2002) 311–316. CrossRef
Biscari, P. and Terentjev, E.M., Nematic membranes: Shape instabilities of closed achiral vesicles. Phys. Rev. E 73 (2006) 051706. CrossRef
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York, USA (1991).
Canham, P.B., 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. CrossRef
Chen, Y.M., The weak solutions to the evolution problems of harmonic maps. Math. Z. 201 (1989) 69–74. CrossRef
Cheng, C.H.A., Coutand, D. and Shkoller, S., Navier-Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 39 (2007) 742–800 (electronic). CrossRef
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)].
Deckelnick, K., Dziuk, G. and Elliott, C.M., Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139–232. CrossRef
Du, Q., Liu, C. and Wang, X., A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198 (2004) 450–468. CrossRef
Du, Q., Liu, C. and Wang, X., Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212 (2006) 757–777. CrossRef
Dziuk, G., Computational parametric Willmore flow. Numer. Math. 111 (2008) 55–80. CrossRef
Evans, E., Bending resistance and chemically induced moments in membrane bilayers. Biophys. J. 14 (1974) 923–931. CrossRef
Fournier, J.B. and Galatoa, P., Sponges, tubules and modulated phases of para-antinematic membranes. J. Phys. II 7 (1997) 1509–1520.
Freire, A., Müller, S. and Struwe, M., Weak compactness of wave maps and harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 725–754. CrossRef
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).
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston, USA (1985).
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.
Helfrich, W., Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C 28 (1973) 693–703. CrossRef
Helfrich, W. and Prost, J., Intrinsic bending force in anisotropic membranes made of chiral molecules. Phys. Rev. A 38 (1988) 3065–3068. CrossRef
Jenkins, J.T., The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math. 32 (1977) 755–764. CrossRef
Johnson, M.A. and Decca, R.S., Dynamics of topological defects in the $l_{\beta^\prime}$ phase of 1,2-dipalmitoyl phosphatidylcholine bilayers. Opt. Commun. 281 (2008) 1870–1875. CrossRef
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.
Lubensky, T. C. and MacKintosh, F.C., Theory of “ripple” phases of bilayers. Phys. Rev. Lett. 71 (1993) 1565–1568. CrossRef
MacKintosh, F.C. and Lubensky, T.C., Orientational order, topology, and vesicle shapes. Phys. Rev. Lett. 67 (1991) 1169–1172. CrossRef
Marrink, S.J., Risselada, J. and Mark, A.E., Simulation of gel phase formation and melting in lipid bilayers using a coarse grained model. Chem. Phys. Lipids 135 (2005) 223–244. CrossRef
Nagle, S.T.-N.J.F., Structure of lipid bilayers. Biochim. Biophys. Acta 1469 (2000) 159–195. CrossRef
Nelson, P. and Powers, T., Rigid chiral membranes. Phys. Rev. Lett. 69 (1992) 3409–3412. CrossRef
Oda, R., Huc, I., Schmutz, M. and Candau, S.J., Tuning bilayer twist using chiral counterions. Nature 399 (1999) 566–569. CrossRef
M.S. Pauletti, Parametric AFEM for geometric evolution equations coupled fluid-membrane interaction. Ph.D. Thesis, University of Maryland, USA (2008).
Rusu, R.E., An algorithm for the elastic flow of surfaces. Interfaces Free Bound. 7 (2005) 229–239. CrossRef
Seifert, U., Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1997) 13–137. CrossRef
Selinger, J.V. and Schnur, J.M., Theory of chiral lipid tubules. Phys. Rev. Lett. 71 (1993) 4091–4094. CrossRef
Steigmann, D., Fluid films with curvature elasticity. Arch. Ration. Mech. Anal. 150 (1999) 127–152. CrossRef
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.
Uchida, N., Dynamics of orientational ordering in fluid membranes. Phys. Rev. E 66 (2002) 040902. CrossRef
E.G. Virga, Variational theories for liquid crystals, Appl. Math. Math. Comput. 8. Chapman & Hall, London, UK (1994).
T.J. Willmore, Riemannian geometry, Oxford Science Publications. The Clarendon Press Oxford University Press, New York, USA (1993).