For a hypersurface in ℝ3, Willmore flow is defined as the L2-gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.
Keywords: Willmore flow, Helfrich flow, axisymmetry, parametric finite elements, stability, tangential movement, spontaneous curvature, ADE model, clamped boundary conditions, Navier boundary conditions, Gaussian curvature energy, line energy
@article{M2AN_2021__55_3_833_0,
author = {Barrett, John W. and Garcke, Harald and N\"urnberg, Robert},
title = {Stable approximations for axisymmetric {Willmore} flow for closed and open surfaces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {833--885},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2021014},
mrnumber = {4253162},
zbl = {1500.65058},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021014/}
}
TY - JOUR AU - Barrett, John W. AU - Garcke, Harald AU - Nürnberg, Robert TI - Stable approximations for axisymmetric Willmore flow for closed and open surfaces JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 833 EP - 885 VL - 55 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021014/ DO - 10.1051/m2an/2021014 LA - en ID - M2AN_2021__55_3_833_0 ER -
%0 Journal Article %A Barrett, John W. %A Garcke, Harald %A Nürnberg, Robert %T Stable approximations for axisymmetric Willmore flow for closed and open surfaces %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 833-885 %V 55 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021014/ %R 10.1051/m2an/2021014 %G en %F M2AN_2021__55_3_833_0
Barrett, John W.; Garcke, Harald; Nürnberg, Robert. Stable approximations for axisymmetric Willmore flow for closed and open surfaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 833-885. doi: 10.1051/m2an/2021014
[1] , and , A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222 (2007) 441–462. | MR | Zbl | DOI
[2] , and , Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput. 31 (2008) 225–253. | MR | Zbl | DOI
[3] , and , Elastic flow with junctions: Variational approximation and applications to nonlinear splines. Math. Models Methods Appl. Sci. 22 (2012) 1250037. | MR | Zbl | DOI
[4] , and , Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves. Numer. Math. 120 (2012) 489–542. | MR | Zbl | DOI
[5] , and , Computational parametric Willmore flow with spontaneous curvature and area difference elasticity effects. SIAM J. Numer. Anal. 54 (2016) 1732–1762. | MR | Zbl | DOI
[6] , and , Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature. IMA J. Numer. Anal. 37 (2017) 1657–1709. | MR | Zbl
[7] , and , Finite element methods for fourth order axisymmetric geometric evolution equations. J. Comput. Phys. 376 (2019) 733–766. | MR | Zbl | DOI
[8] , and , Stable discretizations of elastic flow in Riemannian manifolds. SIAM J. Numer. Anal. 57 (2019) 1987–2018. | MR | Zbl | DOI
[9] , and , Variational discretization of axisymmetric curvature flows. Numer. Math. 141 (2019) 791–837. | MR | Zbl | DOI
[10] , and , Numerical approximation of curve evolutions in Riemannian manifolds. IMA J. Numer. Anal. 40 (2020) 1601–1651. | MR | Zbl | DOI
[11] , and , Parametric finite element approximations of curvature driven interface evolutions, edited by and . In: Vol. 21 of Handb. Numer. Anal.. Elsevier, Amsterdam (2020) 275–423. | MR | Zbl
[12] and , Discrete Willmore flow, edited by In: ACM SIGGRAPH 2005 Courses. ACM, New York, NY, SIGGRAPH ‘05, 5–es.
[13] , The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26 (1970) 61–81. | DOI
[14] , and , Lipid membranes with an edge. Phys. Rev. E 66 (2002) 021607. | DOI
[15] , , , and , A finite element method for surface restoration with smooth boundary conditions. Comput. Aided Geom. Design 21 (2004) 427–445. | MR | Zbl | DOI
[16] and , The effect of spontaneous curvature on a two-phase vesicle. Nonlinearity 28 (2015) 773–793. | MR | Zbl | DOI
[17] , , and , The Willmore flow of tori of revolution. Preprint (2020). | arXiv
[18] , Algorithm 832: UMFPACK V4.3 – an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30 (2004) 196–199. | MR | Zbl | DOI
[19] and , Error analysis for the elastic flow of parametrized curves. Math. Comput. 78 (2009) 645–671. | MR | Zbl | DOI
[20] and , Error analysis for the approximation of axisymmetric Willmore flow by -finite elements. Interfaces Free Bound. 12 (2010) 551–574. | MR | Zbl | DOI
[21] , Computational parametric Willmore flow. Numer. Math. 111 (2008) 55–80. | MR | Zbl | DOI
[22] , Recherches sur la théorie des surfaces élastiques. Veuve Courcier, Paris (1821).
[23] , Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch C 28 (1973) 693–703. | DOI
[24] and , Shape transformations of vesicles with intramembrane domains. Phys. Rev. E 53 (1996) 2670–2683. | DOI
[25] and , Shape equations for axisymmetric vesicles: A clarification. Phys. Rev. E 49 (1994) 4728–4731. | DOI
[26] , Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. 40 (1850) 51–88. | MR | Zbl
[27] , In: Vol. 77 of Differential geometry: Curves – Surfaces – Manifolds. Student Mathematical Library. Amer. Math. Soc. Providence, RI (2015). | MR | Zbl | DOI
[28] and , Min-max theory and the Willmore conjecture. Ann. Math. 179 (2014) 683–782. | MR | Zbl | DOI
[29] and , A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow. Interfaces Free Bound. 4 (2002) 89–109. | MR | Zbl | DOI
[30] , Boundary value problems for variational integrals involving surface curvatures. Quart. Appl. Math. 51 (1993) 363–387. | MR | Zbl | DOI
[31] , Mémoire sur les surfaces élastiques, Mémoires de l’Institut 1812 9 (1814) 167–226.
[32] , An algorithm for the elastic flow of surfaces. Interfaces Free Bound. 7 (2005) 229–239. | MR | Zbl | DOI
[33] , Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1997) 13–137. | DOI
[34] and , Lipid membranes with free edges. Phys. Rev. E 68 (2003) 061915. | DOI
[35] and , Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol. 56 (2008) 347–371. | MR | Zbl | DOI
[36] , Riemannian Geometry. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1993). | MR | Zbl | DOI
Cité par Sources :
John died on 30 June 2019, when this manuscript was nearly completed. We dedicate this article to his memory.
