Free Access
Volume 55, Number 3, May-June 2021
Page(s) 833 - 885
Published online 05 May 2021
  1. J.W. Barrett, H. Garcke and R. Nürnberg, A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222 (2007) 441–462. [CrossRef] [Google Scholar]
  2. J.W. Barrett, H. Garcke and R. Nürnberg, Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput. 31 (2008) 225–253. [CrossRef] [MathSciNet] [Google Scholar]
  3. J.W. Barrett, H. Garcke and R. Nürnberg, Elastic flow with junctions: Variational approximation and applications to nonlinear splines. Math. Models Methods Appl. Sci. 22 (2012) 1250037. [Google Scholar]
  4. J.W. Barrett, H. Garcke and R. Nürnberg, Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves. Numer. Math. 120 (2012) 489–542. [Google Scholar]
  5. J.W. Barrett, H. Garcke and R. Nürnberg, Computational parametric Willmore flow with spontaneous curvature and area difference elasticity effects. SIAM J. Numer. Anal. 54 (2016) 1732–1762. [CrossRef] [Google Scholar]
  6. J.W. Barrett, H. Garcke and R. Nürnberg, Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature. IMA J. Numer. Anal. 37 (2017) 1657–1709. [Google Scholar]
  7. J.W. Barrett, H. Garcke and R. Nürnberg, Finite element methods for fourth order axisymmetric geometric evolution equations. J. Comput. Phys. 376 (2019) 733–766. [Google Scholar]
  8. J.W. Barrett, H. Garcke and R. Nürnberg, Stable discretizations of elastic flow in Riemannian manifolds. SIAM J. Numer. Anal. 57 (2019) 1987–2018. [Google Scholar]
  9. J.W. Barrett, H. Garcke and R. Nürnberg, Variational discretization of axisymmetric curvature flows. Numer. Math. 141 (2019) 791–837. [Google Scholar]
  10. J.W. Barrett, H. Garcke and R. Nürnberg, Numerical approximation of curve evolutions in Riemannian manifolds. IMA J. Numer. Anal. 40 (2020) 1601–1651. [Google Scholar]
  11. J.W. Barrett, H. Garcke and R. Nürnberg, Parametric finite element approximations of curvature driven interface evolutions, edited by A. Bonito and R.H. Nochetto. In: Vol. 21 of Handb. Numer. Anal.. Elsevier, Amsterdam (2020) 275–423. [Google Scholar]
  12. A.I. Bobenko and P. Schröder, Discrete Willmore flow, edited by J. Fujii In: ACM SIGGRAPH 2005 Courses. ACM, New York, NY, SIGGRAPH ‘05, 5–es. [Google Scholar]
  13. 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) 61–81. [CrossRef] [PubMed] [Google Scholar]
  14. R. Capovilla, J. Guven and J.A. Santiago, Lipid membranes with an edge. Phys. Rev. E 66 (2002) 021607. [CrossRef] [Google Scholar]
  15. U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf and R. Rusu, A finite element method for surface restoration with smooth boundary conditions. Comput. Aided Geom. Design 21 (2004) 427–445. [Google Scholar]
  16. G. Cox and J. Lowengrub, The effect of spontaneous curvature on a two-phase vesicle. Nonlinearity 28 (2015) 773–793. [CrossRef] [PubMed] [Google Scholar]
  17. A. Dallacqua, M. Müller, R. Schätzle and A. Spener, The Willmore flow of tori of revolution. Preprint arXiv:2005.13500 (2020). [Google Scholar]
  18. T.A. Davis, Algorithm 832: UMFPACK V4.3 – an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30 (2004) 196–199. [Google Scholar]
  19. K. Deckelnick and G. Dziuk, Error analysis for the elastic flow of parametrized curves. Math. Comput. 78 (2009) 645–671. [Google Scholar]
  20. K. Deckelnick and F. Schieweck, Error analysis for the approximation of axisymmetric Willmore flow by C1-finite elements. Interfaces Free Bound. 12 (2010) 551–574. [Google Scholar]
  21. G. Dziuk, Computational parametric Willmore flow. Numer. Math. 111 (2008) 55–80. [CrossRef] [MathSciNet] [Google Scholar]
  22. S. Germain, Recherches sur la théorie des surfaces élastiques. Veuve Courcier, Paris (1821). [Google Scholar]
  23. W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch C 28 (1973) 693–703. [PubMed] [Google Scholar]
  24. F. Jülicher and R. Lipowsky, Shape transformations of vesicles with intramembrane domains. Phys. Rev. E 53 (1996) 2670–2683. [CrossRef] [Google Scholar]
  25. F. Jülicher and U. Seifert, Shape equations for axisymmetric vesicles: A clarification. Phys. Rev. E 49 (1994) 4728–4731. [CrossRef] [Google Scholar]
  26. G.R. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. 40 (1850) 51–88. [Google Scholar]
  27. W. Kühnel, In: Vol. 77 of Differential geometry: Curves – Surfaces – Manifolds. Student Mathematical Library. Amer. Math. Soc. Providence, RI (2015). [Google Scholar]
  28. F.C. Marques and A. Neves, Min-max theory and the Willmore conjecture. Ann. Math. 179 (2014) 683–782. [Google Scholar]
  29. U.F. Mayer and G. Simonett, 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. [Google Scholar]
  30. J.C.C. Nitsche, Boundary value problems for variational integrals involving surface curvatures. Quart. Appl. Math. 51 (1993) 363–387. [CrossRef] [Google Scholar]
  31. S.D. Poisson, Mémoire sur les surfaces élastiques, Mémoires de l’Institut 1812 9 (1814) 167–226. [Google Scholar]
  32. R.E. Rusu, An algorithm for the elastic flow of surfaces. Interfaces Free Bound. 7 (2005) 229–239. [CrossRef] [MathSciNet] [Google Scholar]
  33. U. Seifert, Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1997) 13–137. [CrossRef] [Google Scholar]
  34. Z.C. Tu and Z.C. Ou-Yang, Lipid membranes with free edges. Phys. Rev. E 68 (2003) 061915. [CrossRef] [Google Scholar]
  35. X. Wang and Q. Du, Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol. 56 (2008) 347–371. [PubMed] [Google Scholar]
  36. T.J. Willmore, Riemannian Geometry. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1993). [Google Scholar]

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