Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T20:03:21.801Z Has data issue: false hasContentIssue false
  • English
  • Français

First variation of the generalcurvature-dependent surface energy

Published online by Cambridge University Press:  22 July 2011

Günay Doğan  and
Ricardo H. Nochetto
Günay Doğan
Affiliation:
Theiss Research, San Diego, CA, USA, and Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, 20899 MD, USA. [email protected].
Ricardo H. Nochetto
Affiliation:
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, 20742 MD, USA. [email protected].
Get access

Abstract

We consider general surface energies, which areweighted integrals over a closed surface with a weight functiondepending on the position, the unit normal andthe mean curvature of the surface. Energiesof this form have applications in many areas, such as materials science,biology and image processing. Often one is interested in findinga surface that minimizes such an energy, which entails finding its firstvariation with respect to perturbations of the surface.We present a concise derivation of the first variation of thegeneral surface energy using tools from shape differential calculus.We first derive a scalar strong form and nexta vector weak form of the first variation. The latter reveals thevariational structure of the first variation, avoids dealingexplicitly with the tangential gradient of the unit normal,and thus can be easily discretized using parametric finite elements.Our results are valid for surfaces in any number of dimensionsand unify all previous results derived for specific examples ofsuch surface energies.

Keywords

Surface energygradient flowmean curvatureWillmore functional
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 1 , January 2012 , pp. 59 - 79
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Almgren, F. and Taylor, J.E., Optimal geometry in equilibrium and growth. Fractals 3 (1995) 713–723. Symposium in Honor of B. Mandelbrot. CrossRef
Almgren, F., Taylor, J.E. and Wang, L., Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993) 387–438. CrossRef
Ambrosio, L., Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191–246.
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2005).
Barrett, J.W., Garcke, H. and Nürnberg, R., Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput. 31 (2008) 225–253. CrossRef
Bauer, M. and Kuwert, E., Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 10 (2003) 553–576. CrossRef
Baumgart, T., Hess, S.T. and Webb, W.W., Image coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature 425 (2003) 821–824. CrossRef
Bellettini, G. and Paolini, M., Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25 (1996) 537–566. CrossRef
Bonito, A., Nochetto, R.H. and Pauletti, M.S., Parametric FEM for geometric biomembranes. J. Comput. Phys. 229 (2010) 3171–3188. CrossRef
Cahn, J.W. and Hoffman, D.W., A vector thermodynamics for anisotropic surfaces. II. Curved and facetted surfaces. Acta Metall. 22 (1974) 1205–1214. CrossRef
T. Chan and L. Vese, A level set algorithm for minimizing the Mumford-Shah functional in image processing, in Proceedings of the 1st IEEE Workshop on Variational and Level Set Methods in Computer Vision (2001) 161–168.
Chen, K., Jayaprakash, C., Pandit, R. and Wenzel, W., Microemulsions: A Landau-Ginzburg theory. Phys. Rev. Lett. 65 (1990) 2736–2739. CrossRef
Cicuta, P., Keller, S.L. and Veatch, S.L., Diffusion of liquid domains in lipid bilayer membranes. J. Phys. Chem. B 111 (2007) 3328–3331. CrossRef
Clarenz, U., Diewald, U., Dziuk, G., Rumpf, M. and Rusu, R., A finite element method for surface restoration with smooth boundary conditions. Comput. Aided Geom. Des. 21 (2004) 427–445. CrossRef
M.C. Delfour and J.-P. Zolésio, Shapes and Geometries, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001).
Döbereiner, H.G., Selchow, O. and Lipowsky, R., Spontaneous curvature of fluid vesicles induced by trans-bilayer sugar asymmetry. Eur. Biophys. J. 28 (1999) 174–178.
Doğan, G., Morin, P. and Nochetto, R.H., A variational shape optimization approach for image segmentation with a Mumford-Shah functional. SIAM J. Sci. Comput. 30 (2008) 3028–3049. CrossRef
Doğan, G., Morin, P., Nochetto, R.H. and Verani, M., Discrete gradient flows for shape optimization and applications. Comput. Meth. Appl. Mech. Eng. 196 (2007) 3898–3914. CrossRef
Droske, M. and Bertozzi, M., Higher-order feature-preserving geometric regularization. SIAM J. Imaging Sci. 3 (2010) 21–51. CrossRef
Dziuk, G., Computational parametric Willmore flow. Numer. Math. 111 (2008) 55–80. CrossRef
G. Dziuk, E. Kuwert and R. Schätzle, Evolution of elastic curves in $\mathbb{R}^n$: existence and computation. SIAM J. Math. Anal. 33 (electronic) (2002) 1228–1245. CrossRef
Elliott, C.M. and Stinner, B., Modeling and computation of two phase geometric biomembranes using surface finite elements. J. Comput. Phys. 229 (2010) 6585–6612. CrossRef
Helfrich, W., Elastic properties of lipid bilayers – theory and possible experiments. Zeitschrift Fur Naturforschung C-A J. Biosc. 28 (1973) 693. CrossRef
M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64 (2003/04) 442–467.
Hintermüller, M. and Ring, W., An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Imaging and Vision 20 (2004) 19–42. Special issue on mathematics and image analysis. CrossRef
Jenkins, J.T., The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math. 32 (1977) 755–764. CrossRef
R. Keriven and O. Faugeras, Variational principles, surface evolution, PDEs, level set methods and the stereo problem. Technical Report 3021, INRIA (1996).
Keriven, R. and Faugeras, O., Variational principles, surface evolution, PDEs, level set methods and the stereo problem. IEEE Trans. Image Process. 7 (1998) 336–344.
Kimmel, R. and Bruckstein, A.M., Regularized Laplacian zero crossings as optimal edge integrators. IJCV 53 (2003) 225–243. CrossRef
Kuwert, E. and Schätzle, R., The Willmore flow with small initial energy. J. Differential Geom. 57 (2001) 409–441. CrossRef
Kuwert, E. and Schätzle, R., Gradient flow for the Willmore functional. Comm. Anal. Geom. 10 (2002) 307–339. CrossRef
Kuwert, E. and Schätzle, R., Removability of point singularities of Willmore surfaces. Ann. Math. 160 (2004) 315–357. CrossRef
Laradji, M. and Mouritsen, O.G., Elastic properties of surfactant monolayers at liquid-liquid interfaces: A molecular dynamics study. J. Chem. Phys. 112 (2000) 8621–8630. CrossRef
M. Leventon, O. Faugeraus and W. Grimson, Level set based segmentation with intensity and curvature priors, in Proceedings of Workshop on Mathematical Methods in Biomedical Image Analysis Proceedings (2000) 4–11.
McFadden, G.B., Wheeler, A.A., Braun, R.J., Coriell, S.R. and Sekerka, R.F., Phase-field models for anisotropic interfaces. Phys. Rev. E 48 (1993) 2016–2024. CrossRef
Melenkevitz, J. and Javadpour, S.H., Phase separation dynamics in mixtures containing surfactants. J. Chem. Phys. 107 (1997) 623–629. CrossRef
Rusu, R., An algorithm for the elastic flow of surfaces. Interfaces and Free Boundaries 7 (2005) 229–239. CrossRef
Seifert, U., Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1997) 13–137. CrossRef
Simon, L., Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1 (1993) 281–326. CrossRef
Simonett, G., The Willmore flow near spheres. Differential Integral Equations 14 (2001) 1005–1014.
J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992).
G. Sundaramoorthi, A. Yezzi, A. Mennucci and G. Sapiro, New possibilities with Sobolev active contours, in Proceedings of the 1st International Conference on Scale Space Methods and Variational Methods in Computer Vision (2007).
Taylor, J.E., Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978) 568–588. CrossRef
Taylor, J.E., Mean curvature and weighted mean curvature. Acta Metall. Mater. 40 (1992) 1475–1485. CrossRef
Taylor, J.E. and Cahn, J.W., Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys. 77 (1994) 183–197. CrossRef
Taylor, J.E. and Cahn, J.W., Diffuse interfaces with sharp corners and facets: Phase field modeling of strongly anisotropic surfaces. Physica D 112 (1998) 381–411. CrossRef
Veatch, S.L. and Keller, S.L., Separation of liquid phases in giant vesicles of ternary mixtures of phospholipids and cholesterol. Biophys. J. 85 (2003) 3074–3083. CrossRef
Wheeler, A.A. and McFadden, G.B., A ξ-vector formulation of anisotropic phase-field models: 3D asymptotics. Eur. J. Appl. Math. 7 (1996) 367–381. CrossRef
T.J. Willmore, Total curvature in Riemannian geometry. Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester (1982).