FIRST VARIATION OF THE GENERAL CURVATURE-DEPENDENT SURFACE ENERGY

We consider general surface energies, which are weighted integrals over a closed surface with a weight function depending on the position, the unit normal and the mean curvature of the surface. Energies of this form have applications in many areas, such as materials science, biology and image processing. Often one is interested in finding a surface that minimizes such an energy, which entails finding its first variation with respect to perturbations of the surface. We present a concise derivation of the first variation of the general surface energy using tools from shape differential calculus. We first derive a scalar strong form and next a vector weak form of the first variation. The latter reveals the variational structure of the first variation, avoids dealing explicitly 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 dimensions and unify all previous results derived for specific examples of such surface energies.

[1]  A. Rau.  Variational Principles , 2021, Classical Mechanics.

[2]  Charles M. Elliott,et al.  Modeling and computation of two phase geometric biomembranes using surface finite elements , 2010, J. Comput. Phys..

[3]  Ricardo H. Nochetto,et al.  Geometrically Consistent Mesh Modification , 2010, SIAM J. Numer. Anal..

[4]  Ricardo H. Nochetto,et al.  Parametric FEM for geometric biomembranes , 2010, J. Comput. Phys..

[5]  Andrea L. Bertozzi,et al.  Higher-Order Feature-Preserving Geometric Regularization , 2010, SIAM J. Imaging Sci..

[6]  Gerhard Dziuk,et al.  Computational parametric Willmore flow , 2008, Numerische Mathematik.

[7]  Harald Garcke,et al.  Parametric Approximation of Willmore Flow and Related Geometric Evolution Equations , 2008, SIAM J. Sci. Comput..

[8]  Ricardo H. Nochetto,et al.  A Variational Shape Optimization Approach for Image Segmentation with a Mumford--Shah Functional , 2008, SIAM J. Sci. Comput..

[9]  Miguel Sebastian Pauletti,et al.  Parametric AFEM for geometric evolution equation and coupled fluid -membrane interaction , 2008 .

[10]  Ricardo H. Nochetto,et al.  Discrete gradient flows for shape optimization and applications , 2007 .

[11]  Guillermo Sapiro,et al.  New Possibilities with Sobolev Active Contours , 2007, International Journal of Computer Vision.

[12]  Morteza Zadimoghaddam,et al.  Minimizing movement , 2007, SODA '07.

[13]  P. Cicuta,et al.  Diffusion of liquid domains in lipid bilayer membranes. , 2006, The journal of physical chemistry. B.

[14]  Raluca E. Rusu.  An algorithm for the elastic flow of surfaces , 2005 .

[15]  Martin Rumpf,et al.  Axioms and variational problems in surface parameterization , 2004, Comput. Aided Geom. Des..

[16]  Reiner Schätzle,et al.  Removability of point singularities of Willmore surfaces , 2004 .

[17]  Martin Rumpf,et al.  A finite element method for surface restoration with smooth boundary conditions , 2004, Comput. Aided Geom. Des..

[18]  Sarah L Veatch,et al.  Separation of liquid phases in giant vesicles of ternary mixtures of phospholipids and cholesterol. , 2003, Biophysical journal.

[19]  Watt W. Webb,et al.  Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension , 2003, Nature.

[20]  Alfred M. Bruckstein,et al.  Regularized Laplacian Zero Crossings as Optimal Edge Integrators , 2003, International Journal of Computer Vision.

[21]  O. Faugeras,et al.  Variational principles, surface evolution, PDE's, level set methods and the stereo problem , 1998, 5th IEEE EMBS International Summer School on Biomedical Imaging, 2002..

[22]  L. Vese,et al.  A level set algorithm for minimizing the Mumford-Shah functional in image processing , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[23]  E. Kuwert,et al.  The Willmore Flow with Small Initial Energy , 2001 .

[24]  Gieri Simonett,et al.  The Willmore flow near spheres , 2001, Differential and Integral Equations.

[25]  O. Faugeras,et al.  Level set based segmentation with intensity and curvature priors , 2000, Proceedings IEEE Workshop on Mathematical Methods in Biomedical Image Analysis. MMBIA-2000 (Cat. No.PR00737).

[26]  M. Laradji,et al.  Elastic properties of surfactant monolayers at liquid–liquid interfaces: A molecular dynamics study , 2000 .

[27]  Reinhard Lipowsky,et al.  Spontaneous curvature of fluid vesicles induced by trans-bilayer sugar asymmetry , 1999, European Biophysics Journal.

[28]  Olivier D. Faugeras,et al.  Variational principles, surface evolution, PDEs, level set methods, and the stereo problem , 1998, IEEE Trans. Image Process..

[29]  John W. Cahn,et al.  Diffuse interfaces with sharp corners and facets: phase field models with strongly anisotropic surfaces , 1998 .

[30]  S. Javadpour,et al.  Phase separation dynamics in mixtures containing surfactants , 1997 .

[31]  Geoffrey B. McFadden,et al.  A ξ-vector formulation of anisotropic phase-field models: 3D asymptotics , 1996, European Journal of Applied Mathematics.

[32]  G. Bellettini,et al.  Anisotropic motion by mean curvature in the context of Finsler geometry , 1996 .

[33]  F. Almgren,et al.  OPTIMAL GEOMETRY IN EQUILIBRIUM AND GROWTH , 1995 .

[34]  John W. Cahn,et al.  Linking anisotropic sharp and diffuse surface motion laws via gradient flows , 1994 .

[35]  Wheeler,et al.  Phase-field models for anisotropic interfaces. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[36]  F. Almgren,et al.  Curvature-driven flows: a variational approach , 1993 .

[37]  J. Taylor,et al.  II—mean curvature and weighted mean curvature , 1992 .

[38]  Chen,et al.  Microemulsions: A Landau-Ginzburg theory. , 1990, Physical review letters.

[39]  J. Taylor,et al.  Crystalline variational problems , 1978 .

[40]  James T. Jenkins,et al.  The Equations of Mechanical Equilibrium of a Model Membrane , 1977 .

[41]  D. W. Hoffman,et al.  A Vector Thermodynamics for Anisotropic Surfaces—II. Curved and Faceted Surfaces , 1974 .

[42]  W. Helfrich.  Elastic Properties of Lipid Bilayers: Theory and Possible Experiments , 1973, Zeitschrift fur Naturforschung. Teil C: Biochemie, Biophysik, Biologie, Virologie.

[43]  D. W. Hoffman,et al.  A Vector Thermodynamics for Anisotropic Surfaces , 1972 .

[44]  Matthew MacDonald,et al.  Shapes and Geometries , 1987 .

[45]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[46]  Michael Hintermüller,et al.  A Second Order Shape Optimization Approach for Image Segmentation , 2004, SIAM J. Appl. Math..

[47]  Michael Hintermüller,et al.  An Inexact Newton-CG-Type Active Contour Approach for the Minimization of the Mumford-Shah Functional , 2004, Journal of Mathematical Imaging and Vision.

[48]  E. Kuwert,et al.  Existence of minimizing Willmore surfaces of prescribed genus , 2003 .

[49]  Gerhard Dziuk,et al.  Evolution of Elastic Curves in Rn: Existence and Computation , 2002, SIAM J. Math. Anal..

[50]  E. Kuwert,et al.  Gradient flow for the Willmore functional , 2002 .

[51]  Udo Seifert,et al.  Configurations of fluid membranes and vesicles , 1997 .

[52]  L. Simon.  Existence of surfaces minimizing the Willmore functional , 1993 .

[53]  Jan Sokolowski,et al.  Introduction to shape optimization , 1992 .

[54]  Bang‐Yen Chen,et al.  TOTAL CURVATURE IN RIEMANNIAN GEOMETRY (Ellis Horwood Series: Mathematics and Its Applications) , 1984 .

[55]  T. Willmore.  Total curvature in Riemannian geometry , 1982 .

[56]  D. W. Hoffman,et al.  A vector thermodynamics for anisotropic surfaces II. Curved and faceted surfaces , 1974 .