Abstract
We propose a new algorithm for the computation of Willmore flow. This is the L 2-gradient flow for the Willmore functional, which is the classical bending energy of a surface. Willmore flow is described by a highly nonlinear system of PDEs of fourth order for the parametrization of the surface. The spatially discrete numerical scheme is stable and consistent. The discretization relies on an adequate calculation of the first variation of the Willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements. The algorithm uses finite elements on surfaces. We give numerical examples and tests for piecewise linear finite elements. A convergence proof for the full algorithm remains an open question.
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References
Barrett J.W., Garcke H., Nürnberg R.: A parametric finite element method for fourth order geometric evolution equations. J. Comp. Phys. 222, 441–467 (2007)
Bobenko A., Schroeder P.: Discrete Willmore flow, SIGGRAPH ’05: ACM SIGGRAPH 2005 (Courses). ACM Press, New York (2005)
Clarenz U., Diewald U., Dziuk G., Rumpf M., Rusu R.: A finite element method for surface restoration with smooth boundary conditions. Comput. Aided Geometr. Des. 21, 427–445 (2004)
Deckelnick K., Dziuk G.: Error analysis for the Willmore flow of graphs. Interfaces Free Boundaries 8, 21–46 (2006)
Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 139–232 (2005)
Diewald, U.: Anisotrope Krümmungsflüsse parametrischer Flächen sowie deren Anwendung in der Flächenverarbeitung, Dissertation University Duisburg-Essen (2005)
Droske M., Rumpf M.: A level set formulation for Willmore flow. Interfaces Free Boundaries 6, 361–378 (2004)
Du Q., Liu C., Wang X.: Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comp. Phys. 212, 757–777 (2006)
Dziuk G.: Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differ. Equ. Calculus Var. Lect. Notes Math. 1357, 142–155 (1988)
Dziuk G.: An algorithm for evolutionary surfaces. Numer. Math. 58, 603–611 (1991)
Dziuk G., Kuwert E., Schätzle R.: Evolution of elastic curves in \({{\mathbb R}^n}\): existence and computation. SIAM J. Math. Anal. 33, 1228–1245 (2002)
Dziuk G., Elliott C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, 262–292 (2006)
Feng X., Prohl A.: Analysis of gradient flow of a regularized Mumford–Shah functional for image segmentation and image inpainting. Math. Model. Numer. Anal. 38, 291–320 (2004)
Francis G., Sullivan J.M., Kusner R.B., Brakke K.A., Hartman C., Chappell G.: The minimax sphere eversion. In: Hege, H.-C., Polthier, K. (eds) Visualization and Mathematics, pp. 3–20. Springer, Heidelberg (1997)
Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg (1998)
Kuwert E., Schätzle R.: The Willmore flow with small initial energy. J. Differ. Geom. 57, 409–441 (2001)
Kuwert E., Schätzle R.: Removability of point singularities of Willmore surfaces. Ann. Math. 159, 1–43 (2004)
Mayer U.F., Simonett G.: A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow. Interfaces Free Boundaries 4, 89–109 (2002)
Rusu R.: An algorithm for the elastic flow of surfaces. Interfaces Free Boundaries 7, 229–239 (2005)
Simonett G.: The Willmore flow near spheres. Differ. Integral Equ. 14, 1005–1014 (2005)
Willmore, T.J.: Riemannian Geometry. Oxford (1993)