×

High order numerical approximation of minimal surfaces. (English) Zbl 1220.65079

Summary: We present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary. The algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the Laplace-Beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface. For the steady state solution we measure the approximation error in a few cases where the exact solution is known. In the framework of parametric interpolation, the choice of interpolation points (mesh nodes) is directly affecting the approximation error, and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth. In our test cases we may achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases, but the rate of convergence greatly differs with different choices of mesh update algorithms. The present work is also of relevance to high order numerical approximation of fluid flow problems involving free surfaces.

MSC:

65K10 Numerical optimization and variational techniques
49Q05 Minimal surfaces and optimization
49M25 Discrete approximations in optimal control
76B07 Free-surface potential flows for incompressible inviscid fluids
76M30 Variational methods applied to problems in fluid mechanics

Software:

Surface Evolver
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] De Gennes, P.-G.; Brochard-Wyart, F.; Quere, D., Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (2004), Springer-Verlag · Zbl 1139.76004
[2] Isenberg, C., The Science of Soap Films and Soap Bubbles (1992), Dover Publications, Inc. · Zbl 0447.76001
[3] Osserman, R., A Survey of Minimal Surfaces (2002), Dover Publications, Inc. · Zbl 0209.52901
[4] Plateau, J. A.F., Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires (1873), Gauthier-Villars · JFM 06.0516.03
[5] Douglas, J., Solution of the problem of Plateau, Proceedings of the National Academy of Sciences, 16, 242-248 (1930)
[6] Radó, T., On Plateau’s problem, Annals of Mathematics, 31, 3, 457-469 (1930) · JFM 56.0437.02
[7] Concus, P., Numerical solution of the minimal surface equation, Mathematics of Computation, 21, 99, 340-350 (1967) · Zbl 0189.16605
[8] Greenspan, D., On approximating extremals of functionals - II theory and generalizations related to boundary value problems for nonlinear differential equations, International Journal of Engineering Science, 5, 7, 571-588 (1967) · Zbl 0171.13704
[9] Elcrat, A. R.; Lancaster, K. E., On the behavior of a non-parametric minimal surface in a non-convex quadrilateral, Archive for Rational Mechanics and Analysis, 94, 3, 209-226 (1986) · Zbl 0619.49018
[10] Hoppe, R. H.W., Multigrid algorithms for variational inequalities, SIAM Journal on Numerical Analysis, 24, 5, 1046-1065 (1987) · Zbl 0628.65046
[11] Dziuk, G.; Hutchinson, J. E., The discrete Plateau problem: algorithm and numerics, Mathematics of Computation, 68, 225, 1-23 (1999) · Zbl 0913.65062
[12] Dziuk, G.; Hutchinson, J. E., The discrete Plateau problem: convergence results, Mathematics of Computation, 68, 226, 519-546 (1999) · Zbl 1043.65126
[13] Brakke, K. A., The surface evolver, Experimental Mathematics, 1, 2, 141-165 (1992) · Zbl 0769.49033
[14] Hinata, M.; Shimasaki, M.; Kiyono, T., Numerical solution of Plateau’s problem by a finite element method, Mathematics of Computation, 28, 125, 45-60 (1974) · Zbl 0275.49032
[15] Wagner, H. J., A contribution to the numerical approximation of minimal surfaces, Computing, 19, 1, 35-58 (1977) · Zbl 0374.49021
[16] Coppin, C.; Greenspan, D., A contribution to the particle modeling of soap films, Applied Mathematics and Computation, 26, 4, 315-331 (1988) · Zbl 0651.65008
[17] Chopp, D. L., Computing minimal surfaces via level set curvature flow, Journal of Computational Physics, 106, 77-91 (1993) · Zbl 0786.65015
[18] Huerta, A.; Rodríguez-Ferran, A., The arbitrary Lagrangian-Eulerian formulation. The arbitrary Lagrangian-Eulerian formulation, Computer Methods in Applied Mechanics and Engineering, 193, 39-41, 4073-4456 (2004)
[19] Willmore, T. J., Riemannian Geometry (1993), Clarendon Press: Clarendon Press New York · Zbl 0797.53002
[20] Fomenko, A. T., The Plateau Problem (1990), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York · Zbl 0729.53001
[21] Struwe, M., Plateau’s Problem and the Calculus of Variations (1988), Princeton University Press: Princeton University Press Princeton · Zbl 0694.49028
[22] Landau, L. D.; Lifshitz, E. M., Fluid Mechanics, Course of Theoretical Physics, Vol. 6 (1987), Butterworth-Heinemann · Zbl 0146.22405
[23] Ho, L. W.; Patera, A. T., Variational formulation of three-dimensional viscous free-surface flows: natural imposition of surface tension boundary conditions, International Journal for Numerical Methods in Fluids, 13, 691-698 (1991) · Zbl 0739.76057
[24] Kreyszig, E., Differential Geometry (1991), Dover Publications, Inc.
[25] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods, (Fundamentals in Single Domains (2006), Springer) · Zbl 0717.76004
[26] Dierkes, U.; Hildebrandt, S.; Küster, A.; Wohlrab, O., Minimal surfaces. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 295 (1992), Springer-Verlag: Springer-Verlag Berlin
[27] Bjøntegaard, T.; Rønquist, E. M.; Tråsdahl, Ø., High order polynomial interpolation of parameterized curves, (Spectral and High Order Methods for Partial Differential Equations. Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol. 76 (2011), Springer), 365-372 · Zbl 1216.65013
[28] Bjøntegaard, T.; Rønquist, E. M., Accurate interface-tracking for arbitrary Lagrangian-Eulerian schemes, Journal of Computational Physics, 228, 12, 4379-4399 (2009) · Zbl 1395.76055
[29] Dziuk, G., An algorithm for evolutionary surfaces, Numerische Mathematik, 58, 1, 603-611 (1991) · Zbl 0714.65092
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.