Home Approximation of the Willmore energy by a discrete geometry model
Article
Licensed
Unlicensed Requires Authentication

Approximation of the Willmore energy by a discrete geometry model

  • Peter Gladbach EMAIL logo and Heiner Olbermann
Published/Copyright: August 25, 2021

Abstract

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of Γ-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.

MSC 2010: 52B10; 49J45; 74S20
  1. Communicated by: Irene Fonseca

References

[1] R. Alicandro, A. Braides and M. Cicalese, Continuum limits of discrete thin films with superlinear growth densities, Calc. Var. Partial Differential Equations 33 (2008), no. 3, 267–297. 10.1007/s00526-008-0159-4Search in Google Scholar

[2] R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math. Anal. 36 (2004), no. 1, 1–37. 10.1137/S0036141003426471Search in Google Scholar

[3] R. Alicandro and M. Cicalese, Variational analysis of the asymptotics of the X Y model, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 501–536. 10.1007/s00205-008-0146-0Search in Google Scholar

[4] R. Alicandro, M. Cicalese and A. Gloria, Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity, Arch. Ration. Mech. Anal. 200 (2011), no. 3, 881–943. 10.1007/s00205-010-0378-7Search in Google Scholar

[5] R. Alicandro, L. De Luca, A. Garroni and M. Ponsiglione, Metastability and dynamics of discrete topological singularities in two dimensions: A Γ-convergence approach, Arch. Ration. Mech. Anal. 214 (2014), no. 1, 269–330. 10.1007/s00205-014-0757-6Search in Google Scholar

[6] X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics, Arch. Ration. Mech. Anal. 164 (2002), no. 4, 341–381. 10.1007/s00205-002-0218-5Search in Google Scholar

[7] A. I. Bobenko, A conformal energy for simplicial surfaces, Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ. 52, Cambridge University, Cambridge (2005), 135–145. Search in Google Scholar

[8] A. I. Bobenko, Surfaces from circles, Discrete Differential Geometry, Oberwolfach Semin. 38, Birkhäuser, Basel (2008), 3–35. 10.1007/978-3-7643-8621-4_1Search in Google Scholar

[9] A. I. Bobenko, J. M. Sullivan and G. M. Ziegler, Discrete Differential Geometry, Springer, Berlin, 2008. 10.1007/978-3-7643-8621-4Search in Google Scholar

[10] J.-D. Boissonnat, R. Dyer and A. Ghosh, Constructing intrinsic Delaunay triangulations of submanifolds, preprint (2013), https://arxiv.org/abs/1303.6493. Search in Google Scholar

[11] A. Braides, Γ-Convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford University, Oxford, 2002. 10.1093/acprof:oso/9780198507840.001.0001Search in Google Scholar

[12] A. Braides and M. S. Gelli, Continuum limits of discrete systems without convexity hypotheses, Math. Mech. Solids 7 (2002), no. 1, 41–66. 10.1177/1081286502007001229Search in Google Scholar

[13] J. Braun and B. Schmidt, On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with 𝑝-growth, Netw. Heterog. Media 8 (2013), no. 4, 879–912. 10.3934/nhm.2013.8.879Search in Google Scholar

[14] B. Buet, G. P. Leonardi and S. Masnou, A varifold approach to surface approximation, Arch. Ration. Mech. Anal. 226 (2017), no. 2, 639–694. 10.1007/s00205-017-1141-0Search in Google Scholar

[15] B. Buet, G. P. Leonardi and S. Masnou, Discretization and approximation of surfaces using varifolds, Geom. Flows 3 (2018), no. 1, 28–56. 10.1515/geofl-2018-0004Search in Google Scholar

[16] B. Buet, G.-P. Leonardi and S. Masnou, Weak and approximate curvatures of a measure: A varifold perspective, preprint (2019), https://arxiv.org/abs/1904.05930. Search in Google Scholar

[17] G. Canevari and A. Segatti, Defects in nematic shells: A Γ-convergence discrete-to-continuum approach, Arch. Ration. Mech. Anal. 229 (2018), no. 1, 125–186. 10.1007/s00205-017-1215-zSearch in Google Scholar

[18] G. Dal Maso, An Introduction to Γ-Convergence, Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser, Boston, 1993. 10.1007/978-1-4612-0327-8Search in Google Scholar

[19] C. Davini and I. Pitacco, Relaxed notions of curvature and a lumped strain method for elastic plates, SIAM J. Numer. Anal. 35 (1998), no. 2, 677–691. 10.1137/S0036142995296102Search in Google Scholar

[20] M. de Berg, O. Cheong, M. van Kreveld and M. Overmars, Computational Geometry. Algorithms and Applications, 3rd ed., Springer, Berlin, 2008. 10.1007/978-3-540-77974-2Search in Google Scholar

[21] E. Grinspun, A. N. Hirani, M. Desbrun and P. Schröder, Discrete shells, Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, ACM, New York (2003), 62–67. Search in Google Scholar

[22] J. E. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J. 35 (1986), no. 1, 45–71. 10.1512/iumj.1986.35.35003Search in Google Scholar

[23] C. Mantegazza, Curvature varifolds with boundary, J. Differential Geom. 43 (1996), no. 4, 807–843. 10.4310/jdg/1214458533Search in Google Scholar

[24] U. Menne, Weakly differentiable functions on varifolds, Indiana Univ. Math. J. 65 (2016), no. 3, 977–1088. 10.1512/iumj.2016.65.5829Search in Google Scholar

[25] M. Meyer, M. Desbrun, P. Schröder and A. H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, Visualization and Mathematics III, Math. Vis., Springer, Berlin (2003), 35–57. 10.1007/978-3-662-05105-4_2Search in Google Scholar

[26] A. Schlömerkemper, Mathematical derivation of the continuum limit of the magnetic force between two parts of a rigid crystalline material, Arch. Ration. Mech. Anal. 176 (2005), no. 2, 227–269. 10.1007/s00205-004-0354-1Search in Google Scholar

[27] A. Schlömerkemper and B. Schmidt, Discrete-to-continuum limit of magnetic forces: Dependence on the distance between bodies, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 589–611. 10.1007/s00205-008-0134-4Search in Google Scholar

[28] B. Schmidt, On the passage from atomic to continuum theory for thin films, Arch. Ration. Mech. Anal. 190 (2008), no. 1, 1–55. 10.1007/s00205-008-0138-0Search in Google Scholar

[29] B. Schmidt and F. Fraternali, Universal formulae for the limiting elastic energy of membrane networks, J. Mech. Phys. Solids 60 (2012), no. 1, 172–180. 10.1016/j.jmps.2011.09.003Search in Google Scholar

[30] H. S. Seung and D. R. Nelson, Defects in flexible membranes with crystalline order, Phys. Rev. A 38 (1988), 1005–1018. 10.1103/PhysRevA.38.1005Search in Google Scholar PubMed

Received: 2020-09-23
Revised: 2021-07-20
Accepted: 2021-07-29
Published Online: 2021-08-25
Published in Print: 2023-04-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 22.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/acv-2020-0094/html
Scroll to top button