Computational Mechanics

, Volume 56, Issue 1, pp 87–95 | Cite as

Tangential differential calculus and the finite element modeling of a large deformation elastic membrane problem

  • Peter Hansbo
  • Mats G. Larson
  • Fredrik Larsson
Original Paper


We develop a finite element method for a large deformation membrane elasticity problem on meshed curved surfaces using a tangential differential calculus approach that avoids the use of classical differential geometric methods. The method is also applied to form finding problems.


Curved membrane Tangential derivative Finite element method Form finding 



This research was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029 and the Swedish Research Council Grants No. 2011-4992 and No. 2013-4708.


  1. 1.
    Ahmad S, Irons BM, Zienkiewicz OC (1970) Analysis of thick and thin shell structures by curved finite elements. Intern J Numer Methods Eng 2(3):419–451CrossRefGoogle Scholar
  2. 2.
    Betsch P, Gruttmann F, Stein E (1996) A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains. Comput Methods Appl Mech Eng 130(1–2):57–79zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bletzinger K-U, Wüchner R, Daoud F, Camprubí N (2005) Computational methods for form finding and optimization of shells and membranes. Comput Methods Appl Mech Eng 194(30–33):3438–3452zbMATHCrossRefGoogle Scholar
  4. 4.
    Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis, 2nd edn. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  5. 5.
    Campello EMB, Pimenta PM, Wriggers P (2003) A triangular finite shell element based on a fully nonlinear shell formulation. Comput Mech 31(6):505–518Google Scholar
  6. 6.
    Chapelle D, Bathe KJ (2003) The finite element analysis of shells-fundamentals. Computational fluid and solid mechanics. Springer, BerlinCrossRefGoogle Scholar
  7. 7.
    Chien D (1995) Numerical evaluation of surface integrals in three dimensions. Math Comput 64(210):727–743zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ciarlet PG (2000) Mathematical elasticity. Vol. III, volume 29 of studies in mathematics and its applications. North-Holland Publishing Co., AmsterdamGoogle Scholar
  9. 9.
    Delfour MC, Zolésio J-P (1995) A boundary differential equation for thin shells. J Differ Equ 119(2):426–449zbMATHCrossRefGoogle Scholar
  10. 10.
    Delfour MC, Zolésio J-P (1997) Differential equations for linear shells: comparison between intrinsic and classical models. In: Advances in mathematical sciences: CRM’s 25 years (Montreal, PQ, 1994), volume 11 of CRM Proceedings, Lecture Notes, American Mathematical Society, Providence, pp 41–124Google Scholar
  11. 11.
    Dziuk G (1991) An algorithm for evolutionary surfaces. Numer Math 58(6):603–611zbMATHMathSciNetGoogle Scholar
  12. 12.
    Monteiro EE, He Q-C, Yvonnet J (2011) Hyperelastic large deformations of two-phase composites with membrane-type interface. Int J Eng Sci 49(9):985–1000zbMATHCrossRefGoogle Scholar
  13. 13.
    Hansbo P, Larson MG (2014) Finite element modeling of a linear membrane shell problem using tangential differential calculus. Comput Methods Appl Mech Eng 270:1–14zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hansbo P, Larson MG, Larsson K (2014) Variational formulation of curved beams in global coordinates. Comput Mech 53(4):611–623zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Hughes TJR, Liu WK (1981) Nonlinear finite element analysis of shells: part I. Three dimensional shells. Comput Methods Appl Mech Eng 26(3):331–362zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Ibrahimbegović A, Gruttmann F (1993) A consistent finite element formulation of nonlinear membrane shell theory with particular reference to elastic rubberlike material. Finite Elem Anal Des 13(1):75–86zbMATHCrossRefGoogle Scholar
  17. 17.
    Klinkel S, Govindjee S (2002) Using finite strain 3D-material models in beam and shell elements. Eng Comput 19(8):902–921zbMATHCrossRefGoogle Scholar
  18. 18.
    Larsson F, Runesson K (2004) Modeling and discretization errors in hyperelasto-(visco-)plasticity with a view to hierarchical modeling. Comput Methods Appl Mech Eng 193(48–51):5283–5300zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Pimenta PM, Campello EMB (2009) Shell curvature as an initial deformation: a geometrically exact finite element approach. Intern J Numer Methods Eng 78(9):1094–1112zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Simo JC, Fox DD, Rifai MS (1989) On a stress resultant geometrically exact shell model. II. The linear theory; computational aspects. Comput Methods Appl Mech Eng 73(1):53–92zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Peter Hansbo
    • 1
  • Mats G. Larson
    • 2
  • Fredrik Larsson
    • 3
  1. 1.Department of Mechanical EngineeringJönköping UniversityJönköpingSweden
  2. 2.Department of Mathematics and Mathematical StatisticsUmeå UniversityUmeåSweden
  3. 3.Department of Applied MechanicsChalmers University of TechnologyGöteborgSweden

Personalised recommendations