Computational Mechanics

, Volume 58, Issue 6, pp 929–935 | Cite as

The nonconforming linear strain tetrahedron for a large deformation elasticity problem

  • Peter HansboEmail author
  • Fredrik Larsson
Original Paper


In this paper we investigate the performance of the nonconforming linear strain tetrahedron element introduced by Hansbo (Comput Methods Appl Mech Eng 200(9–12):1311–1316, 2011; J Numer Methods Eng 91(10):1105–1114, 2012). This approximation uses midpoints of edges on tetrahedra in three dimensions with either point continuity or mean continuity along edges of the tetrahedra. Since it contains (rotated) bilinear terms it performs substantially better than the standard constant strain element in bending. It also allows for under-integration in the form of one point Gauss integration of volumetric terms in near incompressible situations. We combine under-integration of the volumetric terms with houglass stabilization for the isochoric terms.


Nonconforming method Tetrahedral element Nonlinear elasticity 


  1. 1.
    Hansbo P (2011) A nonconforming rotated \(Q_1\) approximation on tetrahedra. Comput Methods Appl Mech Eng 200(9–12):1311–1316MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hansbo P (2012) Nonconforming rotated \(Q_1\) tetrahedral element with explicit time stepping for elastodynamics. Internat J Numer Methods Eng 91(10):1105–1114MathSciNetCrossRefGoogle Scholar
  3. 3.
    Rannacher R, Turek S (1992) Simple nonconforming quadrilateral Stokes element. Numer Methods Partial Differ Equ 8(2):97–111MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Mahnken R, Caylak I (2008) Stabilization of bi-linear mixed finite elements for tetrahedra with enhanced interpolation using volume and area bubble functions. Internat J Numer Methods Eng 75(4):377–413MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonet J, Marriott H, Hassan O (2001) An averaged nodal deformation gradient linear tetrahedral element for large strain explicit dynamic applications. Commun Numer Methods Eng 17(8):551–561Google Scholar
  6. 6.
    Simo JC, Armero F (1992) Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Internat J Numer Methods Eng 33(7):1413–1449MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Glaser S, Armero F (1997) On the formulation of enhanced strain finite elements in finite deformations. Eng Comput 14(7):759–791CrossRefzbMATHGoogle Scholar
  8. 8.
    Reese S, Küssner M, Reddy BD (1999) A new stabilization technique for finite elements in non-linear elasticity. Internat J Numer Methods Eng 44(11):1617–1652MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Reese S, Wriggers P (2000) A stabilization technique to avoid hourglassing in finite elasticity. Internat J Numer Methods Eng 48(1):79–109CrossRefzbMATHGoogle Scholar
  10. 10.
    Hansbo P (1998) A new approach to quadrature for finite elements incorporating hourglass control as a special case. Comput Methods Appl Mech Eng 158(3–4):301–309MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Schulz JC (1985) Finite element hourglassing control. Internat J Numer Methods Eng 21(6):1039–1048CrossRefzbMATHGoogle Scholar
  12. 12.
    Liu WK, Ong JS-J, Uras RA (1985) Finite element stabilization matrices–a unification approach. Comput Methods Appl Mech Eng 53(1):13–46MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu WK, Belytschko T, Ong JS-J, Law SE (1985) Use of stabilization matrices in non-linear analysis. Eng Comput 2(1):47–55CrossRefGoogle Scholar
  14. 14.
    Nadler B, Rubin MB (2003) A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point. Internat J Solids Struct 40(17):4585–4614CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringJönköping UniversityJönköpingSweden
  2. 2.Department of Applied MechanicsChalmers University of TechnologyGöteborgSweden

Personalised recommendations