Numerical evaluation of discontinuous and nonconforming finite element methods in nonlinear solid mechanics

  • Hamid Reza Bayat
  • Julian Krämer
  • Linus Wunderlich
  • Stephan Wulfinghoff
  • Stefanie Reese
  • Barbara Wohlmuth
  • Christian Wieners
Original Paper


This work presents a systematic study of discontinuous and nonconforming finite element methods for linear elasticity, finite elasticity, and small strain plasticity. In particular, we consider new hybrid methods with additional degrees of freedom on the skeleton of the mesh and allowing for a local elimination of the element-wise degrees of freedom. We show that this process leads to a well-posed approximation scheme. The quality of the new methods with respect to locking and anisotropy is compared with standard and in addition locking-free conforming methods as well as established (non-) symmetric discontinuous Galerkin methods with interior penalty. For several benchmark configurations, we show that all methods converge asymptotically for fine meshes and that in many cases the hybrid methods are more accurate for a fixed size of the discrete system.


Discontinuous Galerkin Hybridization Locking Nonconforming methods 



The authors gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft within the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis” in the Projects RE 1057/30-1, WI 1430/8-1 and WO 671/15-1, and partially by WO 671/11-1.


  1. 1.
    Arnold DN (1981) Discretization by finite elements of a model parameter dependent problem. Numer Math 37(3):405–421MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold DN, Brezzi F, Cockburn B, Marini D (2000) Discontinuous Galerkin methods for elliptic problems. In: Cockburn B, Karniadakis GE, Shu C-W (eds) Discontinuous Galerkin methods: theory, computation and applications. Springer, Berlin, pp 89–101CrossRefGoogle Scholar
  3. 3.
    Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 39(5):1749–1779MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babuška I, Suri M (1992) Locking effects in the finite element approximation of elasticity problems. Numer Math 62(1):439–463MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baumann CE, Oden JT (1999) A discontinuous hp finite element method for convection–diffusion problems. Comput Methods Appl Mech Eng 175(3):311–341MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bayat HR, Wulfinghoff S, Reese S, Cavaliere F (2016) The discontinuous Galerkin method with reduced integration scheme for the boundary terms in almost incompressible linear elasticity. PAMM 16(1):189–190CrossRefGoogle Scholar
  7. 7.
    Belytschko T, Liu W, Moran B, Elkhodary K (2014) Nonlinear finite elements for continua and structures. Wiley, HobokenzbMATHGoogle Scholar
  8. 8.
    Braess D (2007) Finite Elements: theory, fast solvers, and applications in solid mechanics, 3rd edn. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  9. 9.
    Bramwell J, Demkowicz L, Gopalakrishnan J, Qiu W (2012) A locking-free \(hp\) DPG method for linear elasticity with symmetric stresses. Numer Math 122(4):671–707MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brenner S (2004) Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems. Math Comput 73(247):1041–1066MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carstensen C, Demowicz L, Gopalakrishnan J (2014) A posteriori error control for DPG methods. SIAM J Numer Anal 52(3):1335–1353MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Carstensen C, Hellwig F (2016) Low-order discontinuous Petrov–Galerkin finite element methods for linear elasticity. SIAM J Numer Anal 54(6):3388–3410MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chavan K, Lamichhane B, Wohlmuth B (2007) Locking-free finite element methods for linear and nonlinear elasticity in 2D and 3D. Comput Methods Appl Mech Eng 196:4075–4086CrossRefzbMATHGoogle Scholar
  14. 14.
    Chen Z, Chen H (2004) Pointwise error estimates of discontinuous Galerkin methods with penalty for second-order elliptic problems. SIAM J Numer Anal 42(3):1146–1166MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Di Pietro DA, Ern A (2012) Mathematical aspects of discontinuous Galerkin methods. Springer, BerlinCrossRefzbMATHGoogle Scholar
  16. 16.
    Di Pietro DA, Ern A (2015) A hybrid high-order locking-free method for linear elasticity on general meshes. Comput Methods Appl Mech Eng 283(Supplement C):1–21MathSciNetCrossRefGoogle Scholar
  17. 17.
    Di Pietro DA, Nicaise S (2013) A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media. Appl Numer Math 63:105–116MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fuentes F, Keith B, Demkowicz L, Le Tallec P (2017) Coupled variational formulations of linear elasticity and the DPG methodology. J Comput Phys 348:715–731MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grieshaber B, McBride A, Reddy B (2015) Uniformly convergent interior penalty methods using multilinear approximations for problems in elasticity. SIAM J Numer Anal 53(5):2255–2278MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hansbo P, Larson MG (2002) Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput Methods Appl Mech Eng 191(17):1895–1908MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Harder C, Madureira A L, Valentin F (2016) A hybrid-mixed method for elasticity. ESAIM: M2AN 50(2):311–336MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Horger T, Reali A, Wohlmuth B, Wunderlich L (2016) Improved approximation of eigenvalues in isogeometric methods for multi-patch geometries and Neumann boundaries. arXiv:1701.06353v1
  23. 23.
    Kabaria H, Lew AJ, Cockburn B (2015) A hybridizable discontinuous Galerkin formulation for non-linear elasticity. Comput Methods Appl Mech Eng 283:303–329MathSciNetCrossRefGoogle Scholar
  24. 24.
    Keith B, Fuentes F, Demkowicz L (2016) The DPG methodology applied to different variational formulations of linear elasticity. Comput Methods Appl Mech Eng 309:579–609MathSciNetCrossRefGoogle Scholar
  25. 25.
    Krämer J, Wieners C, Wohlmuth B, Wunderlich L (2016) A hybrid weakly nonconforming discretization for linear elasticity. PAMM 16(1):849–850CrossRefGoogle Scholar
  26. 26.
    Lamichhane B, Reddy D, Wohlmuth B (2006) Convergence in the incompressible limit of finite element approximations based on the Hu–Washizu formulation. Numer Math 104:151–175MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lehrenfeld C, Schöberl J (2016) High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput Methods Appl Mech Eng 307:339–361MathSciNetCrossRefGoogle Scholar
  28. 28.
    Li J, Melenk JM, Wohlmuth B, Zou J (2010) Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl Numer Math 60(1):19–37MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Liu R, Wheeler M, Dawson C (2009) A three-dimensional nodal-based implementation of a family of discontinuous Galerkin methods for elasticity problems. Comput Struct 87(3–4):141–150CrossRefGoogle Scholar
  30. 30.
    Liu R, Wheeler MF, Yotov I (2013) On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity. Comput Methods Appl Mech Eng 253:219–236MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mergheim J, Kuhl E, Steinmann P (2004) A hybrid discontinuous Galerkin/interface method for the computational modelling of failure. Commun Numer Methods Eng 20(7):511–519MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. In: Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol 36. Springer, pp 9–15Google Scholar
  33. 33.
    Reese S (2002) On the equivalent of mixed element formulations and the concept of reduced integration in large deformation problems. Int J Nonlinear Sci Numer Simul 3(1):1–34MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Reese S, Bayat H, Wulfinghoff S (2017) On an equivalence between a discontinuous Galerkin method and reduced integration with hourglass stabilization for finite elasticity. Comput Methods Appl Mech Eng 325(Supplement C):175–197MathSciNetCrossRefGoogle Scholar
  35. 35.
    Reese S, Küssner M, Reddy BD (1999) A new stabilization technique for finite elements in non-linear elasticity. Int J Numer Methods Eng 44(11):1617–1652MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Reese S, Wriggers P, Reddy BD (2000) A new locking-free brick element technique for large deformation problems in elasticity. Comput Struct 75(3):291–304MathSciNetCrossRefGoogle Scholar
  37. 37.
    Reissner E (1950) On a variational theorem in elasticity. J Math Phys 29(1–4):90–95MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rivière B (2008) Discontinuous Galerkin methods for solving elliptic and parabolic equations. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefzbMATHGoogle Scholar
  39. 39.
    Simo J, Hughes T (1998) Computational inelasticity, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  40. 40.
    Soon S, Cockburn B, Stolarski HK (2009) A hybridizable discontinuous Galerkin method for linear elasticity. Int J Numer Methods Eng 80(8):1058–1092MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Taylor RL (2003) FEAP—a finite element analysis program. Version 7.5 theory manual. University of California, Berkeley.
  42. 42.
    Ten Eyck A, Lew A (2006) Discontinuous Galerkin methods for non-linear elasticity. Int J Numer Methods Eng 67(9):1204–1243MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Washizu K (1968) Variational methods in elasticity and plasticity. Pergamon Press, OxfordzbMATHGoogle Scholar
  44. 44.
    Wieners C (2010) A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing. Comput Vis Sci 13(4):161–175MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Wieners C (2016) The skeleton reduction for finite element substructuring methods. In: Karasözen B, Manguoğlu M, Tezer-Sezgin M, Göktepe S, Uğur Ö (eds) Numerical mathematics and advanced applications ENUMATH 2015. Springer, Cham, pp 133–141Google Scholar
  46. 46.
    Wieners C, Wohlmuth B (2014) Robust operator estimates and the application to substructuring methods for first-order systems. ESAIM: Math Model Numer Anal 48(5):1473–1494MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Wihler T (2006) Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Math Comput 75(255):1087–1102MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Wulfinghoff S, Bayat HR, Alipour A, Reese S (2017) A low-order locking-free hybrid discontinuous Galerkin element formulation for large deformations. Comput Methods Appl Mech Eng 323(Supplement C):353–372MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied MechanicsRWTH AachenAachenGermany
  2. 2.Institute for Numerical Mathematics (M2)TU MünchenGarching b. MünchenGermany
  3. 3.Institute for Applied and Numerical MathematicsKIT KarlsruheKarlsruheGermany

Personalised recommendations