SpringerLink
Log in

Hybrid Discretizations in Solid Mechanics for Non-linear and Non-smooth Problems

  • Chapter
  • First Online:
Non-standard Discretisation Methods in Solid Mechanics

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 98))

  • 842 Accesses

Abstract

We introduce novel hybrid discontinuous Galerkin methods for applications in solid mechanics. Different methods are introduced and numerically evaluated for several benchmark scenarios which show that our new approaches are more efficient and in many applications more robust than lowest order conforming finite elements. We consider different methods with discontinuous ansatz spaces in the cells with different concepts to achieve approximate continuity on cell interfaces. In one approach we select adaptively constraints on the faces. This corresponds to a weakly conforming finite element space defined by primal and dual face degrees of freedom. For the hybrid formulation, the element bubble degrees of freedom can be locally eliminated. Here we show robustness of the hybrid method in the nearly incompressible limit and for thin structures. Non-linear applications including contact, plasticity, and large strain elasticity show the flexibility of this discretization. Then, a locking-free incomplete interior penalty Galerkin (IIPG) variant of the discontinuous Galerkin (DG) method with reduced integration on the boundary terms is introduced. Based on the idea of this element formulation, a novel low-order hybrid DG method for geometrically non-linear problems is proposed which eliminates the locking effects. The drawback is the non-symmetric structure of the stiffness matrix. Next, the symmetric version of the aforementioned element formulation is presented based on a finite element technology with reduced integration and hourglass stabilization. Furthermore, the free (penalty) scalar parameter is transformed to a matrix form that is analytically obtained from the finite element technology. Finally, the IIPG method in combination with a cohesive zone model is applied to model failure at the interface.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Abbas, A. Ern, N. Pignet, Hybrid high-order methods for finite deformations of hyperelastic materials. Comput. Mech. 62(4), 909–928 (2018)

    Article  MathSciNet  Google Scholar 

  2. M. Abbas, A. Ern, N. Pignet, A hybrid high-order method for finite elastoplastic deformations within a logarithmic strain framework. Numer. Methods Eng. 120(3), 303–327 (2019)

    Article  MathSciNet  Google Scholar 

  3. M. Abbas, A. Ern, N. Pignet, A hybrid high-order method for incremental associative plasticity with small deformations. Comput. Methods Appl. Mech. Eng. 346, 891–912 (2019)

    Article  MathSciNet  Google Scholar 

  4. A. Alipour, S. Wulfinghoff, H.R. Bayat, S. Reese, Geometrically nonlinear crystal plasticity implemented into a discontinuous Galerkin element formulation. PAMM 17(1), 753–754 (2017)

    Article  Google Scholar 

  5. A. Alipour, S. Wulfinghoff, H.R. Bayat, S. Reese, B. Svendsen, The concept of control points in hybrid discontinuous Galerkin methods-application to geometrically nonlinear crystal plasticity. Int. J. Numer. Methods Eng. 114(5), 557–579 (2018)

    Article  MathSciNet  Google Scholar 

  6. A. Alipour, S. Wulfinghoff, B. Svendsen, S. Reese, Geometrically nonlinear single crystal viscoplasticity implemented into a hybrid discontinuous Galerkin framework, in Proceedings of the 7th GACM Colloquium on Computational Mechanics (2017)

    Google Scholar 

  7. D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)

    Article  MathSciNet  Google Scholar 

  8. C.E. Baumann, J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 175(3–4), 311–341 (1999)

    Article  MathSciNet  Google Scholar 

  9. N. Baumgarten, C. Wieners, The parallel finite element system M++ with integrated multilevel preconditioning and multilevel Monte Carlo methods. Comput. Math. Appl. (subm.) (2019). Manuscript available at http://www.math.kit.edu/user/~wieners/BaumgartenWieners2019.pdf

  10. H.R. Bayat, Failure modeling of interfaces and sheet metals. Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen (2020). https://doi.org/10.18154/RWTH-2020-04847. https://publications.rwth-aachen.de/record/788898

  11. H.R. Bayat, S. Kastian, S. Wulfinghoff, S. Reese, Discontinuous Galerkin (DG) method in 3D linear elasticity with application in problems with locking. PAMM 17(1), 19–22 (2017)

    Article  Google Scholar 

  12. H.R. Bayat, J. Krämer, L. Wunderlich, S. Wulfinghoff, S. Reese, B. Wohlmuth, C. Wieners, Numerical evaluation of discontinuous and nonconforming finite element methods in nonlinear solid mechanics. Comput. Mech. 62(6), 1413–1427 (2018). https://doi.org/10.1007/s00466-018-1571-z

    Article  MathSciNet  MATH  Google Scholar 

  13. H.R. Bayat, S. Rezaei, T. Brepols, S. Reese, Locking-free interface failure modeling by a cohesive discontinuous Galerkin method for matching and nonmatching meshes. Int. J. Numer. Methods Eng. 121(8), 1762–1790 (2020)

    Google Scholar 

  14. H.R. Bayat, S. Wulfinghoff, S. Kastian, S. Reese, On the use of reduced integration in combination with discontinuous Galerkin discretization: application to volumetric and shear locking problems. Adv. Model. Simul. Eng. Sci. 5(1), 10 (2018). https://doi.org/10.1186/s40323-018-0103-x

    Article  Google Scholar 

  15. H.R. Bayat, S. Wulfinghoff, S. Reese, Application of the discontinuous Galerkin finite element method in small deformation regimes. PAMM 15(1), 171–172 (2015)

    Article  Google Scholar 

  16. H.R. Bayat, S. Wulfinghoff, S. Reese, Discontinuous Galerkin analysis of displacement discontinuities for linear elasticity, in 3rd ECCOMAS Young Investigators Conference and 6th GACM Colloquium on Computational Mechanics, RWTH-2015-04002. Lehrstuhl und Institut für Angewandte Mechanik (2015)

    Google Scholar 

  17. H.R. Bayat, S. Wulfinghoff, S. Reese, F. Cavaliere, The discontinuous Galerkin method with reduced integration scheme for the boundary terms in almost incompressible linear elasticity. PAMM 16(1), 189–190 (2016)

    Article  Google Scholar 

  18. J. Bramwell, L. Demkowicz, J. Gopalakrishnan, W. Qiu, A locking-free \(hp\) DPG method for linear elasticity with symmetric stresses. Numerische Mathematik 122(4), 671–707 (2012)

    Article  MathSciNet  Google Scholar 

  19. S.C. Brenner, Korn’s inequalities for piecewise H1 vector fields. Math. Comput. pp. 1067–1087 (2004)

    Google Scholar 

  20. T. Brepols, Theory and numerics of gradient-extended damage coupled with plasticity. Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen (2018)

    Google Scholar 

  21. L.J. Bridgeman, T. Wihler, Stability and a posteriori error analysis of discontinuous Galerkin methods for linearized elasticity. Comput. Methods Appl. Mech. Eng. 200(13), 1543–1557 (2011)

    Article  MathSciNet  Google Scholar 

  22. F. Chouly, A. Ern, N. Pignet, A hybrid high-order discretization combined with Nitsche’s method for contact and Tresca friction in small strain elasticity (2019). hal.archives-ouvertes.fr/hal-02283418

    Google Scholar 

  23. P. Ciarlet Jr., C.F. Dunkl, S.A. Sauter, A family of Crouzeix-Raviart finite elements in 3D. Anal. Appl. 16(05), 649–691 (2018)

    Article  MathSciNet  Google Scholar 

  24. B. Cockburn, G. Kanschat, D. Schötzau, C. Schwab, Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40(1), 319–343 (2002)

    Article  MathSciNet  Google Scholar 

  25. B. Cockburn, G.E. Karniadakis, C.W. Shu, Discontinuous Galerkin Methods: Theory, Computation and Applications, vol. 11 (Springer Science & Business Media, 2012)

    Google Scholar 

  26. D.A. Di Pietro, A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)

    Article  MathSciNet  Google Scholar 

  27. D.A. Di Pietro, S. Nicaise, A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media. Appl. Numer. Math. 63, 105–116 (2013)

    Article  MathSciNet  Google Scholar 

  28. P. Hansbo, M.G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191(17), 1895–1908 (2002)

    Article  MathSciNet  Google Scholar 

  29. J. Krämer, C. Wieners, B. Wohlmuth, L. Wunderlich, A hybrid weakly nonconforming discretization for linear elasticity. PAMM 16(1), 849–850 (2016)

    Article  Google Scholar 

  30. R. Liu, M. Wheeler, C. Dawson, A three-dimensional nodal-based implementation of a family of discontinuous Galerkin methods for elasticity problems. Comput. Struct. 87(3–4), 141–150 (2009)

    Article  Google Scholar 

  31. D. Maurer, C. Wieners, A parallel block LU decomposition method for distributed finite element matrices. Parallel Comput. 37(12), 742–758 (2011)

    Article  Google Scholar 

  32. M. Paggi, P. Wriggers, Node-to-segment and node-to-surface interface finite elements for fracture mechanics. Comput. Methods Appl. Mech. Eng. 300, 540–560 (2016)

    Article  MathSciNet  Google Scholar 

  33. S. Reese, 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), 1–34 (2002)

    Article  MathSciNet  Google Scholar 

  34. S. Reese, On a consistent hourglass stabilization technique to treat large inelastic deformations and thermo-mechanical coupling in plane strain problems. Int. J. Numer. Methods Eng. 57(8), 1095–1127 (2003)

    Article  Google Scholar 

  35. S. Reese, H. Bayat, S. Wulfinghoff, On an equivalence between a discontinuous Galerkin method and reduced integration with hourglass stabilization for finite elasticity. Comput. Methods Appl. Mech. Eng. 325, 175–197 (2017)

    Google Scholar 

  36. S. Reese, P. Wriggers, A stabilization technique to avoid hourglassing in finite elasticity. Int. J. Numer. Methods Eng. 48(1), 79–109 (2000)

    Article  Google Scholar 

  37. S. Reese, P. Wriggers, B.D. Reddy, A new locking-free brick element technique for large deformation problems in elasticity. Comput. Struct. 75(3), 291–304 (2000)

    Article  MathSciNet  Google Scholar 

  38. S. Rezaei, S. Wulfinghoff, S. Reese, Prediction of fracture and damage in micro/nano coating systems using cohesive zone elements. Int. J. Solids Struct. 121, 62–74 (2017)

    Article  Google Scholar 

  39. J. Schröder, T. Wick, S. Reese, P. Wriggers, R. Müller, S. Kollmannsberger, M. Kästner, A. Schwarz, M. Igelbüscher, N. Viebahn, H.R. Bayat, S. Wulfinghoff, K. Mang, E. Rank, T. Bog, D. DAngella, M. Elhaddad, P. Hennig, A. Düster, W. Garhuom, S. Hubrich, M. Walloth, C. Wollner Winnifried Kuhn, T. Heister, A selection of benchmark problems in solid mechanics and applied mathematics. Archives of Computational Methods in Engineering, pp. 1–39 (2020)

    Google Scholar 

  40. M. Schwarze, S. Reese, A reduced integration solid-shell finite element based on the EAS and the ANS concept - geometrically linear problems. Int. J. Numer. Methods Eng. 80(10), 1322–1355 (2009)

    Article  MathSciNet  Google Scholar 

  41. M. Schwarze, S. Reese, A reduced integration solid-shell finite element based on the EAS and the ANS concept - large deformation problems. Int. J. Numer. Methods Eng. 85, 289–329 (2011)

    Article  MathSciNet  Google Scholar 

  42. R. Shirazi Nejad, C. Wieners, Parallel inelastic heterogeneous multi-scale simulations, in Multi-scale Simulation of Composite Materials (Springer, 2019), pp. 57–96

    Google Scholar 

  43. J. Spahn, H. Andrä, M. Kabel, R. Müller, A multiscale approach for modeling progressive damage of composite materials using fast fourier transforms. Comput. Methods Appl. Mech. Eng. 268, 871–883 (2014)

    Article  MathSciNet  Google Scholar 

  44. R.L. Taylor, FEAP - finite element analysis program (2014). http://projects.ce.berkeley.edu/feap/

  45. C. Wieners, A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing. Comput. Vis. Sci. 13(4), 161–175 (2010)

    Article  MathSciNet  Google Scholar 

  46. S. Wulfinghoff, H.R. Bayat, A. Alipour, S. Reese, Investigation of a locking-free hybrid discontinuous Galerkin element that is very easy to implement into fe-codes. PAMM 17(1), 87–90 (2017)

    Article  Google Scholar 

  47. S. Wulfinghoff, H.R. Bayat, A. Alipour, S. Reese, A low-order locking-free hybrid discontinuous Galerkin element formulation for large deformations. Comput. Methods Appl. Mech. Eng. 323(Supplement C), 353–372 (2017)

    Google Scholar 

Download references

Acknowledgements

Financial support of this work, related to the projects “Hybrid discretizations for nonlinear and nonsmooth problems in solid mechanics” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 255721882 - SPP 1748 is gratefully acknowledged.

Author information

Authors and Affiliations

  1. RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52074, Aachen, Germany

    H. R. Bayat & S. Reese

  2. KIT Karlsruhe, Englerstr. 2, 76131, Karlsruhe, Germany

    J. Krämer & C. Wieners

  3. TU München, Boltzmannstr. 3/III, 85748, Garching b. München, Germany

    B. Wohlmuth & L. Wunderlich

Authors
  1. H. R. Bayat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Krämer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Reese
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Wieners
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. B. Wohlmuth
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. L. Wunderlich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Reese .

Editor information

Editors and Affiliations

  1. Institute of Mechanics, University of Duisburg-Essen, Essen, Germany

    Jörg Schröder

  2. Institute of Continuum Mechanics, Leibniz Universität Hannover, Hannover, Germany

    Peter Wriggers

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bayat, H.R., Krämer, J., Reese, S., Wieners, C., Wohlmuth, B., Wunderlich, L. (2022). Hybrid Discretizations in Solid Mechanics for Non-linear and Non-smooth Problems. In: Schröder, J., Wriggers, P. (eds) Non-standard Discretisation Methods in Solid Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-030-92672-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-92672-4_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-92671-7

  • Online ISBN: 978-3-030-92672-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation