Skip to main content
SpringerLink
Log in

Adaptive refinement for phase-field models of brittle fracture based on Nitsche’s method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A new adaptive refinement strategy for phase-field models of brittle fracture is proposed. The approach provides a computationally efficient solution to the high demand in spatial resolution of phase-field models. The strategy is based on considering two types of elements: h-refined elements along cracks, where more accuracy is needed to capture the solution, and standard elements in the rest of the domain. Continuity between adjacent elements of different type is imposed in weak form by means of Nitsche’s method. The weakly imposition of continuity leads to a very local refinement in a simple way, for any degree of approximation and both in 2D and 3D. The performance of the strategy is assessed for several scenarios in the quasi-static regime, including coalescence and branching of cracks in 2D and a twisting crack in 3D.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29

Similar content being viewed by others

References

  1. Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55:383–405

    Article  MathSciNet  Google Scholar 

  2. Annavarapu C, Hautefeuille M, Dolbow J (2012) A robust Nitsche’s formulation for interface problems. Comput Methods Appl Mech Eng 225–228:44–54

    Article  MathSciNet  Google Scholar 

  3. Becker R, Hansbo P, Stenberg R (2010) A finite element method for domain decomposition with non-matching grids. ESAIM: Math Model Numer Anal 37(2):209–225

    Article  MathSciNet  Google Scholar 

  4. Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95

    Article  MathSciNet  Google Scholar 

  5. Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48:797–826

    Article  MathSciNet  Google Scholar 

  6. Budyn E, Zi G, Moës N, Belytschko T (2004) A method for multiple crack growth in brittle materials without remeshing. Int J Numer Methods Eng 61(10):1741–1770

    Article  Google Scholar 

  7. Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193(12–14):1257–1275

    Article  MathSciNet  Google Scholar 

  8. Geelen RJ, Liu Y, Dolbow J, Rodríguez-Ferran A (2018) An optimization-based phase-field method for continuous-discontinuous crack propagation. Int J Numer Methods Eng 116(1):1–20

    Article  MathSciNet  Google Scholar 

  9. Geelen RJ, Liu Y, Hu T, Tupek M, Dolbow J (2019) A phase-field formulation for dynamic cohesive fracture. Comput Methods Appl Mech Eng 348:680–711

    Article  MathSciNet  Google Scholar 

  10. Griebel M, Schweitzer MA (2003) A particle-partition of unity method. Part V: boundary conditions. In: Hildebrandt Stefan, Karcher Hermann (eds) Geometric analysis and nonlinear partial differential equations. Springer, Berlin, pp 519–542

    Chapter  Google Scholar 

  11. Hansbo P (2005) Nitsche’s method for interface problems in computational mechanics. GAMM-Mitt 28(2):183–206

    Article  MathSciNet  Google Scholar 

  12. Hennig P, Ambati M, De Lorenzis L, Kästner M (2018) Projection and transfer operators in adaptive isogeometric analysis with hierarchical B-splines. Comput Methods Appl Mech Eng 334:313–336

    Article  MathSciNet  Google Scholar 

  13. Hennig P, Müller S, Kästner M (2016) Bézier extraction and adaptive refinement of truncated hierarchical NURBS. Comput Methods Appl Mech Eng 305:316–339

    Article  Google Scholar 

  14. La Spina A, Giacomini M, Huerta A (2020) Hybrid coupling of CG and HDG discretizations based on Nitsche’s method. Comput Mech 65(2):311–330

    Article  MathSciNet  Google Scholar 

  15. Lo YS, Borden MJ, Ravi-Chandar K, Landis CM (2019) A phase-field model for fatigue crack growth. J Mech Phys Solids 132:103684

    Article  MathSciNet  Google Scholar 

  16. Miehe C, Hofacker M, Welschinger F (2010) A phase-field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199:2765–2778

    Article  MathSciNet  Google Scholar 

  17. Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83:1273–1311

    Article  MathSciNet  Google Scholar 

  18. Muixí A, Fernández-Méndez S, Rodríguez-Ferran A (2019) Phase-field solution of brittle fracture with adaptive refinement. https://youtu.be/FiQQe6UpenI

  19. Muixí A, Rodríguez-Ferran A, Fernández-Méndez S (2020) A Hybridizable Discontinuous Galerkin phase-field model for brittle fracture with adaptive refinement. Int J Numer Methods Eng 121(6):1147–1169

    Article  Google Scholar 

  20. Nagaraja S, Elhaddad M, Ambati M, Kollmannsberger S, De Lorenzis L, Rank E (2019) Phase-field modeling of brittle fracture with multi-level \(hp\)-FEM and the finite cell method. Comput Mech 63(6):1283–1300

    Article  MathSciNet  Google Scholar 

  21. Nitsche J (1971) Über ein variationsprinzip zur lösung von Dirichlet-problemen bei verwendung von teilr äumen, die keinen randbedingungen unterworfen sind. Abh Math Se Univ 36(1):9–15

    Article  Google Scholar 

  22. Noii N, Aldakheel F, Wick T, Wriggers P (2019) An adaptive global-local approach for phase-field modeling of anisotropic brittle fracture. Comput Methods Appl Mech Eng https://arxiv.org/pdf/1905.07519.pdf

  23. Patil RU, Mishra BK, Singh IV (2018) A local moving extended phase field method (LMXPFM) for failure analysis of brittle fracture. Comput Methods Appl Mech Eng 342:674–709

    Article  Google Scholar 

  24. Tamayo-Mas E, Rodríguez-Ferran A (2015) A medial-axis-based model for propagating cracks in a regularised bulk. Int J Numer Methods Eng 101(7):489–520

    Article  Google Scholar 

  25. Wu JY, Nguyen VP, Nguyen CT, Sutula D, Bordas S, Sinaie S (2019) Phase field modelling of fracture. Adv Appl Mech 53

Download references

Acknowledgements

This work was supported by the Agència de Gestió d’Ajuts Universitaris i de Recerca training Grant FI-DGR 2017, the DAFOH2 Project (Ministerio de Ciencia e Innovación, MTM2013-46313-R) and the Departament d’Innovació, Universitats i Empresa, Generalitat de Catalunya (2017-SGR-1278).

Author information

Authors and Affiliations

  1. Laboratori de Càlcul Numèric (LaCàN), Universitat Politècnica de Catalunya, Barcelona, 08034, Spain

    Alba Muixí, Sonia Fernández-Méndez & Antonio Rodríguez-Ferran

Authors
  1. Alba Muixí
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sonia Fernández-Méndez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio Rodríguez-Ferran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Rodríguez-Ferran.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Muixí, A., Fernández-Méndez, S. & Rodríguez-Ferran, A. Adaptive refinement for phase-field models of brittle fracture based on Nitsche’s method. Comput Mech 66, 69–85 (2020). https://doi.org/10.1007/s00466-020-01841-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-020-01841-1

Keywords

Search

Navigation