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XLME interpolants, a seamless bridge between XFEM and enriched meshless methods

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Abstract

In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method at a comparable computational cost. In addition, we keep the advantages of the LME shape functions, such as smoothness and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) enrichment with no special treatment of the nodes near the crack such as blending or shifting.

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References

  1. Sukumar N (2004) Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Methods Eng 61(12):2159–2181. doi:10.1002/nme.1193

    Article  MATH  MathSciNet  Google Scholar 

  2. Shannon CE (1948) A mathematical theory of communication. Bell Labs Tech J 27:379–423

    MATH  MathSciNet  Google Scholar 

  3. Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106(4):620–630. doi:10.1103/PhysRev.106.620

    Article  MATH  MathSciNet  Google Scholar 

  4. Jaynes ET (1957) Information theory and statistical mechanics II. Phys Rev 108(2):171–190. doi:10.1103/PhysRev.103.171

    Article  MathSciNet  Google Scholar 

  5. Arroyo M, Ortiz M (2006) Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int J Numer Methods Eng 65:2167–2202. doi:10.1002/nme.1534

    Article  MATH  MathSciNet  Google Scholar 

  6. Shepard D (1968) A two dimensional interpolation function for irregularly spaced data. In: Proceedings of the 23rd National Conference of ACM, pp 517–523. doi:10.1145/800186.810616

  7. Millán D, Rosolen A, Arroyo M (2011) Thin shell analysis from scattered points with maximum-entropy approximants. Int J Numer Methods Eng 85:723–751. doi:10.1002/nme.2992

    Article  MATH  Google Scholar 

  8. Ortiz A, Puso MA, Sukumar N (2010) Maximum-entropy meshfree method for compressible and near-incompressible elasticity. Comput Methods Appl Mech Eng 199:1859–1871. doi:10.1016/j.cma.2010.02.013

    Article  MATH  MathSciNet  Google Scholar 

  9. Ortiz A, Puso MA, Sukumar N (2011) Maximum-entropy meshfree method for incompressible media problems. Finite Elem Anal Des 47:572–585. doi:10.1016/j.finel.2010.12.009

    Article  MathSciNet  Google Scholar 

  10. Cyron CJ, Arroyo M, Ortiz M (2009) Smooth, second order, non-negative meshfree approximants selected by maximum entropy. Int J Numer Methods Eng 79:1605–1632. doi:10.1002/nme.2597

    Article  MATH  MathSciNet  Google Scholar 

  11. Rabczuk T, Belytschko T (2007) A three dimensional large deformation meshfree method for arbitrary evolving cracks. Comput Methods Appl Mech Eng 196:2777–2799. doi:10.1016/j.cma.2006.06.020

    Article  MATH  MathSciNet  Google Scholar 

  12. Rabczuk T, Areias PMA, Belytschko T (2007) A meshfree thin shell method for non-linear dynamic fracture. Int J Numer Methods Eng 72:524–548. doi:10.1002/nme.2013

    Article  MATH  MathSciNet  Google Scholar 

  13. Bordas S, Rabczuk T, Zi G (2008) Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by extrinsic discontinuous enrichment of meshfree methods without asymptotic enrichment. Eng Fract Mech 75:943–960. doi:10.1016/j.engfracmech.2007.05.010

    Article  Google Scholar 

  14. Nguyen VP, Rabczuck T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79:763–813. doi:10.1016/j.matcom.2008.01.003

    Article  MATH  Google Scholar 

  15. Rabczuk T, Bordas S, Zi G (2010) On three-dimensional modelling of crack growth using partition of unity methods. Comput Struct 88:1391–1411. doi:10.1016/j.compstruc.2008.08.010

    Article  Google Scholar 

  16. Ventura G, Xu J, Belytschko T (2002) A vector level set method and new discontinuity approximations for crack growth by EFG. Int J Numer Methods Eng 54:923–944. doi:10.1002/nme.471

    Article  MATH  Google Scholar 

  17. De Luycker E, Benson DJ, Belytschko T, Bazilevs Y, Hsu MC (2011) X-FEM in isogeometric analysis for linear fracture mechanics. Int J Numer Methods Eng 87:541–565. doi:10.1002/nme.3121

    Article  MATH  Google Scholar 

  18. Bordas SPA, Rabczuk T, Nguyen-Xuan H, Nguyen VP, Natarajan S, Bog T, Quan DM, Nguyen VH (2010) Strain smoothing in FEM and XFEM. Comput Struct 88:1419–1443. doi:10.1016/j.compstruc.2008.07.006

    Article  Google Scholar 

  19. Chen L, Rabczuk T, Bordas SPA, Liu GR, Zeng KY, Kerfriden P. Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth. Comput Methods Appl Mech Eng 209–212:250–265. doi:10.1016/j.cma.2011.08.013

  20. Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61:2316–2343. doi:10.1002/nme.1151

    Article  MATH  Google Scholar 

  21. Duflot M, Nguyen-Dang H (2004) A meshless method with enriched weight functions for fatigue crack growth. Int J Numer Methods Eng 59:1945–1961. doi:10.1002/nme.948

    Article  MATH  Google Scholar 

  22. Barbieri E, Petrinic N, Meo M, Tagarielli VL (2012) A new weight-function enrichment in meshless methods for multiple cracks in linear elasticity. Int J Numer Methods Eng 90:177–195. doi:10.1002/nme.3313

    Article  MATH  MathSciNet  Google Scholar 

  23. Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Methods Eng 43(5):839–887

    Article  MATH  MathSciNet  Google Scholar 

  24. Cirak F, Ortiz M, Schröder P (2000) Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. Int J Numer Methods Eng 47(12):2039–2072

    Article  MATH  Google Scholar 

  25. Hughes T, Cottrell J, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195. doi:10.1016/j.cma.2004.10.008

    Article  MATH  MathSciNet  Google Scholar 

  26. Rajan V (1994) Optimality of the Delaunay triangulation in \(R^d\). Discret Comput Geom 12(2):189–202. doi: 10.1007/BF02574375

    Article  MATH  MathSciNet  Google Scholar 

  27. Rabczuk T, Wall WA (2007) Extended finite element and meshfree methods. Technical University of Munich, Munich

    Google Scholar 

  28. Stazi FL, Budyn E, Chessa J, Belytschko T (2003) An extended finite element method with higher-order elements for curved cracks. Comput Mech 31(1–2):38–48. doi:10.1007/s00466-002-0391-2

    Article  MATH  Google Scholar 

  29. Laborde P, Pommier J, Renard Y, Salaün M (2005) High-order extended finite element method for cracked domains. Int J Numer Methods Eng 64(12):354–381. doi:10.1002/nme.1370

    Article  MATH  Google Scholar 

  30. Natarajan S, Mahapatra DR, Bordas SPA (2010) Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework. Int J Numer Methods Eng 83:269–294. doi:10.1002/nme.2798

    MATH  MathSciNet  Google Scholar 

  31. Babuška I, Banerjee U, Osborn JE, Zhang Q (2009) Effect of numerical integration on meshless methods. Comput Meth Appl Mech Eng 198:2886–2897. doi:10.1016/j.cma.2009.04.008

    Article  MATH  Google Scholar 

  32. Béchet E, Minnebo H, Moës N, Burgardt B (2005) Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int J Numer Methods Eng 64:1033–1056. doi:10.1002/nme.1386

    Article  MATH  Google Scholar 

  33. Menk A, Bordas SPA (2011) A robust preconditioning technique for the extended finite element method. Int J Numer Methods Eng 85:1609–1632. doi:10.1002/nme.3032

    Article  MATH  MathSciNet  Google Scholar 

  34. Babuška I, Banerjee U (2011) Stable generalized finite element method (SGFEM). Comput Methods Appl Mech Eng 201–204:91–111. doi:10.1016/j.cma.2011.09.012

    Google Scholar 

  35. Nicaise S, Renard Y, Chahine E (2011) Optimal convergence analysis for the extended finite element method. Int J Numer Methods Eng 86:528548

    Article  MathSciNet  Google Scholar 

  36. Quak W, van den Boogaard AH, González D, Cueto E (2011) A comparative study on the performance of meshless approximations and their integration. Comput Mech 48(2):121–137. doi:10.1007/s00466-011-0577-6

    Google Scholar 

  37. Moës N, Dolbow J, Belytscho T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150

    Google Scholar 

  38. Yau J, Wang S, Corten H (1980) A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J Appl Mech 47:335–341. doi:10.1115/1.3153665

    Google Scholar 

  39. Zehnder A (2010) Fracture mechanics. Cornell University, Lecture Notes

  40. Rabczuk T, Si G (2006) A meshfree method based on the local partition of unity for cohesive cracks. Comput Mech 39(6):743–760. doi:10.1007/s00466-006-0067-4

    Google Scholar 

  41. Anderson TL (1995) Fracture mechanics, fundamentals and applications, 2nd edn. Texas A &M University, College Station

    MATH  Google Scholar 

Download references

Acknowledgments

The first two authors would like to thank the Free State of Thuringia and Bauhaus Research School for financial support during the duration of this project. We would also like to thank the anonymous reviewers for their helpful comments and suggestions. Stéphane Bordas acknowledges support for his time from the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) entitled “Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery”.

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Authors and Affiliations

  1. Institute of Structural Mechanics, Bauhaus University Weimar, Marienstr. 15, 99423 , Weimar, Germany

    F. Amiri, C. Anitescu & T. Rabczuk

  2. Departament de Matemática Aplicada 3, School of Civil Engineering of Barcelona (ETSECCPB), Universitat Politécnica de Catalunya, Barcelona, Spain

    M. Arroyo

  3. Cardiff School of Engineering, Institute of Mechanics and Advanced Materials, Cardiff University, Queen’s Buildings, The Parade, Cardiff, Wales, CF24 3AA, UK

    S. P. A. Bordas

  4. School of Civil, Environmental and Architectural Engineering, Korea University, Seoul, South Korea

    T. Rabczuk

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  1. F. Amiri
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  2. C. Anitescu
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  3. M. Arroyo
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  4. S. P. A. Bordas
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  5. T. Rabczuk
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Correspondence to F. Amiri.

Additional information

S. P. A. Bordas’s ORCID ID is 0000-0001-7622-2193.

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Amiri, F., Anitescu, C., Arroyo, M. et al. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Comput Mech 53, 45–57 (2014). https://doi.org/10.1007/s00466-013-0891-2

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