Skip to main content
SpringerLink
Log in

Independent cover meshless method using a polynomial approximation

  • CompMech
  • Published:
International Journal of Fracture Aims and scope Submit manuscript

Abstract

In this paper, a novel independent cover meshless method (ICMM) is presented for the analyses of two dimensional linear elastic solids and the simulation of crack propagation. In the ICMM, the Delaunay criterion is employed to dynamically construct the non-overlapping independent nodal cover for each discrete node of analysis domain, and the virtual crack closure technique is used to calculate the stress intensity factors. The ICMM has the definite nodal cover (or nodal influence domain) and employs the general polynomial function to interpolate the nodal cover, which results in a simple formulation and numerical implementation, and facilitates the accurate numerical integration of the equilibrium equation and the enforcement of the boundary conditions. It well solves the problem of the general meshless methods interpolated by rational functions, and has a wide application prospective in practical engineering. Several representative numerical examples demonstrate the convergence, accuracy and robustness of the present method.

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

Similar content being viewed by others

References

  • Anderson TL (2005) Fracture mechanics: fundamentals and applications. CRC Press, Boca Raton

    Google Scholar 

  • Augarde CE, Deeks AJ (2008) The use of Timoshenko’s exact solution for a cantilever beam in adaptive analysis. Finite Elem Anal Des 44:595–601

    Article  Google Scholar 

  • Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin method. Int J Numer Methods Eng 37:229–256

    Article  Google Scholar 

  • Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47

    Article  Google Scholar 

  • Belytschko T, Xiao SP (2003) Coupling methods for continuum model with molecular model. Int J Multiscale Comput Eng 1:115–126

    Article  Google Scholar 

  • Bittencourt TN, Wawrzynek PA, Ingraffea AR et al (1996) Quasi-automatic simulation of crack propagation for 2D LEFM problems. Eng Fract Mech 55:321–334

    Article  Google Scholar 

  • Budarapu PR, Gracie R, Bordas SPA, Rabczuk T (2014a) An adaptive multiscale method for quasi-static crack growth. Comput Mech 53:1129–1148

    Article  Google Scholar 

  • Budarapu PR, Gracie R, Yang SW, Zhuang XY, Rabczuk T (2014b) Efficient coarse graining in multiscale modeling of fracture. Theor Appl Fract Mech 69:126–143

    Article  Google Scholar 

  • Budyn ERL (2004) Multiple crack growth by the extended finite element method. Dissertation, Northweastern University

  • Cai YC, Zhu HH (2010) A PU-based meshless Shepard interpolation method satisfying delta property. Eng Anal Bound Elem 34:9–16

    Article  Google Scholar 

  • Cai YC, Zhu HH, Zhuang XY (2013) A continuous/discontinuous deformation analysis (CDDA) method based on deformable blocks for fracture modeling. Front Struct Civ Eng 7:369–378

    Article  Google Scholar 

  • Chen YZ, Hasebe N (1995) New integration scheme for the branch crack problem. Eng Fract Mech 52:791–801

    Article  Google Scholar 

  • Cook RD, Malkus DS, Plesha ME (1989) Concepts and applications of finite element analysis, 3rd edn. Wiley, New York

    Google Scholar 

  • Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Fluid Eng 85:519–525

    Google Scholar 

  • Gu L (2003) Moving kriging interpolation and element-free Galerkin method. Int J Numer Methods Eng 56:1–11

    Article  Google Scholar 

  • Khoei AR, Yasbolaghi R, Biabanaki SOR (2015) A polygonal finite element method for modeling crack propagation with minimum remeshing. Int J Fract 194:123–148

    Article  Google Scholar 

  • Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37:141–158

    Article  Google Scholar 

  • Liu WK, Han W, Lu H, Li S, Cao J (2004) Reproducing kernel element method. Part I: Theoretical formulation. Comput Methods Appl Mech Eng 19:933–951

    Article  Google Scholar 

  • Liu WK, Park HS, Qian D, Karpov EG, Kadowaki H, Wagner GJ (2006) Bridging scale methods for nanomechanics and materials. Comput Methods Appl Mech Eng 195:1407–1421

    Article  Google Scholar 

  • Miller RE, Tadmor EB (2002) The quasicontinuum method: overview, applications and current directions. J Comput Aided Mater Des 9:203–239

    Article  Google Scholar 

  • Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318

    Article  Google Scholar 

  • Nguyena VP, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simulat 79:763–813

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Rabczuk T, Zi G, Bordas S, Hung NX (2010) A simple and robust three-dimensional cracking-particle method without enrichment. Comput Methods Appl Mech Eng 199:2437–2455

    Article  Google Scholar 

  • Riker C, Holzer SM (2009) The mixed-cell-complex partition-of-unity method. Comput Methods Appl Mech Eng 198:1235–1248

    Article  Google Scholar 

  • Rybicki EF, Kanninen MF (1977) A finite element calculation of stress intensity factors by a modified crack closure integral. Eng Fract Mech 9:931–938

    Article  Google Scholar 

  • Sukumar N (2004) Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Math Eng 61:2159–2181

    Article  Google Scholar 

  • Sukumar N, Moranx B, Semenov Y, Belikovk VV (2001) Natural neighbour Galerkin methods. Int J Numer Methods Eng 50:1–27

    Article  Google Scholar 

  • Srawley JE (1976) Wide range stress intensity factor expressions for ASTM E399 standard fracture toughness specimens. Int J Fract 12:475–476

    Google Scholar 

  • Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York

    Google Scholar 

  • Valvo PS (2015) A further step towards a physically consistent virtual crack closure technique. Int J Fract 192:235–244

    Article  Google Scholar 

  • Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Methods Eng 54:1623–1648

    Article  Google Scholar 

  • Wendland H (1999) Meshless Galerkin methods using radial basis functions. Math Comput 68:1521–1531

    Article  Google Scholar 

  • Wu CT, Park CK, Chen JS (2011) A generalized approximation for the meshfree analysis of solids. Int J Numer Methods Eng 85:693–722

    Article  Google Scholar 

  • Xie D, Waas AM, Shahwan KW, Schroeder JA, Boeman RG (2004) Computation of energy release rates for kinking cracks based on virtual crack closure technique. CMES-Comp Model Eng 6:515–524

    Google Scholar 

  • Yang SW, Budarapu PR, Mahapatra DR, Bordas SPA, Zi G, Rabczuk T (2015) A meshless adaptive multiscale method for fracture. Comp Mater Sci 96:382–395

    Article  Google Scholar 

  • Zheng H, Liu F, Du XL (2015) Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method. Comput Methods Appl Mech Eng 295:150–171

    Article  Google Scholar 

  • Zheng H, Liu F, Li CG (2014) The MLS-based numerical manifold method with applications to crack analysis. Int J Fract 190:147–166

    Article  Google Scholar 

  • Zhuang XY, Augarde C, Bordas S (2011) Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling. Int J Numer Methods Eng 86:249–268

    Article  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge the support of Nature Science Foundation of China (NSFC 11472194), and New Century Excellent Talents Project in China (NCET-12-0415).

Author information

Authors and Affiliations

  1. Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai, 200092, China

    Yongchang Cai & Hehua Zhu

Authors
  1. Yongchang Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hehua Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongchang Cai.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cai, Y., Zhu, H. Independent cover meshless method using a polynomial approximation. Int J Fract 203, 63–80 (2017). https://doi.org/10.1007/s10704-016-0110-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10704-016-0110-1

Keywords

Search

Navigation