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The MLS-based numerical manifold method with applications to crack analysis

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Abstract

In order to solve problems, from a continuum point of view and in a unified way, involving continuum and discontinuum deformation, and small deformation and large movement, the numerical manifold method (NMM) introduces two covers, namely the mathematical cover (MC) and the physical cover (PC). This study generates the MC with the influence domains of nodes in the moving least squares (MLS) interpolation instead of commonly-used finite element meshes, significantly simplifying the generation of PCs and the simulation of crack growth. Advantageous over the conventional meshfree method, the MLS-based NMM can naturally treat complex geometry without recourse to those complicated but contrived criteria or operations. Moreover, the treatment of large movement caused by cracking is much easier with the MLS-based NMM than with the FE-based NMM.

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References

  • An XM, Fu GY, Ma GW (2012) A comparison between the NMM and the XFEM in discontinuity modeling. Int J Comput Methods 9:1240030. doi:10.1142/S0219876212400300

    Article  Google Scholar 

  • An XM, Li LX, Ma GW, Zhang HH (2011) Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes. Comput Methods Appl Mech Eng 200:665–674

    Article  Google Scholar 

  • Babuška I, Melenk JM (1996) The partition of unity method. Int J Numer Methods Eng 40:727–758

    Article  Google Scholar 

  • Barbieri E, Petrinic N, Meo M et al (2012) A new weight-function enrichment in meshless methods for multiple cracks in linear elasticity. Int J Numer Methods Eng 90:177–195

    Article  Google Scholar 

  • Belytschko T, Gu L, Lu YY (1994) Fracture and crack growth by element free Galerkin methods. Model Simul Mater Sci Eng 2:519–534

    Article  Google Scholar 

  • Belytschko T, Krongauz Y, Fleming M, Organ D, Liu WK (1996) Smoothing and accelerated computations in the element free Galerkin method. J Comput Appl Math 74:111–126

    Article  Google Scholar 

  • Bordas S, Nguyen PV, Dunant C et al (2007) An extended finite element library. Int J Numer Methods Eng 71:703–732

    Article  Google Scholar 

  • Bordas S, Rabczuk T, Zi G (2008) Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. Eng Fract Mech 75:943–960

    Article  Google Scholar 

  • Chen GQ, Ohnishi Y, Ito T (1998) Development of high-order manifold method. Int J Numer Methods Eng 43:685–712

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Dolbow J, Devan A (2004) Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch test. Int J Numer Methods Eng 59:47–67

  • Fleming M, Chu Y, Moran B, Belytschko T, Lu Y, Gu L (1997) Enriched element-free Galerkin methods for crack tip fields. Int J Numer Methods Eng 40:1483–1504

    Article  Google Scholar 

  • Huang R, Sukumar N, Prévost JH (2003) Modeling quasi-static crack growth with the extended finite element method, part II: numerical applications. Int J Solids Struct 40:7539–7552

    Article  Google Scholar 

  • Jiang QH, Zhou CB, Li DQ (2009) A three-dimensional numerical manifold method based on tetrahedral meshes. Comput Struct 87:880–889

    Article  Google Scholar 

  • Khoei AR, Anahid M, Shahim K (2009) An extended arbitrary Lagrangian–Eulerian finite element method for large deformation of solid mechanics. Comput Methods Appl Mech Eng 44:401–416

    Google Scholar 

  • Kourepinis D, Pearce C, Bićanić N (2010) Higher-order discontinuous modeling of fracturing in concrete using the numerical manifold method. Int J Comput Methods 7. doi:10.1142/S0219876210002076

  • Kurumatani M, Terada K (2009) Finite cover method with multi-cover layers for the analysis of evolving discontinuities in heterogeneous media. Int J Numer Methods Eng 79:1–24

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Li SC, Cheng Y (2005) Enriched meshless manifold method for two-dimensional crack modeling. Theor Appl Fract Mech 44:234–248

    Article  Google Scholar 

  • Lin JS (2003) A mesh-based partition of unity method for discontinuity modeling. Comput Methods Appl Mech Eng 192:1515–1532

    Article  Google Scholar 

  • Liu WK, Sukky Jun S (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38:1655–1679

    Article  Google Scholar 

  • Ma GW, An XM, He L (2010) The numerical manifold method: a review. Int J Comput Methods 7:1–32

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Ning YJ, An XM, Ma GW (2010) Footwall slope stability analysis with the numerical manifold method. Int J Rock Mech Min Sci 48:2039–2071

    Google Scholar 

  • Organ D, Fleming M, Terry T, Belytschko T (1996) Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Comput Mech 18:225–235

    Article  Google Scholar 

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

    Google Scholar 

  • Rabczuk T, Belytschko T (2007) A three dimensional large deformation meshfree method for arbitrary evolving cracks. Comput Methods Appl Mech Eng 196:2777–2799

    Article  Google Scholar 

  • Shi GH (1991) Manifold method of material analysis. In: Transactions of the 9th army conference on applied mathematics and computing. Report No. 92–1, U.S. Army Research Office, Minneapolis, MN, pp 57–76

  • Spivak M (1965) Calculus on manifolds. Benjamin, New York

  • Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190:4081–4193

    Article  Google Scholar 

  • Sumi Y, Yang C, Wang ZN (1996) Morphological aspects of fatigue crack propagation part II: effects of stress biaxiality and welding residual stress. Int J Fract 82:221–235

  • Swenson DV, Kaushik N (1990) Finite element analysis of edge cracking in plates. Eng Fract Mech 37:641–652

    Article  Google Scholar 

  • Szabó B, Babuška I (2011) Introduction to finite element analysis: formulation, verification and validation. Wiley, Chichester

  • Tian R, Yagawa G (2005) Generalized nodes and high-performance elements. Int J Numer Methods Eng 64:2039–2071

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech 24:109–114

    Google Scholar 

  • Wu ZJ, Wong LNY (2012) Frictional crack initiation and propagation analysis using the numerical manifold method. Comput Geotech 39:38–53

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

    Google Scholar 

  • Yang Z (2006) Fully automatic modelling of mixed-mode crack propagation using scaled boundary finite element method[J]. Eng Fract Mech 73:1711–1731

    Article  Google Scholar 

  • Zhang HW, Zhou L (2008) Numerical manifold method for dynamic nonlinear analysis of saturated porous media. Int J Numer Anal Methods Geomech 30:927–951

    Article  Google Scholar 

  • Zheng H, Li JL (2007) A practical solution for KKT systems. Numer Algorithms 46:105–119

    Article  Google Scholar 

  • Zheng H, Liu ZJ, Ge XR (2013) Numerical manifold space of Hermitian form and application to Kirchhoff’s thin plate problems. Int J Numer Methods Eng 95:721–739

    Article  Google Scholar 

  • Zheng H, Xu DD (2014) New strategies for some issues of numerical manifold method in simulation of crack propagation. Int J Numer Methods Eng 97:986–1010

    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

This study is supported by the National Basic Research Program of China (973 Program), under the Grant Nos. 2011CB013505 and 2014CB047100; and the National Natural Science Foundation of China, under the Grant No. 11172313.

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Authors and Affiliations

  1. Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing, 100124, China

    Hong Zheng

  2. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071, China

    Feng Liu & Chunguang Li

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  1. Hong Zheng
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  2. Feng Liu
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  3. Chunguang Li
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Correspondence to Hong Zheng.

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Zheng, H., Liu, F. & Li, C. The MLS-based numerical manifold method with applications to crack analysis. Int J Fract 190, 147–166 (2014). https://doi.org/10.1007/s10704-014-9980-2

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  • DOI: https://doi.org/10.1007/s10704-014-9980-2

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