Skip to main content
SpringerLink
Log in

Quasi-convex reproducing kernel meshfree method

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

Abstract

A quasi-convex reproducing kernel approximation is presented for Galerkin meshfree analysis. In the proposed meshfree scheme, the monomial reproducing conditions are relaxed to maximizing the positivity of the meshfree shape functions and the resulting shape functions are referred as the quasi-convex reproducing kernel shape functions. These quasi-convex meshfree shape functions are still established within the framework of the classical reproducing or consistency conditions, namely the shape functions have similar form as that of the conventional reproducing kernel shape functions. Thus this approach can be conveniently implemented in the standard reproducing kernel meshfree formulation without an overmuch increase of computational effort. Meanwhile, the present formulation enables a straightforward construction of arbitrary higher order shape functions. It is shown that the proposed method yields nearly positive shape functions in the interior problem domain, while in the boundary region the negative effect of the shape functions are also reduced compared with the original meshfree shape functions. Subsequently a Galerkin meshfree analysis is carried out by employing the proposed quasi-convex reproducing kernel shape functions. Numerical results reveal that the proposed method has more favorable accuracy than the conventional reproducing kernel meshfree method, especially for structural vibration analysis.

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
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

  2. Belytschko T, Lu YY, Gu L (1994) Element-free Gakerkin methods. Int J Numer Meth Eng 37:229–256

    Article  MATH  MathSciNet  Google Scholar 

  3. Duarte CAM, Oden JT (1996) H-p clouds and h-p meshless method. Numer Methods Partial Differ Equ 12:673–705

    Article  MATH  MathSciNet  Google Scholar 

  4. Onate E, Idelsohn S, Zienkiewicz OC, Taylor RL (1996) A finite point method in computational mechanics: applications to convective transport and fluid flow. Int J Numer Meth Eng 39:3839–3866

    Article  MATH  MathSciNet  Google Scholar 

  5. Atluri SN, Zhu T (1998) A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117–127

    Article  MATH  MathSciNet  Google Scholar 

  6. De S, Bathe KJ (2000) The method of finite spheres. Comput Mech 25:329–345

    Article  MATH  MathSciNet  Google Scholar 

  7. Monaghan JJ (1992) Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 30:543–574

    Google Scholar 

  8. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu WK, Li S, Belytschko T (1997) Moving least square reproducing kernel method: (I) methodology and convergence. Comput Methods Appl Mech Eng 143:113–154

    Article  MATH  MathSciNet  Google Scholar 

  10. Li S, Liu WK (1996) Moving least-square reproducing kernel method. Part II fourier analysis. Comput Methods Appl Mech Eng 139:159–193

    Article  MATH  Google Scholar 

  11. Chen JS, Han W, You Y, Meng X (2003) A reproducing kernel method with nodal interpolation property. Int J Numer Meth Eng 56:935–960

    Article  MATH  MathSciNet  Google Scholar 

  12. Liu WK, Han W, Lu H, Li S, Cao J (2004) Reproducing kernel element method. Part I: theoretical formulation. Comput Methods Appl Mech Eng 193:933–951

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu GR, Gu YT (2000) A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. J Sound Vib 246:29–46

    Article  Google Scholar 

  15. Kansa EJ (1992) Multiqudrics-a scattered data approximation scheme with applications to computational fluid dynamics-I. Surface approximations and partial derivatives. Comput Math Appl 19:127–145

    Article  MathSciNet  Google Scholar 

  16. Chen JS, Hu W, Hu HY (2008) Reproducing kernel enhanced local radial basis collocation method. Int J Numer Meth Eng 75:600–627

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Arroyo M, Ortiz M (2006) Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int J Numer Meth Eng 65:2167–2202

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  23. Liu WK, Chen Y, Jun S, Chen JS, Belytschko T, Pan C, Uras RA, Chang CT (1996) Overview and applications of the reproducing kernel particle methods. Arch Comput Methods Eng 3:3–80

    Article  MathSciNet  Google Scholar 

  24. Atluri SN, Shen SP (2002) The meshless local Petrov-Galerkin method. Tech Science Press, Duluth

    MATH  Google Scholar 

  25. Babuška I, Banerjee U, Osborn JE (2003) Survey of meshless and generalized finite element methods: a unified approach. Acta Numer 12:1–125

    Article  MATH  MathSciNet  Google Scholar 

  26. Li S, Liu WK (2004) Meshfree particle methods. Springer, New York

    MATH  Google Scholar 

  27. Huerta A, Belytschko T, Fernández-Méndez S, Rabczuk T (2004) Meshfree methods. Encycl Comput Mech 1:279–309

    Google Scholar 

  28. Nguyen VP, Rabczuk T, Bordas S (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79:763–813

    Article  MATH  MathSciNet  Google Scholar 

  29. Liu GR (2009) Mesh free methods: moving beyond the finite element method, 2nd edn. CRC Press, Boca

    Book  Google Scholar 

  30. Zhang X, Liu Y, Ma S (2009) Meshfree methods and their applications. Adv Mech 39:1–36

    Article  MATH  Google Scholar 

  31. Sukumar N, Wright RW (2007) Overview and construction of meshfree basis functions: from moving least squares to entropy approximants. Int J Numer Meth Eng 70:181–205

    Article  MATH  MathSciNet  Google Scholar 

  32. Sukumar N, Wets RB (2007) Deriving the continuity of maximum-entropy basis functions via variational analysis. SIAM J Optim 18:914–925

    Article  MATH  MathSciNet  Google Scholar 

  33. Yaw LL, Sukumar N, Kunnath SK (2009) Meshfree co-rotational formulation for two dimensional continua. Int J Numer Meth Eng 79:979–1003

    Article  MATH  MathSciNet  Google Scholar 

  34. Rosolen A, Millán D, Arroyo M (2010) On the optimum support size in meshfree methods: a variational adaptivity approach with maximum entropy approximants. Int J Numer Meth Eng 82:868–895

    MATH  Google Scholar 

  35. Ortiz A, Puso M, Sukumar N (2010) Maximum-entropy meshfree method for compressible and near-incompressible elasticity. Comput Methods Appl Mech Eng 199:1859–1871

    Article  MATH  MathSciNet  Google Scholar 

  36. Millán D, Rosolen A, Arroyo M (2011) Thin shell analysis from scattered points with maximum-entropy approximants. Int J Numer Meth Eng 85:723–751

    Article  MATH  Google Scholar 

  37. Ortiz A, Puso M, Sukumar N (2011) Maximum-entropy meshfree method for incompressible media problems. Finite Elem Anal Des 47:572–585

    Article  MathSciNet  Google Scholar 

  38. Quaranta G, Kunnath SK, Sukumar N (2012) Maximum-entropy meshfree method for nonlinear static analysis of planar reinforced concrete structures. Eng Struct 42:179–189

    Article  Google Scholar 

  39. Rosolen A, Peco C, Arroyo M (2013) An adaptive meshfree method for phase-field models of biomembranes. Part I: approximation with maximum-entropy approximants. J Comput Phys 249:303–319

    Article  MathSciNet  Google Scholar 

  40. Greco F, Sukumar N (2013) Derivatives of maximum-entropy basis functions on the boundary: theory and computations. Int J Numer Meth Eng 94:1123–1149

    Article  MathSciNet  Google Scholar 

  41. Wu CT, Hu W (2011) Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and near-incompressible solids. Comput Methods Appl Mech Eng 200:2991–3010

    Article  MATH  MathSciNet  Google Scholar 

  42. Wu CT, Koishi M (2012) Three-dimensional meshfree-enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites. Int J Numer Meth Eng 81:1127–1156

    MathSciNet  Google Scholar 

  43. Wu CT, Hu W, Chen JS (2012) A meshfree-enriched finite element method for compressible and near-incompressible elasticity. Int J Numer Meth Eng 90:882–914

    Article  MATH  MathSciNet  Google Scholar 

  44. Hu W, Wu CT, Koishi M (2012) A displacement-based nonlinear finite element formulation using meshfree-enriched triangular elements for the two-dimensional large deformation analysis of elastomers. Finite Elem Anal Des 50:161–172

    Article  MathSciNet  Google Scholar 

  45. Cyron C, Arroyo M, Ortiz M (2009) Smooth, second order, non-negative meshfree approximants selected by maximum entropy. Int J Numer Meth Eng 79:1605–1632

    Article  MATH  MathSciNet  Google Scholar 

  46. González D, Cueto E, Doblaré M (2010) A higher-order method based on local maximum entropy approximation. Int J Numer Meth Eng 83:741–764

    MATH  Google Scholar 

  47. Rosolen A, Millán D, Arroyo M (2013) Second-order convex maximum entropy approximants with applications to high-order PDE. Int J Numer Meth Eng 94:150–182

    Article  Google Scholar 

  48. Bompadre A, Perotti LE, Cyron C, Ortiz M (2012) Convergent meshfree approximation schemes of arbitrary order and smoothness. Comput Methods Appl Mech Eng 221:83–103

    Article  MathSciNet  Google Scholar 

  49. Sukumar N (2013) Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons. Comput Methods Appl Mech Eng 263:27–41

    Article  MATH  MathSciNet  Google Scholar 

  50. Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Meth Eng 38:1655–1679

    Article  MATH  MathSciNet  Google Scholar 

  51. Liu WK, Jun S (1998) Multiple-scale reproducing kernel particle methods for large deformation problems. Int J Numer Meth Eng 41:1339–1362

    Article  MATH  MathSciNet  Google Scholar 

  52. Li S, Liu WK (1998) Synchronized reproducing kernel interpolant via multiple wavelet expansion. Comput Mech 21:28–47

    Google Scholar 

  53. Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity, Part I: formulations. Int J Numer Meth Eng 45:251–288

    Google Scholar 

  54. Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity, part II: applications. Int J Numer Meth Eng 45:289–317

    Article  Google Scholar 

  55. Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of nonlinear structures. Comput Methods Appl Mech Eng 139:195–227

    Article  MATH  MathSciNet  Google Scholar 

  56. Chen JS, Pan C, Wu CT (1997) Large deformation analysis of rubber based on a reproducing kernel particle method. Comput Mech 19:211–227

    Article  MATH  MathSciNet  Google Scholar 

  57. Chen JS, Pan C, Roque CMOL, Wang HP (1998) A Lagrangian reproducing kernel particle method for metal forming analysis. Comput Mech 22:289–307

    Article  MATH  Google Scholar 

  58. Chen JS, Yoon S, Wang HP, Liu WK (2000) An improved reproducing kernel particle method for nearly incompressible finite elasticity. Comput Methods Appl Mech Eng 181:117–145

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgments

The support of this work by the National Natural Science Foundation of China (11222221) is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Xiamen University, Xiamen , 361005, Fujian, China

    Dongdong Wang & Pengjie Chen

Authors
  1. Dongdong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pengjie Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongdong Wang.

Appendix

Appendix

In this appendix, the nodal numberings and locations for the first non-uniform discretizations of Figs. 19, 30, and 34 are listed for reference (Figs. 37, 38, 39; Tables 1, 2, 3).

Fig. 37
figure 37

Nodal numberings for the rod problem with non-uniform discretization (Fig. 19)

Full size image
Fig. 38
figure 38

Nodal numberings for the membrane problem with non-uniform discretization (Fig. 30)

Full size image
Fig. 39
figure 39

Nodal numberings for the cantilever beam problem with non-uniform discretization (Fig. 34)

Full size image
Table 1 Nodal locations for the rod problem with non-uniform dicsretization
Full size table
Table 2 Nodal locations for the membrane problem with non-uniform dicsretization
Full size table
Table 3 Nodal locations for the cantilever beam problem with non-uniform dicsretization
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, D., Chen, P. Quasi-convex reproducing kernel meshfree method. Comput Mech 54, 689–709 (2014). https://doi.org/10.1007/s00466-014-1022-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-014-1022-4

Keywords

Search

Navigation