Corrected Stabilized Non-conforming Nodal Integration in Meshfree Methods

  • Marcus Rüter
  • Michael Hillman
  • Jiun-Shyan Chen
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 89)


A novel approach is presented to correct the error from numerical integration in Galerkin methods for meeting linear exactness. This approach is based on a Ritz projection of the integration error that allows a modified Galerkin discretization of the original weak form to be established in terms of assumed strains. The solution obtained by this method is the correction of the original Galerkin discretization obtained by the inaccurate numerical integration scheme. The proposed method is applied to elastic problems solved by the reproducing kernel particle method (RKPM) with first-order correction of numerical integration. In particular, stabilized non-conforming nodal integration (SNNI) is corrected using modified ansatz functions that fulfill the linear integration constraint and therefore conforming sub-domains are not needed for linear exactness. Illustrative numerical examples are also presented.


Reproducing kernel particle method Stabilized non-conforming nodal integration Integration constraint Strain smoothing 



The support of this work by the US Army Engineer Research and Development Center under the contract W912HZ-07-C-0019:P00001 to the second and third authors and DFG (German Research Foundation) under the grant no. RU 1213/2-1 to the first author is very much appreciated.


  1. 1.
    I. Babuška, U. Banerjee, J.E. Osborn, Q. Li, Quadrature for meshless methods. Int. J. Numer. Methods Eng. 76, 1434–1470 (2008)zbMATHCrossRefGoogle Scholar
  2. 2.
    I. Babuška, U. Banerjee, J.E. Osborn, Q. Zhang, Effect of numerical integration on meshless methods. Comput. Methods Appl. Mech. Eng. 198, 2886–2897 (2009)zbMATHCrossRefGoogle Scholar
  3. 3.
    I. Babuška, J. Whiteman, T. Strouboulis, Finite Elements: An Introduction to the Method and Error Estimation (Oxford University Press, Oxford, 2010)Google Scholar
  4. 4.
    S. Beissel, T. Belytschko, Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 139, 49–74 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    J. Bonet, S. Kulasegaram, Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations. Int. J. Numer. Methods Eng. 47, 1189–1214 (2000)zbMATHCrossRefGoogle Scholar
  6. 6.
    J.S. Chen, M. Hillman, M. Rüter, A unified domain integration method for Galerkin meshfree methods, submitted to Int. J. Numer. Methods. Eng., (2012)Google Scholar
  7. 7.
    J.S. Chen, W. Hu, M.A. Puso, Y. Wu, X. Zhang, Strain smoothing for stabilization and regularization of Galerkin meshfree methods, in Meshfree Methods for Partial Differential Equations III, ed. by M. Griebel, M.A. Schweitzer (Springer, Berlin, 2007), pp. 57–75CrossRefGoogle Scholar
  8. 8.
    J.S. Chen, C.-T. Wu, S. Yoon, Y. You, A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Methods Eng. 50, 435–466 (2001)zbMATHCrossRefGoogle Scholar
  9. 9.
    J.S. Chen, S. Yoon, C.-T. Wu, Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Methods Eng. 53, 2587–2615 (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    J. Dolbow, T. Belytschko, Numerical integration of the Galerkin weak form in meshfree methods. Comput. Mech. 23, 219–230 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    P.C. Guan, S.W. Chi, J.S. Chen, T.R. Slawson, M.J. Roth, Semi-Lagrangian reproducing kernel particle method for fragment-impact problems. Int. J. Impact Eng. 38, 1033–1047 (2011)CrossRefGoogle Scholar
  12. 12.
    J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hambg. 36, 9–15 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    M.A. Puso, J.S. Chen, E. Zywicz, W. Elmer, Meshfree and finite element nodal integration methods. Int. J. Numer. Methods Eng. 74, 416–446 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    E. Stein, M. Rüter, Finite element methods for elasticity with error-controlled discretization and model adaptivity, in Encyclopedia of Computational Mechanics, 2nd edn., ed. by E. Stein, R. de Borst, T.J.R. Hughes (Wiley, Chichester, 2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marcus Rüter
    • 1
  • Michael Hillman
    • 1
  • Jiun-Shyan Chen
    • 1
  1. 1.Department of Civil and Environmental EngineeringUniversity of CaliforniaLos AngelesUSA

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