Convergence and stabilization of stress-point integration in mesh-free and particle methods

Thomas Peter Fries*, Ted Belytschko

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Stress-point integration provides significant reductions in the computational effort of mesh-free Galerkin methods by using fewer integration points, and thus facilitates the use of mesh-free methods in applications where full integration would be prohibitively expensive. The influence of stress-point integration on the convergence and stability properties of mesh-free methods is studied. It is shown by numerical examples that for regular nodal arrangements, good rates of convergence can be achieved. For non-uniform nodal arrangements, stress-point integration is associated with a mild instability which is manifested by small oscillations. Addition of stabilization improves the rates of convergence significantly. The stability properties are investigated by an eigenvalue study of the Laplace operator. It is found that the eigenvalues of the stress-point quadrature models are between those of full integration and nodal integration. Stabilized stress-point integration is proposed in order to improve convergence and stability properties.

Original languageEnglish
Pages (from-to)1067-1087
Number of pages21
JournalInternational Journal for Numerical Methods in Engineering
Issue number7
Publication statusPublished - 14 May 2008


  • Convergence
  • Integration
  • Mesh-free method
  • Meshless method
  • Moving least squares
  • Particle method

ASJC Scopus subject areas

  • Engineering (miscellaneous)
  • Applied Mathematics
  • Computational Mechanics

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