Reproducing kernel particle methods for structural dynamics

Wing Kam Liu*, Sukky Jun, Shaofan Li, Jonathan Adee, Ted Belytschko

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

704 Scopus citations


This paper explores a Reproducing Kernel Particle Method (RKPM) which incorporates several attractive features. The emphasis is away from classical mesh generated elements in favour of a mesh free system which only requires a set of nodes or particles in space. Using a Gaussian function or a cubic spline function, flexible window functions are implemented to provide refinement in the solution process. It also creates the ability to analyse a specific frequency range in dynamic problems reducing the computer time required. This advantage is achieved through an increase in the critical time step when the frequency range is low and a large window is used. The stability of the window function as well as the critical time step formula are investigated to provide insight into RKPMs. The predictions of the theories are confirmed through numerical experiments by performing reconstructions of given functions and solving elastic and elastic–plastic one‐dimensional (1‐D) bar problems for both small and large deformation as well as three 2‐D large deformation non‐linear elastic problems. Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods. In contrast with what has been reported in SPH literature, we do not find any tensile instability with RKPMs.

Original languageEnglish (US)
Pages (from-to)1655-1679
Number of pages25
JournalInternational Journal for Numerical Methods in Engineering
Issue number10
StatePublished - May 30 1995


  • aliasing control
  • correction function
  • elastic‐plastic large deformation
  • smooth particle hydrodynamics
  • tensile instability
  • wavelets

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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