On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods

Hossein Talebi, Cristóbal Samaniego, Esteban Samaniego, Timon Rabczuk

Research output: Contribution to journalArticlepeer-review

56 Citations (Scopus)


Meshfree methods (MMs) such as the element free Galerkin (EFG)method have gained popularity because of some advantages over other numerical methods such as the finite element method (FEM). A group of problems that have attracted a great deal of attention from the EFG method community includes the treatment of large deformations and dealing with strong discontinuities such as cracks. One efficient solution to model cracks is adding special enrichment functions to the standard shape functions such as extended FEM, within the FEM context, and the cracking particles method, based on EFG method. It is well known that explicit time integration in dynamic applications is conditionally stable. Furthermore, in enriched methods, the critical time step may tend to very small values leading to computationally expensive simulations. In this work, we study the stability of enriched MMs and propose two mass-lumping strategies. Then we show that the critical time step for enriched MMs based on lumped mass matrices is of the same order as the critical time step of MMs without enrichment. Moreover, we show that, in contrast to extended FEM, even with a consistent mass matrix, the critical time step does not vanish even when the crack directly crosses a node.

Original languageEnglish
Pages (from-to)1009-1027
Number of pages19
JournalInternational Journal for Numerical Methods in Engineering
Issue number8
Publication statusPublished - 2012 Feb 24


  • Dynamic crack propagation
  • Element free Galerkin method
  • Enrichment
  • Explicit time integration
  • XEFG
  • XFEM

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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