New shape functions for arbitrary discontinuities without additional unknowns

Thomas Peter Fries, Ted Belytschko

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


A method is proposed for arbitrary discontinuities, without the need for a mesh that aligns with the interfaces, and without introducing additional unknowns as in the extended finite element method. The approximation space is built by special shape functions that are able to represent the discontinuity, which is described by the level-set method. The shape functions are constructed by means of the moving least-squares technique. This technique employs special mesh-based weight functions such that the resulting shape functions are discontinuous along the interface. The new shape functions are used only near the interface, and are coupled with standard finite elements, which are employed in the rest of the domain for efficiency. The coupled set of shape functions builds a linear partition of unity that represents the discontinuity. The method is illustrated for linear elastic examples involving strong and weak discontinuities.

Original languageEnglish
Title of host publicationMeshfree Methods for Partial Differential Equations III
Number of pages17
Publication statusPublished - 2007

Publication series

NameLecture Notes in Computational Science and Engineering
ISSN (Print)14397358


  • Discontinuity
  • Moving least-squares
  • Partition of unity

ASJC Scopus subject areas

  • Engineering(all)
  • Computational Mathematics
  • Modelling and Simulation
  • Control and Optimization
  • Discrete Mathematics and Combinatorics

Cite this