An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order

H. Nguyen-Xuan, G. R. Liu, S. Bordas, S. Natarajan, T. Rabczuk

Research output: Contribution to journalArticlepeer-review

185 Citations (Scopus)


This paper presents a singular edge-based smoothed finite element method (sES-FEM) for mechanics problems with singular stress fields of arbitrary order. The sES-FEM uses a basic mesh of three-noded linear triangular (T3) elements and a special layer of five-noded singular triangular elements (sT5) connected to the singular-point of the stress field. The sT5 element has an additional node on each of the two edges connected to the singular-point. It allows us to represent simple and efficient enrichment with desired terms for the displacement field near the singular-point with the satisfaction of partition-of-unity property. The stiffness matrix of the discretized system is then obtained using the assumed displacement values (not the derivatives) over smoothing domains associated with the edges of elements. An adaptive procedure for the sES-FEM is proposed to enhance the quality of the solution with minimized number of nodes. Several numerical examples are provided to validate the reliability of the present sES-FEM method.

Original languageEnglish
Pages (from-to)252-273
Number of pages22
JournalComputer Methods in Applied Mechanics and Engineering
Publication statusPublished - 2013 Jan 1

Bibliographical note

Funding Information:
The support of the Vietnam National Foundation for Science and Technology Development (NAFOSTED; Grant No. 107.02-2012.17) and of the Research Scholar Program from the University of Cincinnati (USA) is gratefully acknowledged.


  • Adaptive finite elements
  • Crack propagation
  • Singular ES-FEM
  • Singularity
  • Smoothed finite element method

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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