INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2009; 00:1–6 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02] On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM) St´ ephane PA Bordas 1,2, ∗ , Sundararajan Natarajan 1,3 , Pierre Kerfriden 1,4 , Charles Augarde 5 , D Roy Mahapatra 6 , Timon Rabczuk 7 , Stefano Dal Pont 8 1 Cardiff School of Engineering Theoretical, Applied and Computational mechanics Cardiff University Queen’s Buildings The Parade CARDIFF CF24 3AA Wales, UK. 2 Professor of Theoretical Applied and Computational Mechanics, Leverhulme/Royal Academy of Engineering Senior Research Fellow 3 Ph.D. Research Student 4 Lecturer in Theoretical Applied and Computational Mechanics 5 Senior Lecturer in Civil Engineering, School of Engineering and Computing Sciences, Durham University, Durham, UK. 6 Assistant Professor, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India. 7 Researcher, University Paris-Est, Laboratoire Central des Ponts et Chausses, BCC-LCPC, 58 bld Lefebvre 75732 Paris, France. 8 Professor, Department of Civil Engineering, Institute for Structural Mechanics, Bauhaus-Universit¨ at, Weimar, Germany. SUMMARY By using the strain smoothing technique proposed by Chen et al. [1] for meshless methods in the context of the finite element method (FEM), Liu et al. [2] developed the Smoothed FEM (SFEM). Although the SFEM is not yet well-understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically the convergence properties in which conditions strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial [cases (a) and (b)], but that non-polynomial enrichment of type (c) lead to inferior methods compared to standard enriched FEM (e.g. XFEM). Copyright c  2009 John Wiley & Sons, Ltd. key words: Smoothed finite element method, boundary integration, eXtended finite element method, strain smoothing, linear elastic fracture mechanics, ∗ Correspondence to: St´ ephane P. A. Bordas, School of Engineering, Theoretical, applied and computational mechanics, Queen’s Building, Room S1.03, Cardiff University, CF24 3AA, Wales, U.K. stephane.bordas@alumni.northwestern.edu. Tel. +44 (0)29 20875941. http://www.researcherid.com/rid/ A-1858-2009, http://www.engin.cf.ac.uk/whoswho/profile.asp?RecordNo=679 Received Copyright c  2009 John Wiley & Sons, Ltd. Revised