8th. World Congress on Computational Mechanics (WCCM8) 5th European Congress on Computational Methods in Applied Sciences and Engineeering (ECCOMAS 2008) June 30 –July 5, 2008 Venice, Italy THE SMOOTHED EXTENDED FINITE ELEMENT METHOD (SmXFEM) * St´ ephane P. A. Bordas 1 , Sundararajan Natarajan, Marc Duflot 2 , Nguyen Xuan Hung and Timon Rabczuk 3 1 Lecturer University of Glasgow, Civil Engineering Rankine Building, G12 8LT, Scotland, UK http://people. civil.gla.ac.uk/ ∼ bordas stephane(dot) bordas(at)alumni. northwestern.edu 2 CENAERO, Belgium marc.duflot@ cenaero.be 5 Senior Lecturer University of Canterbury Mechanical Engineering New Zealand timon.rabczuk@ canterbury.ac.nz Key Words: Extended finite element method, strain smoothing, fracture mechanics, boundary integra- tion, polygonal finite elements ABSTRACT The extended finite element method (XFEM) has emerged as a valid alternative to remeshing for crack propagation problems [1,2] and is now employed with success for three dimensional crack propagation analysis of complex structures [5,6]. The basic idea of XFEM is to add special functions to describe the crack kinematics within the finite elements, so as to avoid the need for a conforming mesh. To introduce the discontinuity, discontinuous functions are added; to help capture the singularity, near-tip fields from the Westergaard asymptotic expansion are used. See [2] for a recent review of the XFEM literature. Recently, strain smoothing has appeared in the finite element literature and resulted in the discovery of the smoothed FEM (SFEM) [3,4]. The idea is to write the strain field as a spatial average of the compat- ible strains and use this “smoothed strain” to obtain the element stiffness matrix. This enables the use of polygonal and very distorted meshes and was shown to yield locking-free results for incompressible 2D and 3D elasticity, elasto-plasticity, plate and shells. In this paper, we combine strain smoothing to the XFEM to obtain the smoothed XFEM (SmXFEM), which shares properties with both the SFEM and XFEM. The integration of the XFEM weak form is performed on the boundary of the split elements, which simplifies implementation, allows dealing with distorted meshes, and arbitrary polygonal meshes. We study the convergence properties of the SmXFEM in the energy norm and analyse its behaviour in incompressible settings.