doi: 10.1111/j.1460-2695.2009.01359.x Development of the extended parametric meshless Galerkin method to predict the crack propagation path in two-dimensional damaged structures M. MUSIVAND-ARZANFUDI and H. HOSSEINI-TOUDESHKY Aerospace Engineering Department and Center of Excellence in Computational Aerospace Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran Received in final form 16 April 2009 ABSTRACT The parametric meshless Galerkin method (PMGM) enhances the promising features of the meshless methods by utilizing the parametric spaces and parametric mapping, and im- proves their efficiency from the practical viewpoints. The computation of meshless shape functions has been usually a time-consuming and complicated task in the meshless meth- ods. In the PMGM, the meshless shape functions are mapped from the parametric space to the physical space, and therefore, the necessary computational time to generate the meshless shape functions is saved. The extended parametric meshless Galerkin method (X-PMGM) even improves the parametric property of the PMGM by incorporating the partition of unity concepts. In this paper, the development of the X-PMGM is extended by incorporating a crack-tip formulation in X-PMGM for fracture analysis and predic- tion of crack propagation path in the damaged structures. In this formulation, meshless shape functions are enriched by discontinuous enrichment function as well as crack-tip enrichment functions. The obtained results show that the predicted crack growth path is in good agreement with the experimental results. Keywords crack propagation; Galerkin; meshless; parametric; partition of unity. NOMENCLATURE a(X) = vector of unknowns of the MLS approximation at sample point X a i (X) = components of the vector a(X) AD = approximation domain ASD = approximation subdomain b j = additional degrees of freedom corresponding to the nodes belonging to the set J C i = stretch of the mapping in ith direction c l 1 k and c l 2 k = additional degrees of freedom corresponding to the nodes belonging to the sets K 1 and K 2 , respectively d = vector of nodal parameters for the MLS approximation d i = nodal parameter of node i for the MLS approximation d mi = size of the influence domain for node i d n = nodal spacing in the MPD in both x and y directions D = constitutive matrix for both plane stress and plane strain conditions DOP = domain of problem Correspondence: H. Hosseini-Toudeshky. E-mail: hosseini@aut.ac.ir 552 c  2009 The Authors. Journal compilation c  2009 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 32, 552–566 Fatigue & Fracture of Engineering Materials & Structures