This paper presents an efficient meshless method for analyzing linear-elastic cracked structures subject to single- or mixed-mode loading conditions. The method involves an element-free Galerkin formulation in conjunction with an exact implementation of essential boundary conditions and a new weight function. The proposed method eliminates the shortcomings of Lagrange multipliers typically used in element-free Galerkin formulations. Numerical examples show that the proposed method yields accurate estimates of stress-intensity factors and near-tip stress field in two-dimensional cracked structures. Since the method is meshless and no element connectivity data are needed, the burdensome remeshing required by finite element method (FEM) is avoided. By sidestepping remeshing requirement, crack-propagation analysis can be dramatically simplified. An example problem on mixed-mode condition is presented to simulate crack propagation. The agreement between the predicted crack trajectories by the proposed meshless method and FEM is excellent. In recent years, a class of Galerkin-based meshfree or meshless methods have been developed that do not require a structured mesh to discretize the problem, such as the element-free Galerkin method, and the reproducing kernel particle method. These methods employ a moving least-squares approximation method that allows resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes. Meshless discretization presents significant advantages for modeling fracture propagation. Since no element connectivity data are needed, the burdensome remeshing required by the finite element method (FEM) is avoided. A growing crack can be modeled by simply extending the free surfaces, which correspond to the crack. Although meshless methods are attractive for simulating crack propagation, because of the versatility, the computational cost of a meshless method typically exceeds the cost of a regular FEM. Also in some cases, the MLS which is the bases of the meshless method may form an ill-conditioned system of equations so that the solution cannot be correctly obtained. Hence, in this paper, we propose an improved element-free Galerkin method based on an improved moving least-square approximation (IMLS) method. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill-conditioned system of equations. Numerical examples are presented to illustrate the computational efficiency and accuracy of the proposed improved element-free Galerkin method.

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