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In this paper, an anisotropic weight function in the elliptic form is introduced for the Element Free Galerkin Method (EFGM). In the circular (isotropic) weight function,each node has one characteristic parameter that determines its domain of influence. In the elliptic weight function,each node has three characteristic parameters that are major influence radius, minor influence radius and the direction of the major influence. Using the elliptic weight function each point of the domain may be affected by a less number of nodes in certain conditions. Thus, the computational cost of the method is decreased. In addition, the dependency of the solution on the method that is used for the enforcement of the essential boundary conditions, decreases. As an application of the proposed elliptic weight function, some examples of elastostatic problems are solved and the results are compared with those available in the literature.<\div>

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