The meshless Shepard and least-squares (MSLS) interpolation is a newly developed partition of unity- (PU-) based method which removes the difficulties with many other meshless methods such as the lack of the Kronecker delta property. The MSLS interpolation is efficient to compute and retain compatibility for any basis function used. In this paper, we extend the MSLS interpolation to the local Petrov-Galerkin weak form and adopt the duo nodal support domain. In the new formulation, there is no need for employing singular weight functions as is required in the original MSLS and also no need for background mesh for integration. Numerical examples demonstrate the effectiveness and robustness of the present method. 1. Introduction Meshless methods have prospered both in theory and application in engineering problems in the past two decades as they offer the possibility of a discretised approach without the occurrence of mesh entanglement requiring remeshing. A wide range of meshless methods have been proposed as outlined in recent surveys [1–3]. Remarkable successes have been reported in applying these methods for analyzing challenging engineering problems, namely, fracture modeling [4–7], plate and shell analysis [8–15], three-dimensional problems [16–18], fluid structure interaction analysis [19], strain localization problems [20], large deformation problems [21], and other applications [22–29]. Some currently popular meshless approximations are the moving least-squares (MLS) approximation, Shepard shape functions, partition of unity (PU), radial basis functions (RBF), reproducing kernel particle method (RKPM), point interpolation (PI), and Kriging interpolation (KI). Among them, the MLS approximation [30] is one of the most widely used approximations at present due to its global continuity, completeness, and robustness. However, the MLS approximation suffers from a number of problems that practically limit its application, namely, the high computational cost in obtaining the shape functions and their derivatives, difficulty in retaining accuracy with respect to nodal arrangement, and also the difficulty with which essential boundary conditions can be imposed due to the lack of the Kronecker delta property. Efforts have been made to address these problems by various means. Breitkopf et al. [31] developed the analytical forms for computing shape functions and diffuse derivatives of shape functions by assuming that some terms are constant and complete derivatives of shape functions. However, these formulations are dependent on the number of nodes and the
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