SUMMARYThis paper presents a new approach in the construction of meshfree approximations as well as the weak Kronecker-delta property at the boundary, referred to as a generalized meshfree (GMF) approximation. The GMF approximation introduces an enriched basis function in the original Shepard's method. This enriched basis function is introduced to meet the linear or higher order reproducing conditions and at the same time to offer great flexibility on the control of the smoothness and convexity of the approximation. The construction of the GMF approximation can be viewed as a special root-finding scheme of constraint equations that enforces that the basis functions are corrected and the reproducing conditions with certain orders are satisfied within a set of nodes. By choosing different basis functions, various convex and nonconvex approximations including moving least-squares (MLS), reproducing kernel (RK), and maximum entropy (ME) approximations can be obtained. Furthermore, the basis function can also be translated or blended with other functions to generate a particular approximation for a special purpose. One application in this paper is to incorporate a blending function at the boundary based on the concept of local convexity for the non-convex approximation, such as MLS, to acquire the weak Kronecker-delta property. To achieve the higher order GMF approximation, two possible methods are also introduced. Several examples are presented to examine the effectiveness of various GMF approximations.

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