The generalized product partition of unity for the meshless methods

The partition of unity is an essential ingredient for meshless methods named by GFEM, PUFEM (partition of unity FEM), XFEM (extended FEM), RKPM (reproducing kernel particle method), RPPM (reproducing polynomial particle method), the method of hp clouds in the literature. There are two popular choices for partition of unity: a piecewise linear FEM mesh and the Shepard-type partition of unity. However, the partition of unity (PU) by a FEM mesh leads to the singular (or nearly singular) matrices and non-smooth approximation functions. The Shepard-type partition of unity requires lengthy computing time and its implementation is difficult. In order to alleviate these difficulties, Oh et al. introduced the smooth piecewise polynomial PU functions with flat-top, that lead to small matrix condition numbers, and almost everywhere partition of unity, that can handle essential boundary conditions. Nevertheless, we could not have the smooth closed form PU functions with flat-top for general polygonal patches (2D) and general polyhedral patches (3D). In this paper, we introduce one of the most simple and efficient partition of unity, called the (generalized) product partition of unity. The product PU functions constructed by this method are the closed form smooth piecewise polynomials with flat-top and could handle background meshes (general polygonal patches as well as general polyhedral patches) arising in practical applications of meshless methods.