Node-based Smoothed Point Interpolation Method (NS-PIM)

Abstract

With the theory given in Chapters 2-5, we are now ready to present a particular S-PIM method: the node-based smoothed point interpolation method or NS-PIM. Because the NS-PIM is the first to be introduced, we shall present it as in detail as possible.The NS-PIM was originally proposed by Liu and Zhang et al. [1][2][3][4] using the generalized gradient smoothing technique [3] and the PIM shape functions [5-13] created using only a small number of local nodes in meshfree settings. The NS-PIM is a typical method created using generalized smoothed Galerkin (GS-Galerkin) weak form based on the normed G space theory that allows the use of discontinuous displacement functions. This chapter formulates the NS-PIM for mechanics problems for 2D and 3D solids. We focus first on 2D cases, because it is much easier to describe and comprehend. We will then extend the formulations to 3D cases by simply highlighting the major differences.The PIM shape function used in the NS-PIM was created using the T-schemes with nodes that can come from more than one cell, and hence are discontinuous in general. The NS-PIM was termed as linearly conforming point interpolation method (or LC-PIM) [1, 2, 4], because it is at least linearly conforming despite the use of discontinuous PIM shape functions. It was later termed as NS-PIM because the smoothing operation is node based. The name of NS-PIM is more convenient for distinguishing from other S-PIM models, many of which are also linearly conforming but use different types of smoothing domains for different properties and features. The NS-PIM was found possessing the following important features.