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.

In Chapter 6, we have presented the node-based smoothed PIM (NS-PIM), and examined the properties of various NS-PIM models. These NS-PIM models are softer models, spatially stable, having nonzero-energy modes, work very well for static mechanics problems, and have very important upper bound and super-convergence properties in the strain energy [1][2][3][4][5][6][7][8]. However, the NS-PIM was found temporally instable [7-10], when applied to solve dynamic problems. Such instability is often observed at higher energy level as spurious (nonphysical) nonzero-energy modes in the free vibration analysis of (stable and constrained) solids. Our studies have found that the cause of such instability is the "overly-soft" nature of these models introduced by the node-based smoothing operations. The authors believe that any method that has upper bound property can have spurious modes at a higher energy level and hence may suffer from the instability for solving dynamic problems. Therefore, for dynamic analysis of solid and structures, we need alternatives or special treatments to install temporal stability for these overly-soft models.In this chapter, we introduce another important S-PIM model called edgebased smoothed point interpolation method or ES-PIM, which has been found both spatially and temporally stable, and works well for both static and dynamic problems [9,10]. It is found also that the ES-PIM can produce much more accurate results compared to the NS-PIM and FEM using the same mesh [9][10][11][12][13][14][15][16][17][18][19][20][21]. The ES-PIM models are usually stiffer models with very "close-to-exact" stiffness. In particular the linear ES-PIM-Tr3 is found as one of the best linear models, and is regarded as a "star performer" among all the linear models.