In the previous chapters, three major models of smoothed point interpolation methods, i.e. NS-PIM/NS-RPIM, ES-PIM/ES-RPIM and CS-PIM/CS-RPIM, have been presented for 2D and 3D solid mechanics problems. It has been found that these three types of S-PIMs have their own attractive properties. The NS-PIMs use the node-based strain smoothing operation, have superconvergence property in energy norm, and can provide upper bound solutions in energy norm, but behave overly-soft and cannot be directly used in dynamic analysis [1-6]. The ES-PIMs use the edge/face-based strain smoothing operation, can provide very accurate and superconvergent results in both displacement and energy norms, produce models with "close-to-exact" stiffness, are both specially and temporally stable, and hence works very well for dynamic analysis [7][8][9][10][11][12]. It is regarded as the "star" performer. The CS-PIMs use the cell-based strain smoothing operation, provide very accurate numerical results with superconvergence property in both displacement and energy norms, possess models with the stiffness being stiffer than the overly-soft NS-PIMs and generally softer than the ES-PIMs, can be both specially and temporally stable and works well also for dynamic analysis [1,[13][14][15].Obtaining the exact solution, at least in a norm, using a discrete numerical method is a very attractive but challenging idea in the field of computational methods. Some effective efforts have been made by Liu's group aiming to obtain the exact solution in a norm using discrete numerical models [16][17][18]. The so-called alpha finite element method using four-node quadrilateral elements (αFEM-Q4) was developed for finding a nearly exact solution in strain energy Smoothed Point Interpolation Methods Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 09/30/15. For personal use only. Smoothed Point Interpolation Methods Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 09/30/15. For personal use only.

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.