ABSTRACT
In Chapters 6 and 7, we introduced the element-free Galerkin (EFG) method and the meshless local Petrov-Galerkin (MLPG) method. Both methods use moving least squares (MLS) approximation for constructing shape functions, and hence they are accompanied by issues related to essential boundary conditions. We also discussed a number of ways to tackle these issues, which require extra efforts both in formulation and computation. The point interpolation method (PIM) was proposed by Liu et al. [1-6] to replace MLS
approximation for creating shape functions in meshfree settings. The major advantages of the PIM, as shown in Chapter 2, are the excellent accuracy in function fitting and the Kronecker delta function property, which allows simple imposition of essential boundary conditions as in the standard finite element method (FEM) (e.g., [51]). In the lengthy process of developing PIM, the battle has been on two fronts. The first front was on how to overcome the problem related to the singular moment matrix using local irregularly distributed nodes. The second one was on how to create weak forms that can always ensure stable and convergent solutions. On the first battle front, two significant advances have been made over the past years, after multiple attempts. The first is the use of RPIM shape functions allowing the use of virtually randomly distributed nodes [7]. The second approach is to use T-Schemes to create polynomial or radial PIM shape functions efficiently with a small number of local nodes selected, based on triangular cells. The second battle is essentially the restoration of the conformability of the PIM methods
caused by the incompatibility of the PIM (or RPIM) shape functions. This incompatibility problem has been a very difficult one to overcome, and lots of efforts have been made during the past years. It is now well resolved by the use of weakened-weak (W2) formulations based on the G space theory [27,28,62]: the generalized smoothed Galerkin (GS-Galerkin) weak form, and the strain-constructed Galerkin (SC-Galerkin) weak forms. The W2 formulations not only solves the compatibility problem effectively, but also offers a variety of ways to implement PIM models with excellent properties (upper bound, lower bound, superconvergence, free of locking, works well with triangular types of mesh, etc.). This chapter details a number of PIMs based on W2 formulations for stress analysis of solids. For convenience and easy reference, we list various PIM methods in Table 8.1.
