ABSTRACT
The boundary element method (BEM) is a numerical technique based on the boundary integral equation (BIE), which has been developed in the 1960s. For many (especially linear) problems, BEM is undoubtedly superior to the ‘‘domain discretization’’ type of methods, such as the finite element method (FEM) and the finite difference method (FDM). In BEM, for example, only the two-dimensional (2D) bounding surface of a three-dimensional (3D) body needs to be discretized. BEM is applicable to all those problems for which the fundamental solution (Green’s functions) is known in a reasonably simple form, preferably in a closed form. The meshfree idea has also been used in BIE, such as the boundary node method (BNM)
by Mukherjee et al. [1-4] and the local boundary integral equation (LBIE) method by Zhu and Atluri [5]. These boundary-type meshfree methods are formulated using the moving least squares (MLS) approximations and techniques of BIE. In BNM, the surface of the problem domain is discretized by properly scattered nodes. The BNM has been applied to 3D problems of both potential theory and the theory of elastostatics [3,4]. Very good results have been obtained in these problems. However, because the shape functions based on the MLS approximation lack the delta function property, extra efforts are needed to satisfy accurately the boundary conditions in BNM. This problem becomes even more serious in BNM because of the large number of boundary conditions, compared with the total number of nodes for the problem. The method used in [2] to impose boundary conditions resulted in a set of system equations that were doubled in number. The advantage of the boundary-type method is therefore eroded and discounted to a certain degree, making BNM computationally much more expensive than the conventional BEM. A boundary-type meshfree method called the boundary point interpolation method
(BPIM) has been formulated [6,7], where the point interpolation method (PIM) [8,9] was used to construct shape functions. It is confirmed that there is no need at all to use MLS in boundary-type meshfree methods, at least for 2D problems. PIM works much more efficiently in constructing shape functions, and all the PIM shape functions possess the Kronecker delta function property. This removes the issue of treatment of boundary conditions, which is especially beneficial to methods based on BIE. The dimension of the equation system of BPIM is equivalent to that of BEM. For 2D problems, BPIM works perfectly well without any special trick and is superior to BNM in simplicity, accuracy, and computational efficiency. For 3D problems, for which 2D shape functions need to be constructed, there could be an issue of singular moment matrices. In such cases, the special techniques discussed in Section 2.5.4 should be applied. A robust way of constructing PIM shape functions is to use radial functions as the basis. The advantage of using a radial function basis is that it guarantees the existence of the inverse moment matrix. The methods formulated are termed as boundary radial PIM (BRPIM) [10]. A good alternative could be the use of T-schemes (Section 1.7.6). This chapter focuses on the introduction of boundary meshfree methods formulated
using both MLS shape functions (BNM) and PIM shape functions (BPIM and BRPIM). These boundary meshfree methods can be formulated in the same manner, except that in
conditions [12,2] are required. Only 2D problems are discussed in this chapter. In all these boundary meshfree methods,
only the boundary of the problem domain is represented by properly scattered nodes. BIE for 2D elastostatics is then discretized using meshfree shape functions based only on a group of arbitrarily distributed boundary nodes that are included in the support domain of a point of interest. For 3D BNM, readers are referred to the work by Chati et al. [3].
