Application of an accelerated boundary-based mesh-free method to two-dimensional problems in potential theory

Abstract

The boundary node method (BNM) is a boundary-only meshfree method based on boundary integral equations (BIE). One drawback of the usual BNM, however, is that it typically requires much more computer time than the usual boundary element method (BEM). The multipole method (MM) has been demonstrated, in the context of the n body problem, and the BEM, to greatly accelerate these methods while still maintaining sufficient accuracy. The present paper explores, for the first time, a coupling of the BNM with the MM (called the BNMM) in the context of 2-D potential theory. Numerical results (for selected problems) from the BNM are compared with those from the BNMM with regard to accuracy and computational efficiency. Introduction Domain-based meshfree methodsMeshfree (also called meshless) methods, that only require points rather than elements to be specified in the physical domain, have tremendous potential advantages over methods such as the finite element method (FEM) that require discretization of a body into elements.The idea of moving least squares (MLS) interpolants, for curve and surface fitting, is described in a book by [21].[29] proposed a coupling of MLS interpolants with Galerkin procedures in order to solve boundary value problems. They called their method the diffuse element method (DEM) and applied it to two-dimensional (2-D) problems in potential theory and linear elasticity.During the relatively short span of less than a decade, great progress has been made in solid mechanics applications of domain-based meshfree methods. Some of these methods are: The element-free Galerkin (EFG) method [4], the reproducing kernel particle method (RKPM) [23], h À p clouds [31] the meshless local Petrov-Galerkin (MLPG) approach [2], the extended finite element method (X-FEM) [12], the finite cloud method (FCM) [1] and the point interpolation method (PIM) [24]. The boundary element methodAnother approach towards alleviating meshing difficulties associated with the FEM is the use of the boundary element method (BEM) (e.g. [26,3,7,6]). The standard BEM is a very mature method for the analysis of a broad class of problems in science and engineering. For linear problems, the BEM has the well known dimensionality advantage in that only the 2-D bounding surface of a 3-D body needs to be meshed when this method is used. Boundary-based meshfree methodsS. Mukherjee, together with his research collaborators, has recently proposed a new computational approach called the boundary node method (BNM) [27,20,9,8,10,11,14]. Other examples of boundary-based meshless methods are the local BIE (LBIE) approach [36,35] and the boundary point interpolation method (BPIM) [18]. The LBIE, however, is not strictly a boundary method since it requires evaluation of integrals over certain surfaces (called L s in [36]) that can be regarded as ''closure surfaces'' of boundary elements. [22] have very recently proposed a boundary only method called the boundary cloud method (BCM). This method is very similar to the BNM in that scattered points are used f...

The extended boundary node method for three-dimensional potential theory

“…[37] where the author states that ''with fast multipole accelerated BIEM, problems of the size of 10 6 unknowns are well within reach even for users of desktop computers''). Also, [38] presents a first attempt at accelerating the standard BNM by the MM. Parallel computing is another promising avenue.…”

“…This problem is very important because it could not be solved by the standard BNM in which curvilinear surface co-ordinates were used and ROIs were truncated at the edges. For this problem, the function u, obtained from (38), is again prescribed on the curved surface, while s = 0 is prescribed on the three flat surfaces.…”