Dual Reciprocity Hybrid Boundary Node Method for Solving Poisson Equations

It has long been claimed that the boundary element method (BEM) is a viable alternative to the domain-type finite element method (FEM) and finite difference method (FDM) due to its advantages in dimensional reducibility and suitability to infinite domain problems. However, it has a major difficulty in handling inhomogeneous terms such as time-dependent and nonlinear problems. It is over 18 years since the dual reciprocity method (DRM) was first proposed by Nardini and Brebbia which provides a very general methodology for obtaining a boundary element solution to wide range of problems. Another obstacle in BEM, just like the FEM, surface mesh or remesh requires costly computation, especially for moving boundary and nonlinear problems. The boundary-type meshless methods such as the hybrid boundary node method (Hybrid-BNM) , the boundary node method (BNM) shown an emerging technique to alleviate these drawbacks. As a truly meshless method, the Hybrid BNM does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. In this paper, based on the radial basis function (RBF) and the Hybrid BNM, it presents an inherently meshless, boundary-only technique, which named dual reciprocity hybrid boundary node method (DRHBNM), for numerical solution of various Poisson equations. In this study, the RBFs are employed to approximation the inhomogeneous terms via dual reciprocity method (DRM), while the general solution is solved by means of Hybrid BNM, in which only requires discrete nodes constructed on the boundary of a domain, several nodes in the domain are needed just for the RBF interpolation. The rigid body movement method is employed to solve the hypersingular integrations. The ‘boundary layer effect’, which is the main drawback of the original Hybrid BNM, has been circumvented by an adaptive integration scheme. The computation results obtained by the present method are shown that high convergence and high accuracy with a small node number are achievable.