RBF-based meshless boundary knot method and boundary particle method

This paper is concerned with the two new boundary-type radial basis function collocation schemes, boundary knot method (BKM) and boundary particle method (BPM). The BKM is developed based on the dual reciprocity theorem, while the BKM employs the multiple reciprocity technique. Unlike the method of fundamental solution, the two methods use the non-singular general solution instead of singular fundamental solution to circumvent the controversial artificial boundary outside physical domain. Compared with the boundary element method, both BKM and BPM are meshless, super-convergent, meshfree, integration-free, symmetric, and mathematically simple collocation techniques for general PDE’s. In particular, the BPM does not require any inner nodes for inhomogeneous problems. In this study, the accuracy and efficiency of the two methods are numerically demonstrated to some 2D, 3D Helmholtz and convection-diffusion problems under complicated geometries. Keyword: boundary knot method; boundary particle method; radial basis function; meshfree; method of fundamental solution, dual reciprocity BEM, multiple reciprocity BEM.