A Hybrid Shapeless Radial Basis Function Applied With the Dual Reciprocity Boundary Element Method

The so-called Dual Reciprocity Boundary Element Method (DRBEM) has been a popular alternative scheme designed to alleviate problems encountered when using the traditional BEM for numerically solving engineering problems that are described by PDEs. The method starts with writing the right-hand-side of Poisson equation as a summation of a pre-chosen multivariate function known as ‘Radial Basis Function (RBF)’. Nevertheless, a common undesirable feature of using RBFs is the appearance of the so-called ‘shape parameter’ whose value greatly affects the solution accuracy. In this work, a new form of RBF containing no shape (so that it can be called ‘shapefree/shapeless’) is invented, proposed and applied in conjunction with DRBEM is validated numerically. The solutions obtained are compared against both exact ones and those presented in literature where appropriate, for validation. It is found that reasonably and comparatively good approximated solutions of PDEs can still be obtained without the difficulty of choosing a good shape for RBF used.