New advances in dual reciprocity and boundary-only RBF methods

TL;DR

This paper introduces significant advances in dual reciprocity and boundary-only RBF methods, notably the boundary knot method (BKM), which is meshless, integration-free, and capable of handling nonlinear PDEs efficiently.

Contribution

It proposes the boundary knot method (BKM) as a novel, meshless, symmetric, and integration-free boundary-only technique, and develops a general RBF methodology based on Green's identity for improved efficiency.

Findings

BKM is meshless and exponential convergent.

BKM can formulate linear models for nonlinear PDEs.

The GSR approach links BEM and RBFs explicitly.

Abstract

This paper made some significant advances in the dual reciprocity and boundary-only RBF techniques. The proposed boundary knot method (BKM) is different from the standard boundary element method in a number of important aspects. Namely, it is truly meshless, exponential convergence, integration-free (of course, no singular integration), boundary-only for general problems, and leads to symmetric matrix under certain conditions (able to be extended to general cases after further modified). The BKM also avoids the artificial boundary in the method of fundamental solution. An amazing finding is that the BKM can formulate linear modeling equations for nonlinear partial differential systems with linear boundary conditions. This merit makes it circumvent all perplexing issues in the iteration solution of nonlinear equations. On the other hand, by analogy with Green's second identity, this paper also presents a general solution RBF (GSR) methodology to construct efficient RBFs in the dual reciprocity and domain-type RBF collocation methods. The GSR approach first establishes an explicit relationship between the BEM and RBF itself on the ground of the weighted residual principle. This paper also discusses the RBF convergence and stability problems within the framework of integral equation theory.