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.
[1]
Jean Duchon,et al.
Splines minimizing rotation-invariant semi-norms in Sobolev spaces
,
1976,
Constructive Theory of Functions of Several Variables.
[2]
W. Chen.
Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique
,
2000
.
[3]
T. Kitagawa,et al.
Asymptotic stability of the fundamental solution method
,
1991
.
[4]
Reinhard E. Piltner,et al.
Recent developments in the Trefftz method for finite element and boundary element applications
,
1995
.
[5]
C. Brebbia,et al.
A new approach to free vibration analysis using boundary elements
,
1983
.
[6]
E. Kansa,et al.
Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations
,
2000
.
[7]
P. W. Partridge,et al.
The dual reciprocity boundary element method
,
1991
.
[8]
Walter Schempp,et al.
Constructive Theory of Functions of Several Variables: Proceedings of a Conference Held at Oberwolfach, Germany, April 25 - May 1, 1976
,
1977,
Constructive Theory of Functions of Several Variables.