http://www.angelfire.com/me/cvewjg/wjg02.pdf

On the optimal shape parameters of radial basis functions used for 2-D meshless methods

https://repository.upenn.edu/cgi/viewcontent.cgi?article=1001&context=meam_papers

A kernel-independent adaptive fast multipole algorithm in two and three dimensions

Fast multipole accelerated boundary integral equation methods

1. Introduction 2. Conventional BEM for potential problems 3. Fast multipole BEM for potential problems 4. Elastostatic problems 5. Stokes flow problems 6. Acoustic wave problems.

/pdf/fast-multipole-boundary-element-method-theory-and-41w29r2035.pdf

Fast multipole boundary element method : theory and applications in engineering

https://msvlab.hre.ntou.edu.tw/mcs2008-dehghan.pdf

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

A numerical method for heat transfer problems using collocation and radial basis functions

Part 1 Introduction to fluid mechanics: basic conservation laws approximate forms of the governing equations special forms of the governing equations. Part 2 Integral equation theory: classification of integral equations method of successive approximations integral equations with degenerate kernels general case of Fredholm's equation systems of integral equations. Part 3 Potential theory: basic concepts of potential theory indirect formulation regularity conditions for exterior problems. Part 4 Numerical solution of potential flow problems: boundary integral equation formulation and numerical solution of selected problems. Part 5 Boundary integral equations for low Reynolds number flow: Greens' identities hydrodynamic single- and double-layer potentials indirect formulation Lyapunov-Tauber theorem for Stokes double-layer potential dynamic properties of the singularities and their distributions. Part 6 The low Reynolds number deformation of viscous drops and gas bubbles: viscous drop deformation compound drop deformation gas bubble deformation. Part 7 Navier-Stokes equations: velocity-pressure formulation velocity-vorticity formulation.

Boundary Integral Methods in Fluid Mechanics

Compactly supported radial basis functions have been used to interpolate the forcing term in the DRM. The resulting DRM matrix is sparse and our approach is very accurate and efficient. Our proposed method is especially attractive for large scale problems. Copyright © 1999 John Wiley & Sons, Ltd.

Dual reciprocity method using compactly supported radial basis functions

We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris, Polyzos and Beskos (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results confirmed by a numerical experiment.

Some comments on the use of radial basis functions in the dual reciprocity method

This work presents a multi-domain decomposition integral equation method for the numerical solution of domain dominant problems, for which it is known that the standard Boundary Element Method (BEM) is in disadvantage in comparison with classical domain schemes, such as Finite Difference (FDM) and Finite Element (FEM) methods. As in the recently developed Green Element Method (GEM), in the present approach the original domain is divided into several subdomains. In each of them the corresponding Green's integral representational formula is applied, and on the interfaces of the adjacent subregions the full matching conditions are imposed.

In contrast with the GEM, where in each subregion the domain integrals are computed by the use of cell integration, here those integrals are transformed into surface integrals at the contour of each subregion via the Dual Reciprocity Method (DRM), using some of the most efficient radial basis functions known in the literature on mathematical interpolation. In the numerical examples presented in the paper, the contour elements are defined in terms of isoparametric linear elements, for which the analytical integrations of the kernels of the integral representation formula are known. As in the FEM and GEM the obtained global matrix system possesses a banded structure. However in contrast with these two methods (GEM and non-Hermitian FEM), here one is able to solve the system for the complete internal nodal variables, i.e. the field variables and their derivatives, without any additional interpolation.

Finally, some examples showing the accuracy, the efficiency, and the flexibility of the method for the solution of the linear and non-linear convection–diffusion equation are presented. Copyright © 1999 John Wiley & Sons, Ltd.

H. Power

Papers

A numerical method for heat transfer problems using collocation and radial basis functions

Boundary Integral Methods in Fluid Mechanics

Dual reciprocity method using compactly supported radial basis functions

Some comments on the use of radial basis functions in the dual reciprocity method

The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems