Radial basis functions (RBFs) are constructed in terms of one-dimensional distance variable and appear to have certain advantages over the traditional coordinates-based functions. In contrast to the traditional meshed-based methods, the RBF collocation methods are mathematically simple and truly meshless, which avoid troublesome mesh generation for high-dimensional problems involving irregular or moving boundary. This opening chapter begins with the introduction to RBF history and its applications in numerical solution of partial differential equations and then gives a general overview of the book.

Recent Advances in Radial Basis Function Collocation Methods

This paper describes an application of the recently proposed modified method of fundamental solutions (MMFS) to potential flow problems. The solution in two-dimensional Cartesian coordinates is represented in terms of the single layer and the double layer fundamental solutions. Collocation is used for the determination of the expansion coefficients. This novel method does not require a fictitious boundary as the conventional method of fundamental solutions (MFS). The source and the collocation points thus coincide on the physical boundary of the system. The desingularised values, consistent with the fundamental solutions used, are deduced from the direct boundary element method (BEM) integral equations by assuming a linear shape of the boundary between the collocation points. The respective values of the derivatives of the fundamental solution in the coordinate directions, as required in potential flow calculations, are calculated indirectly from the considerations of the constant potential field. The normal on the boundary is calculated by parametrisation of its length and the use of the cubic radial basis functions with the second-order polynomial augmentation. The components of the normal are calculated in an analytical way. A numerical example of potential flow around a two-dimensional circular region is presented. The results with the new MMFS are compared with the results of the classical MFS and the analytical solution. It is shown that the MMFS gives better accuracy for the potential, velocity components (partial derivatives of the potential), and absolute value of the velocity as compared with the classical MFS. The results with the single layer fundamental solution are more accurate than the results with the double layer fundamental solution.

https://msvlab.hre.ntou.edu.tw/eabe2009-Saler.pdf

Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions

The satisfactory location for the sources outside the closure of the domain of the problem under consideration remains one of the major issues in the application of the method of fundamental solutions (MFS). In this work we investigate this issue by means of two algorithms, one based on the satisfaction of the boundary conditions and one based on the leave-one-out cross validation algorithm. By applying these algorithms to several numerical examples for the Laplace and biharmonic equations in a variety of geometries in two and three dimensions, we obtain locations of the sources which lead to highly accurate results, at a relatively low cost.

On choosing the location of the sources in the MFS

This paper proposes a novel meshless boundary method called the singular boundary method (SBM). This method is mathematically simple, easy-to-program, and truly meshless. Like the method of fundamental solutions (MFS), the SBM employs the singular fundamental solution of the governing equation of interest as the interpolation basis function. However, unlike the MFS, the source and collocation points of the SBM coincide on the physical boundary without the requirement of introducing fictitious boundary. In order to avoid the singularity at the origin, this method proposes an inverse interpolation technique to evaluate the singular diagonal elements of the MFS coefficient matrix. The SBM is successfully tested on a benchmark problems, which shows that the method has a rapid convergence rate and is numerically stable.

https://www.witpress.com/Secure/elibrary/papers/BE09/BE09010FU1.pdf

A method of fundamental solutions without fictitious boundary

Burton–Miller-type singular boundary method for acoustic radiation and scattering

https://msvlab.hre.ntou.edu.tw/cite/JCP-Young-2005.pdf

Novel meshless method for solving the potential problems with arbitrary domain

Time-harmonic exterior acoustic problems are solved by using a singular meshless method in this paper. It is well known that the source points cannot be located on the real boundary, when the method of fundamental solutions (MFS) is used due to the singularity of the adopted kernel functions. Hence, if the source points are right on the boundary the diagonal terms of the influence matrices cannot be derived. Herein we present an approach to obtain the diagonal terms of the influence matrices of the MFS for the numerical treatment of exterior acoustics. By using the regularization technique to regularize the singularity and hypersingularity of the proposed kernel functions, the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined. We also maintain the prominent features of the MFS, that it is free from mesh, singularity, and numerical integration. The normal derivative of the fundamental solution of the Helmholtz equation is composed of a two-point function, which is one of the radial basis functions. The solution of the problem is expressed in terms of a double-layer potential representation on the physical boundary based on the potential theory. The solutions of three selected examples are used to compare with the results of the exact solution, conventional MFS, boundary element method, and Dirichlet-to-Neumann finite element method. Good numerical performance is demonstrated by close agreement with other solutions.

https://msvlab.hre.ntou.edu.tw/cite/JASA-Young-2006.pdf

D. L. Young

Papers

Novel meshless method for solving the potential problems with arbitrary domain

Singular meshless method using double layer potentials for exterior acoustics

Arbitrary Lagrangian-Eulerian finite element analysis of free surface flow using a velocity-vorticity formulation

A multiple-scale Pascal polynomial for 2D Stokes and inverse Cauchy-Stokes problems

Adaptive multilayer method of fundamental solutions using a weighted greedy QR decomposition for the Laplace equation