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Über dieses Buch

This book surveys the latest advances in radial basis function (RBF) meshless collocation methods which emphasis on recent novel kernel RBFs and new numerical schemes for solving partial differential equations. The RBF collocation methods are inherently free of integration and mesh, and avoid tedious mesh generation involved in standard finite element and boundary element methods. This book focuses primarily on the numerical algorithms, engineering applications, and highlights a large class of novel boundary-type RBF meshless collocation methods. These methods have shown a clear edge over the traditional numerical techniques especially for problems involving infinite domain, moving boundary, thin-walled structures, and inverse problems.

Due to the rapid development in RBF meshless collocation methods, there is a need to summarize all these new materials so that they are available to scientists, engineers, and graduate students who are interest to apply these newly developed methods for solving real world’s problems. This book is intended to meet this need.

Prof. Wen Chen and Dr. Zhuo-Jia Fu work at Hohai University. Prof. C.S. Chen works at the University of Southern Mississippi.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
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.
Wen Chen, Zhuo-Jia Fu, C. S. Chen

Chapter 2. Radial Basis Functions

Abstract
The traditional basis functions in numerical PDEs are mostly coordinate functions, such as polynomial and trigonometric functions, which are computationally expensive in dealing with high dimensional problems due to their dependency on geometric complexity. Alternatively, radial basis functions (RBFs) are constructed in terms of one-dimensional distance variable irrespective of dimensionality of problems and appear to have a clear edge over the traditional basis functions directly in terms of coordinates. In the first part of this chapter, we introduces classical RBFs, such as globally-supported RBFs (Polyharmonic splines, Multiquadratics, Gaussian, etc.), and recently developed RBFs, such as compactly-supported RBFs. Following this, several problem-dependent RBFs, such as fundamental solutions, general solutions, harmonic functions, and particular solutions, are presented. Based on the second Green identity, we propose the kernel RBF-creating strategy to construct the appropriate RBFs.
Wen Chen, Zhuo-Jia Fu, C. S. Chen

Chapter 3. Different Formulations of the Kansa Method: Domain Discretization

Abstract
In contrast to the traditional meshed-based methods such as finite difference, finite element, and boundary element methods, the RBF collocation methods are mathematically very simple to implement and are truly free of troublesome mesh generation for high-dimensional problems involving irregular or moving boundary. This chapter introduces the basic procedure of the Kansa method, the very first domain-type RBF collocation method. Following this, several improved formulations of the Kansa method are described, such as the Hermite collocation method, the modified Kansa method, the method of particular solutions, the method of approximate particular solutions, and the localized RBF methods. Numerical demonstrations show the convergence rate and stability of these domain-type RBF collocation methods for several benchmark examples.
Wen Chen, Zhuo-Jia Fu, C. S. Chen

Chapter 4. Boundary-Type RBF Collocation Methods

Abstract
The mesh generation in the standard BEM is still not trivial as one may imagine, especially for high-dimensional moving boundary problems. To overcome this difficulty, the boundary-type RBF collocation methods have been proposed and endured a fast development in the recent decade thanks to being integration-free, spectral convergence, easy-to-use, and inherently truly meshless. First, this chapter introduces the basic concepts of the method of fundamental solutions (MFS). Then a few recent boundary-type RBF collocation schemes are presented to tackle the issue of the fictitious boundary in the MFS, such as boundary knot method (BKM), regularized meshless method, and singular boundary method. Following this, an improved multiple reciprocity method (MRM), the recursive composite MRM (RC-MRM), is introduced to establish a boundary-only discretization of nonhomogeneous problems. Finally, numerical demonstrations show the convergence rate and stability of these boundary-type RBF collocation methods for several benchmark examples.
Wen Chen, Zhuo-Jia Fu, C. S. Chen

Chapter 5. Open Issues and Perspectives

Abstract
The RBF collocation schemes provide attractive alternatives to traditional mesh-based methods in engineering and science community, particularly, for solving high dimensional, irregular, or moving boundary problems. This chapter discusses some open issues and gives a potential perspective of the RBF collocation methods.
Wen Chen, Zhuo-Jia Fu, C. S. Chen
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