Radial basis function collocation

The aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

In Chapter 11 of this book we will use the thin spline radial function to develop the radial basis functions collocation method (RBFCM). A well known property of radial interpolation is that it renders a convenient way to calculate derivatives of the interpolated function. This is an advantage over other interpolation functions and it is used in other methods such us the dual reciprocity boundary elements [43], collocation techniques [24], RBFCM, etc. For an interpolated function u,... 

O.A. Estrada, I.D. Lopez-Gomez, and T.A. Osswald. Modeling the non-newtonian calendering process using a coupled flow and heat transfer radial basis functions collocation method. Journal of Polymer Technology, 2005. 

In collocation methods, different sets of basis functions can also be used, for example radial basis functions, which are functions that depend only on the euclidean distance between collocation points i and j, ri j. 

While expanding in a basis of orthogonal functions is fairly easily understood, care must be taken in choosing an appropriate set of basis functions. In this case the symmetry of the basis functions chosen must match that of the problem, as seen below. The importance of symmetry in the problem is beautifully presented by the choice of basis for the radial coordinate. Consider two choices of the basis for the radial coordinate - a Fourier basis and a Laguerre basis. That is, the radial functions can be expanded in a basis of Fourier functions (sines and cosines) or Laguerre functions. The collocation points can be loosely thought of as the nodes of the basis functions. 

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