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Classical Planar Grid Generation
Published in Patrick Knupp, Stanly Steinberg, Fundamentals of Grid Generation, 2020
Patrick Knupp, Stanly Steinberg
The transfinite interpolation map discussed in Section 1.5 is an example of a map that solves the basic planar grid-generation problem (the reader should review Section 1.5). Transfinite interpolation maps are useful in their own right and as initial guesses for maps that are computed using iterative algorithms. As previously noted, such maps are limited by their lack of smoothness and potential for folding. Many other approaches to the planar grid-generation problem have been proposed; a survey of some of these approaches follows in the next section.
Constrained univariate and bivariate rational fractal interpolation
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2019
As a natural follow-up, fractal functions have been extended to bivariate interpolation and applied for modeling non-linear objects such as landscapes, metals, rocks, clouds, porous materials, skins of living objects, and so on, see for instance, Mandelbrot [23]. Different types of fractal surfaces via IFS and recurrent IFSs has been taken up in earnest by many authors Bouboulis et al. [24]; Bouboulis and Dalla [25]; Massopust [26]; Metzler and Yun [27]. In references Dalla [28]; Xie and Sun [29], the continuity of the fractal surface is achieved based on the assumption that interpolation points on the boundary are collinear. Construction of fractal interpolation surfaces for arbitrary data on a rectangular grid is given in Feng et al. [30]; Malysz [31]. In these methods, fractal surfaces are generated by creating new points recursively in the given domain. But transfinite interpolation method via blending function schemes Farin [32] is popular in the literature as it use the corresponding univariate interpolants, and the shape properties of transfinite interpolating surface follow trivially from the corresponding shape properties of the networks of boundary curves (univariate interpolants), see Casciola and Romani [33]. Recently, constrained FIS by a plane is introduced by Chand et al. [34]. Thus, using univariate RCFIFs as boundary curves, we construct rational cubic fractal interpolation surface (RCFIS). The parameters of RCFIS are identified so that it stays above a piecewise plane and within a prescribed cuboid whenever the corresponding surface data are constrained in the same manner. The constrained FIS in a cuboid is reported for the first time in the literature. In particular, one can use negative scalings factors for the constrained FIS even though its functional equation is implicit in nature.