Standard and Non-standard CAGD Tools for Isogeometric Analysis: A Tutorial

Abstract

We present a short summary of CAGD tools of main interest in isogeometric analysis, namely Bernstein polynomials and B-splines. Besides their well-known algebraic and geometric properties, we provide a deeper insight why these representations are so popular and efficient by proving that they are optimal bases for the corresponding function spaces. Moreover, we review some generalizations of the B-spline structure in function spaces which extend classical polynomials. Extensions to the bivariate case beyond the straightforward tensor-product case are also presented.

Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement

“…This species will be the topic of the chapter. Several other species can be found in [35,45] and references therein.…”

“…Tensor-product B-splines are currently the most common tool in CAD systems and IgA. It is worth mentioning that there are also many other important extensions of the univariate B-spline concepts to the multivariate setting, not restricted to a tensor-product grid; see, for example, [31,35] and references therein.…”

Rectangular GLT Sequences

“…More precisely, for p, n ≥ 1 and 0 For the formal definition of B-splines, as well as for the proof of the properties listed below, see [26]. For more on spline functions, see [15,27,29].…”

Adaptive refinement with locally linearly independent LR B-splines: Theory and applications

“…Since the '70s, curves and surfaces in engineering are usually expressed by means of computer aided design (CAD) technologies, such as B-splines and non-uniform rational Bsplines (NURBS). Thanks to properties like nonnegativity, local support and partition of unity, they allow for an easy control and modification of the geometries they describe, and this motivates their undisputed success as main modeling tools for objects with complex shapes in engineering; see, e.g., [21,7,18] and references therein. On the other hand, Bsplines also provide a very efficient representation of smooth piecewise polynomial spaces, and so are a popular ingredient in the construction of approximation schemes; see, e.g., [2,22,17] and references therein.…”

“…Throughout the paper, we assume the reader to be familiar with the definition and main properties of (univariate) B-splines, in particular with the knot insertion procedure. An introduction to this topic can be found, e.g., in the review papers [18,17] or in the classical books [2] and [22].…”