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

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IsoGeometric Analysis: A New Paradigm in the Numerical Approximation of PDEs

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2161))

Abstract

We present a detailed summary of the main CAGD tools of interest in IgA: Bernstein polynomials and B-splines. Besides their well-known algebraic and geometric properties, we give a deeper insight into 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 polynomial spaces. Extensions to the bivariate setting beyond the straightforward tensor-product case are discussed as well. In particular, we focus on the triangular setting.

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Notes

  1. 1.

    Bernstein polynomials were introduced in [3] to provide a constructive proof of the Weierstrass theorem, i.e., to explicitly construct a sequence of algebraic polynomials, namely

    $$\displaystyle{B_{p}(\,f,t):=\sum _{ i=0}^{p}f\left ( \frac{i} {p}\right )\binom{p}{i}t^{i}(1 - t)^{p-i},}$$

    which uniformly converges on [0, 1] to any f ∈ C([0, 1]). From the approximation perspective, the above operator did not receive much attention due to its slow convergence. Indeed, it can be proved that (see [3])

    $$\displaystyle{\lim _{p\rightarrow \infty }p[B_{p}(\,f,t_{0}) - f(t_{0})] = \frac{1} {2}t_{0}(1 - t_{0})f''(t_{0})\ \ \mathrm{if}\ \ f''(t_{0})\not =0.}$$

    We refer to the nice paper [27] for a historical summary of the properties of Bernstein polynomials.

  2. 2.

    A set of polynomials with the same properties on a general interval [a, b] can be immediately obtained by the usual change of variable \(t = \frac{x-a} {b-a}\).

  3. 3.

    We introduce this notation for the sake of symmetry with a recurrence relation we need later on in Sect. 4.1.

  4. 4.

    Referring to footnote 1, Sergei N. Bernstein used the Greville abscissae in his proof of the Weierstrass theorem [3].

  5. 5.

    Pierre Bézier was a French engineer working at Renault. His software UNISURF was one of the first CAD systems [4, 41].

  6. 6.

    Paul de Casteljau is a French physicist and mathematician. He worked at Citroën, where he developed his famous algorithm for evaluation of a family of polynomial curves [5, 22]. The same curves were used independently by Pierre Bézier at Renault. The connection between these curves and Bernstein polynomials was not clear in the beginning.

  7. 7.

    It follows that the Bernstein operator B p ( f, ⋅ ), as defined in footnote 1, is shape preserving in the sense that if f is positive and/or monotone and/or convex then B p ( f, ⋅ ) is positive and/or monotone and/or convex as well.

  8. 8.

    B-splines were introduced by Curry and Schoenberg [16] as divided differences of truncated powers, see, e.g., [19, Chap. IX]. Here we present an equivalent definition in terms of a recurrence relation. The original definition can be used to prove properties of the B-splines, but it is not suited for numerical evaluation, see [18]. On the other hand, Definition 6 only uses convex combinations which ensure a numerically stable procedure. Nevertheless, Isaac J. Schoenberg is indisputably considered as the father of splines although other authors considered this concept already earlier, see, e.g., [8, 37, 62].

  9. 9.

    An explicit expression of (40) was given by Thomas N.E. Greville in [34]. According to Isaac J. Schoenberg [61], Thomas N.E. Greville reviewed the paper [61] introducing some elegant simplifications.

  10. 10.

    As in the case of Bernstein polynomials ( footnote 1), the Greville abscissae for B-splines can be used to define an interesting approximating operator, namely the so-called Schoenberg operator, see [61, Sect. 10]:

    $$\displaystyle{Q(f,t):=\sum _{ i=1}^{n}f(\xi _{ i,p}^{{\ast}})B_{ i,\varXi }^{(\,p)}(t).}$$

    Analogously to the Bernstein operator, the Schoenberg operator reproduces polynomials of first degree and it is shape preserving, see Theorems 9 and 10.

  11. 11.

    If τ i in (44) is a knot, then the left or right derivatives of f have to be considered. Actually, a slightly more general form of (44) can be given, see [19, Chap. IX].

  12. 12.

    The right derivative has to be considered where B i, Ξ ( p) is not sufficiently smooth.

  13. 13.

    The formula is well defined for knots having multiplicity at most p. For the more general case, we refer to Definition 11.

  14. 14.

    Like in the case of Bézier curves, the development of a stable evaluation algorithm for B-spline curves is related to the car industry. In the early 1960s, Carl de Boor worked at General Motors Research, where splines were already being employed in automotive design. There he laid the foundations of his famous algorithm, see [18].

  15. 15.

    A multiplicity p is actually sufficient for Bézier extraction.

  16. 16.

    Negative weights could be used as well. Here, we just consider strictly positive weights because they ensure nice properties of the corresponding NURBS basis functions.

  17. 17.

    For a more general approach we refer to [33].

  18. 18.

    This is always the case for an EC space.

  19. 19.

    Beyond the neat theory of GB-splines presented here, more general constructions of B-spline-like functions in EC-spaces can be obtained with less restrictive hypotheses, see [42, 49, 63] and references therein.

  20. 20.

    We use the term degree to stress the similarity with classical polynomial B-splines.

  21. 21.

    GB-splines are defined by successive integration of elements in the bidimensional spaces (71). Therefore, it is natural to start the recurrence relation from degree 1, instead of from degree 0 as is usually done in the polynomial case (see Definition 6).

  22. 22.

    In both the univariate and bivariate case we use the same symbol for the space of algebraic polynomials, \(\mathbb{P}_{p}\), but the meaning will be clear from the context.

  23. 23.

    In the univariate case both approaches lead to the same result, see Sect. 3.2.

  24. 24.

    Powell–Sabin splines were originally developed in [56] with the main aim of drawing contour lines of bivariate functions.

  25. 25.

    Such a choice is always possible: Z i can be selected as the center of the inscribed circle. Usually, the barycenter of T i is also a valid choice (but not always).

  26. 26.

    For boundary edges the subscript j refers to the edge.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Roma, Italy

    Carla Manni & Hendrik Speleers

  2. Departement Computerwetenschappen, Katholieke Universiteit Leuven, Leuven, Belgium

    Hendrik Speleers

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  1. Carla Manni
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  1. IMATI - CNR , Pavia, Italy

    Annalisa Buffa

  2. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giancarlo Sangalli

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Manni, C., Speleers, H. (2016). Standard and Non-standard CAGD Tools for Isogeometric Analysis: A Tutorial. In: Buffa, A., Sangalli, G. (eds) IsoGeometric Analysis: A New Paradigm in the Numerical Approximation of PDEs. Lecture Notes in Mathematics(), vol 2161. Springer, Cham. https://doi.org/10.1007/978-3-319-42309-8_1

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