B-spline-like bases for C 2 cubics on the Powell–Sabin 12-split
The SMAI journal of computational mathematics, Volume S5 (2019) , pp. 129-159.

For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell–Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a Bézier-like manner.

In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for C 0 -, C 1 -, and C 2 -smoothness are derived.

Published online: 2020-01-29
DOI: https://doi.org/10.5802/smai-jcm.56
Classification: 41A15,  65D07,  65D17
Keywords: Stable bases, Powell–Sabin 12-split, Simplex splines, Marsden identity, Quasi-interpolation
@article{SMAI-JCM_2019__S5__129_0,
     author = {Tom Lyche and Georg Muntingh},
     title = {B-spline-like bases for $C^2$ cubics on the Powell--Sabin 12-split},
     journal = {The SMAI journal of computational mathematics},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {S5},
     year = {2019},
     pages = {129-159},
     doi = {10.5802/smai-jcm.56},
     language = {en},
     url = {smai-jcm.centre-mersenne.org/item/SMAI-JCM_2019__S5__129_0/}
}
Lyche, Tom; Muntingh, Georg. B-spline-like bases for $C^2$ cubics on the Powell–Sabin 12-split. The SMAI journal of computational mathematics, Volume S5 (2019) , pp. 129-159. doi : 10.5802/smai-jcm.56. https://smai-jcm.centre-mersenne.org/item/SMAI-JCM_2019__S5__129_0/

[1] P. G. Ciarlet The Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, 2002 | arXiv:https://epubs.siam.org/doi/pdf/10.1137/1.9780898719208 | Article

[2] R. W. Clough; J. L. Tocher Finite element stiffness matrices for analysis of plate bending, Proceedings of the conference on Matrix Methods in Structural Mechanics, Wright-Patterson A.F.B., 1965, pp. 515-546

[3] E. Cohen; T. Lyche; R. F. Riesenfeld A B-spline-like basis for the Powell–Sabin 12-split based on simplex splines, Math. Comput., Volume 82 (2013) no. 283, pp. 1667-1707 | Article | MR 3042581 | Zbl 1278.41005

[4] E. Cohen; R. F. Riesenfeld; G. Elber Geometric modeling with splines: an introduction, A K Peters, 2001 | Article | Zbl 0980.65016

[5] J. A. Cottrell; T. J. R. Hughes; Y. Bazilevs Isogeometric analysis: toward integration of CAD and FEA, John Wiley & Sons, 2009, 360 pages | Zbl 1378.65009

[6] O. Davydov; W. P. Yeo Refinable C 2 piecewise quintic polynomials on Powell–Sabin-12 triangulations, J. Comput. Appl. Math., Volume 240 (2013), pp. 62-73 | Article | MR 2991108 | Zbl 1255.65036

[7] P. Dierckx On calculating normalized Powell–Sabin B-splines, Comput. Aided Geom. Des., Volume 15 (1997) no. 1, pp. 61-78 | Article | MR 1484258 | Zbl 0894.68152

[8] N. Dyn; T. Lyche A Hermite subdivision scheme for the evaluation of the Powell–Sabin 12-split element, Approximation theory IX (Innovations in Applied Mathematics) Volume 2, Vanderbilt University Press, 1998, pp. 33-38 | Zbl 0930.65005

[9] J. Grošelj; H. Speleers Construction and analysis of cubic Powell–Sabin B-splines, Comput. Aided Geom. Des., Volume 57 (2017), pp. 1-22 | Article | MR 3715336 | Zbl 1379.65009

[10] D. Knuth Bracket notation for the “coefficient of” operator, A Classical Mind: Essays in Honour of C. A. R. Hoare, Prentice Hall International (UK) Ltd., 1994 | arXiv:http://arxiv.org/abs/math/9402216

[11] M.-J. Lai; L. L. Schumaker Macro-elements and stable local bases for splines on Powell–Sabin triangulations, Math. Comput., Volume 72 (2003) no. 241, pp. 335-354 | Article | MR 1933824 | Zbl 1009.41007

[12] M.-J. Lai; L. L. Schumaker Spline functions on triangulations, Encyclopedia of Mathematics and Its Applications, Volume 110, Cambridge University Press, 2007, xvi+592 pages | MR 2355272 | Zbl 1185.41001

[13] T. Lyche; J.-L. Merrien Simplex Splines on the Clough–Tocher Element, Comput. Aided Geom. Des., Volume 65 (2018), pp. 76-92 | Article | MR 3849068 | Zbl 07038571

[14] T. Lyche; G. Muntingh A Hermite interpolatory subdivision scheme for C 2 -quintics on the Powell–Sabin 12-split, Comput. Aided Geom. Des., Volume 31 (2014) no. 7-8, pp. 464-474 | Article | MR 3268221 | Zbl 1364.65031

[15] T. Lyche; G. Muntingh Stable Simplex Spline Bases for C 3 Quintics on the Powell–Sabin 12-Split, Constr. Approx., Volume 45 (2016), pp. 1-32 | Article | MR 3590694 | Zbl 1360.41005

[16] C. A. Micchelli On a numerically efficient method for computing multivariate B-splines, Multivariate Approximation Theory: Proceedings of the Conference held at the Mathematical Research Institute at Oberwolfach Black Forest, Birkhäuser, 1979, pp. 211-248 | Article | Zbl 0422.41008

[17] G. Muntingh Notebook: B-spline-like bases for C 2 cubics on the Powell–Sabin 12-split (2019) (https://github.com/georgmuntingh/SSplines/blob/master/examples/C2-cubic.ipynb)

[18] P. Oswald Hierarchical conforming finite element methods for the biharmonic equation, SIAM J. Numer. Anal., Volume 29 (1992) no. 6, pp. 1610-1625 | Article | MR 1191139 | Zbl 0771.65071

[19] Michael J. D. Powell; Malcolm A. Sabin Piecewise quadratic approximations on triangles, ACM Trans. Math. Softw., Volume 3 (1977) no. 4, pp. 316-325 | Article | MR 483304 | Zbl 0375.41010

[20] H. Prautzsch; W. Boehm; M. Paluszny Bézier and B-spline techniques, Mathematics and Visualization, Springer, 2002, xiv+304 pages | Article | Zbl 1033.65008

[21] L. L. Schumaker; T. Sorokina Smooth macro-elements on Powell–Sabin-12 splits, Math. Comput., Volume 75 (2006) no. 254, pp. 711-726 | Article | MR 2196988 | Zbl 1094.65012

[22] H. Speleers A normalized basis for quintic Powell–Sabin splines, Comput. Aided Geom. Des., Volume 27 (2010) no. 6, pp. 438-457 | Article | MR 2657545 | Zbl 1210.65027

[23] I. H. Stangeby Simplex splines on the Powell–Sabin 12-split: components of the finite element method (2018) (Master’s thesis) | arXiv:http://urn.nb.no/URN:NBN:no-66606

[24] A. Ženíšek A general theorem on triangular finite C (m) -elements, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, Volume 8 (1974) no. R-2, pp. 119-127 | Numdam | MR 388731 | Zbl 0321.41003