A local simplex spline basis for C 3 quartic splines on arbitrary triangulations
References
[1]
P. Alfeld, L.L. Schumaker, Smooth macro-elements based on Powell–Sabin triangle splits, Adv. Comput. Math. 16 (2002) 29–46.
[2]
C.K. Chui, R.-H. Wang, Multivariate spline spaces, J. Math. Anal. Appl. 94 (1983) 197–221.
[3]
P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40, SIAM, Philadelphia, 2002.
[4]
R.W. Clough, J.L. Tocher, Finite element stiffness matrices for analysis of plates in bending, in: Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, 1965, pp. 515–545.
[5]
E. Cohen, T. Lyche, R.F. Riesenfeld, A B-spline-like basis for the Powell–Sabin 12-split based on simplex splines, Math. Comput. 82 (2013) 1667–1707.
[6]
J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley Publishing, 2009.
[7]
P. Dierckx, On calculating normalized Powell–Sabin B-splines, Comput. Aided Geom. Des. 15 (1997) 61–78.
[8]
J. Grošelj, A normalized representation of super splines of arbitrary degree on Powell–Sabin triangulations, BIT Numer. Math. 56 (2016) 1257–1280.
[9]
J. Grošelj, M. Knez, Generalized Clough–Tocher splines for CAGD and FEM, Comput. Methods Appl. Mech. Eng. 395 (2022).
[10]
J. Grošelj, H. Speleers, Construction and analysis of cubic Powell–Sabin B-splines, Comput. Aided Geom. Des. 57 (2017) 1–22.
[11]
J. Grošelj, H. Speleers, Super-smooth cubic Powell–Sabin splines on three-directional triangulations: B-spline representation and subdivision, J. Comput. Appl. Math. 386 (2021).
[12]
M.-J. Lai, Geometric interpretation of smoothness conditions of triangular polynomial patches, Comput. Aided Geom. Des. 14 (1997) 191–199.
[13]
M.-J. Lai, L.L. Schumaker, Spline Functions on Triangulations, Encyclopedia of Mathematics and Its Applications, vol. 110, Cambridge University Press, Cambridge, 2007.
[14]
T. Lyche, C. Manni, H. Speleers, Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement, in: T. Lyche, et al. (Eds.), Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, in: Lecture Notes in Mathematics, vol. 2219, Springer, 2018, pp. 1–76.
[15]
T. Lyche, C. Manni, H. Speleers, Construction of C 2 cubic splines on arbitrary triangulations, Found. Comput. Math. 22 (2022) 1309–1350.
[16]
T. Lyche, J.-L. Merrien, Simplex-splines on the Clough–Tocher element, Comput. Aided Geom. Des. 65 (2018) 76–92.
[17]
T. Lyche, J.-L. Merrien, A C 1 simplex-spline basis for the Alfeld split in R s, preprint, 2022.
[18]
T. Lyche, J.-L. Merrien, T. Sauer, Simplex-splines on the Clough–Tocher split with arbitrary smoothness, in: C. Manni, H. Speleers (Eds.), Geometric Challenges in Isogeometric Analysis, in: Springer INdAM Series, vol. 49, Springer, 2022, pp. 85–121.
[19]
T. Lyche, G. Muntingh, Stable simplex spline bases for C 3 quintics on the Powell–Sabin 12-split, Constr. Approx. 45 (2017) 1–32.
[20]
C.A. Micchelli, On a numerically efficient method for computing multivariate B-splines, in: W. Schempp, K. Zeller (Eds.), Multivariate Approximation Theory, in: Internat. Ser. Numer. Math., vol. 51, Birkhäuser, Basel–Boston, 1979, pp. 211–248.
[21]
M.J.D. Powell, M.A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Softw. 3 (1977) 316–325.
[22]
H. Prautzsch, W. Boehm, M. Paluszny, Bézier and B-Spline Techniques, Mathematics and Visualization, Springer–Verlag, Berlin, 2002.
[23]
P. Sablonnière, Composite finite elements of class C k, J. Comput. Appl. Math. 12–13 (1985) 541–550.
[24]
L.L. Schumaker, Spline Functions: Basic Theory, 3rd edition, Cambridge University Press, Cambridge, 2007.
[25]
L.L. Schumaker, T. Sorokina, Smooth macro-elements on Powell–Sabin-12 splits, Math. Comput. 75 (2006) 711–726.
[26]
H.-P. Seidel, Polar forms and triangular B-spline surfaces, in: Blossoming: The New Polar Form Approach to Spline Curves and Surfaces, Siggraph'91, in: Course Notes, vol. 26, World Scientific Publishing Company, 1992, pp. 235–286.
[27]
H. Speleers, A normalized basis for quintic Powell–Sabin splines, Comput. Aided Geom. Des. 27 (2010) 438–457.
[28]
H. Speleers, A normalized basis for reduced Clough–Tocher splines, Comput. Aided Geom. Des. 27 (2010) 700–712.
[29]
H. Speleers, Construction of normalized B-splines for a family of smooth spline spaces over Powell–Sabin triangulations, Constr. Approx. 37 (2013) 41–72.
[30]
H. Speleers, A new B-spline representation for cubic splines over Powell–Sabin triangulations, Comput. Aided Geom. Des. 37 (2015) 42–56.
[31]
R.-H. Wang, Multivariate Spline Functions and Their Applications, Kluwer Academic Publishers, Beijing, 2001.
[32]
R.-H. Wang, X.-Q. Shi, S μ + 1 μ surface interpolations over triangulations, in: A.G. Law, C.L. Wang (Eds.), Approximation, Optimization and Computing: Theory and Applications, Elsevier Science Publishers B.V., 1990, pp. 205–208.
[33]
A. Ženíšek, A general theorem on triangular finite C ( m )-elements, Rev. Fr. Autom. Inform. Rech. Opér., Sér. Rouge 8 (1974) 119–127.
Index Terms
- A local simplex spline basis for C 3 quartic splines on arbitrary triangulations
Index terms have been assigned to the content through auto-classification.
Recommendations
Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations
AbstractWe investigate the construction of C 2 cubic spline quasi-interpolants on a given arbitrary triangulation T to approximate a sufficiently smooth function f. The proposed quasi-interpolants are locally represented in terms of a simplex spline ...
Highlights- We construct three C 2 cubic spline quasi-interpolants over arbitrary triangulations.
- They are represented via a local simplex spline basis defined on the cubic Wang–Shi split.
- Their coefficients are determined by considering a ...
Construction of Cubic Splines on Arbitrary Triangulations
AbstractIn this paper, we address the problem of constructing cubic spline functions on a given arbitrary triangulation . To this end, we endow every triangle of with a Wang–Shi macro-structure. The cubic space on such a refined triangulation has ...
Quartic Beta-splines
Quartic Beta-splines have third-degree arc-length or geometric continuity at simple knots and are determined by three β or shape parameters. We present a general explicit formula for quartic Beta-splines, and determine and illustrate the effects of ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
The Author(s).
Publisher
Elsevier Science Inc.
United States
Publication History
Published: 01 February 2024
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 16 Feb 2025