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Algorithm 999: Computation of Multi-Degree B-Splines

Author:

Hendrik SpeleersAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 45, Issue 4

Article No.: 43, Pages 1 - 15

https://doi.org/10.1145/3321514

Published: 09 December 2019 Publication History

Abstract

Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A MATLAB implementation is provided to illustrate the computation and use of MDB-splines.

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Software for Computation of Multi-Degree B-Splines

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References

[1]

C. V. Beccari, G. Casciola, and S. Morigi. 2017. On multi-degree splines. Computer Aided Geometric Design 58 (2017), 8--23.

[2]

M. J. Borden, M. A. Scott, J. A. Evans, and T. J. R. Hughes. 2011. Isogeometric finite element data structures based on Bézier extraction of NURBS. International Journal for Numerical Methods in Engineering 87 (2011), 15--47.

[3]

B. Buchwald and G. Mühlbach. 2003. Construction of B-splines for generalized spline spaces generated from local ECT-systems. Journal of Computational and Applied Mathematics 159 (2003), 249--267.

Digital Library

[4]

P. L. Butzer, M. Schmidt, and E. L. Stark. 1988. Observations on the history of central B-splines. Archive for History of Exact Sciences 39 (1988), 137--156.

[5]

E. Cohen, R. F. Riesenfeld, and G. Elber. 2001. Geometric Modeling with Splines: An Introduction. CRC Press, Boca Raton, FL.

Digital Library

[6]

J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs. 2009. Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley 8 Sons.

[7]

M. G. Cox. 1972. The numerical evaluation of B-splines. Journal of the Institute of Mathematics and Its Applications 10 (1972), 134--149.

[8]

H. B. Curry and I. J. Schoenberg. 1947. On spline distributions and their limits: The Pólya distribution functions. Bulletin of the American Mathematical Society 53 (1947), 1114.

[9]

C. de Boor. 1972. On calculating with B-splines. Journal of Approximation Theory 6 (1972), 50--62.

[10]

C. de Boor. 1977. Package for calculating with B-splines. SIAM Journal on Numerical Analysis 14 (1977), 441--472.

[11]

C. de Boor. 2001. A Practical Guide to Splines, Revised Edition. Springer-Verlag.

[12]

C. de Boor and A. Pinkus. 2003. The B-spline recurrence relations of Chakalov and of Popoviciu. Journal of Approximation Theory 124 (2003), 115--123.

Digital Library

[13]

X. Li, Z. J. Huang, and Z. Liu. 2012. A geometric approach for multi-degree splines. Journal of Computer Science and Technology 27 (2012), 841--850.

[14]

L. Liu, H. Casquero, H. Gomez, and Y. J. Zhang. 2016. Hybrid-degree weighted T-splines and their application in isogeometric analysis. Computers and Fluids 141 (2016), 42--53.

[15]

G. Nürnberger, L. L. Schumaker, M. Sommer, and H. Strauss. 1984. Generalized Chebyshevian splines. SIAM Journal on Mathematical Analysis 15 (1984), 790--804.

[16]

L. Piegl and W. Tiller. 2012. The NURBS Book. Springer-Verlag.

[17]

D. Schillinger, K. R. Praneeth, and H. N. Lam. 2016. Lagrange extraction and projection for NURBS basis functions: A direct link between isogeometric and standard nodal finite element formulations. International Journal for Numerical Methods in Engineering 108 (2016), 515--534.

[18]

I. J. Schoenberg. 1946. Contributions to the problem of approximation of equidistant data by analytic functions, Part A: On the problem of smoothing or graduation, a first class of analytic approximation formulas. Quarterly of Applied Mathematics 4 (1946), 45--99.

[19]

T. W. Sederberg, J. Zheng, and X. Song. 2003. Knot intervals and multi-degree splines. Computer Aided Geometric Design 20 (2003), 455--468.

Digital Library

[20]

W. Shen and G. Wang. 2010a. A basis of multi-degree splines. Computer Aided Geometric Design 27 (2010), 23--35.

Digital Library

[21]

W. Shen and G. Wang. 2010b. Changeable degree spline basis functions. Journal of Computational and Applied Mathematics 234 (2010), 2516--2529.

Digital Library

[22]

M. Sommer and H. Strauss. 1988. Weak Descartes systems in generalized spline spaces. Constructive Approximation 4 (1988), 133--145.

[23]

D. Toshniwal, H. Speleers, R. R. Hiemstra, and T. J. R. Hughes. 2017. Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis. Computer Methods in Applied Mechanics and Engineering 316 (2017), 1005--1061.

[24]

D. Toshniwal, H. Speleers, R. R. Hiemstra, C. Manni, and T. J. R. Hughes. 2020. Multi-degree B-splines: Algorithmic computation and properties. Computer Aided Geometric Design 76, Article 101792 (2020). https://www.sciencedirect.com/science/article/pii/S0167839619301013.

Cited By

Takacs STyoler S(2025)Multi-resolution isogeometric analysis – efficient adaptivity utilizing the multi-patch structureComputers & Mathematics with Applications10.1016/j.camwa.2024.12.005179(103-125)Online publication date: Feb-2025
https://doi.org/10.1016/j.camwa.2024.12.005
Boushabi MEddargani SIbáñez MLamnii A(2024)Normalized B-spline-like representation for low-degree Hermite osculatory interpolation problemsMathematics and Computers in Simulation10.1016/j.matcom.2024.05.011225(98-110)Online publication date: Nov-2024
https://doi.org/10.1016/j.matcom.2024.05.011
Ma XShen W(2023)Generalized de Boor–Cox Formulas and Pyramids for Multi-Degree Spline Basis FunctionsMathematics10.3390/math1102036711:2(367)Online publication date: 10-Jan-2023
https://doi.org/10.3390/math11020367
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Index Terms

Algorithm 999: Computation of Multi-Degree B-Splines
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials

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Information

Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 45, Issue 4

December 2019

207 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/3375544

Editors:
Zhaojun Bai
University of California at Davis, USA
,
Wolfgang Bangerth
Colorado State University, USA

Issue’s Table of Contents

Copyright © 2019 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 December 2019

Accepted: 01 March 2019

Revised: 01 December 2018

Received: 01 January 2018

Published in TOMS Volume 45, Issue 4

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Ministero dellðIstruzione, dellðUniversità e della Ricerca
Università degli Studi di Roma Tor Vergata

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Cited By

Takacs STyoler S(2025)Multi-resolution isogeometric analysis – efficient adaptivity utilizing the multi-patch structureComputers & Mathematics with Applications10.1016/j.camwa.2024.12.005179(103-125)Online publication date: Feb-2025
https://doi.org/10.1016/j.camwa.2024.12.005
Boushabi MEddargani SIbáñez MLamnii A(2024)Normalized B-spline-like representation for low-degree Hermite osculatory interpolation problemsMathematics and Computers in Simulation10.1016/j.matcom.2024.05.011225(98-110)Online publication date: Nov-2024
https://doi.org/10.1016/j.matcom.2024.05.011
Ma XShen W(2023)Generalized de Boor–Cox Formulas and Pyramids for Multi-Degree Spline Basis FunctionsMathematics10.3390/math1102036711:2(367)Online publication date: 10-Jan-2023
https://doi.org/10.3390/math11020367
Manni CSande ESpeleers H(2023)Outlier-free spline spaces for isogeometric discretizations of biharmonic and polyharmonic eigenvalue problemsComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2023.116314417(116314)Online publication date: Dec-2023
https://doi.org/10.1016/j.cma.2023.116314
Takacs TToshniwal D(2023) Almost- splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems Computer Methods in Applied Mechanics and Engineering10.1016/j.cma.2022.115640403(115640)Online publication date: Jan-2023
https://doi.org/10.1016/j.cma.2022.115640
Grošelj JSpeleers H(2023)Extraction and application of super-smooth cubic B-splines over triangulationsComputer Aided Geometric Design10.1016/j.cagd.2023.102194103(102194)Online publication date: Jun-2023
https://doi.org/10.1016/j.cagd.2023.102194
Xiao ZShen W(2023)An Efficient Algorithm for Degree Reduction of MD-SplinesAdvances in Computer Graphics10.1007/978-3-031-50078-7_1(3-14)Online publication date: 28-Aug-2023
https://dl.acm.org/doi/10.1007/978-3-031-50078-7_1
Speleers H(2022)Algorithm 1020: Computation of Multi-Degree Tchebycheffian B-SplinesACM Transactions on Mathematical Software10.1145/347868648:1(1-31)Online publication date: 16-Feb-2022
https://dl.acm.org/doi/10.1145/3478686
Thomas DEngvall LSchmidt STew KScott M(2022)U-splines: Splines over unstructured meshesComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2022.115515401(115515)Online publication date: Nov-2022
https://doi.org/10.1016/j.cma.2022.115515
Manni CSande ESpeleers H(2022)Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizationsComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2021.114260389(114260)Online publication date: Feb-2022
https://doi.org/10.1016/j.cma.2021.114260
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