Abstract
This paper provides the explicit and optimal quadrature rules for the cubic \(C^1\) spline space, which is the extension of the results in Ait-Haddou et al. (J Comput Appl Math 290:543–552, 2015) for less restricted non-uniform knot values. The rules are optimal in the sense that there exist no other quadrature rules with fewer quadrature points to exactly integrate the functions in the given spline space. The explicit means that the quadrature nodes and weights are derived via an explicit recursive formula. Numerical experiments and the error estimations of the quadrature rules are also presented in the end.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ait-Haddou, R., Barton, M., Calo, V.: Explicit gaussian quadrature rules for \(C^1\) cubic splines with symmetrically stretched knot sequences. J. Comput. Appl. Math. 290, 543–552 (2015)
Atkinson, K.: A Survey of Numerical Methods for the Solution of Fredholm Integral Equations Of the Second-Kind. SIAM, Philadelphia (1989)
Barendrecht, P., Bartoň, M., Kosinka, J.: Efficient quadrature rules for subdivision surfaces in isogeometric analysis. Comput. Methods Appl. Mech. Eng. (2018). https://doi.org/10.1016/j.cma.2018.05.017
Barton, M., Calo, V.: Gaussian quadrature for splines via homotopy continuation: Rules for \(C^2\) cubic splines. J. Comput. Appl. Math. 296, 709–723 (2016)
Barton, M., Calo, V.M.: Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 305, 217–240 (2016)
Boor, C.D.: On calculating with B-splines. J. Approx. Theory 6, 50–62 (1972)
Cottrell, J., Hughes, T., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Hoboken (2009)
Deng, F., Zeng, C., Deng, J.: Boundary-mapping parametrization in isogeometric analysis. Commun. Math. Stat. 4, 203–216 (2016)
Cohen, E., Riesenfeld, R.F., Elber, G.: Geometric Modeling with Splines: An Introduction. A.K. Peters Ltd, Wellesley (2001)
Farin, G., Hoschek, J., Kim, M.: Handbook of Computer Aided Geometric Design. Elsevier, Amsterdam (2002)
Gautschi, W.: Numerical Analysis. Springer, Berlin (1999)
Golub, G., Welsch, J.: Calculation of gauss quadrature rules. Math. Comput. 23, 221–230 (1969)
Hiemstra, R.R., Calabro, F., Schillinger, D., Hughes, T.: Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 316, 966–1004 (2017)
Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)
Hughes, T.J.R., Reali, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 199(5–8), 301–313 (2010)
Jingjing Zhang, X.L.: On the linear independence and partition of unity of arbitrary degree analysis-suitable t-splines. Commun. Math. Stat. 3, 353–364 (2015)
Karlin, S., Studden, W.: Tchebycheff systems: with applications in analysis and statistics. Rev. Inst. Int. Stat. 35, 1093 (1967)
Ma, J., Rokhlin, V., Wandzura, S.: Generalized gaussian quadrature rules for systems of arbitrary functions. Siam J. Numer. Anal. 33, 971–996 (1996)
Micchelli, C., Pinkus, A.: Moment theory for weak chebyshev systems with applications to monosplines, quadrature formulae and best one-sided \(L^1\)-approximation by spline functions with fixed knots. SIAM J. Math. Anal. 8, 206–230 (1977)
Nikolov, G.: On certain definite quadrature formulae. J. Comput. Appl. Math. 75, 329–343 (1996)
Nikolov, G., Simian, C.: Approximation and Computation. Springer, New York (2010)
Qarariyah, A., Deng, F., Yang, T., Deng, J.: Numerical solution for schrödinger eigenvalue problem using isogeometric analysis on implicit domains. Commun. Math. Stat. (2019)
Sloan, I.: A quadrature-based approach to improving the collocation method. Numer. Math. 54, 41–56 (1988)
Solin, P., Segeth, K., Dolezel, I.: Higher-Order Finite Element Methods. CRC Press, Balkema (2003)
Xu, G., Kwok, T.H., Wang, C.: Isogeometric computation reuse method for complex objects with topology-consistent volumetric parameterization. Comput. Aided Des. 91, 1–3 (2017)
Yang, T., Qarariyah, A., Kang, H., Deng, J.: Numerical integration over implicitly defined domains with topological guarantee. Commun. Math. Stat. 7, 459–474 (2019)
Zhang, F., Xu, Y., Chen, F.: Discontinuous galerkin methods for isogeometric analysis for elliptic equations on surfaces. Commun. Math. Stat. 2, 431–461 (2014)
Acknowledgements
The authors are supported by the NSF of China (No.61872328), NKBRPC (2011CB302400), SRF for ROCS SE and the Youth Innovation Promotion Association CAS.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, P., Li, X. Explicit Gaussian Quadrature Rules for \(C^1\) Cubic Splines with Non-uniform Knot Sequences. Commun. Math. Stat. 9, 331–345 (2021). https://doi.org/10.1007/s40304-020-00220-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40304-020-00220-9