Exploring Matrix Generation Strategies in Isogeometric Analysis

  • Angelos Mantzaflaris
  • Bert Jüttler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8177)


An important step in simulation via isogeometric analysis (IGA) is the assembly step, where the coefficients of the final linear system are generated. Typically, these coefficients are integrals of products of shape functions and their derivatives. Similarly to the finite element analysis (FEA), the standard choice for integral evaluation in IGA is Gaussian quadrature. Recent developments propose different quadrature rules, that reduce the number of quadrature points and weights used. We experiment with the existing methods for matrix generation. Furthermore we propose a new, quadrature-free approach, based on interpolation of the geometry factor and fast look-up operations for values of B-spline integrals. Our method builds upon the observation that exact integration is not required to achieve the optimal convergence rate of the solution. In particular, it suffices to generate the linear system within the order of accuracy matching the approximation order of the discretization space. We demonstrate that the best strategy is one that follows the above principle, resulting in expected accuracy and improved computational time.


isogeometric analysis stiffness matrix mass matrix numerical integration quadrature 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Auricchio, F., Calabrò, F., Hughes, T., Reali, A., Sangalli, G.: A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis. Comput. Meth. Appl. Mech. Eng. (2012)Google Scholar
  2. 2.
    Beirão da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for h-p-k-refinement in isogeometric analysis. Numerische Mathematik 118, 271–305 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Costantini, P., Manni, C., Pelosi, F., Sampoli, M.L.: Quasi-interpolation in isogeometric analysis based on generalized B-splines. Comput. Aided Geom. Des. 27(8), 656–668 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    de Boor, C., Fix, G.: Spline approximation by quasi-interpolants. J. Approx. Theory 8, 19–45 (1973)CrossRefzbMATHGoogle Scholar
  5. 5.
    de Falco, C., Reali, A., Vázquez, R.: GeoPDEs: A research tool for isogeometric analysis of PDEs. Advances in Engineering Software 42(12), 1020–1034 (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Farin, G.: Curves and surfaces for CAGD: A practical guide. Morgan Kaufmann Publishers Inc., San Francisco (2002)Google Scholar
  7. 7.
    Gahalaut, K., Tomar, S.: Condition number estimates for matrices arising in the isogeometric discretizations. Technical Report RR-2012-23, Johann Radon Institute for Computational and Applied Mathematics, Linz (December 2012),
  8. 8.
    Großmann, D., Jüttler, B., Schlusnus, H., Barner, J., Vuong, A.-V.: Isogeometric simulation of turbine blades for aircraft engines. Comput. Aided Geom. Des. 29(7), 519–531 (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    He, T.-X.: Eulerian polynomials and B-splines. J. Comput. Appl. Math. 236(15), 3763–3773 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hopkins, T., Wait, R.: Some quadrature rules for Galerkin methods using B-spline basis functions. Comput. Meth. Appl. Mech. Eng. 19(3), 401–416 (1979)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hughes, T., Cottrell, J., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. Mech. Eng. 194(39-41), 4135–4195 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hughes, T., Reali, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Comput. Meth. Appl. Mech. Eng. 199(5-8), 301–313 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kahaner, D., Moler, C., Nash, S.: Numerical methods and software. Prentice-Hall, Inc., Upper Saddle River (1989)zbMATHGoogle Scholar
  14. 14.
    Lyche, T., Schumaker, L.: Local spline approximation methods. J. Approx. Theory 25, 266–279 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Patzák, B., Rypl, D.: Study of computational efficiency of numerical quadrature schemes in the isogeometric analysis. In: Proc. of the 18th Int’l Conf. Engineering Mechanics, EM 2012, pp. 1135–1143 (2012)Google Scholar
  16. 16.
    Strang, G.: Approximation in the finite element method. Numerische Mathematik 19, 81–98 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)zbMATHGoogle Scholar
  18. 18.
    Stroud, A., Secrest, D.: Gaussian quadrature formulas. Prentice-Hall Series in Automatic Computation. Prentice-Hall, Englewood Cliffs (1966)zbMATHGoogle Scholar
  19. 19.
    Vermeulen, A.H., Bartels, R.H., Heppler, G.R.: Integrating products of B-splines. SIAM J. on Sci. and Stat. Computing 13(4), 1025–1038 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Wang, R.-H., Xu, Y., Xu, Z.-Q.: Eulerian numbers: A spline perspective. J. Math. Anal. and Appl. 370(2), 486–490 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Beirão da Veiga, B.Y.L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16(07), 1031–1090 (2006)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Angelos Mantzaflaris
    • 1
  • Bert Jüttler
    • 2
  1. 1.RICAMAustrian Academy of SciencesLinzAustria
  2. 2.AGJohannes Kepler UniversityLinzAustria

Personalised recommendations