Construction of Smooth Isogeometric Function Spaces on Singularly Parameterized Domains

  • Thomas TakacsEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9213)


We aim at constructing a smooth basis for isogeometric function spaces on domains of reduced geometric regularity. In this context an isogeometric function is the composition of a piecewise rational function with the inverse of a piecewise rational geometry parameterization. We consider two types of singular parameterizations, domains where a part of the boundary is mapped onto one point and domains where the two parameter lines at a corner of the parameter domain are collinear in the physical domain. We locally map a singular tensor-product patch of arbitrary degree onto a triangular patch, thus splitting the parameterization into a singular bilinear mapping and a regular mapping on a triangular domain. This construction yields an isogeometric function space of prescribed smoothness. Generalizations to higher dimensions are also possible and are briefly discussed in the final section.



The work presented here is partially supported by the Italian MIUR through the FIRB “Futuro in Ricerca” Grant RBFR08CZ0S and by the European Research Council through the FP7 Ideas Consolidator Grant HIgeoM. This support is gratefully acknowledged.


  1. 1.
    Beirão da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for h-p-k-refinement in isogeometric analysis. Numer. Math. 118, 271–305 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benson, D.J., Bazilevs, Y., Hsu, M.C., Hughes, T.J.R.: Isogeometric shell analysis: the Reissner-Mindlin shell. Comput. Meth. Appl. Mech. Eng. 199(5–8), 276–289 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of structural vibrations. Comput. Meth. Appl. Mech. Eng. 195(41–43), 5257–5296 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Farin, G.E.: NURBS: From Projective Geometry to Practical Use. Ak Peters Series. A.K Peters, Natick (1999)zbMATHGoogle Scholar
  5. 5.
    Hu, S.-M.: Conversion between triangular and rectangular Bézier patches. Comput. Aided Geom. Des. 18(7), 667–671 (2001)zbMATHCrossRefGoogle Scholar
  6. 6.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Kiendl, J., Bazilevs, Y., Hsu, M.-C., Wüchner, R., Bletzinger, K.-U.: The bending strip method for isogeometric analysis of Kirchhoff-Love shell structures comprised of multiple patches. Comput. Meth. Appl. Mech. Eng. 199(37–40), 2403–2416 (2010)zbMATHCrossRefGoogle Scholar
  8. 8.
    Kiendl, J., Bletzinger, K.-U., Linhard, J., Wüchner, R.: Isogeometric shell analysis with Kirchhoff-Love elements. Comput. Meth. Appl. Mech. Eng. 198(49–52), 3902–3914 (2009)zbMATHCrossRefGoogle Scholar
  9. 9.
    Lu, J.: Circular element: isogeometric elements of smooth boundary. Comput. Meth. Appl. Mech. Eng. 198(30–32), 2391–2402 (2009)zbMATHCrossRefGoogle Scholar
  10. 10.
    Piegl, L., Tiller, W.: The NURBS book. Springer, London (1995)zbMATHCrossRefGoogle Scholar
  11. 11.
    Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Springer, New York (2002)zbMATHCrossRefGoogle Scholar
  12. 12.
    Schumaker, L.L.: Spline Functions: Basic Theory. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  13. 13.
    Simpson, R.N., Scott, M.A., Taus, M., Thomas, D.C., Lian, H.: Acoustic isogeometric boundary element analysis. Comput. Meth. Appl. Mech. Eng. 269, 265–290 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Takacs, T., Jüttler, B.: Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis. Comput. Meth. Appl. Mech. Eng. 200(49–52), 3568–3582 (2011)zbMATHCrossRefGoogle Scholar
  15. 15.
    Takacs, T., Jüttler, B.: \({H}^2\) regularity properties of singular parameterizations in isogeometric analysis. Graph. Model. 74(6), 361–372 (2012)CrossRefGoogle Scholar
  16. 16.
    Takacs, T., Jüttler, B., Scherzer, O.: Derivatives of isogeometric functions on \(n\)-dimensional rational patches in \(\mathbb{R}^d\). Comput. Aided Geom. Des. 31(78), 567–581 (2014). Recent Trends in Theoretical and Applied GeometryCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PaviaPaviaItaly

Personalised recommendations