Algorithms and Data Structures for Truncated Hierarchical B–splines

  • Gábor Kiss
  • Carlotta Giannelli
  • Bert Jüttler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8177)


Tensor–product B–spline surfaces are commonly used as standard modeling tool in Computer Aided Geometric Design and for numerical simulation in Isogeometric Analysis. However, when considering tensor–product grids, there is no possibility of a localized mesh refinement without propagation of the refinement outside the region of interest. The recently introduced truncated hierarchical B–splines (THB–splines) [5] provide the possibility of a local and adaptive refinement procedure, while simultaneously preserving the partition of unity property. We present an effective implementation of the fundamental algorithms needed for the manipulation of THB–spline representations based on standard data structures. By combining a quadtree data structure — which is used to represent the nested sequence of subdomains — with a suitable data structure for sparse matrices, we obtain an efficient technique for the construction and evaluation of THB–splines.


hierarchical tensor–product B–splines truncated basis THB–splines isogeometric analysis local refinement 


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  1. 1.
    Bornemann, P.B., Cirak, F.: A subdivision–based implementation of the hierarchical B–spline finite element method. Comput. Methods Appl. Mech. Engrg. 253, 584–598 (2012)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Deng, J., Chen, F., Feng, Y.: Dimensions of spline spaces over T–meshes. J. Comput. Appl. Math. 194, 267–283 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Design 30, 331–356 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Forsey, D.R., Bartels, R.H.: Hierarchical B–spline refinement. Comput. Graphics 22, 205–212 (1988)CrossRefGoogle Scholar
  5. 5.
    Giannelli, C., Jüttler, B., Speleers, H.: THB–splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Design 29, 485–498 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comp. Math. (to appear, 2013)Google Scholar
  7. 7.
    Giannelli, C., Jüttler, B.: Bases and dimensions of bivariate hierarchical tensor–product splines. J. Comput. Appl. Math. 239, 162–178 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gilbert, J.R., Moler, C., Schreiber, R.: Sparse matrices in MATLAB: design and implementation. SIAM J. Matrix Anal. Appl. 13, 333–356 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Gonzalez-Ochoa, C., Peters, J.: Localized–hierarchy surface splines (LeSS). In: Proceedings of the 1999 Symposium on Interactive 3D Graphics, pp. 7–15. ACM, New York (1999)CrossRefGoogle Scholar
  10. 10.
    Greiner, G., Hormann, K.: Interpolating and approximating scattered 3D–Data with hierarchical tensor product B–splines. In: Méhauté, A.L., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods. In Innovations in Applied Mathematics, pp. 163–172. Vanderbilt University Press, Nashville (1997)Google Scholar
  11. 11.
    Kraft, R.: Adaptive and linearly independent multilevel B–splines. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 209–218. Vanderbilt University Press, Nashville (1997)Google Scholar
  12. 12.
    Kraft, R.: Adaptive und linear unabhängige Multilevel B–Splines und ihre Anwendungen. PhD Thesis, Universität Stuttgart (1998)Google Scholar
  13. 13.
    Lee, S., Wolberg, G., Shin, S.Y.: Scattered data interpolation with multilevel B–splines. IEEE Trans. on Visualization and Computer Graphics 3, 228–244 (1997)CrossRefGoogle Scholar
  14. 14.
    Schillinger, D., Dedè, L., Scott, M.A., Evans, J.A., Borden, M.J., Rank, E., Hughes, T.J.R.: An isogeometric design–through–analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T–spline CAD surfaces. Comput. Methods Appl. Mech. Engrg., 249–252, 116–150 (2012)Google Scholar
  15. 15.
    Schumaker, L.L., Wang, L.: Approximation power of polynomial splines on T–meshes. Comput. Aided Geom. Design 29, 599–612 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T–splines and T–NURCCS. ACM Trans. Graphics 22, 477–484 (2003)CrossRefGoogle Scholar
  17. 17.
    Stollnitz, E.J., DeRose, T.D., Salesin, D.H.: Wavelets For Computer Graphics: Theory and Application, 1st edn. Morgan Kaufmann Publishers, Inc. (1996)Google Scholar
  18. 18.
    Vuong, A.-V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 200, 3554–3567 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Yvart, A., Hahmann, S.: Hierarchical triangular splines. ACM Trans. Graphics 24, 1374–1391 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Gábor Kiss
    • 1
  • Carlotta Giannelli
    • 2
  • Bert Jüttler
    • 2
  1. 1.Doctoral Program “Computational Mathematics”Johannes Kepler University LinzLinzAustria
  2. 2.Institute of Applied GeometryJohannes Kepler University LinzLinzAustria

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