A Versatile Strategy for the Implementation of Adaptive Splines

  • Andrea BressanEmail author
  • Dominik Mokriš
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10521)


This paper presents an implementation framework for spline spaces over T-meshes (and their d-dimensional analogs). The aim is to share code between the implementations of several spline spaces. This is achieved by reducing evaluation to a generalized Bézier extraction.

The approach was tested by implementing hierarchical B-splines, truncated hierarchical B-splines, decoupled hierarchical B-splines (a novel variation presented here), truncated B-splines for partially nested refinement and hierarchical LR-splines.


Implementation Bézier extraction THB-splines LR-splines 



The authors have been supported by the Austrian Science Fund (FWF, NFN S117 “Geometry + Simulation”) and by the Seventh Framework Programme of the EU (project EXAMPLE, GA No. 324340). This support is gratefully acknowledged. The authors would also like to thank Dr. Rafael Vázquez for commenting on an earlier version of this paper and to the reviewers for their valuable suggestions.


  1. 1.
    Ayachit, U.: The paraview guide: a parallel visualization application (2015)Google Scholar
  2. 2.
    Borden, M.J., Scott, M.A., Evans, J.A., Hughes, T.J.R.: Isogeometric finite element data structures based on Bézier extraction of NURBS. Int. J. Numer. Methods Eng. 87(1–5), 15–47 (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bracco, C., Lyche, T., Manni, C., Roman, F., Speleers, H.: Generalized spline spaces over T-meshes: dimension formula and locally refined generalized B-splines. Appl. Math. Comput. 272(part 1), 187–198 (2016)MathSciNetGoogle Scholar
  4. 4.
    Bressan, A.: Some properties of LR-splines. Comput. Aided. Geom. Des. 30(8), 778–794 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bressan, A., Jüttler, B.: A hierarchical construction of LR meshes in 2D. Comput. Aided Geom. Des. 37, 9–24 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brovka, M., López, J., Escobar, J., Montenegro, R., Cascón, J.: A simple strategy for defining polynomial spline spaces over hierarchical T-meshes. Comput. Aided Des. 72, 140–156 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Buchegger, F., Jüttler, B., Mantzaflaris, A.: Adaptively refined multi-patch B-splines with enhanced smoothness. Appl. Math. Comput. 272(part 1), 159–172 (2016)MathSciNetGoogle Scholar
  8. 8.
    Buffa, A., Garau, E.M.: Refinable spaces and local approximation estimates for hierarchical splines. IMA J. Numer. Anal. 37(3), 1125–1149 (2017)MathSciNetGoogle Scholar
  9. 9.
    Buffa, A., Giannelli, C.: Adaptive isogeometric methods with hierarchical splines: error estimator and convergence. Math. Models Methods Appl. Sci. 26(1), 1–25 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Collin, A., Sangalli, G., Takacs, T.: Analysis-suitable \({G}^1\) multi-patch parametrizations for \({C}^1\) isogeometric spaces. Comput. Aided Geom. Des. 47, 93–113 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Da Veiga, L.B., Buffa, A., Sangalli, G., Vázquez, R.: Analysis-suitable T-splines of arbitrary degree: definition, linear independence and approximation properties. Math. Models Methods Appl. Sci. 23(11), 1979–2003 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Danial, A.: CLOC: Count Lines of Code (2006–2017).
  13. 13.
    Deng, J., Chen, F., Li, X., Hu, C., Tong, W., Yang, Z., Feng, Y.: Polynomial splines over hierarchical T-meshes. Graph. Models 70(4), 76–86 (2008)CrossRefGoogle Scholar
  14. 14.
    Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided. Geom. Des. 30(3), 331–356 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Evans, E.J., Scott, M.A., Li, X., Thomas, D.C.: Hierarchical T-splines: analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis. Comput. Methods Appl. Mech. Eng. 284, 1–20 (2015)CrossRefGoogle Scholar
  16. 16.
    Forsey, D.R., Bartels, R.H.: Hierarchical B-spline refinement. In: SIGGRAPH Computer Graphics, vol. 22, no. 4, pp. 205–212 (1988)Google Scholar
  17. 17.
    Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29(7), 485–498 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comput. Math. 40(2), 459–490 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Geometry + simulation modules (G+Smo): Open source C++ library for isogeometric analysis (2016).
  20. 20.
    GoTools: Collection of C++ libraries connected to geometry (2016).
  21. 21.
    Guennebaud, G., Jacob, B., et al.: Eigen v3 (2010).
  22. 22.
    Hennig, P., Müller, S., Kästner, M.: Bézier extraction and adaptive refinement of truncated hierarchical NURBS. Comput. Methods Appl. Mech. Eng. 305, 316–339 (2016)CrossRefGoogle Scholar
  23. 23.
    Hennig, P., Kästner, M., Morgenstern, P., Peterseim, D.: Adaptive mesh refinement strategies in isogeometric analysis-a computational comparison. Comput. Methods Appl. Mech. Eng. 316, 424–448 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    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(39–41), 4135–4195 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Johannessen, K.A., Remonato, F., Kvamsdal, T.: On the similarities and differences between classical hierarchical, truncated hierarchical and LR B-splines. Comput. Methods Appl. Mech. Eng. 291, 64–101 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Jüttler, B., Langer, U., Mantzaflaris, A., Moore, S.E., Zulehner, W.: Geometry + simulation modules: Implementing isogeometric analysis. PAMM 14(1), 961–962 (2014)CrossRefGoogle Scholar
  27. 27.
    Kang, H., Xu, J., Chen, F., Deng, J.: A new basis for PHT-splines. Graph. Models 82, 149–159 (2015)CrossRefGoogle Scholar
  28. 28.
    Kapl, M., Vitrih, V., Jüttler, B., Birner, K.: Isogeometric analysis with geometrically continuous functions on two-patch geometries. Comput. Math. Appl. 70(7), 1518–1538 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kiss, G., Giannelli, C., Jüttler, B.: Algorithms and data structures for truncated hierarchical B-splines. In: Floater, M., Lyche, T., Mazure, M.-L., Mørken, K., Schumaker, L.L. (eds.) MMCS 2012. LNCS, vol. 8177, pp. 304–323. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54382-1_18 CrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    Li, X., Zheng, J., Sederberg, T.W., Hughes, T.J.R., Scott, M.A.: On linear independence of T-spline blending functions. Comput. Aided Geom. Des. 29(1), 63–76 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mokriš, D., Jüttler, B.: TDHB-splines: the truncated decoupled basis of hierarchical tensor-product splines. Comput. Aided Geom. Des. 31(7–8), 531–544 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mokriš, D., Jüttler, B., Giannelli, C.: On the completeness of hierarchical tensor product B-splines. J. Comput. Appl. Math. 271, 53–70 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Morgenstern, P.: Globally structured three-dimensional analysis-suitable T-splines: definition, linear independence and \(m\)-graded local refinement. SIAM J. Numer. Anal. 54(4), 2163–2186 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Morgenstern, P., Peterseim, D.: Analysis-suitable adaptive T-mesh refinement with linear complexity. Comput. Aided. Geom. Des. 34, 50–66 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rabut, C.: Locally tensor product functions. Numer. Algorithms 39(1–3), 329–348 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Scott, M.A., Borden, M.J., Verhoosel, C.V., Sederberg, T.W., Hughes, T.J.R.: Isogeometric finite element data structures based on Bézier extraction of T-splines. Int. J. Numer. Methods Eng. 88(2), 126–156 (2011)CrossRefzbMATHGoogle Scholar
  38. 38.
    Sederberg, T.W., Cardon, D.L., Finnigan, G.T., North, N.S., Zheng, J., Lyche, T.: T-spline simplification and local refinement. ACM Trans. Graph. 23(3), 276–283 (2004)CrossRefGoogle Scholar
  39. 39.
    Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCs. ACM Trans. Graph. 22(3), 477–484 (2003)CrossRefGoogle Scholar
  40. 40.
    Thibault, W.C., Naylor, B.F.: Set operations on polyhedra using binary space partitioning trees. In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1987, pp. 153–162. ACM, New York (1987)Google Scholar
  41. 41.
    Toth, C.D., O’Rourke, J., Goodman, J.E.: Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (2004)zbMATHGoogle Scholar
  42. 42.
    Vázquez, R., Garau, E.: Algorithms for the implementation of adaptive isogeometric methods using hierarchical splines. Tech. report 16–08, IMATI-CNR, Pavia, July 2016Google Scholar
  43. 43.
    Zore, U.: Constructions and properties of adaptively refined multilevel spline spaces. Dissertation, Johannes Kepler University Linz (2016).

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway
  2. 2.Institute of Applied GeometryJohannes Kepler University LinzLinzAustria

Personalised recommendations