Skip to main content
SpringerLink
Log in

AS++ T-splines: arbitrary degree, nestedness and approximation

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Bi-cubic analysis-suitable++ T-splines (AS++ T-splines) (Li in Comput Methods Appl Mech Eng 333:462–474, 2018) are T-splines defined on less restricted T-meshes than analysis-suitable T-splines (AS T-splines), which are both important tools in isogeometric analysis (IGA). In this paper, we generalize the bi-cubic AS++ T-splines to arbitrary degrees and describe some important mathematical properties. Specifically, we develop the conditions under which an AS++ T-spline space belongs to another AS++ T-spline space. This result provides the foundation for the optimized local refinement (Zhang in Comput Methods Appl Mech Eng 342:32–45, 2018) and also is one of the keys for AS++ T-spline approximation. In the end, the optimal approximation properties of the associated T-spline space are developed for arbitrary AS++ T-spline space with the assumption of existence of dual basis, which is automatically satisfied for bi-cubic AS++ T-spline spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Antolin, P., Buffa, A., Martinelli, M.: Isogeometric analysis on V-reps: first results. Comput. Methods Appl. Mech. Eng. 355, 976–1002 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antolin, P., Buffa, A., Coradello, L.: A hierarchical approach to the a posteriori error estimation of isogeometric Kirchhoff plates and Kirchhoff-Love shells. Comput. Methods Appl. Mech. Eng. 363, 112919 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bazilevs, Y., da Veiga, L.B., Cottrell, J., Hughes, T., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16, 1031–1090 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bazilevs, Y., Calo, V.M., Cottrell, J.A., Evans, J.A., Hughes, T.J., Lipton, S., Scott, M.A., Sederberg, T.W.: Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Eng. 199(5–8), 229–263 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beck, J., Sangalli, G., Tamellini, L.: A sparse-grid isogeometric solver. Comput. Methods Appl. Mech. Eng. 335, 128–151 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benson, D.J., Bazilevs, Y., De Luycker, E., Hsu, M.C., Scott, M.A., Hughes, T.J., Belytschko, T.: A generalized finite element formulation for arbitrary basis functions: from isogeometric analysis to XFEM. Int. J. Numer. Meth. Eng. 83, 765–785 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J., Landis, C.M.: A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217–220, 77–95 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bressan, A., Sande, E.: Approximation in fem, dg and iga: a theoretical comparison. Numer. Math. 143, 923–942 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bressan, A., Takacs, S.: Sum factorization techniques in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 352, 437–460 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bressan, A., Buffa, A., Sangalli, G.: Characterization of analysis-suitable t-splines. Comput. Aided Geom. Des. 39, 17–49 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Buffa, A., Cho, D., Sangalli, G.: Linear independence of the T-spline blending functions associated with some particular T-meshes. Comput. Methods Appl. Mech. Eng. 199(23–24), 1437–1445 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buffa, A., Dolz, J., Kurz, S., Schops, S., Vazquez, R., Wolf, F.: Multipatch approximation of the de rham sequence and its traces in isogeometric analysis. Numer. Math. 144, 201–236 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Casquero, H., Wei, X., Toshniwal, D., Li, A., Hughes, T.J., Kiendl, J., Zhang, Y.J.: Seamless integration of design and Kirchhoff-Love shell analysis using analysis-suitable unstructured T-splines. Comput. Methods Appl. Mech. Eng. 360, 112765 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Coradello, L., Antolin, P., Vzquez, R., Buffa, A.: Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach. Comput. Methods Appl. Mech. Eng. 364, 112925 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. da Veiga, L.B., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for h-p-k-refinement in isogeometric analysis. Numer. Math. 118(2), 271–305 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. da Veiga, L.B., Buffa, A., Sangalli, G.: Analysis-suitable T-splines are dual-compatible. Comput. Methods Appl. Mech. Eng. 249–252, 42–51 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. da Veiga, L.B., Buffa, A., Sangalli, G., Vazquez, R.: Analysis-suitable T-splines of arbitrary degree: definition and properties. Math. Models Methods Appl. Sci. 23, 1979–2003 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dimitri, R., Lorenzis, L.D., Scott, M.A., Wriggers, P., Taylor, R.L., Zavarise, G.: Isogeometric large deformation frictionless contact using T-splines. Comput. Methods Appl. Mech. Eng. 269, 394–414 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dörfel, M., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local h-refinement with T-splines. Comput. Methods Appl. Mech. Eng. 199(5–8), 264–275 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Fahrendorf, F., Lorenzis, L.D., Gomez, H.: Reduced integration at superconvergent points in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 328, 390–410 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Feischl, M., Gantner, G., Haberl, A., Praetorius, D.: Optimal convergence for adaptive iga boundary element methods for weakly-singular integral equations. Numer. Math. 136, 147–182 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Garcia, D., Pardo, D., Dalcin, L., Calo, V.M.: Refined isogeometric analysis for a preconditioned conjugate gradient solver. Comput. Methods Appl. Mech. Eng. 335, 490–509 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Guo, Y., Heller, J., Hughes, T.J., Ruess, M., Schillinger, D.: Variationally consistent isogeometric analysis of trimmed thin shells at finite deformations, based on the STEP exchange format. Comput. Methods Appl. Mech. Eng. 336, 39–79 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hiemstra, R.R., Sangalli, G., Tani, M., Calabr, F., Hughes, T.J.: Fast formation and assembly of finite element matrices with application to isogeometric linear elasticity. Comput. Methods Appl. Mech. Eng. 355, 234–260 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hughes, T.J., 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ipson, H.: T-spline merging, Master’s thesis, Brigham Young University (April 2005)

  27. Kamensky, D., Bazilevs, Y.: tIGAr: automating isogeometric analysis with FEniCS. Comput. Methods Appl. Mech. Eng. 344, 477–498 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kargaran, S., Jttler, B., Kleiss, S., Mantzaflaris, A., Takacs, T.: Overlapping multi-patch structures in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 356, 325–353 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, X., Scott, M.A.: Analysis-suitable T-splines: characterization, refinablility and approximation. Math. Models Methods Appl. Sci. 24(06), 1141–1164 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, X., Zhang, J.: Analysis-suitable++ T-splines: linear independence and approximation. Comput. Methods Appl. Mech. Eng. 333, 462–474 (2018)

    Article  MATH  Google Scholar 

  31. Li, X., Zheng, J., Sederberg, T.W., Hughes, T.J., Scott, M.A.: On the linear independence of T-splines blending functions. Comput. Aided Geom. Des. 29, 63–76 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, X., Wei, X., Zhang, Y.: Hybrid non-uniform recursive subdivision with improved convergence rates. Comput. Methods Appl. Mech. Eng. 352, 606–624 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu, L., Zhang, Y., Hughes, T.J., Scott, M.A., Sederberg, T.W.: Volumetric T-spline construction using boolean operations. Eng. Comput. 30, 425–439 (2014)

    Article  Google Scholar 

  34. Mazza, M., Manni, C., Ratnani, A., Serra-Capizzano, S., Speleers, H.: Isogeometric analysis for 2D and 3D curl-div problems: spectral symbols and fast iterative solvers. Comput. Methods Appl. Mech. Eng. 344, 970–997 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Miao, D., Zou, Z., Scott, M.A., Borden, M.J., Thomas, D.C.: Isogeometric Bezier dual mortaring: the enriched Bezier dual basis with application to second- and fourth-order problems. Comput. Methods Appl. Mech. Eng. 363, 112900 (2020)

    Article  MATH  Google Scholar 

  36. Patrizi, F., Manni, C., Pelosi, F., Speleers, H.: Adaptive refinement with locally linearly independent LR B-splines: theory and applications. Comput. Methods Appl. Mech. Eng. 369, 113230 (2020)

    Article  MathSciNet  Google Scholar 

  37. Pegolotti, L., Ded, L., Quarteroni, A.: Isogeometric analysis of the electrophysiology in the human heart: numerical simulation of the bidomain equations on the atria. Comput. Methods Appl. Mech. Eng. 343, 52–73 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  38. Puzyrev, V., Deng, Q., Calo, V.: Spectral approximation properties of isogeometric analysis with variable continuity. Comput. Methods Appl. Mech. Eng. 334, 22–39 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Sande, E., Manni, C., Speleers, H.: Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis. Numer. Math. 144, 889–929 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schillinger, D., Dede, L., Scott, M.A., Evans, J.A., Borden, M.J., Rank, E., Hughes, T.J.: 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. Eng. 249–252, 116–150 (2014)

    MathSciNet  MATH  Google Scholar 

  41. Schumaker, L.L.: Spline Functions: Basic Theory. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  42. Scott, M.A., Li, X., Sederberg, T.W., Hughes, T.J.: Local refinement of analysis-suitable T-splines. Comput. Methods Appl. Mech. Eng. 213–216, 206–222 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Scott, M.A., Simpson, R.N., Evans, J.A., Lipton, S., Bordas, S.P.A., Hughes, T.J., Sederberg, T.W.: Isogeometric boundary element analysis using unstructured T-splines. Comput. Methods Appl. Mech. Eng. 254, 197–221 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCSs. ACM Trans. Graph. 22(3), 477–484 (2003)

    Article  Google Scholar 

  45. 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)

    Article  Google Scholar 

  46. Sederberg, T.W., Finnigan, G.T., Li, X., Lin, H., Ipson, H.: Watertight trimmed NURBS. ACM Trans. Graph. 27, 1–8 (2008)

    Article  Google Scholar 

  47. Taus, M., Rodin, G.J., Hughes, T.J., Scott, M.A.: Isogeometric boundary element methods and patch tests for linear elastic problems: formulation, numerical integration, and applications. Comput. Methods Appl. Mech. Eng. 357, 112591 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  48. Verhoosel, C.V., Scott, M.A., de Borst, R., Hughes, T.J.: An isogeometric approach to cohesive zone modeling. Int. J. Numer. Meth. Eng. 87, 336–360 (2011)

    Article  MATH  Google Scholar 

  49. Wei, X., Zhang, Y.J., Toshniwal, D., Speleers, H., Li, X., Manni, C., Evans, J.A., Hughes, T.J.: Blended B-spline construction on unstructured quadrilateral and hexahedral meshes with optimal convergence rates in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 341, 609–639 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  50. Xu, J., Vilanova, G., Gomez, H.: Phase-field model of vascular tumor growth: Three-dimensional geometry of the vascular network and integration with imaging data. Comput. Methods Appl. Mech. Eng. 359, 112648 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhang, J., Li, X.: Local refinement of analysis-suitable++ T-splines. Comput. Methods Appl. Mech. Eng. 342, 32–45 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zimmermann, C., Toshniwal, D., Landis, C.M., Hughes, T.J., Mandadapu, K.K., Sauer, R.A.: An isogeometric finite element formulation for phase transitions on deforming surfaces. Comput. Methods Appl. Mech. Eng. 351, 441–477 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Xiliang Li is partially supported by the NSF of China (No.11971273) and the NSF of Shandong Province (ZR2018MA004). Xin Li is supported by the NSF of China (No.61872328), SRF for ROCS SE, and the Youth Innovation Promotion Association CAS.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, 264003, People’s Republic of China

    Xiliang Li

  2. School of Mathematical Science, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Xin Li

Authors
  1. Xiliang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Li, X. AS++ T-splines: arbitrary degree, nestedness and approximation. Numer. Math. 148, 795–816 (2021). https://doi.org/10.1007/s00211-021-01214-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-021-01214-7

Mathematics Subject Classification

Search

Navigation