Skip to main content
SpringerLink
Log in

Isogeometric Analysis Using Basis Functions of Splines Spaces over Hierarchical T-meshes with High Level Differences

  • Published:
Communications in Mathematics and Statistics Aims and scope Submit manuscript

Abstract

By using the method of space mapping, basis functions of biquadratic polynomial spline spaces over the hierarchical T-meshes without limitation for level difference can be constructed. In this paper, the basis functions defined over hierarchical T-meshes with high level differences are adopted for the application in the isogeometric analysis problems with rapidly changing local features. Without subdividing redundant cells to ensure the level difference of the adjacent cells, the refinement becomes more local, and fewer cells are subdivided for each refinement of the hierarchical T-mesh. Therefore, the dimension of the biquadratic polynomial spline space over the hierarchical T-mesh can be reduced, the superfluous control points or coefficients can be avoided, and the quantity of calculations can be decreased. Numerical examples show that these basis functions can work well on physical domains with different boundaries for the application in IGA.

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Cottrell, J., Huges, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009)

    Book  MATH  Google Scholar 

  3. Deng, J., Chen, F., Feng, Y.: Dimensions of spline spaces over T-meshes. J. Comput. Appl. Math. 194(2), 267–283 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deng, J., Chen, F., Li, X., Changqi, H., Tong, W., Yang, Z., Feng, Y.: Polynomial splines over hierarchical T-meshes. Graph. Models 70(4), 76–86 (2008)

    Article  Google Scholar 

  5. Deng, J., Chen, F., Jin, L.: Dimensions of biquadratic spline spaces over T-meshes. J. Comput. Appl. Math. 238, 68–94 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deng, F., Zeng, C., Meng, W., Deng, J.: Bases of biquadratic polynomial spline spaces over hierarchical T-meshes. J. Comput. Math. 35(1), 91–120 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Des. 30(3), 331–356 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dörfel, M.R., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local \(h\)-refinement with T-splines. Comput. Methods Appl. Mech. Eng. 199(5–8), 264–275 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MATH  Google Scholar 

  10. Forsey, D.R., Bartels, R.H.: Hierarchical B-spline refinement. ACM Siggraph Comput. Graph. 22(4), 205–212 (1988)

    Article  Google Scholar 

  11. Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29(7), 485–498 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comput. Math. 40(2), 459–490 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Giannelli, C., Jüttler, B., Kleiss, S.K., Mantzaflaris, A., Simeon, B., Špeh, J.: THB-splines: an effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis. Comput. Methods Appl. Mech. Eng. 299, 337–365 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Greiner, G., Hormann, K.: Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines. In: Surface Fitting and Multiresolution Methods, vol 1, pp. 163–172. Vanderbilt University (1996)

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Johannessen, K.A., Kvamsdal, T., Dokken, T.: Isogeometric analysis using LR B-splines. Comput. Methods Appl. Mech. Eng. 269, 471–514 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kiss, G., Giannelli, C., Zore, U., Jüttler, B., Großmann, D., Barner, J.: Adaptive CAD model (re-)construction with THB-splines. Graph. Models 76(5), 273–288 (2014)

    Article  Google Scholar 

  18. Kleiss, S.K., Jüttler, B., Zulehner, W.: Enhancing isogeometric analysis by a finite element-based local refinement strategy. Comput. Methods Appl. Mech. Eng. 213, 168–182 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, H., Yang, Y., Liu, Y., Xiaoming, F.: Simultaneous interior and boundary optimization of volumetric domain parameterizations for IGA. Comput. Aided Geom. Des. 79, 101851 (2020)

    Article  MathSciNet  Google Scholar 

  22. Liu, J., Deng, F., Deng, J.: Space mapping of spline spaces over hierarchical T-meshes. arXiv:2103.11688

  23. Meng, W., Deng, J., Chen, F.: Dimension of spline spaces with highest order smoothness over hierarchical T-meshes. Comput. Aided Geom. Des. 30(1), 20–34 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Marussig, B., Hiemstra, R., Hughes, T.J.R.: Improved conditioning of isogeometric analysis matrices for trimmed geometries. Comput. Methods Appl. Mech. Eng. 334, 79–110 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ni, Q., Wang, X., Deng, J.: Modified PHT-splines. Comput. Aided Geom. Des. 73, 37–53 (2019)

    Article  Google Scholar 

  26. Ni, Q., Wang, X., Deng, J.: Modified basis functions for MPHT-splines. J. Comput. Appl. Math. 375, 112817 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schillinger, D., Dede, 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. Eng. 249, 116–150 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. 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 

  30. Speleers, H.: Hierarchical spline spaces: quasi-interpolants and local approximation estimates. Adv. Comput. Math. 43(2), 235–255 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  32. Speleers, H., Manni, C.: Effortless quasi-interpolation in hierarchical spaces. Numer. Math. 132(1), 155–184 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tian, L., Chen, F., Qiang, D.: Adaptive finite element methods for elliptic equations over hierarchical T-meshes. J. Comput. Appl. Math. 236(5), 878–891 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Vuong, A.-V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200(49–52), 3554–3567 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, P., Jinlan, X., Deng, J., Chen, F.: Adaptive isogeometric analysis using rational PHT-splines. Comput. Aided Des. 43(11), 1438–1448 (2011)

    Article  Google Scholar 

  36. Wang, J., Yang, Z., Jin, L., Deng, J., Chen, F.: Adaptive surface reconstruction based on implicit PHT-splines. In: Proceedings of the 14th ACM Symposium on Solid and Physical Modeling

  37. Zeng, C., Deng, F., Li, X., Deng, J.: Dimensions of biquadratic and bicubic spline spaces over hierarchical T-meshes. J. Comput. Appl. Math. 287, 162–178 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zeng, C., Deng, F., Deng, J.: Bicubic hierarchical B-splines: dimensions, completeness, and bases. Comput. Aided Geom. Des. 38, 1–23 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are supported by the NSF of China (Nos. 11771420, 12001197).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Hefei Normal University, Hefei, 230601, Anhui, People’s Republic of China

    Jingjing Liu

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

    Jingjing Liu, Huanhuan Ma & Jiansong Deng

  3. School of Mathematics and Statistics, North China University of Water Resources and Electric Power, ZhengZhou, 450045, Henan, People’s Republic of China

    Fang Deng

Authors
  1. Jingjing Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fang Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huanhuan Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiansong Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiansong Deng.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Deng, F., Ma, H. et al. Isogeometric Analysis Using Basis Functions of Splines Spaces over Hierarchical T-meshes with High Level Differences. Commun. Math. Stat. (2023). https://doi.org/10.1007/s40304-022-00324-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40304-022-00324-4

Keywords

Mathematics Subject Classification

Search

Navigation