SpringerLink
Log in

THU-Splines: Highly Localized Refinement on Smooth Unstructured Splines

  • Conference paper
  • First Online:
Geometric Challenges in Isogeometric Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 49))

Abstract

We present a novel method named truncated hierarchical unstructured splines (THU-splines) that supports both local h-refinement and unstructured quadrilateral meshes. In a THU-spline construction, an unstructured quadrilateral mesh is taken as the input control mesh, where the degenerated-patch method [20] is adopted in irregular regions to define \(C^1\)-continuous bicubic splines, whereas regular regions only involve \(C^2\) B-splines. Irregular regions are then smoothly joined with regular regions through the truncation mechanism [31], leading to a globally smooth spline construction. Subsequently, local refinement is performed following the truncated hierarchical B-spline construction [11] to achieve a flexible refinement without propagating to unanticipated regions. Challenges lie in refining transition regions where a mixed types of splines play a role. THU-spline basis functions are globally \(C^1\)-continuous and are non-negative everywhere except near extraordinary vertices, where slight negativity is inevitable to retain refinability of the spline functions defined using the degenerated-patch method. Such functions also have a finite representation that can be easily integrated with existing finite element or isogeometric codes through Bézier extraction.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (France)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 128.39
Price includes VAT (France)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 168.79
Price includes VAT (France)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
EUR 168.79
Price includes VAT (France)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    An extraordinary vertex is an interior vertex shared by other than four quadrilateral faces.

  2. 2.

    We use “vertex”, “edge” and “face” to emphasize mesh connectivity (or topology), whereas using “point” to carry the position or geometry information. “Element” and “face” are used interchangeably.

  3. 3.

    We only focus on methods based on quadrilaterals.

References

  1. 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, 112,919 (2020)

    Google Scholar 

  2. 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  Google Scholar 

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

    Google Scholar 

  4. Bercovier, M., Matskewich, T.: Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes. Springer International Publishing (2017)

    Google Scholar 

  5. Boehm, W.: Inserting new knots into B-spline curves. Comput. Aided Des. 12, 199–201 (1980)

    Article  Google Scholar 

  6. Boehm, W., Müller, A.: On de Casteljau’s algorithm. Comput. Aided Geomet. Design 16, 587–605 (1999)

    Article  MathSciNet  Google Scholar 

  7. Bornermann, P.B., Cirak, F.: A subdivision-based implementation of the hierarchical B-spline finite element method. Comput. Methods Appl. Mech. Eng. 253, 584–598 (2013)

    Article  MathSciNet  Google Scholar 

  8. Casquero, H., Wei, X., Toshniwal, D., Li, A., Hughes, T.J.R., Kiendl, J., Zhang, Y.: Seamless integration of design and Kirchhoff-Love shell analysis using analysis-suitable unstructured T-splines. Comput. Methods Appl. Mech. Eng. 360, 112,765 (2020)

    Google Scholar 

  9. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogemetric Analysis: Toward Integration of CAD and FEA. Wiley, New York (2009)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. 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, 4135–4195 (2005)

    Article  MathSciNet  Google Scholar 

  13. Kapl, M., Vitrih, V., Jüttler, B., Birner, K.: Isogeometric analysis with geometrically continuous functions on two-patch geometries. Comput. Math. Appl. 70, 1518–1538 (2015)

    Article  MathSciNet  Google Scholar 

  14. Kraft, R.: Adaptive and linearly independent multilevel B-splines. In: Méhauté, A.L., Rabut, C., Schumaker, L.L., (eds.) Surface Fitting and Multiresolution Methods, pp. 209–218. Vanderbilt University Press (1997)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. 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  Google Scholar 

  17. Lyche, T., Manni, C., Speleers, H.: Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In: Lyche, T., Manni, C., Speleers, H. (eds.), Splines and PDEs: From Approximation Theory to Numerical Linear Algebra. Lecture Notes in Mathematics, vol. 2219, pp. 1–76. Springer International Publishing (2018)

    Google Scholar 

  18. Majeed, M., Cirak, F.: Isogeometric analysis using manifold-based smooth basis functions. Comput. Methods Appl. Mech. Eng. 316, 547–567 (2017)

    Article  MathSciNet  Google Scholar 

  19. Nguyen, T., Peters, J.: Refinable \(C^1\) spline elements for irregular quad layout. Comput. Aided Geometr. Design 43, 123–130 (2016)

    Article  Google Scholar 

  20. Reif, U.: A refineable space of smooth spline surfaces of arbitrary topological genus. J. Approx. Theory 90, 174–199 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Scott, M.A., Thomas, D.C., Evans, E.J.: Isogeometric spline forests. Comput. Methods Appl. Mech. Eng. 269, 222–264 (2014)

    Article  MathSciNet  Google Scholar 

  24. 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, 276–283 (2004)

    Article  Google Scholar 

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

    Article  Google Scholar 

  26. Toshniwal, D., Speleers, H., Hughes, T.J.R.: Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: geometric design and isogeometric analysis considerations. Comput. Methods Appl. Mech. Eng. 327, 411–458 (2017)

    Article  MathSciNet  Google Scholar 

  27. 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, 3554–3567 (2011)

    Article  MathSciNet  Google Scholar 

  28. Wei, X., Li, X., Zhang, Y., Hughes, T.J.R.: Tuned hybrid nonuniform subdivision surfaces with optimal convergence rates. Int. J. Numer. Meth. Eng. 122, 2117–2144 (2021)

    Article  MathSciNet  Google Scholar 

  29. Wei, X., Zhang, Y., Hughes, T.J.R., Scott, M.A.: Truncated hierarchical Catmull-Clark subdivision with local refinement. Comput. Methods Appl. Mech. Eng. 291, 1–20 (2015)

    Article  MathSciNet  Google Scholar 

  30. Wei, X., Zhang, Y., Hughes, T.J.R., Scott, M.A.: Extended truncated hierarchical Catmull-Clark subdivision. Comput. Methods Appl. Mech. Eng. 299, 316–336 (2016)

    Article  MathSciNet  Google Scholar 

  31. Wei, X., Zhang, Y., Toshniwal, D., Speleers, H., Li, X., Manni, C., Evans, J.A., Hughes, T.J.R.: 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  Google Scholar 

Download references

Acknowledgements

X. Wei is partially supported by the ERC AdG project CHANGE n. 694515, as well as the Swiss National Science Foundation project HOGAEMS n.200021_188589.

Author information

Authors and Affiliations

  1. Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    **aodong Wei

Authors
  1. **aodong Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to **aodong Wei .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Carla Manni

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Hendrik Speleers

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wei, X. (2022). THU-Splines: Highly Localized Refinement on Smooth Unstructured Splines. In: Manni, C., Speleers, H. (eds) Geometric Challenges in Isogeometric Analysis. Springer INdAM Series, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-92313-6_13

Download citation

Publish with us

Policies and ethics

Search

Navigation