Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

An Adaptive Collocation Method with Weighted Extended PHT-Splines

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper presents an adaptive collocation method with weighted extended PHT-splines. The authors modify the classification rules for basis functions based on the relation between the basis vertices and the computational domain. The Gaussian points are chosen to be collocation points since PHT-splines are C1 continuous. The authors also provide relocation techniques to resolve the mismatch problem between the number of basis functions and the number of interpolation conditions. Compared to the traditional Greville collocation method, the new approach has improved accuracy with fewer oscillations. Several numerical examples are also provided to test our the proposed approach.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hughes T J, Cottrell J A, and Bazilevs Y, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195.

    MathSciNet  MATH  Google Scholar 

  2. Bazilevs Y, Calo V M, Cottrell J A, et al., Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5–8): 229–263.

    MathSciNet  MATH  Google Scholar 

  3. Dörfel M, Jüttler B, and Simeon B, Adaptive isogeometric analysis by local h-refinement with T-splines, Computer Methods in Applied Mechanics and Engineering, 2010, 199(5–8): 264–275.

    MathSciNet  MATH  Google Scholar 

  4. Li X, Zheng J, Sederberg T W, et al., On the linear independence of T-splines blending functions, Computer Aided Geometric Design, 2012, 29(1): 63–76.

    MathSciNet  MATH  Google Scholar 

  5. Li X and Scott M, Analysis-suitable T-splines: Characterization, refineability, and approximation, Mathematical Models and Methods in Applied Sciences, 2014, 24(06): 1141–1164.

    MathSciNet  MATH  Google Scholar 

  6. Scott M A, Li X, Sederberg T W, et al., Local refinement of analysis-suitable T-splines, Computer Methods in Applied Mechanics and Engineering, 2012, 213–216(1): 206–222.

    MathSciNet  MATH  Google Scholar 

  7. Li X and Zhang J, AS++ T-splines: Linear independence and approximation, Computer Methods in Applied Mechanics and Engineering, 2018, 333(1): 462–474.

    MathSciNet  MATH  Google Scholar 

  8. Deng J, Chen F, Li X, et al., Polynomial splines over hierarchical T-meshes, Graphical models, 2008, 70(4): 76–86.

    Google Scholar 

  9. Nguyen-Thanh N, Nguyen-Xuan H, Bordas S P, et al., Isogeometric analysisusing polynomial splines over hierarchical T-meshes for two-dimensional elastic solids, Computer Methods in Applied Mechanics and Engineering, 2011, 200(21–22): 1892–1908.

    MathSciNet  MATH  Google Scholar 

  10. Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, et al., Rotation free isogeometric thin shell analysis using PHT-splines, Computer Methods in Applied Mechanics and Engineering, 2011, 200(47–48): 3410–3424.

    MathSciNet  MATH  Google Scholar 

  11. Wang P, Xu J, Deng J, et al., Adaptive isogeometric analysis using rational PHT-splines, Computer-Aided Design, 2011, 43(11): 1438–1448.

    Google Scholar 

  12. Ni Q, Wang X, and Deng J, Modified PHT-splines, Computer Aided Geometric Design, 2019, 73: 37–53.

    MathSciNet  MATH  Google Scholar 

  13. Vuong A-V, Giannelli C, Jüttler B, et al., A hierarchical approach to adaptive local refinement in isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 2011, 200(49–52): 3554–3567.

    MathSciNet  MATH  Google Scholar 

  14. Evans E J, Scott M A, Li X, et al., Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 2015, 284(1): 1–20.

    MathSciNet  MATH  Google Scholar 

  15. Schillinger D, Dedé L, Scott M A, et al., An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Computer Methods in Applied Mechanics and Engineering, 2012, 249–252(1): 116–150.

    MathSciNet  MATH  Google Scholar 

  16. Giannelli C, Jüttler B, Kleiss S K, et al., THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 2016, 299(1): 337–365.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Johannessen K A, Kvamsdal T, and Dokken T, Isogeometric analysis using LR B-splines, Computer Methods in Applied Mechanics and Engineering, 2014, 269(1): 471–514.

    MathSciNet  MATH  Google Scholar 

  19. Xu G, Sun N, Xu J, et al., A unified approach to construct generalized B-Splines for isogeometric applications, Journal of System Science and Complexity, 2017, 30: 983–998.

    MathSciNet  MATH  Google Scholar 

  20. Cohen E, Martin T, Kirby R M, et al., Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 2010, 199(5–8): 334–356.

    MathSciNet  MATH  Google Scholar 

  21. Nian X and Chen F, Planar domain parameterization for isogeometric analysis based on Teichmller mapping, Computer Methods in Applied Mechanics and Engineering, 2016, 311(1): 41–55.

    MathSciNet  MATH  Google Scholar 

  22. Xu G, Mourrain B, Duvigneau R, et al., Analysis-suitable volume parameterization of multiblock computational domain in isogeometric applications, Computer-Aided Design, 2013, 45(2): 395–404.

    MathSciNet  Google Scholar 

  23. Takacs T and Jüttler B, Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 2011, 200(49–52): 3568–3582.

    MathSciNet  MATH  Google Scholar 

  24. Takacs T and Jüttler B, H2 regularity properties of singular parameterizations in isogeometric analysis, Graphical models, 2012, 74(6): 361–372.

    Google Scholar 

  25. Wu M and Wang X, A H1-integrability condition of surfaces with singular parametrizations in isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 2018, 332(15): 136–156.

    MathSciNet  MATH  Google Scholar 

  26. Höllig K, Reif U, and Wipper J, Weighted extended B-spline approximation of Dirichlet problems, SIAM Journal on Numerical Analysis, 2001, 39(2): 442–462.

    MathSciNet  MATH  Google Scholar 

  27. Liu L, Zhang Y, and Wei X, Weighted T-splines with application in reparameterizing trimmed NURBS surfaces, Computer Methods in Applied Mechanics and Engineering, 2015, 295(1): 108–126.

    MathSciNet  MATH  Google Scholar 

  28. Marussig B, Zechner J, Beer G, et al., Stable isogeometric analysis of trimmed geometries, Computer Methods in Applied Mechanics and Engineering, 2017, 316(1): 497–521.

    MathSciNet  MATH  Google Scholar 

  29. Gautschi W and Notaris S E, Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type, Journal of Computational and Applied Mathematics, 1989, 25(2): 199–224.

    MathSciNet  MATH  Google Scholar 

  30. Yang T, Qarariyah A, Kang H, et al., Numerical integration over implicitly defined domains with topological guarantee, Communications in Mathematics and Statistics, 2019, 7: 459–474.

    MathSciNet  MATH  Google Scholar 

  31. Anitescu C, Jia Y, Zhang Y J, et al., An isogeometric collocation method using superconvergent points, Computer Methods in Applied Mechanics and Engineering, 2015, 284(1): 1073–1097.

    MathSciNet  MATH  Google Scholar 

  32. Kiendl J, Marino E, and Lorenzis D L, Isogeometric collocation for the Reissner-Mindlin shell problem, Computer Methods in Applied Mechanics and Engineering, 2017, 325(1): 645–665.

    MathSciNet  MATH  Google Scholar 

  33. Manni C, Reali A, and Speleers H, Isogeometric collocation methods with generalized B-splines, Computers & Mathematics with Applications, 2015, 70(7): 1659–1675.

    MathSciNet  MATH  Google Scholar 

  34. Lin H, Hu Q, and Xiong Y, Consistency and convergence properties of the isogeometric collocation method, Computer Methods in Applied Mechanics and Engineering, 2013, 267(1): 471–486.

    MathSciNet  MATH  Google Scholar 

  35. Jia Y, Anitescu C, Zhang Y J, et al., An adaptive isogeometric analysis collocation method with a recovery-based error estimator, Computer Methods in Applied Mechanics and Engineering, 2019, 345(1): 52–74.

    MathSciNet  MATH  Google Scholar 

  36. Deng J, Chen F, and Feng Y, Dimensions of spline spaces over T-meshes, Journal of Computational and Applied Mathematics, 2006, 194(2): 267–283.

    MathSciNet  MATH  Google Scholar 

  37. Sederberg T W, Zheng J, Bakenov A, et al., T-splines and T-NURCCs, ACM Transactions on Graphics, 2003, 22(3): 477–484.

    Google Scholar 

  38. Gomes A, Voiculescu I Jorge J, et al., Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, Springer Science & Business Media, London, 2009.

    MATH  Google Scholar 

  39. Apprich C, Höllig K, Hörner J, et al., Collocation with WEB-splines, Advances in Computational Mathematics, 2016, 42(4): 823–842.

    MathSciNet  MATH  Google Scholar 

  40. Qarariyah A, Deng F, Yang T, et al., Isogeometric analysis on implicit domains using weighted extended PHT-splines, Journal of Computational and Applied Mathematics, 2019, 350: 353–371.

    MathSciNet  MATH  Google Scholar 

  41. Höllig K, Finite Element Methods with B-Splines, SIAM, Philadelphia, 2003.

    MATH  Google Scholar 

  42. De Boor C, A Practical Guide to Splines, Springer-Verlag, New York, 1978.

    MATH  Google Scholar 

  43. Piegl L and Tiller W, The NURBS Book, Springer Science & Business Media, Berlin, 2012.

    MATH  Google Scholar 

  44. Auricchio F, Da Veiga L B, Hughes T J, et al., Isogeometric collocation methods, Mathematical Models and Methods in Applied Sciences, 2010, 20(11): 2075–2107.

    MathSciNet  MATH  Google Scholar 

  45. Gomez H and Lorenzis D L, The variational collocation method, Computer Methods in Applied Mechanics and Engineering, 2016, 309(1): 152–181.

    MathSciNet  MATH  Google Scholar 

  46. De Boor C and Swartz B, Collocation at Gaussian points, SIAM Journal on Numerical Analysis, 1973, 10(4): 582–606.

    MathSciNet  MATH  Google Scholar 

  47. Dörfler W, A convergent adaptive algorithm for Poisson’s equation, SIAM Journal on Numerical Analysis, 1996, 33(3): 1106–1124.

    MathSciNet  MATH  Google Scholar 

  48. Cheney E W and Light W A, A Course in Approximation Theory, American Mathematical Society, Providence, Rhode Island, 2009.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Qian Ni & Jiansong Deng

  2. School of Mathematics, Hefei University of Technology, Hefei, 230009, China

    Xuhui Wang

Authors
  1. Qian Ni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiansong Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuhui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuhui Wang.

Additional information

This research was supported by the National Natural Science Fondation of China under Grant Nos. 11601114, 11771420, 61772167.

This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ni, Q., Deng, J. & Wang, X. An Adaptive Collocation Method with Weighted Extended PHT-Splines. J Syst Sci Complex 34, 47–67 (2021). https://doi.org/10.1007/s11424-020-9390-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-020-9390-7

Keywords

Search

Navigation