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Simplex-Splines on the Clough-Tocher Split with Arbitrary Smoothness

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Geometric Challenges in Isogeometric Analysis

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

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Abstract

The space of piecewise polynomials of smoothness r and degree 3r is considered on the Clough-Tocher split of a triangle. For any \(r\ge 1\) we give a basis of simplex splines for this space, then a Marsden-like identity, which is proved explicitly for \(r\le 3\) and symbolically for \(4 \le r\le 6\). In addition, generalizing results for \(r=1\), we prove for \(r=2,3\) a geometry independent bound for the condition number in the infinity norm of this basis, and conditions to connect two triangles with smoothness r.

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References

  1. Alfeld, P.: A bivariate \(C^2\) Clough-Tocher scheme. Comput. Aided Geom. Des. 1, 257–267 (1984)

    Article  Google Scholar 

  2. Alfeld, P., Schumaker, L.L.: Smooth macro-elements based on Powell-Sabin triangle splits. Adv. Comput. Math. 16, 29–46 (2002)

    Article  MathSciNet  Google Scholar 

  3. Cao, J., Chen, Z., Wei, X., Zhang, Y.J.: A finite element framework based on bivariate simplex splines on triangle configurations. Comput. Methods Appl. Mech. Eng. 357, 112–598 (2019)

    Google Scholar 

  4. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)

    Book  Google Scholar 

  5. Clough, R.W., Tocher, J.L.: Finite element stiffness matrices for analysis of plates in bending. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics, pp. 515–545. Wright-Patterson Air Force Base (1965)

    Google Scholar 

  6. Cohen, E., Lyche, T., Riesenfeld, R.F.: A B-spline-like basis for the Powell-Sabin 12-split based on simplex splines. Math. Comput. 82, 1667–1707 (2013)

    Article  MathSciNet  Google Scholar 

  7. Cohen, E., Riesenfeld, R.F., Elber, G.: Geometric Modeling with Splines. A K Peters Ltd., Natick (2001)

    Book  Google Scholar 

  8. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley Publishing (2009)

    Google Scholar 

  9. Davydov, O., Yeo, W.P.: Refinable \(C^2\) piecewise quintic polynomials on Powell-Sabin-12 triangulations. J. Comput. Appl. Math. 240, 62–73 (2013)

    Article  MathSciNet  Google Scholar 

  10. Dierckx, P.: On calculating normalized Powell-Sabin B-splines. Comput. Aided Geom. Des. 15, 61–78 (1997)

    Article  MathSciNet  Google Scholar 

  11. Farin, G.: A modified Clough-Tocher interpolant. Comput. Aided Geom. Des. 2, 19–27 (1985)

    Article  MathSciNet  Google Scholar 

  12. Grošelj, J.: A normalized representation of super splines of arbitrary degree on Powell-Sabin triangulations. BIT Numer. Math. 56, 1257–1280 (2016)

    Article  MathSciNet  Google Scholar 

  13. Grošelj, J., Speleers, H.: Construction and analysis of cubic Powell-Sabin B-splines. Comput. Aided Geom. Des. 57, 1–22 (2017)

    Article  MathSciNet  Google Scholar 

  14. Laghchim-Lahlou, M., Sablonnière, P.: Triangular finite elements of HCT type and class \(C^p\). Adv. Comput. Math. 2, 101–122 (1994)

    Article  MathSciNet  Google Scholar 

  15. Lai, M.J., Schumaker, L.L.: Macro-elements and stable local bases for splines on Clough-Tocher triangulations. Numerische Mathematik 88, 105–119 (2001)

    Article  MathSciNet  Google Scholar 

  16. Lai, M.J., Schumaker, L.L.: Macro-elements and stable local bases for splines on Powell-Sabin triangulations. Math. Comput. 72, 335–354 (2003)

    Article  MathSciNet  Google Scholar 

  17. Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations, Encyclopedia of Mathematics and its Applications, vol. 110. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  18. Lyche, T., Merrien, J.L.: Simplex-splines on the Clough-Tocher element. Comput Aided Geom Des 65, 76–92 (2018)

    Article  MathSciNet  Google Scholar 

  19. Lyche, T., Muntingh, G.: Stable simplex spline bases for \(C^3\) quintics on the Powell-Sabin 12-split. Constr. Approx 45, 1–32 (2017)

    Article  MathSciNet  Google Scholar 

  20. Mann, S.: Cubic precision Clough-Tocher interpolation. Comput. Aided Geom. Des. 16, 85–88 (1999)

    Article  MathSciNet  Google Scholar 

  21. Micchelli, C.A.: On a numerically efficient method for computing multivariate \(B\)-splines. In: Schempp, W., Zeller, K., (eds.), Multivariate Approximation Theory, International Series of Numerical Mathematics, vol. 51, pp. 211–248. Birkhäuser, Basel-Boston (1979)

    Google Scholar 

  22. Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977)

    Article  MathSciNet  Google Scholar 

  23. Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Mathematics and Visualization, Springer, Berlin (2002)

    Book  Google Scholar 

  24. Sablonnière, P.: Composite finite elements of class \(C^k\). J. Comput. Appl. Math. 12–13, 541–550 (1985)

    Article  Google Scholar 

  25. Schumaker, L.L., Sorokina, T.: Smooth macro-elements on Powell-Sabin-12 splits. Math. Comput. 75, 711–726 (2006)

    Article  MathSciNet  Google Scholar 

  26. Speleers, H.: A normalized basis for quintic Powell-Sabin splines. Comput. Aided Geom. Des. 27, 438–457 (2010)

    Article  MathSciNet  Google Scholar 

  27. Speleers, H.: A normalized basis for reduced Clough-Tocher splines. Comput. Aided Geom. Des. 27, 700–712 (2010)

    Article  MathSciNet  Google Scholar 

  28. Speleers, H.: Construction of normalized B-splines for a family of smooth spline spaces over Powell-Sabin triangulations. Constr. Approx. 37, 41–72 (2013)

    Article  MathSciNet  Google Scholar 

  29. Speleers, H.: A family of smooth quasi-interpolants defined over Powell-Sabin triangulations. Constr. Approx. 41, 297–324 (2015)

    Article  MathSciNet  Google Scholar 

  30. Speleers, H.: A new B-spline representation for cubic splines over Powell-Sabin triangulations. Comput. Aided Geom. Des. 37, 42–56 (2015)

    Article  MathSciNet  Google Scholar 

  31. Speleers, H., Manni, C.: Optimizing domain parameterization in isogeometric analysis based on Powell-Sabin splines. J. Comput. Appl. Math. 289, 68–86 (2015)

    Article  MathSciNet  Google Scholar 

  32. Speleers, H., Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis with Powell-Sabin splines for advection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 221–222, 132–148 (2012)

    Article  MathSciNet  Google Scholar 

  33. Toshniwal, D., Hughes, T.J.R.: Polynomial splines of non-uniform degree on triangulations: Combinatorial bounds on the dimension. Comput. Aided Geom. Des. 75, 101–763 (2019)

    Google Scholar 

  34. Wang, C., Xia, S., Wang, X., Qian, X.: Isogeometric shape optimization on triangulations. Comput. Methods Appl. Mech. Eng. 331, 585–622 (2018)

    Article  MathSciNet  Google Scholar 

  35. Wang, R.H., Shi, X.Q.: \(S_{\mu +1}^{\mu }\) surface interpolations over triangulations. In: Law, A.G., Wang, C.L. (eds.), Approximation, Optimization and Computing: Theory and Applications, pp. 205–208. Elsevier Science Publishers B.V. (1990)

    Google Scholar 

  36. Ženíšek, A.: A general theorem on triangular finite \(C^{(m)}\)-elements. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974)

    Google Scholar 

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Acknowledgements

We thank the referees for their extremely thorough and careful review and for pointing out a flaw in the first version of the paper.

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Authors and Affiliations

  1. University of Oslo, Oslo, Norway

    Tom Lyche

  2. University of Rennes, INSA Rennes, CNRS, Rennes, France

    Jean-Louis Merrien

  3. University of Passau, Passau, Germany

    Tomas Sauer

Authors
  1. Tom Lyche
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  2. Jean-Louis Merrien
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  3. Tomas Sauer
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Corresponding author

Correspondence to Tom Lyche .

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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

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Lyche, T., Merrien, JL., Sauer, T. (2022). Simplex-Splines on the Clough-Tocher Split with Arbitrary Smoothness. 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_5

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