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Local Hierarchical h-Refinements in IgA Based on Generalized B-Splines

  • Conference paper
Mathematical Methods for Curves and Surfaces (MMCS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8177))

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Abstract

In this paper we construct multilevel representations in terms of a hierarchy of tensor-product generalized B-splines. These representations combine the positive properties of a non-rational model with the possibility of dealing with local refinements. We discuss their use in the context of isogeometric analysis.

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References

  1. Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. de Boor, C.: A Practical Guide to Splines, revised edn. Springer (2001)

    Google Scholar 

  4. Carnicer, J.M., Mainar, E., Peña, J.M.: Critical length for design purposes and Extended Chebyshev spaces. Constr. Approx. 20, 55–71 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Costantini, P.: Curve and surface construction using variable degree polynomial splines. Comput. Aided Geom. Design 17, 419–446 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Costantini, P., Lyche, T., Manni, C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Costantini, P., Manni, C.: Geometric construction of generalized cubic splines. Rend. Matem. Appl. 26, 327–338 (2006)

    MATH  MathSciNet  Google Scholar 

  8. Costantini, P., Manni, C., Pelosi, F., Sampoli, M.L.: Quasi-interpolation in isogeometric analysis based on generalized B-splines. Comput. Aided Geom. Design 27, 656–668 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons (2009)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comput. Math. (to appear, 2013)

    Google Scholar 

  15. Gould, P.L.: Introduction to Linear Elasticity. Springer, Berlin (1999)

    Google Scholar 

  16. Grinspun, E., Krysl, P., Schröder, P.: CHARMS: a simple framework for adaptive simulation. ACM Trans. Graphics 21, 281–290 (2002)

    Article  Google Scholar 

  17. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg. 194, 4135–4195 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Koch, P.E., Lyche, T.: Interpolation with exponential B-splines in tension. In: Farin, G., Hagen, H., Noltemeier, H., Knödel, W. (eds.) Geometric Modelling, pp. 173–190. Springer (1993)

    Google Scholar 

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

    Google Scholar 

  20. Krysl, P., Grinspun, E., Schröder, P.: Natural hierarchical refinement for finite element methods. Int. J. Numer. Meth. Eng. 56, 1109–1124 (2003)

    Article  MATH  Google Scholar 

  21. Kvasov, B.I., Sattayatham, P.: GB-splines of arbitrary order. J. Comput. Appl. Math. 104, 63–88 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mainar, E., Peña, J.M., Sánchez-Reyes, J.: Shape preserving alternatives to the rational Bézier model. Comput. Aided Geom. Design 18, 37–60 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Manni, C., Pelosi, F., Sampoli, M.L.: Generalized B-splines as a tool in isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 200, 867–881 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis in advection-diffusion problems: tension splines approximation. J. Comput. Appl. Math. 236, 511–528 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Marušic, M., Rogina, M.: Sharp error bounds for interpolating splines in tension. J. Comput. Appl. Math. 61, 205–223 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mazure, M.L.: Chebyshev-Bernstein bases. Comput. Aided Geom. Design 16, 649–669 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mazure, M.L.: How to build all Chebyshevian spline spaces good for geometric design? Numer. Math. 119, 517–556 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  28. Schumaker, L.L.: Spline Functions: Basic Theory, 3rd edn., Cambridge U.P. (2007)

    Google Scholar 

  29. Speleers, H., Dierckx, P., Vandewalle, S.: Quasi-hierarchical Powell-Sabin B-splines. Comput. Aided Geom. Design 26, 174–191 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Speleers, H., Dierckx, P., Vandewalle, S.: On the local approximation power of quasi-hierarchical Powell-Sabin splines. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, J.-L., Mørken, K., Schumaker, L.L. (eds.) MMCS 2008. LNCS, vol. 5862, pp. 419–433. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  31. Speleers, H., Manni, C., Pelosi, F.: From NURBS to NURPS geometries. Comput. Methods Appl. Mech. Engrg. 255, 238–254 (2013)

    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. Engrg. 221–222, 132–148 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  34. Wang, G., Fang, M.: Unified and extended form of three types of splines. J. Comput. Appl. Math. 216, 498–508 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Roma, Italy

    Carla Manni & Francesca Pelosi

  2. Departement Computerwetenschappen, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001, Leuven, Belgium

    Hendrik Speleers

Authors
  1. Carla Manni
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  2. Francesca Pelosi
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  3. Hendrik Speleers
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Editor information

Editors and Affiliations

  1. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Norway

    Michael Floater

  2. DMA, University of Oslo, P.O.Box 1053, 0316, Blindern, Oslo, Norway

    Tom Lyche

  3. Laboratoire LJK, Université Joseph Fourier, BP 53, 38041, Grenoble Cedex 9, France

    Marie-Laurence Mazure

  4. University of Oslo, Norway

    Knut Mørken

  5. Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, USA

    Larry L. Schumaker

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Manni, C., Pelosi, F., Speleers, H. (2014). Local Hierarchical h-Refinements in IgA Based on Generalized B-Splines. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol 8177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54382-1_20

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  • DOI: https://doi.org/10.1007/978-3-642-54382-1_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-54381-4

  • Online ISBN: 978-3-642-54382-1

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