Local and Adaptive Refinement with Hierarchical B-splines

Carlotta Giannelli; Bert Jüttler

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 3, page 735-740
  • ISSN: 0392-4041

Abstract

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Adaptive spline models for geometric modeling and spline-based PDEs solvers have recently attracted increasing attention both in the context of computer aided geometric design and isogeometric analysis. In particular, approximation spaces defined over extensions of tensor-product meshes which allow axis aligned segments with T-junctions are currently receiving particular attention. In this short paper we review some recent results concerning the characterization of the space spanned by the hierarchical B-spline basis. In addition, we formulate a refinement algorithm which allows us to satisfy the conditions needed for this characterization.

How to cite

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Giannelli, Carlotta, and Jüttler, Bert. "Local and Adaptive Refinement with Hierarchical B-splines." Bollettino dell'Unione Matematica Italiana 6.3 (2013): 735-740. <http://eudml.org/doc/294045>.

@article{Giannelli2013,
abstract = {Adaptive spline models for geometric modeling and spline-based PDEs solvers have recently attracted increasing attention both in the context of computer aided geometric design and isogeometric analysis. In particular, approximation spaces defined over extensions of tensor-product meshes which allow axis aligned segments with T-junctions are currently receiving particular attention. In this short paper we review some recent results concerning the characterization of the space spanned by the hierarchical B-spline basis. In addition, we formulate a refinement algorithm which allows us to satisfy the conditions needed for this characterization.},
author = {Giannelli, Carlotta, Jüttler, Bert},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {735-740},
publisher = {Unione Matematica Italiana},
title = {Local and Adaptive Refinement with Hierarchical B-splines},
url = {http://eudml.org/doc/294045},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Giannelli, Carlotta
AU - Jüttler, Bert
TI - Local and Adaptive Refinement with Hierarchical B-splines
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/10//
PB - Unione Matematica Italiana
VL - 6
IS - 3
SP - 735
EP - 740
AB - Adaptive spline models for geometric modeling and spline-based PDEs solvers have recently attracted increasing attention both in the context of computer aided geometric design and isogeometric analysis. In particular, approximation spaces defined over extensions of tensor-product meshes which allow axis aligned segments with T-junctions are currently receiving particular attention. In this short paper we review some recent results concerning the characterization of the space spanned by the hierarchical B-spline basis. In addition, we formulate a refinement algorithm which allows us to satisfy the conditions needed for this characterization.
LA - eng
UR - http://eudml.org/doc/294045
ER -

References

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  1. BUFFA, A., CHO, D. and SANGALLI, G., Linear independence of the T-spline blending functions associated with some particular T-meshes, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1437-1445. Zbl1231.65027MR2630153DOI10.1016/j.cma.2009.12.004
  2. DÖRFEL, M.R., JÜTTLER, B. and SIMEON, B., Adaptive Isogeometric Analysis by Local h-Refinement with T-splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 264-275. MR2576760DOI10.1016/j.cma.2008.07.012
  3. FORSEY, D.R. and BARTELS, R.H., Hierarchical B-spline refinement, Comput. Graphics, 22 (1988), 205-212. 
  4. FORSEY, D.R. and BARTELS, R.H., Surface fitting with hierarchical splines, ACM Trans. Graphics, 14 (1995), 134-161. 
  5. GIANNELLI, C., JÜTTLER, B. and SPELEERS, H., THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Design, 29 (2012), 485-498. Zbl1252.65030MR2925951DOI10.1016/j.cagd.2012.03.025
  6. GIANNELLI, C., JÜTTLER, B. and SPELEERS, H., Strongly stable bases for adaptively refined multilevel spline spaces, Adv. Comp. Math., to appear (2013). MR3194714DOI10.1007/s10444-013-9315-2
  7. GIANNELLI, C. and JÜTTLER, B., Bases and dimensions of bivariate hierarchical tensor-product splines, J. Comput. Appl. Math., 239 (2013), 162-178. Zbl1259.41014MR2991965DOI10.1016/j.cam.2012.09.031
  8. HUGHES, T.J.R., COTTRELL, J.A. and BAZILEVS, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4135-4195. Zbl1151.74419MR2152382DOI10.1016/j.cma.2004.10.008
  9. KRAFT, R., Adaptive and linearly independent multilevel B-splines, in: LE MÉHAUTÉ A., RABUT C. and SCHUMAKER L.L. (Eds.), Surface Fitting and Multiresolution Methods, Vanderbilt University Press, Nashville (1997), 209-218. Zbl0937.65014MR1659975
  10. KRAFT, R., Adaptive und linear unabhängige Multilevel B-Splines und ihre Anwendungen, PhD Thesis, Universität Stuttgart (1998). Zbl0903.68195MR1641654
  11. VUONG, A.-V., GIANNELLI, C., JÜTTLER, B. and SIMEON, B., A hierarchical approach to adaptive local refinement in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 200 (2011), 3554-3567. Zbl1239.65013MR2851579DOI10.1016/j.cma.2011.09.004

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