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Manifold-Based B-Splines on Unstructured Meshes

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Isogeometric Analysis and Applications 2018 (IGAA 2018)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 133))

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Abstract

We introduce new manifold-based splines that are able to exactly reproduce B-splines on unstructured surface meshes. Such splines can be used in isogeometric analysis (IGA) to represent smooth surfaces of arbitrary topology. Since prevalent computer-aided design (CAD) models are composed of tensor-product B-spline patches, any IGA suitable construction should be able to reproduce B-splines. To achieve this goal, we focus on univariate manifold-based constructions that can reproduce B-splines. The manifold-based splines are constructed by smoothly blending together polynomial interpolants defined on overlapping charts. The proposed constructions are able to reproduce B-splines in regular parts of the mesh, with no extraordinary vertices, and polynomial basis functions in the remaining parts of the mesh. We study and compare analytically and numerically the finite element convergence of several univariate constructions. The obtained results directly carry over to the tensor-product case.

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Notes

  1. 1.

    The index s for the reference element is usually dropped because all of them can be assumed to have the same domain.

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Acknowledgements

The authors would like to acknowledge the kind hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), where some of this research was carried out as part of the thematic programme Numerical Analysis of Complex PDE Models in the Sciences.

Author information

Authors and Affiliations

  1. Department of Engineering, University of Cambridge, Cambridge, UK

    Qiaoling Zhang & Fehmi Cirak

  2. Institute of Applied Geometry, Johannes Kepler University, Linz, Austria

    Thomas Takacs

Authors
  1. Qiaoling Zhang
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  2. Thomas Takacs
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  3. Fehmi Cirak
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Corresponding author

Correspondence to Fehmi Cirak .

Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Harald van Brummelen

  2. Department of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Cornelis Vuik

  3. Department of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Matthias Möller

  4. Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Clemens Verhoosel

  5. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Bernd Simeon

  6. Institut für Angewandte Geometrie, Johannes Kepler Universität Linz, Linz, Austria

    Bert Jüttler

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Zhang, Q., Takacs, T., Cirak, F. (2021). Manifold-Based B-Splines on Unstructured Meshes. In: van Brummelen, H., Vuik, C., Möller, M., Verhoosel, C., Simeon, B., Jüttler, B. (eds) Isogeometric Analysis and Applications 2018. IGAA 2018. Lecture Notes in Computational Science and Engineering, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-030-49836-8_12

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