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Parametric finite elements with bijective mappings

BIT Numerical Mathematics volume 57, pages 1185–1203 (2017)Cite this article

Abstract

The discretization of the computational domain plays a central role in the finite element method. In the standard discretization the domain is triangulated with a mesh and its boundary is approximated by a polygon. The boundary approximation induces a geometry-related error which influences the accuracy of the solution. To control this geometry-related error, iso-parametric finite elements and iso-geometric analysis allow for high order approximation of smooth boundary features. We present an alternative approach which combines parametric finite elements with smooth bijective mappings leaving the choice of approximation spaces free. Our approach allows to represent arbitrarily complex geometries on coarse meshes with curved edges, regardless of the domain boundary complexity. The main idea is to use a bijective mapping for automatically warping the volume of a simple parameterization domain to the complex computational domain, thus creating a curved mesh of the latter. Numerical examples provide evidence that our method has lower approximation error for domains with complex shapes than the standard finite element method, because we are able to solve the problem directly on the exact domain without having to approximate it.

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Acknowledgements

This work was supported by the SNF under Project Number 200020_156178, SCCER-SoE, and SCCER-FURIES. The method presented in this paper is implemented within the MOONoLith software library. We thank the anonymous reviewers for their useful comments which helped improving both content and presentation of this paper.

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Authors and Affiliations

  1. Università della Svizzera italiana, Via Giuseppe Buffi 13, 6900, Lugano, Switzerland

    Patrick Zulian, Teseo Schneider, Kai Hormann & Rolf Krause

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  1. Patrick Zulian
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  2. Teseo Schneider
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  3. Kai Hormann
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  4. Rolf Krause
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Correspondence to Patrick Zulian.

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Zulian, P., Schneider, T., Hormann, K. et al. Parametric finite elements with bijective mappings. Bit Numer Math 57, 1185–1203 (2017). https://doi.org/10.1007/s10543-017-0669-6

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  • DOI: https://doi.org/10.1007/s10543-017-0669-6

Keywords

  • Finite element method
  • Discretization
  • Barycentric coordinates
  • Bijective mapping
  • Super parametric
  • Parametrization

Mathematics Subject Classification

  • 35-XX
  • 51-XX
  • 65N30
  • 74-XX