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Numerical Integration on Hyperrectangles in Isoparametric Unfitted Finite Elements

  • Fabian Heimann
  • Christoph LehrenfeldEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

We consider the recently introduced idea of isoparametric unfitted finite element methods and extend it from simplicial meshes to quadrilateral and hexahedral meshes. The concept of the isoparametric unfitted finite element method is the construction of a mapping from a reference configuration to a higher order accurate configuration where the reference configuration is much more accessible for higher order quadrature. The mapping is based on a level set description of the geometry and the reference configuration is a lowest order level set approximation. On simplices this results in a piecewise planar and continuous approximation of the interface. With a simple geometry decomposition quadrature rules can easily be applied based on a tesselation. On hyperrectangles the reference configuration corresponds to the zero level of a multilinear level set function which is not piecewise planar. In this work we explain how to achieve higher order accurate quadrature with only positive quadrature weights also in this case.

Notes

Acknowledgements

The authors gratefully acknowledge funding by the German Science Foundation (DFG) within the project “LE 3726/1-1” and suggestions on a former version of this paper by Hans-Georg Raumer and an anonymous reviewer.

References

  1. 1.
    P. Bastian, C. Engwer, An unfitted finite element method using discontinuous Galerkin. Int. J. Numer. Meth. Eng. 79, 1557–1576 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    E. Burman, S. Claus, P. Hansbo, M.G. Larson, A. Massing, CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Meth. Eng. 104, 472–501 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    E. Burman, P. Hansbo, M. Larson, A cut finite element method with boundary value correction. Math. Comput. 87, 633–657 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    K.W. Cheng, T.-P. Fries, Higher-order XFEM for curved strong and weak discontinuities. Int. J. Numer. Meth. Eng. 82, 564–590 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    K. Dréau, N. Chevaugeon, N. Moës, Studied X-FEM enrichment to handle material interfaces with higher order finite element. Comput. Meth. Appl. Mech. Eng. 199, 1922–1936 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    C. Engwer, A. Nüßing, Geometric integration over irregular domains with topologic guarantees (2016). arXiv:1601.03597Google Scholar
  7. 7.
    T.-P. Fries, T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Meth. Eng. 84, 253–304 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    T.-P. Fries, S. Omerovi, Higher-order accurate integration of implicit geometries. Int. J. Numer. Meth. Eng. 106, 323–371 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    J. Grande, C. Lehrenfeld, A. Reusken, Analysis of a high order trace finite element method for PDEs on level set surfaces (2016). arXiv:1611.01100Google Scholar
  10. 10.
    P. Lederer, C.-M. Pfeiler, C. Wintersteiger, C. Lehrenfeld, Higher order unfitted FEM for Stokes interface problems. Proc. Appl. Math. Mech. 16, 7–10 (2016)CrossRefGoogle Scholar
  11. 11.
    C. Lehrenfeld, A higher order isoparametric fictitious domain method for level set domains (2016). arXiv:1612.02561Google Scholar
  12. 12.
    C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings. Comput. Methods Appl. Mech. Eng. 300, 716–733 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    C. Lehrenfeld, A. Reusken, Analysis of a high-order unfitted finite element method for elliptic interface problems. IMA J. Numer. Anal. (2017).Google Scholar
  14. 14.
    B. Müller, F. Kummer, M. Oberlack, Highly accurate surface and volume integration on implicit domains by means of moment-fitting. Int. J. Numer. Meth. Eng. 96, 512–528 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M.A. Olshanskii, A. Reusken, J. Grande, A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47, 3339–3358 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    J. Parvizian, A. Düster, E. Rank, Finite cell method. Comput. Mech. 41, 121–133 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Saye, High-order quadrature method for implicitly defined surfaces and volumes in hyperrectangles. SIAM J. Sci. Comput. 37, A993–A1019 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Numerische und Angewandte MathematikUniversity of GöttingenGöttingenGermany

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