Skip to main content

Numerical Integration on Hyperrectangles in Isoparametric Unfitted Finite Elements

  • Conference paper
  • First Online:
Numerical Mathematics and Advanced Applications ENUMATH 2017 (ENUMATH 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 126))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P. Bastian, C. Engwer, An unfitted finite element method using discontinuous Galerkin. Int. J. Numer. Meth. Eng. 79, 1557–1576 (2009)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  3. E. Burman, P. Hansbo, M. Larson, A cut finite element method with boundary value correction. Math. Comput. 87, 633–657 (2018)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  6. C. Engwer, A. Nüßing, Geometric integration over irregular domains with topologic guarantees (2016). arXiv:1601.03597

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  8. T.-P. Fries, S. Omerovi, Higher-order accurate integration of implicit geometries. Int. J. Numer. Meth. Eng. 106, 323–371 (2016)

    Article  MathSciNet  Google Scholar 

  9. J. Grande, C. Lehrenfeld, A. Reusken, Analysis of a high order trace finite element method for PDEs on level set surfaces (2016). arXiv:1611.01100

    Google Scholar 

  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)

    Article  Google Scholar 

  11. C. Lehrenfeld, A higher order isoparametric fictitious domain method for level set domains (2016). arXiv:1612.02561

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  16. J. Parvizian, A. Düster, E. Rank, Finite cell method. Comput. Mech. 41, 121–133 (2007)

    Article  MathSciNet  Google Scholar 

  17. R. Saye, High-order quadrature method for implicitly defined surfaces and volumes in hyperrectangles. SIAM J. Sci. Comput. 37, A993–A1019 (2015)

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christoph Lehrenfeld .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Heimann, F., Lehrenfeld, C. (2019). Numerical Integration on Hyperrectangles in Isoparametric Unfitted Finite Elements. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_16

Download citation

Publish with us

Policies and ethics