Skip to main content
SpringerLink
Log in

The intrinsic partition of unity method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A method is presented which enables the global enrichment of the approximation space without introducing additional unknowns. Only one shape function per node is used. The shape functions are constructed by means of the moving least-squares method with an intrinsic basis vector and weight functions based on finite element shape functions. The enrichment is achieved through the intrinsic basis. By using polynomials in the intrinsic basis, optimal rates of convergence can be achieved even on distorted elements. Special enrichment functions can be chosen to enhance accuracy for solutions that are not polynomial in character. Results are presented which show optimal convergence on randomly distorted elements and improved accuracy for the oscillatory solution of the Helmholtz equation.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, Chichester

    MATH  Google Scholar 

  2. Zienkiewicz OC, Taylor RL (2000) The finite element method, vols. 1–3. Butterworth-Heinemann, Oxford

    Google Scholar 

  3. Babuška I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40:727–758

    Article  Google Scholar 

  4. Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314

    Article  MATH  Google Scholar 

  5. Strouboulis T, Babuška I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181:43–69

    Article  MATH  Google Scholar 

  6. Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190:4081–4193

    Article  MATH  Google Scholar 

  7. Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50:993–1013

    Article  MATH  Google Scholar 

  8. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150

    Article  MATH  Google Scholar 

  9. Oden JT, Duarte CA, Zienkiewicz OC (1998) A new cloud-based hp finite element method. Comput Methods Appl Mech Eng 153:117–126

    Article  MATH  MathSciNet  Google Scholar 

  10. Tian R, Yagawa G (2005) Generalized nodes and high-performance elements. Int J Numer Methods Eng 64: 2039–2071

    Article  MATH  Google Scholar 

  11. Chen G, Ohnishi Y, Ito T (1998) Development of high-order manifold method. Int J Numer Methods Eng 43:685–712

    Article  MATH  Google Scholar 

  12. Belytschko T, Gu L, Lu YY (1994) Fracture and crack growth by element free galerkin methods. Model Simul Mater Sci Eng 2:519–534

    Article  Google Scholar 

  13. Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37:141–158

    Article  MATH  MathSciNet  Google Scholar 

  14. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47

    Article  MATH  Google Scholar 

  15. Fries TP, Matthies HG (2003) Classification and overview of meshfree methods. Informatikbericht-Nr. 2003-03, Technical University Braunschweig, (http://opus.tu-bs.de/opus/volltexte/2003/418/), Brunswick

  16. Fries TP, Belytschko T (2006) The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int J Numer Methods Eng. DOI 10.1002/nme.1761

  17. Cho YS, Jun S, Im S, Kim HG (2005) An improved interface element with variable nodes for non-matching finite element meshes. Comput Methods Appl Mech Eng 194: 3022–3046

    Article  MATH  Google Scholar 

  18. Cho YS, Im S (2006) MLS-based variable-node elements compatible with quadratic interpolation. Part I: Formulation and application for non-matching meshes. Int J Numer Methods Eng 65:494–516

    Article  MATH  Google Scholar 

  19. Cho YS, Im S (2006) MLS-based variable-node elements compatible with quadratic interpolation. Part II: Application for finite crack element. Int J Numer Methods Eng 65: 517–547

    Article  MATH  Google Scholar 

  20. Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256

    Article  MATH  MathSciNet  Google Scholar 

  21. Huerta A, Fernández-Méndez S (2000) Enrichment and coupling of the finite element and meshless methods. Int J Numer Methods Eng 48:1615–1636

    Article  MATH  Google Scholar 

  22. Brooks AN, Hughes TJR (1982) Streamline upwind/ Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32: 199–259

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, 60202, IL, USA

    Thomas-Peter Fries & Ted Belytschko

Authors
  1. Thomas-Peter Fries
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ted Belytschko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas-Peter Fries.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fries, TP., Belytschko, T. The intrinsic partition of unity method. Comput Mech 40, 803–814 (2007). https://doi.org/10.1007/s00466-006-0142-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-006-0142-x

Keywords

Search

Navigation