Skip to main content
SpringerLink
Log in

Preconditioned Conjugate Gradient Solvers for the Generalized Finite Element Method

  • Chapter
  • First Online:
Meshfree Methods for Partial Differential Equations IX (IWMMPDE 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 129))

Abstract

This paper focuses on preconditioners for the conjugate gradient method and their applications to the Generalized FEM with global-local enrichments (GFEMgl) and the Stable GFEMgl. The preconditioners take advantage of the hierarchical struture of the matrices in these methods and the fact that most of the matrix does not change when simulating for example, the evolution of interfaces and fractures. The performance of the conjugate gradient method with the proposed preconditioner is investigated. A 3-D fracture problem is adopted for the numerical experiments.

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

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 29.99
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. I. Babuška, U. Banerjee, Stable generalized finite element method (SGFEM). Tech. Report ICES REPORT 11–07, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, April 2011

    Google Scholar 

  2. I. Babuška, U. Banerjee, Stable generalized finite element method (SGFEM). Comput. Methods Appl. Mech. Eng. 201–204, 91–111 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. I. Babuška, G. Caloz, J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31(4), 945–981 (1994)

    Article  MathSciNet  Google Scholar 

  5. E. Béchet, H. Minnebo, N. Moës, B. Burgardt, Improved implementation and robustness study of the x-fem for stress analysis around cracks. Int. J. Numer. Methods Eng. 64, 1033–1056 (2005)

    Article  Google Scholar 

  6. T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–620 (1999)

    Article  Google Scholar 

  7. L. Berger-Vergiat, H. Waisman, B. Hiriyur, R. Tuminaro, D. Keyes, Inexact Schwarz-AMG preconditioners for crack problems modeled by XFEM. Int. J. Numer. Methods Eng. 90, 311–328 (2012)

    Article  Google Scholar 

  8. P. Bochev, R.B. Lehoucq, On the finite element solution of the pure Neumann problem. SIAM Rev. 47(1), 50–66 (2005)

    Article  MathSciNet  Google Scholar 

  9. A.Th. Diamantoudis, G.N. Labeas, Stress intensity factors of semi-elliptical surface cracks in pressure vessels by global-local finite element methodology. Eng. Fract. Mech. 72, 1299–1312 (2005)

    Article  Google Scholar 

  10. C.A. Duarte, D.-J. Kim, Analysis and applications of a generalized finite element method with global-local enrichment functions. Comput. Methods Appl. Mech. Eng. 197(6–8), 487–504 (2008)

    Article  MathSciNet  Google Scholar 

  11. C.A.M. Duarte, J.T. Oden, An hp adaptive method using clouds. Comput. Methods Appl. Mech. Eng. 139, 237–262 (1996)

    Article  Google Scholar 

  12. C.A. Duarte, I. Babuška, J.T. Oden, Generalized finite element methods for three dimensional structural mechanics problems. Comput. Struct. 77, 215–232 (2000)

    Article  MathSciNet  Google Scholar 

  13. C.A. Duarte, O.N. Hamzeh, T.J. Liszka, W.W. Tworzydlo, A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comput. Methods Appl. Mech. Eng. 190(15–17), 2227–2262 (2001)

    Article  Google Scholar 

  14. C.A. Duarte, D.-J. Kim, I. Babuška, A global-local approach for the construction of enrichment functions for the generalized FEM and its application to three-dimensional cracks, in Advances in Meshfree Techniques, ed. by V.M.A. Leitão, C.J.S. Alves, C.A. Duarte. Computational Methods in Applied Sciences, vol. 5 (Springer, Dordrecht, 2007), pp. 1–26

    Google Scholar 

  15. V. Gupta, Improved conditioning and accuracy of a two-scale generalized finite element method for fracture mechanics. Ph.D. thesis, University of Illinois at Urbana-Champaign, 2014

    Google Scholar 

  16. V. Gupta, D.-J. Kim, C.A. Duarte, Analysis and improvements of global-local enrichments for the generalized finite element method. Comput. Methods Appl. Mech. Eng. 245–246, 47–62 (2012)

    Article  MathSciNet  Google Scholar 

  17. V. Gupta, C.A. Duarte, I. Babuška, U. Banerjee, A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics. Comput. Methods Appl. Mech. Eng. 266, 23–39 (2013)

    Article  MathSciNet  Google Scholar 

  18. V. Gupta, C.A. Duarte, I. Babuška, U. Banerjee, Stable GFEM (SGFEM): improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics. Comput. Methods Appl. Mech. Eng. 289, 355–386 (2015)

    Article  MathSciNet  Google Scholar 

  19. M.T. Heath, Scientific Computing: An Introductory Survey, 2nd edn. McGraw-Hill Series in Computer Science (McGraw-Hill, Boston, MA, 2002)

    Google Scholar 

  20. K. Kergrene, I. Babuska, U. Banerjee, Stable generalized finite element method and associated iterative schemes; application to interface problems. Comput. Methods Appl. Mech. Eng. 305, 1–36 (2016)

    Article  MathSciNet  Google Scholar 

  21. D.-J. Kim, C.A. Duarte, J.P. Pereira, Analysis of interacting cracks using the generalized finite element method with global-local enrichment functions. J. Appl. Mech. 75(5), 1–12 (2008)

    Google Scholar 

  22. D.-J. Kim, J.P. Pereira, C.A. Duarte, Analysis of three-dimensional fracture mechanics problems: a two-scale approach using coarse generalized FEM meshes. Int. J. Numer. Methods Eng. 81(3), 335–365 (2010)

    MATH  Google Scholar 

  23. D.-J. Kim, S.-G. Hong, C.A. Duarte, Generalized finite element analysis using the preconditioned conjugate gradient method. Appl. Math. Modell. 39(19), 5837–5848 (2015)

    Article  MathSciNet  Google Scholar 

  24. A. Kuzmin, M. Luisier, O. Schenk, Fast methods for computing selected elements of the greens function in massively parallel nanoelectronic device simulations. Euro-Par 2013 Parallel Process. 8097, 533–544 (2013)

    Google Scholar 

  25. M. Malekan, F.B. Barros, Well-conditioning global-local analysis using stable generalized/extended finite element method for linear elastic fracture mechanics. Comput. Mech. 58(5), 819–831 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. A. Menk, S. Bordas, A robust preconditioning technique for the extended finite element method. Int. J. Numer. Methods Eng. 85(13), 1609–1632 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. A.K. Noor, Global-local methodologies and their applications to nonlinear analysis. Finite Elem. Anal. Des. 2, 333–346 (1986)

    Article  Google Scholar 

  30. J.T. Oden, C.A. Duarte, Clouds, cracks and FEMs, in Recent Developments in Computational and Applied Mechanics, ed. by B.D. Reddy (International Center for Numerical Methods in Engineering, CIMNE, Barcelona, 1997), pp. 302–321. http://gfem.cee.illinois.edu/papers/jMartin_color.pdf

    Google Scholar 

  31. J.T. Oden, C.A. Duarte, O.C. Zienkiewicz, A new cloud-based hp finite element method. Comput. Methods Appl. Mech. Eng. 153, 117–126 (1998)

    Article  MathSciNet  Google Scholar 

  32. J.P. Pereira, D.-J. Kim, C.A. Duarte, A two-scale approach for the analysis of propagating three-dimensional fractures. Comput. Mech. 49(1), 99–121 (2012)

    Article  MathSciNet  Google Scholar 

  33. O. Schenk, K. Gärtner, Solving unsymmetric sparse systems of linear equations with PARDISO. J. Futur. Gener. Comput. Syst. 20(9), 475–487 (2004)

    Article  Google Scholar 

  34. M.A. Schweitzer, Generalizations of the finite element method. Cen. Eur. J. Math. 10, 3–24 (2012)

    Article  MathSciNet  Google Scholar 

  35. J.R. Shewchuk, An introduction to the conjugate gradient method without the agonizing pain (1994)

    Google Scholar 

  36. H. Waisman, L. Berger-Vergiat, An adaptive domain decomposition preconditioner for crack propagation problems modeled by XFEM. Int. J. Multiscale Comput. Eng. 11(6), 633–654 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

T.B. Fillmore and C.A. Duarte gratefully acknowledge the research funding under contract number AF Sub OSU 60038238 provided by the Collaborative Center in Structural Sciences (C2S2) at the Ohio State University, supported by the U.S. Air Force Research Laboratory.

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Travis B. Fillmore & Carlos Armando Duarte

  2. Pacific Northwest National Laboratory, Richland, WA, USA

    Varun Gupta

Authors
  1. Travis B. Fillmore
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Varun Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carlos Armando Duarte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Armando Duarte .

Editor information

Editors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Michael Griebel

  2. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Marc Alexander Schweitzer

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Fillmore, T.B., Gupta, V., Duarte, C.A. (2019). Preconditioned Conjugate Gradient Solvers for the Generalized Finite Element Method. In: Griebel, M., Schweitzer, M. (eds) Meshfree Methods for Partial Differential Equations IX. IWMMPDE 2017. Lecture Notes in Computational Science and Engineering, vol 129. Springer, Cham. https://doi.org/10.1007/978-3-030-15119-5_1

Download citation

Publish with us

Policies and ethics

Search

Navigation