Skip to main content
SpringerLink
Log in

Robust Discretization of Nonlocal Models Related to Peridynamics

  • Conference paper
  • First Online:
Meshfree Methods for Partial Differential Equations VII

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 100))

Abstract

We study some nonlocal models related to peridynamics. These models are parameterized by a horizon that serves as a length scale measuring the range of nonlocal interactions. We focus on robust numerical schemes that can approximate both nonlocal models and their local limits as the horizon parameter vanishes. A representative linear model is used as an illustration. We show the lack of robustness of some standard numerical methods and describe a remedy to get asymptotically compatible schemes by utilizing elements of the recently developed nonlocal vector calculus and nonlocal calculus of variations. Such findings may be useful to ongoing research on modeling and simulations of nonlocal and multiscale problems.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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. E. Askari, F. Bobaru, R.B. Lehoucq, M.L. Parks, S.A. Silling, O. Weckner, Peridynamics for multiscale materials modeling. J. Phys.: Conf. Ser. 125, 012078 (2008)

    Google Scholar 

  2. Z.P. Bazant, T. Belytschko, T.-P. Chang, Continuum theory for strain-softening. J. Eng. Mech. 110(12), 1666–1692 (1984)

    Article  Google Scholar 

  3. Z.P. Bazant, M. Jirásek, Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128(11), 1119–1149 (2002)

    Article  Google Scholar 

  4. J.S. Chen, C.-T.g. Wu, T. Belytschko, Regularization of material instabilities by meshfree approximations with intrinsic length scales. Int. J. Numer. Methods Eng. 47(7), 1303–1322 (2000)

    Google Scholar 

  5. Q. Du, M. Gunzburger, R.B. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 56, 676–696 (2012)

    MathSciNet  Google Scholar 

  6. Q. Du, M. Gunzburger, R.B. Lehoucq, K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity. J. Elast. 113, 193–217 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Q. Du, M. Gunzburger, R.B. Lehoucq, K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23, 493–540 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Q. Du, Z. Huang, R. Lehoucq, Nonlocal convection-diffusion volume-constrained problems and jump processes. Discret. Contin. Dyn. Syst. B 19, 961–977 (2014)

    Article  MathSciNet  Google Scholar 

  9. Q. Du, L. Ju, L. Tian, K. Zhou, A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models. Math. Comput. 82, 1889–1922 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Q. Du, L. Tian, X. Zhao, A convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models. SIAM J. Numer. Anal. 51, 1211–1234 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Q. Du, K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory. Math. Model. Numer. Anal. 45, 217–234 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. E. Emmrich, O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5(4), 851–864 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. B. Kilic, E. Madenci, Coupling of peridynamic theory and the finite element method. J. Mech. Mater. Struct. 5(5), 707–733 (2010)

    Article  Google Scholar 

  14. W.K. Liu, Y. Chen, S. Jun, J.S. Chen, T. Belytschko, C. Pan, R.A. Uras, C.T. Chang, Overview and applications of the reproducing kernel particle methods. Arch. Comput. Methods Eng. 3(1), 3–80 (1996)

    Article  MathSciNet  Google Scholar 

  15. W.K. Liu, S. Li, T. Belytschko, Moving least-square reproducing kernel methods (i) methodology and convergence. Comput. Methods Appl. Mech. Eng. 143(1), 113–154 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. T. Mengesha, Q. Du, Analysis of a scalar nonlocal peridynamic model with a sign changing kernel. Discret. Contin. Dyn. Syst. B 18(5), 1415–1437 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Mengesha, Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation. J. Elast. 116(1), 27–51 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Mengesha, Q. Du, The bond-based peridynamic system with dirichlet type volume constraint. Proc. R. Soc. Edinb. A 144, 161–186 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Monaghan, Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 30, 543–574 (1992)

    Article  Google Scholar 

  20. J. Monaghan, Smoothed particle hydrodynamics. Rep. Prog. Phys. 68(8), 1703 (2005)

    Google Scholar 

  21. S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48(1), 175–209 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. S.A. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83(17), 1526–1535 (2005)

    Article  Google Scholar 

  23. S.A. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88(2), 151–184 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. S.A. Silling, R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elast. 93(1), 13–37 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. S.A. Silling, O. Weckner, E. Askari, F. Bobaru, Crack nucleation in a peridynamic solid. Int. J. Fract. 162(1), 219–227 (2010)

    Article  MATH  Google Scholar 

  26. X. Tian, Q. Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51, 3458–3482 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  27. X. Tian, Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52(4), 1641–1665 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  28. K. Zhou, Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48(5), 1759–1780 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Max Gunzburger, Rich Lehoucq, Tadele Mengesha, Michael Parks, Stewart Silling, Florin Bobaru, John Foster, Erdogan Madenci and John Mitchell for their discussions on the subject. This work is supported in part by the U.S. NSF grant DMS-1318586, and AFOSR MURI center for material failure prediction through peridynamics.

Author information

Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Qiang Du & Xiaochuan Tian

Authors
  1. Qiang Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaochuan Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiang Du .

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

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Du, Q., Tian, X. (2015). Robust Discretization of Nonlocal Models Related to Peridynamics. In: Griebel, M., Schweitzer, M. (eds) Meshfree Methods for Partial Differential Equations VII. Lecture Notes in Computational Science and Engineering, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-319-06898-5_6

Download citation

Publish with us

Policies and ethics

Search

Navigation