Advertisement

Numerical Tools for Improved Convergence of Meshfree Peridynamic Discretizations

  • Pablo Seleson
  • David J. Littlewood
Living reference work entry

Abstract

Peridynamic models have been employed to simulate a broad range of engineering applications concerning material failure and damage, with the majority of these simulations using a meshfree discretization. This chapter reviews that meshfree discretization, related issues present in peridynamic convergence studies, and possible remedies proposed in the literature. In particular, we discuss two numerical tools, partial-volume algorithms and influence functions, to improve the convergence behavior of numerical solutions in peridynamics. Numerical studies in this chapter involve static and dynamic simulations for linear elastic state-based peridynamic problems.

Keywords

Peridynamics Meshfree discretization Partial volumes Influence functions Convergence Statics Dynamics 

Notes

Acknowledgments

This study was supported in part by the Householder Fellowship which is jointly funded by: the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program, under award number ERKJE45, and the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory (ORNL), which is operated by UT-Battelle, LLC., for the U.S. Department of Energy under Contract DE-AC05-00OR22725; and by the Laboratory Directed Research and Development program at Sandia National Laboratories, which is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

References

  1. E. Askari, F. Bobaru, R.B. Lehoucq, M.L. Parks, S.A. Silling, O. Weckner, Peridynamics for multiscale materials modeling, in SciDAC 2008, 13–17 July, Seattle, Journal of Physics: Conference Series, vol. 125, IOP Publishing, 2008. 012078Google Scholar
  2. F. Bobaru, Y.D. Ha, Adaptive refinement and multiscale modeling in 2D peridynamics. Int. J. Multiscale Comput. Eng. 9, 635–659 (2011)CrossRefGoogle Scholar
  3. F. Bobaru, Y.D. Ha, W. Hu, Damage progression from impact in layered glass modeled with peridynamics. Centr. Eur. J. Eng. 2, 551–561 (2012)Google Scholar
  4. F. Bobaru, M. Yang, L.F. Alves, S.A. Silling, E. Askari, J. Xu, Convergence, adaptive refinement, and scaling in 1D peridynamics. Int. J. Numer. Meth. Eng. 77, 852–877 (2009)CrossRefGoogle Scholar
  5. X. Chen, M. Gunzburger, Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Comput. Methods Appl. Mech. Eng. 200, 1237–1250 (2011)MathSciNetCrossRefGoogle Scholar
  6. D. De Meo, N. Zhu, E. Oterkus, Peridynamic modeling of granular fracture in polycrystalline materials. J. Eng. Mater. Technol. 138, 041008–041008–16 (2016)CrossRefGoogle Scholar
  7. Q. Du, X. Tian, Asymptotically compatible schemes for peridynamics based on numerical quadratures, in Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition, Montreal, 2014. Paper No. IMECE2014-39620Google Scholar
  8. W. Gerstle, N. Sau, S. Silling, Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007)CrossRefGoogle Scholar
  9. M. Ghajari, L. Iannucci, P. Curtis, A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput. Methods Appl. Mech. Eng. 276, 431–452 (2014)MathSciNetCrossRefGoogle Scholar
  10. Y.D. Ha, F. Bobaru, Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162, 229–244 (2010)CrossRefGoogle Scholar
  11. Y.D. Ha, F. Bobaru, Characteristics of dynamic brittle fracture captured with peridynamics. Eng. Fract. Mech. 78, 1156–1168 (2011)CrossRefGoogle Scholar
  12. W. Hu, Y. D. Ha, F. Bobaru, Numerical integration in peridynamics, Technical report, University of Nebraska-Lincoln, Sept. 2010Google Scholar
  13. W. Hu, Y.D. Ha, F. Bobaru, Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput. Methods Appl. Mech. Eng. 217–220, 247–261 (2012)MathSciNetCrossRefGoogle Scholar
  14. B. Kilic, A. Agwai, E. Madenci, Peridynamic theory for progressive damage prediction in center-cracked composite laminates. Compos. Struct. 90, 141–151 (2009)CrossRefGoogle Scholar
  15. B. Kilic, E. Madenci, Prediction of crack paths in a quenched glass plate by using peridynamic theory. Int. J. Fract. 156, 165–177 (2009)CrossRefGoogle Scholar
  16. D.J. Littlewood, A nonlocal approach to modeling crack nucleation in AA 7075-T651, in Proceedings of the ASME 2011 International Mechanical Engineering Congress and Exposition, Denver, 2011. Paper No. IMECE2011-64236Google Scholar
  17. D.J. Littlewood, K. Mish, K. Pierson, Peridynamic simulation of damage evolution for structural health monitoring, in Proceedings of the ASME 2012 International Mechanical Engineering Congress and Exposition, Houston, 2012. Paper No. IMECE2012-86400Google Scholar
  18. E. Oterkus, E. Madenci, Peridynamics for failure prediction in composites, in 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, 2012Google Scholar
  19. E. Oterkus, E. Madenci, O.Weckner, S. Silling, P. Bogert, A. Tessler, Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot. Compos. Struct. 94, 839–850 (2012)Google Scholar
  20. M.L. Parks, R.B. Lehoucq, S.J. Plimpton, S.A. Silling, Implementing peridynamics within a molecular dynamics code. Comput. Phys. Commun. 179, 777–783 (2008)CrossRefGoogle Scholar
  21. M.L. Parks, P. Seleson, S.J. Plimpton, S.A. Silling, R.B. Lehoucq, Peridynamics with LAMMPS: a user guide v0.3 beta, Report SAND2011-8523, Sandia National Laboratories, Albuquerque and Livermore, 2011Google Scholar
  22. M.L. Parks, D.J. Littlewood, J.A. Mitchell, S.A. Silling, Peridigm users’ guide v1.0.0, Report SAND2012-7800, Sandia National Laboratories, Albuquerque and Livermore, 2012Google Scholar
  23. P.D. Seleson, Peridynamic Multiscale Models for the Mechanics of Materials: Constitutive Relations, Upscaling from Atomistic Systems, and Interface Problems, PhD thesis, Florida State University, 2010. Electronic Theses, Treatises and Dissertations. Paper 273Google Scholar
  24. P. Seleson, Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Comput. Methods Appl. Mech. Eng. 282, 184–217 (2014)MathSciNetCrossRefGoogle Scholar
  25. P. Seleson, D.J. Littlewood, Convergence studies in meshfree peridynamic simulations. Comput. Math. Appl. 71, 2432–2448 (2016)MathSciNetCrossRefGoogle Scholar
  26. P. Seleson, M.L. Parks, On the role of the influence function in the peridynamic theory. Int. J. Multiscale Comput. Eng. 9, 689–706 (2011)CrossRefGoogle Scholar
  27. S.A. Silling, Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)MathSciNetCrossRefGoogle Scholar
  28. S.A. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005)CrossRefGoogle Scholar
  29. S.A. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)MathSciNetCrossRefGoogle Scholar
  30. S.A. Silling, O. Weckner, E. Askari, F. Bobaru, Crack nucleation in a peridynamic solid. Int. J. Fract. 162, 219–227 (2010)CrossRefGoogle Scholar
  31. M.R. Tupek, J.J. Rimoli, R. Radovitzky, An approach for incorporating classical continuum damage models in state-based peridynamics. Comput. Methods Appl. Mech. Eng. 263, 20–26 (2013)MathSciNetCrossRefGoogle Scholar
  32. O. Weckner, R. Abeyaratne, The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005)MathSciNetCrossRefGoogle Scholar
  33. O. Weckner, G. Brunk, M.A. Epton, S.A. Silling, E. Askari, Green’s functions in non-local three-dimensional linear elasticity. Proc. R. Soc. A 465, 3463–3487 (2009)MathSciNetCrossRefGoogle Scholar
  34. J. Xu, A. Askari, O. Weckner, S. Silling, Peridynamic analysis of impact damage in composite laminates. J. Aerosp. Eng. Spec. Issue Impact Mech. Compos. Mater. Aerosp. Appl. 21, 187–194 (2008)Google Scholar

Copyright information

© Springer International Publishing AG (outside the USA) 2018

Authors and Affiliations

  1. 1.Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  2. 2.Center for Computing ResearchSandia National LaboratoriesAlbuquerqueUSA

Personalised recommendations