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Peridynamics and Nonlocal Diffusion Models: Fast Numerical Methods

  • Hong WangEmail author
Reference work entry

Abstract

We outline the recent developments of fast numerical methods for linear nonlocal diffusion and peridynamic models in one and two space dimensions. We show how the analysis was carried out to take full advantage of the structure of the stiffness matrices of the numerical methods in its storage, evaluation, and assembly and in the efficient solution of the corresponding numerical schemes. This significantly reduces the computational complexity and storage of the numerical methods over conventional ones, without using any lossy compression. For instance, we would use the same numerical quadratures for conventional methods to evaluate the singular integrals in the stiffness matrices, except that we only need to evaluate O(N) of them instead of O(N2) of them. Numerical results are presented to show the utility of these fast methods.

Keywords

Peridynamics Nonlocal diffusion model Fast numerical method Toplitz matrix 

Notes

Acknowledgements

This work was supported in part by the OSD/ARO MURI under grant W911NF-15-1-0562 and by the National Science Foundation under grants DMS-1620194 and DMS-1216923.

References

  1. D. Applebaum, Lévy Processes and Stochastic Calculus (Cambridge University Press, Cambridge/New York , 2009)CrossRefGoogle Scholar
  2. H.G. Chan, M.K. Ng, Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38, 427–482 (1996)MathSciNetCrossRefGoogle Scholar
  3. S. Chen, F. Liu, X. Jiang, I. Turner, V. Anh, A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients. Appl. Math. Comp. 257, 591–601 (2015)MathSciNetCrossRefGoogle Scholar
  4. 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
  5. K. Dayal, K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006)MathSciNetCrossRefGoogle Scholar
  6. D. del-Castillo-Negrete, B.A. Carreras, V.E. Lynch, Fractional diffusion in plasma turbulence. Phys. Plasmas 11, 3854 (2004)CrossRefGoogle Scholar
  7. M. D’Elia, R. Lehoucq, M. Gunzburger, Q. Du, Finite range jump processes and volume-constrained diffusion problems, Sandia National Labs SAND, 2014–2584 (Sandia National Laboratories, Albuquerque/Livermore, 2014)Google Scholar
  8. Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54, 667–696 (2012)MathSciNetCrossRefGoogle 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. Comp. 82, 1889–1922 (2013)MathSciNetCrossRefGoogle Scholar
  10. E. Emmrich, O. Weckner, The peridynamic equation and its spatial discretisation. Math. Model. Anal. 12, 17–27 (2007)MathSciNetCrossRefGoogle Scholar
  11. W. Gerstle, N. Sau, S. Silling, Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007)CrossRefGoogle Scholar
  12. 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
  13. Y.D. Ha, F. Bobaru, Characteristics of dynamic brittle fracture captured with peridynamics. Eng. Fract. Mech. 78, 1156–1168 (2011)CrossRefGoogle Scholar
  14. J. Jia, C. Wang, H. Wang, A fast locally refined method for a space-fractional diffusion equation. # IE0147, ICFDA’14 Catania, 23–25 June 2014. Copyright 2014 IEEE ISBN:978-1-4799-2590-2Google Scholar
  15. J. Jia, H. Wang, A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh. J. Comput. Phys. 299, 842–862 (2015)MathSciNetCrossRefGoogle Scholar
  16. X. Lai, B. Ren, H. Fan, S. Li, C.T. Wu, R.A. Regueiro, L. Liu, Peridynamics simulations of geomaterial fragmentation by impulse loads. Int. J. Numer. Anal. Meth. Geomech 39, 1304–1330 (2015)CrossRefGoogle Scholar
  17. M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus (De Gruyter, Berlin, 2011)CrossRefGoogle Scholar
  18. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRefGoogle Scholar
  19. B. Øksendal, Stochastic Differential Equations: An Introduction with Applications (Springer, Heidelberg, 2010)zbMATHGoogle Scholar
  20. E. Oterkus, E. Madenci, O. Weckner, S.A. Silling, P. Bogert, 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)CrossRefGoogle Scholar
  21. M.L. Parks, R.B. Lehoucq, S.J. Plimpton, S.A. Silling, Implementing peridynamics within a molecular dynamics code. Comp. Phys. Comm. 179, 777–783 (2008)CrossRefGoogle Scholar
  22. M.L. Parks, P. Seleson, S.J. Plimpton, S.A. Silling, R.B. Lehoucq, Peridynamics with LAMMPS: a user guide v0.3 beta, SAND report 2011–8523 (Sandia National Laboratories, Albuquerque/Livermore, 2011)Google Scholar
  23. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)zbMATHGoogle 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, Q. Du, M.L. Parks, On the consistency between nearest-neighbor peridynamic discretizations and discretized classical elasticity models. Comput. Methods Appl. Mech. Eng. 311, 698–722 (2016)MathSciNetCrossRefGoogle Scholar
  26. P. Seleson, D. Littlewood, Convergence studies in meshfree peridynamic simulations. Comput. Math. Appl. 71, 2432–2448 (2016)MathSciNetCrossRefGoogle Scholar
  27. P. Seleson, M.L. Parks, On the role of the infuence function in the peridynamic theory. Int. J. Multiscale Comput. Eng. 9, 689–706 (2011)CrossRefGoogle Scholar
  28. P. Seleson, M.L. Parks, M. Gunzburger, R.B. Lehoucq, Peridynamics as an upscaling of molecular dynamics. Multiscale Model. Simul. 8, 204–227 (2009)MathSciNetCrossRefGoogle Scholar
  29. S.A. Silling, Reformulation of elasticity theory for discontinuous and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)MathSciNetCrossRefGoogle Scholar
  30. S.A. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005)CrossRefGoogle Scholar
  31. S.A. Silling, M. Epton, O. Wecker, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)MathSciNetCrossRefGoogle Scholar
  32. S.A. Silling, O. Weckner, E. Askari, F. Bobaru, Crack nucleation in a peridynamic solid. Int. J. Fract. 162, 219–227 (2010)CrossRefGoogle Scholar
  33. S. Sun, V. Sundararaghavan, A peridynamic implementation of crystal plasticity. Int. J. Solids Struct. 51, 3350–3360 (2014)CrossRefGoogle Scholar
  34. H. Tian, H. Wang, W. Wang, An efficient collocation method for a non-local diffusion model. Int. J. Numer. Anal. Model. 10, 815–825 (2013)MathSciNetzbMATHGoogle Scholar
  35. 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)MathSciNetCrossRefGoogle Scholar
  36. C. Wang, H. Wang, A fast collocation method for a variable-coefficient nonlocal diffusion model. J. Comput. Phys. 330, 114–126 (2017)MathSciNetCrossRefGoogle Scholar
  37. H. Wang, T.S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34, A2444–A2458 (2012)MathSciNetCrossRefGoogle Scholar
  38. H.Wang, N. Du, Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J. Comput. Phys. 258, 305–318 (2013)MathSciNetCrossRefGoogle Scholar
  39. H. Wang, H. Tian, A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model. J. Comput. Phys. 231, 7730–7738 (2012)MathSciNetCrossRefGoogle Scholar
  40. H. Wang, H. Tian, A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model. Comput. Methods Appl. Mech. Eng. 273, 19–36 (2014)MathSciNetCrossRefGoogle Scholar
  41. H. Wang, K. Wang, T. Sircar, A direct \(O(N\log ^2 N)\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)Google Scholar
  42. Q. Yang, I. Turner, F. Liu, M. Ilis, Novel numerical methods for solving the time-space fractional diffusion equation in 2D. SIAM Sci. Comput. 33, 1159–1180 (2011)MathSciNetCrossRefGoogle Scholar
  43. Q. Yang, I. Turner, T. Moroney F. Liu, A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction-diffusion equations. Appl. Math. Model. 38, 3755–3762 (2014)MathSciNetCrossRefGoogle Scholar
  44. X. Zhang, H. Wang, A fast method for a steady-state bond-based peridynamic model. Comput. Methods Appl. Mech. Eng. 311, 280–303 (2016)CrossRefGoogle Scholar
  45. X. Zhang, M. Gunzburger, L. Ju, Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems. Comput. Methods Appl. Mech. Eng. 299, 401–420 (2016)MathSciNetCrossRefGoogle Scholar
  46. K. Zhou, Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary condition. SIAM J. Numer. Anal. 48, 1759–1780 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA

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