Journal of Scientific Computing

, Volume 74, Issue 2, pp 728–742 | Cite as

A Fast Finite Difference Method for a Continuous Static Linear Bond-Based Peridynamics Model of Mechanics

  • Zhengguang Liu
  • Xiaoli Li


The peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. Several numerical methods such as finite element method and collocation method have been developed and analyzed in many articles. However, there is no theory to give a finite difference method because the model does not involve spatial derivatives of the displacement field. Here, we consider a finite difference scheme to solve a continuous static bond-based peridynamics model of mechanics based on its equivalent partial integro-differential equations. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization respectively. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from \(O(N^{3})\) required by traditional methods to O(Nlog\(^{2}N)\) and the memory requirement from \(O(N^{2})\) to O(N) without using any lossy compression, where N is the number of unknowns. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.


Peridynamic Finite difference method Topelitz Fast Fourier transform 

Mathematics Subject Classification

65M06 65M12 65M15 26A33 



We would like to acknowledge the assistance of volunteers in putting together this example manuscript and supplement.


  1. 1.
    Bobaru, F., Yang, M., Alves, L.F., Silling, S.A., Askari, E., Xu, J.: Convergence, adaptive refinement, and scaling in 1D peridynamics. Int. J. Numer. Methods Eng. 77, 852–877 (2009)CrossRefMATHGoogle Scholar
  2. 2.
    Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer Science & Business Media, Berlin (2012)MATHGoogle Scholar
  3. 3.
    Casey, J., Krishnaswamy, S.: A characterization of internally constrained thermoelastic materials. Math. Mech. Solids 3, 71–89 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, X., Gunzburger, M.: Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Comput. Methods Appl. Mech. Eng. 200, 1237–1250 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dan, G., Rand, O.: Dynamic thermoelastic coupling effects in a rod. Aiaa J. 33, 776–778 (2015)Google Scholar
  6. 6.
    Ding, H., Li, C., Chen, Y.: High-order algorithms for Riesz derivative and their applications (ii). J. Comput. Phys. 293, 218–237 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Du, Q., Zhou, K.: Mathematical analysis for the peridynamic nonlocal continuum theory. ESAIM Math. Model. Numer. Anal. 45, 217–234 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gray, R.M.: Toeplitz and Circulant Matrices: A Review. Foundations and Trends in Communications and Information Theory 2(3), 155–239 (2006)Google Scholar
  9. 9.
    Hosseini-Tehrani, P., Eslami, M.R.: Bem analysis of thermal and mechanical shock in a two-dimensional finite domain considering coupled thermoelasticity. Eng. Anal. Bound. Elem. 24, 249–257 (2000)CrossRefMATHGoogle Scholar
  10. 10.
    Li, C., Zeng, F.: Numerical Methods for Fractional Calculus, vol. 24. CRC Press, Boca Raton (2015)MATHGoogle Scholar
  11. 11.
    Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys 225, 1533–1552 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Liu, W., Long, X.: A new nonconforming finite element with a conforming finite element approximation for a coupled continuum pipe-flow/darcy model in karst aquifers. Numer. Methods Partial Differ. Equ. 32, 778–798 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Liu, W., Kang, Z., Rui, H.: Finite volume element approximation of the coupled continuum pipe-flow/darcy model for flows in karst aquifers. Numer. Methods Partial Differ. Equ. 30, 376–392 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Liu, Z., Cheng, A., Wang, H.: An hp-Galerkin method with fast solution for linear peridynamic models in one dimension. Comput. Math. Appl. 73, 1546–1565 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Madenci, E., Oterkus, E.: Peridynamic Theory and Its Applications. Springer, New York (2014)CrossRefMATHGoogle Scholar
  16. 16.
    Nickell, R.E., Sackman, J.L.: Approximate solutions in linear, coupled thermoelasticity. J. Appl. Mech. 35, 255–266 (1968)Google Scholar
  17. 17.
    Seleson, P., Littlewood, D.J.: Convergence studies in meshfree peridynamic simulations. Comput. Math. Appl. 71, 2432–2448 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Silling, S.A.: Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005)CrossRefGoogle Scholar
  21. 21.
    Silling, S.A., Zimmermann, M., Abeyaratne, R.: Deformation of a peridynamic bar. J. Elast. 73, 173–190 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Taylor, M.J.: Numerical Simulation of Thermo-elasticity, Inelasticity and Rupture in Membrane Theory. Dissertations and Theses-Gradworks (2008)Google Scholar
  23. 23.
    Tian, X., Du, Q.: Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51, 3458–3482 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wang, H., Tian, H.: A fast galerkin method with efficient matrix assembly and storage for a peridynamic model. J. Comput. Phys. 231, 7730–7738 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wang, H., Tian, H.: 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)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zhou, K., Du, Q.: Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48, 1759–1780 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina

Personalised recommendations