Recent Progress in Mathematical and Computational Aspects of Peridynamics

  • Marta D’EliaEmail author
  • Qiang Du
  • Max Gunzburger
Reference work entry


Recent developments in the mathematical and computational aspects of the nonlocal peridynamic model for material mechanics are provided. Based on a recently developed vector calculus for nonlocal operators, a mathematical framework is constructed that has proved useful for the mathematical analyses of peridynamic models and for the development of finite element discretizations of those models. A specific class of discretization algorithms referred to as asymptotically compatible schemes is discussed; this class consists of methods that converge to the proper limits as grid sizes and nonlocal effects tend to zero. Then, the multiscale nature of peridynamics is discussed including how, as a single model, it can account for phenomena occurring over a wide range of scales. The use of this feature of the model is shown to result in efficient finite element implementations. In addition, the mathematical and computational frameworks developed for peridynamic simulation problems are shown to extend to control, coefficient identification, and obstacle problems.


Peridynamics Nonlocal vector calculus Variational forms Multiscale methods Finite element method Optimal control Obstacle problems 



MD: supported by the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4). Sandia National Laboratories 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 US Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.

QD: supported in part by the US NSF grant DMS-1558744, the AFOSR MURI Center for Material Failure Prediction Through Peridynamics, and the OSD/ARO/MURI W911NF-15-1-0562 on Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications.

MG: supported by the US NSF grant DMS-1315259, US Department of Energy Office of Science grant DE-SC0009324, US Air Force Office of Scientific Research grant FA9550-15-1-0001, and DARPA Equips program through the Oak Ridge National Laboratory.


  1. B. Aksoylu, T. Mengesha, Results on nonlocal boundary value problems. Numer. Func. Anal. Optim. 31, 1301–1317 (2010)MathSciNetCrossRefGoogle Scholar
  2. B. Aksoylu, M. Parks, Variational theory and domain decomposition for nonlocal problems. Appl. Math. Comp. 217, 6498–6515 (2011)MathSciNetCrossRefGoogle Scholar
  3. B. Aksoylu, Z. Unlu, Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM J. Numer. Anal. 52(2), 653–677 (2014). Scholar
  4. E. Askari, F. Bobaru, R. Lehoucq, M. Parks, S. Silling, O. Weckner, Peridynamics for multiscale materials modeling. J. Phys. Conf. Ser. 125, 012078 (2008)CrossRefGoogle Scholar
  5. F. Bobaru, M. Duangpanya, The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Transf. 53, 4047–4059 (2010)CrossRefGoogle Scholar
  6. F. Bobaru, S. Silling, Peridynamic 3D problems of nanofiber networks and carbon nanotube-reinforced composites, in Materials and Design: Proceeding of International Conference on Numerical Methods in Industrial Forming Processes (American Institute of Physics, 2004), pp. 1565–1570Google Scholar
  7. F. Bobaru, S. Silling, H. Jiang, Peridynamic fracture and damage modeling of membranes and nanofiber networks, in Proceeding of XI International Conference on Fracture, Turin, 2005, pp. 1–6Google Scholar
  8. F. Bobaru, M. Yang, L. Alves, S. Silling, E. Askari, J. Xu, Convergence, adaptive refinement, and scaling in 1d peridynamics. Inter. J. Numer. Meth. Engrg. 77, 852–877 (2009)CrossRefGoogle Scholar
  9. X. Chen, M. Gunzburger, Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Comput. Meth. Appl. Mech. Engrg. 200, 1237–1250 (2011) with X. Chen.MathSciNetCrossRefGoogle Scholar
  10. M. D’Elia, M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66, 1245–1260 (2013)MathSciNetCrossRefGoogle Scholar
  11. M. D’Elia, M. Gunzburger, Optimal distributed control of nonlocal steady diffusion problems. SIAM J. Cont. Optim. 52, 243–273 (2014)MathSciNetCrossRefGoogle Scholar
  12. M. D’Elia, M. Gunzburger, Identification of the diffusion parameter in nonlocal steady diffusion problems. Appl. Math. Optim. 73, 227–249 (2016)MathSciNetCrossRefGoogle Scholar
  13. Q. Du, Nonlocal calculus of variations and well-posedness of peridynamics, in Handbook of Peridynamic Modeling, chapter 3 (CRC Press, Boca Raton, 2016a), pp. 61–86Google Scholar
  14. Q. Du, Local limits and asymptotically compatible discretizations, in Handbook of Peridynamic Modeling, chapter 4 (CRC Press, Boca Raton, 2016b), pp. 87–108Google Scholar
  15. Q. Du, Z. Huang, Numerical solution of a scalar one-dimensional monotonicity-preserving nonlocal nonlinear conservation law. J. Math. Res. Appl. 37, 1–18 (2017)MathSciNetzbMATHGoogle Scholar
  16. 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 IMECE 2014–39620, 2014Google Scholar
  17. Q. Du, K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory. Math. Model. Numer. Anal. 45, 217–234 (2011)MathSciNetCrossRefGoogle Scholar
  18. Q. Du, Z. Zhou, Multigrid finite element method for nonlocal diffusion equations with a fractional kernel (2016, preprint)Google Scholar
  19. Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 56, 676–696 (2012a)MathSciNetzbMATHGoogle Scholar
  20. Q. Du J. Kamm, R. Lehoucq, M. Parks, A new approach to nonlocal nonlinear coservation laws. SIAM J. Appl. Math.72, 464–487 (2012b)MathSciNetCrossRefGoogle Scholar
  21. Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Mod. Meth. Appl. Sci. 23, 493–540 (2013a)MathSciNetCrossRefGoogle Scholar
  22. Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity. J. Elast. 113, 193–217, (2013b)MathSciNetCrossRefGoogle Scholar
  23. 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 (2013c)MathSciNetCrossRefGoogle Scholar
  24. 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 (2013d)MathSciNetCrossRefGoogle Scholar
  25. Q. Du, Z. Huang, R. Lehoucq, Nonlocal convection-diffusion volume-constrained problems and jump processes. Disc. Cont. Dyn. Sys. B 19, 373–389 (2014)MathSciNetCrossRefGoogle Scholar
  26. Q. Du, Y. Tao, X. Tian, J. Yang, Robust a posteriori stress analysis for approximations of nonlocal models via nonlocal gradient. Comput. Meth. Appl. Mech. Eng. 310, 605–627 (2016)MathSciNetCrossRefGoogle Scholar
  27. Q. Du, Z. Huang, P. Lefloch, Nonlocal conservation laws. I. A new class of monotonicity-preserving models. SIAM J. Numer. Anal. 55(5), 2465–2489 (2017)zbMATHGoogle Scholar
  28. 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, 851–864 (2007)MathSciNetCrossRefGoogle Scholar
  29. W. Gerstle, N. Sau, Peridynamic modeling of concrete structures, in Proceeding of 5th International Conference on Fracture Mechanics of Concrete Structures, Ia-FRAMCOS, vol. 2, 2004, pp. 949–956Google Scholar
  30. W. Gerstle, N. Sau, S. Silling, Peridynamic modeling of plain and reinforced concrete structures, in SMiRT18: 18th International Conference on Structural Mechanics in Reactor Technology, Beijing, 2005Google Scholar
  31. Q. Guan, M. Gunzburger, Analysis and approximation of a nonlocal obstacle problem. J. Comput. Appl. Math. 313, 102–118 (2017)MathSciNetCrossRefGoogle Scholar
  32. D. Littlewood, Simulation of dynamic fracture using peridynamics, finite element analysis, and contact, in Proceeding of ASME 2010 International Mechanical Engineering Congress and Exposition, Vancouver, 2010Google Scholar
  33. T. Mengesha, Q. Du, Analysis of a scalar nonlocal peridynamic model with a sign changing kernel. Disc. Cont. Dyn. Syst. B 18, 1415–1437 (2013)MathSciNetCrossRefGoogle Scholar
  34. T. Mengesha, Q. Du, The bond-based peridynamic system with Dirichlet type volume constraint. Proc. R. Soc. Edinb. A 144, 161–186 (2014a)MathSciNetCrossRefGoogle Scholar
  35. T. Mengesha, Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation. J. Elast. 116, 27–51 (2014b)MathSciNetCrossRefGoogle Scholar
  36. T. Mengesha, Q. Du, Characterization of function spaces of vector fields via nonlocal derivatives and an application in peridynamics. Nonlinear Anal. A Theory Meth. Appl. 140, 82–111 (2016)CrossRefGoogle Scholar
  37. M. Parks, R. Lehoucq, S. Plimpton, S. Silling, Implementing peridynamics within a molecular dynamics code. Comput. Phys. Commun. 179, 777–783 (2008)CrossRefGoogle Scholar
  38. M. Parks, D. Littlewood, J. Mitchell, S. Silling, Peridigm Users’ Guide, Sandia report 2012–7800, Albuquerque, 2012Google Scholar
  39. P. Seleson, M. Parks, M. Gunzburger, R. Lehoucq, Peridynamics as an upscaling of molecular dynamics. Mult. Model. Simul. 8, 204–227 (2009)MathSciNetCrossRefGoogle Scholar
  40. P. Seleson, Q. Du, M. Parks, On the consistency between nearest-neighbor peridynamic discretizations and discretized classical elasticity models. Comput. Meth. Appl. Mech. Engrg. 311, 698–722 (2016)MathSciNetCrossRefGoogle Scholar
  41. S. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)MathSciNetCrossRefGoogle Scholar
  42. S. Silling, Dynamic fracture modeling with a meshfree peridynamic code, in Computational Fluid and Solid Mechanics (Elsevier, Amsterdam, 2003), pp. 641–644Google Scholar
  43. S. Silling, Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)MathSciNetCrossRefGoogle Scholar
  44. S. Silling, E. Askari, Peridynamic modeling of impact damage, in PVP-Vol. 489 (ASME, New York, 2004), pp. 197–205Google Scholar
  45. S. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 8, 1526–1535 (2005)CrossRefGoogle Scholar
  46. S. Silling, F. Bobaru, Peridynamic modeling of membranes and fibers. Int. J. Nonlinear Mech. 40, 395–409 (2005)CrossRefGoogle Scholar
  47. S. Silling, R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008)MathSciNetCrossRefGoogle Scholar
  48. S. Silling, R. Lehoucq, Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010)CrossRefGoogle Scholar
  49. S. Silling, M. Zimmermann, R. Abeyaratne, Deformation of a peridynamic bar. J. Elast. 73, 173–190 (2003)MathSciNetCrossRefGoogle Scholar
  50. S. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)MathSciNetCrossRefGoogle Scholar
  51. S. Silling, D. Littlewood, P. Seleson, Variable horizon in a peridynamic medium, Technical report No. SAND2014-19088 (Sandia National Laboratories, Albuquerque, 2014)Google Scholar
  52. 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
  53. X. Tian, Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52, 1641–1665 (2014)MathSciNetCrossRefGoogle Scholar
  54. X. Tian, Q. Du, Nonconforming discontinuous Galerkin methods for nonlocal variational problems. SIAM J. Numer. Anal. 53(2), 762–781 (2015)MathSciNetCrossRefGoogle Scholar
  55. X. Tian, Q. Du, Trace theorems for some nonlocal energy spaces with heterogeneous localization. SIAM J. Math. Anal. 49(2), 1621–1644 (2017)MathSciNetCrossRefGoogle Scholar
  56. H. Tian, L. Ju, Q. Du, Nonlocal convection-diffusion problems and finite element approximations. Comput. Meth. Appl. Mech. Engrg. 289, 60–78 (2015)MathSciNetCrossRefGoogle Scholar
  57. X. Tian, Q. Du, M. Gunzburger, Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains. Adv. Comput. Math. 42, 1363–1380 (2016)MathSciNetCrossRefGoogle Scholar
  58. H. Tian, L. Ju, Q. Du, A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization. Comput. Methods Appl. Mech. Eng. 320, 46–67 (2017)MathSciNetCrossRefGoogle Scholar
  59. H. Wang, H. Tian, A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model. J. Comput. Phys. 240, 49–57 (2012)MathSciNetCrossRefGoogle Scholar
  60. 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
  61. O. Weckner, E. Emmrich, Numerical simulation of the dynamics of a nonlocal, inhomogeneous, infinite bar. J. Comput. Appl. Mech. 6, 311–319 (2005)MathSciNetzbMATHGoogle Scholar
  62. F. Xu, M. Gunzburger, J. Burkardt, Q. Du, A multiscale implementation based on adaptive mesh refinement for the nonlocal peridynamics model in one dimension. Multiscale Model. Simul. 14, 398–429 (2016a)MathSciNetCrossRefGoogle Scholar
  63. F. Xu, M. Gunzburger, J. Burkardt, A multiscale method for nonlocal mechanics and diffusion and for the approximation of discontinuous functions, Comput. Meth. Appl. Mech. Engrg. 307, 117–143 (2016b)MathSciNetCrossRefGoogle Scholar
  64. X. Zhang, M. Gunzburger, L. Ju, Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems. Comput. Meth. Appl. Mech. Engrg. 299, 401–420 (2016a)MathSciNetCrossRefGoogle Scholar
  65. X. Zhang, M. Gunzburger, L. Ju, Quadrature rules for finite element approximations of 1D nonlocal problems. J. Comput. Phys. 310, 213–236 (2016b)MathSciNetCrossRefGoogle Scholar
  66. K. Zhou, Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48, 1759–1780 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Optimization and Uncertainty Quantification Department Center for Computing ResearchSandia National LaboratoriesAlbuquerqueUSA
  2. 2.Department of Applied Physics and Applied MathematicsColumbia UniversityNew YorkUSA
  3. 3.Department of Scientific ComputingFlorida State UniversityTallahasseeUSA

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