Superposition-based coupling of peridynamics and finite element method

  • Wei Sun
  • Jacob FishEmail author
Original Paper


A superposition-based coupling of peridynamics (PD) and finite element method (FEM) for static and quasi-static problems is developed. The proposed coupling approach is based on partial superposition of nonlocal PD and local FEM solutions subjected to appropriate homogeneous boundary conditions that enforce solution continuity. The noteworthy features of the proposed PD–FEM superposition approach are: (1) it is free of blending parameters and (2) it preserves the standard computational structure of its two constituents, i.e., discrete weak form of the FEM and the strong form mesh-free style of PD. The performance of the proposed superposition approach is studied for several one- and two- dimensional problems.


Peridynamics Superposition Local-nonlocal coupling 



The first author thanks the Chinese Scholarship Council (CSC) (No. 201760210218) for providing the funding that enabled Wei Sun to visit Columbia University to do the work presented in this paper. The second author gratefully acknowledge the support from the Office of Naval Research under Grant N00014-17-1-2085.


  1. 1.
    Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209MathSciNetzbMATHGoogle Scholar
  2. 2.
    Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamics states and constitutive modeling. J Elast 88(2):151–184MathSciNetzbMATHGoogle Scholar
  3. 3.
    Silling SA, Lehoucq RB (2008) Convergence of peridynamics to classical elasticity theory. J Elast 93(1):13–37MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162(1–2):229–244zbMATHGoogle Scholar
  5. 5.
    Madenci E, Oterkus E (2014) Peridynamics theory and its applications. Springer, New YorkzbMATHGoogle Scholar
  6. 6.
    Foster JT, Silling SA, Chen W (2011) An energy based failure criterion for use with peridynamic states. Int J Multiscale Comput Eng 9(6):675–687Google Scholar
  7. 7.
    Macek RW, Silling SA (2007) Peridynamics via finite element analysis. Finite Elements Anal Design 43(14):1169–1178MathSciNetGoogle Scholar
  8. 8.
    Belytschko T, Fish J, Engelmann BE (1988) A finite element with embedded localization zones. Comput Methods Appl Mech Eng 70(1):59–89zbMATHGoogle Scholar
  9. 9.
    Fish J, Belytschko T (1988) Elements with embedded localization zones for large deformation problems. Comput Struct 30(1–2):247–256zbMATHGoogle Scholar
  10. 10.
    Silling SA, Littlewood D, Seleso P (2015) Variable horizon in a peridynamic medium. J Mech Mater Struct 10(5):591–612MathSciNetGoogle Scholar
  11. 11.
    Littlewood DJ, Silling SA, Mitchell JA, et al (2015) Strong local-nonlocal coupling for integrated fracture modeling. Sandia Report SAND2015-7998, Sandia National Laboratories, vol 3Google Scholar
  12. 12.
    Lubineau G, Azdoud Y, Han F, Rey C, Askari A (2012) A morphing strategy to couple non-local to local continuum mechanics. J Mech Phys Solids 60(6):1088–1102MathSciNetGoogle Scholar
  13. 13.
    Azdoud Y, Han F, Lubineau G (2014) The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture. Comput Mech 54(3):711–722MathSciNetzbMATHGoogle Scholar
  14. 14.
    Han F, Lubineau G, Azdoud Y (2016) Adaptive coupling between damage mechanics and peridynamics: a route for objective simulation of material degradation up to complete failure. J Mech Phys Solids 94:453–472MathSciNetGoogle Scholar
  15. 15.
    Han F, Lubineau G (2012) Coupling of nonlocal and local continuum models by the Arlequin approach. Int J Numer Methods Eng 89(6):671–685MathSciNetzbMATHGoogle Scholar
  16. 16.
    Seleson P, Beneddine S, Prudhomme S (2013) A force-based coupling scheme for peridynamics and classical elasticity. Comput Mater Sci 66:34–49Google Scholar
  17. 17.
    Seleson P, Ha YD, Beneddine S (2015) Concurrent coupling of bond-based peridynamics and the Navier equation of classical elasticity by blending. Int J Multiscale Comput Eng 13(2):91–113Google Scholar
  18. 18.
    Fish J, Nuggehally MA, Shephard MS et al (2007) Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. Comput Methods Appl Mech Eng 196(45–48):4548–4560MathSciNetzbMATHGoogle Scholar
  19. 19.
    Badia S, Bochev P, Lehoucq R et al (2007) A force-based blending model for atomistic-to-continuum coupling. Int J Multiscale Comput Eng 5(5):387–406Google Scholar
  20. 20.
    Galvanetto U, Mudric T, Shojaei A, Zaccariotto M (2016) An effective way to couple FEM meshes and peridynamics grids for the solution of static equilibrium problems. Mech Res Commun 76:41–47Google Scholar
  21. 21.
    Zaccariotto M, Tomasi D, Galvanetto U (2017) An enhanced coupling of PD grids to FE meshes. Mech Res Commun 84:125–135Google Scholar
  22. 22.
    Zaccariotto M, Mudric T, Tomasi D, Shojaei A, Galvanetto U (2018) Coupling of FEM meshes with peridynamics grids. Comput Methods Appl Mech Eng 330:471–497MathSciNetGoogle Scholar
  23. 23.
    Kulkarni S, Tabarraei A (2018) An analytical study of wave propagation in a peridynamic bar with nonuniform discretization. Eng Fract Mech 190:347–366Google Scholar
  24. 24.
    Shojaei A, Mossaiby F, Zaccariotto M, Galvanetto U (2018) An adaptive multi-grid peridynamic method for dynamic fracture analysis. Int J Mech Sci 144:600–617Google Scholar
  25. 25.
    Bie YH, Cui XY, Li ZC (2018) A coupling approach of state-based peridynamics with node-based smoothed finite element method. Comput Methods Appl Mech Eng 331:675–700MathSciNetGoogle Scholar
  26. 26.
    Kilic B, Madenci E (2010) Coupling of peridynamic theory and the finite element method. J Mech Mater Struct 5(5):707–733Google Scholar
  27. 27.
    Liu W, Hong J-W (2012) A coupling approach of discretized peridynamics with finite element method. Comput Methods Appl Mech Eng 245–246:163–175MathSciNetzbMATHGoogle Scholar
  28. 28.
    D’Elia M, Perego M, Bochev P, Littlewood D (2016) A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions. Comput Math Appl 71(11):2218–2230MathSciNetGoogle Scholar
  29. 29.
    Yu Y, Bargos FF, You H, Parks ML, Bittencourt ML, Karniadakis GE (2018) A partitioned coupling framework for peridynamics and classical theory: analysis and simulations. Comput Methods Appl Mech Eng. MathSciNetGoogle Scholar
  30. 30.
    Lindsay P, Parks ML, Prakash A (2016) Enabling fast, stable and accurate peridynamic computations using multi-time-step integration. Comput Methods Appl Mech Eng 306:382–405MathSciNetGoogle Scholar
  31. 31.
    Oterkus E, Madenci E, Weckner O, Silling S, Bogert P, Tessler A (2012) Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot. Compos Struct 94(3):839–850Google Scholar
  32. 32.
    Madenci E, Barut A, Dorduncu M, Phan ND (2018) Coupling of peridynamics with finite elements without an overlap zone. In: American Institute of Aeronautics and AstronauticsGoogle Scholar
  33. 33.
    Madenci E, Barut A, Futch M (2016) Peridynamic differential operator and its applications. Comput Methods Appl Mech Eng 304:408–451MathSciNetGoogle Scholar
  34. 34.
    Madenci E, Dorduncu M, Barut A, Phan N (2018) Weak form of peridynamics for nonlocal essential and natural boundary conditions. Comput Methods Appl Mech Eng 337:598–631MathSciNetGoogle Scholar
  35. 35.
    Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17–18):1526–1535Google Scholar
  36. 36.
    Shojaei A, Mudric T, Zaccariotto M, Galvanetto U (2016) A coupled meshless finite point/peridynamics method for 2D dynamic fracture analysis. Int J Mech Sci 119:419–431Google Scholar
  37. 37.
    Shojaei A, Zaccariotto M, Galvanetto U (2017) Coupling of 2D discretized peridynamics with a meshless method based on classical elasticity using switching of nodal behaviour. Eng Comput 34(5):1334–1366Google Scholar
  38. 38.
    Fan H, Li S (2017) A peridynamics-SPH modeling and simulation of blast fragmentation of soil under buried explosive loads. Comput Methods Appl Mech Eng 318:349–381MathSciNetGoogle Scholar
  39. 39.
    Seleson P, Gunzburger M, Parks ML (2013) Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains. Comput Methods Appl Mech Eng 266:185–204MathSciNetzbMATHGoogle Scholar
  40. 40.
    Nicely C, Tang S, Qian D (2018) Nonlocal matching boundary conditions for non-ordinary peridynamics with correspondence material model. Comput Methods Appl Mech Eng 338:463–490MathSciNetGoogle Scholar
  41. 41.
    Sun W, Fish J, Dhia HB (2018) A variant of the s-version of the finite element method for concurrent multiscale coupling. Int J Multiscale Comput Eng 16(2):21–40Google Scholar
  42. 42.
    Fish J (1992a) The s-version of the finite element method. Comput Struct 43(3):539–547zbMATHGoogle Scholar
  43. 43.
    Fish J (1992b) Hierarchical modelling of discontinuous fields. Int J Numer Methods Eng 8(7):443–453zbMATHGoogle Scholar
  44. 44.
    Fish J, Markolefas S (1993) Adaptive s-method for linear elastostatics. Comput Methods Appl Mech Eng 104(3):363–396zbMATHGoogle Scholar
  45. 45.
    Fish J, Suvorov A, Belsky V (1997) Hierarchical composite grid method for global-local analysis of laminated composite shells. Appl Numer Math 23(2):241–258MathSciNetzbMATHGoogle Scholar
  46. 46.
    Fan R, Fish J (2008) The rs-method for material failure simulations. Int J Numer Methods Eng 73(11):1607–1623MathSciNetzbMATHGoogle Scholar
  47. 47.
    Yang J, Fish J (2015a) Adaptive delamination analysis. Int J Numer Methods Eng 104(11):1008–1037MathSciNetzbMATHGoogle Scholar
  48. 48.
    Yang J, Fish J (2015b) On the equivalence between the s-method, the XFEM and the ply-by-ply discretization for delamination analyses of laminated composites. Int J Fract 191(1—-2):107–129Google Scholar
  49. 49.
    Dhia HB (1998) Multiscale mechanical problems: the Arlequin method. Comptes Rendus de l’Académie des Sciences Série IIb 326:899–904zbMATHGoogle Scholar
  50. 50.
    Dhia HB, Rateau G (2005) The Arlequin method as a flexible engineering design tool. Int J Numer Methods Eng 62(11):1442–1462zbMATHGoogle Scholar
  51. 51.
    Dhia HB (2008) Further insights by theoretical investigations of the multiscale Arlequin method. Int J Multiscale Comput Eng 6(3):1–18Google Scholar
  52. 52.
    Bobaru F, Ha YD (2011) Adaptive refinement and multiscale modelling in 2D peridynamics. Int J Multiscale Comput Eng 9(6):635–659Google Scholar
  53. 53.
    Bobaru F, Yang M, Alves LF, Silling SA, Askari E, Xu J (2009) Convergence, adaptive refinement, and scaling in 1D peridynamics. Int J Numer Methods Eng 77(6):852–877zbMATHGoogle Scholar
  54. 54.
    Huang D, Lu G, Qiao P (2015) An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis. Int J Mech Sci 94–95:111–122Google Scholar
  55. 55.
    Seleson P, Parks M (2011) On the role of the influence function in the peridynamic theory. Int J Multiscale Comput Eng 9(6):689–706Google Scholar
  56. 56.
    Silling SA, Zimmermann M, Abeyaratne R (2003) Deformation of a peridynamic bar. J Elast 73(1–3):173–190MathSciNetzbMATHGoogle Scholar
  57. 57.
    Weckner O, Abeyaratne R (2005) The effect of long-range forces on the dynamics of a bar. J Mech Phys Solids 53(3):705–728MathSciNetzbMATHGoogle Scholar
  58. 58.
    Mikata Y (2012) Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod. Int J Solids Struct 49(20):2887–2897Google Scholar
  59. 59.
    Seleson P, Littlewood DJ (2016) Convergence studies in meshfree peridynamic simulations. Comput Math Appl 71(11):2432–2448MathSciNetGoogle Scholar
  60. 60.
    Parks ML, Seleson P, Plimpton SJ, Lehoucq RB, Silling SA (2010) Peridynamics with LAMMPS: a User Guide v0.2 Beta. Technical Report Sandia Report, Sandia National LaboratoriesGoogle Scholar
  61. 61.
    Seleson P (2014) Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Comput Methods Appl Mech Eng 282:184–217MathSciNetzbMATHGoogle Scholar
  62. 62.
    Lehoucq R, Silling S (2008) Force flux and the peridynamic stress tensor. J Mech Phys Solids 56(4):1566–1577MathSciNetzbMATHGoogle Scholar
  63. 63.
    Gerstle WH, Sau N, Silling SA (2005) Peridynamic modeling of plain and reinforced concrete structures. In: Presented at the 18th international conference on structural mechanics in reactor technology, Beijing, ChinaGoogle Scholar
  64. 64.
    Madenci E, Oterkus S (2016) Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening. J Mech Phys Solids 86:192–219MathSciNetGoogle Scholar
  65. 65.
    Le QV, Bobaru F (2018) Surface corrections for peridynamic models in elasticity and fracture. Comput Mech 61(4):499–518MathSciNetzbMATHGoogle Scholar
  66. 66.
    Miller RE, Tadmor EB (2009) A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model Simul Mater Sci Eng 17(5):1–51Google Scholar
  67. 67.
    Kilic B, Madenci E (2010) An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 53(3):194–204Google Scholar
  68. 68.
    Shizhong Q (1988) An adaptive dynamic relaxation method for nonlinear problems. Comput Struct 30(4):855–859zbMATHGoogle Scholar
  69. 69.
    Breitenfeld MS, Geubelle PH, Weckner O, Silling SA (2014) Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput Methods Appl Mech Eng 272:233–250MathSciNetzbMATHGoogle Scholar
  70. 70.
    Silling SA (2017) Stability of peridynamic correspondence material models and their particle discretizations. Comput Methods Appl Mech Eng 322:42–57MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Hydroscience and EngineeringBeijingChina
  2. 2.Civil Engineering and Engineering MechanicsNew YorkUSA

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