A non-ordinary state-based Godunov-peridynamics formulation for strong shocks in solids

  • Guohua Zhou
  • Michael HillmanEmail author


The theory and meshfree implementation of peridynamics has been proposed to model problems involving transient strong discontinuities such as dynamic fracture and fragment-impact problems. For effective application of numerical methods to these events, essential shock physics and Gibbs instability should be addressed. The Godunov scheme for shock treatment has been shown to be an effective approach for tackling these two issues but has not been considered yet for peridynamics. This work introduces a physics-based shock modeling formulation for non-ordinary state-based peridynamics, in which the Godunov scheme is introduced by embedding the Riemann solution into the force state, resulting in a shock formulation free of tuneable parameters. Several benchmark problems are solved to demonstrate the effectiveness of the proposed formulation for modeling problems involving shocks in solids.


Peridynamics Meshfree Shockwaves Godunov scheme 



Both authors greatly acknowledge the support of this work by The Pennsylvania State University.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. 1.
    Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2):151–184MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162(1–2):229–244CrossRefzbMATHGoogle Scholar
  4. 4.
    Bobaru F, Zhang G (2015) Why do cracks branch? A peridynamic investigation of dynamic brittle fracture. Int J Fract 196(1–2):59–98CrossRefGoogle Scholar
  5. 5.
    Bobaru F, Ha YD, Hu W (2012) Damage progression from impact in layered glass modeled with peridynamics. Cent Eur J Eng 2(4):551–561Google Scholar
  6. 6.
    Gerstle W, Sau N, Silling S (2007) Peridynamic modeling of concrete structures. Nucl Eng Des 237(12–13):1250–1258CrossRefGoogle Scholar
  7. 7.
    Tupek MR, Rimoli JJ, Radovitzky R (2013) An approach for incorporating classical continuum damage models in state-based peridynamics. Comput Methods Appl Mech Eng 263:20–26MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Methods Eng 81(10):1242–1258zbMATHGoogle Scholar
  9. 9.
    Warren TL, Silling SA, Askari A, Weckner O, Epton MA, Xu J (2009) A non-ordinary state-based peridynamic method to model solid material deformation and fracture. Int J Solids Struct 46(5):1186–1195CrossRefzbMATHGoogle Scholar
  10. 10.
    Ren B, Fan H, Bergel GL, Regueiro RA, Lai X, Li S (2014) A peridynamics-SPH coupling approach to simulate soil fragmentation induced by shock waves. Comput Mech 55(2):287–302MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lai X, Liu L, Li S, Zeleke M, Liu Q, Wang Z (2018) A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials. Int J Impact Eng 111:130–146CrossRefGoogle Scholar
  12. 12.
    Silling SA, Parks ML, Kamm JR, Weckner O, Rassaian M (2017) Modeling shockwaves and impact phenomena with Eulerian peridynamics. Int J Impact Eng 107:47–57CrossRefGoogle Scholar
  13. 13.
    Godunov SK (1969) A finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Sb Math 47:271–306Google Scholar
  14. 14.
    Cockburn B, Shu CW (1998) The Runge–Kutta discontinuous Galerkin method for conservation laws V. J Comput Phys 141(2):199–224MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Inutsuka SI (2002) Reformulation of smoothed particle hydrodynamics with Riemann solver. J Comput Phys 179(1):238–267CrossRefzbMATHGoogle Scholar
  16. 16.
    Löhner R, Sacco C, Oñate E, Idelsohn SR (2002) A finite point method for compressible flow. Int J Numer Methods Eng 53(8):1765–1779CrossRefzbMATHGoogle Scholar
  17. 17.
    Dukowicz JK, Cline MC, Addessio FL (1989) A general topology Godunov method. J Comput Phys 82(1):29–63MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hietel D, Steiner K, Struckmeier J (2000) A finite volume particle method for compressible flows. Math Models Methods Appl Sci 10:1363–1382MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chiu EK, Wang Q, Jameson A (2011) A conservative meshless scheme: general order formulation and application to Euler equations. In: AIAA 2011-651 49th Aerospace Sciences MeetingGoogle Scholar
  20. 20.
    Luo H, Baum JD, Loehner R (1994) Edge-based finite element scheme for the Euler equations. AIAA J 32(6):1183–1190CrossRefzbMATHGoogle Scholar
  21. 21.
    Praveen C (2004) A positive meshless method for hyperbolic equations. Technical Report, Department of Aerospace Engineering, Indian Institute of ScienceGoogle Scholar
  22. 22.
    Ma ZH, Wang H, Qian L (2014) A meshless method for compressible flows with the HLLC Riemann solver 44(0):1–24 arXiv preprint arXiv:1402.2690
  23. 23.
    van Leer B (1979) Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J Comput Phys 32(1):101–136CrossRefzbMATHGoogle Scholar
  24. 24.
    Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8–9):1081–1106MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139(1–4):195–227MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50(2):435–466CrossRefzbMATHGoogle Scholar
  27. 27.
    Roth MJ, Chen JS, Danielson KT, Slawson TR (2016) Hydrodynamic meshfree method for high-Rate solid dynamics using a Rankine–Hugoniot enhancement in a Riemann-SCNI framework. Int J Numer Methods Eng 108:1525–1549MathSciNetCrossRefGoogle Scholar
  28. 28.
    Roth MJ, Chen JS, Slawson TR, Danielson KT (2016) Stable and flux-conserved meshfree formulation to model shocks. Comput Mech 57(5):773–792MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Seleson P, Littlewood DJ (2016) Convergence studies in meshfree peridynamic simulations. Comput Math Appl 71(11):2432–2448MathSciNetCrossRefGoogle Scholar
  30. 30.
    LeVeque RJ (2002) Finite volume methods for hyperbolic problems, vol 31. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  31. 31.
    Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17–18):1526–1535CrossRefGoogle Scholar
  32. 32.
    Dukowicz JK (1985) A general, non-iterative Riemann solver for Godunov’s method. J Comput Phys 61(1):119–137MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Silling SA, Lehoucq RB (2008) Convergence of peridynamics to classical elasticity theory. J Elast 93(1):13MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    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):852–877CrossRefzbMATHGoogle Scholar
  35. 35.
    Davison L (2008) Fundamentals of shock wave propagation in solids. Springer, BerlinzbMATHGoogle Scholar
  36. 36.
    Marsh SA (1980) LASL shock Hugoniot data. Technical report, University of California Press, BerkleyGoogle Scholar

Copyright information

© OWZ 2019

Authors and Affiliations

  1. 1.Optimal Inc.PlymouthUSA
  2. 2.The Pennsylvania State UniversityUniversity ParkUSA

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