Structural and Multidisciplinary Optimization

, Volume 51, Issue 3, pp 585–598 | Cite as

Adjoint design sensitivity analysis of dynamic crack propagation using peridynamic theory

  • Min-Yeong Moon
  • Jae-Hyun Kim
  • Youn Doh Ha
  • Seonho ChoEmail author


Based on the peridynamics of the reformulated continuum theory, an adjoint design sensitivity analysis (DSA) method is developed for the solution of dynamic crack propagation problems using the explicit scheme of time integration. Non-shape DSA problems are considered for the dynamic crack propagation including the successive branching of cracks. The adjoint variable method is generally suitable for path-independent problems but employed in this bond-based peridynamics since its path is readily available. Since both original and adjoint systems possess time-reversal symmetry, the trajectories of systems are symmetric about the u-axis. We take advantage of the time-reversal symmetry for the efficient and concurrent computation of original and adjoint systems. Also, to improve the numerical efficiency of large scale problems, a parallel computation scheme is employed using a binary space decomposition method. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It is demonstrated that the peridynamic adjoint sensitivity involving history-dependent variables can be accurate only if the path of the adjoint response analysis is identical to that of the original response.


Peridynamic theory Dynamic crack propagation Adjoint variable method Design sensitivity analysis Path dependent problem Parallel computation 



This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2010-0018282). The support is gratefully acknowledged. The authors would also like to thank Ms. Inyoung Cho at Korea University for editing assistance.


  1. Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity with a new discontinuous enrichment. Int J Numer Methods Eng 58:1873–1905CrossRefzbMATHGoogle Scholar
  2. Berger MJ, Bokhari SH (1987) A partition strategy for non-uniform problems on multiprocessors. IEEE Trans Comput C-36:570–580CrossRefGoogle Scholar
  3. 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 77:852–877CrossRefzbMATHGoogle Scholar
  4. Bowden FP, Brunton JH, Field JE, Heyes AD (1967) Controlled fracture of brittle solids and interruption of electrical current. Nature 216:38–42CrossRefGoogle Scholar
  5. Camacho GT, Ortiz M (1996) Computational modeling of impact damage in brittle materials. Int J Solids Struct 33:2899–2938CrossRefzbMATHGoogle Scholar
  6. Chen G, Rahman S, Park YH (2001) Shape sensitivity analysis in mixed-mode fracture mechanics. Comput Mech 27:282–291CrossRefzbMATHGoogle Scholar
  7. Cho S, Choi KK (2000) Design sensitivity analysis and optimization of non-linear transient dynamics Part I-sizing design. Int J Numer Methods Eng 48:351–373CrossRefzbMATHGoogle Scholar
  8. Choi KK, Kim NH (2004) Structural Sensitivity Analysis and Optimization, vol 1. Springer, New YorkGoogle Scholar
  9. Gao Z, Ma Y, Zhuang H (2008) Optimal shape design for the time-dependent Navier-Stokes flow. Int J Numer Methods Fluids 57:1505–1526CrossRefzbMATHMathSciNetGoogle Scholar
  10. Ha YD, Boraru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162:229–244CrossRefzbMATHGoogle Scholar
  11. Hsieh CC, Arora JS (1984) Design sensitivity analysis and optimization of dynamic response. Comput Methods Appl Mech Eng 43:195–219CrossRefzbMATHGoogle Scholar
  12. Kilic Bahattin (2008) Peridynamic theory for progressive failure prediction in homogeneous and heterogeneous materials, Mechanical Engineering. The University of Arizona, ProQuestGoogle Scholar
  13. Lamb JSW, Roberts JAG (1998) Time-reversal symmetry in dynamical systems: A survey. Physica D: Nonlinear Phenomena 112:1–39CrossRefzbMATHMathSciNetGoogle Scholar
  14. Nam KH, et al. (2012) Patterning by controlled cracking. Nature 485:221–224CrossRefGoogle Scholar
  15. Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three dimensional crack propagation analysis. Int J Numer Methods Eng 44:1267–1282CrossRefzbMATHGoogle Scholar
  16. Parks ML, Lehoucq RB, Plimpton SJ, Silling SA (2008) Implementing peridynamics within a molecular dynamics code. Comput Phys Commun 179:777–783CrossRefzbMATHGoogle Scholar
  17. Ravi-Chandar K, Knauss WG (1984) An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int J Fract 26:41–154Google Scholar
  18. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range force. J Mech Phys Solid 48:175–209CrossRefzbMATHMathSciNetGoogle Scholar
  19. Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Mech 83:1526–2535Google Scholar
  20. Steven H (1994) Strogatz, Nonlinear Dynamics and Chaos. Perseus Publishing, CambridgeGoogle Scholar
  21. Tortorelli DA, Haber RB, Lu SCY (1989) Design sensitivity analysis for nonlinear and thermal systems. Comput Methods Appl Mech Eng 77:61–77CrossRefzbMATHMathSciNetGoogle Scholar
  22. Tortorelli DA, Haber RB, Lu SCY (1991) Adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems. AIAA J 29(2):253–263CrossRefGoogle Scholar
  23. Tsay JJ, Arora JS (1990) Nonlinear structural design sensitivity analysis for path dependent problems, part 1: General theory. Comput Methods Appl Mech Eng 81:183–208CrossRefzbMATHMathSciNetGoogle Scholar
  24. Xu X-P, Needleman A (1994) Numerical simulations of fast crack growth in brittle solid. J Mech Phys Solid 42(9):1397–1434CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Min-Yeong Moon
    • 1
  • Jae-Hyun Kim
    • 2
  • Youn Doh Ha
    • 3
  • Seonho Cho
    • 2
    Email author
  1. 1.Department of Mechanical and Industrial EngineeringUniversity of IowaIowa CityUSA
  2. 2.Seoul National UniversitySeoulRepublic of Korea
  3. 3.Kunsan National UniversityKunsanRepublic of Korea

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