Skip to main content
SpringerLink
Log in

Recent Progress in Mathematical and Computational Aspects of Peridynamics

  • Reference work entry
  • First Online:
Handbook of Nonlocal Continuum Mechanics for Materials and Structures

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 799.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 999.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • B. Aksoylu, T. Mengesha, Results on nonlocal boundary value problems. Numer. Func. Anal. Optim. 31, 1301–1317 (2010)

    Article  MathSciNet  Google Scholar 

  • B. Aksoylu, M. Parks, Variational theory and domain decomposition for nonlocal problems. Appl. Math. Comp. 217, 6498–6515 (2011)

    Article  MathSciNet  Google Scholar 

  • B. Aksoylu, Z. Unlu, Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM J. Numer. Anal. 52(2), 653–677 (2014). https://doi.org/10.1137/13092407X

    Article  MathSciNet  Google Scholar 

  • E. Askari, F. Bobaru, R. Lehoucq, M. Parks, S. Silling, O. Weckner, Peridynamics for multiscale materials modeling. J. Phys. Conf. Ser. 125, 012078 (2008)

    Article  Google Scholar 

  • F. Bobaru, M. Duangpanya, The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Transf. 53, 4047–4059 (2010)

    Article  Google Scholar 

  • 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. 1565–1570

    Google Scholar 

  • 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. 1–6

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • M. D’Elia, M. Gunzburger, Optimal distributed control of nonlocal steady diffusion problems. SIAM J. Cont. Optim. 52, 243–273 (2014)

    Article  MathSciNet  Google Scholar 

  • M. D’Elia, M. Gunzburger, Identification of the diffusion parameter in nonlocal steady diffusion problems. Appl. Math. Optim. 73, 227–249 (2016)

    Article  MathSciNet  Google Scholar 

  • Q. Du, Nonlocal calculus of variations and well-posedness of peridynamics, in Handbook of Peridynamic Modeling, chapter 3 (CRC Press, Boca Raton, 2016a), pp. 61–86

    Google Scholar 

  • Q. Du, Local limits and asymptotically compatible discretizations, in Handbook of Peridynamic Modeling, chapter 4 (CRC Press, Boca Raton, 2016b), pp. 87–108

    Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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, 2014

    Google Scholar 

  • Q. Du, K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory. Math. Model. Numer. Anal. 45, 217–234 (2011)

    Article  MathSciNet  Google Scholar 

  • Q. Du, Z. Zhou, Multigrid finite element method for nonlocal diffusion equations with a fractional kernel (2016, preprint)

    Google Scholar 

  • Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 56, 676–696 (2012a)

    MathSciNet  MATH  Google Scholar 

  • Q. Du J. Kamm, R. Lehoucq, M. Parks, A new approach to nonlocal nonlinear coservation laws. SIAM J. Appl. Math.72, 464–487 (2012b)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Q. Du, Z. Huang, R. Lehoucq, Nonlocal convection-diffusion volume-constrained problems and jump processes. Disc. Cont. Dyn. Sys. B 19, 373–389 (2014)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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. 949–956

    Google Scholar 

  • 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, 2005

    Google Scholar 

  • Q. Guan, M. Gunzburger, Analysis and approximation of a nonlocal obstacle problem. J. Comput. Appl. Math. 313, 102–118 (2017)

    Article  MathSciNet  Google Scholar 

  • 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, 2010

    Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • T. Mengesha, Q. Du, The bond-based peridynamic system with Dirichlet type volume constraint. Proc. R. Soc. Edinb. A 144, 161–186 (2014a)

    Article  MathSciNet  Google Scholar 

  • T. Mengesha, Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation. J. Elast. 116, 27–51 (2014b)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  Google Scholar 

  • M. Parks, R. Lehoucq, S. Plimpton, S. Silling, Implementing peridynamics within a molecular dynamics code. Comput. Phys. Commun. 179, 777–783 (2008)

    Article  Google Scholar 

  • M. Parks, D. Littlewood, J. Mitchell, S. Silling, Peridigm Users’ Guide, Sandia report 2012–7800, Albuquerque, 2012

    Google Scholar 

  • P. Seleson, M. Parks, M. Gunzburger, R. Lehoucq, Peridynamics as an upscaling of molecular dynamics. Mult. Model. Simul. 8, 204–227 (2009)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • S. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)

    Article  MathSciNet  Google Scholar 

  • S. Silling, Dynamic fracture modeling with a meshfree peridynamic code, in Computational Fluid and Solid Mechanics (Elsevier, Amsterdam, 2003), pp. 641–644

    Google Scholar 

  • S. Silling, Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)

    Article  MathSciNet  Google Scholar 

  • S. Silling, E. Askari, Peridynamic modeling of impact damage, in PVP-Vol. 489 (ASME, New York, 2004), pp. 197–205

    Google Scholar 

  • S. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 8, 1526–1535 (2005)

    Article  Google Scholar 

  • S. Silling, F. Bobaru, Peridynamic modeling of membranes and fibers. Int. J. Nonlinear Mech. 40, 395–409 (2005)

    Article  Google Scholar 

  • S. Silling, R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008)

    Article  MathSciNet  Google Scholar 

  • S. Silling, R. Lehoucq, Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010)

    Article  Google Scholar 

  • S. Silling, M. Zimmermann, R. Abeyaratne, Deformation of a peridynamic bar. J. Elast. 73, 173–190 (2003)

    Article  MathSciNet  Google Scholar 

  • S. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)

    Article  MathSciNet  Google Scholar 

  • S. Silling, D. Littlewood, P. Seleson, Variable horizon in a peridynamic medium, Technical report No. SAND2014-19088 (Sandia National Laboratories, Albuquerque, 2014)

    Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • X. Tian, Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52, 1641–1665 (2014)

    Article  MathSciNet  Google Scholar 

  • X. Tian, Q. Du, Nonconforming discontinuous Galerkin methods for nonlocal variational problems. SIAM J. Numer. Anal. 53(2), 762–781 (2015)

    Article  MathSciNet  Google Scholar 

  • X. Tian, Q. Du, Trace theorems for some nonlocal energy spaces with heterogeneous localization. SIAM J. Math. Anal. 49(2), 1621–1644 (2017)

    Article  MathSciNet  Google Scholar 

  • H. Tian, L. Ju, Q. Du, Nonlocal convection-diffusion problems and finite element approximations. Comput. Meth. Appl. Mech. Engrg. 289, 60–78 (2015)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • O. Weckner, R. Abeyaratne, The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005)

    Article  MathSciNet  Google Scholar 

  • O. Weckner, E. Emmrich, Numerical simulation of the dynamics of a nonlocal, inhomogeneous, infinite bar. J. Comput. Appl. Mech. 6, 311–319 (2005)

    MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • X. Zhang, M. Gunzburger, L. Ju, Quadrature rules for finite element approximations of 1D nonlocal problems. J. Comput. Phys. 310, 213–236 (2016b)

    Article  MathSciNet  Google Scholar 

  • K. Zhou, Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48, 1759–1780 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Optimization and Uncertainty Quantification Department Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Marta D’Elia

  2. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, USA

    Qiang Du

  3. Department of Scientific Computing, Florida State University, Tallahassee, FL, USA

    Max Gunzburger

Authors
  1. Marta D’Elia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiang Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Max Gunzburger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marta D’Elia .

Editor information

Editors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, USA

    George Z. Voyiadjis

Rights and permissions

Reprints and permissions

Copyright information

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

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

D’Elia, M., Du, Q., Gunzburger, M. (2019). Recent Progress in Mathematical and Computational Aspects of Peridynamics. In: Voyiadjis, G. (eds) Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-58729-5_30

Download citation

Publish with us

Policies and ethics

Search

Navigation