Optimization-Based Coupling of Local and Nonlocal Models: Applications to Peridynamics

  • Marta D’Elia
  • Pavel Bochev
  • David J. Littlewood
  • Mauro Perego
Reference work entry


Nonlocal continuum theories for mechanics can capture strong nonlocal effects due to long-range forces in their governing equations. When these effects cannot be neglected, nonlocal models are more accurate than partial differential equations (PDEs); however, the accuracy comes at the price of a prohibitive computational cost, making local-to-nonlocal (LtN) coupling strategies mandatory.

In this chapter, we review the state of the art of LtN methods where the efficiency of PDEs is combined with the accuracy of nonlocal models. Then, we focus on optimization-based coupling strategies that couch the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the local and nonlocal problem domains, and the virtual controls are the nonlocal volume constraint and the local boundary condition. The strategy is described in the context of nonlocal and local elasticity and illustrated by numerical tests on three-dimensional realistic geometries. Additional numerical tests also prove the consistency of the method via patch tests.


Optimization-based coupling methods Local-nonlocal coupling Nonlocal elasticity Classical elasticity Peridynamics Domain decomposition Finite element method Particle methods 



This material is based upon work supported by the US DOE’s Laboratory Directed Research and Development (LDRD) program at Sandia National Laboratories and the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research. Part of this research was carried under the auspices of 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. SAND2017-3003 B.


  1. A. Abdulle, O. Jecker, A. Shapeev, An optimization based coupling method for multiscale problems. Technical Report 36.2015, EPFL, Mathematics Institute of Computational Science and Engineering, Lausanne, Dec 2015Google Scholar
  2. Y. Azdoud, F. Han, G. Lubineau, A morphing framework to couple non-local and local anisotropic continua. Int. J. Solids Struct. 50(9), 1332–1341 (2013)CrossRefGoogle Scholar
  3. M. D’Elia, P. Bochev, Optimization-based coupling of nonlocal and local diffusion models, in Proceedings of the Fall 2014 Materials Research Society Meeting, ed. by R. Lipton. MRS Symposium Proceedings (Cambridge University Press, Boston, 2014)Google Scholar
  4. M. D’Elia, P. Bochev, Formulation, analysis and computation of an optimization-based local-to-nonlocal coupling method. Technical Report SAND2017–1029J, Sandia National Laboratories, 2016Google Scholar
  5. M. D’Elia, M. Perego, P. Bochev, D. Littlewood, A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions. Comput. Math. Appl. 71(11), 2218–2230 (2016)MathSciNetCrossRefGoogle Scholar
  6. H.B. Dhia, G. Rateau, The Arlequin method as a flexible engineering design tool. Int. J. Numer. Methods Eng. 62(11), 1442–1462 (2005)CrossRefGoogle Scholar
  7. M. Di Paola, G. Failla, M. Zingales, Physically-based approach to the mechanics of strong non-local linear elasticity theory. J. Elast. 97(2), 103–130 (2009)MathSciNetCrossRefGoogle Scholar
  8. M. Discacciati, P. Gervasio, A. Quarteroni, The interface control domain decomposition (ICDD) method for elliptic problems. SIAM J. Control. Optim. 51(5), 3434–3458 (2013)MathSciNetCrossRefGoogle Scholar
  9. Q. Du, Optimization based nonoverlapping domain decomposition algorithms and their convergence. SIAM J. Numer. Anal. 39(3), 1056–1077 (2001)MathSciNetCrossRefGoogle Scholar
  10. Q. Du, M.D. Gunzburger, A gradient method approach to optimization-based multidisciplinary simulations and nonoverlapping domain decomposition algorithms. SIAM J. Numer. Anal. 37(5), 1513–1541 (2000)MathSciNetCrossRefGoogle Scholar
  11. Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54(4), 667–696 (2012)MathSciNetCrossRefGoogle Scholar
  12. P. Gervasio, J.-L. Lions, A. Quarteroni, Heterogeneous coupling by virtual control methods. Numerische Mathematik 90, 241–264 (2001). MathSciNetCrossRefGoogle Scholar
  13. M.D. Gunzburger, H.K. Lee, An optimization-based domain decomposition method for the Navier-Stokes equations. SIAM J. Numer. Anal. 37(5), 1455–1480 (2000)MathSciNetCrossRefGoogle Scholar
  14. M.D. Gunzburger, J.S. Peterson, H. Kwon, An optimization based domain decomposition method for partial differential equations. Comput. Math. Appl. 37(10), 77–93 (1999)MathSciNetCrossRefGoogle Scholar
  15. M.D. Gunzburger, M. Heinkenschloss, H.K. Lee, Solution of elliptic partial differential equations by an optimization-based domain decomposition method. Appl. Math. Comput. 113(2–3), 111–139 (2000)MathSciNetzbMATHGoogle Scholar
  16. F. Han, G. Lubineau, Coupling of nonlocal and local continuum models by the Arlequin approach. Int. J. Numer. Methods Eng. 89(6), 671–685 (2012)MathSciNetCrossRefGoogle Scholar
  17. P. Kuberry, H. Lee, A decoupling algorithm for fluid-structure interaction problems based on optimization. Comput. Methods Appl. Mech. Eng. 267, 594–605 (2013)MathSciNetCrossRefGoogle Scholar
  18. D.J. Littlewood, Roadmap for peridynamic software implementation. Report SAND2015-9013, Sandia National Laboratories, Albuquerque, 2015Google Scholar
  19. G. Lubineau, Y. Azdoud, F. Han, C. Rey, A. Askari, A morphing strategy to couple non-local to local continuum mechanics. J. Mech. Phys. Solids 60(6), 1088–1102 (2012)MathSciNetCrossRefGoogle Scholar
  20. D. Olson, P. Bochev, M. Luskin, A. Shapeev, Development of an optimization-based atomistic-to-continuum coupling method, in Proceedings of LSSC 2013, ed. by I. Lirkov, S. Margenov, J. Wasniewski. Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2014a)Google Scholar
  21. D. Olson, P. Bochev, M. Luskin, A. Shapeev, An optimization-based atomistic-to-continuum coupling method. SIAM J. Numer. Anal. 52(4), 2183–2204 (2014b)MathSciNetCrossRefGoogle Scholar
  22. M.L. Parks, D.J. Littlewood, J.A. Mitchell, S.A. Silling, Peridigm Users’ Guide v1.0.0. SAND Report 2012-7800, Sandia National Laboratories, Albuquerque, 2012Google Scholar
  23. A.G. Salinger, R.A. Bartlett, Q. Chen, X. Gao, G.A. Hansen, I. Kalashnikova, A. Mota, R.P. Muller, E. Nielsen, J.T. Ostien, R.P. Pawlowski, E.T. Phipps, W. Sun, Albany: a component–based partial differential equation code built on Trilinos. SAND Report 2013-8430J, Sandia National Laboratories, Albuquerque, 2013Google Scholar
  24. P. Seleson, D.J. Littlewood, Convergence studies in meshfree peridynamic simulations. Comput. Math. Appl. 71(11), 2432–2448 (2016)MathSciNetCrossRefGoogle Scholar
  25. P. Seleson, M.L. Parks, On the role of the influence function in the peridynamic theory. Int. J. Multiscale Comput. Eng. 9, 689–706 (2011)CrossRefGoogle Scholar
  26. P. Seleson, S. Beneddine, S. Prudhomme, A force-based coupling scheme for peridynamics and classical elasticity. Comput. Mater. Sci. 66, 34–49 (2013)CrossRefGoogle Scholar
  27. S.A. Silling, R.B. Lehoucq, Peridynamic theory of solid mechanics, in Advances in Applied Mechanics, vol. 44 (Elsevier, San Diego, 2010), pp. 73–168Google Scholar
  28. S.A. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)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

  • Marta D’Elia
    • 1
  • Pavel Bochev
    • 2
  • David J. Littlewood
    • 2
  • Mauro Perego
    • 2
  1. 1.Optimization and Uncertainty Quantification Department Center for Computing ResearchSandia National LaboratoriesAlbuquerqueUSA
  2. 2.Center for Computing ResearchSandia National LaboratoriesAlbuquerqueUSA

Personalised recommendations