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

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

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

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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

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