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

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Handbook of Nonlocal Continuum Mechanics for Materials and Structures

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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.

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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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Authors and Affiliations

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

    Marta D’Elia

  2. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Pavel Bochev, David J. Littlewood & Mauro Perego

Authors
  1. Marta D’Elia
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  2. Pavel Bochev
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  3. David J. Littlewood
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  4. Mauro Perego
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Corresponding author

Correspondence to Marta D’Elia .

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Editors and Affiliations

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

    George Z. Voyiadjis

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D’Elia, M., Bochev, P., J. Littlewood, D., Perego, M. (2019). Optimization-Based Coupling of Local and Nonlocal Models: Applications to 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_31

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