Optimization-Based Coupling of Local and Nonlocal Models: Applications to Peridynamics
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.
KeywordsOptimization-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.
- 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
- 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
- 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
- D.J. Littlewood, Roadmap for peridynamic software implementation. Report SAND2015-9013, Sandia National Laboratories, Albuquerque, 2015Google Scholar
- 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
- 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
- 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
- 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