Analysis of an optimization-based atomistic-to-continuum coupling method for point defects
Olson, Derek1 ; Shapeev, Alexander V.2 ; Bochev, Pavel B.3 ; Luskin, Mitchell4
1 University of Minnesota, MN 55455, USA. DO was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program
2 Skolkovo Institute of Science and Technology, Skolkovo, Russie. AS was supported in part by the AFOSR Award FA9550-12-1-0187
3 Sandia National Laboratories, Computational Mathematics, P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320, USA
4 University of Minnesota, MN 55455, USA. ML was supported in part by the NSF PIRE Grant OISE-0967140, NSF Grant 1310835, DOE Award DE-SC0012733
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 1-41.

We formulate and analyze an optimization-based Atomistic-to-Continuum (AtC) coupling method for problems with point defects. Application of a potential-based atomistic model near the defect core enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the atomistic and continuum force balance equations acting independently in their domains. We prove that the optimization problem is well-posed and establish error estimates.

DOI : 10.1051/m2an/2015023
Classification : 65N99, 65G99, 73S10
Mots-clés : Atomistic-to-continuum coupling, atomic lattice, constrained optimization, point defect
Olson, Derek 1 ; Shapeev, Alexander V. 2 ; Bochev, Pavel B. 3 ; Luskin, Mitchell 4

1 University of Minnesota, MN 55455, USA. DO was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program
2 Skolkovo Institute of Science and Technology, Skolkovo, Russie. AS was supported in part by the AFOSR Award FA9550-12-1-0187
3 Sandia National Laboratories, Computational Mathematics, P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320, USA
4 University of Minnesota, MN 55455, USA. ML was supported in part by the NSF PIRE Grant OISE-0967140, NSF Grant 1310835, DOE Award DE-SC0012733 
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Olson, Derek; Shapeev, Alexander V.; Bochev, Pavel B.; Luskin, Mitchell. Analysis of an optimization-based atomistic-to-continuum coupling method for point defects. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 1-41. doi : 10.1051/m2an/2015023. http://www.numdam.org/articles/10.1051/m2an/2015023/

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