Analysis of an optimization-based atomistic-to-continuum coupling method for point defects^∗

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
olso4056@umn.edu
² Skolkovo Institute of Science and Technology, Skolkovo, Russie. AS was supported in part by the AFOSR Award FA9550-12-1-0187
a.shapeev@skoltech.ru
³ Sandia National Laboratories, Computational Mathematics, P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320, USA
pbboche@sandia.gov This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR). Part of this research was carried under the auspices of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4)
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
luskin@umn.edu

Received: 29 October 2014
Revised: 11 March 2015

Abstract

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.