Browse > Article
http://dx.doi.org/10.12989/csm.2018.7.3.255

Stability of matching boundary conditions for diatomic chain and square lattice  

Ji, Songsong (Department of Mechanics and Engineering Science, College of Engineering, Peking University)
Tang, Shaoqiang (HEDPS, CAPT and LTCS, College of Engineering, Peking University)
Publication Information
Coupled systems mechanics / v.7, no.3, 2018 , pp. 255-268 More about this Journal
Abstract
Stability of MBC1, a specific matching boundary condition, is proved for atomic simulations of a diatomic chain. The boundary condition and the Newton equations that govern the atomic dynamics form a coupled system. Energy functions that decay along with time are constructed for both the boundary with the same type atoms and the one with different type atoms. For a nonlinear chain, MBC1 is also shown stable. Numerical verifications are presented. Moreover, MBC1 is proved to be stable for a two dimensional square lattice.
Keywords
stability; matching boundary condition; diatomic chain; two-dimensional square lattice; energy function;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Karpov, E.G., Wagner, G.J. and Liu W.K. (2005), "A Green's function approach to deriving nonreflecting boundary conditions in molecular dynamics simulations", Int. J. Numer. Meth. Eng., 62, 1250-1262.   DOI
2 Li, X.T. (2008), "Radiation boundary conditions for acoustic and elastic calculations", J. Comput. Phys., 227(24), 10078-10093.   DOI
3 Li, X.T. (2009), "On the stability of boundary conditions for molecular dynamics", J. Comput. Appl. Math., 231(2), 493-505.   DOI
4 Liu, W.K., Karpov, E.G. and Park, H.S. (2005), Nano Mechanics and Materials: Theory, Multiscale Methods and Applications, John Wiley, New York, U.S.A.
5 Tang, S. and Ji, S. (2014), "Stability of atomic simulations with matching boundary conditions", Adv. Appl. Math. Mech., 6(5), 539-551.   DOI
6 Tang, S. (2008), "A finite difference approach with velocity interfacial conditions for multiscale computations of crystalline solids", J. Comput. Phys., 227, 4038-4062.   DOI
7 Tang, S., Hou, T.Y. and Liu, W.K. (2006a), "A mathematical framework of the bridging scale method", Int. J. Numer. Meth. Eng., 65, 1688-1713.   DOI
8 Tang, S., Hou, T.Y. and Liu, W.K. (2006b), "A pseudo-spectral multiscale method: interfacial conditions and coarse grid equations", J. Comput. Phys., 213, 57-85.   DOI
9 Trefethen, L.N. (1985), "Stability of finite-difference models containing two boundaries or interfaces", Math. Comput., 45(172), 279-300.   DOI
10 Thirunavukkarasu, S. and Guddati, M.N. (2011). "Absorbing boundary conditions for time harmonic wave propagation in discretized domains", Comput. Meth. Appl. Mech. Eng., 200(33-36), 2483-2497.   DOI
11 Wagner, G.J. and Liu, W.K. (2003), "Coupling of atomistic and continuum simulations using a bridging scale decomposition", J. Comput. Phys., 190(1), 249-274.   DOI
12 Wang, X. and Tang, S. (2010), "Matching boundary conditions for diatomic chains", Comput. Mech., 46, 813-826.   DOI
13 Wang, X. (2010), "Matching boundary conditions for atomic simulations of crystalline solids", Ph.D. Dissertation, Tsinghua University, Beijing, China.
14 Wang, X. and Tang, S. (2013), "Matching boundary conditions for lattice dynamics", Int. J. Numer. Meth. Eng., 93, 1255-1285.   DOI
15 Zhang, W.S., Chung, E.T. and Wang, C.W. (2014), "Stability for imposing absorbing boundary conditions in the finite element simulation of acoustic wave propagation", Appl. Numer. Math., 32(1), 1-20.   DOI
16 Adelman, S.A. and Doll, J.D. (1974), "Generalized Langevin equation approach for atom/solid-surface scattering: Collinear atom/harmonic chain model", J. Chem. Phys., 61, 4242-4245.   DOI
17 Baffet, D. and Givoli, D. (2011), "On the stability of the high-order higdon absorbing boundary conditions", Appl. Numer. Math., 61(6), 768-784.   DOI
18 Berenger, J.P. (1994), "A perfectly matched layer for the absorption of electromagnetic waves", J. Comput. Phys., 114, 185-200.   DOI
19 Cai, W., De Koning, M., Bulatov, V.V. and Yip, S. (2000), "Minimizing boundary reflections in coupled-domain simulations", Phys. Rev. Lett., 85(15), 3213-3216.   DOI
20 Dreher, M. and Tang, S. (2008), "Time history interfacial conditions in multiscale computations of lattice oscillations", Comput. Mech., 41(5), 683-698.   DOI
21 Engquist, B. and Majda, A. (1979), "Variational boundary conditions for molecular dynamics simulations: Treatment of the loading condition", Commun. Pure Appl. Math., 32, 313-357.   DOI
22 Eriksson, S. and Nordstrom, J. (2017), "Exact non-reflecting boundary conditions revisited: Well-posedness and stability", Foundat. Comput. Math., 17(4), 957-986.   DOI
23 Fang, M. (2012), "Boundary treatments and statistical convergence of particle simulations", Ph.D. Dissertation, Peking University, Beijing, China.