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Stability of Atomic Simulations with Matching Boundary Conditions

Published online by Cambridge University Press:  03 June 2015

Shaoqiang Tang  and
Songsong Ji
Shaoqiang Tang*
Affiliation:
HEDPS, CAPT and LTCS, College of Engineering, Peking University, Beijing 100871, China
Songsong Ji*
Affiliation:
HEDPS, CAPT and LTCS, College of Engineering, Peking University, Beijing 100871, China
*
Corresponding author. Email: [email protected]
Email: [email protected]
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Abstract

We explore the stability of matching boundary conditions in one space dimension, which were proposed recently for atomic simulations (Wang and Tang, Int. J. Numer. Mech. Eng., 93 (2013), pp. 1255–1285). For a finite segment of the linear harmonic chain, we construct explicit energy functionals that decay along with time. For a nonlinear atomic chain with its nonlinearity vanished around the boundaries, an energy functional is constructed for the first order matching boundary condition. Numerical verifications are also presented.

Keywords

70-0865L9934B60Stabilitymatching boundary conditionatomic simulation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 5 , October 2014 , pp. 539 - 551
Copyright
Copyright © Global-Science Press 2014

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References

[1]Adelman, S. A. and Doll, J. D., Generalized Langevin equation appraochfor atom/solid-surface scattering: collinear atom/harmonic chain model, J. Chem. Phys., 61 (1974), pp. 4242–4245.Google Scholar
[2]Cai, W., De Koning, M., Bulatov, V. V. and Yip, S., Minimizing boundary reflections in coupled-domain simulations, Phys. Rev. Lett., 85 (2000), pp. 3213–3216.CrossRefGoogle ScholarPubMed
[3]Dreher, M. and Tang, S., Time history interfacial conditions in multiscale computations of lattice oscillations, Comput. Mech., 41 (2008), pp. 683–698.Google Scholar
[4]Fang, M., Boundary Treatments and Statistical Convergence of Particle Simulations, PhD Thesis, Peking University, Beijing, 2012.Google Scholar
[5]Fang, M., Tang, S., Li, Z. and Wang, X., Artificial boundary conditions for atomic simulations of face-centered-cubic lattice, Comput. Mech., 50 (2012), pp. 645–655.Google Scholar
[6]Fang, M., Wang, X., Li, Z. and Tang, S., Matching boundary conditions for scalar waves in face-centered-cubic lattice, Adv. Appl. Math. Mech., 5 (2013), pp. 337–350.Google Scholar
[7]Karpov, E. G., Park, H. S., Liu, W. K. and Dorofeev, D. L., On the modelling of chaotic thermal motion in solids, Preprint.Google Scholar
[8]Liu, W. K., Karpov, E. G and Park, H. S., Nano-Mechanics and Materials: Theory, Multiscale Methods and Applications, Wiley, 2005.Google Scholar
[9]Pang, G. and Tang, S., Time history kernel junctions for square lattice, Comput. Mech., 48 (2011), pp. 699–711.Google Scholar
[10]Savadatti, S. and Guddati, M., Absorbing boundary conditions for scalar waves in anisotropic media, part 1: time harmonic modeling, J. Comput. Phys., 229 (2010), pp. 6696–6714.Google Scholar
[11]Tang, S., A finite difference approach with velocity interfacial conditions for multiscale computations of crystalline solids, J. Comput. Phys., 227 (2008), pp. 4038–4062.Google Scholar
[12]Tang, S. and Fang, M., Unstable surface modes infinite chain computations: deficiency of reflection coefficient approach, Commun. Comput. Phys., 8 (2010), pp. 143–158.Google Scholar
[13]Tang, S., Hou, T. Y. and Liu, W. K., A mathematical framework of the bridging scale method, Int. J. Numer. Methods Eng., 65 (2006), pp. 1688–1713.Google Scholar
[14]Tang, S., Hou, T. Y. and Liu, W. K., A pseudo-spectral multiscale method: interfacial conditions and coarse grid equations, J. Comput. Phys., 213 (2006), pp. 57–85.Google Scholar
[15]Trefethen, L. N., Stability of finite-difference models containing two boundaries or interfaces, Math. Comput., 45 (1985), pp. 279–300.Google Scholar
[16]To, A. and Li, S., Perfectly matched multiscale simulations, Phys. Rev. B, 72 (2005), 035414.Google Scholar
[17]Wagner, G. J. and Liu, W. k., Coupling of atomistic and continuum simulations using a bridging scale decomposition, J. Comput. Phys., 190 (2003), pp. 249–274.Google Scholar
[18]Wang, X. and Tang, S., Matching boundary conditions for lattice dynamics, Int. J. Numer. Methods Eng., 93 (2013), pp. 1255–1285.Google Scholar
[19]Yong, W. and Zhu, Y., Convergence of finite difference method for nonlinear movable boundary problem, Science in China, Series A, 8 (1989), pp. 785–795 (in Chinese).Google Scholar